Cohomology on Sites

Introduction

In this document we work out some topics on cohomology of sheaves. We work out what happens for sheaves on sites, although often we will simply duplicate the discussion, the constructions, and the proofs from the topological case in the case. Basic references are [SGA4], [Godement] and [Iversen].

Cohomology of sheaves

Let \(\mathcal{C}\) be a site, see Sites, Definition 00VH. Let \(\mathcal{F}\) be an abelian sheaf on \(\mathcal{C}\). We know that the category of abelian sheaves on \(\mathcal{C}\) has enough injectives, see Injectives, Theorem 01DP. Hence we can choose an injective resolution \(\mathcal{F}[0] \to \mathcal{I}^\bullet\). For any object \(U\) of the site \(\mathcal{C}\) we define [071C]\[\begin{equation} H^i(U, \mathcal{F}) = H^i(\Gamma(U, \mathcal{I}^\bullet)) \end{equation}\] to be the \(i\)th cohomology group of the abelian sheaf \(\mathcal{F}\) over the object \(U\). In other words, these are the right derived functors of the functor \(\mathcal{F} \mapsto \mathcal{F}(U)\). The family of functors \(H^i(U, -)\) forms a universal \(\delta\)-functor \(\textit{Ab}(\mathcal{C}) \to \textit{Ab}\).

It sometimes happens that the site \(\mathcal{C}\) does not have a final object. In this case we define the global sections of a presheaf of sets \(\mathcal{F}\) over \(\mathcal{C}\) to be the set [071D]\[\begin{equation} \Gamma(\mathcal{C}, \mathcal{F}) = \Mor_{\textit{PSh}(\mathcal{C})}(e, \mathcal{F}) \end{equation}\] where \(e\) is a final object in the category of presheaves on \(\mathcal{C}\). In this case, given an abelian sheaf \(\mathcal{F}\) on \(\mathcal{C}\), we define the \(i\)th cohomology group of \(\mathcal{F}\) on \(\mathcal{C}\) as follows [071E]\[\begin{equation} H^i(\mathcal{C}, \mathcal{F}) = H^i(\Gamma(\mathcal{C}, \mathcal{I}^\bullet)) \end{equation}\] in other words, it is the \(i\)th right derived functor of the global sections functor. The family of functors \(H^i(\mathcal{C}, -)\) forms a universal \(\delta\)-functor \(\textit{Ab}(\mathcal{C}) \to \textit{Ab}\).

Let \(f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\) be a morphism of topoi, see Sites, Definition 00XA. With \(\mathcal{F}[0] \to \mathcal{I}^\bullet\) as above we define [071F]\[\begin{equation} R^if_*\mathcal{F} = H^i(f_*\mathcal{I}^\bullet) \end{equation}\] to be the \(i\)th higher direct image of \(\mathcal{F}\). These are the right derived functors of \(f_*\). The family of functors \(R^if_*\) forms a universal \(\delta\)-functor from \(\textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})\).

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site, see Modules on Sites, Definition 03AD. Let \(\mathcal{F}\) be an \(\mathcal{O}\)-module. We know that the category of \(\mathcal{O}\)-modules has enough injectives, see Injectives, Theorem 01DU. Hence we can choose an injective resolution \(\mathcal{F}[0] \to \mathcal{I}^\bullet\). For any object \(U\) of the site \(\mathcal{C}\) we define [071G]\[\begin{equation} H^i(U, \mathcal{F}) = H^i(\Gamma(U, \mathcal{I}^\bullet)) \end{equation}\] to be the the \(i\)th cohomology group of \(\mathcal{F}\) over \(U\). The family of functors \(H^i(U, -)\) forms a universal \(\delta\)-functor \(\textit{Mod}(\mathcal{O}) \to \text{Mod}_{\mathcal{O}(U)}\). Similarly [071H]\[\begin{equation} H^i(\mathcal{C}, \mathcal{F}) = H^i(\Gamma(\mathcal{C}, \mathcal{I}^\bullet)) \end{equation}\] is the \(i\)th cohomology group of \(\mathcal{F}\) on \(\mathcal{C}\). The family of functors \(H^i(\mathcal{C}, -)\) forms a universal \(\delta\)-functor \(\textit{Mod}(\mathcal{O}) \to \text{Mod}_{\Gamma(\mathcal{C}, \mathcal{O})}\).

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) be a morphism of ringed topoi, see Modules on Sites, Definition 01D3. With \(\mathcal{F}[0] \to \mathcal{I}^\bullet\) as above we define [071I]\[\begin{equation} R^if_*\mathcal{F} = H^i(f_*\mathcal{I}^\bullet) \end{equation}\] to be the \(i\)th higher direct image of \(\mathcal{F}\). These are the right derived functors of \(f_*\). The family of functors \(R^if_*\) forms a universal \(\delta\)-functor from \(\textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}')\).

Derived functors

We briefly explain an approach to right derived functors using resolution functors. Namely, suppose that \((\mathcal{C}, \mathcal{O})\) is a ringed site. In this chapter we will write \[K(\mathcal{O}) = K(\textit{Mod}(\mathcal{O})) \quad \text{and} \quad D(\mathcal{O}) = D(\textit{Mod}(\mathcal{O}))\] and similarly for the bounded versions for the triangulated categories introduced in Derived Categories, Definition 013H and Definition 05RU. By Derived Categories, Remark 0142 there exists a resolution functor \[j = j_{(\mathcal{C}, \mathcal{O})} : K^{+}(\textit{Mod}(\mathcal{O})) \longrightarrow K^{+}(\mathcal{I})\] where \(\mathcal{I}\) is the strictly full additive subcategory of \(\textit{Mod}(\mathcal{O})\) which consists of injective \(\mathcal{O}\)-modules. For any left exact functor \(F : \textit{Mod}(\mathcal{O}) \to \mathcal{B}\) into any abelian category \(\mathcal{B}\) we will denote \(RF\) the right derived functor of Derived Categories, Section 0156 constructed using the resolution functor \(j\) just described: [05U5]\[\begin{equation} RF = F \circ j' : D^{+}(\mathcal{O}) \longrightarrow D^{+}(\mathcal{B}) \end{equation}\] see Derived Categories, Lemma 05TN for notation. Note that we may think of \(RF\) as defined on \(\textit{Mod}(\mathcal{O})\), \(\text{Comp}^{+}(\textit{Mod}(\mathcal{O}))\), or \(K^{+}(\mathcal{O})\) depending on the situation. According to Derived Categories, Definition 015A we obtain the \(i\)the right derived functor [05U6]\[\begin{equation} R^iF = H^i \circ RF : \textit{Mod}(\mathcal{O}) \longrightarrow \mathcal{B} \end{equation}\] so that \(R^0F = F\) and \(\{R^iF, \delta\}_{i \geq 0}\) is universal \(\delta\)-functor, see Derived Categories, Lemma 015B.

Here are two special cases of this construction. Given a ring \(R\) we write \(K(R) = K(\text{Mod}_R)\) and \(D(R) = D(\text{Mod}_R)\) and similarly for the bounded versions. For any object \(U\) of \(\mathcal{C}\) have a left exact functor \(\Gamma(U, -) : \textit{Mod}(\mathcal{O}) \longrightarrow \text{Mod}_{\mathcal{O}(U)}\) which gives rise to \[R\Gamma(U, -) : D^{+}(\mathcal{O}) \longrightarrow D^{+}(\mathcal{O}(U))\] by the discussion above. Note that \(H^i(U, -) = R^i\Gamma(U, -)\) is compatible with (071G) above. We similarly have \[R\Gamma(\mathcal{C}, -) : D^{+}(\mathcal{O}) \longrightarrow D^{+}(\Gamma(\mathcal{C}, \mathcal{O}))\] compatible with (071H). If \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) is a morphism of ringed topoi then we get a left exact functor \(f_* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}')\) which gives rise to derived pushforward \[Rf_* : D^{+}(\mathcal{O}) \to D^+(\mathcal{O}')\] The \(i\)th cohomology sheaf of \(Rf_*\mathcal{F}^\bullet\) is denoted \(R^if_*\mathcal{F}^\bullet\) and called the \(i\)th higher direct image in accordance with (071I). The displayed functors above are exact functor of derived categories.

First cohomology and torsors

Definition

Let \(\mathcal{C}\) be a site. Let \(\mathcal{G}\) be a sheaf of (possibly non-commutative) groups on \(\mathcal{C}\). A pseudo torsor, or more precisely a pseudo \(\mathcal{G}\)-torsor, is a sheaf of sets \(\mathcal{F}\) on \(\mathcal{C}\) endowed with an action \(\mathcal{G} \times \mathcal{F} \to \mathcal{F}\) such that

1. whenever \(\mathcal{F}(U)\) is nonempty the action \(\mathcal{G}(U) \times \mathcal{F}(U) \to \mathcal{F}(U)\) is simply transitive.

A morphism of pseudo \(\mathcal{G}\)-torsors \(\mathcal{F} \to \mathcal{F}'\) is a morphism of sheaves of sets compatible with the \(\mathcal{G}\)-actions. A torsor, or more precisely a \(\mathcal{G}\)-torsor, is a pseudo \(\mathcal{G}\)-torsor such that in addition

1. for every \(U \in \Ob(\mathcal{C})\) there exists a covering \(\{U_i \to U\}_{i \in I}\) of \(U\) such that \(\mathcal{F}(U_i)\) is nonempty for all \(i \in I\).

A morphism of \(\mathcal{G}\)-torsors is a morphism of pseudo \(\mathcal{G}\)-torsors. The trivial \(\mathcal{G}\)-torsor is the sheaf \(\mathcal{G}\) endowed with the obvious left \(\mathcal{G}\)-action.

It is clear that a morphism of torsors is automatically an isomorphism.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{G}\) be a sheaf of (possibly non-commutative) groups on \(\mathcal{C}\). A \(\mathcal{G}\)-torsor \(\mathcal{F}\) is trivial if and only if \(\Gamma(\mathcal{C}, \mathcal{F}) \not = \emptyset\).

Proof

Omitted.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{H}\) be an abelian sheaf on \(\mathcal{C}\). There is a canonical bijection between the set of isomorphism classes of \(\mathcal{H}\)-torsors and \(H^1(\mathcal{C}, \mathcal{H})\).

Proof

Let \(\mathcal{F}\) be a \(\mathcal{H}\)-torsor. Consider the free abelian sheaf \(\mathbf{Z}[\mathcal{F}]\) on \(\mathcal{F}\). It is the sheafification of the rule which associates to \(U \in \Ob(\mathcal{C})\) the collection of finite formal sums \(\sum n_i[s_i]\) with \(n_i \in \mathbf{Z}\) and \(s_i \in \mathcal{F}(U)\). There is a natural map \[\sigma : \mathbf{Z}[\mathcal{F}] \longrightarrow \underline{\mathbf{Z}}\] which to a local section \(\sum n_i[s_i]\) associates \(\sum n_i\). The kernel of \(\sigma\) is generated by sections of the form \([s] - [s']\). There is a canonical map \(a : \Ker(\sigma) \to \mathcal{H}\) which maps \([s] - [s'] \mapsto h\) where \(h\) is the local section of \(\mathcal{H}\) such that \(h \cdot s' = s\). Consider the pushout diagram \[\xymatrix{ 0 \ar[r] & \Ker(\sigma) \ar[r] \ar[d]^a & \mathbf{Z}[\mathcal{F}] \ar[r] \ar[d] & \underline{\mathbf{Z}} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{H} \ar[r] & \mathcal{E} \ar[r] & \underline{\mathbf{Z}} \ar[r] & 0 }\] Here \(\mathcal{E}\) is the extension obtained by pushout. From the long exact cohomology sequence associated to the lower short exact sequence we obtain an element \(\xi = \xi_\mathcal{F} \in H^1(\mathcal{C}, \mathcal{H})\) by applying the boundary operator to \(1 \in H^0(\mathcal{C}, \underline{\mathbf{Z}})\).

Conversely, given \(\xi \in H^1(\mathcal{C}, \mathcal{H})\) we can associate to \(\xi\) a torsor as follows. Choose an embedding \(\mathcal{H} \to \mathcal{I}\) of \(\mathcal{H}\) into an injective abelian sheaf \(\mathcal{I}\). We set \(\mathcal{Q} = \mathcal{I}/\mathcal{H}\) so that we have a short exact sequence \[\xymatrix{ 0 \ar[r] & \mathcal{H} \ar[r] & \mathcal{I} \ar[r] & \mathcal{Q} \ar[r] & 0 }\] The element \(\xi\) is the image of a global section \(q \in H^0(\mathcal{C}, \mathcal{Q})\) because \(H^1(\mathcal{C}, \mathcal{I}) = 0\) (see Derived Categories, Lemma 015B). Let \(\mathcal{F} \subset \mathcal{I}\) be the subsheaf (of sets) of sections that map to \(q\) in the sheaf \(\mathcal{Q}\). It is easy to verify that \(\mathcal{F}\) is a \(\mathcal{H}\)-torsor.

We omit the verification that the two constructions given above are mutually inverse.

First cohomology and extensions

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}\)-modules on \(\mathcal{C}\). There is a canonical bijection \[\Ext^1_{\textit{Mod}(\mathcal{O})}(\mathcal{O}, \mathcal{F}) \longrightarrow H^1(\mathcal{C}, \mathcal{F})\] which associates to the extension \[0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{O} \to 0\] the image of \(1 \in \Gamma(\mathcal{C}, \mathcal{O})\) in \(H^1(\mathcal{C}, \mathcal{F})\).

Proof

Let us construct the inverse of the map given in the lemma. Let \(\xi \in H^1(\mathcal{C}, \mathcal{F})\). Choose an injection \(\mathcal{F} \subset \mathcal{I}\) with \(\mathcal{I}\) injective in \(\textit{Mod}(\mathcal{O})\). Set \(\mathcal{Q} = \mathcal{I}/\mathcal{F}\). By the long exact sequence of cohomology, we see that \(\xi\) is the image of a section \(\tilde \xi \in \Gamma(\mathcal{C}, \mathcal{Q}) = \Hom_\mathcal{O}(\mathcal{O}, \mathcal{Q})\). Now, we just form the pullback \[\xymatrix{ 0 \ar[r] & \mathcal{F} \ar[r] \ar@{=}[d] & \mathcal{E} \ar[r] \ar[d] & \mathcal{O} \ar[r] \ar[d]^{\tilde \xi} & 0 \\ 0 \ar[r] & \mathcal{F} \ar[r] & \mathcal{I} \ar[r] & \mathcal{Q} \ar[r] & 0 }\] see Homology, Section 010I.

The following lemma will be superseded by the more general Lemma 03FD.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}\)-modules on \(\mathcal{C}\). Let \(\mathcal{F}_{ab}\) denote the underlying sheaf of abelian groups. Then there is a functorial isomorphism \[H^1(\mathcal{C}, \mathcal{F}_{ab}) = H^1(\mathcal{C}, \mathcal{F})\] where the left hand side is cohomology computed in \(\textit{Ab}(\mathcal{C})\) and the right hand side is cohomology computed in \(\textit{Mod}(\mathcal{O})\).

Proof

Let \(\underline{\mathbf{Z}}\) denote the constant sheaf \(\mathbf{Z}\). As \(\textit{Ab}(\mathcal{C}) = \textit{Mod}(\underline{\mathbf{Z}})\) we may apply Lemma 03F1 twice, and it follows that we have to show \[\Ext^1_{\textit{Mod}(\mathcal{O})}(\mathcal{O}, \mathcal{F}) = \Ext^1_{\textit{Mod}(\underline{\mathbf{Z}})}( \underline{\mathbf{Z}}, \mathcal{F}_{ab}).\] Suppose that \(0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{O} \to 0\) is an extension in \(\textit{Mod}(\mathcal{O})\). Then we can use the obvious map of abelian sheaves \(1 : \underline{\mathbf{Z}} \to \mathcal{O}\) and pullback to obtain an extension \(\mathcal{E}_{ab}\), like so: \[\xymatrix{ 0 \ar[r] & \mathcal{F}_{ab} \ar[r] \ar@{=}[d] & \mathcal{E}_{ab} \ar[r] \ar[d] & \underline{\mathbf{Z}} \ar[r] \ar[d]^{1} & 0 \\ 0 \ar[r] & \mathcal{F} \ar[r] & \mathcal{E} \ar[r] & \mathcal{O} \ar[r] & 0 }\] The converse is a little more fun. Suppose that \(0 \to \mathcal{F}_{ab} \to \mathcal{E}_{ab} \to \underline{\mathbf{Z}} \to 0\) is an extension in \(\textit{Mod}(\underline{\mathbf{Z}})\). Since \(\underline{\mathbf{Z}}\) is a flat \(\underline{\mathbf{Z}}\)-module we see that the sequence \[0 \to \mathcal{F}_{ab} \otimes_{\underline{\mathbf{Z}}} \mathcal{O} \to \mathcal{E}_{ab} \otimes_{\underline{\mathbf{Z}}} \mathcal{O} \to \underline{\mathbf{Z}} \otimes_{\underline{\mathbf{Z}}} \mathcal{O} \to 0\] is exact, see Modules on Sites, Lemma 03EX. Of course \(\underline{\mathbf{Z}} \otimes_{\underline{\mathbf{Z}}} \mathcal{O} = \mathcal{O}\). Hence we can form the pushout via the (\(\mathcal{O}\)-linear) multiplication map \(\mu : \mathcal{F} \otimes_{\underline{\mathbf{Z}}} \mathcal{O} \to \mathcal{F}\) to get an extension of \(\mathcal{O}\) by \(\mathcal{F}\), like this \[\xymatrix{ 0 \ar[r] & \mathcal{F}_{ab} \otimes_{\underline{\mathbf{Z}}} \mathcal{O} \ar[r] \ar[d]^\mu & \mathcal{E}_{ab} \otimes_{\underline{\mathbf{Z}}} \mathcal{O} \ar[r] \ar[d] & \mathcal{O} \ar[r] \ar@{=}[d] & 0 \\ 0 \ar[r] & \mathcal{F} \ar[r] & \mathcal{E} \ar[r] & \mathcal{O} \ar[r] & 0 }\] which is the desired extension. We omit the verification that these constructions are mutually inverse.

First cohomology and invertible sheaves

The Picard group of a ringed site is defined in Modules on Sites, Section 0408.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a locally ringed site. There is a canonical isomorphism \[H^1(\mathcal{C}, \mathcal{O}^*) = \Pic(\mathcal{O}).\] of abelian groups.

Proof

Let \(\mathcal{L}\) be an invertible \(\mathcal{O}\)-module. Consider the presheaf \(\mathcal{L}^*\) defined by the rule \[U \longmapsto \{s \in \mathcal{L}(U) \text{ such that } \mathcal{O}_U \xrightarrow{s \cdot -} \mathcal{L}_U \text{ is an isomorphism}\}\] This presheaf satisfies the sheaf condition. Moreover, if \(f \in \mathcal{O}^*(U)\) and \(s \in \mathcal{L}^*(U)\), then clearly \(fs \in \mathcal{L}^*(U)\). By the same token, if \(s, s' \in \mathcal{L}^*(U)\) then there exists a unique \(f \in \mathcal{O}^*(U)\) such that \(fs = s'\). Moreover, the sheaf \(\mathcal{L}^*\) has sections locally by Modules on Sites, Lemma 0B8Q. In other words we see that \(\mathcal{L}^*\) is a \(\mathcal{O}^*\)-torsor. Thus we get a map \[\begin{matrix} \text{set of invertible sheaves on }(\mathcal{C}, \mathcal{O}) \\ \text{ up to isomorphism} \end{matrix} \longrightarrow \begin{matrix} \text{set of }\mathcal{O}^*\text{-torsors} \\ \text{ up to isomorphism} \end{matrix}\] We omit the verification that this is a homomorphism of abelian groups. By Lemma 03AJ the right hand side is canonically bijective to \(H^1(\mathcal{C}, \mathcal{O}^*)\). Thus we have to show this map is injective and surjective.

Injective. If the torsor \(\mathcal{L}^*\) is trivial, this means by Lemma 03AI that \(\mathcal{L}^*\) has a global section. Hence this means exactly that \(\mathcal{L} \cong \mathcal{O}\) is the neutral element in \(\Pic(\mathcal{O})\).

Surjective. Let \(\mathcal{F}\) be an \(\mathcal{O}^*\)-torsor. Consider the presheaf of sets \[\mathcal{L}_1 : U \longmapsto (\mathcal{F}(U) \times \mathcal{O}(U))/\mathcal{O}^*(U)\] where the action of \(f \in \mathcal{O}^*(U)\) on \((s, g)\) is \((fs, f^{-1}g)\). Then \(\mathcal{L}_1\) is a presheaf of \(\mathcal{O}\)-modules by setting \((s, g) + (s', g') = (s, g + (s'/s)g')\) where \(s'/s\) is the local section \(f\) of \(\mathcal{O}^*\) such that \(fs = s'\), and \(h(s, g) = (s, hg)\) for \(h\) a local section of \(\mathcal{O}\). We omit the verification that the sheafification \(\mathcal{L} = \mathcal{L}_1^\#\) is an invertible \(\mathcal{O}\)-module whose associated \(\mathcal{O}^*\)-torsor \(\mathcal{L}^*\) is isomorphic to \(\mathcal{F}\).

Locality of cohomology

The following lemma says there is no ambiguity in defining the cohomology of a sheaf \(\mathcal{F}\) over an object of the site.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(U\) be an object of \(\mathcal{C}\).

1. If \(\mathcal{I}\) is an injective \(\mathcal{O}\)-module then \(\mathcal{I}|_U\) is an injective \(\mathcal{O}_U\)-module.

2. For any sheaf of \(\mathcal{O}\)-modules \(\mathcal{F}\) we have \(H^p(U, \mathcal{F}) = H^p(\mathcal{C}/U, \mathcal{F}|_U)\).

Proof

Recall that the functor \(j_U^{-1}\) of restriction to \(U\) is a right adjoint to the functor \(j_{U!}\) of extension by \(0\), see Modules on Sites, Section 03DH. Moreover, \(j_{U!}\) is exact. Hence (1) follows from Homology, Lemma 015Z.

By definition \(H^p(U, \mathcal{F}) = H^p(\mathcal{I}^\bullet(U))\) where \(\mathcal{F} \to \mathcal{I}^\bullet\) is an injective resolution in \(\textit{Mod}(\mathcal{O})\). By the above we see that \(\mathcal{F}|_U \to \mathcal{I}^\bullet|_U\) is an injective resolution in \(\textit{Mod}(\mathcal{O}_U)\). Hence \(H^p(U, \mathcal{F}|_U)\) is equal to \(H^p(\mathcal{I}^\bullet|_U(U))\). Of course \(\mathcal{F}(U) = \mathcal{F}|_U(U)\) for any sheaf \(\mathcal{F}\) on \(\mathcal{C}\). Hence the equality in (2).

The following lemma will be used to see what happens if we change a partial universe, or to compare cohomology of the small and big étale sites.

Lemma

Let \(\mathcal{C}\) and \(\mathcal{D}\) be sites. Let \(u : \mathcal{C} \to \mathcal{D}\) be a functor. Assume \(u\) satisfies the hypotheses of Sites, Lemma 00XU. Let \(g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\) be the associated morphism of topoi. For any abelian sheaf \(\mathcal{F}\) on \(\mathcal{D}\) we have isomorphisms \[R\Gamma(\mathcal{C}, g^{-1}\mathcal{F}) = R\Gamma(\mathcal{D}, \mathcal{F}),\] in particular \(H^p(\mathcal{C}, g^{-1}\mathcal{F}) = H^p(\mathcal{D}, \mathcal{F})\) and for any \(U \in \Ob(\mathcal{C})\) we have isomorphisms \[R\Gamma(U, g^{-1}\mathcal{F}) = R\Gamma(u(U), \mathcal{F}),\] in particular \(H^p(U, g^{-1}\mathcal{F}) = H^p(u(U), \mathcal{F})\). All of these isomorphisms are functorial in \(\mathcal{F}\).

Proof

Since it is clear that \(\Gamma(\mathcal{C}, g^{-1}\mathcal{F}) = \Gamma(\mathcal{D}, \mathcal{F})\) by hypothesis (e), it suffices to show that \(g^{-1}\) transforms injective abelian sheaves into injective abelian sheaves. As usual we use Homology, Lemma 015Z to see this. The left adjoint to \(g^{-1}\) is \(g_! = f^{-1}\) with the notation of Sites, Lemma 00XU which is an exact functor. Hence the lemma does indeed apply.

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}\)-modules. Let \(\varphi : U \to V\) be a morphism of \(\mathcal{C}\). Then there is a canonical restriction mapping [01FV]\[\begin{equation} H^n(V, \mathcal{F}) \longrightarrow H^n(U, \mathcal{F}), \quad \xi \longmapsto \xi|_U \end{equation}\] functorial in \(\mathcal{F}\). Namely, choose any injective resolution \(\mathcal{F} \to \mathcal{I}^\bullet\). The restriction mappings of the sheaves \(\mathcal{I}^p\) give a morphism of complexes \[\Gamma(V, \mathcal{I}^\bullet) \longrightarrow \Gamma(U, \mathcal{I}^\bullet)\] The LHS is a complex representing \(R\Gamma(V, \mathcal{F})\) and the RHS is a complex representing \(R\Gamma(U, \mathcal{F})\). We get the map on cohomology groups by applying the functor \(H^n\). As indicated we will use the notation \(\xi \mapsto \xi|_U\) to denote this map. Thus the rule \(U \mapsto H^n(U, \mathcal{F})\) is a presheaf of \(\mathcal{O}\)-modules. This presheaf is customarily denoted \(\underline{H}^n(\mathcal{F})\). We will give another interpretation of this presheaf in Lemma 03AY.

The following lemma says that it is possible to kill higher cohomology classes by going to a covering.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}\)-modules. Let \(U\) be an object of \(\mathcal{C}\). Let \(n > 0\) and let \(\xi \in H^n(U, \mathcal{F})\). Then there exists a covering \(\{U_i \to U\}\) of \(\mathcal{C}\) such that \(\xi|_{U_i} = 0\) for all \(i \in I\).

Proof

Let \(\mathcal{F} \to \mathcal{I}^\bullet\) be an injective resolution. Then \[H^n(U, \mathcal{F}) = \frac{\Ker(\mathcal{I}^n(U) \to \mathcal{I}^{n + 1}(U))} {\Im(\mathcal{I}^{n - 1}(U) \to \mathcal{I}^n(U))}.\] Pick an element \(\tilde \xi \in \mathcal{I}^n(U)\) representing the cohomology class in the presentation above. Since \(\mathcal{I}^\bullet\) is an injective resolution of \(\mathcal{F}\) and \(n > 0\) we see that the complex \(\mathcal{I}^\bullet\) is exact in degree \(n\). Hence \(\Im(\mathcal{I}^{n - 1} \to \mathcal{I}^n) = \Ker(\mathcal{I}^n \to \mathcal{I}^{n + 1})\) as sheaves. Since \(\tilde \xi\) is a section of the kernel sheaf over \(U\) we conclude there exists a covering \(\{U_i \to U\}\) of the site such that \(\tilde \xi|_{U_i}\) is the image under \(d\) of a section \(\xi_i \in \mathcal{I}^{n - 1}(U_i)\). By our definition of the restriction \(\xi|_{U_i}\) as corresponding to the class of \(\tilde \xi|_{U_i}\) we conclude.

Lemma

Let \(f : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D})\) be a morphism of ringed sites corresponding to the continuous functor \(u : \mathcal{D} \to \mathcal{C}\). For any \(\mathcal{F} \in \Ob(\textit{Mod}(\mathcal{O}_\mathcal{C}))\) the sheaf \(R^if_*\mathcal{F}\) is the sheaf associated to the presheaf \[V \longmapsto H^i(u(V), \mathcal{F})\]

Proof

Let \(\mathcal{F} \to \mathcal{I}^\bullet\) be an injective resolution. Then \(R^if_*\mathcal{F}\) is by definition the \(i\)th cohomology sheaf of the complex \[f_*\mathcal{I}^0 \to f_*\mathcal{I}^1 \to f_*\mathcal{I}^2 \to \ldots\] By definition of the abelian category structure on \(\mathcal{O}_\mathcal{D}\)-modules this cohomology sheaf is the sheaf associated to the presheaf \[V \longmapsto \frac{\Ker(f_*\mathcal{I}^i(V) \to f_*\mathcal{I}^{i + 1}(V))} {\Im(f_*\mathcal{I}^{i - 1}(V) \to f_*\mathcal{I}^i(V))}\] and this is obviously equal to \[\frac{\Ker(\mathcal{I}^i(u(V)) \to \mathcal{I}^{i + 1}(u(V)))} {\Im(\mathcal{I}^{i - 1}(u(V)) \to \mathcal{I}^i(u(V)))}\] which is equal to \(H^i(u(V), \mathcal{F})\) and we win.

The Čech complex and Čech cohomology

Let \(\mathcal{C}\) be a category. Let \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) be a family of morphisms with fixed target, see Sites, Definition 0396. Assume that all fibre products \(U_{i_0} \times_U \ldots \times_U U_{i_p}\) exist in \(\mathcal{C}\). Let \(\mathcal{F}\) be an abelian presheaf on \(\mathcal{C}\). Set \[\check{\mathcal{C}}^p(\mathcal{U}, \mathcal{F}) = \prod\nolimits_{(i_0, \ldots, i_p) \in I^{p + 1}} \mathcal{F}(U_{i_0} \times_U \ldots \times_U U_{i_p}).\] This is an abelian group. For \(s \in \check{\mathcal{C}}^p(\mathcal{U}, \mathcal{F})\) we denote \(s_{i_0\ldots i_p}\) its value in the factor \(\mathcal{F}(U_{i_0} \times_U \ldots \times_U U_{i_p})\). We define \[d : \check{\mathcal{C}}^p(\mathcal{U}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^{p + 1}(\mathcal{U}, \mathcal{F})\] by the formula [03AL]\[\begin{equation} d(s)_{i_0\ldots i_{p + 1}} = \sum\nolimits_{j = 0}^{p + 1} (-1)^j s_{i_0\ldots \hat i_j \ldots i_{p + 1}} |_{U_{i_0} \times_U \ldots \times_U U_{i_{p + 1}}} \end{equation}\] where the restriction is via the projection map \[U_{i_0} \times_U \ldots \times_U U_{i_{p + 1}} \longrightarrow U_{i_0} \times_U \ldots \times_U \widehat{U_{i_j}} \times_U \ldots \times_U U_{i_{p + 1}}.\] It is straightforward to see that \(d \circ d = 0\). In other words \(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})\) is a complex.

Definition

Let \(\mathcal{C}\) be a category. Let \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) be a family of morphisms with fixed target such that all fibre products \(U_{i_0} \times_U \ldots \times_U U_{i_p}\) exist in \(\mathcal{C}\). Let \(\mathcal{F}\) be an abelian presheaf on \(\mathcal{C}\). The complex \(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})\) is the Čech complex associated to \(\mathcal{F}\) and the family \(\mathcal{U}\). Its cohomology groups \(H^i(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}))\) are called the Čech cohomology groups of \(\mathcal{F}\) with respect to \(\mathcal{U}\). They are denoted \(\check H^i(\mathcal{U}, \mathcal{F})\).

We observe that any covering \(\{U_i \to U\}\) of a site \(\mathcal{C}\) is a family of morphisms with fixed target to which the definition applies.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{F}\) be an abelian presheaf on \(\mathcal{C}\). The following are equivalent

1. \(\mathcal{F}\) is an abelian sheaf on \(\mathcal{C}\) and

2. for every covering \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) of the site \(\mathcal{C}\) the natural map \[\mathcal{F}(U) \to \check{H}^0(\mathcal{U}, \mathcal{F})\] (see Sites, Section 00W1) is bijective.

Proof

This is true since the sheaf condition is exactly that \(\mathcal{F}(U) \to \check{H}^0(\mathcal{U}, \mathcal{F})\) is bijective for every covering of \(\mathcal{C}\).

Let \(\mathcal{C}\) be a category. Let \(\mathcal{U} = \{U_i \to U\}_{i\in I}\) be a family of morphisms of \(\mathcal{C}\) with fixed target such that all fibre products \(U_{i_0} \times_U \ldots \times_U U_{i_p}\) exist in \(\mathcal{C}\). Let \(\mathcal{V} = \{V_j \to V\}_{j\in J}\) be another. Let \(f : U \to V\), \(\alpha : I \to J\) and \(f_i : U_i \to V_{\alpha(i)}\) be a morphism of families of morphisms with fixed target, see Sites, Section 00VS. In this case we get a map of Čech complexes [03F4]\[\begin{equation} \varphi : \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \end{equation}\] which in degree \(p\) is given by \[\varphi(s)_{i_0 \ldots i_p} = (f_{i_0} \times \ldots \times f_{i_p})^*s_{\alpha(i_0) \ldots \alpha(i_p)}\]

Čech cohomology as a functor on presheaves

Warning: In this section we work exclusively with abelian presheaves on a category. The results are completely wrong in the setting of sheaves and categories of sheaves!

Let \(\mathcal{C}\) be a category. Let \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) be a family of morphisms with fixed target such that all fibre products \(U_{i_0} \times_U \ldots \times_U U_{i_p}\) exist in \(\mathcal{C}\). Let \(\mathcal{F}\) be an abelian presheaf on \(\mathcal{C}\). The construction \[\mathcal{F} \longmapsto \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})\] is functorial in \(\mathcal{F}\). In fact, it is a functor [03AP]\[\begin{equation} \check{\mathcal{C}}^\bullet(\mathcal{U}, -) : \textit{PAb}(\mathcal{C}) \longrightarrow \text{Comp}^{+}(\textit{Ab}) \end{equation}\] see Derived Categories, Definition 013H for notation. Recall that the category of bounded below complexes in an abelian category is an abelian category, see Homology, Lemma 0114.

Lemma

The functor given by Equation (03AP) is an exact functor (see Homology, Lemma 010N).

Proof

For any object \(W\) of \(\mathcal{C}\) the functor \(\mathcal{F} \mapsto \mathcal{F}(W)\) is an additive exact functor from \(\textit{PAb}(\mathcal{C})\) to \(\textit{Ab}\). The terms \(\check{\mathcal{C}}^p(\mathcal{U}, \mathcal{F})\) of the complex are products of these exact functors and hence exact. Moreover a sequence of complexes is exact if and only if the sequence of terms in a given degree is exact. Hence the lemma follows.

Lemma

Let \(\mathcal{C}\) be a category. Let \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) be a family of morphisms with fixed target such that all fibre products \(U_{i_0} \times_U \ldots \times_U U_{i_p}\) exist in \(\mathcal{C}\). The functors \(\mathcal{F} \mapsto \check{H}^n(\mathcal{U}, \mathcal{F})\) form a \(\delta\)-functor from the abelian category \(\textit{PAb}(\mathcal{C})\) to the category of \(\mathbf{Z}\)-modules (see Homology, Definition 010Q).

Proof

By Lemma 03AQ a short exact sequence of abelian presheaves \(0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\) is turned into a short exact sequence of complexes of \(\mathbf{Z}\)-modules. Hence we can use Homology, Lemma 0117 to get the boundary maps \(\delta_{\mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3} : \check{H}^n(\mathcal{U}, \mathcal{F}_3) \to \check{H}^{n + 1}(\mathcal{U}, \mathcal{F}_1)\) and a corresponding long exact sequence. We omit the verification that these maps are compatible with maps between short exact sequences of presheaves.

Lemma

Let \(\mathcal{C}\) be a category. Let \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) be a family of morphisms with fixed target such that all fibre products \(U_{i_0} \times_U \ldots \times_U U_{i_p}\) exist in \(\mathcal{C}\). Consider the chain complex \(\mathbf{Z}_{\mathcal{U}, \bullet}\) of abelian presheaves \[\ldots \to \bigoplus_{i_0i_1i_2} \mathbf{Z}_{U_{i_0} \times_U U_{i_1} \times_U U_{i_2}} \to \bigoplus_{i_0i_1} \mathbf{Z}_{U_{i_0} \times_U U_{i_1}} \to \bigoplus_{i_0} \mathbf{Z}_{U_{i_0}} \to 0 \to \ldots\] where the last nonzero term is placed in degree \(0\) and where the map \[\mathbf{Z}_{U_{i_0} \times_U \ldots \times_U U_{i_{p + 1}}} \longrightarrow \mathbf{Z}_{U_{i_0} \times_U \ldots \widehat{U_{i_j}} \ldots \times_U U_{i_{p + 1}}}\] is given by \((-1)^j\) times the canonical map. Then there is an isomorphism \[\Hom_{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_{\mathcal{U}, \bullet}, \mathcal{F}) = \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})\] functorial in \(\mathcal{F} \in \Ob(\textit{PAb}(\mathcal{C}))\).

Proof

This is a tautology based on the fact that \[\begin{align*} \Hom_{\textit{PAb}(\mathcal{C})}( \bigoplus_{i_0 \ldots i_p} \mathbf{Z}_{U_{i_0} \times_U \ldots \times_U U_{i_p}}, \mathcal{F}) & = \prod_{i_0 \ldots i_p} \Hom_{\textit{PAb}(\mathcal{C})}( \mathbf{Z}_{U_{i_0} \times_U \ldots \times_U U_{i_p}}, \mathcal{F}) \\ & = \prod_{i_0 \ldots i_p} \mathcal{F}(U_{i_0} \times_U \ldots \times_U U_{i_p}) \end{align*}\] see Modules on Sites, Lemma 03A8.

Lemma

Let \(\mathcal{C}\) be a category. Let \(\mathcal{U} = \{f_i : U_i \to U\}_{i \in I}\) be a family of morphisms with fixed target such that all fibre products \(U_{i_0} \times_U \ldots \times_U U_{i_p}\) exist in \(\mathcal{C}\). The chain complex \(\mathbf{Z}_{\mathcal{U}, \bullet}\) of presheaves of Lemma 03AS above is exact in positive degrees, i.e., the homology presheaves \(H_i(\mathbf{Z}_{\mathcal{U}, \bullet})\) are zero for \(i > 0\).

Proof

Let \(V\) be an object of \(\mathcal{C}\). We have to show that the chain complex of abelian groups \(\mathbf{Z}_{\mathcal{U}, \bullet}(V)\) is exact in degrees \(> 0\). This is the complex \[\xymatrix{ \ldots \ar[d] \\ \bigoplus_{i_0i_1i_2} \mathbf{Z}[ \Mor_\mathcal{C}(V, U_{i_0} \times_U U_{i_1} \times_U U_{i_2}) ] \ar[d] \\ \bigoplus_{i_0i_1} \mathbf{Z}[ \Mor_\mathcal{C}(V, U_{i_0} \times_U U_{i_1}) ] \ar[d] \\ \bigoplus_{i_0} \mathbf{Z}[ \Mor_\mathcal{C}(V, U_{i_0}) ] \ar[d] \\ 0 }\] For any morphism \(\varphi : V \to U\) denote \(\Mor_\varphi(V, U_i) = \{\varphi_i : V \to U_i \mid f_i \circ \varphi_i = \varphi\}\). We will use a similar notation for \(\Mor_\varphi(V, U_{i_0} \times_U \ldots \times_U U_{i_p})\). Note that composing with the various projection maps between the fibred products \(U_{i_0} \times_U \ldots \times_U U_{i_p}\) preserves these morphism sets. Hence we see that the complex above is the same as the complex \[\xymatrix{ \ldots \ar[d] \\ \bigoplus_\varphi \bigoplus_{i_0i_1i_2} \mathbf{Z}[ \Mor_\varphi(V, U_{i_0} \times_U U_{i_1} \times_U U_{i_2}) ] \ar[d] \\ \bigoplus_\varphi \bigoplus_{i_0i_1} \mathbf{Z}[ \Mor_\varphi(V, U_{i_0} \times_U U_{i_1}) ] \ar[d] \\ \bigoplus_\varphi \bigoplus_{i_0} \mathbf{Z}[ \Mor_\varphi(V, U_{i_0}) ] \ar[d] \\ 0 }\] Next, we make the remark that we have \[\Mor_\varphi(V, U_{i_0} \times_U \ldots \times_U U_{i_p}) = \Mor_\varphi(V, U_{i_0}) \times \ldots \times \Mor_\varphi(V, U_{i_p})\] Using this and the fact that \(\mathbf{Z}[A] \oplus \mathbf{Z}[B] = \mathbf{Z}[A \amalg B]\) we see that the complex becomes \[\xymatrix{ \ldots \ar[d] \\ \bigoplus_\varphi \mathbf{Z}\left[ \coprod_{i_0i_1i_2} \Mor_\varphi(V, U_{i_0}) \times \Mor_\varphi(V, U_{i_1}) \times \Mor_\varphi(V, U_{i_2}) \right] \ar[d] \\ \bigoplus_\varphi \mathbf{Z}\left[ \coprod_{i_0i_1} \Mor_\varphi(V, U_{i_0}) \times \Mor_\varphi(V, U_{i_1}) \right] \ar[d] \\ \bigoplus_\varphi \mathbf{Z}\left[ \coprod_{i_0} \Mor_\varphi(V, U_{i_0}) \right] \ar[d] \\ 0 }\] Finally, on setting \(S_\varphi = \coprod_{i \in I} \Mor_\varphi(V, U_i)\) we see that we get \[\bigoplus\nolimits_\varphi \left(\ldots \to \mathbf{Z}[S_\varphi \times S_\varphi \times S_\varphi] \to \mathbf{Z}[S_\varphi \times S_\varphi] \to \mathbf{Z}[S_\varphi] \to 0 \to \ldots \right)\] Thus we have simplified our task. Namely, it suffices to show that for any nonempty set \(S\) the (extended) complex of free abelian groups \[\ldots \to \mathbf{Z}[S \times S \times S] \to \mathbf{Z}[S \times S] \to \mathbf{Z}[S] \xrightarrow{\Sigma} \mathbf{Z} \to 0 \to \ldots\] is exact in all degrees. To see this fix an element \(s \in S\), and use the homotopy \[n_{(s_0, \ldots, s_p)} \longmapsto n_{(s, s_0, \ldots, s_p)}\] with obvious notations.

Lemma

Let \(\mathcal{C}\) be a category. Let \(\mathcal{U} = \{f_i : U_i \to U\}_{i \in I}\) be a family of morphisms with fixed target such that all fibre products \(U_{i_0} \times_U \ldots \times_U U_{i_p}\) exist in \(\mathcal{C}\). Let \(\mathcal{O}\) be a presheaf of rings on \(\mathcal{C}\). The chain complex \[\mathbf{Z}_{\mathcal{U}, \bullet} \otimes_{p, \mathbf{Z}} \mathcal{O}\] is exact in positive degrees. Here \(\mathbf{Z}_{\mathcal{U}, \bullet}\) is the chain complex of Lemma 03AS, and the tensor product is over the constant presheaf of rings with value \(\mathbf{Z}\).

Proof

Let \(V\) be an object of \(\mathcal{C}\). In the proof of Lemma 03AT we saw that \(\mathbf{Z}_{\mathcal{U}, \bullet}(V)\) is isomorphic as a complex to a direct sum of complexes which are homotopic to \(\mathbf{Z}\) placed in degree zero. Hence also \(\mathbf{Z}_{\mathcal{U}, \bullet}(V) \otimes_\mathbf{Z} \mathcal{O}(V)\) is isomorphic as a complex to a direct sum of complexes which are homotopic to \(\mathcal{O}(V)\) placed in degree zero. Or you can use Modules on Sites, Lemma 03EZ, which applies since the presheaves \(\mathbf{Z}_{\mathcal{U}, i}\) are flat, and the proof of Lemma 03AT shows that \(H_0(\mathbf{Z}_{\mathcal{U}, \bullet})\) is a flat presheaf also.

Lemma

Let \(\mathcal{C}\) be a category. Let \(\mathcal{U} = \{f_i : U_i \to U\}_{i \in I}\) be a family of morphisms with fixed target such that all fibre products \(U_{i_0} \times_U \ldots \times_U U_{i_p}\) exist in \(\mathcal{C}\). The Čech cohomology functors \(\check{H}^p(\mathcal{U}, -)\) are canonically isomorphic as a \(\delta\)-functor to the right derived functors of the functor \[\check{H}^0(\mathcal{U}, -) : \textit{PAb}(\mathcal{C}) \longrightarrow \textit{Ab}.\] Moreover, there is a functorial quasi-isomorphism \[\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \longrightarrow R\check{H}^0(\mathcal{U}, \mathcal{F})\] where the right hand side indicates the derived functor \[R\check{H}^0(\mathcal{U}, -) : D^{+}(\textit{PAb}(\mathcal{C})) \longrightarrow D^{+}(\mathbf{Z})\] of the left exact functor \(\check{H}^0(\mathcal{U}, -)\).

Proof

Note that the category of abelian presheaves has enough injectives, see Injectives, Proposition 01DK. Note that \(\check{H}^0(\mathcal{U}, -)\) is a left exact functor from the category of abelian presheaves to the category of \(\mathbf{Z}\)-modules. Hence the derived functor and the right derived functor exist, see Derived Categories, Section 0156.

Let \(\mathcal{I}\) be an injective abelian presheaf. In this case the functor \(\Hom_{\textit{PAb}(\mathcal{C})}(-, \mathcal{I})\) is exact on \(\textit{PAb}(\mathcal{C})\). By Lemma 03AS we have \[\Hom_{\textit{PAb}(\mathcal{C})}( \mathbf{Z}_{\mathcal{U}, \bullet}, \mathcal{I}) = \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}).\] By Lemma 03AT we have that \(\mathbf{Z}_{\mathcal{U}, \bullet}\) is exact in positive degrees. Hence by the exactness of Hom into \(\mathcal{I}\) mentioned above we see that \(\check{H}^i(\mathcal{U}, \mathcal{I}) = 0\) for all \(i > 0\). Thus the \(\delta\)-functor \((\check{H}^n, \delta)\) (see Lemma 03AR) satisfies the assumptions of Homology, Lemma 010T, and hence is a universal \(\delta\)-functor.

By Derived Categories, Lemma 015B also the sequence \(R^i\check{H}^0(\mathcal{U}, -)\) forms a universal \(\delta\)-functor. By the uniqueness of universal \(\delta\)-functors, see Homology, Lemma 010U we conclude that \(R^i\check{H}^0(\mathcal{U}, -) = \check{H}^i(\mathcal{U}, -)\). This is enough for most applications and the reader is suggested to skip the rest of the proof.

Let \(\mathcal{F}\) be any abelian presheaf on \(\mathcal{C}\). Choose an injective resolution \(\mathcal{F} \to \mathcal{I}^\bullet\) in the category \(\textit{PAb}(\mathcal{C})\). Consider the double complex \(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet)\) with terms \(\check{\mathcal{C}}^p(\mathcal{U}, \mathcal{I}^q)\). Next, consider the total complex \(\text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet))\) associated to this double complex, see Homology, Section 0FNB. There is a map of complexes \[\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \longrightarrow \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet))\] coming from the maps \(\check{\mathcal{C}}^p(\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}^p(\mathcal{U}, \mathcal{I}^0)\) and there is a map of complexes \[\check{H}^0(\mathcal{U}, \mathcal{I}^\bullet) \longrightarrow \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet))\] coming from the maps \(\check{H}^0(\mathcal{U}, \mathcal{I}^q) \to \check{\mathcal{C}}^0(\mathcal{U}, \mathcal{I}^q)\). Both of these maps are quasi-isomorphisms by an application of Homology, Lemma 0133. Namely, the columns of the double complex are exact in positive degrees because the Čech complex as a functor is exact (Lemma 03AQ) and the rows of the double complex are exact in positive degrees since as we just saw the higher Čech cohomology groups of the injective presheaves \(\mathcal{I}^q\) are zero. Since quasi-isomorphisms become invertible in \(D^{+}(\mathbf{Z})\) this gives the last displayed morphism of the lemma. We omit the verification that this morphism is functorial.

Čech cohomology and cohomology

The relationship between cohomology and Čech cohomology comes from the fact that the Čech cohomology of an injective abelian sheaf is zero. To see this we note that an injective abelian sheaf is an injective abelian presheaf and then we apply results in Čech cohomology in the preceding section.

Lemma

Let \(\mathcal{C}\) be a site. An injective abelian sheaf is also injective as an object in the category \(\textit{PAb}(\mathcal{C})\).

Proof

Apply Homology, Lemma 015Z to the categories \(\mathcal{A} = \textit{Ab}(\mathcal{C})\), \(\mathcal{B} = \textit{PAb}(\mathcal{C})\), the inclusion functor and sheafification. (See Modules on Sites, Section 03CM to see that all assumptions of the lemma are satisfied.)

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) be a covering of \(\mathcal{C}\). Let \(\mathcal{I}\) be an injective abelian sheaf, i.e., an injective object of \(\textit{Ab}(\mathcal{C})\). Then \[\check{H}^p(\mathcal{U}, \mathcal{I}) = \left\{ \begin{matrix} \mathcal{I}(U) & \text{if} & p = 0 \\ 0 & \text{if} & p > 0 \end{matrix} \right.\]

Proof

By Lemma 03F6 we see that \(\mathcal{I}\) is an injective object in \(\textit{PAb}(\mathcal{C})\). Hence we can apply Lemma 03AU (or its proof) to see the vanishing of higher Čech cohomology group. For the zeroth see Lemma 03AN.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) be a covering of \(\mathcal{C}\). There is a transformation \[\check{\mathcal{C}}^\bullet(\mathcal{U}, -) \longrightarrow R\Gamma(U, -)\] of functors \(\textit{Ab}(\mathcal{C}) \to D^{+}(\mathbf{Z})\). In particular this gives a transformation of functors \(\check{H}^p(\mathcal{U}, \mathcal{F}) \to H^p(U, \mathcal{F})\) for \(\mathcal{F}\) ranging over \(\textit{Ab}(\mathcal{C})\).

Proof

Let \(\mathcal{F}\) be an abelian sheaf. Choose an injective resolution \(\mathcal{F} \to \mathcal{I}^\bullet\). Consider the double complex \(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet)\) with terms \(\check{\mathcal{C}}^p(\mathcal{U}, \mathcal{I}^q)\). Next, consider the associated total complex \(\text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet))\), see Homology, Definition 012Z. There is a map of complexes \[\alpha : \Gamma(U, \mathcal{I}^\bullet) \longrightarrow \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet))\] coming from the maps \(\mathcal{I}^q(U) \to \check{H}^0(\mathcal{U}, \mathcal{I}^q)\) and a map of complexes \[\beta : \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \longrightarrow \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet))\] coming from the map \(\mathcal{F} \to \mathcal{I}^0\). We can apply Homology, Lemma 0133 to see that \(\alpha\) is a quasi-isomorphism. Namely, Lemma 03AW implies that the \(q\)th row of the double complex \(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet)\) is a resolution of \(\Gamma(U, \mathcal{I}^q)\). Hence \(\alpha\) becomes invertible in \(D^{+}(\mathbf{Z})\) and the transformation of the lemma is the composition of \(\beta\) followed by the inverse of \(\alpha\). We omit the verification that this is functorial.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{G}\) be an abelian sheaf on \(\mathcal{C}\). Let \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) be a covering of \(\mathcal{C}\). The map \[\check{H}^1(\mathcal{U}, \mathcal{G}) \longrightarrow H^1(U, \mathcal{G})\] is injective and identifies \(\check{H}^1(\mathcal{U}, \mathcal{G})\) via the bijection of Lemma 03AJ with the set of isomorphism classes of \(\mathcal{G}|_U\)-torsors which restrict to trivial torsors over each \(U_i\).

Proof

To see this we construct an inverse map. Namely, let \(\mathcal{F}\) be a \(\mathcal{G}|_U\)-torsor on \(\mathcal{C}/U\) whose restriction to \(\mathcal{C}/U_i\) is trivial. By Lemma 03AI this means there exists a section \(s_i \in \mathcal{F}(U_i)\). On \(U_{i_0} \times_U U_{i_1}\) there is a unique section \(s_{i_0i_1}\) of \(\mathcal{G}\) such that \(s_{i_0i_1} \cdot s_{i_0}|_{U_{i_0} \times_U U_{i_1}} = s_{i_1}|_{U_{i_0} \times_U U_{i_1}}\). An easy computation shows that \(s_{i_0i_1}\) is a Čech cocycle and that its class is well defined (i.e., does not depend on the choice of the sections \(s_i\)). The inverse maps the isomorphism class of \(\mathcal{F}\) to the cohomology class of the cocycle \((s_{i_0i_1})\). We omit the verification that this map is indeed an inverse.

Lemma

Let \(\mathcal{C}\) be a site. Consider the functor \(i : \textit{Ab}(\mathcal{C}) \to \textit{PAb}(\mathcal{C})\). It is a left exact functor with right derived functors given by \[R^pi(\mathcal{F}) = \underline{H}^p(\mathcal{F}) : U \longmapsto H^p(U, \mathcal{F})\] see discussion in Section 01FU.

Proof

It is clear that \(i\) is left exact. Choose an injective resolution \(\mathcal{F} \to \mathcal{I}^\bullet\). By definition \(R^pi\) is the \(p\)th cohomology presheaf of the complex \(\mathcal{I}^\bullet\). In other words, the sections of \(R^pi(\mathcal{F})\) over an object \(U\) of \(\mathcal{C}\) are given by \[\frac{\Ker(\mathcal{I}^n(U) \to \mathcal{I}^{n + 1}(U))} {\Im(\mathcal{I}^{n - 1}(U) \to \mathcal{I}^n(U))}.\] which is the definition of \(H^p(U, \mathcal{F})\).

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) be a covering of \(\mathcal{C}\). For any abelian sheaf \(\mathcal{F}\) there is a spectral sequence \((E_r, d_r)_{r \geq 0}\) with \[E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F}))\] converging to \(H^{p + q}(U, \mathcal{F})\). This spectral sequence is functorial in \(\mathcal{F}\).

Proof

This is a Grothendieck spectral sequence (see Derived Categories, Lemma 015N) for the functors \[i : \textit{Ab}(\mathcal{C}) \to \textit{PAb}(\mathcal{C}) \quad\text{and}\quad \check{H}^0(\mathcal{U}, - ) : \textit{PAb}(\mathcal{C}) \to \textit{Ab}.\] Namely, we have \(\check{H}^0(\mathcal{U}, i(\mathcal{F})) = \mathcal{F}(U)\) by Lemma 03AN. We have that \(i(\mathcal{I})\) is Čech acyclic by Lemma 03AW. And we have that \(\check{H}^p(\mathcal{U}, -) = R^p\check{H}^0(\mathcal{U}, -)\) as functors on \(\textit{PAb}(\mathcal{C})\) by Lemma 03AU. Putting everything together gives the lemma.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) be a covering. Let \(\mathcal{F} \in \Ob(\textit{Ab}(\mathcal{C}))\). Assume that \(H^i(U_{i_0} \times_U \ldots \times_U U_{i_p}, \mathcal{F}) = 0\) for all \(i > 0\), all \(p \geq 0\) and all \(i_0, \ldots, i_p \in I\). Then \(\check{H}^p(\mathcal{U}, \mathcal{F}) = H^p(U, \mathcal{F})\).

Proof

We will use the spectral sequence of Lemma 03AZ. The assumptions mean that \(E_2^{p, q} = 0\) for all \((p, q)\) with \(q \not = 0\). Hence the spectral sequence degenerates at \(E_2\) and the result follows.

Lemma

Let \(\mathcal{C}\) be a site. Let \[0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0\] be a short exact sequence of abelian sheaves on \(\mathcal{C}\). Let \(U\) be an object of \(\mathcal{C}\). If there exists a cofinal system of coverings \(\mathcal{U}\) of \(U\) such that \(\check{H}^1(\mathcal{U}, \mathcal{F}) = 0\), then the map \(\mathcal{G}(U) \to \mathcal{H}(U)\) is surjective.

Proof

Take an element \(s \in \mathcal{H}(U)\). Choose a covering \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) such that (a) \(\check{H}^1(\mathcal{U}, \mathcal{F}) = 0\) and (b) \(s|_{U_i}\) is the image of a section \(s_i \in \mathcal{G}(U_i)\). Since we can certainly find a covering such that (b) holds it follows from the assumptions of the lemma that we can find a covering such that (a) and (b) both hold. Consider the sections \[s_{i_0i_1} = s_{i_1}|_{U_{i_0} \times_U U_{i_1}} - s_{i_0}|_{U_{i_0} \times_U U_{i_1}}.\] Since \(s_i\) lifts \(s\) we see that \(s_{i_0i_1} \in \mathcal{F}(U_{i_0} \times_U U_{i_1})\). By the vanishing of \(\check{H}^1(\mathcal{U}, \mathcal{F})\) we can find sections \(t_i \in \mathcal{F}(U_i)\) such that \[s_{i_0i_1} = t_{i_1}|_{U_{i_0} \times_U U_{i_1}} - t_{i_0}|_{U_{i_0} \times_U U_{i_1}}.\] Then clearly the sections \(s_i - t_i\) satisfy the sheaf condition and glue to a section of \(\mathcal{G}\) over \(U\) which maps to \(s\). Hence we win.

Lemma

(Variant of Cohomology, Lemma 01EV.) Let \(\mathcal{C}\) be a site. Let \(\text{Cov}_\mathcal{C}\) be the set of coverings of \(\mathcal{C}\) (see Sites, Definition 00VH). Let \(\mathcal{B} \subset \Ob(\mathcal{C})\), and \(\text{Cov} \subset \text{Cov}_\mathcal{C}\) be subsets. Let \(\mathcal{F}\) be an abelian sheaf on \(\mathcal{C}\). Assume that

1. For every \(\mathcal{U} \in \text{Cov}\), \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) we have \(U, U_i \in \mathcal{B}\) and every \(U_{i_0} \times_U \ldots \times_U U_{i_p} \in \mathcal{B}\).

2. For every \(U \in \mathcal{B}\) the coverings of \(U\) occurring in \(\text{Cov}\) is a cofinal system of coverings of \(U\).

3. For every \(\mathcal{U} \in \text{Cov}\) we have \(\check{H}^p(\mathcal{U}, \mathcal{F}) = 0\) for all \(p > 0\).

Then \(H^p(U, \mathcal{F}) = 0\) for all \(p > 0\) and any \(U \in \mathcal{B}\).

Proof

Let \(\mathcal{F}\) and \(\text{Cov}\) be as in the lemma. We will indicate this by saying “\(\mathcal{F}\) has vanishing higher Čech cohomology for any \(\mathcal{U} \in \text{Cov}\)”. Choose an embedding \(\mathcal{F} \to \mathcal{I}\) into an injective abelian sheaf. By Lemma 03AW \(\mathcal{I}\) has vanishing higher Čech cohomology for any \(\mathcal{U} \in \text{Cov}\). Let \(\mathcal{Q} = \mathcal{I}/\mathcal{F}\) so that we have a short exact sequence \[0 \to \mathcal{F} \to \mathcal{I} \to \mathcal{Q} \to 0.\] By Lemma 03F8 and our assumption (2) this sequence gives rise to an exact sequence \[0 \to \mathcal{F}(U) \to \mathcal{I}(U) \to \mathcal{Q}(U) \to 0.\] for every \(U \in \mathcal{B}\). Hence for any \(\mathcal{U} \in \text{Cov}\) we get a short exact sequence of Čech complexes \[0 \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}) \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{Q}) \to 0\] since each term in the Čech complex is made up out of a product of values over elements of \(\mathcal{B}\) by assumption (1). In particular we have a long exact sequence of Čech cohomology groups for any covering \(\mathcal{U} \in \text{Cov}\). This implies that \(\mathcal{Q}\) is also an abelian sheaf with vanishing higher Čech cohomology for all \(\mathcal{U} \in \text{Cov}\).

Next, we look at the long exact cohomology sequence \[\xymatrix{ 0 \ar[r] & H^0(U, \mathcal{F}) \ar[r] & H^0(U, \mathcal{I}) \ar[r] & H^0(U, \mathcal{Q}) \ar[lld] \\ & H^1(U, \mathcal{F}) \ar[r] & H^1(U, \mathcal{I}) \ar[r] & H^1(U, \mathcal{Q}) \ar[lld] \\ & \ldots & \ldots & \ldots \\ }\] for any \(U \in \mathcal{B}\). Since \(\mathcal{I}\) is injective we have \(H^n(U, \mathcal{I}) = 0\) for \(n > 0\) (see Derived Categories, Lemma 015B). By the above we see that \(H^0(U, \mathcal{I}) \to H^0(U, \mathcal{Q})\) is surjective and hence \(H^1(U, \mathcal{F}) = 0\). Since \(\mathcal{F}\) was an arbitrary abelian sheaf with vanishing higher Čech cohomology for all \(\mathcal{U} \in \text{Cov}\) we conclude that also \(H^1(U, \mathcal{Q}) = 0\) since \(\mathcal{Q}\) is another of these sheaves (see above). By the long exact sequence this in turn implies that \(H^2(U, \mathcal{F}) = 0\). And so on and so forth.

Second cohomology and gerbes

Let \(p : \mathcal{S} \to \mathcal{C}\) be a gerbe over a site all of whose automorphism groups are commutative. In this situation the first and second cohomology groups of the sheaf of automorphisms (Stacks, Lemma 0CJY) controls the existence of objects.

The following lemma will be made obsolete by a more complete discussion of this relationship we will add in the future.

Lemma

Let \(\mathcal{C}\) be a site. Let \(p : \mathcal{S} \to \mathcal{C}\) be a gerbe over a site whose automorphism sheaves are abelian. Let \(\mathcal{G}\) be the sheaf of abelian groups constructed in Stacks, Lemma 0CJY. Let \(U\) be an object of \(\mathcal{C}\) such that

1. there exists a cofinal system of coverings \(\{U_i \to U\}\) of \(U\) in \(\mathcal{C}\) such that \(H^1(U_i, \mathcal{G}) = 0\) and \(H^1(U_i \times_U U_j, \mathcal{G}) = 0\) for all \(i, j\), and

2. \(H^2(U, \mathcal{G}) = 0\).

Then there exists an object of \(\mathcal{S}\) lying over \(U\).

Proof

By Stacks, Definition 06NZ there exists a covering \(\mathcal{U} = \{U_i \to U\}\) and \(x_i\) in \(\mathcal{S}\) lying over \(U_i\). Write \(U_{ij} = U_i \times_U U_j\). By (1) after refining the covering we may assume that \(H^1(U_i, \mathcal{G}) = 0\) and \(H^1(U_{ij}, \mathcal{G}) = 0\). Consider the sheaf \[\mathcal{F}_{ij} = \mathit{Isom}(x_i|_{U_{ij}}, x_j|_{U_{ij}})\] on \(\mathcal{C}/U_{ij}\). Since \(\mathcal{G}|_{U_{ij}} = \mathit{Aut}(x_i|_{U_{ij}})\) we see that there is an action \[\mathcal{G}|_{U_{ij}} \times \mathcal{F}_{ij} \to \mathcal{F}_{ij}\] by precomposition. It is clear that \(\mathcal{F}_{ij}\) is a pseudo \(\mathcal{G}|_{U_{ij}}\)-torsor and in fact a torsor because any two objects of a gerbe are locally isomorphic. By our choice of the covering and by Lemma 03AJ these torsors are trivial (and hence have global sections by Lemma 03AI). In other words, we can choose isomorphisms \[\varphi_{ij} : x_i|_{U_{ij}} \longrightarrow x_j|_{U_{ij}}\] To find an object \(x\) over \(U\) we are going to massage our choice of these \(\varphi_{ij}\) to get a descent datum (which is necessarily effective as \(p : \mathcal{S} \to \mathcal{C}\) is a stack). Namely, the obstruction to being a descent datum is that the cocycle condition may not hold. Namely, set \(U_{ijk} = U_i \times_U U_j \times_U U_k\). Then we can consider \[g_{ijk} = \varphi_{ik}^{-1}|_{U_{ijk}} \circ \varphi_{jk}|_{U_{ijk}} \circ \varphi_{ij}|_{U_{ijk}}\] which is an automorphism of \(x_i\) over \(U_{ijk}\). Thus we may and do consider \(g_{ijk}\) as a section of \(\mathcal{G}\) over \(U_{ijk}\). A computation (omitted) shows that \((g_{i_0i_1i_2})\) is a \(2\)-cocycle in the Čech complex \({\check C}^\bullet(\mathcal{U}, \mathcal{G})\) of \(\mathcal{G}\) with respect to the covering \(\mathcal{U}\). By the spectral sequence of Lemma 03AZ and since \(H^1(U_i, \mathcal{G}) = 0\) for all \(i\) we see that \({\check H}^2(\mathcal{U}, \mathcal{G}) \to H^2(U, \mathcal{G})\) is injective. Hence \((g_{i_0i_1i_2})\) is a coboundary by our assumption that \(H^2(U, \mathcal{G}) = 0\). Thus we can find sections \(g_{ij} \in \mathcal{G}(U_{ij})\) such that \(g_{ik}^{-1}|_{U_{ijk}} g_{jk}|_{U_{ijk}} g_{ij}|_{U_{ijk}} = g_{ijk}\) for all \(i, j, k\). After replacing \(\varphi_{ij}\) by \(\varphi_{ij}g_{ij}^{-1}\) we see that \(\varphi_{ij}\) gives a descent datum on the objects \(x_i\) over \(U_i\) and the proof is complete.

Cohomology of modules

Everything that was said for cohomology of abelian sheaves goes for cohomology of modules, since the two agree.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. An injective sheaf of modules is also injective as an object in the category \(\textit{PMod}(\mathcal{O})\).

Proof

Apply Homology, Lemma 015Z to the categories \(\mathcal{A} = \textit{Mod}(\mathcal{O})\), \(\mathcal{B} = \textit{PMod}(\mathcal{O})\), the inclusion functor and sheafification. (See Modules on Sites, Section 03CX to see that all assumptions of the lemma are satisfied.)

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Consider the functor \(i : \textit{Mod}(\mathcal{C}) \to \textit{PMod}(\mathcal{C})\). It is a left exact functor with right derived functors given by \[R^pi(\mathcal{F}) = \underline{H}^p(\mathcal{F}) : U \longmapsto H^p(U, \mathcal{F})\] see discussion in Section 01FU.

Proof

It is clear that \(i\) is left exact. Choose an injective resolution \(\mathcal{F} \to \mathcal{I}^\bullet\) in \(\textit{Mod}(\mathcal{O})\). By definition \(R^pi\) is the \(p\)th cohomology presheaf of the complex \(\mathcal{I}^\bullet\). In other words, the sections of \(R^pi(\mathcal{F})\) over an object \(U\) of \(\mathcal{C}\) are given by \[\frac{\Ker(\mathcal{I}^n(U) \to \mathcal{I}^{n + 1}(U))} {\Im(\mathcal{I}^{n - 1}(U) \to \mathcal{I}^n(U))}.\] which is the definition of \(H^p(U, \mathcal{F})\).

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) be a covering of \(\mathcal{C}\). Let \(\mathcal{I}\) be an injective \(\mathcal{O}\)-module, i.e., an injective object of \(\textit{Mod}(\mathcal{O})\). Then \[\check{H}^p(\mathcal{U}, \mathcal{I}) = \left\{ \begin{matrix} \mathcal{I}(U) & \text{if} & p = 0 \\ 0 & \text{if} & p > 0 \end{matrix} \right.\]

Proof

Lemma 03AS gives the first equality in the following sequence of equalities \[\begin{align*} \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}) & = \Mor_{\textit{PAb}(\mathcal{C})}( \mathbf{Z}_{\mathcal{U}, \bullet}, \mathcal{I}) \\ & = \Mor_{\textit{PMod}(\mathbf{Z})}( \mathbf{Z}_{\mathcal{U}, \bullet}, \mathcal{I}) \\ & = \Mor_{\textit{PMod}(\mathcal{O})}( \mathbf{Z}_{\mathcal{U}, \bullet} \otimes_{p, \mathbf{Z}} \mathcal{O}, \mathcal{I}) \end{align*}\] The third equality by Modules on Sites, Lemma 03CU. By Lemma 03FB we see that \(\mathcal{I}\) is an injective object in \(\textit{PMod}(\mathcal{O})\). Hence \(\Hom_{\textit{PMod}(\mathcal{O})}(-, \mathcal{I})\) is an exact functor. By Lemma 03F5 we see the vanishing of higher Čech cohomology groups. For the zeroth see Lemma 03AN.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}\). Let \(\mathcal{F}\) be an \(\mathcal{O}\)-module, and denote \(\mathcal{F}_{ab}\) the underlying sheaf of abelian groups. Then we have \[H^i(\mathcal{C}, \mathcal{F}_{ab}) = H^i(\mathcal{C}, \mathcal{F})\] and for any object \(U\) of \(\mathcal{C}\) we also have \[H^i(U, \mathcal{F}_{ab}) = H^i(U, \mathcal{F}).\] Here the left hand side is cohomology computed in \(\textit{Ab}(\mathcal{C})\) and the right hand side is cohomology computed in \(\textit{Mod}(\mathcal{O})\).

Proof

By Derived Categories, Lemma 015B the \(\delta\)-functor \((\mathcal{F} \mapsto H^p(U, \mathcal{F}))_{p \geq 0}\) is universal. The functor \(\textit{Mod}(\mathcal{O}) \to \textit{Ab}(\mathcal{C})\), \(\mathcal{F} \mapsto \mathcal{F}_{ab}\) is exact. Hence \((\mathcal{F} \mapsto H^p(U, \mathcal{F}_{ab}))_{p \geq 0}\) is a \(\delta\)-functor also. Suppose we show that \((\mathcal{F} \mapsto H^p(U, \mathcal{F}_{ab}))_{p \geq 0}\) is also universal. This will imply the second statement of the lemma by uniqueness of universal \(\delta\)-functors, see Homology, Lemma 010U. Since \(\textit{Mod}(\mathcal{O})\) has enough injectives, it suffices to show that \(H^i(U, \mathcal{I}_{ab}) = 0\) for any injective object \(\mathcal{I}\) in \(\textit{Mod}(\mathcal{O})\), see Homology, Lemma 010T.

Let \(\mathcal{I}\) be an injective object of \(\textit{Mod}(\mathcal{O})\). Apply Lemma 03F9 with \(\mathcal{F} = \mathcal{I}\), \(\mathcal{B} = \mathcal{C}\) and \(\text{Cov} = \text{Cov}_\mathcal{C}\). Assumption (3) of that lemma holds by Lemma 03FC. Hence we see that \(H^i(U, \mathcal{I}_{ab}) = 0\) for every object \(U\) of \(\mathcal{C}\).

If \(\mathcal{C}\) has a final object then this also implies the first equality. If not, then according to Sites, Lemma 03CI we see that the ringed topos \((\Sh(\mathcal{C}), \mathcal{O})\) is equivalent to a ringed topos where the underlying site does have a final object. Hence the lemma follows.

Lemma

Let \(\mathcal{C}\) be a site. Let \(I\) be a set. For \(i \in I\) let \(\mathcal{F}_i\) be an abelian sheaf on \(\mathcal{C}\). Let \(U \in \Ob(\mathcal{C})\). The canonical map \[H^p(U, \prod\nolimits_{i \in I} \mathcal{F}_i) \longrightarrow \prod\nolimits_{i \in I} H^p(U, \mathcal{F}_i)\] is an isomorphism for \(p = 0\) and injective for \(p = 1\).

Proof

The statement for \(p = 0\) is true because the product of sheaves is equal to the product of the underlying presheaves, see Sites, Lemma 00W2. Proof for \(p = 1\). Set \(\mathcal{F} = \prod \mathcal{F}_i\). Let \(\xi \in H^1(U, \mathcal{F})\) map to zero in \(\prod H^1(U, \mathcal{F}_i)\). By locality of cohomology, see Lemma 01FW, there exists a covering \(\mathcal{U} = \{U_j \to U\}\) such that \(\xi|_{U_j} = 0\) for all \(j\). By Lemma 0A6G this means \(\xi\) comes from an element \(\check \xi \in \check H^1(\mathcal{U}, \mathcal{F})\). Since the maps \(\check H^1(\mathcal{U}, \mathcal{F}_i) \to H^1(U, \mathcal{F}_i)\) are injective for all \(i\) (by Lemma 0A6G), and since the image of \(\xi\) is zero in \(\prod H^1(U, \mathcal{F}_i)\) we see that the image \(\check \xi_i = 0\) in \(\check H^1(\mathcal{U}, \mathcal{F}_i)\). However, since \(\mathcal{F} = \prod \mathcal{F}_i\) we see that \(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})\) is the product of the complexes \(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}_i)\), hence by Homology, Lemma 060K we conclude that \(\check \xi = 0\) as desired.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(a : U' \to U\) be a monomorphism in \(\mathcal{C}\). Then for any injective \(\mathcal{O}\)-module \(\mathcal{I}\) the restriction mapping \(\mathcal{I}(U) \to \mathcal{I}(U')\) is surjective.

Proof

Let \(j : \mathcal{C}/U \to \mathcal{C}\) and \(j' : \mathcal{C}/U' \to \mathcal{C}\) be the localization morphisms (Modules on Sites, Section 03DH). Since \(j_!\) is a left adjoint to restriction we see that for any sheaf \(\mathcal{F}\) of \(\mathcal{O}\)-modules \[\Hom_\mathcal{O}(j_!\mathcal{O}_U, \mathcal{F}) = \Hom_{\mathcal{O}_U}(\mathcal{O}_U, \mathcal{F}|_U) = \mathcal{F}(U)\] Similarly, the sheaf \(j'_!\mathcal{O}_{U'}\) represents the functor \(\mathcal{F} \mapsto \mathcal{F}(U')\). Moreover below we describe a canonical map of \(\mathcal{O}\)-modules \[j'_!\mathcal{O}_{U'} \longrightarrow j_!\mathcal{O}_U\] which corresponds to the restriction mapping \(\mathcal{F}(U) \to \mathcal{F}(U')\) via Yoneda’s lemma (Categories, Lemma 001P). It suffices to prove the displayed map of modules is injective, see Homology, Lemma 0136.

To construct our map it suffices to construct a map between the presheaves which assign to an object \(V\) of \(\mathcal{C}\) the \(\mathcal{O}(V)\)-module \[\bigoplus\nolimits_{\varphi' \in \Mor_\mathcal{C}(V, U')} \mathcal{O}(V) \quad\text{and}\quad \bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{O}(V)\] see Modules on Sites, Lemma 03DI. We take the map which maps the summand corresponding to \(\varphi'\) to the summand corresponding to \(\varphi = a \circ \varphi'\) by the identity map on \(\mathcal{O}(V)\). As \(a\) is a monomorphism, this map is injective. As sheafification is exact, the result follows.

Totally acyclic sheaves

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(K\) be a presheaf of sets on \(\mathcal{C}\) (we intentionally use a roman capital here to distinguish from abelian sheaves). Given a sheaf of \(\mathcal{O}\)-modules \(\mathcal{F}\) we set \[\mathcal{F}(K) = \Mor_{\textit{PSh}(\mathcal{C})}(K, \mathcal{F}) = \Mor_{\Sh(\mathcal{C})}(K^\#, \mathcal{F})\] The functor \(\mathcal{F} \mapsto \mathcal{F}(K)\) is a left exact functor \(\textit{Mod}(\mathcal{O}) \to \textit{Ab}\) hence we have its right derived functors. We will denote these \(H^p(K, \mathcal{F})\) so that \(H^0(K, \mathcal{F}) = \mathcal{F}(K)\).

Here are some observations:

1. Since \(\mathcal{F}(K) = \mathcal{F}(K^\#)\), we have \(H^p(K, \mathcal{F}) = H^p(K^\#, \mathcal{F})\). Allowing \(K\) to be a presheaf in the definition above is a purely notational convenience.

2. Suppose that \(K = h_U\) or \(K = h_U^\#\) for some object \(U\) of \(\mathcal{C}\). Then \(H^p(K, \mathcal{F}) = H^p(U, \mathcal{F})\), because \(\Mor_{\Sh(\mathcal{C})}(h_U^\#, \mathcal{F}) = \mathcal{F}(U)\), see Sites, Section 00WO.

3. If \(\mathcal{O} = \mathbf{Z}\) (the constant sheaf), then the cohomology groups are functors \(H^p(K, - ) : \textit{Ab}(\mathcal{C}) \to \textit{Ab}\) since \(\textit{Mod}(\mathcal{O}) = \textit{Ab}(\mathcal{C})\) in this case.

We can translate some of our already proven results using this language.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(K\) be a presheaf of sets on \(\mathcal{C}\). Let \(\mathcal{F}\) be an \(\mathcal{O}\)-module and denote \(\mathcal{F}_{ab}\) the underlying sheaf of abelian groups. Then \(H^p(K, \mathcal{F}) = H^p(K, \mathcal{F}_{ab})\).

Proof

We may replace \(K\) by its sheafification and assume \(K\) is a sheaf. Note that both \(H^p(K, \mathcal{F})\) and \(H^p(K, \mathcal{F}_{ab})\) depend only on the topos, not on the underlying site. Hence by Sites, Lemma 03CI we may replace \(\mathcal{C}\) by a “larger” site such that \(K = h_U\) for some object \(U\) of \(\mathcal{C}\). In this case the result follows from Lemma 03FD.

Lemma

Let \(\mathcal{C}\) be a site. Let \(K' \to K\) be a map of presheaves of sets on \(\mathcal{C}\) whose sheafification is surjective. Set \(K'_p = K' \times_K \ldots \times_K K'\) (\(p + 1\)-factors). For every abelian sheaf \(\mathcal{F}\) there is a spectral sequence with \(E_1^{p, q} = H^q(K'_p, \mathcal{F})\) converging to \(H^{p + q}(K, \mathcal{F})\).

Proof

Since sheafification is exact, we see that \((K_p')^\#\) is equal to \((K')^\# \times_{K^\#} \ldots \times_{K^\#} (K')^\#\) (\(p + 1\)-factors). Thus we may replace \(K\) and \(K'\) by their sheafifications and assume \(K' \to K\) is a surjective map of sheaves. After replacing \(\mathcal{C}\) by a “larger” site as in Sites, Lemma 03CI we may assume that \(K, K'\) are objects of \(\mathcal{C}\) and that \(\mathcal{U} = \{K' \to K\}\) is a covering. Then we have the Čech to cohomology spectral sequence of Lemma 03AZ whose \(E_1\) page is as indicated in the statement of the lemma.

Lemma

Let \(\mathcal{C}\) be a site. Let \(K\) be a sheaf of sets on \(\mathcal{C}\). Consider the morphism of topoi \(j : \Sh(\mathcal{C}/K) \to \Sh(\mathcal{C})\), see Sites, Lemma 0791. Then \(j^{-1}\) preserves injectives and \(H^p(K, \mathcal{F}) = H^p(\mathcal{C}/K, j^{-1}\mathcal{F})\) for any abelian sheaf \(\mathcal{F}\) on \(\mathcal{C}\).

Proof

By Sites, Lemmas 04GZ and 0791 the morphism of topoi \(j\) is equivalent to a localization. Hence this follows from Lemma 03F3.

Keeping in mind Lemma 079Y we see that the following definition is the “correct one” also for sheaves of modules on ringed sites.

Definition

Let \(\mathcal{C}\) be a site. We say an abelian sheaf \(\mathcal{F}\) is totally acyclic¹ if for every sheaf of sets \(K\) we have \(H^p(K, \mathcal{F}) = 0\) for all \(p \geq 1\).

It is clear that being totally acyclic is an intrinsic property, i.e., preserved under equivalences of topoi. A totally acyclic sheaf has vanishing higher cohomology on all objects of the site, but in general the condition of being totally acyclic is strictly stronger. Here is a characterization of totally acyclic sheaves which is sometimes useful.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{F}\) be an abelian sheaf. If

1. \(H^p(U, \mathcal{F}) = 0\) for \(p > 0\) and \(U \in \Ob(\mathcal{C})\), and

2. for every surjection \(K' \to K\) of sheaves of sets the extended Čech complex \[0 \to H^0(K, \mathcal{F}) \to H^0(K', \mathcal{F}) \to H^0(K' \times_K K', \mathcal{F}) \to \ldots\] is exact,

then \(\mathcal{F}\) is totally acyclic (and the converse holds too).

Proof

By assumption (1) we have \(H^p(h_U^\#, \mathcal{F}) = 0\) for all \(p > 0\) and all objects \(U\) of \(\mathcal{C}\). Note that if \(K = \coprod K_i\) is a coproduct of sheaves of sets on \(\mathcal{C}\) then \(H^p(K, \mathcal{F}) = \prod H^p(K_i, \mathcal{F})\). For any sheaf of sets \(K\) there exists a surjection \[K' = \coprod h_{U_i}^\# \longrightarrow K\] see Sites, Lemma 00WS. Thus we conclude that: (*) for every sheaf of sets \(K\) there exists a surjection \(K' \to K\) of sheaves of sets such that \(H^p(K', \mathcal{F}) = 0\) for \(p > 0\). We claim that (*) and condition (2) imply that \(\mathcal{F}\) is totally acyclic. Note that conditions (*) and (2) only depend on \(\mathcal{F}\) as an object of the topos \(\Sh(\mathcal{C})\) and not on the underlying site. (We will not use property (1) in the rest of the proof.)

We are going to prove by induction on \(n \geq 0\) that (*) and (2) imply the following induction hypothesis \(IH_n\): \(H^p(K, \mathcal{F}) = 0\) for all \(0 < p \leq n\) and all sheaves of sets \(K\). Note that \(IH_0\) holds. Assume \(IH_n\). Pick a sheaf of sets \(K\). Pick a surjection \(K' \to K\) such that \(H^p(K', \mathcal{F}) = 0\) for all \(p > 0\). We have a spectral sequence with \[E_1^{p, q} = H^q(K'_p, \mathcal{F})\] converging to \(H^{p + q}(K, \mathcal{F})\), see Lemma 079Z. By \(IH_n\) we see that \(E_1^{p, q} = 0\) for \(0 < q \leq n\) and by assumption (2) we see that \(E_2^{p, 0} = 0\) for \(p > 0\). Finally, we have \(E_1^{0, q} = 0\) for \(q > 0\) because \(H^q(K', \mathcal{F}) = 0\) by choice of \(K'\). Hence we conclude that \(H^{n + 1}(K, \mathcal{F}) = 0\) because all the terms \(E_2^{p, q}\) with \(p + q = n + 1\) are zero.

The Leray spectral sequence

The key to proving the existence of the Leray spectral sequence is the following lemma.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi. Then for any injective object \(\mathcal{I}\) in \(\textit{Mod}(\mathcal{O}_\mathcal{C})\) the pushforward \(f_*\mathcal{I}\) is totally acyclic.

Proof

Let \(K\) be a sheaf of sets on \(\mathcal{D}\). By Modules on Sites, Lemma 03CR we may replace \(\mathcal{C}\), \(\mathcal{D}\) by “larger” sites such that \(f\) comes from a morphism of ringed sites induced by a continuous functor \(u : \mathcal{D} \to \mathcal{C}\) such that \(K = h_V\) for some object \(V\) of \(\mathcal{D}\).

Thus we have to show that \(H^q(V, f_*\mathcal{I})\) is zero for \(q > 0\) and all objects \(V\) of \(\mathcal{D}\) when \(f\) is given by a morphism of ringed sites. Let \(\mathcal{V} = \{V_j \to V\}\) be any covering of \(\mathcal{D}\). Since \(u\) is continuous we see that \(\mathcal{U} = \{u(V_j) \to u(V)\}\) is a covering of \(\mathcal{C}\). Then we have an equality of Čech complexes \[\check{\mathcal{C}}^\bullet(\mathcal{V}, f_*\mathcal{I}) = \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I})\] by the definition of \(f_*\). By Lemma 03FC we see that the cohomology of this complex is zero in positive degrees. We win by Lemma 03F9.

For flat morphisms the functor \(f_*\) preserves injective modules. In particular the functor \(f_* : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})\) always transforms injective abelian sheaves into injective abelian sheaves.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi. If \(f\) is flat, then \(f_*\mathcal{I}\) is an injective \(\mathcal{O}_\mathcal{D}\)-module for any injective \(\mathcal{O}_\mathcal{C}\)-module \(\mathcal{I}\).

Proof

In this case the functor \(f^*\) is exact, see Modules on Sites, Lemma 04JC. Hence the result follows from Homology, Lemma 015Z.

Lemma

Let \((\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})\) be a ringed topos. A totally acyclic sheaf is right acyclic for the following functors:

1. the functor \(H^0(U, -)\) for any object \(U\) of \(\mathcal{C}\),

2. the functor \(\mathcal{F} \mapsto \mathcal{F}(K)\) for any presheaf of sets \(K\),

3. the functor \(\Gamma(\mathcal{C}, -)\) of global sections,

4. the functor \(f_*\) for any morphism \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) of ringed topoi.

Proof

Part (2) is the definition of a totally acyclic sheaf. Part (1) is a consequence of (2) as pointed out in the discussion following the definition of totally acyclic sheaves. Part (3) is a special case of (2) where \(K = e\) is the final object of \(\Sh(\mathcal{C})\).

To prove (4) we may assume, by Modules on Sites, Lemma 03CR that \(f\) is given by a morphism of sites. In this case we see that \(R^if_*\), \(i > 0\) of a totally acyclic sheaf are zero by the description of higher direct images in Lemma 072W.

Remark

As a consequence of the results above we find that Derived Categories, Lemma 015M applies to a number of situations. For example, given a morphism \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) of ringed topoi we have \[R\Gamma(\mathcal{D}, Rf_*\mathcal{F}) = R\Gamma(\mathcal{C}, \mathcal{F})\] for any sheaf of \(\mathcal{O}_\mathcal{C}\)-modules \(\mathcal{F}\). Namely, for an injective \(\mathcal{O}_\mathcal{C}\)-module \(\mathcal{I}\) the \(\mathcal{O}_\mathcal{D}\)-module \(f_*\mathcal{I}\) is totally acyclic by Lemma 072Z and a totally acyclic sheaf is acyclic for \(\Gamma(\mathcal{D}, -)\) by Lemma 0731.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi. Let \(\mathcal{F}^\bullet\) be a bounded below complex of \(\mathcal{O}_\mathcal{C}\)-modules. There is a spectral sequence \[E_2^{p, q} = H^p(\mathcal{D}, R^qf_*(\mathcal{F}^\bullet))\] converging to \(H^{p + q}(\mathcal{C}, \mathcal{F}^\bullet)\).

Proof

This is just the Grothendieck spectral sequence Derived Categories, Lemma 015N coming from the composition of functors \(\Gamma(\mathcal{C}, -) = \Gamma(\mathcal{D}, -) \circ f_*\). To see that the assumptions of Derived Categories, Lemma 015N are satisfied, see Lemmas 072Z and 0731.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi. Let \(\mathcal{F}\) be an \(\mathcal{O}_\mathcal{C}\)-module.

1. If \(R^qf_*\mathcal{F} = 0\) for \(q > 0\), then \(H^p(\mathcal{C}, \mathcal{F}) = H^p(\mathcal{D}, f_*\mathcal{F})\) for all \(p\).

2. If \(H^p(\mathcal{D}, R^qf_*\mathcal{F}) = 0\) for all \(q\) and \(p > 0\), then \(H^q(\mathcal{C}, \mathcal{F}) = H^0(\mathcal{D}, R^qf_*\mathcal{F})\) for all \(q\).

Proof

These are two simple conditions that force the Leray spectral sequence to degenerate. You can also prove these facts directly (without using the spectral sequence) which is a good exercise in cohomology of sheaves.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) and \(g : (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) \to (\Sh(\mathcal{E}), \mathcal{O}_\mathcal{E})\) be morphisms of ringed topoi. Let \(\mathcal{F}\) be an \(\mathcal{O}_\mathcal{C}\)-module. There is a spectral sequence with \[E_2^{p, q} = R^pg_*(R^qf_*\mathcal{F})\] converging to \(R^{p + q}(g \circ f)_*\mathcal{F}\). This spectral sequence is functorial in \(\mathcal{F}\), and there is a version for bounded below complexes of \(\mathcal{O}_\mathcal{C}\)-modules.

Proof

This is a Grothendieck spectral sequence for composition of functors, see Derived Categories, Lemma 015N and Lemmas 072Z and 0731.

The base change map

In this section we construct the base change map in some cases; the general case is treated in Remark 07A7. The discussion in this section avoids using derived pullback by restricting to the case of a base change by a flat morphism of ringed sites. Before we state the result, let us discuss flat pullback on the derived category. Suppose \(g : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) is a flat morphism of ringed topoi. By Modules on Sites, Lemma 04JC the functor \(g^* : \textit{Mod}(\mathcal{O}_\mathcal{D}) \to \textit{Mod}(\mathcal{O}_\mathcal{C})\) is exact. Hence it has a derived functor \[g^* : D(\mathcal{O}_\mathcal{D}) \to D(\mathcal{O}_\mathcal{C})\] which is computed by simply pulling back a representative of a given object in \(D(\mathcal{O}_\mathcal{D})\), see Derived Categories, Lemma 015F. It preserves the bounded (above, below) subcategories. Hence as indicated we denote this functor by \(g^*\) rather than \(Lg^*\).

Lemma

Let \[\xymatrix{ (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_{g'} \ar[d]_{f'} & (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^f \\ (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^g & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) }\] be a commutative diagram of ringed topoi. Let \(\mathcal{F}^\bullet\) be a bounded below complex of \(\mathcal{O}_\mathcal{C}\)-modules. Assume both \(g\) and \(g'\) are flat. Then there exists a canonical base change map \[g^*Rf_*\mathcal{F}^\bullet \longrightarrow R(f')_*(g')^*\mathcal{F}^\bullet\] in \(D^{+}(\mathcal{O}_{\mathcal{D}'})\).

Proof

Choose injective resolutions \(\mathcal{F}^\bullet \to \mathcal{I}^\bullet\) and \((g')^*\mathcal{F}^\bullet \to \mathcal{J}^\bullet\). By Lemma 0730 we see that \((g')_*\mathcal{J}^\bullet\) is a complex of injectives representing \(R(g')_*(g')^*\mathcal{F}^\bullet\). Hence by Derived Categories, Lemmas 013P and 013S the arrow \(\beta\) in the diagram \[\xymatrix{ (g')_*(g')^*\mathcal{F}^\bullet \ar[r] & (g')_*\mathcal{J}^\bullet \\ \mathcal{F}^\bullet \ar[u]^{adjunction} \ar[r] & \mathcal{I}^\bullet \ar[u]_\beta }\] exists and is unique up to homotopy. Pushing down to \(\mathcal{D}\) we get \[f_*\beta : f_*\mathcal{I}^\bullet \longrightarrow f_*(g')_*\mathcal{J}^\bullet = g_*(f')_*\mathcal{J}^\bullet\] By adjunction of \(g^*\) and \(g_*\) we get a map of complexes \(g^*f_*\mathcal{I}^\bullet \to (f')_*\mathcal{J}^\bullet\). Note that this map is unique up to homotopy since the only choice in the whole process was the choice of the map \(\beta\) and everything was done on the level of complexes.

Cohomology and colimits

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{I} \to \textit{Mod}(\mathcal{O})\), \(i \mapsto \mathcal{F}_i\) be a diagram over the index category \(\mathcal{I}\), see Categories, Section 002D. For each \(i\) there is a canonical map \(\mathcal{F}_i \to \colim_i \mathcal{F}_i\) which induces a map on cohomology. Hence we get a canonical map \[\colim_i H^p(U, \mathcal{F}_i) \longrightarrow H^p(U, \colim_i \mathcal{F}_i)\] for every \(p \geq 0\) and every object \(U\) of \(\mathcal{C}\). These maps are in general not isomorphisms, even for \(p = 0\).

The following lemma is the analogue of Sites, Lemma 0738 for cohomology.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\text{Cov}_\mathcal{C}\) be the set of coverings of \(\mathcal{C}\) (see Sites, Definition 00VH). Let \(\mathcal{B} \subset \Ob(\mathcal{C})\), and \(\text{Cov} \subset \text{Cov}_\mathcal{C}\) be subsets. Assume that

1. For every \(\mathcal{U} \in \text{Cov}\) we have \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) with \(I\) finite, \(U, U_i \in \mathcal{B}\) and every \(U_{i_0} \times_U \ldots \times_U U_{i_p} \in \mathcal{B}\).

2. For every \(U \in \mathcal{B}\) the coverings of \(U\) occurring in \(\text{Cov}\) is a cofinal system of coverings of \(U\).

Then the map \[\colim_i H^p(U, \mathcal{F}_i) \longrightarrow H^p(U, \colim_i \mathcal{F}_i)\] is an isomorphism for every \(p \geq 0\), every \(U \in \mathcal{B}\), and every filtered diagram \(\mathcal{I} \to \textit{Ab}(\mathcal{C})\).

Proof

To prove the lemma we will argue by induction on \(p\). Note that we require in (1) the coverings \(\mathcal{U} \in \text{Cov}\) to be finite, so that all the elements of \(\mathcal{B}\) are quasi-compact. Hence (2) and (1) imply that any \(U \in \mathcal{B}\) satisfies the hypothesis of Sites, Lemma 0738 (4). Thus we see that the result holds for \(p = 0\). Now we assume the lemma holds for \(p\) and prove it for \(p + 1\).

Choose a filtered diagram \(\mathcal{F} : \mathcal{I} \to \textit{Ab}(\mathcal{C})\), \(i \mapsto \mathcal{F}_i\). Since \(\textit{Ab}(\mathcal{C})\) has functorial injective embeddings, see Injectives, Theorem 01DP, we can find a morphism of filtered diagrams \(\mathcal{F} \to \mathcal{I}\) such that each \(\mathcal{F}_i \to \mathcal{I}_i\) is an injective map of abelian sheaves into an injective abelian sheaf. Denote \(\mathcal{Q}_i\) the cokernel so that we have short exact sequences \[0 \to \mathcal{F}_i \to \mathcal{I}_i \to \mathcal{Q}_i \to 0.\] Since colimits of sheaves are the sheafification of colimits on the level of presheaves, since sheafification is exact, and since filtered colimits of abelian groups are exact (see Algebra, Lemma 00DB), we see the sequence \[0 \to \colim_i \mathcal{F}_i \to \colim_i \mathcal{I}_i \to \colim_i \mathcal{Q}_i \to 0.\] is also a short exact sequence. We claim that \(H^q(U, \colim_i \mathcal{I}_i) = 0\) for all \(U \in \mathcal{B}\) and all \(q \geq 1\). Accepting this claim for the moment consider the diagram \[\xymatrix{ \colim_i H^p(U, \mathcal{I}_i) \ar[d] \ar[r] & \colim_i H^p(U, \mathcal{Q}_i) \ar[d] \ar[r] & \colim_i H^{p + 1}(U, \mathcal{F}_i) \ar[d] \ar[r] & 0 \ar[d] \\ H^p(U, \colim_i \mathcal{I}_i) \ar[r] & H^p(U, \colim_i \mathcal{Q}_i) \ar[r] & H^{p + 1}(U, \colim_i \mathcal{F}_i) \ar[r] & 0 }\] The zero at the lower right corner comes from the claim and the zero at the upper right corner comes from the fact that the sheaves \(\mathcal{I}_i\) are injective. The top row is exact by an application of Algebra, Lemma 00DB. Hence by the snake lemma we deduce the result for \(p + 1\).

It remains to show that the claim is true. We will use Lemma 03F9. By the result for \(p = 0\) we see that for \(\mathcal{U} \in \text{Cov}\) we have \[\check{\mathcal{C}}^\bullet(\mathcal{U}, \colim_i \mathcal{I}_i) = \colim_i \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}_i)\] because all the \(U_{j_0} \times_U \ldots \times_U U_{j_p}\) are in \(\mathcal{B}\). By Lemma 03AW each of the complexes in the colimit of Čech complexes is acyclic in degree \(\geq 1\). Hence by Algebra, Lemma 00DB we see that also the Čech complex \(\check{\mathcal{C}}^\bullet(\mathcal{U}, \colim_i \mathcal{I}_i)\) is acyclic in degrees \(\geq 1\). In other words we see that \(\check{H}^p(\mathcal{U}, \colim_i \mathcal{I}_i) = 0\) for all \(p \geq 1\). Thus the assumptions of Lemma 03F9 are satisfied and the claim follows.

Lemma

Let \(\mathcal{C}\) be a site. Let \(S \subset \Ob(\Sh(\mathcal{C}))\) be a subset. Denote \(*\) the final object of \(\Sh(\mathcal{C})\). Assume

1. for some \(K \in S\) the map \(K \to *\) is surjective,

2. given a surjective map of sheaves \(\mathcal{F} \to K\) with \(K \in S\) there exists a \(K' \in S\) and a map \(K' \to \mathcal{F}\) such that the composition \(K' \to K\) is surjective,

3. given \(K, K' \in S\) there is a surjection \(K'' \to K \times K'\) with \(K'' \in S\),

4. given \(a, b : K \to K'\) with \(K, K' \in S\) there exists a surjection \(K'' \to \text{Equalizer}(a, b)\) with \(K'' \in S\), and

5. every \(K \in S\) is quasi-compact (Sites, Definition 0D07).

Then for all \(p \geq 0\) the map \[\colim_\lambda H^p(\mathcal{C}, \mathcal{F}_\lambda) \longrightarrow H^p(\mathcal{C}, \colim_\lambda \mathcal{F}_\lambda)\] is an isomorphism for every filtered diagram \(\Lambda \to \textit{Ab}(\mathcal{C})\), \(\lambda \mapsto \mathcal{F}_\lambda\).

Proof

We will prove this by induction on \(p\). The base case \(p = 0\) follows from Sites, Lemma 0GMR part (4). We check the assumptions hold, but we urge the reader to skip this part. Suppose \(\mathcal{F} \to *\) is surjective. Choose \(K \in S\) and \(K \to *\) surjective as in (1). Then \(\mathcal{F} \times K \to K\) is surjective. Choose \(K' \to \mathcal{F} \times K\) with \(K' \in S\) and \(K' \to K\) surjective as in (2). Then there is a map \(K' \to \mathcal{F}\) and \(K' \to *\) is surjective. Hence Sites, Lemma 0GMR assumption (4)(a) is satisfied. By Sites, Lemma 0GMP, assumptions (3) and (5) we see that \(K \times K\) is quasi-compact for all \(K \in S\). Hence Sites, Lemma 0GMR assumption (4)(b) is satisfied. This finishes the proof of the base case.

Induction step. Assume the result holds for \(H^p\) for \(p \leq p_0\) and for all topoi \(\Sh(\mathcal{C})\) such that a set \(S \subset \Ob(\Sh(\mathcal{C}))\) can be found satisfying (1) – (5). Arguing exactly as in the proof of Lemma 0739 we see that it suffices to show: given a filtered colimit \(\mathcal{I} = \colim \mathcal{I}_\lambda\) with \(\mathcal{I}_\lambda\) injective abelian sheaves, we have \(H^{p_0 + 1}(\mathcal{C}, \mathcal{I}) = 0\). Choose \(K \to *\) surjective with \(K \in S\) as in (1). Denote \(K^n\) the \(n\)-fold self product of \(K\). Consider the spectral sequence \[E_1^{p, q} = H^q(K^{p + 1}, \mathcal{I}) \Rightarrow H^{p + q}(*, \mathcal{I}) = H^{p + q}(\mathcal{C}, \mathcal{I})\] of Lemma 079Z. Recall that \(H^q(K^{p + 1}, \mathcal{F}) = H^q(\mathcal{C}/K^{p + 1}, j^{-1}\mathcal{F})\), for any abelian sheaf on \(\mathcal{C}\), see Lemma 07A0. We have \(j^{-1}\mathcal{I} = \colim j^{-1}\mathcal{I}_\lambda\) as \(j^{-1}\) commutes with colimits. The restrictions \(j^{-1}\mathcal{I}_\lambda\) are injective abelian sheaves on \(\mathcal{C}/K^{p + 1}\) by Lemma 03F3. Below we will show that the induction hypothesis applies to \(\mathcal{C}/K^{p + 1}\) and hence we see that \(H^q(K^{p + 1}, \mathcal{I}) = \colim H^q(K^{p + 1}, \mathcal{I}_\lambda) = 0\) for \(q < p_0 + 1\) (vanishing as \(\mathcal{I}_\lambda\) is injective). It follows that \[H^{p_0 + 1}(\mathcal{C}, \mathcal{I}) = H^{p_0 + 1}\left(\ldots \to H^0(K^{p_0}, \mathcal{I}) \to H^0(K^{p_0 + 1}, \mathcal{I}) \to H^0(K^{p_0 + 2}, \mathcal{I}) \to \ldots\right)\] Again using the induction hypothesis, the complex depicted on the right hand side is the colimit over \(\Lambda\) of the complexes \[\ldots \to H^0(K^{p_0}, \mathcal{I}_\lambda) \to H^0(K^{p_0 + 1}, \mathcal{I}_\lambda) \to H^0(K^{p_0 + 2}, \mathcal{I}_\lambda) \to \ldots\] These complexes are exact as \(\mathcal{I}_\lambda\) is an injective abelian sheaf (follows from the spectral sequence for example). Since filtered colimits are exact in the category of abelian groups we obtain the desired vanishing.

We still have to show that the induction hypothesis applies to the site \(\mathcal{C}/K^n\) for all \(n \geq 1\). Recall that \(\Sh(\mathcal{C}/K^n) = \Sh(\mathcal{C})/K^n\), see Sites, Lemma 0791. Thus we may work in \(\Sh(\mathcal{C})/K^n\). Denote \(S_n \subset \Ob(\Sh(\mathcal{C}/K^n))\) the set of objects of the form \(K' \to K^n\). We check each property in turn:

1. By (3) and induction there exists a surjection \(K' \to K^n\) with \(K' \in S\). Then \((K' \to K^n) \to (K^n \to K^n)\) is a surjection in \(\Sh(\mathcal{C})/K^n\) and \(K^n \to K^n\) is the final object of \(\Sh(\mathcal{C})/K^n\). Hence (1) holds for \(S_n\),

2. Property (2) for \(S_n\) is an immediate consequence of (2) for \(S\).

3. Let \(a : K_1 \to K^n\) and \(b : K_2 \to K^n\) be in \(S_n\). Then \((K_1 \to K^n) \times (K_2 \to K^n)\) is the object \(K_1 \times_{K^n} K_2 \to K^n\) of \(\Sh(\mathcal{C})/K^n\). The subsheaf \(K_1 \times_{K^n} K_2 \subset K_1 \times K_2\) is the equalizer of \(a \circ \text{pr}_1\) and \(b \circ \text{pr}_2\). Write \(a = (a_1, \ldots, a_n)\) and \(b = (b_1, \ldots, b_n)\). Pick \(K_3 \to K_1 \times K_2\) surjective with \(K_3 \in S\); this is possible by assumption (3) for \(\mathcal{C}\). Pick \[K_4 \longrightarrow \text{Equalizer}(K_3 \to K_1 \times K_2 \xrightarrow{a_1, b_1} K)\] surjective with \(K_4 \in S\). This is possible by assumption (4) for \(\mathcal{C}\). Pick \[K_5 \longrightarrow \text{Equalizer}(K_4 \to K_1 \times K_2 \xrightarrow{a_2, b_2} K)\] surjective with \(K_5 \in S\). Again this is possible. Continue in this fashion until we get \[K_{3 + n} \longrightarrow \text{Equalizer}(K_{2 + n} \to K_1 \times K_2 \xrightarrow{a_n, b_n} K)\] surjective with \(K_{3 + n} \in S\). By construction \(K_{3 + n} \to K_1 \times_{K^n} K_2\) is surjective. Hence \((K_{3 + n} \to K^n)\) is in \(S_n\) and surjects onto the product \((K_1 \to K^n) \times (K_2 \to K^n)\). Thus (3) holds for \(S_n\).

4. Property (4) for \(S_n\) is an immediate consequence of property (4) for \(S\).

5. Property (5) for \(S_n\) is a consequence of property (5) for \(S\). Namely, an object \(\mathcal{F} \to K^n\) of \(\Sh(\mathcal{C})/K^n\) corresponds to a quasi-compact object of \(\Sh(\mathcal{C}/K^n)\) if and only if \(\mathcal{F}\) is a quasi-compact object of \(\Sh(\mathcal{C})\).

This finishes the proof of the lemma.

Remark

Let \(\mathcal{C}\) be a site. Let \(\mathcal{B} \subset \Ob(\mathcal{C})\) be a subset. Let \(S \subset \Ob(\Sh(\mathcal{C}))\) be the set of sheaves \(K\) which have the form \[K = \coprod\nolimits_{i = 1, \ldots, n} h_{U_i}^\#\] with \(U_1, \ldots, U_n \in \mathcal{B}\). Then we can ask: when does this set satisfy the assumptions of Lemma 0GN3? One answer is that it suffices if

1. for some \(n \geq 0\), \(U_1, \ldots, U_n \in \mathcal{B}\) the map \(\coprod_{i = 1, \ldots, n} h_{U_i}^\# \to *\) is surjective,

2. every covering of \(U \in \mathcal{B}\) can be refined by a covering of the form \(\{U_i \to U\}_{i = 1, \ldots, n}\) with \(U_i \in \mathcal{B}\),

3. given \(U, U' \in \mathcal{B}\) there exist \(n \geq 0\), \(U_1, \ldots, U_n \in \mathcal{B}\), maps \(U_i \to U\) and \(U_i \to U'\) such that \(\coprod_{i = 1, \ldots, n} h_{U_i}^\# \to h_U^\# \times h_{U'}^\#\) is surjective,

4. given morphisms \(a, b : U \to U'\) in \(\mathcal{C}\) with \(U, U' \in \mathcal{B}\), there exist \(U_1, \ldots, U_n \in \mathcal{B}\), maps \(U_i \to U\) equalizing \(a, b\) such that \(\coprod_{i = 1, \ldots, n} h_{U_i}^\# \to \text{Equalizer}(h_a^\#, h_b^\# : h_U^\# \to h_{U'}^\#)\) is surjective.

We omit the detailed verification, except to mention that part (2) above insures that every element of \(\mathcal{B}\) is quasi-compact and hence every \(K \in S\) is quasi-compact as well by Sites, Lemma 0GMQ.

Lemma

Let \(f : \mathcal{C} \to \mathcal{D}\) be a morphism of sites corresponding to the continuous functor \(u : \mathcal{D} \to \mathcal{C}\). Let \((\mathcal{F}_i, \varphi_{ii'})\) be a system of abelian sheaves on \(\mathcal{C}\). Set \(\mathcal{F} = \colim \mathcal{F}_i\). Let \(p \geq 0\) be an integer. Denote \(\mathcal{B}\) the set of \(V \in \Ob(\mathcal{D})\) such that \(H^p(u(V), \mathcal{F}) = \colim H^p(u(V), \mathcal{F}_i)\). If every object of \(\mathcal{D}\) has a covering by elements of \(\mathcal{B}\), then \(R^pf_*\mathcal{F} = \colim R^pf_*\mathcal{F}_i\).

Proof

Recall that \(R^pf_*\mathcal{F}\) is the sheafification of the presheaf \(\mathcal{G}\) sending \(V\) to \(H^p(u(V), \mathcal{F})\), see Lemma 072W. Similarly, \(R^pf_*\mathcal{F}_i\) is the sheafification of the presheaf \(\mathcal{G}_i\) sending \(V\) to \(H^p(u(V), \mathcal{F}_i)\). Recall that sheafification is the left adjoint to the inclusion from sheaves to presheaves, see Sites, Section 00W1. Hence sheafification commutes with colimits, see Categories, Lemma 0038. Hence it suffices to show that the map of presheaves (with colimit in the category of presheaves) \[\colim \mathcal{G}_i \longrightarrow \mathcal{G}\] induces an isomorphism on sheafifications. This follows from Sites, Lemma 0H6Y and our assumption on \(\mathcal{B}\).

Lemma

Let \(\mathcal{I}\) be a cofiltered index category and let \((\mathcal{C}_i, f_a)\) be an inverse system of sites over \(\mathcal{I}\) as in Sites, Situation 0A34. Set \(\mathcal{C} = \colim \mathcal{C}_i\) as in Sites, Lemmas 09YL and 0A35. Moreover, assume given

1. an abelian sheaf \(\mathcal{F}_i\) on \(\mathcal{C}_i\) for all \(i \in \Ob(\mathcal{I})\),

2. for \(a : j \to i\) a map \(\varphi_a : f_a^{-1}\mathcal{F}_i \to \mathcal{F}_j\) of abelian sheaves on \(\mathcal{C}_j\)

such that \(\varphi_c = \varphi_b \circ f_b^{-1}\varphi_a\) whenever \(c = a \circ b\). Then there exists a map of systems \((\mathcal{F}_i, \varphi_a) \to (\mathcal{G}_i, \psi_a)\) such that \(\mathcal{F}_i \to \mathcal{G}_i\) is injective and \(\mathcal{G}_i\) is an injective abelian sheaf.

Proof

For each \(i\) we pick an injection \(\mathcal{F}_i \to \mathcal{A}_i\) where \(\mathcal{A}_i\) is an injective abelian sheaf on \(\mathcal{C}_i\). Then we can consider the family of maps \[\gamma_i : \mathcal{F}_i \longrightarrow \prod\nolimits_{b : k \to i} f_{b, *}\mathcal{A}_k = \mathcal{G}_i\] where the component maps are the maps adjoint to the maps \(f_b^{-1}\mathcal{F}_i \to \mathcal{F}_k \to \mathcal{A}_k\). For \(a : j \to i\) in \(\mathcal{I}\) there is a canonical map \[\psi_a : f_a^{-1}\mathcal{G}_i \to \mathcal{G}_j\] whose components are the canonical maps \(f_b^{-1}f_{a \circ b, *}\mathcal{A}_k \to f_{b, *}\mathcal{A}_k\) for \(b : k \to j\). Thus we find an injection \((\gamma_i) : (\mathcal{F}_i, \varphi_a) \to (\mathcal{G}_i, \psi_a)\) of systems of abelian sheaves. Note that \(\mathcal{G}_i\) is an injective sheaf of abelian groups on \(\mathcal{C}_i\), see Lemma 0730 and Homology, Lemma 0137. This finishes the construction.

Lemma

In the situation of Lemma 0EXZ set \(\mathcal{F} = \colim f_i^{-1}\mathcal{F}_i\). Let \(i \in \Ob(\mathcal{I})\), \(X_i \in \text{Ob}(\mathcal{C}_i)\). Then \[\colim_{a : j \to i} H^p(u_a(X_i), \mathcal{F}_j) = H^p(u_i(X_i), \mathcal{F})\] for all \(p \geq 0\).

Proof

The case \(p = 0\) is Sites, Lemma 09YN.

Choose \((\mathcal{F}_i, \varphi_a) \to (\mathcal{G}_i, \psi_a)\) as in Lemma 0EXZ. Arguing exactly as in the proof of Lemma 0739 we see that it suffices to prove that \(H^p(X, \colim f_i^{-1}\mathcal{G}_i) = 0\) for \(p > 0\).

Set \(\mathcal{G} = \colim f_i^{-1}\mathcal{G}_i\). To show vanishing of cohomology of \(\mathcal{G}\) on every object of \(\mathcal{C}\) we show that the Čech cohomology of \(\mathcal{G}\) for any covering \(\mathcal{U}\) of \(\mathcal{C}\) is zero (Lemma 03F9). The covering \(\mathcal{U}\) comes from a covering \(\mathcal{U}_i\) of \(\mathcal{C}_i\) for some \(i\). We have \[\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{G}) = \colim_{a : j \to i} \check{\mathcal{C}}^\bullet(u_a(\mathcal{U}_i), \mathcal{G}_j)\] by the case \(p = 0\). The right hand side is acyclic in positive degrees as a filtered colimit of acyclic complexes by Lemma 03AW. See Algebra, Lemma 00DB.

Flat resolutions

In this section we redo the arguments of Cohomology, Section 06Y7 in the setting of ringed sites and ringed topoi.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{G}^\bullet\) be a complex of \(\mathcal{O}\)-modules. The functors \[K(\textit{Mod}(\mathcal{O})) \longrightarrow K(\textit{Mod}(\mathcal{O})), \quad \mathcal{F}^\bullet \longmapsto \text{Tot}(\mathcal{G}^\bullet \otimes_\mathcal{O} \mathcal{F}^\bullet)\] and \[K(\textit{Mod}(\mathcal{O})) \longrightarrow K(\textit{Mod}(\mathcal{O})), \quad \mathcal{F}^\bullet \longmapsto \text{Tot}(\mathcal{F}^\bullet \otimes_\mathcal{O} \mathcal{G}^\bullet)\] are exact functors of triangulated categories.

Proof

This follows from Derived Categories, Remark 0G6E.

Definition

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. A complex \(\mathcal{K}^\bullet\) of \(\mathcal{O}\)-modules is called K-flat if for every acyclic complex \(\mathcal{F}^\bullet\) of \(\mathcal{O}\)-modules the complex \[\text{Tot}(\mathcal{F}^\bullet \otimes_\mathcal{O} \mathcal{K}^\bullet)\] is acyclic.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{K}^\bullet\) be a K-flat complex. Then the functor \[K(\textit{Mod}(\mathcal{O})) \longrightarrow K(\textit{Mod}(\mathcal{O})), \quad \mathcal{F}^\bullet \longmapsto \text{Tot}(\mathcal{F}^\bullet \otimes_\mathcal{O} \mathcal{K}^\bullet)\] transforms quasi-isomorphisms into quasi-isomorphisms.

Proof

Follows from Lemma 06YM and the fact that quasi-isomorphisms are characterized by having acyclic cones.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(U\) be an object of \(\mathcal{C}\). If \(\mathcal{K}^\bullet\) is a K-flat complex of \(\mathcal{O}\)-modules, then \(\mathcal{K}^\bullet|_U\) is a K-flat complex of \(\mathcal{O}_U\)-modules.

Proof

Let \(\mathcal{G}^\bullet\) be an exact complex of \(\mathcal{O}_U\)-modules. Since \(j_{U!}\) is exact (Modules on Sites, Lemma 03DJ) and \(\mathcal{K}^\bullet\) is a K-flat complex of \(\mathcal{O}\)-modules we see that the complex \[j_{U!}(\text{Tot}(\mathcal{G}^\bullet \otimes_{\mathcal{O}_U} \mathcal{K}^\bullet|_U)) = \text{Tot}(j_{U!}\mathcal{G}^\bullet \otimes_\mathcal{O} \mathcal{K}^\bullet)\] is exact. Here the equality comes from Modules on Sites, Lemma 0E8I and the fact that \(j_{U!}\) commutes with direct sums (as a left adjoint). We conclude because \(j_{U!}\) reflects exactness by Modules on Sites, Lemma 0E8G.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. If \(\mathcal{K}^\bullet\), \(\mathcal{L}^\bullet\) are K-flat complexes of \(\mathcal{O}\)-modules, then \(\text{Tot}(\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{L}^\bullet)\) is a K-flat complex of \(\mathcal{O}\)-modules.

Proof

Follows from the isomorphism \[\text{Tot}(\mathcal{M}^\bullet \otimes_\mathcal{O} \text{Tot}(\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{L}^\bullet)) = \text{Tot}(\text{Tot}(\mathcal{M}^\bullet \otimes_\mathcal{O} \mathcal{K}^\bullet) \otimes_\mathcal{O} \mathcal{L}^\bullet)\] and the definition.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \((\mathcal{K}_1^\bullet, \mathcal{K}_2^\bullet, \mathcal{K}_3^\bullet)\) be a distinguished triangle in \(K(\textit{Mod}(\mathcal{O}))\). If two out of three of \(\mathcal{K}_i^\bullet\) are K-flat, so is the third.

Proof

Follows from Lemma 06YM and the fact that in a distinguished triangle in \(K(\textit{Mod}(\mathcal{O}))\) if two out of three are acyclic, so is the third.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(0 \to \mathcal{K}_1^\bullet \to \mathcal{K}_2^\bullet \to \mathcal{K}_3^\bullet \to 0\) be a short exact sequence of complexes such that the terms of \(\mathcal{K}_3^\bullet\) are flat \(\mathcal{O}\)-modules. If two out of three of \(\mathcal{K}_i^\bullet\) are K-flat, so is the third.

Proof

By Modules on Sites, Lemma 03EX for every complex \(\mathcal{L}^\bullet\) we obtain a short exact sequence \[0 \to \text{Tot}(\mathcal{L}^\bullet \otimes_\mathcal{O} \mathcal{K}_1^\bullet) \to \text{Tot}(\mathcal{L}^\bullet \otimes_\mathcal{O} \mathcal{K}_2^\bullet) \to \text{Tot}(\mathcal{L}^\bullet \otimes_\mathcal{O} \mathcal{K}_3^\bullet) \to 0\] of complexes. Hence the lemma follows from the long exact sequence of cohomology sheaves and the definition of K-flat complexes.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. A bounded above complex of flat \(\mathcal{O}\)-modules is K-flat.

Proof

Let \(\mathcal{K}^\bullet\) be a bounded above complex of flat \(\mathcal{O}\)-modules. Let \(\mathcal{L}^\bullet\) be an acyclic complex of \(\mathcal{O}\)-modules. Note that \(\mathcal{L}^\bullet = \colim_m \tau_{\leq m}\mathcal{L}^\bullet\) where we take termwise colimits. Hence also \[\text{Tot}(\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{L}^\bullet) = \colim_m \text{Tot}( \mathcal{K}^\bullet \otimes_\mathcal{O} \tau_{\leq m}\mathcal{L}^\bullet)\] termwise. Hence to prove the complex on the left is acyclic it suffices to show each of the complexes on the right is acyclic. Since \(\tau_{\leq m}\mathcal{L}^\bullet\) is acyclic this reduces us to the case where \(\mathcal{L}^\bullet\) is bounded above. In this case the spectral sequence of Homology, Lemma 0132 has \[{}'E_1^{p, q} = H^p(\mathcal{L}^\bullet \otimes_\mathcal{O} \mathcal{K}^q)\] which is zero as \(\mathcal{K}^q\) is flat and \(\mathcal{L}^\bullet\) acyclic. Hence we win.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{K}_1^\bullet \to \mathcal{K}_2^\bullet \to \ldots\) be a system of K-flat complexes. Then \(\colim_i \mathcal{K}_i^\bullet\) is K-flat.

Proof

Because we are taking termwise colimits it is clear that \[\colim_i \text{Tot}( \mathcal{F}^\bullet \otimes_\mathcal{O} \mathcal{K}_i^\bullet) = \text{Tot}(\mathcal{F}^\bullet \otimes_\mathcal{O} \colim_i \mathcal{K}_i^\bullet)\] Hence the lemma follows from the fact that filtered colimits are exact.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. For any complex \(\mathcal{G}^\bullet\) of \(\mathcal{O}\)-modules there exists a commutative diagram of complexes of \(\mathcal{O}\)-modules \[\xymatrix{ \mathcal{K}_1^\bullet \ar[d] \ar[r] & \mathcal{K}_2^\bullet \ar[d] \ar[r] & \ldots \\ \tau_{\leq 1}\mathcal{G}^\bullet \ar[r] & \tau_{\leq 2}\mathcal{G}^\bullet \ar[r] & \ldots }\] with the following properties: (1) the vertical arrows are quasi-isomorphisms and termwise surjective, (2) each \(\mathcal{K}_n^\bullet\) is a bounded above complex whose terms are direct sums of \(\mathcal{O}\)-modules of the form \(j_{U!}\mathcal{O}_U\), and (3) the maps \(\mathcal{K}_n^\bullet \to \mathcal{K}_{n + 1}^\bullet\) are termwise split injections whose cokernels are direct sums of \(\mathcal{O}\)-modules of the form \(j_{U!}\mathcal{O}_U\). Moreover, the map \(\colim \mathcal{K}_n^\bullet \to \mathcal{G}^\bullet\) is a quasi-isomorphism.

Proof

The existence of the diagram and properties (1), (2), (3) follows immediately from Modules on Sites, Lemma 03EW and Derived Categories, Lemma 06XX. The induced map \(\colim \mathcal{K}_n^\bullet \to \mathcal{G}^\bullet\) is a quasi-isomorphism because filtered colimits are exact.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. For any complex \(\mathcal{G}^\bullet\) there exists a \(K\)-flat complex \(\mathcal{K}^\bullet\) whose terms are flat \(\mathcal{O}\)-modules and a quasi-isomorphism \(\mathcal{K}^\bullet \to \mathcal{G}^\bullet\) which is termwise surjective.

Proof

Choose a diagram as in Lemma 077J. Each complex \(\mathcal{K}_n^\bullet\) is a bounded above complex of flat modules, see Modules on Sites, Lemma 03EV. Hence \(\mathcal{K}_n^\bullet\) is K-flat by Lemma 06YQ. Thus \(\colim \mathcal{K}_n^\bullet\) is K-flat by Lemma 06YR. The induced map \(\colim \mathcal{K}_n^\bullet \to \mathcal{G}^\bullet\) is a quasi-isomorphism and termwise surjective by construction. Property (3) of Lemma 077J shows that \(\colim \mathcal{K}_n^m\) is a direct sum of flat modules and hence flat which proves the final assertion.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\alpha : \mathcal{P}^\bullet \to \mathcal{Q}^\bullet\) be a quasi-isomorphism of K-flat complexes of \(\mathcal{O}\)-modules. For every complex \(\mathcal{F}^\bullet\) of \(\mathcal{O}\)-modules the induced map \[\text{Tot}(\text{id}_{\mathcal{F}^\bullet} \otimes \alpha) : \text{Tot}(\mathcal{F}^\bullet \otimes_\mathcal{O} \mathcal{P}^\bullet) \longrightarrow \text{Tot}(\mathcal{F}^\bullet \otimes_\mathcal{O} \mathcal{Q}^\bullet)\] is a quasi-isomorphism.

Proof

Choose a quasi-isomorphism \(\mathcal{K}^\bullet \to \mathcal{F}^\bullet\) with \(\mathcal{K}^\bullet\) a K-flat complex, see Lemma 06YS. Consider the commutative diagram \[\xymatrix{ \text{Tot}(\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{P}^\bullet) \ar[r] \ar[d] & \text{Tot}(\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{Q}^\bullet) \ar[d] \\ \text{Tot}(\mathcal{F}^\bullet \otimes_\mathcal{O} \mathcal{P}^\bullet) \ar[r] & \text{Tot}(\mathcal{F}^\bullet \otimes_\mathcal{O} \mathcal{Q}^\bullet) }\] The result follows as by Lemma 06YP the vertical arrows and the top horizontal arrow are quasi-isomorphisms.

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{F}^\bullet\) be an object of \(D(\mathcal{O})\). Choose a K-flat resolution \(\mathcal{K}^\bullet \to \mathcal{F}^\bullet\), see Lemma 06YS. By Lemma 06YM we obtain an exact functor of triangulated categories \[K(\mathcal{O}) \longrightarrow K(\mathcal{O}), \quad \mathcal{G}^\bullet \longmapsto \text{Tot}(\mathcal{G}^\bullet \otimes_\mathcal{O} \mathcal{K}^\bullet)\] By Lemma 06YP this functor induces a functor \(D(\mathcal{O}) \to D(\mathcal{O})\) simply because \(D(\mathcal{O})\) is the localization of \(K(\mathcal{O})\) at quasi-isomorphisms. As the category of \(K\)-flat resolutions of \(\mathcal{F}^\bullet\) is cofiltered and as we have Lemma 06YT the resulting functor (up to isomorphism) does not depend on the choice of \(\mathcal{K}^\bullet\).

Definition

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{F}^\bullet\) be an object of \(D(\mathcal{O})\). The derived tensor product \[- \otimes_\mathcal{O}^{\mathbf{L}} \mathcal{F}^\bullet : D(\mathcal{O}) \longrightarrow D(\mathcal{O})\] is the exact functor of triangulated categories described above.

It is clear from our explicit constructions that there is a canonical isomorphism \[\mathcal{F}^\bullet \otimes_\mathcal{O}^{\mathbf{L}} \mathcal{G}^\bullet \cong \mathcal{G}^\bullet \otimes_\mathcal{O}^{\mathbf{L}} \mathcal{F}^\bullet\] for \(\mathcal{G}^\bullet\) and \(\mathcal{F}^\bullet\) in \(D(\mathcal{O})\). Here we use sign rules as given in More on Algebra, Section 0FNG. Hence when we write \(\mathcal{F}^\bullet \otimes_\mathcal{O}^{\mathbf{L}} \mathcal{G}^\bullet\) we will usually be agnostic about which variable we are using to define the derived tensor product with.

Definition

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{F}\), \(\mathcal{G}\) be \(\mathcal{O}\)-modules. The Tor’s of \(\mathcal{F}\) and \(\mathcal{G}\) are defined by the formula \[\text{Tor}_p^\mathcal{O}(\mathcal{F}, \mathcal{G}) = H^{-p}(\mathcal{F} \otimes_\mathcal{O}^\mathbf{L} \mathcal{G})\] with derived tensor product as defined above.

This definition implies that for every short exact sequence of \(\mathcal{O}\)-modules \(0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\) we have a long exact cohomology sequence \[\xymatrix{ \mathcal{F}_1 \otimes_\mathcal{O} \mathcal{G} \ar[r] & \mathcal{F}_2 \otimes_\mathcal{O} \mathcal{G} \ar[r] & \mathcal{F}_3 \otimes_\mathcal{O} \mathcal{G} \ar[r] & 0 \\ \text{Tor}_1^\mathcal{O}(\mathcal{F}_1, \mathcal{G}) \ar[r] & \text{Tor}_1^\mathcal{O}(\mathcal{F}_2, \mathcal{G}) \ar[r] & \text{Tor}_1^\mathcal{O}(\mathcal{F}_3, \mathcal{G}) \ar[ull] }\] for every \(\mathcal{O}\)-module \(\mathcal{G}\). This will be called the long exact sequence of \(\text{Tor}\) associated to the situation.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{F}\) be an \(\mathcal{O}\)-module. The following are equivalent

1. \(\mathcal{F}\) is a flat \(\mathcal{O}\)-module, and

2. \(\text{Tor}_1^\mathcal{O}(\mathcal{F}, \mathcal{G}) = 0\) for every \(\mathcal{O}\)-module \(\mathcal{G}\).

Proof

If \(\mathcal{F}\) is flat, then \(\mathcal{F} \otimes_\mathcal{O} -\) is an exact functor and the satellites vanish. Conversely assume (2) holds. Then if \(\mathcal{G} \to \mathcal{H}\) is injective with cokernel \(\mathcal{Q}\), the long exact sequence of \(\text{Tor}\) shows that the kernel of \(\mathcal{F} \otimes_\mathcal{O} \mathcal{G} \to \mathcal{F} \otimes_\mathcal{O} \mathcal{H}\) is a quotient of \(\text{Tor}_1^\mathcal{O}(\mathcal{F}, \mathcal{Q})\) which is zero by assumption. Hence \(\mathcal{F}\) is flat.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{K}^\bullet\) be a K-flat, acyclic complex with flat terms. Then \(\mathcal{F} = \Ker(\mathcal{K}^n \to \mathcal{K}^{n + 1})\) is a flat \(\mathcal{O}\)-module.

Proof

Observe that \[\ldots \to \mathcal{K}^{n - 2} \to \mathcal{K}^{n - 1} \to \mathcal{F} \to 0\] is a flat resolution of our module \(\mathcal{F}\). Since a bounded above complex of flat modules is K-flat (Lemma 06YQ) we may use this resolution to compute \(\text{Tor}_i(\mathcal{F}, \mathcal{G})\) for any \(\mathcal{O}\)-module \(\mathcal{G}\). On the one hand \(\mathcal{K}^\bullet \otimes_\mathcal{O}^\mathbf{L} \mathcal{G}\) is zero in \(D(\mathcal{O})\) because \(\mathcal{K}^\bullet\) is acyclic and on the other hand it is represented by \(\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{G}\). Hence we see that \[\mathcal{K}^{n - 3} \otimes_\mathcal{O} \mathcal{G} \to \mathcal{K}^{n - 2} \otimes_\mathcal{O} \mathcal{G} \to \mathcal{K}^{n - 1} \otimes_\mathcal{O} \mathcal{G}\] is exact. Thus \(\text{Tor}_1(\mathcal{F}, \mathcal{G}) = 0\) and we conclude by Lemma 08FG.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(a : \mathcal{K}^\bullet \to \mathcal{L}^\bullet\) be a map of complexes of \(\mathcal{O}\)-modules. If \(\mathcal{K}^\bullet\) is K-flat, then there exist a complex \(\mathcal{N}^\bullet\) and maps of complexes \(b : \mathcal{K}^\bullet \to \mathcal{N}^\bullet\) and \(c : \mathcal{N}^\bullet \to \mathcal{L}^\bullet\) such that

1. \(\mathcal{N}^\bullet\) is K-flat,

2. \(c\) is a quasi-isomorphism,

3. \(a\) is homotopic to \(c \circ b\).

If the terms of \(\mathcal{K}^\bullet\) are flat, then we may choose \(\mathcal{N}^\bullet\), \(b\), and \(c\) such that the same is true for \(\mathcal{N}^\bullet\).

Proof

We will use that the homotopy category \(K(\textit{Mod}(\mathcal{O}))\) is a triangulated category, see Derived Categories, Proposition 014S. Choose a distinguished triangle \(\mathcal{K}^\bullet \to \mathcal{L}^\bullet \to \mathcal{C}^\bullet \to \mathcal{K}^\bullet[1]\). Choose a quasi-isomorphism \(\mathcal{M}^\bullet \to \mathcal{C}^\bullet\) with \(\mathcal{M}^\bullet\) K-flat with flat terms, see Lemma 06YS. By the axioms of triangulated categories, we may fit the composition \(\mathcal{M}^\bullet \to \mathcal{C}^\bullet \to \mathcal{K}^\bullet[1]\) into a distinguished triangle \(\mathcal{K}^\bullet \to \mathcal{N}^\bullet \to \mathcal{M}^\bullet \to \mathcal{K}^\bullet[1]\). By Lemma 07A3 we see that \(\mathcal{N}^\bullet\) is K-flat. Again using the axioms of triangulated categories, we can choose a map \(\mathcal{N}^\bullet \to \mathcal{L}^\bullet\) fitting into the following morphism of distinguished triangles \[\xymatrix{ \mathcal{K}^\bullet \ar[r] \ar[d] & \mathcal{N}^\bullet \ar[r] \ar[d] & \mathcal{M}^\bullet \ar[r] \ar[d] & \mathcal{K}^\bullet[1] \ar[d] \\ \mathcal{K}^\bullet \ar[r] & \mathcal{L}^\bullet \ar[r] & \mathcal{C}^\bullet \ar[r] & \mathcal{K}^\bullet[1] }\] Since two out of three of the arrows are quasi-isomorphisms, so is the third arrow \(\mathcal{N}^\bullet \to \mathcal{L}^\bullet\) by the long exact sequences of cohomology associated to these distinguished triangles (or you can look at the image of this diagram in \(D(\mathcal{O})\) and use Derived Categories, Lemma 014A if you like). This finishes the proof of (1), (2), and (3). To prove the final assertion, we may choose \(\mathcal{N}^\bullet\) such that \(\mathcal{N}^n \cong \mathcal{M}^n \oplus \mathcal{K}^n\), see Derived Categories, Lemma 0G6C. Hence we get the desired flatness if the terms of \(\mathcal{K}^\bullet\) are flat.

Derived pullback

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')\) be a morphism of ringed topoi. We can use K-flat resolutions to define a derived pullback functor \[Lf^* : D(\mathcal{O}') \to D(\mathcal{O})\]

Lemma

Let \(f : (\Sh(\mathcal{C}'), \mathcal{O}') \to (\Sh(\mathcal{C}), \mathcal{O})\) be a morphism of ringed topoi. Let \(\mathcal{K}^\bullet\) be a K-flat complex of \(\mathcal{O}\)-modules whose terms are flat \(\mathcal{O}\)-modules. Then \(f^*\mathcal{K}^\bullet\) is a K-flat complex of \(\mathcal{O}'\)-modules whose terms are flat \(\mathcal{O}'\)-modules.

Proof

The terms \(f^*\mathcal{K}^n\) are flat \(\mathcal{O}'\)-modules by Modules on Sites, Lemma 05VD. Choose a diagram \[\xymatrix{ \mathcal{K}_1^\bullet \ar[d] \ar[r] & \mathcal{K}_2^\bullet \ar[d] \ar[r] & \ldots \\ \tau_{\leq 1}\mathcal{K}^\bullet \ar[r] & \tau_{\leq 2}\mathcal{K}^\bullet \ar[r] & \ldots }\] as in Lemma 077J. We will use all of the properties stated in the lemma without further mention. Each \(\mathcal{K}_n^\bullet\) is a bounded above complex of flat modules, see Modules on Sites, Lemma 03EV. Consider the short exact sequence of complexes \[0 \to \mathcal{M}^\bullet \to \colim \mathcal{K}_n^\bullet \to \mathcal{K}^\bullet \to 0\] defining \(\mathcal{M}^\bullet\). By Lemmas 06YQ and 06YR the complex \(\colim \mathcal{K}_n^\bullet\) is K-flat and by Modules on Sites, Lemma 03EU it has flat terms. By Modules on Sites, Lemma 03EY \(\mathcal{M}^\bullet\) has flat terms, by Lemma 0G7B \(\mathcal{M}^\bullet\) is K-flat, and by the long exact cohomology sequence \(\mathcal{M}^\bullet\) is acyclic (because the second arrow is a quasi-isomorphism). The pullback \(f^*(\colim \mathcal{K}_n^\bullet) = \colim f^*\mathcal{K}_n^\bullet\) is a colimit of bounded above complexes of flat \(\mathcal{O}'\)-modules and hence is K-flat (by the same lemmas as above). The pullback of our short exact sequence \[0 \to f^*\mathcal{M}^\bullet \to f^*(\colim \mathcal{K}_n^\bullet) \to f^*\mathcal{K}^\bullet \to 0\] is a short exact sequence of complexes by Modules on Sites, Lemma 0G6R. Hence by Lemma 0G7B it suffices to show that \(f^*\mathcal{M}^\bullet\) is K-flat. This reduces us to the case discussed in the next paragraph.

Assume \(\mathcal{K}^\bullet\) is acyclic as well as K-flat and with flat terms. Then Lemma 0G7C guarantees that all terms of \(\tau_{\leq n}\mathcal{K}^\bullet\) are flat \(\mathcal{O}\)-modules. We choose a diagram as above and we will use all the properties proven above for this diagram. Denote \(\mathcal{M}_n^\bullet\) the kernel of the map of complexes \(\mathcal{K}_n^\bullet \to \tau_{\leq n}\mathcal{K}^\bullet\) so that we have short exact sequences of complexes \[0 \to \mathcal{M}_n^\bullet \to \mathcal{K}_n^\bullet \to \tau_{\leq n}\mathcal{K}^\bullet \to 0\] By Modules on Sites, Lemma 03EY we see that the terms of the complex \(\mathcal{M}_n^\bullet\) are flat. Hence we see that \(\mathcal{M} = \colim \mathcal{M}_n^\bullet\) is a filtered colimit of bounded above complexes of flat modules in this case. Thus \(f^*\mathcal{M}^\bullet\) is K-flat (same argument as above) and we win.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')\) be a morphism of ringed topoi. There exists an exact functor \[Lf^* : D(\mathcal{O}') \longrightarrow D(\mathcal{O})\] of triangulated categories so that \(Lf^*\mathcal{K}^\bullet = f^*\mathcal{K}^\bullet\) for any K-flat complex \(\mathcal{K}^\bullet\) with flat terms and in particular for any bounded above complex of flat \(\mathcal{O}'\)-modules.

Proof

To see this we use the general theory developed in Derived Categories, Section 05S7. Set \(\mathcal{D} = K(\mathcal{O}')\) and \(\mathcal{D}' = D(\mathcal{O})\). Let us write \(F : \mathcal{D} \to \mathcal{D}'\) the exact functor of triangulated categories defined by the rule \(F(\mathcal{G}^\bullet) = f^*\mathcal{G}^\bullet\). We let \(S\) be the set of quasi-isomorphisms in \(\mathcal{D} = K(\mathcal{O}')\). This gives a situation as in Derived Categories, Situation 05S8 so that Derived Categories, Definition 05S9 applies. We claim that \(LF\) is everywhere defined. This follows from Derived Categories, Lemma 06XN with \(\mathcal{P} \subset \Ob(\mathcal{D})\) the collection of K-flat complexes \(\mathcal{K}^\bullet\) with flat terms. Namely, (1) follows from Lemma 06YS and to see (2) we have to show that for a quasi-isomorphism \(\mathcal{K}_1^\bullet \to \mathcal{K}_2^\bullet\) between elements of \(\mathcal{P}\) the map \(f^*\mathcal{K}_1^\bullet \to f^*\mathcal{K}_2^\bullet\) is a quasi-isomorphism. To see this write this as \[f^{-1}\mathcal{K}_1^\bullet \otimes_{f^{-1}\mathcal{O}'} \mathcal{O} \longrightarrow f^{-1}\mathcal{K}_2^\bullet \otimes_{f^{-1}\mathcal{O}'} \mathcal{O}\] The functor \(f^{-1}\) is exact, hence the map \(f^{-1}\mathcal{K}_1^\bullet \to f^{-1}\mathcal{K}_2^\bullet\) is a quasi-isomorphism. The complexes \(f^{-1}\mathcal{K}_1^\bullet\) and \(f^{-1}\mathcal{K}_2^\bullet\) are K-flat complexes of \(f^{-1}\mathcal{O}'\)-modules by Lemma 0G7E because we can consider the morphism of ringed topoi \((\Sh(\mathcal{C}), f^{-1}\mathcal{O}') \to (\Sh(\mathcal{C}'), \mathcal{O}')\). Hence Lemma 06YT guarantees that the displayed map is a quasi-isomorphism. Thus we obtain a derived functor \[LF : D(\mathcal{O}') = S^{-1}\mathcal{D} \longrightarrow \mathcal{D}' = D(\mathcal{O})\] see Derived Categories, Equation (05SW). Finally, Derived Categories, Lemma 06XN also guarantees that \(LF(\mathcal{K}^\bullet) = F(\mathcal{K}^\bullet) = f^*\mathcal{K}^\bullet\) when \(\mathcal{K}^\bullet\) is in \(\mathcal{P}\). The proof is finished by observing that bounded above complexes of flat modules are in \(\mathcal{P}\) by Lemma 06YQ.

Lemma

Consider morphisms of ringed topoi \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) and \(g : (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) \to (\Sh(\mathcal{E}), \mathcal{O}_\mathcal{E})\). Then \(Lf^* \circ Lg^* = L(g \circ f)^*\) as functors \(D(\mathcal{O}_\mathcal{E}) \to D(\mathcal{O}_\mathcal{C})\).

Proof

Let \(E\) be an object of \(D(\mathcal{O}_\mathcal{E})\). We may represent \(E\) by a K-flat complex \(\mathcal{K}^\bullet\) with flat terms, see Lemma 06YS. By construction \(Lg^*E\) is computed by \(g^*\mathcal{K}^\bullet\), see Lemma 06YY. By Lemma 0G7E the complex \(g^*\mathcal{K}^\bullet\) is K-flat with flat terms. Hence \(Lf^*Lg^*E\) is represented by \(f^*g^*\mathcal{K}^\bullet\). Since also \(L(g \circ f)^*E\) is represented by \((g \circ f)^*\mathcal{K}^\bullet = f^*g^*\mathcal{K}^\bullet\) we conclude.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) be a morphism of ringed topoi. There is a canonical bifunctorial isomorphism \[Lf^*( \mathcal{F}^\bullet \otimes_{\mathcal{O}'}^{\mathbf{L}} \mathcal{G}^\bullet ) = Lf^*\mathcal{F}^\bullet \otimes_{\mathcal{O}}^{\mathbf{L}} Lf^*\mathcal{G}^\bullet\] for \(\mathcal{F}^\bullet, \mathcal{G}^\bullet \in \Ob(D(\mathcal{O}'))\).

Proof

By our construction of derived pullback in Lemma 06YY. and the existence of resolutions in Lemma 06YS we may replace \(\mathcal{F}^\bullet\) and \(\mathcal{G}^\bullet\) by complexes of \(\mathcal{O}'\)-modules which are K-flat and have flat terms. In this case \(\mathcal{F}^\bullet \otimes_{\mathcal{O}'}^{\mathbf{L}} \mathcal{G}^\bullet\) is just the total complex associated to the double complex \(\mathcal{F}^\bullet \otimes_{\mathcal{O}'} \mathcal{G}^\bullet\). The complex \(\text{Tot}(\mathcal{F}^\bullet \otimes_{\mathcal{O}'} \mathcal{G}^\bullet)\) is K-flat with flat terms by Lemma 07A2 and Modules on Sites, Lemma 0G6Q. Hence the isomorphism of the lemma comes from the isomorphism \[\text{Tot}(f^*\mathcal{F}^\bullet \otimes_{\mathcal{O}} f^*\mathcal{G}^\bullet) \longrightarrow f^*\text{Tot}(\mathcal{F}^\bullet \otimes_{\mathcal{O}'} \mathcal{G}^\bullet)\] whose constituents are the isomorphisms \(f^*\mathcal{F}^p \otimes_{\mathcal{O}} f^*\mathcal{G}^q \to f^*(\mathcal{F}^p \otimes_{\mathcal{O}'} \mathcal{G}^q)\) of Modules on Sites, Lemma 03EL.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')\) be a morphism of ringed topoi. There is a canonical bifunctorial isomorphism \[\mathcal{F}^\bullet \otimes_\mathcal{O}^{\mathbf{L}} Lf^*\mathcal{G}^\bullet = \mathcal{F}^\bullet \otimes_{f^{-1}\mathcal{O}'}^{\mathbf{L}} f^{-1}\mathcal{G}^\bullet\] for \(\mathcal{F}^\bullet\) in \(D(\mathcal{O})\) and \(\mathcal{G}^\bullet\) in \(D(\mathcal{O}')\).

Proof

Let \(\mathcal{F}\) be an \(\mathcal{O}\)-module and let \(\mathcal{G}\) be an \(\mathcal{O}'\)-module. Then \(\mathcal{F} \otimes_{\mathcal{O}} f^*\mathcal{G} = \mathcal{F} \otimes_{f^{-1}\mathcal{O}'} f^{-1}\mathcal{G}\) because \(f^*\mathcal{G} = \mathcal{O} \otimes_{f^{-1}\mathcal{O}'} f^{-1}\mathcal{G}\). The lemma follows from this and the definitions.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{K}^\bullet\) be a complex of \(\mathcal{O}\)-modules.

1. If \(\mathcal{K}^\bullet\) is K-flat, then for every point \(p\) of the site \(\mathcal{C}\) the complex of \(\mathcal{O}_p\)-modules \(\mathcal{K}_p^\bullet\) is K-flat in the sense of More on Algebra, Definition 06XZ

2. If \(\mathcal{C}\) has enough points, then the converse is true.

Proof

Proof of (2). If \(\mathcal{C}\) has enough points and \(\mathcal{K}_p^\bullet\) is K-flat for all points \(p\) of \(\mathcal{C}\) then we see that \(\mathcal{K}^\bullet\) is K-flat because \(\otimes\) and direct sums commute with taking stalks and because we can check exactness at stalks, see Modules on Sites, Lemma 05V3.

Proof of (1). Assume \(\mathcal{K}^\bullet\) is K-flat. Choose a quasi-isomorphism \(a : \mathcal{L}^\bullet \to \mathcal{K}^\bullet\) such that \(\mathcal{L}^\bullet\) is K-flat with flat terms, see Lemma 06YS. Any pullback of \(\mathcal{L}^\bullet\) is K-flat, see Lemma 0G7E. In particular the stalk \(\mathcal{L}_p^\bullet\) is a K-flat complex of \(\mathcal{O}_p\)-modules. Thus the cone \(C(a)\) on \(a\) is a K-flat (Lemma 07A3) acyclic complex of \(\mathcal{O}\)-modules and it suffuces to show the stalk of \(C(a)\) is K-flat (by More on Algebra, Lemma 06Y2). Thus we may assume that \(\mathcal{K}^\bullet\) is K-flat and acyclic.

Assume \(\mathcal{K}^\bullet\) is acyclic and K-flat. Before continuing we replace the site \(\mathcal{C}\) by another one as in Sites, Lemma 03CI to insure that \(\mathcal{C}\) has all finite limits. This implies the category of neighbourhoods of \(p\) is filtered (Sites, Lemma 00YB) and the colimit defining the stalk of a sheaf is filtered. Let \(M\) be a finitely presented \(\mathcal{O}_p\)-module. It suffices to show that \(\mathcal{K}^\bullet \otimes_{\mathcal{O}_p} M\) is acyclic, see More on Algebra, Lemma 0E8F. Since \(\mathcal{O}_p\) is the filtered colimit of \(\mathcal{O}(U)\) where \(U\) runs over the neighbourhoods of \(p\), we can find a neighbourhood \((U, x)\) of \(p\) and a finitely presented \(\mathcal{O}(U)\)-module \(M'\) whose base change to \(\mathcal{O}_p\) is \(M\), see Algebra, Lemma 05N7. By Lemma 0E8K we may replace \(\mathcal{C}, \mathcal{O}, \mathcal{K}^\bullet\) by \(\mathcal{C}/U, \mathcal{O}_U, \mathcal{K}^\bullet|_U\). We conclude that we may assume there exists an \(\mathcal{O}\)-module \(\mathcal{F}\) such that \(M \cong \mathcal{F}_p\). Since \(\mathcal{K}^\bullet\) is K-flat and acyclic, we see that \(\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{F}\) is acyclic (as it computes the derived tensor product by definition). Taking stalks is an exact functor, hence we get that \(\mathcal{K}^\bullet \otimes_{\mathcal{O}_p} M\) is acyclic as desired.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')\) be a morphism of ringed topoi. If \(\mathcal{C}\) has enough points, then the pullback of a K-flat complex of \(\mathcal{O}'\)-modules is a K-flat complex of \(\mathcal{O}\)-modules.

Proof

This follows from Lemma 0DEN, Modules on Sites, Lemma 05V5, and More on Algebra, Lemma 06Y1.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi. Let \(\mathcal{K}^\bullet\) and \(\mathcal{M}^\bullet\) be complexes of \(\mathcal{O}_\mathcal{D}\)-modules. The diagram \[\xymatrix{ Lf^*(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} \mathcal{M}^\bullet) \ar[r] \ar[d] & Lf^*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}} \mathcal{M}^\bullet) \ar[d] \\ Lf^*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} Lf^*\mathcal{M}^\bullet \ar[d] & f^*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}} \mathcal{M}^\bullet) \ar[d] \\ f^*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} f^*\mathcal{M}^\bullet \ar[r] & \text{Tot}(f^*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} f^*\mathcal{M}^\bullet) }\] commutes.

Proof

We will use the existence of K-flat resolutions with flat terms (Lemma 06YS), we will use that derived pullback is computed by such complexes (Lemma 06YY), and that pullbacks preserve these properties (Lemma 0G7E). If we choose such resolutions \(\mathcal{P}^\bullet \to \mathcal{K}^\bullet\) and \(\mathcal{Q}^\bullet \to \mathcal{M}^\bullet\), then we see that \[\xymatrix{ Lf^*\text{Tot}(\mathcal{P}^\bullet \otimes_{\mathcal{O}_\mathcal{D}} \mathcal{Q}^\bullet) \ar[r] \ar[d] & Lf^*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}} \mathcal{M}^\bullet) \ar[d] \\ f^*\text{Tot}(\mathcal{P}^\bullet \otimes_{\mathcal{O}_\mathcal{D}} \mathcal{Q}^\bullet) \ar[d] \ar[r] & f^*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}} \mathcal{M}^\bullet) \ar[d] \\ \text{Tot}(f^*\mathcal{P}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} f^*\mathcal{Q}^\bullet) \ar[r] & \text{Tot}(f^*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} f^*\mathcal{M}^\bullet) }\] commutes. However, now the left hand side of the diagram is the left hand side of the diagram by our choice of \(\mathcal{P}^\bullet\) and \(\mathcal{Q}^\bullet\) and Lemma 07A2.

Cohomology of unbounded complexes

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. The category \(\textit{Mod}(\mathcal{O})\) is a Grothendieck abelian category: it has all colimits, filtered colimits are exact, and it has a generator, namely \[\bigoplus\nolimits_{U \in \Ob(\mathcal{C})} j_{U!}\mathcal{O}_U,\] see Modules on Sites, Section 03D9 and Lemmas 03EV and 03EW. By Injectives, Theorem 079P for every complex \(\mathcal{F}^\bullet\) of \(\mathcal{O}\)-modules there exists an injective quasi-isomorphism \(\mathcal{F}^\bullet \to \mathcal{I}^\bullet\) to a K-injective complex of \(\mathcal{O}\)-modules and moreover this embedding can be chosen functorial in \(\mathcal{F}^\bullet\). It follows from Derived Categories, Lemma 070K that

1. any exact functor \(F : K(\textit{Mod}(\mathcal{O})) \to \mathcal{D}\) into a trianguated category \(\mathcal{D}\) has a right derived functor \(RF : D(\mathcal{O}) \to \mathcal{D}\),

2. for any additive functor \(F : \textit{Mod}(\mathcal{O}) \to \mathcal{A}\) into an abelian category \(\mathcal{A}\) we consider the exact functor \(F : K(\textit{Mod}(\mathcal{O})) \to K(\mathcal{A})\) induced by \(F\) and we obtain a right derived functor \(RF : D(\mathcal{O}) \to D(\mathcal{A})\).

By construction we have \(RF(\mathcal{F}^\bullet) = F(\mathcal{I}^\bullet)\) where \(\mathcal{F}^\bullet \to \mathcal{I}^\bullet\) is as above.

Here are some examples of the above:

1. The functor \(\Gamma(\mathcal{C}, -) : \textit{Mod}(\mathcal{O}) \to \text{Mod}_{\Gamma(\mathcal{C}, \mathcal{O})}\) gives rise to \[R\Gamma(\mathcal{C}, -) : D(\mathcal{O}) \longrightarrow D(\Gamma(\mathcal{C}, \mathcal{O}))\] We shall use the notation \(H^i(\mathcal{C}, K) = H^i(R\Gamma(\mathcal{C}, K))\) for cohomology.

2. For an object \(U\) of \(\mathcal{C}\) we consider the functor \(\Gamma(U, -) : \textit{Mod}(\mathcal{O}) \to \text{Mod}_{\Gamma(U, \mathcal{O})}\). This gives rise to \[R\Gamma(U, -) : D(\mathcal{O}) \to D(\Gamma(U, \mathcal{O}))\] We shall use the notation \(H^i(U, K) = H^i(R\Gamma(U, K))\) for cohomology.

3. For a morphism of ringed topoi \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) we consider the functor \(f_* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}')\) which gives rise to the total direct image \[Rf_* : D(\mathcal{O}) \longrightarrow D(\mathcal{O}')\] on unbounded derived categories.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) be a morphism of ringed topoi. The functor \(Rf_*\) defined above and the functor \(Lf^*\) defined in Lemma 06YY are adjoint: \[\Hom_{D(\mathcal{O})}(Lf^*\mathcal{G}^\bullet, \mathcal{F}^\bullet) = \Hom_{D(\mathcal{O}')}(\mathcal{G}^\bullet, Rf_*\mathcal{F}^\bullet)\] bifunctorially in \(\mathcal{F}^\bullet \in \Ob(D(\mathcal{O}))\) and \(\mathcal{G}^\bullet \in \Ob(D(\mathcal{O}'))\).

Proof

This follows formally from the fact that \(Rf_*\) and \(Lf^*\) exist, see Derived Categories, Lemma 09T5.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) and \(g : (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) \to (\Sh(\mathcal{E}), \mathcal{O}_\mathcal{E})\) be morphisms of ringed topoi. Then \(Rg_* \circ Rf_* = R(g \circ f)_*\) as functors \(D(\mathcal{O}_\mathcal{C}) \to D(\mathcal{O}_\mathcal{E})\).

Proof

By Lemma 07A6 we see that \(Rg_* \circ Rf_*\) is adjoint to \(Lf^* \circ Lg^*\). We have \(Lf^* \circ Lg^* = L(g \circ f)^*\) by Lemma 0D6D and hence by uniqueness of adjoint functors we have \(Rg_* \circ Rf_* = R(g \circ f)_*\).

Remark

The construction of unbounded derived functor \(Lf^*\) and \(Rf_*\) allows one to construct the base change map in full generality. Namely, suppose that \[\xymatrix{ (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_{g'} \ar[d]_{f'} & (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^f \\ (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^g & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) }\] is a commutative diagram of ringed topoi. Let \(K\) be an object of \(D(\mathcal{O}_\mathcal{C})\). Then there exists a canonical base change map \[Lg^*Rf_*K \longrightarrow R(f')_*L(g')^*K\] in \(D(\mathcal{O}_{\mathcal{D}'})\). Namely, this map is adjoint to a map \(L(f')^*Lg^*Rf_*K \to L(g')^*K\). Since \(L(f')^* \circ Lg^* = L(g')^* \circ Lf^*\) we see this is the same as a map \(L(g')^*Lf^*Rf_*K \to L(g')^*K\) which we can take to be \(L(g')^*\) of the adjunction map \(Lf^*Rf_*K \to K\).

Remark

Consider a commutative diagram \[\xymatrix{ (\Sh(\mathcal{B}'), \mathcal{O}_{\mathcal{B}'}) \ar[r]_k \ar[d]_{f'} & (\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B}) \ar[d]^f \\ (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]^l \ar[d]_{g'} & (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^g \\ (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^m & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) \\ }\] of ringed topoi. Then the base change maps of Remark 07A7 for the two squares compose to give the base change map for the outer rectangle. More precisely, the composition \[\begin{align*} Lm^* \circ R(g \circ f)_* & = Lm^* \circ Rg_* \circ Rf_* \\ & \to Rg'_* \circ Ll^* \circ Rf_* \\ & \to Rg'_* \circ Rf'_* \circ Lk^* \\ & = R(g' \circ f')_* \circ Lk^* \end{align*}\] is the base change map for the rectangle. We omit the verification.

Remark

Consider a commutative diagram \[\xymatrix{ (\Sh(\mathcal{C}''), \mathcal{O}_{\mathcal{C}''}) \ar[r]_{g'} \ar[d]_{f''} & (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_g \ar[d]_{f'} & (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^f \\ (\Sh(\mathcal{D}''), \mathcal{O}_{\mathcal{D}''}) \ar[r]^{h'} & (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^h & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) }\] of ringed topoi. Then the base change maps of Remark 07A7 for the two squares compose to give the base change map for the outer rectangle. More precisely, the composition \[\begin{align*} L(h \circ h')^* \circ Rf_* & = L(h')^* \circ Lh^* \circ Rf_* \\ & \to L(h')^* \circ Rf'_* \circ Lg^* \\ & \to Rf''_* \circ L(g')^* \circ Lg^* \\ & = Rf''_* \circ L(g \circ g')^* \end{align*}\] is the base change map for the rectangle. We omit the verification.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi. Let \(\mathcal{K}^\bullet\) be a complex of \(\mathcal{O}_\mathcal{C}\)-modules. The diagram \[\xymatrix{ Lf^*f_*\mathcal{K}^\bullet \ar[r] \ar[d] & f^*f_*\mathcal{K}^\bullet \ar[d] \\ Lf^*Rf_*\mathcal{K}^\bullet \ar[r] & \mathcal{K}^\bullet }\] coming from \(Lf^* \to f^*\) on complexes, \(f_* \to Rf_*\) on complexes, and adjunction \(Lf^* \circ Rf_* \to \text{id}\) commutes in \(D(\mathcal{O}_\mathcal{C})\).

Proof

We will use the existence of K-flat resolutions and K-injective resolutions, see Lemmas 06YS, 06YY, and 0G7E and the discussion above. Choose a quasi-isomorphism \(\mathcal{K}^\bullet \to \mathcal{I}^\bullet\) where \(\mathcal{I}^\bullet\) is K-injective as a complex of \(\mathcal{O}_\mathcal{C}\)-modules. Choose a quasi-isomorphism \(\mathcal{Q}^\bullet \to f_*\mathcal{I}^\bullet\) where \(\mathcal{Q}^\bullet\) is a K-flat complex of \(\mathcal{O}_\mathcal{D}\)-modules with flat terms. We can choose a K-flat complex of \(\mathcal{O}_\mathcal{D}\)-modules \(\mathcal{P}^\bullet\) with flat terms and a diagram of morphisms of complexes \[\xymatrix{ \mathcal{P}^\bullet \ar[r] \ar[d] & f_*\mathcal{K}^\bullet \ar[d] \\ \mathcal{Q}^\bullet \ar[r] & f_*\mathcal{I}^\bullet }\] commutative up to homotopy where the top horizontal arrow is a quasi-isomorphism. Namely, we can first choose such a diagram for some complex \(\mathcal{P}^\bullet\) because the quasi-isomorphisms form a multiplicative system in the homotopy category of complexes and then we can choose a resolution of \(\mathcal{P}^\bullet\) by a K-flat complex with flat terms. Taking pullbacks we obtain a diagram of morphisms of complexes \[\xymatrix{ f^*\mathcal{P}^\bullet \ar[r] \ar[d] & f^*f_*\mathcal{K}^\bullet \ar[d] \ar[r] & \mathcal{K}^\bullet \ar[d] \\ f^*\mathcal{Q}^\bullet \ar[r] & f^*f_*\mathcal{I}^\bullet \ar[r] & \mathcal{I}^\bullet }\] commutative up to homotopy. The outer rectangle witnesses the truth of the statement in the lemma.

Remark

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi. The adjointness of \(Lf^*\) and \(Rf_*\) allows us to construct a relative cup product \[Rf_*K \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} Rf_*L \longrightarrow Rf_*(K \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} L)\] in \(D(\mathcal{O}_\mathcal{D})\) for all \(K, L\) in \(D(\mathcal{O}_\mathcal{C})\). Namely, this map is adjoint to a map \(Lf^*(Rf_*K \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} Rf_*L) \to K \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} L\) for which we can take the composition of the isomorphism \(Lf^*(Rf_*K \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} Rf_*L) = Lf^*Rf_*K \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} Lf^*Rf_*L\) (Lemma 07A4) with the map \(Lf^*Rf_*K \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} Lf^*Rf_*L \to K \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} L\) coming from the counit \(Lf^* \circ Rf_* \to \text{id}\).

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{A} \subset \textit{Ab}(\mathcal{C})\) denote the Serre subcategory consisting of torsion abelian sheaves. Then the functor \(D(\mathcal{A}) \to D_\mathcal{A}(\mathcal{C})\) is an equivalence.

Proof

A key observation is that an injective abelian sheaf \(\mathcal{I}\) is divisible. Namely, if \(s \in \mathcal{I}(U)\) is a local section, then we interpret \(s\) as a map \(s : j_{U!}\mathbf{Z} \to \mathcal{I}\) and we apply the defining property of an injective object to the injective map of sheaves \(n : j_{U!}\mathbf{Z} \to j_{U!}\mathbf{Z}\) to see that there exists an \(s' \in \mathcal{I}(U)\) with \(ns' = s\).

For a sheaf \(\mathcal{F}\) denote \(\mathcal{F}_{tor}\) its torsion subsheaf. We claim that if \(\mathcal{I}^\bullet\) is a complex of injective abelian sheaves whose cohomology sheaves are torsion, then \[\mathcal{I}^\bullet_{tor} \to \mathcal{I}^\bullet\] is a quasi-isomorphism. Namely, by flatness of \(\mathbf{Q}\) over \(\mathbf{Z}\) we have \[H^p(\mathcal{I}^\bullet) \otimes_\mathbf{Z} \mathbf{Q} = H^p(\mathcal{I}^\bullet \otimes_\mathbf{Z} \mathbf{Q})\] which is zero because the cohomology sheaves are torsion. By divisibility (shown above) we see that \(\mathcal{I}^\bullet \to \mathcal{I}^\bullet \otimes_\mathbf{Z} \mathbf{Q}\) is surjective with kernel \(\mathcal{I}^\bullet_{tor}\). The claim follows from the long exact sequence of cohomology sheaves associated to the short exact sequence you get.

To prove the lemma we will construct right adjoint \(T : D(\mathcal{C}) \to D(\mathcal{A})\). Namely, given \(K\) in \(D(\mathcal{C})\) we can represent \(K\) by a K-injective complex \(\mathcal{I}^\bullet\) whose cohomology sheaves are injective, see Injectives, Theorem 079P. Then we set \(T(K) = \mathcal{I}^\bullet_{tor}\), in other words, \(T\) is the right derived functor of taking torsion. The functor \(T\) is a right adjoint to \(i : D(\mathcal{A}) \to D_\mathcal{A}(\mathcal{C})\). This readily follows from the observation that if \(\mathcal{F}^\bullet\) is a complex of torsion sheaves, then \[\Hom_{K(\mathcal{A})}(\mathcal{F}^\bullet, I^\bullet_{tor}) = \Hom_{K(\textit{Ab}(\mathcal{C}))}(\mathcal{F}^\bullet, I^\bullet)\] in particular \(\mathcal{I}^\bullet_{tor}\) is a K-injective complex of \(\mathcal{A}\). Some details omitted; in case of doubt, it also follows from the more general Derived Categories, Lemma 09T5. Our claim above gives that \(L = T(i(L))\) for \(L\) in \(D(\mathcal{A})\) and \(i(T(K)) = K\) if \(K\) is in \(D_\mathcal{A}(\mathcal{C})\). Using Categories, Lemma 07RB the result follows.

Some properties of K-injective complexes

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(U\) be an object of \(\mathcal{C}\). Denote \(j : (\Sh(\mathcal{C}/U), \mathcal{O}_U) \to (\Sh(\mathcal{C}), \mathcal{O})\) the corresponding localization morphism. The pullback functor \(j^*\) is exact as it is just the restriction functor. Thus derived pullback \(Lj^*\) is computed on any complex by simply restricting the complex. We often simply denote the corresponding functor \[D(\mathcal{O}) \to D(\mathcal{O}_U), \quad E \mapsto j^*E = E|_U\] Similarly, extension by zero \(j_! : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})\) (see Modules on Sites, Definition 04IX) is an exact functor (Modules on Sites, Lemma 03DJ). Thus it induces a functor \[j_! : D(\mathcal{O}_U) \to D(\mathcal{O}), \quad F \mapsto j_!F\] by simply applying \(j_!\) to any complex representing the object \(F\).

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(U\) be an object of \(\mathcal{C}\). The restriction of a K-injective complex of \(\mathcal{O}\)-modules to \(\mathcal{C}/U\) is a K-injective complex of \(\mathcal{O}_U\)-modules.

Proof

Follows immediately from Derived Categories, Lemma 08BJ and the fact that the restriction functor has the exact left adjoint \(j_!\). See discussion above.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(U \in \Ob(\mathcal{C})\). For \(K\) in \(D(\mathcal{O})\) we have \(H^p(U, K) = H^p(\mathcal{C}/U, K|_{\mathcal{C}/U})\).

Proof

Let \(\mathcal{I}^\bullet\) be a K-injective complex of \(\mathcal{O}\)-modules representing \(K\). Then \[H^q(U, K) = H^q(\Gamma(U, \mathcal{I}^\bullet)) = H^q(\Gamma(\mathcal{C}/U, \mathcal{I}^\bullet|_{\mathcal{C}/U}))\] by construction of cohomology. By Lemma 08FI the complex \(\mathcal{I}^\bullet|_{\mathcal{C}/U}\) is a K-injective complex representing \(K|_{\mathcal{C}/U}\) and the lemma follows.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(K\) be an object of \(D(\mathcal{O})\). The sheafification of \[U \mapsto H^q(U, K) = H^q(\mathcal{C}/U, K|_{\mathcal{C}/U})\] is the \(q\)th cohomology sheaf \(H^q(K)\) of \(K\).

Proof

The equality \(H^q(U, K) = H^q(\mathcal{C}/U, K|_{\mathcal{C}/U})\) holds by Lemma 0D6F. Choose a K-injective complex \(\mathcal{I}^\bullet\) representing \(K\). Then \[H^q(U, K) = \frac{\Ker(\mathcal{I}^q(U) \to \mathcal{I}^{q + 1}(U))} {\Im(\mathcal{I}^{q - 1}(U) \to \mathcal{I}^q(U))}.\] by our construction of cohomology. Since \(H^q(K) = \Ker(\mathcal{I}^q \to \mathcal{I}^{q + 1})/ \Im(\mathcal{I}^{q - 1} \to \mathcal{I}^q)\) the result is clear.

Lemma

Let \(f : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D})\) be a morphism of ringed sites corresponding to the continuous functor \(u : \mathcal{D} \to \mathcal{C}\). Given \(V \in \mathcal{D}\), set \(U = u(V)\) and denote \(g : (\mathcal{C}/U, \mathcal{O}_U) \to (\mathcal{D}/V, \mathcal{O}_V)\) the induced morphism of ringed sites (Modules on Sites, Lemma 04J0). Then \((Rf_*E)|_{\mathcal{D}/V} = Rg_*(E|_{\mathcal{C}/U})\) for \(E\) in \(D(\mathcal{O}_\mathcal{C})\).

Proof

Represent \(E\) by a K-injective complex \(\mathcal{I}^\bullet\) of \(\mathcal{O}_\mathcal{C}\)-modules. Then \(Rf_*(E) = f_*\mathcal{I}^\bullet\) and \(Rg_*(E|_{\mathcal{C}/U}) = g_*(\mathcal{I}^\bullet|_{\mathcal{C}/U})\) by Lemma 08FI. Since it is clear that \((f_*\mathcal{F})|_{\mathcal{D}/V} = g_*(\mathcal{F}|_{\mathcal{C}/U})\) for any sheaf \(\mathcal{F}\) on \(\mathcal{C}\) (see Modules on Sites, Lemma 04J0 or the more basic Sites, Lemma 03CF) the result follows.

Lemma

Let \(f : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D})\) be a morphism of ringed sites corresponding to the continuous functor \(u : \mathcal{D} \to \mathcal{C}\). Then \(R\Gamma(\mathcal{D}, -) \circ Rf_* = R\Gamma(\mathcal{C}, -)\) as functors \(D(\mathcal{O}_\mathcal{C}) \to D(\Gamma(\mathcal{O}_\mathcal{D}))\). More generally, for \(V \in \mathcal{D}\) with \(U = u(V)\) we have \(R\Gamma(U, -) = R\Gamma(V, -) \circ Rf_*\).

Proof

Consider the punctual topos \(pt\) endowed with \(\mathcal{O}_{pt}\) given by the ring \(\Gamma(\mathcal{O}_\mathcal{D})\). There is a canonical morphism \((\mathcal{D}, \mathcal{O}_\mathcal{D}) \to (pt, \mathcal{O}_{pt})\) of ringed topoi inducing the identification on global sections of structure sheaves. Then \(D(\mathcal{O}_{pt}) = D(\Gamma(\mathcal{O}_\mathcal{D}))\). The assertion \(R\Gamma(\mathcal{D}, -) \circ Rf_* = R\Gamma(\mathcal{C}, -)\) follows from Lemma 0D6E applied to \[(\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D}) \to (pt, \mathcal{O}_{pt})\] The second (more general) statement follows from the first statement after applying Lemma 0D6G.

Lemma

Let \(f : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D})\) be a morphism of ringed sites corresponding to the continuous functor \(u : \mathcal{D} \to \mathcal{C}\). Let \(K\) be in \(D(\mathcal{O}_\mathcal{C})\). Then \(H^i(Rf_*K)\) is the sheaf associated to the presheaf \[V \mapsto H^i(u(V), K) = H^i(V, Rf_*K)\]

Proof

The equality \(H^i(u(V), K) = H^i(V, Rf_*K)\) follows upon taking cohomology from the second statement in Lemma 0D6H. Then the statement on sheafification follows from Lemma 0BKV.

Lemma

Let \((\mathcal{C}, \mathcal{O}_\mathcal{C})\) be a ringed site. Let \(K\) be an object of \(D(\mathcal{O}_\mathcal{C})\) and denote \(K_{ab}\) its image in \(D(\underline{\mathbf{Z}}_\mathcal{C})\).

1. There is a canonical map \(R\Gamma(\mathcal{C}, K) \to R\Gamma(\mathcal{C}, K_{ab})\) which is an isomorphism in \(D(\textit{Ab})\).

2. For any \(U \in \mathcal{C}\) there is a canonical map \(R\Gamma(U, K) \to R\Gamma(U, K_{ab})\) which is an isomorphism in \(D(\textit{Ab})\).

3. Let \(f : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D})\) be a morphism of ringed sites. There is a canonical map \(Rf_*K \to Rf_*(K_{ab})\) which is an isomorphism in \(D(\underline{\mathbf{Z}}_\mathcal{D})\).

Proof

The map is constructed as follows. Choose a K-injective complex \(\mathcal{I}^\bullet\) representing \(K\). Choose a quasi-isomorpism \(\mathcal{I}^\bullet \to \mathcal{J}^\bullet\) where \(\mathcal{J}^\bullet\) is a K-injective complex of abelian groups. Then the map in (1) is given by \(\Gamma(\mathcal{C}, \mathcal{I}^\bullet) \to \Gamma(\mathcal{C}, \mathcal{J}^\bullet)\) (2) is given by \(\Gamma(U, \mathcal{I}^\bullet) \to \Gamma(U, \mathcal{J}^\bullet)\) and the map in (3) is given by \(f_*\mathcal{I}^\bullet \to f_*\mathcal{J}^\bullet\). To show that these maps are isomorphisms, it suffices to prove they induce isomorphisms on cohomology groups and cohomology sheaves. By Lemmas 0D6F and 0D6I it suffices to show that the map \[H^0(\mathcal{C}, K) \longrightarrow H^0(\mathcal{C}, K_{ab})\] is an isomorphism. Observe that \[H^0(\mathcal{C}, K) = \Hom_{D(\mathcal{O}_\mathcal{C})}(\mathcal{O}_\mathcal{C}, K)\] and similarly for the other group. Choose any complex \(\mathcal{K}^\bullet\) of \(\mathcal{O}_\mathcal{C}\)-modules representing \(K\). By construction of the derived category as a localization we have \[\Hom_{D(\mathcal{O}_\mathcal{C})}(\mathcal{O}_\mathcal{C}, K) = \colim_{s : \mathcal{F}^\bullet \to \mathcal{O}_\mathcal{C}} \Hom_{K(\mathcal{O}_\mathcal{C})}(\mathcal{F}^\bullet, \mathcal{K}^\bullet)\] where the colimit is over quasi-isomorphisms \(s\) of complexes of \(\mathcal{O}_\mathcal{C}\)-modules. Similarly, we have \[\Hom_{D(\underline{\mathbf{Z}}_\mathcal{C})} (\underline{\mathbf{Z}}_\mathcal{C}, K) = \colim_{s : \mathcal{G}^\bullet \to \underline{\mathbf{Z}}_\mathcal{C}} \Hom_{K(\underline{\mathbf{Z}}_\mathcal{C})} (\mathcal{G}^\bullet, \mathcal{K}^\bullet)\] Next, we observe that the quasi-isomorphisms \(s : \mathcal{G}^\bullet \to \underline{\mathbf{Z}}_\mathcal{C}\) with \(\mathcal{G}^\bullet\) bounded above complex of flat \(\underline{\mathbf{Z}}_\mathcal{C}\)-modules is cofinal in the system. (This follows from Modules on Sites, Lemma 03EW and Derived Categories, Lemma 05T7; see discussion in Section 06YL.) Hence we can construct an inverse to the map \(H^0(\mathcal{C}, K) \longrightarrow H^0(\mathcal{C}, K_{ab})\) by representing an element \(\xi \in H^0(\mathcal{C}, K_{ab})\) by a pair \[(s : \mathcal{G}^\bullet \to \underline{\mathbf{Z}}_\mathcal{C}, a : \mathcal{G}^\bullet \to \mathcal{K}^\bullet)\] with \(\mathcal{G}^\bullet\) a bounded above complex of flat \(\underline{\mathbf{Z}}_\mathcal{C}\)-modules and sending this to \[(\mathcal{G}^\bullet \otimes_{\underline{\mathbf{Z}}_\mathcal{C}} \mathcal{O}_\mathcal{C} \to \mathcal{O}_\mathcal{C}, \mathcal{G}^\bullet \otimes_{\underline{\mathbf{Z}}_\mathcal{C}} \mathcal{O}_\mathcal{C} \to \mathcal{K}^\bullet)\] The only thing to note here is that the first arrow is a quasi-isomorphism by Lemmas 06YT and 06YQ. We omit the detailed verification that this construction is indeed an inverse.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(U\) be an object of \(\mathcal{C}\). Denote \(j : (\Sh(\mathcal{C}/U), \mathcal{O}_U) \to (\Sh(\mathcal{C}), \mathcal{O})\) the corresponding localization morphism. The restriction functor \(D(\mathcal{O}) \to D(\mathcal{O}_U)\) is a right adjoint to extension by zero \(j_! : D(\mathcal{O}_U) \to D(\mathcal{O})\).

Proof

We have to show that \[\Hom_{D(\mathcal{O})}(j_!E, F) = \Hom_{D(\mathcal{O}_U)}(E, F|_U)\] Choose a complex \(\mathcal{E}^\bullet\) of \(\mathcal{O}_U\)-modules representing \(E\) and choose a K-injective complex \(\mathcal{I}^\bullet\) representing \(F\). By Lemma 08FI the complex \(\mathcal{I}^\bullet|_U\) is K-injective as well. Hence we see that the formula above becomes \[\Hom_{D(\mathcal{O})}(j_!\mathcal{E}^\bullet, \mathcal{I}^\bullet) = \Hom_{D(\mathcal{O}_U)}(\mathcal{E}^\bullet, \mathcal{I}^\bullet|_U)\] which holds as \(|_U\) and \(j_!\) are adjoint functors (Modules on Sites, Lemma 03DI) and Derived Categories, Lemma 070I.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(U \in \Ob(\mathcal{C})\). For \(L\) in \(D(\mathcal{O}_U)\) and \(K\) in \(D(\mathcal{O})\) we have \(j_!L \otimes_\mathcal{O}^\mathbf{L} K = j_!(L \otimes_{\mathcal{O}_U}^\mathbf{L} K|_U)\).

Proof

Represent \(L\) by a complex of \(\mathcal{O}_U\)-modules and \(K\) by a K-flat complexe of \(\mathcal{O}\)-modules and apply Modules on Sites, Lemma 0E8I. Details omitted.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a flat morphism of ringed topoi. If \(\mathcal{I}^\bullet\) is a K-injective complex of \(\mathcal{O}_\mathcal{C}\)-modules, then \(f_*\mathcal{I}^\bullet\) is K-injective as a complex of \(\mathcal{O}_\mathcal{D}\)-modules.

Proof

This is true because \[\Hom_{K(\mathcal{O}_\mathcal{D})}(\mathcal{F}^\bullet, f_*\mathcal{I}^\bullet) = \Hom_{K(\mathcal{O}_\mathcal{C})}(f^*\mathcal{F}^\bullet, \mathcal{I}^\bullet)\] by Modules on Sites, Lemma 03D7 and the fact that \(f^*\) is exact as \(f\) is assumed to be flat.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O} \to \mathcal{O}'\) be a map of sheaves of rings. If \(\mathcal{I}^\bullet\) is a K-injective complex of \(\mathcal{O}\)-modules, then \(\SheafHom_\mathcal{O}(\mathcal{O}', \mathcal{I}^\bullet)\) is a K-injective complex of \(\mathcal{O}'\)-modules.

Proof

This is true because \(\Hom_{K(\mathcal{O}')}(\mathcal{G}^\bullet, \Hom_\mathcal{O}(\mathcal{O}', \mathcal{I}^\bullet)) = \Hom_{K(\mathcal{O})}(\mathcal{G}^\bullet, \mathcal{I}^\bullet)\) by Modules on Sites, Lemma 0932.

Localization and cohomology

Let \(\mathcal{C}\) be a site. Let \(f : X \to Y\) be a morphism of \(\mathcal{C}\). Then we obtain a morphism of topoi \[j_{X/Y} : \Sh(\mathcal{C}/X) \longrightarrow \Sh(\mathcal{C}/Y)\] See Sites, Sections 00XZ and 04IM. Some questions about cohomology are easier for this type of morphisms of topoi. Here is an example where we get a trivial type of base change theorem.

Lemma

Let \(\mathcal{C}\) be a site. Let \[\xymatrix{ X' \ar[d] \ar[r] & X \ar[d] \\ Y' \ar[r] & Y }\] be a cartesian diagram of \(\mathcal{C}\). Then we have \(j_{Y'/Y}^{-1} \circ Rj_{X/Y, *} = Rj_{X'/Y', *} \circ j_{X'/X}^{-1}\) as functors \(D(\mathcal{C}/X) \to D(\mathcal{C}/Y')\).

Proof

Let \(E \in D(\mathcal{C}/X)\). Choose a K-injective complex \(\mathcal{I}^\bullet\) of abelian sheaves on \(\mathcal{C}/X\) representing \(E\). By Lemma 08FI we see that \(j_{X'/X}^{-1}\mathcal{I}^\bullet\) is K-injective too. Hence we may compute \(Rj_{X'/Y'}(j_{X'/X}^{-1}E)\) by \(j_{X'/Y', *}j_{X'/X}^{-1}\mathcal{I}^\bullet\). Thus we see that the equality holds by Sites, Lemma 0EYV.

If we have a ringed site \((\mathcal{C}, \mathcal{O})\) and a morphism \(f : X \to Y\) of \(\mathcal{C}\), then \(j_{X/Y}\) becomes a morphism of ringed topoi \[j_{X/Y} : (\Sh(\mathcal{C}/X), \mathcal{O}_X) \longrightarrow (\Sh(\mathcal{C}/Y), \mathcal{O}_Y)\] See Modules on Sites, Lemma 04IY.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \[\xymatrix{ X' \ar[d] \ar[r] & X \ar[d] \\ Y' \ar[r] & Y }\] be a cartesian diagram of \(\mathcal{C}\). Then we have \(j_{Y'/Y}^* \circ Rj_{X/Y, *} = Rj_{X'/Y', *} \circ j_{X'/X}^*\) as functors \(D(\mathcal{O}_X) \to D(\mathcal{O}_{Y'})\).

Proof

Since \(j_{Y'/Y}^{-1}\mathcal{O}_Y = \mathcal{O}_{Y'}\) we have \(j_{Y'/Y}^* = Lj_{Y'/Y}^* = j_{Y'/Y}^{-1}\). Similarly we have \(j_{X'/X}^* = Lj_{X'/X}^* = j_{X'/X}^{-1}\). Thus by Lemma 0D6J it suffices to prove the result on derived categories of abelian sheaves which we did in Lemma 0EZ0.

Inverse systems and cohomology

We prove some results on inverse systems of sheaves of modules.

Lemma

Let \(I\) be an ideal of a ring \(A\). Let \(\mathcal{C}\) be a site. Let \[\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1\] be an inverse system of sheaves of \(A\)-modules on \(\mathcal{C}\) such that \(\mathcal{F}_n = \mathcal{F}_{n + 1}/I^n\mathcal{F}_{n + 1}\). Let \(p \geq 0\). Assume \[\bigoplus\nolimits_{n \geq 0} H^{p + 1}(\mathcal{C}, I^n\mathcal{F}_{n + 1})\] satisfies the ascending chain condition as a graded \(\bigoplus_{n \geq 0} I^n/I^{n + 1}\)-module. Then the inverse system \(M_n = H^p(\mathcal{C}, \mathcal{F}_n)\) satisfies the Mittag-Leffler condition².

Proof

Set \(N_n = H^{p + 1}(\mathcal{C}, I^n\mathcal{F}_{n + 1})\) and let \(\delta_n : M_n \to N_n\) be the boundary map on cohomology coming from the short exact sequence \(0 \to I^n\mathcal{F}_{n + 1} \to \mathcal{F}_{n + 1} \to \mathcal{F}_n \to 0\). Then \(\bigoplus \Im(\delta_n) \subset \bigoplus N_n\) is a graded submodule. Namely, if \(s \in M_n\) and \(f \in I^m\), then we have a commutative diagram \[\xymatrix{ 0 \ar[r] & I^n\mathcal{F}_{n + 1} \ar[d]_f \ar[r] & \mathcal{F}_{n + 1} \ar[d]_f \ar[r] & \mathcal{F}_n \ar[d]_f \ar[r] & 0 \\ 0 \ar[r] & I^{n + m}\mathcal{F}_{n + m + 1} \ar[r] & \mathcal{F}_{n + m + 1} \ar[r] & \mathcal{F}_{n + m} \ar[r] & 0 }\] The middle vertical map is given by lifting a local section of \(\mathcal{F}_{n + 1}\) to a section of \(\mathcal{F}_{n + m + 1}\) and then multiplying by \(f\); similarly for the other vertical arrows. We conclude that \(\delta_{n + m}(fs) = f \delta_n(s)\). By assumption we can find \(s_j \in M_{n_j}\), \(j = 1, \ldots, N\) such that \(\delta_{n_j}(s_j)\) generate \(\bigoplus \Im(\delta_n)\) as a graded module. Let \(n > c = \max(n_j)\). Let \(s \in M_n\). Then we can find \(f_j \in I^{n - n_j}\) such that \(\delta_n(s) = \sum f_j \delta_{n_j}(s_j)\). We conclude that \(\delta(s - \sum f_j s_j) = 0\), i.e., we can find \(s' \in M_{n + 1}\) mapping to \(s - \sum f_js_j\) in \(M_n\). It follows that \[\Im(M_{n + 1} \to M_{n - c}) = \Im(M_n \to M_{n - c})\] Namely, the elements \(f_js_j\) map to zero in \(M_{n - c}\). This proves the lemma.

Lemma

Let \(I\) be an ideal of a ring \(A\). Let \(\mathcal{C}\) be a site. Let \[\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1\] be an inverse system of \(A\)-modules on \(\mathcal{C}\) such that \(\mathcal{F}_n = \mathcal{F}_{n + 1}/I^n\mathcal{F}_{n + 1}\). Let \(p \geq 0\). Given \(n\) define \[N_n = \bigcap\nolimits_{m \geq n} \Im\left( H^{p + 1}(\mathcal{C}, I^n\mathcal{F}_{m + 1}) \to H^{p + 1}(\mathcal{C}, I^n\mathcal{F}_{n + 1}) \right)\] If \(\bigoplus N_n\) satisfies the ascending chain condition as a graded \(\bigoplus_{n \geq 0} I^n/I^{n + 1}\)-module, then the inverse system \(M_n = H^p(\mathcal{C}, \mathcal{F}_n)\) satisfies the Mittag-Leffler condition³.

Proof

The proof is exactly the same as the proof of Lemma 0GYQ. In fact, the result will follow from the arguments given there as soon as we show that \(\bigoplus N_n\) is a graded \(\bigoplus_{n \geq 0} I^n/I^{n + 1}\)-submodule of \(\bigoplus H^{p + 1}(\mathcal{C}, I^n\mathcal{F}_{n + 1})\) and that the boundary maps \(\delta_n : M_n \to H^{p + 1}(\mathcal{C}, I^n\mathcal{F}_{n + 1})\) have image contained in \(N_n\).

Suppose that \(\xi \in N_n\) and \(f \in I^k\). Choose \(m \gg n + k\). Choose \(\xi' \in H^{p + 1}(\mathcal{C}, I^n\mathcal{F}_{m + 1})\) lifting \(\xi\). We consider the diagram \[\xymatrix{ 0 \ar[r] & I^n\mathcal{F}_{m + 1} \ar[d]_f \ar[r] & \mathcal{F}_{m + 1} \ar[d]_f \ar[r] & \mathcal{F}_n \ar[d]_f \ar[r] & 0 \\ 0 \ar[r] & I^{n + k}\mathcal{F}_{m + 1} \ar[r] & \mathcal{F}_{m + 1} \ar[r] & \mathcal{F}_{n + k} \ar[r] & 0 }\] constructed as in the proof of Lemma 0GYQ. We get an induced map on cohomology and we see that \(f \xi' \in H^{p + 1}(\mathcal{C}, I^{n + k}\mathcal{F}_{m + 1})\) maps to \(f \xi\). Since this is true for all \(m \gg n + k\) we see that \(f\xi\) is in \(N_{n + k}\) as desired.

To see the boundary maps \(\delta_n\) have image contained in \(N_n\) we consider the diagrams \[\xymatrix{ 0 \ar[r] & I^n\mathcal{F}_{m + 1} \ar[d] \ar[r] & \mathcal{F}_{m + 1} \ar[d] \ar[r] & \mathcal{F}_n \ar[d] \ar[r] & 0 \\ 0 \ar[r] & I^n\mathcal{F}_{n + 1} \ar[r] & \mathcal{F}_{n + 1} \ar[r] & \mathcal{F}_n \ar[r] & 0 }\] for \(m \geq n\). Looking at the induced maps on cohomology we conclude.

Lemma

Let \(I\) be an ideal of a ring \(A\). Let \(\mathcal{C}\) be a site. Let \[\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1\] be an inverse system of sheaves of \(A\)-modules on \(\mathcal{C}\) such that \(\mathcal{F}_n = \mathcal{F}_{n + 1}/I^n\mathcal{F}_{n + 1}\). Let \(p \geq 0\). Assume \[\bigoplus\nolimits_{n \geq 0} H^p(\mathcal{C}, I^n\mathcal{F}_{n + 1})\] satisfies the ascending chain condition as a graded \(\bigoplus_{n \geq 0} I^n/I^{n + 1}\)-module. Then the limit topology on \(M = \lim H^p(\mathcal{C}, \mathcal{F}_n)\) is the \(I\)-adic topology.

Proof

Set \(F^n = \Ker(M \to H^p(\mathcal{C}, \mathcal{F}_n))\) for \(n \geq 1\) and \(F^0 = M\). Observe that \(I F^n \subset F^{n + 1}\). In particular \(I^n M \subset F^n\). Hence the \(I\)-adic topology is finer than the limit topology. For the converse, we will show that given \(n\) there exists an \(m \geq n\) such that \(F^m \subset I^nM\)⁴. We have injective maps \[F^n/F^{n + 1} \longrightarrow H^p(\mathcal{C}, \mathcal{F}_{n + 1})\] whose image is contained in the image of \(H^p(\mathcal{C}, I^n\mathcal{F}_{n + 1}) \to H^p(\mathcal{C}, \mathcal{F}_{n + 1})\). Denote \[E_n \subset H^p(\mathcal{C}, I^n\mathcal{F}_{n + 1})\] the inverse image of \(F^n/F^{n + 1}\). Then \(\bigoplus E_n\) is a graded \(\bigoplus I^n/I^{n + 1}\)-submodule of \(\bigoplus H^p(\mathcal{C}, I^n\mathcal{F}_{n + 1})\) and \(\bigoplus E_n \to \bigoplus F^n/F^{n + 1}\) is a homomorphism of graded modules; details omitted. By assumption \(\bigoplus E_n\) is generated by finitely many homogeneous elements over \(\bigoplus I^n/I^{n + 1}\). Since \(E_n \to F^n/F^{n + 1}\) is surjective, we see that the same thing is true of \(\bigoplus F^n/F^{n + 1}\). Hence we can find \(r\) and \(c_1, \ldots, c_r \geq 0\) and \(a_i \in F^{c_i}\) whose images in \(\bigoplus F^n/F^{n + 1}\) generate. Set \(c = \max(c_i)\).

For \(n \geq c\) we claim that \(I F^n = F^{n + 1}\). The claim shows that \(F^{n + c} = I^nF^c \subset I^nM\) as desired. To prove the claim suppose \(a \in F^{n + 1}\). The image of \(a\) in \(F^{n + 1}/F^{n + 2}\) is a linear combination of our \(a_i\). Therefore \(a - \sum f_i a_i \in F^{n + 2}\) for some \(f_i \in I^{n + 1 - c_i}\). Since \(I^{n + 1 - c_i} = I \cdot I^{n - c_i}\) as \(n \geq c_i\) we can write \(f_i = \sum g_{i, j} h_{i, j}\) with \(g_{i, j} \in I\) and \(h_{i, j}a_i \in F^n\). Thus we see that \(F^{n + 1} = F^{n + 2} + IF^n\). A simple induction argument gives \(F^{n + 1} = F^{n + e} + IF^n\) for all \(e > 0\). It follows that \(IF^n\) is dense in \(F^{n + 1}\). Choose generators \(k_1, \ldots, k_r\) of \(I\) and consider the continuous map \[u : (F^n)^{\oplus r} \longrightarrow F^{n + 1},\quad (x_1, \ldots, x_r) \mapsto \sum k_i x_i\] (in the limit topology). By the above the image of \((F^m)^{\oplus r}\) under \(u\) is dense in \(F^{m + 1}\) for all \(m \geq n\). By the open mapping lemma (More on Algebra, Lemma 0CQW) we find that \(u\) is open. Hence \(u\) is surjective. Hence \(IF^n = F^{n + 1}\) for \(n \geq c\). This concludes the proof.

Lemma

Let \(I\) be an ideal of a ring \(A\). Let \(\mathcal{C}\) be a site. Let \[\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1\] be an inverse system of sheaves of \(A\)-modules on \(\mathcal{C}\) such that \(\mathcal{F}_n = \mathcal{F}_{n + 1}/I^n\mathcal{F}_{n + 1}\). Let \(p \geq 0\). Given \(n\) define \[N_n = \bigcap\nolimits_{m \geq n} \Im\left( H^p(\mathcal{C}, I^n\mathcal{F}_{m + 1}) \to H^p(\mathcal{C}, I^n\mathcal{F}_{n + 1}) \right)\] If \(\bigoplus N_n\) satisfies the ascending chain condition as a graded \(\bigoplus_{n \geq 0} I^n/I^{n + 1}\)-module, then the limit topology on \(M = \lim H^p(\mathcal{C}, \mathcal{F}_n)\) is the \(I\)-adic topology.

Proof

The proof is exactly the same as the proof of Lemma 0GYS. In fact, the result will follow from the arguments given there as soon as we show that \(\bigoplus N_n\) is a graded \(\bigoplus_{n \geq 0} I^n/I^{n + 1}\)-submodule of \(\bigoplus H^{p + 1}(\mathcal{C}, I^n\mathcal{F}_{n + 1})\) and that \(F^n/F^{n + 1} \subset H^p(\mathcal{C}, \mathcal{F}_{n + 1})\) is contained in the image of \(N_n \to H^p(\mathcal{C}, \mathcal{F}_{n + 1})\). In the proof of Lemma 0GYR we have seen the statement on the module structure.

Let \(t \in F^n\). Choose an element \(s \in H^p(\mathcal{C}, I^n\mathcal{F}_{n + 1})\) which maps to the image of \(t\) in \(H^p(\mathcal{C}, \mathcal{F}_{n + 1})\). We have to show that \(s\) is in \(N_n\). Now \(F^n\) is the kernel of the map from \(M \to H^p(\mathcal{C}, \mathcal{F}_n)\) hence for all \(m \geq n\) we can map \(t\) to an element \(t_m \in H^p(\mathcal{C}, \mathcal{F}_{m + 1})\) which maps to zero in \(H^p(\mathcal{C}, \mathcal{F}_n)\). Consider the cohomology sequence \[H^{p - 1}(\mathcal{C}, \mathcal{F}_n) \to H^p(\mathcal{C}, I^n\mathcal{F}_{m + 1}) \to H^p(\mathcal{C}, \mathcal{F}_{m + 1}) \to H^p(\mathcal{C}, \mathcal{F}_n)\] coming from the short exact sequence \(0 \to I^n\mathcal{F}_{m + 1} \to \mathcal{F}_{m + 1} \to \mathcal{F}_n \to 0\). We can choose \(s_m \in H^p(\mathcal{C}, I^n\mathcal{F}_{m + 1})\) mapping to \(t_m\). Comparing the sequence above with the one for \(m = n\) we see that \(s_m\) maps to \(s\) up to an element in the image of \(H^{p - 1}(\mathcal{C}, \mathcal{F}_n) \to H^p(\mathcal{C}, I^n\mathcal{F}_{n + 1})\). However, this map factors through the map \(H^p(\mathcal{C}, I^n\mathcal{F}_{m + 1}) \to H^p(\mathcal{C}, I^n\mathcal{F}_{n + 1})\) and we see that \(s\) is in the image as desired.

Derived and homotopy limits

Let \(\mathcal{C}\) be a site. Consider the category \(\mathcal{C} \times \mathbf{N}\) with \(\Mor((U, n), (V, m)) = \emptyset\) if \(n > m\) and \(\Mor((U, n), (V, m)) = \Mor(U, V)\) else. We endow this with the structure of a site by letting coverings be families \(\{(U_i, n) \to (U, n)\}\) such that \(\{U_i \to U\}\) is a covering of \(\mathcal{C}\). Then the reader verifies immediately that sheaves on \(\mathcal{C} \times \mathbf{N}\) are the same thing as inverse systems of sheaves on \(\mathcal{C}\). In particular \(\textit{Ab}(\mathcal{C} \times \mathbf{N})\) is inverse systems of abelian sheaves on \(\mathcal{C}\). Consider now the functor \[\lim : \textit{Ab}(\mathcal{C} \times \mathbf{N}) \to \textit{Ab}(\mathcal{C})\] which takes an inverse system to its limit. This is nothing but \(g_*\) where \(g : \Sh(\mathcal{C} \times \mathbf{N}) \to \Sh(\mathcal{C})\) is the morphism of topoi associated to the continuous and cocontinuous functor \(\mathcal{C} \times \mathbf{N} \to \mathcal{C}\). (Observe that \(g^{-1}\) assigns to a sheaf on \(\mathcal{C}\) the corresponding constant inverse system.)

By the general machinery explained above we obtain a derived functor \[R\lim = Rg_* : D(\mathcal{C} \times \mathbf{N}) \to D(\mathcal{C}).\] As indicated this functor is often denoted \(R\lim\).

On the other hand, the continuous and cocontinuous functors \(\mathcal{C} \to \mathcal{C} \times \mathbf{N}\), \(U \mapsto (U, n)\) define morphisms of topoi \(i_n : \Sh(\mathcal{C}) \to \Sh(\mathcal{C} \times \mathbf{N})\). Of course \(i_n^{-1}\) is the functor which picks the \(n\)th term of the inverse system. Thus there are transformations of functors \(i_{n + 1}^{-1} \to i_n^{-1}\). Hence given \(K \in D(\mathcal{C} \times \mathbf{N})\) we get \(K_n = i_n^{-1}K \in D(\mathcal{C})\) and maps \(K_{n + 1} \to K_n\). In Derived Categories, Definition 08TC we have defined the notion of a homotopy limit \[R\lim K_n \in D(\mathcal{C})\] We claim the two notions agree (as far as it makes sense).

Lemma

Let \(\mathcal{C}\) be a site. Let \(K\) be an object of \(D(\mathcal{C} \times \mathbf{N})\). Set \(K_n = i_n^{-1}K\) as above. Then \[R\lim K \cong R\lim K_n\] in \(D(\mathcal{C})\).

Proof

To calculate \(R\lim\) on an object \(K\) of \(D(\mathcal{C} \times \mathbf{N})\) we choose a K-injective representative \(\mathcal{I}^\bullet\) whose terms are injective objects of \(\textit{Ab}(\mathcal{C} \times \mathbf{N})\), see Injectives, Theorem 079P. We may and do think of \(\mathcal{I}^\bullet\) as an inverse system of complexes \((\mathcal{I}_n^\bullet)\) and then we see that \[R\lim K = \lim \mathcal{I}_n^\bullet\] where the right hand side is the termwise inverse limit.

Let \(\mathcal{J} = (\mathcal{J}_n)\) be an injective object of \(\textit{Ab}(\mathcal{C} \times \mathbf{N})\). The morphisms \((U, n) \to (U, n + 1)\) are monomorphisms of \(\mathcal{C} \times \mathbf{N}\), hence \(\mathcal{J}(U, n + 1) \to \mathcal{J}(U, n)\) is surjective (Lemma 093X). It follows that \(\mathcal{J}_{n + 1} \to \mathcal{J}_n\) is surjective as a map of presheaves.

Note that the functor \(i_n^{-1}\) has an exact left adjoint \(i_{n, !}\). Namely, \(i_{n, !}\mathcal{F}\) is the inverse system \(\ldots 0 \to 0 \to \mathcal{F} \to \ldots \to \mathcal{F}\). Thus the complexes \(i_n^{-1}\mathcal{I}^\bullet = \mathcal{I}_n^\bullet\) are K-injective by Derived Categories, Lemma 08BJ.

Because we chose our K-injective complex to have injective terms we conclude that \[0 \to \lim \mathcal{I}_n^\bullet \to \prod \mathcal{I}_n^\bullet \to \prod \mathcal{I}_n^\bullet \to 0\] is a short exact sequence of complexes of abelian sheaves as it is a short exact sequence of complexes of abelian presheaves. Moreover, the products in the middle and the right represent the products in \(D(\mathcal{C})\), see Injectives, Lemma 07D9 and its proof (this is where we use that \(\mathcal{I}_n^\bullet\) is K-injective). Thus \(R\lim K\) is a homotopy limit of the inverse system \((K_n)\) by definition of homotopy limits in triangulated categories.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. The functors \(R\Gamma(\mathcal{C}, -)\) and \(R\Gamma(U, -)\) for \(U \in \Ob(\mathcal{C})\) commute with \(R\lim\). Moreover, there are short exact sequences \[0 \to R^1\lim H^{m - 1}(U, K_n) \to H^m(U, R\lim K_n) \to \lim H^m(U, K_n) \to 0\] for any inverse system \((K_n)\) in \(D(\mathcal{O})\) and \(m \in \mathbf{Z}\). Similar for \(H^m(\mathcal{C}, R\lim K_n)\).

Proof

The first statement follows from Injectives, Lemma 08U1. Then we may apply More on Algebra, Remark 08U5 to \(R\lim R\Gamma(U, K_n) = R\Gamma(U, R\lim K_n)\) to get the short exact sequences.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')\) be a morphism of ringed topoi. Then \(Rf_*\) commutes with \(R\lim\), i.e., \(Rf_*\) commutes with derived limits.

Proof

Let \((K_n)\) be an inverse system of objects of \(D(\mathcal{O})\). By induction on \(n\) we may choose actual complexes \(\mathcal{K}_n^\bullet\) of \(\mathcal{O}\)-modules and maps of complexes \(\mathcal{K}_{n + 1}^\bullet \to \mathcal{K}_n^\bullet\) representing the maps \(K_{n + 1} \to K_n\) in \(D(\mathcal{O})\). In other words, there exists an object \(K\) in \(D(\mathcal{C} \times \mathbf{N})\) whose associated inverse system is the given one. Next, consider the commutative diagram \[\xymatrix{ \Sh(\mathcal{C} \times \mathbf{N}) \ar[r]_g \ar[d]_{f \times 1} & \Sh(\mathcal{C}) \ar[d]_f \\ \Sh(\mathcal{C}' \times \mathbf{N}) \ar[r]^{g'} & \Sh(\mathcal{C}') }\] of morphisms of topoi. It follows that \(R\lim R(f \times 1)_*K = Rf_* R\lim K\). Working through the definitions and using Lemma 0941 we obtain that \(R\lim (Rf_*K_n) = Rf_*(R\lim K_n)\).

Alternate proof in case \(\mathcal{C}\) has enough points. Consider the defining distinguished triangle \[R\lim K_n \to \prod K_n \to \prod K_n\] in \(D(\mathcal{O})\). Applying the exact functor \(Rf_*\) we obtain the distinguished triangle \[Rf_*(R\lim K_n) \to Rf_*\left(\prod K_n\right) \to Rf_*\left(\prod K_n\right)\] in \(D(\mathcal{O}')\). Thus we see that it suffices to prove that \(Rf_*\) commutes with products in the derived category (which are not just given by products of complexes, see Injectives, Lemma 07D9). However, since \(Rf_*\) is a right adjoint by Lemma 07A6 this follows formally (see Categories, Lemma 0038). Caution: Note that we cannot apply Categories, Lemma 0038 directly as \(R\lim K_n\) is not a limit in \(D(\mathcal{O})\).

Remark

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \((K_n)\) be an inverse system in \(D(\mathcal{O})\). Set \(K = R\lim K_n\). For each \(n\) and \(m\) let \(\mathcal{H}^m_n = H^m(K_n)\) be the \(m\)th cohomology sheaf of \(K_n\) and similarly set \(\mathcal{H}^m = H^m(K)\). Let us denote \(\underline{\mathcal{H}}^m_n\) the presheaf \[U \longmapsto \underline{\mathcal{H}}^m_n(U) = H^m(U, K_n)\] Similarly we set \(\underline{\mathcal{H}}^m(U) = H^m(U, K)\). By Lemma 0BKV we see that \(\mathcal{H}^m_n\) is the sheafification of \(\underline{\mathcal{H}}^m_n\) and \(\mathcal{H}^m\) is the sheafification of \(\underline{\mathcal{H}}^m\). Here is a diagram \[\xymatrix{ K \ar@{=}[d] & \underline{\mathcal{H}}^m \ar[d] \ar[r] & \mathcal{H}^m \ar[d] \\ R\lim K_n & \lim \underline{\mathcal{H}}^m_n \ar[r] & \lim \mathcal{H}^m_n }\] In general it may not be the case that \(\lim \mathcal{H}^m_n\) is the sheafification of \(\lim \underline{\mathcal{H}}^m_n\). If \(U \in \mathcal{C}\), then we have short exact sequences [0BKX]\[\begin{equation} 0 \to R^1\lim \underline{\mathcal{H}}^{m - 1}_n(U) \to \underline{\mathcal{H}}^m(U) \to \lim \underline{\mathcal{H}}^m_n(U) \to 0 \end{equation}\] by Lemma 0D6K.

The following lemma applies to an inverse system of quasi-coherent modules with surjective transition maps on an algebraic space or an algebraic stack.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \((\mathcal{F}_n)\) be an inverse system of \(\mathcal{O}\)-modules. Let \(\mathcal{B} \subset \Ob(\mathcal{C})\) be a subset. Assume

1. every object of \(\mathcal{C}\) has a covering whose members are elements of \(\mathcal{B}\),

2. \(H^p(U, \mathcal{F}_n) = 0\) for \(p > 0\) and \(U \in \mathcal{B}\),

3. the inverse system \(\mathcal{F}_n(U)\) has vanishing \(R^1\lim\) for \(U \in \mathcal{B}\).

Then \(R\lim \mathcal{F}_n = \lim \mathcal{F}_n\) and we have \(H^p(U, \lim \mathcal{F}_n) = 0\) for \(p > 0\) and \(U \in \mathcal{B}\).

Proof

Set \(K_n = \mathcal{F}_n\) and \(K = R\lim \mathcal{F}_n\). Using the notation of Remark 0BKW and assumption (2) we see that for \(U \in \mathcal{B}\) we have \(\underline{\mathcal{H}}_n^m(U) = 0\) when \(m \not = 0\) and \(\underline{\mathcal{H}}_n^0(U) = \mathcal{F}_n(U)\). From Equation (0BKX) and assumption (3) we see that \(\underline{\mathcal{H}}^m(U) = 0\) when \(m \not = 0\) and equal to \(\lim \mathcal{F}_n(U)\) when \(m = 0\). Sheafifying using (1) we find that \(\mathcal{H}^m = 0\) when \(m \not = 0\) and equal to \(\lim \mathcal{F}_n\) when \(m = 0\). Hence \(K = \lim \mathcal{F}_n\). Since \(H^m(U, K) = \underline{\mathcal{H}}^m(U) = 0\) for \(m > 0\) (see above) we see that the second assertion holds.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \((K_n)\) be an inverse system in \(D(\mathcal{O})\). Let \(V \in \Ob(\mathcal{C})\) and \(m \in \mathbf{Z}\). Assume there exist an integer \(n(V)\) and a cofinal system \(\text{Cov}_V\) of coverings of \(V\) such that for \(\{V_i \to V\} \in \text{Cov}_V\)

1. \(R^1\lim H^{m - 1}(V_i, K_n) = 0\), and

2. \(H^m(V_i, K_n) \to H^m(V_i, K_{n(V)})\) is injective for \(n \geq n(V)\).

Then the map on sections \(H^m(R\lim K_n)(V) \to H^m(K_{n(V)})(V)\) is injective.

Proof

Let \(\gamma \in H^m(R\lim K_n)(V)\) map to zero in \(H^m(K_{n(V)})(V)\). Since \(H^m(R\lim K_n)\) is the sheafification of \(U \mapsto H^m(U, R\lim K_n)\) (by Lemma 0BKV) we can choose \(\{V_i \to V\} \in \text{Cov}_V\) and elements \(\tilde\gamma_i \in H^m(V_i, R\lim K_n)\) mapping to \(\gamma|_{V_i}\). Then \(\tilde\gamma_i\) maps to \(\tilde\gamma_{i, n(V)} \in H^m(V_i, K_{n(V)})\). Using that \(H^m(K_{n(V)})\) is the sheafification of \(U \mapsto H^m(U, K_{n(V)})\) (by Lemma 0BKV again) we see that after replacing \(\{V_i \to V\}\) by a refinement we may assume that \(\tilde\gamma_{i, n(V)} = 0\) for all \(i\). For this covering we consider the short exact sequences \[0 \to R^1\lim H^{m - 1}(V_i, K_n) \to H^m(V_i, R\lim K_n) \to \lim H^m(V_i, K_n) \to 0\] of Lemma 0D6K. By assumption (1) the group on the left is zero and by assumption (2) the group on the right maps injectively into \(H^m(V_i, K_{n(V)})\). We conclude \(\tilde\gamma_i = 0\) and hence \(\gamma = 0\) as desired.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(E \in D(\mathcal{O})\). Let \(\mathcal{B} \subset \Ob(\mathcal{C})\) be a subset. Assume

1. every object of \(\mathcal{C}\) has a covering whose members are elements of \(\mathcal{B}\), and

2. for every \(V \in \mathcal{B}\) there exist a function \(p(V, -) : \mathbf{Z} \to \mathbf{Z}\) and a cofinal system \(\text{Cov}_V\) of coverings of \(V\) such that \[H^p(V_i, H^{m - p}(E)) = 0\] for all \(\{V_i \to V\} \in \text{Cov}_V\) and all integers \(p, m\) satisfying \(p > p(V, m)\).

Then the map \(E \to R\lim \tau_{\geq -n} E\) of Derived Categories, Remark 0H72 is an isomorphism in \(D(\mathcal{O})\).

Proof

Set \(K_n = \tau_{\geq -n}E\) and \(K = R\lim K_n\). The canonical map \(E \to K\) comes from the canonical maps \(E \to K_n = \tau_{\geq -n}E\). We have to show that \(E \to K\) induces an isomorphism \(H^m(E) \to H^m(K)\) of cohomology sheaves. In the rest of the proof we fix \(m\). If \(n \geq -m\), then the map \(E \to \tau_{\geq -n}E = K_n\) induces an isomorphism \(H^m(E) \to H^m(K_n)\). To finish the proof it suffices to show that for every \(V \in \mathcal{B}\) there exists an integer \(n(V) \geq -m\) such that the map \(H^m(K)(V) \to H^m(K_{n(V)})(V)\) is injective. Namely, then the composition \[H^m(E)(V) \to H^m(K)(V) \to H^m(K_{n(V)})(V)\] is a bijection and the second arrow is injective, hence the first arrow is bijective. By property (1) this will imply \(H^m(E) \to H^m(K)\) is an isomorphism. Set \[n(V) = 1 + \max\{-m, p(V, m - 1) - m, -1 + p(V, m) - m, -2 + p(V, m + 1) - m\}.\] so that in any case \(n(V) \geq -m\). Claim: the maps \[H^{m - 1}(V_i, K_{n + 1}) \to H^{m - 1}(V_i, K_n) \quad\text{and}\quad H^m(V_i, K_{n + 1}) \to H^m(V_i, K_n)\] are isomorphisms for \(n \geq n(V)\) and \(\{V_i \to V\} \in \text{Cov}_V\). The claim implies conditions (1) and (2) of Lemma 0D6L are satisfied and hence implies the desired injectivity. Recall (Derived Categories, Remark 08J5) that we have distinguished triangles \[H^{-n - 1}(E)[n + 1] \to K_{n + 1} \to K_n \to H^{-n - 1}(E)[n + 2]\] Looking at the associated long exact cohomology sequence the claim follows if \[H^{m + n}(V_i, H^{-n - 1}(E)),\quad H^{m + n + 1}(V_i, H^{-n - 1}(E)),\quad H^{m + n + 2}(V_i, H^{-n - 1}(E))\] are zero for \(n \geq n(V)\) and \(\{V_i \to V\} \in \text{Cov}_V\). This follows from our choice of \(n(V)\) and the assumption in the lemma.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(E \in D(\mathcal{O})\). Let \(\mathcal{B} \subset \Ob(\mathcal{C})\) be a subset. Assume

1. every object of \(\mathcal{C}\) has a covering whose members are elements of \(\mathcal{B}\), and

2. for every \(V \in \mathcal{B}\) there exist an integer \(d_V \geq 0\) and a cofinal system \(\text{Cov}_V\) of coverings of \(V\) such that \[H^p(V_i, H^q(E)) = 0 \text{ for } \{V_i \to V\} \in \text{Cov}_V,\ p > d_V, \text{ and }q < 0\]

Then the map \(E \to R\lim \tau_{\geq -n} E\) of Derived Categories, Remark 0H72 is an isomorphism in \(D(\mathcal{O})\).

Proof

This follows from Lemma 0D6M with \(p(V, m) = d_V + \max(0, m)\).

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(E \in D(\mathcal{O})\). Assume there exists a function \(p(-) : \mathbf{Z} \to \mathbf{Z}\) and a subset \(\mathcal{B} \subset \Ob(\mathcal{C})\) such that

1. every object of \(\mathcal{C}\) has a covering whose members are elements of \(\mathcal{B}\),

2. \(H^p(V, H^{m - p}(E)) = 0\) for \(p > p(m)\) and \(V \in \mathcal{B}\).

Then the map \(E \to R\lim \tau_{\geq -n} E\) of Derived Categories, Remark 0H72 is an isomorphism in \(D(\mathcal{O})\).

Proof

Apply Lemma 0D6M with \(p(V, m) = p(m)\) and \(\text{Cov}_V\) equal to the set of coverings \(\{V_i \to V\}\) with \(V_i \in \mathcal{B}\) for all \(i\).

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(E \in D(\mathcal{O})\). Assume there exists an integer \(d \geq 0\) and a subset \(\mathcal{B} \subset \Ob(\mathcal{C})\) such that

1. every object of \(\mathcal{C}\) has a covering whose members are elements of \(\mathcal{B}\),

2. \(H^p(V, H^q(E)) = 0\) for \(p > d\), \(q < 0\), and \(V \in \mathcal{B}\).

Then the map \(E \to R\lim \tau_{\geq -n} E\) of Derived Categories, Remark 0H72 is an isomorphism in \(D(\mathcal{O})\).

Proof

Apply Lemma 0D6N with \(d_V = d\) and \(\text{Cov}_V\) equal to the set of coverings \(\{V_i \to V\}\) with \(V_i \in \mathcal{B}\) for all \(i\).

The lemmas above can be used to compute cohomology in certain situations.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(K\) be an object of \(D(\mathcal{O})\). Let \(\mathcal{B} \subset \Ob(\mathcal{C})\) be a subset. Assume

1. every object of \(\mathcal{C}\) has a covering whose members are elements of \(\mathcal{B}\),

2. \(H^p(U, H^q(K)) = 0\) for all \(p > 0\), \(q \in \mathbf{Z}\), and \(U \in \mathcal{B}\).

Then \(H^q(U, K) = H^0(U, H^q(K))\) for \(q \in \mathbf{Z}\) and \(U \in \mathcal{B}\).

Proof

Observe that \(K = R\lim \tau_{\geq -n} K\) by Lemma 0D6P with \(d = 0\). Let \(U \in \mathcal{B}\). By Equation (0BKX) we get a short exact sequence \[0 \to R^1\lim H^{q - 1}(U, \tau_{\geq -n}K) \to H^q(U, K) \to \lim H^q(U, \tau_{\geq -n}K) \to 0\] Condition (2) implies \(H^q(U, \tau_{\geq -n} K) = H^0(U, H^q(\tau_{\geq -n} K))\) for all \(q\) by using the spectral sequence of Derived Categories, Lemma 015J. The spectral sequence converges because \(\tau_{\geq -n}K\) is bounded below. If \(n > -q\) then we have \(H^q(\tau_{\geq -n}K) = H^q(K)\). Thus the systems on the left and the right of the displayed short exact sequence are eventually constant with values \(H^0(U, H^{q - 1}(K))\) and \(H^0(U, H^q(K))\) and the lemma follows.

Here is another case where we can describe the derived limit.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \((K_n)\) be an inverse system of objects of \(D(\mathcal{O})\). Let \(\mathcal{B} \subset \Ob(\mathcal{C})\) be a subset. Assume

1. every object of \(\mathcal{C}\) has a covering whose members are elements of \(\mathcal{B}\),

2. for all \(U \in \mathcal{B}\) and all \(q \in \mathbf{Z}\) we have

   1. \(H^p(U, H^q(K_n)) = 0\) for \(p > 0\),

   2. the inverse system \(H^0(U, H^q(K_n))\) has vanishing \(R^1\lim\).

Then \(H^q(R\lim K_n) = \lim H^q(K_n)\) for \(q \in \mathbf{Z}\).

Proof

Set \(K = R\lim K_n\). We will use notation as in Remark 0BKW. Let \(U \in \mathcal{B}\). By Lemma 0BKZ and (2)(a) we have \(H^q(U, K_n) = H^0(U, H^q(K_n))\). Using that the functor \(R\Gamma(U, -)\) commutes with derived limits we have \[H^q(U, K) = H^q(R\lim R\Gamma(U, K_n)) = \lim H^0(U, H^q(K_n))\] where the final equality follows from More on Algebra, Remark 08U5 and assumption (2)(b). Thus \(H^q(U, K)\) is the inverse limit of the sections of the sheaves \(H^q(K_n)\) over \(U\). Since \(\lim H^q(K_n)\) is a sheaf we find using assumption (1) that \(H^q(K)\), which is the sheafification of the presheaf \(U \mapsto H^q(U, K)\), is equal to \(\lim H^q(K_n)\). This proves the lemma.

Producing K-injective resolutions

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{F}^\bullet\) be a complex of \(\mathcal{O}\)-modules. The category \(\textit{Mod}(\mathcal{O})\) has enough injectives, hence we can use Derived Categories, Lemma 070F to produce a diagram \[\xymatrix{ \ldots \ar[r] & \tau_{\geq -2}\mathcal{F}^\bullet \ar[r] \ar[d] & \tau_{\geq -1}\mathcal{F}^\bullet \ar[d] \\ \ldots \ar[r] & \mathcal{I}_2^\bullet \ar[r] & \mathcal{I}_1^\bullet }\] in the category of complexes of \(\mathcal{O}\)-modules such that

1. the vertical arrows are quasi-isomorphisms,

2. \(\mathcal{I}_n^\bullet\) is a bounded below complex of injectives,

3. the arrows \(\mathcal{I}_{n + 1}^\bullet \to \mathcal{I}_n^\bullet\) are termwise split surjections.

The category of \(\mathcal{O}\)-modules has limits (they are computed on the level of presheaves), hence we can form the termwise limit \(\mathcal{I}^\bullet = \lim_n \mathcal{I}_n^\bullet\). By Derived Categories, Lemmas 070J and 070L this is a K-injective complex. In general the canonical map [070P]\[\begin{equation} \mathcal{F}^\bullet \to \mathcal{I}^\bullet \end{equation}\] may not be a quasi-isomorphism. In the following lemma we describe some conditions under which it is.

Lemma

In the situation described above, denote \(\mathcal{H}^m = H^m(\mathcal{F}^\bullet)\) the \(m\)th cohomology sheaf. Let \(\mathcal{B} \subset \Ob(\mathcal{C})\) be a subset. Let \(d \in \mathbf{N}\). Assume

1. every object of \(\mathcal{C}\) has a covering whose members are elements of \(\mathcal{B}\),

2. for every \(U \in \mathcal{B}\) we have \(H^p(U, \mathcal{H}^q) = 0\) for \(p > d\) and \(q < 0\)⁵.

Then (070P) is a quasi-isomorphism.

Proof

By Derived Categories, Lemma 070M it suffices to show that the map \(\mathcal{F}^\bullet \to R\lim \tau_{\geq -n} \mathcal{F}^\bullet\) is an isomorphism. This follows from Lemma 0D6P.

Here is a technical lemma about cohomology sheaves of termwise limits of inverse systems of complexes of modules. We should avoid using this lemma as much as possible and instead use arguments with derived inverse limits.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \((\mathcal{F}_n^\bullet)\) be an inverse system of complexes of \(\mathcal{O}\)-modules. Let \(m \in \mathbf{Z}\). Suppose given \(\mathcal{B} \subset \Ob(\mathcal{C})\) and an integer \(n_0\) such that

1. every object of \(\mathcal{C}\) has a covering whose members are elements of \(\mathcal{B}\),

2. for every \(U \in \mathcal{B}\)

   1. the systems of abelian groups \(\mathcal{F}_n^{m - 2}(U)\) and \(\mathcal{F}_n^{m - 1}(U)\) have vanishing \(R^1\lim\) (for example these have the Mittag-Leffler property),

   2. the system of abelian groups \(H^{m - 1}(\mathcal{F}_n^\bullet(U))\) has vanishing \(R^1\lim\) (for example it has the Mittag-Leffler property), and

   3. we have \(H^m(\mathcal{F}_n^\bullet(U)) = H^m(\mathcal{F}_{n_0}^\bullet(U))\) for all \(n \geq n_0\).

Then the maps \(H^m(\mathcal{F}^\bullet) \to \lim H^m(\mathcal{F}_n^\bullet) \to H^m(\mathcal{F}_{n_0}^\bullet)\) are isomorphisms of sheaves where \(\mathcal{F}^\bullet = \lim \mathcal{F}_n^\bullet\) is the termwise inverse limit.

Proof

Let \(U \in \mathcal{B}\). Note that \(H^m(\mathcal{F}^\bullet(U))\) is the cohomology of \[\lim_n \mathcal{F}_n^{m - 2}(U) \to \lim_n \mathcal{F}_n^{m - 1}(U) \to \lim_n \mathcal{F}_n^m(U) \to \lim_n \mathcal{F}_n^{m + 1}(U)\] in the third spot from the left. By assumptions (2)(a) and (2)(b) we may apply More on Algebra, Lemma 0918 to conclude that \[H^m(\mathcal{F}^\bullet(U)) = \lim H^m(\mathcal{F}_n^\bullet(U))\] By assumption (2)(c) we conclude \[H^m(\mathcal{F}^\bullet(U)) = H^m(\mathcal{F}_n^\bullet(U))\] for all \(n \geq n_0\). By assumption (1) we conclude that the sheafification of \(U \mapsto H^m(\mathcal{F}^\bullet(U))\) is equal to the sheafification of \(U \mapsto H^m(\mathcal{F}_n^\bullet(U))\) for all \(n \geq n_0\). Thus the inverse system of sheaves \(H^m(\mathcal{F}_n^\bullet)\) is constant for \(n \geq n_0\) with value \(H^m(\mathcal{F}^\bullet)\) which proves the lemma.

Bounded cohomological dimension

In this section we ask when a functor \(Rf_*\) has bounded cohomological dimension. This is a rather subtle question when we consider unbounded complexes.

Situation

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}\). Let \(\mathcal{A} \subset \textit{Mod}(\mathcal{O})\) be a weak Serre subcategory. We assume the following is true: there exists a subset \(\mathcal{B} \subset \Ob(\mathcal{C})\) such that

1. every object of \(\mathcal{C}\) has a covering whose members are in \(\mathcal{B}\), and

2. for every \(V \in \mathcal{B}\) there exists an integer \(d_V\) and a cofinal system \(\text{Cov}_V\) of coverings of \(V\) such that \[H^p(V_i, \mathcal{F}) = 0 \text{ for } \{V_i \to V\} \in \text{Cov}_V,\ p > d_V, \text{ and } \mathcal{F} \in \Ob(\mathcal{A})\]

Lemma

In Situation 0D6R for any \(E \in D_\mathcal{A}(\mathcal{O})\) the map \(E \to R\lim \tau_{\geq -n} E\) of Derived Categories, Remark 0H72 is an isomorphism in \(D(\mathcal{O})\).

Proof

Follows immediately from Lemma 0D6N.

Lemma

In Situation 0D6R let \((K_n)\) be an inverse system in \(D_\mathcal{A}^+(\mathcal{O})\). Assume that for every \(j\) the inverse system \((H^j(K_n))\) in \(\mathcal{A}\) is eventually constant with value \(\mathcal{H}^j\). Then \(H^j(R\lim K_n) = \mathcal{H}^j\) for all \(j\).

Proof

Let \(V \in \mathcal{B}\). Let \(\{V_i \to V\}\) be in the set \(\text{Cov}_V\) of Situation 0D6R. Because \(K_n\) is bounded below there is a spectral sequence \[E_2^{p, q} = H^p(V_i, H^q(K_n))\] converging to \(H^{p + q}(V_i, K_n)\). See Derived Categories, Lemma 015J. Observe that \(E_2^{p, q} = 0\) for \(p > d_V\) by assumption. Pick \(n_0\) such that \[\begin{matrix} \mathcal{H}^{j + 1} & = & H^{j + 1}(K_n), \\ \mathcal{H}^j & = & H^j(K_n), \\ \ldots, \\ \mathcal{H}^{j - d_V - 2} & = & H^{j - d_V - 2}(K_n) \end{matrix}\] for all \(n \geq n_0\). Comparing the spectral sequences above for \(K_n\) and \(K_{n_0}\), we see that for \(n \geq n_0\) the cohomology groups \(H^{j - 1}(V_i, K_n)\) and \(H^j(V_i, K_n)\) are independent of \(n\). It follows that the map on sections \(H^j(R\lim K_n)(V) \to H^j(K_n)(V)\) is injective for \(n\) large enough (depending on \(V\)), see Lemma 0D6L. Since every object of \(\mathcal{C}\) can be covered by elements of \(\mathcal{B}\), we conclude that the map \(H^j(R\lim K_n) \to \mathcal{H}^j\) is injective.

Surjectivity is shown in a similar manner. Namely, pick \(U \in \Ob(\mathcal{C})\) and \(\gamma \in \mathcal{H}^j(U)\). We want to lift \(\gamma\) to a section of \(H^j(R\lim K_n)\) after replacing \(U\) by the members of a covering. Hence we may assume \(U = V \in \mathcal{B}\) by property (1) of Situation 0D6R. Pick \(n_0\) such that \[\begin{matrix} \mathcal{H}^{j + 1} & = & H^{j + 1}(K_n), \\ \mathcal{H}^j & = & H^j(K_n), \\ \ldots, \\ \mathcal{H}^{j - d_V - 2} & = & H^{j - d_V - 2}(K_n) \end{matrix}\] for all \(n \geq n_0\). Choose an element \(\{V_i \to V\}\) of \(\text{Cov}_V\) such that \(\gamma|_{V_i} \in \mathcal{H}^j(V_i) = H^j(K_{n_0})(V_i)\) lifts to an element \(\gamma_{n_0, i} \in H^j(V_i, K_{n_0})\). This is possible because \(H^j(K_{n_0})\) is the sheafification of \(U \mapsto H^j(U, K_{n_0})\) by Lemma 0BKV. By the discussion in the first paragraph of the proof we have that \(H^{j - 1}(V_i, K_n)\) and \(H^j(V_i, K_n)\) are independent of \(n \geq n_0\). Hence \(\gamma_{n_0, i}\) lifts to an element \(\gamma_i \in H^j(V_i, R\lim K_n)\) by Lemma 0D6K. This finishes the proof.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')\) be a morphism of ringed topoi. Let \(\mathcal{A} \subset \textit{Mod}(\mathcal{O})\) and \(\mathcal{A}' \subset \textit{Mod}(\mathcal{O}')\) be weak Serre subcategories. Assume there is an integer \(N\) such that

1. \(\mathcal{C}, \mathcal{O}, \mathcal{A}\) satisfy the assumption of Situation 0D6R,

2. \(\mathcal{C}', \mathcal{O}', \mathcal{A}'\) satisfy the assumption of Situation 0D6R,

3. \(R^pf_*\mathcal{F} \in \Ob(\mathcal{A}')\) for \(p \geq 0\) and \(\mathcal{F} \in \Ob(\mathcal{A})\),

4. \(R^pf_*\mathcal{F} = 0\) for \(p > N\) and \(\mathcal{F} \in \Ob(\mathcal{A})\),

Then for \(K\) in \(D_\mathcal{A}(\mathcal{O})\) we have

1. \(Rf_*K\) is in \(D_{\mathcal{A}'}(\mathcal{O}')\),

2. the map \(H^j(Rf_*K) \to H^j(Rf_*(\tau_{\geq -n}K))\) is an isomorphism for \(j \geq N - n\).

Proof

By Lemma 0D6S we have \(K = R\lim \tau_{\geq -n}K\). By Lemma 0A07 we have \(Rf_*K = R\lim Rf_*\tau_{\geq -n}K\). The complexes \(Rf_*\tau_{\geq -n}K\) are bounded below. The spectral sequence \[E_2^{p, q} = R^pf_*H^q(\tau_{\geq -n}K)\] converging to \(H^{p + q}(Rf_*\tau_{\geq -n}K)\) (Derived Categories, Lemma 015J) and assumption (3) show that \(Rf_*\tau_{\geq -n}K\) lies in \(D^+_{\mathcal{A}'}(\mathcal{O}')\), see Homology, Lemma 012W. Observe that for \(m \geq n\) the map \[Rf_*(\tau_{\geq -m}K) \longrightarrow Rf_*(\tau_{\geq -n}K)\] induces an isomorphism on cohomology sheaves in degrees \(j \geq -n + N\) by the spectral sequences above. Hence we may apply Lemma 0D6T to conclude.

It turns out that we sometimes need a variant of the lemma above where the assumptions are slightly different.

Situation

Let \(f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')\) be a morphism of ringed sites. Let \(u : \mathcal{C}' \to \mathcal{C}\) be the corresponding continuous functor of sites. Let \(\mathcal{A} \subset \textit{Mod}(\mathcal{O})\) be a weak Serre subcategory. We assume the following is true: there exists a subset \(\mathcal{B}' \subset \Ob(\mathcal{C}')\) such that

1. every object of \(\mathcal{C}'\) has a covering whose members are in \(\mathcal{B}'\), and

2. for every \(V' \in \mathcal{B}'\) there exists an integer \(d_{V'}\) and a cofinal system \(\text{Cov}_{V'}\) of coverings of \(V'\) such that \[H^p(u(V'_i), \mathcal{F}) = 0 \text{ for } \{V'_i \to V'\} \in \text{Cov}_{V'},\ p > d_{V'}, \text{ and } \mathcal{F} \in \Ob(\mathcal{A})\]

Lemma

Let \(f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')\) be a morphism of ringed sites. assume moreover there is an integer \(N\) such that

1. \(\mathcal{C}, \mathcal{O}, \mathcal{A}\) satisfy the assumption of Situation 0D6R,

2. \(f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')\) and \(\mathcal{A}\) satisfy the assumption of Situation 0D6V,

3. \(R^pf_*\mathcal{F} = 0\) for \(p > N\) and \(\mathcal{F} \in \Ob(\mathcal{A})\),

Then for \(K\) in \(D_\mathcal{A}(\mathcal{O})\) the map \(H^j(Rf_*K) \to H^j(Rf_*(\tau_{\geq -n}K))\) is an isomorphism for \(j \geq N - n\).

Proof

Let \(K\) be in \(D_\mathcal{A}(\mathcal{O})\). By Lemma 0D6S we have \(K = R\lim \tau_{\geq -n}K\). By Lemma 0A07 we have \(Rf_*K = R\lim Rf_*(\tau_{\geq -n}K)\). Let \(V' \in \mathcal{B}'\) and let \(\{V'_i \to V'\}\) be an element of \(\text{Cov}_{V'}\). Then we consider \[H^j(V'_i, Rf_*K) = H^j(u(V'_i), K) \quad\text{and}\quad H^j(V'_i, Rf_*(\tau_{\geq -n}K)) = H^j(u(V'_i), \tau_{\geq -n}K)\] The assumption in Situation 0D6V implies that the last group is independent of \(n\) for \(n\) large enough depending on \(j\) and \(d_{V'}\). Some details omitted. We apply this for \(j\) and \(j - 1\) and via Lemma 0D6K this gives that \[H^j(V'_i, Rf_*K) = \lim H^j(V'_i, Rf_*(\tau_{\geq -n} K))\] and the system on the right is constant for \(n\) larger than a constant depending only on \(d_{V'}\) and \(j\). Thus Lemma 0D6L implies that \[H^j(Rf_*K)(V') \longrightarrow \left(\lim H^j(Rf_*(\tau_{\geq -n}K))\right)(V')\] is injective. Since the elements \(V' \in \mathcal{B}'\) cover every object of \(\mathcal{C}'\) we conclude that the map \(H^j(Rf_*K) \to \lim H^j(Rf_*(\tau_{\geq -n}K))\) is injective. The spectral sequence \[E_2^{p, q} = R^pf_*H^q(\tau_{\geq -n}K)\] converging to \(H^{p + q}(Rf_*(\tau_{\geq -n}K))\) (Derived Categories, Lemma 015J) and assumption (3) show that \(H^j(Rf_*(\tau_{\geq -n}K))\) is constant for \(n \geq N - j\). Hence \(H^j(Rf_*K) \to H^j(Rf_*(\tau_{\geq -n}K))\) is injective for \(j \geq N - n\).

Thus we proved the lemma with “isomorphism” in the last line of the lemma replaced by “injective”. However, now choose \(j\) and \(n\) with \(j \geq N - n\). Then consider the distinguished triangle \[\tau_{\leq -n - 1}K \to K \to \tau_{\geq -n}K \to (\tau_{\leq -n - 1}K)[1]\] See Derived Categories, Remark 08J5. Since \(\tau_{\geq -n}\tau_{\leq -n -1}K = 0\), the injectivity already proven for \(\tau_{-n - 1}K\) implies \[0 = H^j(Rf_*(\tau_{\leq -n - 1}K)) = H^{j + 1}(Rf_*(\tau_{\leq -n - 1}K)) = H^{j + 2}(Rf_*(\tau_{\leq -n - 1}K)) = \ldots\] By the long exact cohomology sequence associated to the distinguished triangle \[Rf_*(\tau_{\leq -n - 1}K) \to Rf_*K \to Rf_*(\tau_{\geq -n}K) \to Rf_*(\tau_{\leq -n - 1}K)[1]\] this implies that \(H^j(Rf_*K) \to H^j(Rf_*(\tau_{\geq -n}K))\) is an isomorphism.

Mayer-Vietoris

For the usual statement and proof of Mayer-Vietoris, please see Cohomology, Section 01E9.

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Consider a commutative diagram \[\xymatrix{ E \ar[d] \ar[r] & Y \ar[d] \\ Z \ar[r] & X }\] in the category \(\mathcal{C}\). In this situation, given an object \(K\) of \(D(\mathcal{O})\) we get what looks like the beginning of a distinguished triangle \[R\Gamma(X, K) \to R\Gamma(Z, K) \oplus R\Gamma(Y, K) \to R\Gamma(E, K)\] In the following lemma we make this more precise.

Lemma

In the situation above, choose a K-injective complex \(\mathcal{I}^\bullet\) of \(\mathcal{O}\)-modules representing \(K\). Using \(-1\) times the canonical map for one of the four arrows we get maps of complexes \[\mathcal{I}^\bullet(X) \xrightarrow{\alpha} \mathcal{I}^\bullet(Z) \oplus \mathcal{I}^\bullet(Y) \xrightarrow{\beta} \mathcal{I}^\bullet(E)\] with \(\beta \circ \alpha = 0\). Thus we obtain a canonical map \[c^K_{X, Z, Y, E} : \mathcal{I}^\bullet(X) \longrightarrow C(\beta)^\bullet[-1]\] This map is canonical in the sense that a different choice of K-injective complex representing \(K\) determines an isomorphic arrow in the derived category of abelian groups. If \(c^K_{X, Z, Y, E}\) is an isomorphism, then using its inverse we obtain a canonical distinguished triangle \[R\Gamma(X, K) \to R\Gamma(Z, K) \oplus R\Gamma(Y, K) \to R\Gamma(E, K) \to R\Gamma(X, K)[1]\] All of these constructions are functorial in \(K\).

Proof

This lemma proves itself. For example, if \(\mathcal{J}^\bullet\) is a second K-injective complex representing \(K\), then we can choose a quasi-isomorphism \(\mathcal{I}^\bullet \to \mathcal{J}^\bullet\) which determines quasi-isomorphisms between all the complexes in sight. Details omitted. For the construction of cones and the relationship with distinguished triangles see Derived Categories, Sections 014D and 014P.

Lemma

In the situation above, let \(K_1 \to K_2 \to K_3 \to K_1[1]\) be a distinguished triangle in \(D(\mathcal{O})\). If \(c^{K_i}_{X, Z, Y, E}\) is a quasi-isomorphism for two \(i\) out of \(\{1, 2, 3\}\), then it is a quasi-isomorphism for the third \(i\).

Proof

By rotating the triangle we may assume \(c^{K_1}_{X, Z, Y, E}\) and \(c^{K_2}_{X, Z, Y, E}\) are quasi-isomorphisms. Choose a map \(f : \mathcal{I}^\bullet_1 \to \mathcal{I}^\bullet_2\) of K-injective complexes of \(\mathcal{O}\)-modules representing \(K_1 \to K_2\). Then \(K_3\) is represented by the K-injective complex \(C(f)^\bullet\), see Derived Categories, Lemma 090X. Then the morphism \(c^{K_3}_{X, Z, Y, E}\) is an isomorphism as it is the third leg in a map of distinguished triangles in \(K(\textit{Ab})\) whose other two legs are quasi-isomorphisms. Some details omitted; use Derived Categories, Lemma 014A.

Let us give a criterion for when this does produce a distinguished triangle.

Lemma

In the situation above assume

1. \(h_X^\# = h_Y^\# \amalg_{h_E^\#} h_Z^\#\), and

2. \(h_E^\# \to h_Y^\#\) is injective.

Then the construction of Lemma 0F16 produces a distinguished triangle \[R\Gamma(X, K) \to R\Gamma(Z, K) \oplus R\Gamma(Y, K) \to R\Gamma(E, K) \to R\Gamma(X, K)[1]\] functorial for \(K\) in \(D(\mathcal{C})\).

Proof

We can represent \(K\) by a K-injective complex whose terms are injective abelian sheaves, see Section 07A5. Thus it suffices to show: if \(\mathcal{I}\) is an injective abelian sheaf, then \[0 \to \mathcal{I}(X) \to \mathcal{I}(Z) \oplus \mathcal{I}(Y) \to \mathcal{I}(E) \to 0\] is a short exact sequence. The first arrow is injective because by condition (1) the map \(h_Y \amalg h_Z \to h_X\) becomes surjective after sheafification, which means that \(\{Y \to X, Z \to X\}\) can be refined by a covering of \(X\). The last arrow is surjective because \(\mathcal{I}(Y) \to \mathcal{I}(E)\) is surjective. Namely, we have \(\mathcal{I}(E) = \Hom(\mathbf{Z}_E^\#, \mathcal{I})\), \(\mathcal{I}(Y) = \Hom(\mathbf{Z}_Y^\#, \mathcal{I})\), the map \(\mathbf{Z}_E^\# \to \mathbf{Z}_Y^\#\) is injective by (2), and \(\mathcal{I}\) is an injective abelian sheaf. Please compare with Modules on Sites, Section 03CQ. Finally, suppose we have \(s \in \mathcal{I}(Y)\) and \(t \in \mathcal{F}(Z)\) mapping to the same element of \(\mathcal{I}(E)\). Then \(s\) and \(t\) define a map \[s \amalg t : h_Y^\# \amalg h_Z^\# \longrightarrow \mathcal{I}\] which by assumption factors through \(h_Y^\# \amalg_{h_E^\#} h_Z^\#\). Thus by assumption (1) we obtain a unique map \(h_X^\# \to \mathcal{I}\) which corresponds to an element of \(\mathcal{I}(X)\) restricting to \(s\) on \(Y\) and \(t\) on \(Z\).

Lemma

Let \(\mathcal{C}\) be a site. Consider a commutative diagram \[\xymatrix{ \mathcal{D} \ar[r] \ar[d] & \mathcal{F} \ar[d] \\ \mathcal{E} \ar[r] & \mathcal{G} }\] of presheaves of sets on \(\mathcal{C}\) and assume that

1. \(\mathcal{G}^\# = \mathcal{E}^\# \amalg_{\mathcal{D}^\#} \mathcal{F}^\#\), and

2. \(\mathcal{D}^\# \to \mathcal{F}^\#\) is injective.

Then there is a canonical distinguished triangle \[R\Gamma(\mathcal{G}, K) \to R\Gamma(\mathcal{E}, K) \oplus R\Gamma(\mathcal{F}, K) \to R\Gamma(\mathcal{D}, K) \to R\Gamma(\mathcal{G}, K)[1]\] functorial in \(K \in D(\mathcal{C})\) where \(R\Gamma(\mathcal{G}, -)\) is the cohomology discussed in Section 079X.

Proof

Since sheafification is exact and since \(R\Gamma(\mathcal{G}, -) = R\Gamma(\mathcal{G}^\#, -)\) we may assume \(\mathcal{D}, \mathcal{E}, \mathcal{F}, \mathcal{G}\) are sheaves of sets. Moreover, the cohomology \(R\Gamma(\mathcal{G}, -)\) only depends on the topos, not on the underlying site. Hence by Sites, Lemma 03CI we may replace \(\mathcal{C}\) by a “larger” site with a subcanonical topology such that \(\mathcal{G} = h_X\), \(\mathcal{F} = h_Y\), \(\mathcal{E} = h_Z\), and \(\mathcal{D} = h_E\) for some objects \(X, Y, Z, E\) of \(\mathcal{C}\). In this case the result follows from Lemma 0EVY.

Comparing two topologies

Let \(\mathcal{C}\) be a category. Let \(\text{Cov}(\mathcal{C}) \supset \text{Cov}'(\mathcal{C})\) be two ways to endow \(\mathcal{C}\) with the structure of a site. Denote \(\tau\) the topology corresponding to \(\text{Cov}(\mathcal{C})\) and \(\tau'\) the topology corresponding to \(\text{Cov}'(\mathcal{C})\). Then the identity functor on \(\mathcal{C}\) defines a morphism of sites \[\epsilon : \mathcal{C}_\tau \longrightarrow \mathcal{C}_{\tau'}\] where \(\epsilon_*\) is the identity functor on underlying presheaves and where \(\epsilon^{-1}\) is the \(\tau\)-sheafification of a \(\tau'\)-sheaf. See Sites, Examples 0EWI and 0EWJ. In the situation above we have the following

1. \(\epsilon_* : \Sh(\mathcal{C}_\tau) \to \Sh(\mathcal{C}_{\tau'})\) is fully faithful and \(\epsilon^{-1} \circ \epsilon_* = \text{id}\),

2. \(\epsilon_* : \textit{Ab}(\mathcal{C}_\tau) \to \textit{Ab}(\mathcal{C}_{\tau'})\) is fully faithful and \(\epsilon^{-1} \circ \epsilon_* = \text{id}\),

3. \(R\epsilon_* : D(\mathcal{C}_\tau) \to D(\mathcal{C}_{\tau'})\) is fully faithful and \(\epsilon^{-1} \circ R\epsilon_* = \text{id}\),

4. if \(\mathcal{O}\) is a sheaf of rings for the \(\tau\)-topology, then \(\mathcal{O}\) is also a sheaf for the \(\tau'\)-topology and \(\epsilon\) becomes a flat morphism of ringed sites \[\epsilon : (\mathcal{C}_\tau, \mathcal{O}_\tau) \longrightarrow (\mathcal{C}_{\tau'}, \mathcal{O}_{\tau'})\]

5. \(\epsilon_* : \textit{Mod}(\mathcal{O}_\tau) \to \textit{Mod}(\mathcal{O}_{\tau'})\) is fully faithful and \(\epsilon^* \circ \epsilon_* = \text{id}\)

6. \(R\epsilon_* : D(\mathcal{O}_\tau) \to D(\mathcal{O}_{\tau'})\) is fully faithful and \(\epsilon^* \circ R\epsilon_* = \text{id}\).

Here are some explanations.

Ad (1). Let \(\mathcal{F}\) be a sheaf of sets in the \(\tau\)-topology. Then \(\epsilon_*\mathcal{F}\) is just \(\mathcal{F}\) viewed as a sheaf in the \(\tau'\)-topology. Applying \(\epsilon^{-1}\) means taking the \(\tau\)-sheafification of \(\mathcal{F}\), which doesn’t do anything as \(\mathcal{F}\) is already a \(\tau\)-sheaf. Thus \(\epsilon^{-1}(\epsilon_*\mathcal{F})) = \mathcal{F}\). The fully faithfulness follows by Categories, Lemma 07RB.

Ad (2). This is a consequence of (1) since pullback and pushforward of abelian sheaves is the same as doing those operations on the underlying sheaves of sets.

Ad (3). Let \(K\) be an object of \(D(\mathcal{C}_\tau)\). To compute \(R\epsilon_*K\) we choose a K-injective complex \(\mathcal{I}^\bullet\) representing \(K\) and we set \(R\epsilon_*K = \epsilon_*\mathcal{I}^\bullet\). Since \(\epsilon^{-1} : D(\mathcal{C}_{\tau'}) \to D(\mathcal{C}_\tau)\) is computed on an object \(L\) by applying the exact functor \(\epsilon^{-1}\) to any complex of abelian sheaves representing \(L\), we find that \(\epsilon^{-1}R\epsilon_*K\) is represented by \(\epsilon^{-1}\epsilon_*\mathcal{I}^\bullet\). By Part (1) we have \(\mathcal{I}^\bullet = \epsilon^{-1}\epsilon_*\mathcal{I}^\bullet\). In other words, we have \(\epsilon^{-1} \circ R\epsilon_* = \text{id}\) and we conclude as before.

Ad (4). Observe that \(\epsilon^{-1}\mathcal{O}_{\tau'} = \mathcal{O}_\tau\), see discussion in part (1). Hence \(\epsilon\) is a flat morphism of ringed sites, see Modules on Sites, Definition 04JB. Not only that, it is moreover clear that \(\epsilon^* = \epsilon^{-1}\) on \(\mathcal{O}_{\tau'}\)-modules (the pullback as a module has the same underlying abelian sheaf as the pullback of the underlying abelian sheaf).

Ad (5). This is clear from (2) and what we said in (4).

Ad (6). This is analogous to (3). We omit the details.

Formalities on cohomological descent

In this section we discuss only to what extent a morphism of ringed topoi determines an embedding from the derived category downstairs to the derived category upstairs. Here is a typical result.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi. Consider the full subcategory \(D' \subset D(\mathcal{O}_\mathcal{D})\) consisting of objects \(K\) such that \[K \longrightarrow Rf_*Lf^*K\] is an isomorphism. Then \(D'\) is a saturated triangulated strictly full subcategory of \(D(\mathcal{O}_\mathcal{D})\) and the functor \(Lf^* : D' \to D(\mathcal{O}_\mathcal{C})\) is fully faithful.

Proof

See Derived Categories, Definition 05RB for the definition of saturated in this setting. See Derived Categories, Lemma 05QX for a discussion of triangulated subcategories. The canonical map of the lemma is the unit of the adjoint pair of functors \((Lf^*, Rf_*)\), see Lemma 07A6. Having said this the proof that \(D'\) is a saturated triangulated subcategory is omitted; it follows formally from the fact that \(Lf^*\) and \(Rf_*\) are exact functors of triangulated categories. The final part follows formally from fact that \(Lf^*\) and \(Rf_*\) are adjoint; compare with Categories, Lemma 07RB.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi. Consider the full subcategory \(D' \subset D(\mathcal{O}_\mathcal{C})\) consisting of objects \(K\) such that \[Lf^*Rf_*K \longrightarrow K\] is an isomorphism. Then \(D'\) is a saturated triangulated strictly full subcategory of \(D(\mathcal{O}_\mathcal{C})\) and the functor \(Rf_* : D' \to D(\mathcal{O}_\mathcal{D})\) is fully faithful.

Proof

See Derived Categories, Definition 05RB for the definition of saturated in this setting. See Derived Categories, Lemma 05QX for a discussion of triangulated subcategories. The canonical map of the lemma is the counit of the adjoint pair of functors \((Lf^*, Rf_*)\), see Lemma 07A6. Having said this the proof that \(D'\) is a saturated triangulated subcategory is omitted; it follows formally from the fact that \(Lf^*\) and \(Rf_*\) are exact functors of triangulated categories. The final part follows formally from fact that \(Lf^*\) and \(Rf_*\) are adjoint; compare with Categories, Lemma 07RB.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi. Let \(K\) be an object of \(D(\mathcal{O}_\mathcal{C})\). Assume

1. \(f\) is flat,

2. \(K\) is bounded below,

3. \(f^*Rf_*H^q(K) \to H^q(K)\) is an isomorphism.

Then \(f^*Rf_*K \to K\) is an isomorphism.

Proof

Observe that \(f^*Rf_*K \to K\) is an isomorphism if and only if it is an isomorphism on cohomology sheaves \(H^j\). Observe that \(H^j(f^*Rf_*K) = f^*H^j(Rf_*K) = f^*H^j(Rf_*\tau_{\leq j}K) = H^j(f^*Rf_*\tau_{\leq j}K)\). Hence we may assume that \(K\) is bounded. Then property (3) tells us the cohomology sheaves are in the triangulated subcategory \(D' \subset D(\mathcal{O}_\mathcal{C})\) of Lemma 0D7R. Hence \(K\) is in it too.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi. Let \(K\) be an object of \(D(\mathcal{O}_\mathcal{D})\). Assume

1. \(f\) is flat,

2. \(K\) is bounded below,

3. \(H^q(K) \to Rf_*f^*H^q(K)\) is an isomorphism.

Then \(K \to Rf_*f^*K\) is an isomorphism.

Proof

Observe that \(K \to Rf_*f^*K\) is an isomorphism if and only if it is an isomorphism on cohomology sheaves \(H^j\). Observe that \(H^j(Rf_*f^*K) = H^j(Rf_*\tau_{\leq j}f^*K) = H^j(Rf_*f^*\tau_{\leq j}K)\). Hence we may assume that \(K\) is bounded. Then property (3) tells us the cohomology sheaves are in the triangulated subcategory \(D' \subset D(\mathcal{O}_\mathcal{D})\) of Lemma 0D7Q. Hence \(K\) is in it too.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')\) be a morphism of ringed topoi. Let \(\mathcal{A} \subset \textit{Mod}(\mathcal{O})\) and \(\mathcal{A}' \subset \textit{Mod}(\mathcal{O}')\) be weak Serre subcategories. Assume

1. \(f\) is flat,

2. \(f^*\) induces an equivalence of categories \(\mathcal{A}' \to \mathcal{A}\),

3. \(\mathcal{F}' \to Rf_*f^*\mathcal{F}'\) is an isomorphism for \(\mathcal{F}' \in \Ob(\mathcal{A}')\).

Then \(f^* : D_{\mathcal{A}'}^+(\mathcal{O}') \to D_\mathcal{A}^+(\mathcal{O})\) is an equivalence of categories with quasi-inverse given by \(Rf_* : D_\mathcal{A}^+(\mathcal{O}) \to D_{\mathcal{A}'}^+(\mathcal{O}')\).

Proof

By assumptions (2) and (3) and Lemmas 0D7T and 0D7Q we see that \(f^* : D_{\mathcal{A}'}^+(\mathcal{O}') \to D_\mathcal{A}^+(\mathcal{O})\) is fully faithful. Let \(\mathcal{F} \in \Ob(\mathcal{A})\). Then we can write \(\mathcal{F} = f^*\mathcal{F}'\). Then \(Rf_*\mathcal{F} = Rf_* f^*\mathcal{F}' = \mathcal{F}'\). In particular, we have \(R^pf_*\mathcal{F} = 0\) for \(p > 0\) and \(f_*\mathcal{F} \in \Ob(\mathcal{A}')\). Thus for any \(K \in D^+_\mathcal{A}(\mathcal{O})\) we see, using the spectral sequence \(E_2^{p, q} = R^pf_*H^q(K)\) converging to \(R^{p + q}f_*K\), that \(Rf_*K\) is in \(D^+_{\mathcal{A}'}(\mathcal{O}')\). Of course, it also follows from Lemmas 0D7S and 0D7R that \(Rf_* : D_\mathcal{A}^+(\mathcal{O}) \to D_{\mathcal{A}'}^+(\mathcal{O}')\) is fully faithful. Since \(f^*\) and \(Rf_*\) are adjoint we then get the result of the lemma, for example by Categories, Lemma 07RB.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')\) be a morphism of ringed topoi. Let \(\mathcal{A} \subset \textit{Mod}(\mathcal{O})\) and \(\mathcal{A}' \subset \textit{Mod}(\mathcal{O}')\) be weak Serre subcategories. Assume

1. \(f\) is flat,

2. \(f^*\) induces an equivalence of categories \(\mathcal{A}' \to \mathcal{A}\),

3. \(\mathcal{F}' \to Rf_*f^*\mathcal{F}'\) is an isomorphism for \(\mathcal{F}' \in \Ob(\mathcal{A}')\),

4. \(\mathcal{C}, \mathcal{O}, \mathcal{A}\) satisfy the assumption of Situation 0D6R,

5. \(\mathcal{C}', \mathcal{O}', \mathcal{A}'\) satisfy the assumption of Situation 0D6R.

Then \(f^* : D_{\mathcal{A}'}(\mathcal{O}') \to D_\mathcal{A}(\mathcal{O})\) is an equivalence of categories with quasi-inverse given by \(Rf_* : D_\mathcal{A}(\mathcal{O}) \to D_{\mathcal{A}'}(\mathcal{O}')\).

Proof

Since \(f^*\) is exact, it is clear that \(f^*\) defines a functor \(f^* : D_{\mathcal{A}'}(\mathcal{O}') \to D_\mathcal{A}(\mathcal{O})\) as in the statement of the lemma and that moreover this functor commutes with the truncation functors \(\tau_{\geq -n}\). We already know that \(f^*\) and \(Rf_*\) are quasi-inverse equivalence on the corresponding bounded below categories, see Lemma 0D7U. By Lemma 0D6U with \(N = 0\) we see that \(Rf_*\) indeed defines a functor \(Rf_* : D_\mathcal{A}(\mathcal{O}) \to D_{\mathcal{A}'}(\mathcal{O}')\) and that moreover this functor commutes with the truncation functors \(\tau_{\geq -n}\). Thus for \(K\) in \(D_\mathcal{A}(\mathcal{O})\) the map \(f^*Rf_*K \to K\) is an isomorphism as this is true on trunctions. Similarly, for \(K'\) in \(D_{\mathcal{A}'}(\mathcal{O}')\) the map \(K' \to Rf_*f^*K'\) is an isomorphism as this is true on trunctions. This finishes the proof.

Lemma

Let \(f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')\) be a morphism of ringed sites. Let \(\mathcal{A} \subset \textit{Mod}(\mathcal{O})\) and \(\mathcal{A}' \subset \textit{Mod}(\mathcal{O}')\) be weak Serre subcategories. Assume

1. \(f\) is flat,

2. \(f^*\) induces an equivalence of categories \(\mathcal{A}' \to \mathcal{A}\),

3. \(\mathcal{F}' \to Rf_*f^*\mathcal{F}'\) is an isomorphism for \(\mathcal{F}' \in \Ob(\mathcal{A}')\),

4. \(\mathcal{C}, \mathcal{O}, \mathcal{A}\) satisfy the assumption of Situation 0D6R,

5. \(f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')\) and \(\mathcal{A}\) satisfy the assumption of Situation 0D6V.

Then \(f^* : D_{\mathcal{A}'}(\mathcal{O}') \to D_\mathcal{A}(\mathcal{O})\) is an equivalence of categories with quasi-inverse given by \(Rf_* : D_\mathcal{A}(\mathcal{O}) \to D_{\mathcal{A}'}(\mathcal{O}')\).

Proof

The proof of this lemma is exactly the same as the proof of Lemma 0D7V except the reference to Lemma 0D6U is replaced by a reference to Lemma 0D6W.

Comparing two topologies, II

Let \(\mathcal{C}\) be a category. Let \(\text{Cov}(\mathcal{C}) \supset \text{Cov}'(\mathcal{C})\) be two ways to endow \(\mathcal{C}\) with the structure of a site. Denote \(\tau\) the topology corresponding to \(\text{Cov}(\mathcal{C})\) and \(\tau'\) the topology corresponding to \(\text{Cov}'(\mathcal{C})\). Then the identity functor on \(\mathcal{C}\) defines a morphism of sites \[\epsilon : \mathcal{C}_\tau \longrightarrow \mathcal{C}_{\tau'}\] where \(\epsilon_*\) is the identity functor on underlying presheaves and where \(\epsilon^{-1}\) is the \(\tau\)-sheafification of a \(\tau'\)-sheaf (hence clearly exact). Let \(\mathcal{O}\) be a sheaf of rings for the \(\tau\)-topology. Then \(\mathcal{O}\) is also a sheaf for the \(\tau'\)-topology and \(\epsilon\) becomes a morphism of ringed sites \[\epsilon : (\mathcal{C}_\tau, \mathcal{O}_\tau) \longrightarrow (\mathcal{C}_{\tau'}, \mathcal{O}_{\tau'})\] For more discussion, see Section 0EWK.

Lemma

With \(\epsilon : (\mathcal{C}_\tau, \mathcal{O}_\tau) \to (\mathcal{C}_{\tau'}, \mathcal{O}_{\tau'})\) as above. Let \(\mathcal{B} \subset \Ob(\mathcal{C})\) be a subset. Let \(\mathcal{A} \subset \textit{PMod}(\mathcal{O})\) be a full subcategory. Assume

1. every object of \(\mathcal{A}\) is a sheaf for the \(\tau\)-topology,

2. \(\mathcal{A}\) is a weak Serre subcategory of \(\textit{Mod}(\mathcal{O}_\tau)\),

3. every object of \(\mathcal{C}\) has a \(\tau'\)-covering whose members are elements of \(\mathcal{B}\), and

4. for every \(U \in \mathcal{B}\) we have \(H^p_\tau(U, \mathcal{F}) = 0\), \(p > 0\) for all \(\mathcal{F} \in \mathcal{A}\).

Then \(\mathcal{A}\) is a weak Serre subcategory of \(\textit{Mod}(\mathcal{O}_{\tau'})\) and there is an equivalence of triangulated categories \(D_\mathcal{A}(\mathcal{O}_\tau) = D_\mathcal{A}(\mathcal{O}_{\tau'})\) given by \(\epsilon^*\) and \(R\epsilon_*\).

Proof

Since \(\epsilon^{-1}\mathcal{O}_{\tau'} = \mathcal{O}_\tau\) we see that \(\epsilon\) is a flat morphism of ringed sites and that in fact \(\epsilon^{-1} = \epsilon^*\) on sheaves of modules. By property (1) we can think of every object of \(\mathcal{A}\) as a sheaf of \(\mathcal{O}_\tau\)-modules and as a sheaf of \(\mathcal{O}_{\tau'}\)-modules. In other words, we have fully faithful inclusion functors \[\mathcal{A} \to \textit{Mod}(\mathcal{O}_\tau) \to \textit{Mod}(\mathcal{O}_{\tau'})\] To avoid confusion we will denote \(\mathcal{A}' \subset \textit{Mod}(\mathcal{O}_{\tau'})\) the image of \(\mathcal{A}\). Then it is clear that \(\epsilon_* : \mathcal{A} \to \mathcal{A}'\) and \(\epsilon^* : \mathcal{A}' \to \mathcal{A}\) are quasi-inverse equivalences (see discussion preceding the lemma and use that objects of \(\mathcal{A}'\) are sheaves in the \(\tau\) topology).

Conditions (3) and (4) imply that \(R^p\epsilon_*\mathcal{F} = 0\) for \(p > 0\) and \(\mathcal{F} \in \Ob(\mathcal{A})\). This is true because \(R^p\epsilon_*\) is the sheaf associated to the presheave \(U \mapsto H^p_\tau(U, \mathcal{F})\), see Lemma 072W. Thus any exact complex in \(\mathcal{A}\) (which is the same thing as an exact complex in \(\textit{Mod}(\mathcal{O}_\tau)\) whose terms are in \(\mathcal{A}\), see Homology, Lemma 0754) remains exact upon applying the functor \(\epsilon_*\).

Consider an exact sequence \[\mathcal{F}'_0 \to \mathcal{F}'_1 \to \mathcal{F}'_2 \to \mathcal{F}'_3 \to \mathcal{F}'_4\] in \(\textit{Mod}(\mathcal{O}_{\tau'})\) with \(\mathcal{F}'_0, \mathcal{F}'_1, \mathcal{F}'_3, \mathcal{F}'_4\) in \(\mathcal{A}'\). Apply the exact functor \(\epsilon^*\) to get an exact sequence \[\epsilon^*\mathcal{F}'_0 \to \epsilon^*\mathcal{F}'_1 \to \epsilon^*\mathcal{F}'_2 \to \epsilon^*\mathcal{F}'_3 \to \epsilon^*\mathcal{F}'_4\] in \(\textit{Mod}(\mathcal{O}_\tau)\). Since \(\mathcal{A}\) is a weak Serre subcategory and since \(\epsilon^*\mathcal{F}'_0, \epsilon^*\mathcal{F}'_1, \epsilon^*\mathcal{F}'_3, \epsilon^*\mathcal{F}'_4\) are in \(\mathcal{A}\), we conclude that \(\epsilon^*\mathcal{F}'_2\) is in \(\mathcal{A}\) by Homology, Definition 02MO. Consider the map of sequences \[\xymatrix{ \mathcal{F}'_0 \ar[r] \ar[d] & \mathcal{F}'_1 \ar[r] \ar[d] & \mathcal{F}'_2 \ar[r] \ar[d] & \mathcal{F}'_3 \ar[r] \ar[d] & \mathcal{F}'_4 \ar[d] \\ \epsilon_*\epsilon^*\mathcal{F}'_0 \ar[r] & \epsilon_*\epsilon^*\mathcal{F}'_1 \ar[r] & \epsilon_*\epsilon^*\mathcal{F}'_2 \ar[r] & \epsilon_*\epsilon^*\mathcal{F}'_3 \ar[r] & \epsilon_*\epsilon^*\mathcal{F}'_4 }\] The lower row is exact by the discussion in the preceding paragraph. The vertical arrows with index \(0\), \(1\), \(3\), \(4\) are isomorphisms by the discussion in the first paragraph. By the \(5\) lemma (Homology, Lemma 05QB) we find that \(\mathcal{F}'_2 \cong \epsilon_*\epsilon^*\mathcal{F}'_2\) and hence \(\mathcal{F}'_2\) is in \(\mathcal{A}'\). In this way we see that \(\mathcal{A}'\) is a weak Serre subcategory of \(\textit{Mod}(\mathcal{O}_{\tau'})\), see Homology, Definition 02MO.

At this point it makes sense to talk about the derived categories \(D_\mathcal{A}(\mathcal{O}_\tau)\) and \(D_{\mathcal{A}'}(\mathcal{O}_{\tau'})\), see Derived Categories, Section 06UP. To finish the proof we show that conditions (1) – (5) of Lemma 0D7W apply. We have already seen (1), (2), (3) above. Note that since every object has a \(\tau'\)-covering by objects of \(\mathcal{B}\), a fortiori every object has a \(\tau\)-covering by objects of \(\mathcal{B}\). Hence condition (4) of Lemma 0D7W is satisfied. Similarly, condition (5) is satisfied as well.

Lemma

With \(\epsilon : (\mathcal{C}_\tau, \mathcal{O}_\tau) \to (\mathcal{C}_{\tau'}, \mathcal{O}_{\tau'})\) as above. Let \(A\) be a set and for \(\alpha \in A\) let \[\xymatrix{ E_\alpha \ar[d] \ar[r] & Y_\alpha \ar[d] \\ Z_\alpha \ar[r] & X_\alpha }\] be a commutative diagram in the category \(\mathcal{C}\). Assume that

1. a \(\tau'\)-sheaf \(\mathcal{F}'\) is a \(\tau\)-sheaf if \(\mathcal{F}'(X_\alpha) = \mathcal{F}'(Z_\alpha) \times_{\mathcal{F}'(E_\alpha)} \mathcal{F}'(Y_\alpha)\) for all \(\alpha\),

2. for \(K'\) in \(D(\mathcal{O}_{\tau'})\) in the essential image of \(R\epsilon_*\) the maps \(c^{K'}_{X_\alpha, Z_\alpha, Y_\alpha, E_\alpha}\) of Lemma 0F16 are isomorphisms for all \(\alpha\).

Then \(K' \in D^+(\mathcal{O}_{\tau'})\) is in the essential image of \(R\epsilon_*\) if and only if the maps \(c^{K'}_{X_\alpha, Z_\alpha, Y_\alpha, E_\alpha}\) are isomorphisms for all \(\alpha\).

Proof

The “only if” direction is implied by assumption (2). On the other hand, if \(K'\) has a unique nonzero cohomology sheaf, then the “if” direction follows from assumption (1). In general we will use an induction argument to prove the “if” direction. Let us say an object \(K'\) of \(D^+(\mathcal{O}_{\tau'})\) satisfies (P) if the maps \(c^{K'}_{X_\alpha, Z_\alpha, Y_\alpha, E_\alpha}\) are isomorphisms for all \(\alpha \in A\).

Namely, let \(K'\) be an object of \(D^+(\mathcal{O}_{\tau'})\) satisfying (P). Choose a distinguished triangle \[K' \to R\epsilon_*\epsilon^{-1}K' \to M' \to K'[1]\] in \(D^+(\mathcal{O}_{\tau'})\) where the first arrow is the adjuntion map. By (2) and Lemma 0EWP we see that \(M'\) has (P). On the other hand, applying \(\epsilon^{-1}\) and using that \(\epsilon^{-1}R\epsilon_* = \text{id}\) by Section 0EWK we find that \(\epsilon^{-1}M' = 0\). In the next paragraph we will show \(M' = 0\) which finishes the proof.

Let \(K'\) be an object of \(D^+(\mathcal{O}_{\tau'})\) satisfying (P) with \(\epsilon^{-1}K' = 0\). We will show \(K' = 0\). Namely, given \(n \in \mathbf{Z}\) such that \(H^i(K') = 0\) for \(i < n\) we will show that \(H^n(K') = 0\). For \(\alpha \in A\) we have a distinguished triangle \[R\Gamma_{\tau'}(X_\alpha, K') \to R\Gamma_{\tau'}(Z_\alpha, K') \oplus R\Gamma_{\tau'}(Y_\alpha, K') \to R\Gamma_{\tau'}(E_\alpha, K') \to R\Gamma_{\tau'}(X_\alpha, K')[1]\] by Lemma 0F16. Taking cohomology in degree \(n\) and using the assumed vanishing of cohomology sheaves of \(K'\) we obtain an exact sequence \[0 \to H^n_{\tau'}(X_\alpha, K') \to H^n_{\tau'}(Z_\alpha, K') \oplus H^n_{\tau'}(Y_\alpha, K') \to H^n_{\tau'}(E_\alpha, K')\] which is the same as the exact sequence \[0 \to \Gamma(X_\alpha, H^n(K')) \to \Gamma(Z_\alpha, H^n(K')) \oplus \Gamma(Y_\alpha, H^n(K')) \to \Gamma(E_\alpha, H^n(K'))\] We conclude that \(H^n(K')\) is a \(\tau\)-sheaf by assumption (1). However, since the \(\tau\)-sheafification \(\epsilon^{-1}H^n(K') = H^n(\epsilon^{-1}K')\) is \(0\) as \(\epsilon^{-1}K' = 0\) we conclude that \(H^n(K') = 0\) as desired.

Lemma

With \(\epsilon : (\mathcal{C}_\tau, \mathcal{O}_\tau) \to (\mathcal{C}_{\tau'}, \mathcal{O}_{\tau'})\) as above. Let \[\xymatrix{ E \ar[d] \ar[r] & Y \ar[d] \\ Z \ar[r] & X }\] be a commutative diagram in the category \(\mathcal{C}\) such that

1. \(h_X^\# = h_Y^\# \amalg_{h_E^\#} h_Z^\#\), and

2. \(h_E^\# \to h_Y^\#\) is injective

where \({}^\#\) denotes \(\tau\)-sheafification. Then for \(K' \in D(\mathcal{O}_{\tau'})\) in the essential image of \(R\epsilon_*\) the map \(c^{K'}_{X, Z, Y, E}\) of Lemma 0F16 (using the \(\tau'\)-topology) is an isomorphism.

Proof

This helper lemma is an almost immediate consequence of Lemma 0EVY and we strongly urge the reader to skip the proof. Say \(K' = R\epsilon_*K\). Choose a K-injective complex of \(\mathcal{O}_\tau\)-modules \(\mathcal{J}^\bullet\) representing \(K\). Then \(\epsilon_*\mathcal{J}^\bullet\) is a K-injective complex of \(\mathcal{O}_{\tau'}\)-modules representing \(K'\), see Lemma 093Y. Next, \[0 \to \mathcal{J}^\bullet(X) \xrightarrow{\alpha} \mathcal{J}^\bullet(Z) \oplus \mathcal{J}^\bullet(Y) \xrightarrow{\beta} \mathcal{J}^\bullet(E) \to 0\] is a short exact sequence of complexes of abelian groups, see Lemma 0EVY and its proof. Since this is the same as the sequence of complexes of abelian groups which is used to define \(c^{K'}_{X, Z, Y, E}\), we conclude.

Comparing cohomology

We develop some general theory which will help us compare cohomology in different topologies. Given \(\mathcal{C}\), \(\tau\), and \(\tau'\) as in Section 0EWK and a morphism \(f : X \to Y\) in \(\mathcal{C}\) we obtain a commutative diagram of morphisms of topoi [0EZ2]\[\begin{equation} \vcenter{ \xymatrix{ \Sh(\mathcal{C}_\tau/X) \ar[r]_{f_\tau} \ar[d]_{\epsilon_X} & \Sh(\mathcal{C}_\tau/Y) \ar[d]^{\epsilon_Y} \\ \Sh(\mathcal{C}_{\tau'}/X) \ar[r]^{f_{\tau'}} & \Sh(\mathcal{C}_{\tau'}/Y) } } \end{equation}\] Here the morphism \(\epsilon_X\), resp. \(\epsilon_Y\) is the comparison morphism of Section 0EWK for the category \(\mathcal{C}/X\) endowed with the two topologies \(\tau\) and \(\tau'\). The morphisms \(f_\tau\) and \(f_{\tau'}\) are “relocalization” morphisms (Sites, Lemma 03EH). The commutativity of the diagram is a special case of Sites, Lemma 03CF (applied with \(\mathcal{C} = \mathcal{C}_\tau/Y\), \(\mathcal{D} = \mathcal{C}_{\tau'}/Y\), \(u = \text{id}\), \(U = X\), and \(V = X\)). We also get \(\epsilon_{X, *} \circ f_\tau^{-1} = f_{\tau'}^{-1} \circ \epsilon_{Y, *}\) either from the lemma or because it is obvious.

Situation

With \(\mathcal{C}\), \(\tau\), and \(\tau'\) as in Section 0EWK. Assume we are given a subset \(\mathcal{P} \subset \text{Arrows}(\mathcal{C})\) and for every object \(X\) of \(\mathcal{C}\) we are given a weak Serre subcategory \(\mathcal{A}'_X \subset \textit{Ab}(\mathcal{C}_{\tau'}/X)\). We make the following assumption:

1. given \(f : X \to Y\) in \(\mathcal{P}\) and \(Y' \to Y\) general, then \(X \times_Y Y'\) exists and \(X \times_Y Y' \to Y'\) is in \(\mathcal{P}\),

2. \(f_{\tau'}^{-1}\) sends \(\mathcal{A}'_Y\) into \(\mathcal{A}'_X\) for any morphism \(f : X \to Y\) of \(\mathcal{C}\),

3. given \(X\) in \(\mathcal{C}\) and \(\mathcal{F}'\) in \(\mathcal{A}'_X\), then \(\mathcal{F}'\) satisfies the sheaf condition for \(\tau\)-coverings, i.e., \(\mathcal{F}' = \epsilon_{X, *}\epsilon_X^{-1}\mathcal{F}'\),

4. if \(f : X \to Y\) in \(\mathcal{P}\) and \(\mathcal{F}' \in \Ob(\mathcal{A}'_X)\), then \(R^if_{\tau', *}\mathcal{F}' \in \Ob(\mathcal{A}'_Y)\) for \(i \geq 0\).

5. if \(\{U_i \to U\}_{i \in I}\) is a \(\tau\)-covering, then there exist

   1. a \(\tau'\)-covering \(\{V_j \to U\}_{j \in J}\),

   2. a \(\tau\)-covering \(\{f_j : W_j \to V_j\}\) consisting of a single \(f_j \in \mathcal{P}\), and

   3. a \(\tau'\)-covering \(\{W_{jk} \to W_j\}_{k \in K_j}\)

   such that \(\{W_{jk} \to U\}_{j \in J, k \in K_j}\) is a refinement of \(\{U_i \to U\}_{i \in I}\).

Lemma

In Situation 0EZ3 for \(X\) in \(\mathcal{C}\) denote \(\mathcal{A}_X\) the objects of \(\textit{Ab}(\mathcal{C}_\tau/X)\) of the form \(\epsilon_X^{-1}\mathcal{F}'\) with \(\mathcal{F}'\) in \(\mathcal{A}'_X\). Then

1. for \(\mathcal{F}\) in \(\textit{Ab}(\mathcal{C}_\tau/X)\) we have \(\mathcal{F} \in \mathcal{A}_X \Leftrightarrow \epsilon_{X, *}\mathcal{F} \in \mathcal{A}'_X\), and

2. \(f_\tau^{-1}\) sends \(\mathcal{A}_Y\) into \(\mathcal{A}_X\) for any morphism \(f : X \to Y\) of \(\mathcal{C}\).

Proof

Part (1) follows from (0EZ6) and part (2) follows from (0EZ5) and the commutativity of (0EZ2) which gives \(\epsilon_X^{-1} \circ f_{\tau'}^{-1} = f_\tau^{-1} \circ \epsilon_Y^{-1}\).

Our next goal is to prove Lemmas 0EZH and 0EZG. We will do this by an induction argument using the following induction hypothesis.

\((V_n)\) For \(X\) in \(\mathcal{C}\) and \(\mathcal{F}\) in \(\mathcal{A}_X\) we have \(R^i\epsilon_{X, *}\mathcal{F} = 0\) for \(1 \leq i \leq n\).

Lemma

In Situation 0EZ3 assume \((V_n)\) holds. For \(f : X \to Y\) in \(\mathcal{P}\) and \(\mathcal{F}\) in \(\mathcal{A}_X\) we have \(R^if_{\tau', *}\epsilon_{X, *}\mathcal{F} = \epsilon_{Y, *}R^if_{\tau, *}\mathcal{F}\) for \(i \leq n\).

Proof

We will use the commutative diagram (0EZ2) without further mention. In particular have \[Rf_{\tau', *}R\epsilon_{X, *}\mathcal{F} = R\epsilon_{Y, *}Rf_{\tau, *}\mathcal{F}\] Assumption \((V_n)\) tells us that \(\epsilon_{X, *}\mathcal{F} \to R\epsilon_{X, *}\mathcal{F}\) is an isomorphism in degrees \(\leq n\). Hence \(Rf_{\tau', *}\epsilon_{X, *}\mathcal{F} \to Rf_{\tau', *}R\epsilon_{X, *}\mathcal{F}\) is an isomorphism in degrees \(\leq n\). We conclude that \[R^if_{\tau', *}\epsilon_{X, *}\mathcal{F} \to H^i(R\epsilon_{Y, *}Rf_{\tau, *}\mathcal{F})\] is an isomorphism for \(i \leq n\). We will prove the lemma by looking at the second page of the spectral sequence of Lemma 0734 for \(R\epsilon_{Y, *}Rf_{\tau, *}\mathcal{F}\). Here is a picture: \[\begin{matrix} \ldots & \ldots & \ldots & \ldots \\ \epsilon_{Y, *}R^2f_{\tau, *}\mathcal{F} & R^1\epsilon_{Y, *}R^2f_{\tau, *}\mathcal{F} & R^2\epsilon_{Y, *}R^2f_{\tau, *}\mathcal{F} & \ldots \\ \epsilon_{Y, *}R^1f_{\tau, *}\mathcal{F} & R^1\epsilon_{Y, *}R^1f_{\tau, *}\mathcal{F} & R^2\epsilon_{Y, *}R^1f_{\tau, *}\mathcal{F} & \ldots \\ \epsilon_{Y, *}f_{\tau, *}\mathcal{F} & R^1\epsilon_{Y, *}f_{\tau, *}\mathcal{F} & R^2\epsilon_{Y, *}f_{\tau, *}\mathcal{F} & \ldots \end{matrix}\] Let \((C_m)\) be the hypothesis: \(R^if_{\tau', *}\epsilon_{X, *}\mathcal{F} = \epsilon_{Y, *}R^if_{\tau, *}\mathcal{F}\) for \(i \leq m\). Observe that \((C_0)\) holds. We will show that \((C_{m - 1}) \Rightarrow (C_m)\) for \(m < n\). Namely, if \((C_{m - 1})\) holds, then for \(n \geq p > 0\) and \(q \leq m - 1\) we have \[\begin{align*} R^p\epsilon_{Y, *}R^qf_{\tau, *}\mathcal{F} & = R^p\epsilon_{Y, *} \epsilon_Y^{-1} \epsilon_{Y, *} R^qf_{\tau, *}\mathcal{F} \\ & = R^p\epsilon_{Y, *} \epsilon_Y^{-1}R^qf_{\tau', *}\epsilon_{X, *}\mathcal{F} = 0 \end{align*}\] First equality as \(\epsilon_Y^{-1}\epsilon_{Y, *} = \text{id}\), the second by \((C_{m - 1})\), and the final by by \((V_n)\) because \(\epsilon_Y^{-1}R^qf_{\tau', *}\epsilon_{X, *}\mathcal{F}\) is in \(\mathcal{A}_Y\) by (0EZ7). Looking at the spectral sequence we see that \(E_2^{0, m} = \epsilon_{Y, *}R^mf_{\tau, *}\mathcal{F}\) is the only nonzero term \(E_2^{p, q}\) with \(p + q = m\). Recall that \(\text{d}_r^{p, q} : E_r^{p, q} \to E_r^{p + r, q - r + 1}\). Hence there are no nonzero differentials \(\text{d}_r^{p, q}\), \(r \geq 2\) either emanating or entering this spot. We conclude that \(H^m(R\epsilon_{Y, *}Rf_{\tau, *}\mathcal{F}) = \epsilon_{Y, *}R^mf_{\tau, *}\mathcal{F}\) which implies \((C_m)\) by the discussion above.

Finally, assume \((C_{n - 1})\). The same analysis shows that \(E_2^{0, n} = \epsilon_{Y, *}R^nf_{\tau, *}\mathcal{F}\) is the only nonzero term \(E_2^{p, q}\) with \(p + q = n\). We do still have no nonzero differentials entering this spot, but there can be a nonzero differential emanating it. Namely, the map \(d_{n + 1}^{0, n} : \epsilon_{Y, *}R^nf_{\tau, *}\mathcal{F} \to R^{n + 1}\epsilon_{Y, *}f_{\tau, *}\mathcal{F}\). We conclude that there is an exact sequence \[0 \to R^nf_{\tau', *}\epsilon_{X, *}\mathcal{F} \to \epsilon_{Y, *}R^nf_{\tau, *}\mathcal{F} \to R^{n + 1}\epsilon_{Y, *}f_{\tau, *}\mathcal{F}\] By (0EZ7) and (0EZ6) the sheaf \(R^nf_{\tau', *}\epsilon_{X, *}\mathcal{F}\) satisfies the sheaf property for \(\tau\)-coverings as does \(\epsilon_{Y, *}R^nf_{\tau, *}\mathcal{F}\) (use the description of \(\epsilon_*\) in Section 0EWK). However, the \(\tau\)-sheafification of the \(\tau'\)-sheaf \(R^{n + 1}\epsilon_{Y, *}f_{\tau, *}\mathcal{F}\) is zero (by locality of cohomology; use Lemmas 01FW and 072W). Thus \(R^nf_{\tau', *}\epsilon_{X, *}\mathcal{F} \to \epsilon_{Y, *}R^nf_{\tau, *}\mathcal{F}\) has to be an isomorphism and the proof is complete.

If \(E'\), resp. \(E\) is an object of \(D(\mathcal{C}_{\tau'}/X)\), resp. \(D(\mathcal{C}_\tau/X)\) then we will write \(H^n_{\tau'}(U, E')\), resp. \(H^n_\tau(U, E)\) for the cohomology of \(E'\), resp. \(E\) over an object \(U\) of \(\mathcal{C}/X\).

Lemma

In Situation 0EZ3 if \((V_n)\) holds, then for \(X\) in \(\mathcal{C}\) and \(L \in D(\mathcal{C}_{\tau'}/X)\) with \(H^i(L) = 0\) for \(i < 0\) and \(H^i(L)\) in \(\mathcal{A}'_X\) for \(0 \leq i \leq n\) we have \(H^n_{\tau'}(X, L) = H^n_\tau(X, \epsilon_X^{-1}L)\).

Proof

By Lemma 0D6H we have \(H^n_\tau(X, \epsilon_X^{-1}L) = H^n_{\tau'}(X, R\epsilon_{X, *}\epsilon_X^{-1}L)\). There is a spectral sequence \[E_2^{p, q} = R^p\epsilon_{X, *}\epsilon_X^{-1}H^q(L)\] converging to \(H^{p + q}(R\epsilon_{X, *}\epsilon_X^{-1}L)\). By \((V_n)\) we have the vanishing of \(E_2^{p, q}\) for \(0 < p \leq n\) and \(0 \leq q \leq n\). Thus \(E_2^{0, q} = \epsilon_{X, *}\epsilon_X^{-1}H^q(L) = H^q(L)\) are the only nonzero terms \(E_2^{p, q}\) with \(p + q \leq n\). It follows that the map \[L \longrightarrow R\epsilon_{X, *}\epsilon_X^{-1}L\] is an isomorphism in degrees \(\leq n\) (small detail omitted). Hence we find that \(H^i_{\tau'}(X, L) = H^i_{\tau'}(X, R\epsilon_{X, *}\epsilon_X^{-1}L)\) for \(i \leq n\). Thus the lemma is proved.

Lemma

In Situation 0EZ3 if \((V_n)\) holds, then for \(X\) in \(\mathcal{C}\) and \(\mathcal{F}\) in \(\mathcal{A}_X\) the map \(H^{n + 1}_{\tau'}(X, \epsilon_{X, *}\mathcal{F}) \to H^{n + 1}_\tau(X, \mathcal{F})\) is injective with image those classes which become trivial on a \(\tau'\)-covering of \(X\).

Proof

Recall that \(\epsilon_X^{-1}\epsilon_{X, *}\mathcal{F} = \mathcal{F}\) hence the map is given by pulling back cohomology classes by \(\epsilon_X\). The Leray spectral sequence (Lemma 0732) \[E_2^{p, q} = H^p_{\tau'}(X, R^q\epsilon_{X, *}\mathcal{F}) \Rightarrow H^{p + q}_\tau(X, \mathcal{F})\] combined with the assumed vanishing gives an exact sequence \[0 \to H^{n + 1}_{\tau'}(X, \epsilon_{X, *}\mathcal{F}) \to H^{n + 1}_\tau(X, \mathcal{F}) \to H^0_{\tau'}(X, R^{n + 1}\epsilon_{X, *}\mathcal{F})\] This is a restatement of the lemma.

Lemma

In Situation 0EZ3 let \(f : X \to Y\) be in \(\mathcal{P}\) such that \(\{X \to Y\}\) is a \(\tau\)-covering. Let \(\mathcal{F}'\) be in \(\mathcal{A}'_Y\). If \(n \geq 0\) and \[\theta \in \text{Equalizer}\left( \xymatrix{ H^{n + 1}_{\tau'}(X, \mathcal{F}') \ar@<1ex>[r] \ar@<-1ex>[r] & H^{n + 1}_{\tau'}(X \times_Y X, \mathcal{F}') } \right)\] then there exists a \(\tau'\)-covering \(\{Y_i \to Y\}\) such that \(\theta\) restricts to zero in \(H^{n + 1}_{\tau'}(Y_i \times_Y X, \mathcal{F}')\).

Proof

Observe that \(X \times_Y X\) exists by (0EZ4). For \(Z\) in \(\mathcal{C}/Y\) denote \(\mathcal{F}'|_Z\) the restriction of \(\mathcal{F}'\) to \(\mathcal{C}_{\tau'}/Z\). Recall that \(H^{n + 1}_{\tau'}(X, \mathcal{F}') = H^{n + 1}(\mathcal{C}_{\tau'}/X, \mathcal{F}'|_X)\), see Lemma 03F3. The lemma asserts that the image \(\overline{\theta} \in H^0(Y, R^{n + 1}f_{\tau', *}\mathcal{F}'|_X)\) of \(\theta\) is zero. Consider the cartesian diagram \[\xymatrix{ X \times_Y X \ar[d]_{\text{pr}_1} \ar[r]_{\text{pr}_2} & X \ar[d]^f \\ X \ar[r]^f & Y }\] By trivial base change (Lemma 0EZ0) we have \[f_{\tau'}^{-1}R^{n + 1}f_{\tau', *}(\mathcal{F}'|_X) = R^{n + 1}\text{pr}_{1, \tau', *}(\mathcal{F}'|_{X \times_Y X})\] If \(\text{pr}_1^{-1}\theta = \text{pr}_2^{-1}\theta\), then the section \(f_{\tau'}^{-1}\overline{\theta}\) of \(f_{\tau'}^{-1}R^{n + 1}f_{\tau', *}(\mathcal{F}'|_X)\) is zero, because it is clear that \(\text{pr}_1^{-1}\theta\) maps to the zero element in \(H^0(X, R^{n + 1}\text{pr}_{1, \tau', *}(\mathcal{F}'|_{X \times_Y X}))\). By (0EZ5) we have \(\mathcal{F}'|_X\) in \(\mathcal{A}'_X\). Thus \(\mathcal{G}' = R^{n + 1}f_{\tau', *}(\mathcal{F}'|_X)\) is an object of \(\mathcal{A}'_Y\) by (0EZ7). Thus \(\mathcal{G}'\) satisfies the sheaf property for \(\tau\)-coverings by (0EZ6). Since \(\{X \to Y\}\) is a \(\tau\)-covering we conclude that restriction \(\mathcal{G}'(Y) \to \mathcal{G}'(X)\) is injective. It follows that \(\overline{\theta}\) is zero.

Lemma

In Situation 0EZ3 we have \((V_n) \Rightarrow (V_{n + 1})\).

Proof

Let \(X\) in \(\mathcal{C}\) and \(\mathcal{F}\) in \(\mathcal{A}_X\). Let \(\xi \in H^{n + 1}_\tau(U, \mathcal{F})\) for some \(U/X\). We have to show that \(\xi\) restricts to zero on the members of a \(\tau'\)-covering of \(U\). See Lemma 072W. It follows from this that we may replace \(U\) by the members of a \(\tau'\)-covering of \(U\).

By locality of cohomology (Lemma 01FW) we can choose a \(\tau\)-covering \(\{U_i \to U\}\) such that \(\xi\) restricts to zero on \(U_i\). Choose \(\{V_j \to V\}\), \(\{f_j : W_j \to V_j\}\), and \(\{W_{jk} \to W_j\}\) as in (0EZ8). After replacing both \(U\) by \(V_j\) and \(\mathcal{F}\) by its restriction to \(\mathcal{C}_\tau/V_j\), which is allowed by (0EZ4), we reduce to the case discussed in the next paragraph.

Here \(f : X \to Y\) is an element of \(\mathcal{P}\) such that \(\{X \to Y\}\) is a \(\tau\)-covering, \(\mathcal{F}\) is an object of \(\mathcal{A}_Y\), and \(\xi \in H^{n + 1}_\tau(Y, \mathcal{F})\) is such that there exists a \(\tau'\)-covering \(\{X_i \to X\}_{i \in I}\) such that \(\xi\) restricts to zero on \(X_i\) for all \(i \in I\). Problem: show that \(\xi\) restricts to zero on a \(\tau'\)-covering of \(Y\).

By Lemma 0EZC there exists a unique \(\tau'\)-cohomology class \(\theta \in H^{n + 1}_{\tau'}(X, \epsilon_{X, *}\mathcal{F})\) whose image is \(\xi|_X\). Since \(\xi|_X\) pulls back to the same class on \(X \times_Y X\) via the two projections, we find that the same is true for \(\theta\) (by uniqueness). By Lemma 0EZD we see that after replacing \(Y\) by the members of a \(\tau'\)-covering, we may assume that \(\theta = 0\). Consequently, we may assume that \(\xi|_X\) is zero.

Let \(f : X \to Y\) be an element of \(\mathcal{P}\) such that \(\{X \to Y\}\) is a \(\tau\)-covering, \(\mathcal{F}\) is an object of \(\mathcal{A}_Y\), and \(\xi \in H^{n + 1}_\tau(Y, \mathcal{F})\) maps to zero in \(H^{n + 1}_\tau(X, \mathcal{F})\). Problem: show that \(\xi\) restricts to zero on a \(\tau'\)-covering of \(Y\).

The assumptions tell us \(\xi\) maps to zero under the map \[\mathcal{F} \longrightarrow Rf_{\tau, *}f_\tau^{-1}\mathcal{F}\] Use Lemma 0D6H. A simple argument using the distinguished triangle of truncations (Derived Categories, Remark 08J5) shows that \(\xi\) maps to zero under the map \[\mathcal{F} \longrightarrow \tau_{\leq n}Rf_{\tau, *}f_\tau^{-1}\mathcal{F}\] We will compare this with the map \(\epsilon_{Y, *}\mathcal{F} \to K\) where \[K = \tau_{\leq n}Rf_{\tau', *}f_{\tau'}^{-1}\epsilon_{Y, *}\mathcal{F} = \tau_{\leq n}Rf_{\tau', *}\epsilon_{X, *}f_{\tau}^{-1}\mathcal{F}\] The equality \(\epsilon_{X, *} f_\tau^{-1} = f_{\tau'}^{-1} \epsilon_{Y, *}\) is a property of (0EZ2). Consider the map \[Rf_{\tau', *}\epsilon_{X, *}f_{\tau}^{-1}\mathcal{F} \longrightarrow Rf_{\tau', *}R\epsilon_{X, *}f_{\tau}^{-1}\mathcal{F} = R\epsilon_{Y, *}Rf_{\tau, *}f_\tau^{-1}\mathcal{F}\] used in the proof of Lemma 0EZA which induces by adjunction a map \[\epsilon_Y^{-1} Rf_{\tau', *}\epsilon_{X, *}f_{\tau}^{-1}\mathcal{F} \to Rf_{\tau, *}f_\tau^{-1}\mathcal{F}\] Taking trunctions we find a map \[\epsilon_Y^{-1}K \longrightarrow \tau_{\leq n}Rf_{\tau, *}f_\tau^{-1}\mathcal{F}\] which is an isomorphism by Lemma 0EZA; the lemma applies because \(f_\tau^{-1}\mathcal{F}\) is in \(\mathcal{A}_X\) by Lemma 0EZ9. Choose a distinguished triangle \[\epsilon_{Y, *}\mathcal{F} \to K \to L \to \epsilon_{Y, *}\mathcal{F}[1]\] The map \(\mathcal{F} \to f_{\tau, *}f_\tau^{-1}\mathcal{F}\) is injective as \(\{X \to Y\}\) is a \(\tau\)-covering. Thus \(\epsilon_{Y, *}\mathcal{F} \to \epsilon_{Y, *}f_{\tau, *}f_\tau^{-1}\mathcal{F} = f_{\tau', *}f_{\tau'}^{-1}\epsilon_{Y, *}\mathcal{F}\) is injective too. Hence \(L\) only has nonzero cohomology sheaves in degrees \(0, \ldots, n\). As \(f_{\tau', *}f_{\tau'}^{-1}\epsilon_{Y, *}\mathcal{F}\) is in \(\mathcal{A}'_Y\) by (0EZ5) and (0EZ7) we conclude that \[H^0(L) = \Coker(\epsilon_{Y, *}\mathcal{F} \to f_{\tau', *}f_{\tau'}^{-1}\epsilon_{Y, *}\mathcal{F})\] is in the weak Serre subcategory \(\mathcal{A}'_Y\). For \(1 \leq i \leq n\) we see that \(H^i(L) = R^if_{\tau', *}f_{\tau'}^{-1}\epsilon_{Y, *}\mathcal{F}\) is in \(\mathcal{A}'_Y\) by (0EZ5) and (0EZ7). Pulling back the distinguished triangle above by \(\epsilon_Y\) we get the distinguished triangle \[\mathcal{F} \to \tau_{\leq n}Rf_{\tau, *}f_\tau^{-1}\mathcal{F} \to \epsilon_Y^{-1}L \to \mathcal{F}[1]\] Since \(\xi\) maps to zero in the middle term we find that \(\xi\) is the image of an element \(\xi' \in H^n_\tau(Y, \epsilon_Y^{-1}L)\). By Lemma 0EZB we have \[H^n_{\tau'}(Y, L) = H^n_\tau(Y, \epsilon_Y^{-1}L),\] Thus we may lift \(\xi'\) to an element of \(H^n_{\tau'}(Y, L)\) and take the boundary into \(H^{n + 1}_{\tau'}(Y, \epsilon_{Y, *}\mathcal{F})\) to see that \(\xi\) is in the image of the canonical map \(H^{n + 1}_{\tau'}(Y, \epsilon_{Y, *}\mathcal{F}) \to H^{n + 1}_\tau(Y, \mathcal{F})\). By locality of cohomology for \(H^{n + 1}_{\tau'}(Y,\epsilon_{Y, *}\mathcal{F})\), see Lemma 01FW, we conclude.

Lemma

In Situation 0EZ3 we have that \((V_n)\) is true for all \(n\). Moreover:

1. For \(X\) in \(\mathcal{C}\) and \(K' \in D^+_{\mathcal{A}'_X}(\mathcal{C}_{\tau'}/X)\) the map \(K' \to R\epsilon_{X, *}(\epsilon_X^{-1}K')\) is an isomorphism.

2. For \(f : X \to Y\) in \(\mathcal{P}\) and \(K' \in D^+_{\mathcal{A}'_X}(\mathcal{C}_{\tau'}/X)\) we have \(Rf_{\tau', *}K' \in D^+_{\mathcal{A}'_Y}(\mathcal{C}_{\tau'}/Y)\) and \(\epsilon_Y^{-1}(Rf_{\tau', *}K') = Rf_{\tau, *}(\epsilon_X^{-1}K')\).

Proof

Observe that \((V_0)\) holds as it is the empty condition. Then we get \((V_n)\) for all \(n\) by Lemma 0EZE.

Proof of (1). The object \(K = \epsilon_X^{-1}K'\) has cohomology sheaves \(H^i(K) = \epsilon_X^{-1}H^i(K')\) in \(\mathcal{A}_X\). Hence the spectral sequence \[E_2^{p, q} = R^p\epsilon_{X, *} H^q(K) \Rightarrow H^{p + q}(R\epsilon_{X, *}K)\] degenerates by \((V_n)\) for all \(n\) and we find \[H^n(R\epsilon_{X, *}K) = \epsilon_{X, *}H^n(K) = \epsilon_{X, *}\epsilon_X^{-1}H^n(K') = H^n(K').\] again because \(H^n(K')\) is in \(\mathcal{A}'_X\). Thus the canonical map \(K' \to R\epsilon_{X, *}(\epsilon_X^{-1}K')\) is an isomorphism.

Proof of (2). Using the spectral sequence \[E_2^{p, q} = R^pf_{\tau', *}H^q(K') \Rightarrow R^{p + q}f_{\tau', *}K'\] the fact that \(R^pf_{\tau', *}H^q(K')\) is in \(\mathcal{A}'_Y\) by (0EZ7), the fact that \(\mathcal{A}'_Y\) is a weak Serre subcategory of \(\textit{Ab}(\mathcal{C}_{\tau'}/Y)\), and Homology, Lemma 012W we conclude that \(Rf_{\tau', *}K' \in D^+_{\mathcal{A}'_Y}(\mathcal{C}_{\tau'}/Y)\). To finish the proof we have to show the base change map \[\epsilon_Y^{-1}(Rf_{\tau', *}K') \longrightarrow Rf_{\tau, *}(\epsilon_X^{-1}K')\] is an isomorphism. Comparing the spectral sequence above to the spectral sequence \[E_2^{p, q} = R^pf_{\tau, *}H^q(\epsilon_X^{-1}K') \Rightarrow R^{p + q}f_{\tau, *}\epsilon_X^{-1}K'\] we reduce this to the case where \(K'\) has a single nonzero cohomology sheaf \(\mathcal{F}'\) in \(\mathcal{A}'_X\); details omitted. Then Lemma 0EZA gives \(\epsilon_Y^{-1}R^if_{\tau', *}\mathcal{F}' = R^if_{\tau, *}\epsilon_X^{-1}\mathcal{F}'\) for all \(i\) and the proof is complete.

Lemma

In Situation 0EZ3. For any \(X\) in \(\mathcal{C}\) the category \(\mathcal{A}_X \subset \textit{Ab}(\mathcal{C}_\tau/X)\) is a weak Serre subcategory and the functor \[R\epsilon_{X, *} : D^+_{\mathcal{A}_X}(\mathcal{C}_\tau/X) \longrightarrow D^+_{\mathcal{A}'_X}(\mathcal{C}_{\tau'}/X)\] is an equivalence with quasi-inverse given by \(\epsilon_X^{-1}\).

Proof

We need to check the conditions listed in Homology, Lemma 0754 for \(\mathcal{A}_X\). If \(\varphi : \mathcal{F} \to \mathcal{G}\) is a map in \(\mathcal{A}_X\), then \(\epsilon_{X, *}\varphi : \epsilon_{X, *}\mathcal{F} \to \epsilon_{X, *}\mathcal{G}\) is a map in \(\mathcal{A}'_X\). Hence \(\Ker(\epsilon_{X, *}\varphi)\) and \(\Coker(\epsilon_{X, *}\varphi)\) are objects of \(\mathcal{A}'_X\) as this is a weak Serre subcategory of \(\textit{Ab}(\mathcal{C}_{\tau'}/X)\). Applying \(\epsilon_X^{-1}\) we obtain an exact sequence \[0 \to \epsilon_X^{-1}\Ker(\epsilon_{X, *}\varphi) \to \mathcal{F} \to \mathcal{G} \to \epsilon_X^{-1}\Coker(\epsilon_{X, *}\varphi) \to 0\] and we see that \(\Ker(\varphi)\) and \(\Coker(\varphi)\) are in \(\mathcal{A}_X\). Finally, suppose that \[0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\] is a short exact sequence in \(\textit{Ab}(\mathcal{C}_\tau/X)\) with \(\mathcal{F}_1\) and \(\mathcal{F}_3\) in \(\mathcal{A}_X\). Then applying \(\epsilon_{X, *}\) we obtain an exact sequence \[0 \to \epsilon_{X, *}\mathcal{F}_1 \to \epsilon_{X, *}\mathcal{F}_2 \to \epsilon_{X, *}\mathcal{F}_3 \to R^1\epsilon_{X, *}\mathcal{F}_1 = 0\] Vanishing by Lemma 0EZF. Hence \(\epsilon_{X, *}\mathcal{F}_2\) is in \(\mathcal{A}'_X\) as this is a weak Serre subcategory of \(\textit{Ab}(\mathcal{C}_{\tau'}/X)\). Pulling back by \(\epsilon_X\) we conclude that \(\mathcal{F}_2\) is in \(\mathcal{A}_X\).

Thus \(\mathcal{A}_X\) is a weak Serre subcategory of \(\textit{Ab}(\mathcal{C}_\tau/X)\) and it makes sense to consider the category \(D^+_{\mathcal{A}_X}(\mathcal{C}_\tau/X)\). Observe that \(\epsilon_X^{-1} : \mathcal{A}'_X \to \mathcal{A}_X\) is an equivalence and that \(\mathcal{F}' \to R\epsilon_{X, *}\epsilon_X^{-1}\mathcal{F}'\) is an isomorphism for \(\mathcal{F}'\) in \(\mathcal{A}'_X\) since we have \((V_n)\) for all \(n\) by Lemma 0EZF. Thus we conclude by Lemma 0D7U.

Lemma

In Situation 0EZ3. Let \(X\) be in \(\mathcal{C}\).

1. for \(\mathcal{F}'\) in \(\mathcal{A}'_X\) we have \(H^n_{\tau'}(X, \mathcal{F}') = H^n_\tau(X, \epsilon_X^{-1}\mathcal{F}')\),

2. for \(K' \in D^+_{\mathcal{A}'_X}(\mathcal{C}_{\tau'}/X)\) we have \(H^n_{\tau'}(X, K') = H^n_\tau(X, \epsilon_X^{-1}K')\).

Proof

This follows from Lemma 0EZF by Remark 08J6.

Cohomology on Hausdorff and locally quasi-compact spaces

We continue our convention to say “Hausdorff and locally quasi-compact” instead of saying “locally compact” as is often done in the literature. Let \(\textit{LC}\) denote the category whose objects are Hausdorff and locally quasi-compact topological spaces and whose morphisms are continuous maps.

Lemma

The category \(\textit{LC}\) has fibre products and a final object and hence has arbitrary finite limits. Given morphisms \(X \to Z\) and \(Y \to Z\) in \(\textit{LC}\) with \(X\) and \(Y\) quasi-compact, then \(X \times_Z Y\) is quasi-compact.

Proof

The final object is the singleton space. Given morphisms \(X \to Z\) and \(Y \to Z\) of \(\textit{LC}\) the fibre product \(X \times_Z Y\) is a subspace of \(X \times Y\). Hence \(X \times_Z Y\) is Hausdorff as \(X \times Y\) is Hausdorff by Topology, Section 08ZD.

If \(X\) and \(Y\) are quasi-compact, then \(X \times Y\) is quasi-compact by Topology, Theorem 08ZU. Since \(X \times_Z Y\) is a closed subset of \(X \times Y\) (Topology, Lemma 08ZH) we find that \(X \times_Z Y\) is quasi-compact by Topology, Lemma 005C.

Finally, returning to the general case, if \(x \in X\) and \(y \in Y\) we can pick quasi-compact neighbourhoods \(x \in E \subset X\) and \(y \in F \subset Y\) and we find that \(E \times_Z F\) is a quasi-compact neighbourhood of \((x, y)\) by the result above. Thus \(X \times_Z Y\) is an object of \(\textit{LC}\) by Topology, Lemma 08ZR.

We can endow \(\textit{LC}\) with a stronger topology than the usual one.

Definition

Let \(\{f_i : X_i \to X\}\) be a family of morphisms with fixed target in the category \(\textit{LC}\). We say this family is a qc covering⁶ if for every \(x \in X\) there exist \(i_1, \ldots, i_n \in I\) and quasi-compact subsets \(E_j \subset X_{i_j}\) such that \(\bigcup f_{i_j}(E_j)\) is a neighbourhood of \(x\).

Observe that an open covering \(X = \bigcup U_i\) of an object of \(\textit{LC}\) gives a qc covering \(\{U_i \to X\}\) because \(X\) is locally quasi-compact. We start with the obligatory lemma.

Lemma

Let \(X\) be a Hausdorff and locally quasi-compact space, in other words, an object of \(\textit{LC}\).

1. If \(X' \to X\) is an isomorphism in \(\textit{LC}\) then \(\{X' \to X\}\) is a qc covering.

2. If \(\{f_i : X_i \to X\}_{i\in I}\) is a qc covering and for each \(i\) we have a qc covering \(\{g_{ij} : X_{ij} \to X_i\}_{j\in J_i}\), then \(\{X_{ij} \to X\}_{i \in I, j\in J_i}\) is a qc covering.

3. If \(\{X_i \to X\}_{i\in I}\) is a qc covering and \(X' \to X\) is a morphism of \(\textit{LC}\) then \(\{X' \times_X X_i \to X'\}_{i\in I}\) is a qc covering.

Proof

Part (1) holds by the remark above that open coverings are qc coverings.

Proof of (2). Let \(x \in X\). Choose \(i_1, \ldots, i_n \in I\) and \(E_a \subset X_{i_a}\) quasi-compact such that \(\bigcup f_{i_a}(E_a)\) is a neighbourhood of \(x\). For every \(e \in E_a\) we can find a finite subset \(J_e \subset J_{i_a}\) and quasi-compact \(F_{e, j} \subset X_{i_a j}\), \(j \in J_e\) such that \(\bigcup g_{i_a j}(F_{e, j})\) is a neighbourhood of \(e\). Since \(E_a\) is quasi-compact we find a finite collection \(e_1, \ldots, e_{m_a}\) such that \[E_a \subset \bigcup\nolimits_{k = 1, \ldots, m_a} \bigcup\nolimits_{j \in J_{e_k}} g_{i_a j}(F_{e_k, j})\] Then we find that \[\bigcup\nolimits_{a = 1, \ldots, n} \bigcup\nolimits_{k = 1, \ldots, m_a} \bigcup\nolimits_{j \in J_{e_k}} f_{i_a}(g_{i_a j}(F_{e_k, j}))\] is a neighbourhood of \(x\).

Proof of (3). Let \(x' \in X'\) be a point. Let \(x \in X\) be its image. Choose \(i_1, \ldots, i_n \in I\) and quasi-compact subsets \(E_j \subset X_{i_j}\) such that \(\bigcup f_{i_j}(E_j)\) is a neighbourhood of \(x\). Choose a quasi-compact neighbourhood \(F \subset X'\) of \(x'\) which maps into the quasi-compact neighbourhood \(\bigcup f_{i_j}(E_j)\) of \(x\). Then \(F \times_X E_j \subset X' \times_X X_{i_j}\) is a quasi-compact subset and \(F\) is the image of the map \(\coprod F \times_X E_j \to F\). Hence the base change is a qc covering and the proof is finished.

Since all objects of \(\textit{LC}\) are Hausdorff any morphism \(f : X \to Y\) of \(\textit{LC}\) is a separated continuous map of topological spaces. Hence \(f\) is a proper map of topological spaces if and only if \(f\) is universally closed. See discussion in Topology, Section 005M.

Lemma

Let \(f : X \to Y\) be a morphism of \(\textit{LC}\). If \(f\) is proper and surjective, then \(\{f : X \to Y\}\) is a qc covering.

Proof

Let \(y \in Y\) be a point. For each \(x \in X_y\) choose a quasi-compact neighbourhood \(E_x \subset X\). Choose \(x \in U_x \subset E_x\) open. Since \(f\) is proper the fibre \(X_y\) is quasi-compact and we find \(x_1, \ldots, x_n \in X_y\) such that \(X_y \subset U_{x_1} \cup \ldots \cup U_{x_n}\). We claim that \(f(E_{x_1}) \cup \ldots \cup f(E_{x_n})\) is a neighbourhood of \(y\). Namely, as \(f\) is closed (Topology, Theorem 005R) we see that \(Z = f(X \setminus (U_{x_1} \cup \ldots \cup U_{x_n}))\) is a closed subset of \(Y\) not containing \(y\). As \(f\) is surjective we see that \(Y \setminus Z\) is contained in \(f(E_{x_1}) \cup \ldots \cup f(E_{x_n})\) as desired.

Besides some set theoretic issues Lemma 09X1 shows that \(\textit{LC}\) with the collection of qc coverings forms a site. We will denote this site (suitably modified to overcome the set theoretical issues) \(\textit{LC}_{qc}\).

Remark

The category \(\textit{LC}\) is a “big” category as its objects form a proper class. Similarly, the coverings form a proper class. Let us define the size of a topological space \(X\) to be the cardinality of the set of points of \(X\). Choose a function \(Bound\) on cardinals, for example as in Sets, Equation (046U). Finally, let \(S_0\) be an initial set of objects of \(\textit{LC}\), for example \(S_0 = \{(\mathbf{R}, \text{euclidean topology})\}\). Exactly as in Sets, Lemma 000J we can choose a limit ordinal \(\alpha\) such that \(\textit{LC}_\alpha = \textit{LC} \cap V_\alpha\) contains \(S_0\) and is preserved under all countable limits and colimits which exist in \(\textit{LC}\). Moreover, if \(X \in \textit{LC}_\alpha\) and if \(Y \in \textit{LC}\) and \(\text{size}(Y) \leq Bound(\text{size}(X))\), then \(Y\) is isomorphic to an object of \(\textit{LC}_\alpha\). Next, we apply Sets, Lemma 000X to choose set \(\text{Cov}\) of qc covering on \(\textit{LC}_\alpha\) such that every qc covering in \(\textit{LC}_\alpha\) is combinatorially equivalent to a covering this set. In this way we obtain a site \((\textit{LC}_\alpha, \text{Cov})\) which we will denote \(\textit{LC}_{qc}\).

There is a second topology on the site \(\textit{LC}_{qc}\) of Remark 09X2. Namely, given an object \(X\) we can consider all coverings \(\{X_i \to X\}\) of \(\textit{LC}_{qc}\) such that \(X_i \to X\) is an open immersion. We denote this site \(\textit{LC}_{Zar}\). The identity functor \(\textit{LC}_{Zar} \to \textit{LC}_{qc}\) is continuous and defines a morphism of sites \[\epsilon : \textit{LC}_{qc} \longrightarrow \textit{LC}_{Zar}\] See Section 0EWK. For a Hausdorff and locally quasi-compact topological space \(X\), more precisely for \(X \in \Ob(\textit{LC}_{qc})\), we denote the induced morphism \[\epsilon_X : \textit{LC}_{qc}/X \longrightarrow \textit{LC}_{Zar}/X\] (see Sites, Lemma 03CF). Let \(X_{Zar}\) be the site whose objects are opens of \(X\), see Sites, Example 00VJ. There is a morphism of sites \[\pi_X : \textit{LC}_{Zar}/X \longrightarrow X_{Zar}\] given by the continuous functor \(X_{Zar} \to \textit{LC}_{Zar}/X\), \(U \mapsto U\). Namely, \(X_{Zar}\) has fibre products and a final object and the functor above commutes with these and Sites, Proposition 00X6 applies. We often think of \(\pi\) as a morphism of topoi \[\pi_X : \Sh(\textit{LC}_{Zar}/X) \longrightarrow \Sh(X)\] using the equality \(\Sh(X_{Zar}) = \Sh(X)\).

Lemma

Let \(X\) be an object of \(\textit{LC}_{qc}\). Let \(\mathcal{F}\) be a sheaf on \(X\). The rule \[\textit{LC}_{qc}/X \longrightarrow \textit{Sets},\quad (f : Y \to X) \longmapsto \Gamma(Y, f^{-1}\mathcal{F})\] is a sheaf and a fortiori also a sheaf on \(\textit{LC}_{Zar}/X\). This sheaf is equal to \(\pi_X^{-1}\mathcal{F}\) on \(\textit{LC}_{Zar}/X\) and \(\epsilon_X^{-1}\pi_X^{-1}\mathcal{F}\) on \(\textit{LC}_{qc}/X\).

Proof

Denote \(\mathcal{G}\) the presheaf given by the formula in the lemma. Of course the pullback \(f^{-1}\) in the formula denotes usual pullback of sheaves on topological spaces. It is immediate from the definitions that \(\mathcal{G}\) is a sheaf for the Zar topology.

Let \(Y \to X\) be a morphism in \(\textit{LC}_{qc}\). Let \(\mathcal{V} = \{g_i : Y_i \to Y\}_{i \in I}\) be a qc covering. To prove \(\mathcal{G}\) is a sheaf for the qc topology it suffices to show that \(\mathcal{G}(Y) \to H^0(\mathcal{V}, \mathcal{G})\) is an isomorphism, see Sites, Section 00W1. We first point out that the map is injective as a qc covering is surjective and we can detect equality of sections at stalks (use Sheaves, Lemmas 0079 and 008G). Thus \(\mathcal{G}\) is a separated presheaf on \(\textit{LC}_{qc}\) hence it suffices to show that any element \((s_i) \in H^0(\mathcal{V}, \mathcal{G})\) maps to an element in the image of \(\mathcal{G}(Y)\) after replacing \(\mathcal{V}\) by a refinement (Sites, Theorem 00WB).

Identifying sheaves on \(Y_{i, Zar}\) and sheaves on \(Y_i\) we find that \(\mathcal{G}|_{Y_{i, Zar}}\) is the pullback of \(f^{-1}\mathcal{F}\) under the continuous map \(g_i : Y_i \to Y\). Thus we can choose an open covering \(Y_i = \bigcup V_{ij}\) such that for each \(j\) there is an open \(W_{ij} \subset Y\) and a section \(t_{ij} \in \mathcal{G}(W_{ij})\) such that \(V_{ij}\) maps into \(W_{ij}\) and such that \(s|_{V_{ij}}\) is the pullback of \(t_{ij}\). In other words, after refining the covering \(\{Y_i \to Y\}\) we may assume there are opens \(W_i \subset Y\) such that \(Y_i \to Y\) factors through \(W_i\) and sections \(t_i\) of \(\mathcal{G}\) over \(W_i\) which restrict to the given sections \(s_i\). Moreover, if \(y \in Y\) is in the image of both \(Y_i \to Y\) and \(Y_j \to Y\), then the images \(t_{i, y}\) and \(t_{j, y}\) in the stalk \(f^{-1}\mathcal{F}_y\) agree (because \(s_i\) and \(s_j\) agree over \(Y_i \times_Y Y_j\)). Thus for \(y \in Y\) there is a well defined element \(t_y\) of \(f^{-1}\mathcal{F}_y\) agreeing with \(t_{i, y}\) whenever \(y\) is in the image of \(Y_i \to Y\). We will show that the element \((t_y)\) comes from a global section of \(f^{-1}\mathcal{F}\) over \(Y\) which will finish the proof of the lemma.

It suffices to show that this is true locally on \(Y\), see Sheaves, Section 007X. Let \(y_0 \in Y\). Pick \(i_1, \ldots, i_n \in I\) and quasi-compact subsets \(E_j \subset Y_{i_j}\) such that \(\bigcup g_{i_j}(E_j)\) is a neighbourhood of \(y_0\). Let \(V \subset Y\) be an open neighbourhood of \(y_0\) contained in \(\bigcup g_{i_j}(E_j)\) and contained in \(W_{i_1} \cap \ldots \cap W_{i_n}\). Since \(t_{i_1, y_0} = \ldots = t_{i_n, y_0}\), after shrinking \(V\) we may assume the sections \(t_{i_j}|_V\), \(j = 1, \ldots, n\) of \(f^{-1}\mathcal{F}\) agree. As \(V \subset \bigcup g_{i_j}(E_j)\) we see that \((t_y)_{y \in V}\) comes from this section.

We still have to show that \(\mathcal{G}\) is equal to \(\epsilon_X^{-1}\pi_X^{-1}\mathcal{F}\) on \(\textit{LC}_{qc}\), resp. \(\pi_X^{-1}\mathcal{F}\) on \(\textit{LC}_{Zar}\). In both cases the pullback is defined by taking the presheaf \[(f : Y \to X) \longmapsto \colim_{f(Y) \subset U \subset X} \mathcal{F}(U)\] and then sheafifying. Sheafifying in the Zar topology exactly produces our sheaf \(\mathcal{G}\) and the fact that \(\mathcal{G}\) is a qc sheaf, shows that it works as well in the qc topology.

Let \(X \in \Ob(\textit{LC}_{Zar})\) and let \(\mathcal{H}\) be an abelian sheaf on \(\textit{LC}_{Zar}/X\). Then we will write \(H^n_{Zar}(U, \mathcal{H})\) for the cohomology of \(\mathcal{H}\) over an object \(U\) of \(\textit{LC}_{Zar}/X\).

Lemma

Let \(X\) be an object of \(\textit{LC}_{Zar}\). Then

1. for \(\mathcal{F} \in \textit{Ab}(X)\) we have \(H^n_{Zar}(X, \pi_X^{-1}\mathcal{F}) = H^n(X, \mathcal{F})\),

2. \(\pi_{X, *} : \textit{Ab}(\textit{LC}_{Zar}/X) \to \textit{Ab}(X)\) is exact,

3. the unit \(\text{id} \to \pi_{X, *} \circ \pi_X^{-1}\) of the adjunction is an isomorphism, and

4. for \(K \in D(X)\) the canonical map \(K \to R\pi_{X, *} \pi_X^{-1}K\) is an isomorphism.

Let \(f : X \to Y\) be a morphism of \(\textit{LC}_{Zar}\). Then

1. there is a commutative diagram \[\xymatrix{ \Sh(\textit{LC}_{Zar}/X) \ar[r]_{f_{Zar}} \ar[d]_{\pi_X} & \Sh(\textit{LC}_{Zar}/Y) \ar[d]^{\pi_Y} \\ \Sh(X_{Zar}) \ar[r]^f & \Sh(Y_{Zar}) }\] of topoi,

2. for \(L \in D^+(Y)\) we have \(H^n_{Zar}(X, \pi_Y^{-1}L) = H^n(X, f^{-1}L)\),

3. if \(f\) is proper, then we have

   1. \(\pi_Y^{-1} \circ f_* = f_{Zar, *} \circ \pi_X^{-1}\) as functors \(\Sh(X) \to \Sh(\textit{LC}_{Zar}/Y)\),

   2. \(\pi_Y^{-1} \circ Rf_* = Rf_{Zar, *} \circ \pi_X^{-1}\) as functors \(D^+(X) \to D^+(\textit{LC}_{Zar}/Y)\).

Proof

Proof of (1). The equality \(H^n_{Zar}(X, \pi_X^{-1}\mathcal{F}) = H^n(X, \mathcal{F})\) is a general fact coming from the trivial observation that coverings of \(X\) in \(\textit{LC}_{Zar}\) are the same thing as open coverings of \(X\). The reader who wishes to see a detailed proof should apply Lemma 03YU to the functor \(X_{Zar} \to \textit{LC}_{Zar}\).

Proof of (2). This is true because \(\pi_{X, *} = \tau_X^{-1}\) for some morphism of topoi \(\tau_X : \Sh(X_{Zar}) \to \Sh(\textit{LC}_{Zar})\) as follows from Sites, Lemma 00XU applied to the functor \(X_{Zar} \to \textit{LC}_{Zar}/X\) used to define \(\pi_X\).

Proof of (3). This is true because \(\tau_X^{-1} \circ \pi_X^{-1}\) is the identity functor by Sites, Lemma 00XU. Or you can deduce it from the explicit description of \(\pi_X^{-1}\) in Lemma 09X3.

Proof of (4). Apply (3) to an complex of abelian sheaves representing \(K\).

Proof of (5). The morphism of topoi \(f_{Zar}\) comes from an application of Sites, Lemma 03EH and in our case comes from the continuous functor \(Z/Y \mapsto Z \times_Y X/X\) by Sites, Lemma 09W9. The diagram commutes simply because the corresponding continuous functors compose correctly (see Sites, Lemma 03CB).

Proof of (6). We have \(H^n_{Zar}(X, \pi_Y^{-1}\mathcal{G}) = H^n_{Zar}(X, f_{Zar}^{-1}\pi_Y^{-1}\mathcal{G})\) for \(\mathcal{G}\) in \(\textit{Ab}(Y)\), see Lemma 03F3. This is equal to \(H^n_{Zar}(X, \pi_X^{-1}f^{-1}\mathcal{G})\) by the commutativity of the diagram in (5). Hence we conclude by (1) in the case \(L\) consists of a single sheaf in degree \(0\). The general case follows by representing \(L\) by a bounded below complex of abelian sheaves.

Proof of (7a). Let \(\mathcal{F}\) be a sheaf on \(X\). Let \(g : Z \to Y\) be an object of \(\textit{LC}_{Zar}/Y\). Consider the fibre product \[\xymatrix{ Z' \ar[r]_{f'} \ar[d]_{g'} & Z \ar[d]^g \\ X \ar[r]^f & Y }\] Then we have \[(f_{Zar, *}\pi_X^{-1}\mathcal{F})(Z/Y) = (\pi_X^{-1}\mathcal{F})(Z'/X) = \Gamma(Z', (g')^{-1}\mathcal{F}) = \Gamma(Z, f'_*(g')^{-1}\mathcal{F})\] the second equality by Lemma 09X3. On the other hand \[(\pi_Y^{-1}f_*\mathcal{F})(Z/Y) = \Gamma(Z, g^{-1}f_*\mathcal{F})\] again by Lemma 09X3. Hence by proper base change for sheaves of sets (Cohomology, Lemma 0D90) we conclude the two sets are canonically isomorphic. The isomorphism is compatible with restriction mappings and defines an isomorphism \(\pi_Y^{-1}f_*\mathcal{F} = f_{Zar, *}\pi_X^{-1}\mathcal{F}\). Thus an isomorphism of functors \(\pi_Y^{-1} \circ f_* = f_{Zar, *} \circ \pi_X^{-1}\).

Proof of (7b). Let \(K \in D^+(X)\). By Lemma 0D6I the \(n\)th cohomology sheaf of \(Rf_{Zar, *}\pi_X^{-1}K\) is the sheaf associated to the presheaf \[(g : Z \to Y) \longmapsto H^n_{Zar}(Z', \pi_X^{-1}K)\] with notation as above. Observe that \[\begin{align*} H^n_{Zar}(Z', \pi_X^{-1}K) & = H^n(Z', (g')^{-1}K) \\ & = H^n(Z, Rf'_*(g')^{-1}K) \\ & = H^n(Z, g^{-1}Rf_*K) \\ & = H^n_{Zar}(Z, \pi_Y^{-1}Rf_*K) \end{align*}\] The first equality is (6) applied to \(K\) and \(g' : Z' \to X\). The second equality is Leray for \(f' : Z' \to Z\) (Cohomology, Lemma 01EZ). The third equality is the proper base change theorem (Cohomology, Theorem 09V6). The fourth equality is (6) applied to \(g : Z \to Y\) and \(Rf_*K\). Thus \(Rf_{Zar, *}\pi_X^{-1}K\) and \(\pi_Y^{-1}Rf_*K\) have the same cohomology sheaves. We omit the verification that the canonical base change map \(\pi_Y^{-1}Rf_*K \to Rf_{Zar, *}\pi_X^{-1}K\) induces this isomorphism.

In the situation of Lemma 09X3 the composition of \(\epsilon\) and \(\pi\) and the equality \(\Sh(X) = \Sh(X_{Zar})\) determine a morphism of topoi \[a_X : \Sh(\textit{LC}_{qc}/X) \longrightarrow \Sh(X)\]

Lemma

Let \(f : X \to Y\) be a morphism of \(\textit{LC}_{qc}\). Then there are commutative diagrams of topoi \[\vcenter{ \xymatrix{ \Sh(\textit{LC}_{qc}/X) \ar[r]_{f_{qc}} \ar[d]_{\epsilon_X} & \Sh(\textit{LC}_{qc}/Y) \ar[d]^{\epsilon_Y} \\ \Sh(\textit{LC}_{Zar}/X) \ar[r]^{f_{Zar}} & \Sh(\textit{LC}_{Zar}/Y) } } \quad\text{and}\quad \vcenter{ \xymatrix{ \Sh(\textit{LC}_{qc}/X) \ar[r]_{f_{qc}} \ar[d]_{a_X} & \Sh(\textit{LC}_{qc}/Y) \ar[d]^{a_Y} \\ \Sh(X) \ar[r]^f & \Sh(Y) } }\] with \(a_X = \pi_X \circ \epsilon_X\), \(a_Y = \pi_Y \circ \epsilon_Y\). If \(f\) is proper, then \(a_Y^{-1} \circ f_* = f_{qc, *} \circ a_X^{-1}\).

Proof

The morphism of topoi \(f_{qc}\) is the one from Sites, Lemma 03EH which in our case comes from the continuous functor \(Z/Y \mapsto Z \times_Y X/X\), see Sites, Lemma 09W9. The diagram on the left commutes because the corresponding continuous functors compose correctly (see Sites, Lemma 03CB). The diagram on the right commutes because the one on the left does and because of part (5) of Lemma 0DCU.

Proof of the final assertion. The reader may repeat the proof of part (7a) of Lemma 0DCU; we will instead deduce this from it. As \(\epsilon_{Y, *}\) is the identity functor on underlying presheaves, it reflects isomorphisms. The description in Lemma 09X3 shows that \(\epsilon_{Y, *} \circ a_Y^{-1} = \pi_Y^{-1}\) and similarly for \(X\). To show that the canonical map \(a_Y^{-1}f_*\mathcal{F} \to f_{qc, *}a_X^{-1}\mathcal{F}\) is an isomorphism, it suffices to show that \[\pi_Y^{-1}f_*\mathcal{F} = \epsilon_{Y, *}a_Y^{-1}f_*\mathcal{F} \to \epsilon_{Y, *}f_{qc, *}a_X^{-1}\mathcal{F} = f_{Zar, *}\epsilon_{X, *}a_X^{-1}\mathcal{F} = f_{Zar, *}\pi_X^{-1}\mathcal{F}\] is an isomorphism. This is part (7a) of Lemma 0DCU.

Lemma

Consider the comparison morphism \(\epsilon : \textit{LC}_{qc} \to \textit{LC}_{Zar}\). Let \(\mathcal{P}\) denote the class of proper maps of topological spaces. For \(X\) in \(\textit{LC}_{Zar}\) denote \(\mathcal{A}'_X \subset \textit{Ab}(\textit{LC}_{Zar}/X)\) the full subcategory consisting of sheaves of the form \(\pi_X^{-1}\mathcal{F}\) with \(\mathcal{F}\) in \(\textit{Ab}(X)\). Then (0EZ4), (0EZ5), (0EZ6), (0EZ7), and (0EZ8) of Situation 0EZ3 hold.

Proof

We first show that \(\mathcal{A}'_X \subset \textit{Ab}(\textit{LC}_{Zar}/X)\) is a weak Serre subcategory by checking conditions (1), (2), (3), and (4) of Homology, Lemma 0754. Parts (1), (2), (3) are immediate as \(\pi_X^{-1}\) is exact and fully faithful by Lemma 0DCU part (3). If \(0 \to \pi_X^{-1}\mathcal{F} \to \mathcal{G} \to \pi_X^{-1}\mathcal{F}' \to 0\) is a short exact sequence in \(\textit{Ab}(\textit{LC}_{Zar}/X)\) then \(0 \to \mathcal{F} \to \pi_{X, *}\mathcal{G} \to \mathcal{F}' \to 0\) is exact by Lemma 0DCU part (2). Hence \(\mathcal{G} = \pi_X^{-1}\pi_{X, *}\mathcal{G}\) is in \(\mathcal{A}'_X\) which checks the final condition.

Property (0EZ4) holds by Lemma 09WZ and the fact that the base change of a proper map is a proper map (see Topology, Theorem 005R and Lemma 0CY4).

Property (0EZ5) follows from the commutative diagram (5) in Lemma 0DCU.

Property (0EZ6) is Lemma 09X3.

Property (0EZ7) is Lemma 0DCU part (7)(b).

Proof of (0EZ8). Suppose given a qc covering \(\{U_i \to U\}\). For \(u \in U\) pick \(i_1, \ldots, i_m \in I\) and quasi-compact subsets \(E_j \subset U_{i_j}\) such that \(\bigcup f_{i_j}(E_j)\) is a neighbourhood of \(u\). Observe that \(Y = \coprod_{j = 1, \ldots, m} E_j \to U\) is proper as a continuous map between Hausdorff quasi-compact spaces (Topology, Lemma 08YD). Choose an open neighbourhood \(u \in V\) contained in \(\bigcup f_{i_j}(E_j)\). Then \(Y \times_U V \to V\) is a surjective proper morphism and hence a \(qc\) covering by Lemma 09X5. Since we can do this for every \(u \in U\) we see that (0EZ8) holds.

Lemma

With notation as above.

1. For \(X \in \Ob(\textit{LC}_{qc})\) and an abelian sheaf \(\mathcal{F}\) on \(X\) we have \(\epsilon_{X, *}a_X^{-1}\mathcal{F} = \pi_X^{-1}\mathcal{F}\) and \(R^i\epsilon_{X, *}(a_X^{-1}\mathcal{F}) = 0\) for \(i > 0\).

2. For a proper morphism \(f : X \to Y\) in \(\textit{LC}_{qc}\) and abelian sheaf \(\mathcal{F}\) on \(X\) we have \(a_Y^{-1}(R^if_*\mathcal{F}) = R^if_{qc, *}(a_X^{-1}\mathcal{F})\) for all \(i\).

3. For \(X \in \Ob(\textit{LC}_{qc})\) and \(K\) in \(D^+(X)\) the map \(\pi_X^{-1}K \to R\epsilon_{X, *}(a_X^{-1}K)\) is an isomorphism.

4. For a proper morphism \(f : X \to Y\) in \(\textit{LC}_{qc}\) and \(K\) in \(D^+(X)\) we have \(a_Y^{-1}(Rf_*K) = Rf_{qc, *}(a_X^{-1}K)\).

Proof

By Lemma 0EZI the lemmas in Section 0EZ1 all apply to our current setting. To translate the results observe that the category \(\mathcal{A}_X\) of Lemma 0EZ9 is the essential image of \(a_X^{-1} : \textit{Ab}(X) \to \textit{Ab}(\textit{LC}_{qc}/X)\).

Part (1) is equivalent to \((V_n)\) for all \(n\) which holds by Lemma 0EZF.

Part (2) follows by applying \(\epsilon_Y^{-1}\) to the conclusion of Lemma 0EZA.

Part (3) follows from Lemma 0EZF part (1) because \(\pi_X^{-1}K\) is in \(D^+_{\mathcal{A}'_X}(\textit{LC}_{Zar}/X)\) and \(a_X^{-1} = \epsilon_X^{-1} \circ \pi_X^{-1}\).

Part (4) follows from Lemma 0EZF part (2) for the same reason.

Lemma

Let \(X\) be an object of \(\textit{LC}_{qc}\). For \(K \in D^+(X)\) the map \[K \longrightarrow Ra_{X, *}a_X^{-1}K\] is an isomorphism with \(a_X : \Sh(\textit{LC}_{qc}/X) \to \Sh(X)\) as above.

Proof

We first reduce the statement to the case where \(K\) is given by a single abelian sheaf. Namely, represent \(K\) by a bounded below complex \(\mathcal{F}^\bullet\). By the case of a sheaf we see that \(\mathcal{F}^n = a_{X, *} a_X^{-1} \mathcal{F}^n\) and that the sheaves \(R^qa_{X, *}a_X^{-1}\mathcal{F}^n\) are zero for \(q > 0\). By Leray’s acyclicity lemma (Derived Categories, Lemma 015E) applied to \(a_X^{-1}\mathcal{F}^\bullet\) and the functor \(a_{X, *}\) we conclude. From now on assume \(K = \mathcal{F}\).

By Lemma 09X3 we have \(a_{X, *}a_X^{-1}\mathcal{F} = \mathcal{F}\). Thus it suffices to show that \(R^qa_{X, *}a_X^{-1}\mathcal{F} = 0\) for \(q > 0\). For this we can use \(a_X = \pi_X \circ \epsilon_X\) and the Leray spectral sequence Lemma 0734. By Lemma 0DCY we have \(R^i\epsilon_{X, *}(a_X^{-1}\mathcal{F}) = 0\) for \(i > 0\) and \(\epsilon_{X, *}a_X^{-1}\mathcal{F} = \pi_X^{-1}\mathcal{F}\). By Lemma 0DCU we have \(R^j\pi_{X, *}(\pi_X^{-1}\mathcal{F}) = 0\) for \(j > 0\). This concludes the proof.

Lemma

With \(X \in \Ob(\textit{LC}_{qc})\) and \(a_X : \Sh(\textit{LC}_{qc}/X) \to \Sh(X)\) as above:

1. for an abelian sheaf \(\mathcal{F}\) on \(X\) we have \(H^n(X, \mathcal{F}) = H^n_{qc}(X, a_X^{-1}\mathcal{F})\),

2. for \(K \in D^+(X)\) we have \(H^n(X, K) = H^n_{qc}(X, a_X^{-1}K)\).

For example, if \(A\) is an abelian group, then we have \(H^n(X, \underline{A}) = H^n_{qc}(X, \underline{A})\).

Proof

This follows from Lemma 0D91 by Remark 08J6.

Spectral sequences for Ext

In this section we collect various spectral sequences that come up when considering the Ext functors. For any pair of complexes \(\mathcal{G}^\bullet, \mathcal{F}^\bullet\) of complexes of modules on a ringed site \((\mathcal{C}, \mathcal{O})\) we denote \[\Ext^n_\mathcal{O}(\mathcal{G}^\bullet, \mathcal{F}^\bullet) = \Hom_{D(\mathcal{O})}(\mathcal{G}^\bullet, \mathcal{F}^\bullet[n])\] according to our general conventions in Derived Categories, Section 06XP.

Example

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{K}^\bullet\) be a bounded above complex of \(\mathcal{O}\)-modules. Let \(\mathcal{F}\) be an \(\mathcal{O}\)-module. Then there is a spectral sequence with \(E_2\)-page \[E_2^{i, j} = \Ext_\mathcal{O}^i(H^{-j}(\mathcal{K}^\bullet), \mathcal{F}) \Rightarrow \Ext_\mathcal{O}^{i + j}(\mathcal{K}^\bullet, \mathcal{F})\] and another spectral sequence with \(E_1\)-page \[E_1^{i, j} = \Ext_\mathcal{O}^j(\mathcal{K}^{-i}, \mathcal{F}) \Rightarrow \Ext_\mathcal{O}^{i + j}(\mathcal{K}^\bullet, \mathcal{F}).\] To construct these spectral sequences choose an injective resolution \(\mathcal{F} \to \mathcal{I}^\bullet\) and consider the two spectral sequences coming from the double complex \(\Hom_\mathcal{O}(\mathcal{K}^\bullet, \mathcal{I}^\bullet)\), see Homology, Section 012X.

Cup product

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(K, M\) be objects of \(D(\mathcal{O})\). Set \(A = \Gamma(\mathcal{C}, \mathcal{O})\). The (global) cup product in this setting is a map \[R\Gamma(\mathcal{C}, K) \otimes_A^\mathbf{L} R\Gamma(\mathcal{C}, M) \longrightarrow R\Gamma(\mathcal{C}, K \otimes_\mathcal{O}^\mathbf{L} M)\] in \(D(A)\). We define it as the relative cup product for the morphism of ringed topoi \((\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(pt), A)\) as in Remark 0B6C.

Let us formulate and prove a natural compatibility of the relative cup product. Namely, suppose that we have a morphism \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) of ringed topoi. Let \(\mathcal{K}^\bullet\) and \(\mathcal{M}^\bullet\) be complexes of \(\mathcal{O}_\mathcal{C}\)-modules. There is a naive cup product \[\text{Tot}( f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}} f_*\mathcal{M}^\bullet) \longrightarrow f_*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} \mathcal{M}^\bullet)\] We claim that this is related to the relative cup product.

Lemma

In the situation above the following diagram commutes \[\xymatrix{ f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} f_*\mathcal{M}^\bullet \ar[r] \ar[d] & Rf_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} Rf_*\mathcal{M}^\bullet \ar[d]^{\text{Remark \href{sites-cohomology.html#sites-cohomology-remark-cup-product}{0B6C}}} \\ \text{Tot}( f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}} f_*\mathcal{M}^\bullet) \ar[d]_{\text{naive cup product}} & Rf_*(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} \mathcal{M}^\bullet) \ar[d] \\ f_*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} \mathcal{M}^\bullet) \ar[r] & Rf_*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} \mathcal{M}^\bullet) }\]

Proof

By the construction in Remark 0B6C we see that going around the diagram clockwise the map \[f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} f_*\mathcal{M}^\bullet \longrightarrow Rf_*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} \mathcal{M}^\bullet)\] is adjoint to the map \[\begin{align*} Lf^*(f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} f_*\mathcal{M}^\bullet) & = Lf^*f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} Lf^*f_*\mathcal{M}^\bullet \\ & \to Lf^*Rf_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} Lf^*Rf_*\mathcal{M}^\bullet \\ & \to \mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} \mathcal{M}^\bullet \\ & \to \text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} \mathcal{M}^\bullet) \end{align*}\] By Lemma 0FPI this is also equal to \[\begin{align*} Lf^*(f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} f_*\mathcal{M}^\bullet) & = Lf^*f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} Lf^*f_*\mathcal{M}^\bullet \\ & \to f^*f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} f^*f_*\mathcal{M}^\bullet \\ & \to \mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} \mathcal{M}^\bullet \\ & \to \text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} \mathcal{M}^\bullet) \end{align*}\] Going around anti-clockwise we obtain the map adjoint to the map \[\begin{align*} Lf^*(f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} f_*\mathcal{M}^\bullet) & \to Lf^*\text{Tot}( f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}} f_*\mathcal{M}^\bullet) \\ & \to Lf^*f_*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} \mathcal{M}^\bullet) \\ & \to Lf^*Rf_*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} \mathcal{M}^\bullet) \\ & \to \text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} \mathcal{M}^\bullet) \end{align*}\] By Lemma 0FPI this is also equal to \[\begin{align*} Lf^*(f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} f_*\mathcal{M}^\bullet) & \to Lf^*\text{Tot}( f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}} f_*\mathcal{M}^\bullet) \\ & \to Lf^*f_*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} \mathcal{M}^\bullet) \\ & \to f^*f_*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} \mathcal{M}^\bullet) \\ & \to \text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} \mathcal{M}^\bullet) \end{align*}\] Now the proof is finished by a contemplation of the diagram \[\xymatrix{ Lf^*(f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} f_*\mathcal{M}^\bullet) \ar[d] \ar[rr] & & Lf^*f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} Lf^*f_*\mathcal{M}^\bullet \ar[d] \\ Lf^*\text{Tot}( f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}} f_*\mathcal{M}^\bullet) \ar[d]_{naive} \ar[r] & f^*\text{Tot}( f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}} f_*\mathcal{M}^\bullet) \ar[ldd]^{naive} \ar[dd] & f^*f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} f^*f_*\mathcal{M}^\bullet \ar[dd] \ar[ldd] \\ Lf^*f_*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} \mathcal{M}^\bullet) \ar[d] \\ f^*f_*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} \mathcal{M}^\bullet) \ar[rd] & \text{Tot}(f^*f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} f^*f_*\mathcal{M}^\bullet) \ar[d] & \mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} \mathcal{M}^\bullet \ar[ld] \\ & \text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} \mathcal{M}^\bullet) }\] All of the polygons in this diagram commute. The top one commutes by Lemma 0FPH. The square with the two naive cup products commutes because \(Lf^* \to f^*\) is functorial in the complex of modules. Similarly with the square involving the two maps \(\mathcal{A}^\bullet \otimes^\mathbf{L} \mathcal{B}^\bullet \to \text{Tot}(\mathcal{A}^\bullet \otimes \mathcal{B}^\bullet)\). Finally, the commutativity of the remaining square is true on the level of complexes and may be viewed as the definition of the naive cup product (by the adjointness of \(f^*\) and \(f_*\)). The proof is finished because going around the diagram on the outside are the two maps given above.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')\) be a morphism of ringed topoi. The relative cup product of Remark 0B6C is associative in the sense that the diagram \[\xymatrix{ Rf_*K \otimes_{\mathcal{O}'}^\mathbf{L} Rf_*L \otimes_{\mathcal{O}'}^\mathbf{L} Rf_*M \ar[r] \ar[d] & Rf_*(K \otimes_\mathcal{O}^\mathbf{L} L) \otimes_{\mathcal{O}'}^\mathbf{L} Rf_*M \ar[d] \\ Rf_*K \otimes_{\mathcal{O}'}^\mathbf{L} Rf_*(L \otimes_\mathcal{O}^\mathbf{L} M) \ar[r] & Rf_*(K \otimes_\mathcal{O}^\mathbf{L} L \otimes_\mathcal{O}^\mathbf{L} M) }\] is commutative in \(D(\mathcal{O}')\) for all \(K, L, M\) in \(D(\mathcal{O})\).

Proof

Going around either side we obtain the map adjoint to the obvious map \[\begin{align*} Lf^*(Rf_*K \otimes_{\mathcal{O}'}^\mathbf{L} Rf_*L \otimes_{\mathcal{O}'}^\mathbf{L} Rf_*M) & = Lf^*(Rf_*K) \otimes_\mathcal{O}^\mathbf{L} Lf^*(Rf_*L) \otimes_\mathcal{O}^\mathbf{L} Lf^*(Rf_*M) \\ & \to K \otimes_\mathcal{O}^\mathbf{L} L \otimes_\mathcal{O}^\mathbf{L} M \end{align*}\] in \(D(\mathcal{O})\).

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')\) be a morphism of ringed topoi. The relative cup product of Remark 0B6C is commutative in the sense that the diagram \[\xymatrix{ Rf_*K \otimes_{\mathcal{O}'}^\mathbf{L} Rf_*L \ar[r] \ar[d]_\psi & Rf_*(K \otimes_\mathcal{O}^\mathbf{L} L) \ar[d]^{Rf_*\psi} \\ Rf_*L \otimes_{\mathcal{O}'}^\mathbf{L} Rf_*K \ar[r] & Rf_*(L \otimes_\mathcal{O}^\mathbf{L} K) }\] is commutative in \(D(\mathcal{O}')\) for all \(K, L\) in \(D(\mathcal{O})\). Here \(\psi\) is the commutativity constraint on the derived category (Lemma 0FPT).

Proof

Omitted.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')\) and \(f' : (\Sh(\mathcal{C}'), \mathcal{O}') \to (\Sh(\mathcal{C}''), \mathcal{O}'')\) be morphisms of ringed topoi. The relative cup product of Remark 0B6C is compatible with compositions in the sense that the diagram \[\xymatrix{ R(f' \circ f)_*K \otimes_{\mathcal{O}''}^\mathbf{L} R(f' \circ f)_*L \ar@{=}[rr] \ar[d] & & Rf'_*Rf_*K \otimes_{\mathcal{O}''}^\mathbf{L} Rf'_*Rf_*L \ar[d] \\ R(f' \circ f)_*(K \otimes_\mathcal{O}^\mathbf{L} L) \ar@{=}[r] & Rf'_*Rf_*(K \otimes_\mathcal{O}^\mathbf{L} L) & Rf'_*(Rf_*K \otimes_{\mathcal{O}'}^\mathbf{L} Rf_*L) \ar[l] }\] is commutative in \(D(\mathcal{O}'')\) for all \(K, L\) in \(D(\mathcal{O})\).

Proof

This is true because going around the diagram either way we obtain the map adjoint to the map \[\begin{align*} & L(f' \circ f)^*\left(R(f' \circ f)_*K \otimes_{\mathcal{O}''}^\mathbf{L} R(f' \circ f)_*L\right) \\ & = L(f' \circ f)^*R(f' \circ f)_*K \otimes_\mathcal{O}^\mathbf{L} L(f' \circ f)^*R(f' \circ f)_*L \\ & \to K \otimes_\mathcal{O}^\mathbf{L} L \end{align*}\] in \(D(\mathcal{O})\). To see this one uses that the composition of the counits like so \[L(f' \circ f)^*R(f' \circ f)_* = Lf^* L(f')^* Rf'_* Rf_* \to Lf^* Rf_* \to \text{id}\] is the counit for \(L(f' \circ f)^*\) and \(R(f' \circ f)_*\). See Categories, Lemma 0DV0.

Lemma

Consider a commutative square \[\xymatrix{ (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_{g'} \ar[d]_{f'} & (\Sh(\mathcal{C}), \mathcal{O}_{\mathcal{C}}) \ar[d]^f \\ (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^g & (\Sh(\mathcal{D}), \mathcal{O}_{\mathcal{D}}) }\] of ringed topoi. Let \(K, L\) in \(D(\mathcal{O}_{\mathcal{C}})\). The relative cup product is compatible with the square in the sense that the diagram \[\xymatrix{ Lg^*(Rf_*K \otimes_{\mathcal{O}_{\mathcal{D}}}^\mathbf{L} Rf_*L) \ar[r] \ar@{=}[d] & Lg^*(Rf_*(K \otimes_{\mathcal{O}_{\mathcal{C}}}^\mathbf{L} L)) \ar[d] \\ Lg^*Rf_*K \otimes_{\mathcal{O}_{\mathcal{D}'}}^\mathbf{L} Lg^*Rf_*L \ar[d] & R(f')_*L(g')^*(K \otimes_{\mathcal{O}_{\mathcal{C}}}^\mathbf{L} L) \ar@{=}[d] \\ R(f')_*L(g')^*K \otimes_{\mathcal{O}_{\mathcal{D}'}}^\mathbf{L} R(f')_*L(g')^*L \ar[r] & R(f')_*(L(g')^*K \otimes_{\mathcal{O}_{\mathcal{C}'}}^\mathbf{L} L(g')^*L) }\] is commutative in \(D(\mathcal{O}_{\mathcal{D}'})\). The horizontal arrows are given by the relative cup product (Remark 0B6C) and the vertical arrows are given by the base change map (Remark 07A7) and Lemma 07A4.

Proof

Omitted.

Hom complexes

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{L}^\bullet\) and \(\mathcal{M}^\bullet\) be two complexes of \(\mathcal{O}\)-modules. We construct a complex of \(\mathcal{O}\)-modules \(\SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{M}^\bullet)\). Namely, for each \(n\) we set \[\SheafHom^n(\mathcal{L}^\bullet, \mathcal{M}^\bullet) = \prod\nolimits_{n = p + q} \SheafHom_\mathcal{O}(\mathcal{L}^{-q}, \mathcal{M}^p)\] It is a good idea to think of \(\SheafHom^n\) as the sheaf of \(\mathcal{O}\)-modules of all \(\mathcal{O}\)-linear maps from \(\mathcal{L}^\bullet\) to \(\mathcal{M}^\bullet\) (viewed as graded \(\mathcal{O}\)-modules) which are homogeneous of degree \(n\). In this terminology, we define the differential by the rule \[\text{d}(f) = \text{d}_\mathcal{M} \circ f - (-1)^n f \circ \text{d}_\mathcal{L}\] for \(f \in \SheafHom^n_\mathcal{O}(\mathcal{L}^\bullet, \mathcal{M}^\bullet)\). We omit the verification that \(\text{d}^2 = 0\). This construction is a special case of Differential Graded Algebra, Example 09L9. It follows immediately from the construction that we have [0A8Y]\[\begin{equation} H^n(\Gamma(U, \SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{M}^\bullet))) = \Hom_{K(\mathcal{O}_U)}(\mathcal{L}^\bullet|_U, \mathcal{M}^\bullet[n]|_U) \end{equation}\] for all \(n \in \mathbf{Z}\) and every \(U \in \Ob(\mathcal{C})\). Similarly, we have [0A8Z]\[\begin{equation} H^n(\Gamma(\mathcal{C}, \SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{M}^\bullet))) = \Hom_{K(\mathcal{O})}(\mathcal{L}^\bullet, \mathcal{M}^\bullet[n]) \end{equation}\] for the complex of global sections.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Given complexes \(\mathcal{K}^\bullet, \mathcal{L}^\bullet, \mathcal{M}^\bullet\) of \(\mathcal{O}\)-modules there is an isomorphism \[\SheafHom^\bullet(\mathcal{K}^\bullet, \SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{M}^\bullet)) = \SheafHom^\bullet(\text{Tot}(\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{L}^\bullet), \mathcal{M}^\bullet)\] of complexes of \(\mathcal{O}\)-modules functorial in \(\mathcal{K}^\bullet, \mathcal{L}^\bullet, \mathcal{M}^\bullet\).

Proof

Omitted. Hint: This is proved in exactly the same way as More on Algebra, Lemma 0A5Y.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Given complexes \(\mathcal{K}^\bullet, \mathcal{L}^\bullet, \mathcal{M}^\bullet\) of \(\mathcal{O}\)-modules there is a canonical morphism \[\text{Tot}\left( \SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{M}^\bullet) \otimes_\mathcal{O} \SheafHom^\bullet(\mathcal{K}^\bullet, \mathcal{L}^\bullet) \right) \longrightarrow \SheafHom^\bullet(\mathcal{K}^\bullet, \mathcal{M}^\bullet)\] of complexes of \(\mathcal{O}\)-modules.

Proof

Omitted. Hint: This is proved in exactly the same way as More on Algebra, Lemma 0A8I.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Given complexes \(\mathcal{K}^\bullet, \mathcal{L}^\bullet, \mathcal{M}^\bullet\) of \(\mathcal{O}\)-modules there is a canonical morphism \[\text{Tot}\left( \mathcal{K}^\bullet \otimes_\mathcal{O} \SheafHom^\bullet(\mathcal{M}^\bullet, \mathcal{L}^\bullet) \right) \longrightarrow \SheafHom^\bullet(\mathcal{M}^\bullet, \text{Tot}(\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{L}^\bullet))\] of complexes of \(\mathcal{O}\)-modules functorial in all three complexes.

Proof

Omitted. Hint: This is proved in exactly the same way as More on Algebra, Lemma 0BYM.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Given complexes \(\mathcal{K}^\bullet, \mathcal{L}^\bullet\) of \(\mathcal{O}\)-modules there is a canonical morphism \[\mathcal{K}^\bullet \longrightarrow \SheafHom^\bullet(\mathcal{L}^\bullet, \text{Tot}(\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{L}^\bullet))\] of complexes of \(\mathcal{O}\)-modules functorial in both complexes.

Proof

Omitted. Hint: This is proved in exactly the same way as More on Algebra, Lemma 0A62.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Given complexes \(\mathcal{K}^\bullet, \mathcal{L}^\bullet, \mathcal{M}^\bullet\) of \(\mathcal{O}\)-modules there is a canonical morphism \[\text{Tot}(\SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{M}^\bullet) \otimes_\mathcal{O} \mathcal{K}^\bullet) \longrightarrow \SheafHom^\bullet(\SheafHom^\bullet(\mathcal{K}^\bullet, \mathcal{L}^\bullet), \mathcal{M}^\bullet)\] of complexes of \(\mathcal{O}\)-modules functorial in all three complexes.

Proof

Omitted. Hint: This is proved in exactly the same way as More on Algebra, Lemma 0A60.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(L\) and \(M\) be objects of \(D(\mathcal{O})\). Let \(\mathcal{I}^\bullet\) be a K-injective complex of \(\mathcal{O}\)-modules representing \(M\). Let \(\mathcal{L}^\bullet\) be a complex of \(\mathcal{O}\)-modules representing \(L\). Then \[H^0(\Gamma(U, \SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{I}^\bullet))) = \Hom_{D(\mathcal{O}_U)}(L|_U, M|_U)\] for all \(U \in \Ob(\mathcal{C})\). Similarly, \(H^0(\Gamma(\mathcal{C}, \SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{I}^\bullet))) = \Hom_{D(\mathcal{O})}(L, M)\).

Proof

We have \[\begin{align*} H^0(\Gamma(U, \SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{I}^\bullet))) & = \Hom_{K(\mathcal{O}_U)}(\mathcal{L}^\bullet|_U, \mathcal{I}^\bullet|_U) \\ & = \Hom_{D(\mathcal{O}_U)}(L|_U, M|_U) \end{align*}\] The first equality is (0A8Y). The second equality is true because \(\mathcal{I}^\bullet|_U\) is K-injective by Lemma 08FI. The proof of the last equation is similar except that it uses (0A8Z).

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \((\mathcal{I}')^\bullet \to \mathcal{I}^\bullet\) be a quasi-isomorphism of K-injective complexes of \(\mathcal{O}\)-modules. Let \((\mathcal{L}')^\bullet \to \mathcal{L}^\bullet\) be a quasi-isomorphism of complexes of \(\mathcal{O}\)-modules. Then \[\SheafHom^\bullet(\mathcal{L}^\bullet, (\mathcal{I}')^\bullet) \longrightarrow \SheafHom^\bullet((\mathcal{L}')^\bullet, \mathcal{I}^\bullet)\] is a quasi-isomorphism.

Proof

Let \(M\) be the object of \(D(\mathcal{O})\) represented by \(\mathcal{I}^\bullet\) and \((\mathcal{I}')^\bullet\). Let \(L\) be the object of \(D(\mathcal{O})\) represented by \(\mathcal{L}^\bullet\) and \((\mathcal{L}')^\bullet\). By Lemma 0A94 we see that the sheaves \[H^0(\SheafHom^\bullet(\mathcal{L}^\bullet, (\mathcal{I}')^\bullet)) \quad\text{and}\quad H^0(\SheafHom^\bullet((\mathcal{L}')^\bullet, \mathcal{I}^\bullet))\] are both equal to the sheaf associated to the presheaf \[U \longmapsto \Hom_{D(\mathcal{O}_U)}(L|_U, M|_U)\] Thus the map is a quasi-isomorphism.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{I}^\bullet\) be a K-injective complex of \(\mathcal{O}\)-modules. Let \(\mathcal{L}^\bullet\) be a K-flat complex of \(\mathcal{O}\)-modules. Then \(\SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{I}^\bullet)\) is a K-injective complex of \(\mathcal{O}\)-modules.

Proof

Namely, if \(\mathcal{K}^\bullet\) is an acyclic complex of \(\mathcal{O}\)-modules, then \[\begin{align*} \Hom_{K(\mathcal{O})}(\mathcal{K}^\bullet, \SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{I}^\bullet)) & = H^0(\Gamma(\mathcal{C}, \SheafHom^\bullet(\mathcal{K}^\bullet, \SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{I}^\bullet)))) \\ & = H^0(\Gamma(\mathcal{C}, \SheafHom^\bullet(\text{Tot}( \mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{L}^\bullet), \mathcal{I}^\bullet))) \\ & = \Hom_{K(\mathcal{O})}( \text{Tot}(\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{L}^\bullet), \mathcal{I}^\bullet) \\ & = 0 \end{align*}\] The first equality by (0A8Z). The second equality by Lemma 0A90. The third equality by (0A8Z). The final equality because \(\text{Tot}(\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{L}^\bullet)\) is acyclic because \(\mathcal{L}^\bullet\) is K-flat (Definition 06YN) and because \(\mathcal{I}^\bullet\) is K-injective.

Internal hom in the derived category

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(L, M\) be objects of \(D(\mathcal{O})\). We would like to construct an object \(R\SheafHom(L, M)\) of \(D(\mathcal{O})\) such that for every third object \(K\) of \(D(\mathcal{O})\) there exists a canonical bijection [08J8]\[\begin{equation} \Hom_{D(\mathcal{O})}(K, R\SheafHom(L, M)) = \Hom_{D(\mathcal{O})}(K \otimes_\mathcal{O}^\mathbf{L} L, M) \end{equation}\] Observe that this formula defines \(R\SheafHom(L, M)\) up to unique isomorphism by the Yoneda lemma (Categories, Lemma 001P).

To construct such an object, choose a K-injective complex of \(\mathcal{O}\)-modules \(\mathcal{I}^\bullet\) representing \(M\) and any complex of \(\mathcal{O}\)-modules \(\mathcal{L}^\bullet\) representing \(L\). Then we set \[R\SheafHom(L, M) = \SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{I}^\bullet)\] where the right hand side is the complex of \(\mathcal{O}\)-modules constructed in Section 0A8X. This is well defined by Lemma 0A95. We get a functor \[D(\mathcal{O})^{opp} \times D(\mathcal{O}) \longrightarrow D(\mathcal{O}), \quad (K, L) \longmapsto R\SheafHom(K, L)\] As a prelude to proving (08J8) we compute the cohomology groups of \(R\SheafHom(K, L)\).

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(L, M\) be objects of \(D(\mathcal{O})\). For every object \(U\) of \(\mathcal{C}\) we have \[H^0(U, R\SheafHom(L, M)) = \Hom_{D(\mathcal{O}_U)}(L|_U, M|_U)\] and we have \(H^0(\mathcal{C}, R\SheafHom(L, M)) = \Hom_{D(\mathcal{O})}(L, M)\).

Proof

Choose a K-injective complex \(\mathcal{I}^\bullet\) of \(\mathcal{O}\)-modules representing \(M\) and a K-flat complex \(\mathcal{L}^\bullet\) representing \(L\). Then \(\SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{I}^\bullet)\) is K-injective by Lemma 0A96. Hence we can compute cohomology over \(U\) by simply taking sections over \(U\) and the result follows from Lemma 0A94.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(K, L, M\) be objects of \(D(\mathcal{O})\). With the construction as described above there is a canonical isomorphism \[R\SheafHom(K, R\SheafHom(L, M)) = R\SheafHom(K \otimes_\mathcal{O}^\mathbf{L} L, M)\] in \(D(\mathcal{O})\) functorial in \(K, L, M\) which recovers (08J8) on taking \(H^0(\mathcal{C}, -)\).

Proof

Choose a K-injective complex \(\mathcal{I}^\bullet\) representing \(M\) and a K-flat complex of \(\mathcal{O}\)-modules \(\mathcal{L}^\bullet\) representing \(L\). For any complex of \(\mathcal{O}\)-modules \(\mathcal{K}^\bullet\) we have \[\SheafHom^\bullet(\mathcal{K}^\bullet, \SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{I}^\bullet)) = \SheafHom^\bullet( \text{Tot}(\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{L}^\bullet), \mathcal{I}^\bullet)\] by Lemma 0A90. Note that the left hand side represents \(R\SheafHom(K, R\SheafHom(L, M))\) (use Lemma 0A96) and that the right hand side represents \(R\SheafHom(K \otimes_\mathcal{O}^\mathbf{L} L, M)\). This proves the displayed formula of the lemma. Taking global sections and using Lemma 08JA we obtain (08J8).

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(K, L\) be objects of \(D(\mathcal{O})\). The construction of \(R\SheafHom(K, L)\) commutes with restrictions, i.e., for every object \(U\) of \(\mathcal{C}\) we have \(R\SheafHom(K|_U, L|_U) = R\SheafHom(K, L)|_U\).

Proof

This is clear from the construction and Lemma 08FI.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. The bifunctor \(R\SheafHom(- , -)\) transforms distinguished triangles into distinguished triangles in both variables.

Proof

This follows from the observation that the assignment \[(\mathcal{L}^\bullet, \mathcal{M}^\bullet) \longmapsto \SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{M}^\bullet)\] transforms a termwise split short exact sequences of complexes in either variable into a termwise split short exact sequence. Details omitted.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(K, L, M\) be objects of \(D(\mathcal{O})\). There is a canonical morphism \[R\SheafHom(L, M) \otimes_\mathcal{O}^\mathbf{L} K \longrightarrow R\SheafHom(R\SheafHom(K, L), M)\] in \(D(\mathcal{O})\) functorial in \(K, L, M\).

Proof

Choose a K-injective complex \(\mathcal{I}^\bullet\) representing \(M\), a K-injective complex \(\mathcal{J}^\bullet\) representing \(L\), and a K-flat complex \(\mathcal{K}^\bullet\) representing \(K\). The map is defined using the map \[\text{Tot}(\SheafHom^\bullet(\mathcal{J}^\bullet, \mathcal{I}^\bullet) \otimes_\mathcal{O} \mathcal{K}^\bullet) \longrightarrow \SheafHom^\bullet(\SheafHom^\bullet(\mathcal{K}^\bullet, \mathcal{J}^\bullet), \mathcal{I}^\bullet)\] of Lemma 0A92. By our particular choice of complexes the left hand side represents \(R\SheafHom(L, M) \otimes_\mathcal{O}^\mathbf{L} K\) and the right hand side represents \(R\SheafHom(R\SheafHom(K, L), M)\). We omit the proof that this is functorial in all three objects of \(D(\mathcal{O})\).

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Given \(K, L, M\) in \(D(\mathcal{O})\) there is a canonical morphism \[R\SheafHom(L, M) \otimes_\mathcal{O}^\mathbf{L} R\SheafHom(K, L) \longrightarrow R\SheafHom(K, M)\] in \(D(\mathcal{O})\).

Proof

Choose a K-injective complex \(\mathcal{I}^\bullet\) representing \(M\), a K-injective complex \(\mathcal{J}^\bullet\) representing \(L\), and any complex of \(\mathcal{O}\)-modules \(\mathcal{K}^\bullet\) representing \(K\). By Lemma 0A91 there is a map of complexes \[\text{Tot}\left( \SheafHom^\bullet(\mathcal{J}^\bullet, \mathcal{I}^\bullet) \otimes_\mathcal{O} \SheafHom^\bullet(\mathcal{K}^\bullet, \mathcal{J}^\bullet) \right) \longrightarrow \SheafHom^\bullet(\mathcal{K}^\bullet, \mathcal{I}^\bullet)\] The complexes of \(\mathcal{O}\)-modules \(\SheafHom^\bullet(\mathcal{J}^\bullet, \mathcal{I}^\bullet)\), \(\SheafHom^\bullet(\mathcal{K}^\bullet, \mathcal{J}^\bullet)\), and \(\SheafHom^\bullet(\mathcal{K}^\bullet, \mathcal{I}^\bullet)\) represent \(R\SheafHom(L, M)\), \(R\SheafHom(K, L)\), and \(R\SheafHom(K, M)\). If we choose a K-flat complex \(\mathcal{H}^\bullet\) and a quasi-isomorphism \(\mathcal{H}^\bullet \to \SheafHom^\bullet(\mathcal{K}^\bullet, \mathcal{J}^\bullet)\), then there is a map \[\text{Tot}\left( \SheafHom^\bullet(\mathcal{J}^\bullet, \mathcal{I}^\bullet) \otimes_\mathcal{O} \mathcal{H}^\bullet \right) \longrightarrow \text{Tot}\left( \SheafHom^\bullet(\mathcal{J}^\bullet, \mathcal{I}^\bullet) \otimes_\mathcal{O} \SheafHom^\bullet(\mathcal{K}^\bullet, \mathcal{J}^\bullet) \right)\] whose source represents \(R\SheafHom(L, M) \otimes_\mathcal{O}^\mathbf{L} R\SheafHom(K, L)\). Composing the two displayed arrows gives the desired map. We omit the proof that the construction is functorial.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Given \(K, L, M\) in \(D(\mathcal{O})\) there is a canonical morphism \[K \otimes_\mathcal{O}^\mathbf{L} R\SheafHom(M, L) \longrightarrow R\SheafHom(M, K \otimes_\mathcal{O}^\mathbf{L} L)\] in \(D(\mathcal{O})\) functorial in \(K, L, M\).

Proof

Choose a K-flat complex \(\mathcal{K}^\bullet\) representing \(K\), and a K-injective complex \(\mathcal{I}^\bullet\) representing \(L\), and choose any complex of \(\mathcal{O}\)-modules \(\mathcal{M}^\bullet\) representing \(M\). Choose a quasi-isomorphism \(\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_X} \mathcal{I}^\bullet) \to \mathcal{J}^\bullet\) where \(\mathcal{J}^\bullet\) is K-injective. Then we use the map \[\text{Tot}\left( \mathcal{K}^\bullet \otimes_\mathcal{O} \SheafHom^\bullet(\mathcal{M}^\bullet, \mathcal{I}^\bullet) \right) \to \SheafHom^\bullet(\mathcal{M}^\bullet, \text{Tot}(\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{I}^\bullet)) \to \SheafHom^\bullet(\mathcal{M}^\bullet, \mathcal{J}^\bullet)\] where the first map is the map from Lemma 0BYT.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Given \(K, L\) in \(D(\mathcal{O})\) there is a canonical morphism \[K \longrightarrow R\SheafHom(L, K \otimes_\mathcal{O}^\mathbf{L} L)\] in \(D(\mathcal{O})\) functorial in both \(K\) and \(L\).

Proof

Choose a K-flat complex \(\mathcal{K}^\bullet\) representing \(K\) and any complex of \(\mathcal{O}\)-modules \(\mathcal{L}^\bullet\) representing \(L\). Choose a K-injective complex \(\mathcal{J}^\bullet\) and a quasi-isomorphism \(\text{Tot}(\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{L}^\bullet) \to \mathcal{J}^\bullet\). Then we use \[\mathcal{K}^\bullet \to \SheafHom^\bullet(\mathcal{L}^\bullet, \text{Tot}(\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{L}^\bullet)) \to \SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{J}^\bullet)\] where the first map comes from Lemma 0A93.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(L\) be an object of \(D(\mathcal{O})\). Set \(L^\vee = R\SheafHom(L, \mathcal{O})\). For \(M\) in \(D(\mathcal{O})\) there is a canonical map [08JE]\[\begin{equation} M \otimes^\mathbf{L}_\mathcal{O} L^\vee \longrightarrow R\SheafHom(L, M) \end{equation}\] which induces a canonical map \[H^0(\mathcal{C}, M \otimes_\mathcal{O}^\mathbf{L} L^\vee) \longrightarrow \Hom_{D(\mathcal{O})}(L, M)\] functorial in \(M\) in \(D(\mathcal{O})\).

Proof

The map (08JE) is a special case of Lemma 0A98 using the identification \(M = R\SheafHom(\mathcal{O}, M)\).

Remark

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi. Let \(K, L\) be objects of \(D(\mathcal{O}_\mathcal{C})\). We claim there is a canonical map \[Rf_*R\SheafHom(L, K) \longrightarrow R\SheafHom(Rf_*L, Rf_*K)\] Namely, by (08J8) this is the same thing as a map \(Rf_*R\SheafHom(L, K) \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} Rf_*L \to Rf_*K\). For this we can use the composition \[Rf_*R\SheafHom(L, K) \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} Rf_*L \to Rf_*(R\SheafHom(L, K) \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} L) \to Rf_*K\] where the first arrow is the relative cup product (Remark 0B6C) and the second arrow is \(Rf_*\) applied to the canonical map \(R\SheafHom(L, K) \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} L \to K\) coming from Lemma 0A98 (with \(\mathcal{O}_\mathcal{C}\) in one of the spots).

Remark

Let \(h : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')\) be a morphism of ringed topoi. Let \(K, L\) be objects of \(D(\mathcal{O}')\). We claim there is a canonical map \[Lh^*R\SheafHom(K, L) \longrightarrow R\SheafHom(Lh^*K, Lh^*L)\] in \(D(\mathcal{O})\). Namely, by (08J8) proved in Lemma 08J9 such a map is the same thing as a map \[Lh^*R\SheafHom(K, L) \otimes^\mathbf{L} Lh^*K \longrightarrow Lh^*L\] The source of this arrow is \(Lh^*(\SheafHom(K, L) \otimes^\mathbf{L} K)\) by Lemma 07A4 hence it suffices to construct a canonical map \[R\SheafHom(K, L) \otimes^\mathbf{L} K \longrightarrow L.\] For this we take the arrow corresponding to \[\text{id} : R\SheafHom(K, L) \longrightarrow R\SheafHom(K, L)\] via (08J8).

Remark

Suppose that \[\xymatrix{ (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_h \ar[d]_{f'} & (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^f \\ (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^g & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) }\] is a commutative diagram of ringed topoi. Let \(K, L\) be objects of \(D(\mathcal{O}_\mathcal{C})\). We claim there exists a canonical base change map \[Lg^*Rf_*R\SheafHom(K, L) \longrightarrow R(f')_*R\SheafHom(Lh^*K, Lh^*L)\] in \(D(\mathcal{O}_{\mathcal{D}'})\). Namely, we take the map adjoint to the composition \[\begin{align*} L(f')^*Lg^*Rf_*R\SheafHom(K, L) & = Lh^*Lf^*Rf_*R\SheafHom(K, L) \\ & \to Lh^*R\SheafHom(K, L) \\ & \to R\SheafHom(Lh^*K, Lh^*L) \end{align*}\] where the first arrow uses the adjunction mapping \(Lf^*Rf_* \to \text{id}\) and the second arrow is the canonical map constructed in Remark 08JF.

Global derived hom

Let \((\Sh(\mathcal{C}), \mathcal{O})\) be a ringed topos. Let \(K, L \in D(\mathcal{O})\). Using the construction of the internal hom in the derived category we obtain a well defined object \[R\Hom_\mathcal{O}(K, L) = R\Gamma(\mathcal{C}, R\SheafHom(K, L))\] in \(D(\Gamma(\mathcal{C}, \mathcal{O}))\). By Lemma 08JA we have \[H^0(R\Hom_\mathcal{O}(K, L)) = \Hom_{D(\mathcal{O})}(K, L)\] and \[H^p(R\Hom_\mathcal{O}(K, L)) = \Ext_{D(\mathcal{O})}^p(K, L)\] If \(f : (\mathcal{C}', \mathcal{O}') \to (\mathcal{C}, \mathcal{O})\) is a morphism of ringed topoi, then there is a canonical map \[R\Hom_\mathcal{O}(K, L) \longrightarrow R\Hom_{\mathcal{O}'}(Lf^*K, Lf^*L)\] in \(D(\Gamma(\mathcal{O}))\) by taking global sections of the map defined in Remark 08JF.

Derived lower shriek

In this section we study morphisms \(g\) of ringed topoi where besides \(Lg^*\) and \(Rg_*\) there also exists a derived functor \(Lg_!\).

Lemma

Let \(u : \mathcal{C} \to \mathcal{D}\) be a continuous and cocontinuous functor of sites. Let \(g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\) be the corresponding morphism of topoi. Let \(\mathcal{O}_\mathcal{D}\) be a sheaf of rings and let \(\mathcal{I}\) be an injective \(\mathcal{O}_\mathcal{D}\)-module. Then \(H^p(U, g^{-1}\mathcal{I}) = 0\) for all \(p > 0\) and \(U \in \Ob(\mathcal{C})\).

Proof

The vanishing of the lemma follows from Lemma 03F9 if we can prove vanishing of all higher Čech cohomology groups \(\check H^p(\mathcal{U}, g^{-1}\mathcal{I})\) for any covering \(\mathcal{U} = \{U_i \to U\}\) of \(\mathcal{C}\). Since \(u\) is continuous, \(u(\mathcal{U}) = \{u(U_i) \to u(U)\}\) is a covering of \(\mathcal{D}\), and \(u(U_{i_0} \times_U \ldots \times_U U_{i_n}) = u(U_{i_0}) \times_{u(U)} \ldots \times_{u(U)} u(U_{i_n})\). Thus we have \[\check H^p(\mathcal{U}, g^{-1}\mathcal{I}) = \check H^p(u(\mathcal{U}), \mathcal{I})\] because \(g^{-1} = u^p\) by Sites, Lemma 00XR. Since \(\mathcal{I}\) is an injective \(\mathcal{O}_\mathcal{D}\)-module these Čech cohomology groups vanish, see Lemma 03FC.

Lemma

Let \(u : \mathcal{C} \to \mathcal{D}\) be a continuous and cocontinuous functor of sites. Let \(g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\) be the corresponding morphism of topoi. Let \(\mathcal{O}_\mathcal{D}\) be a sheaf of rings and set \(\mathcal{O}_\mathcal{C} = g^{-1}\mathcal{O}_\mathcal{D}\). The functor \(g_! : \textit{Mod}(\mathcal{O}_\mathcal{C}) \to \textit{Mod}(\mathcal{O}_\mathcal{D})\) (see Modules on Sites, Lemma 0797) has a left derived functor \[Lg_! : D(\mathcal{O}_\mathcal{C}) \longrightarrow D(\mathcal{O}_\mathcal{D})\] which is left adjoint to \(g^*\). Moreover, for \(U \in \Ob(\mathcal{C})\) we have \[Lg_!(j_{U!}\mathcal{O}_U) = g_!j_{U!}\mathcal{O}_U = j_{u(U)!} \mathcal{O}_{u(U)}.\] where \(j_{U!}\) and \(j_{u(U)!}\) are extension by zero associated to the localization morphism \(j_U : \mathcal{C}/U \to \mathcal{C}\) and \(j_{u(U)} : \mathcal{D}/u(U) \to \mathcal{D}\).

Proof

We are going to use Derived Categories, Proposition 0794 to construct \(Lg_!\). To do this we have to verify assumptions (1), (2), (3), (4), and (5) of that proposition. First, since \(g_!\) is a left adjoint we see that it is right exact and commutes with all colimits, so (5) holds. Conditions (3) and (4) hold because the category of modules on a ringed site is a Grothendieck abelian category. Let \(\mathcal{P} \subset \Ob(\textit{Mod}(\mathcal{O}_\mathcal{C}))\) be the collection of \(\mathcal{O}_\mathcal{C}\)-modules which are direct sums of modules of the form \(j_{U!}\mathcal{O}_U\). Note that \(g_!j_{U!}\mathcal{O}_U = j_{u(U)!} \mathcal{O}_{u(U)}\), see proof of Modules on Sites, Lemma 0797. Every \(\mathcal{O}_\mathcal{C}\)-module is a quotient of an object of \(\mathcal{P}\), see Modules on Sites, Lemma 03EW. Thus (1) holds. Finally, we have to prove (2). Let \(\mathcal{K}^\bullet\) be a bounded above acyclic complex of \(\mathcal{O}_\mathcal{C}\)-modules with \(\mathcal{K}^n \in \mathcal{P}\) for all \(n\). We have to show that \(g_!\mathcal{K}^\bullet\) is exact. To do this it suffices to show, for every injective \(\mathcal{O}_\mathcal{D}\)-module \(\mathcal{I}\) that \[\Hom_{D(\mathcal{O}_\mathcal{D})}( g_!\mathcal{K}^\bullet, \mathcal{I}[n]) = 0\] for all \(n \in \mathbf{Z}\). Since \(\mathcal{I}\) is injective we have \[\begin{align*} \Hom_{D(\mathcal{O}_\mathcal{D})}( g_!\mathcal{K}^\bullet, \mathcal{I}[n]) & = \Hom_{K(\mathcal{O}_\mathcal{D})}( g_!\mathcal{K}^\bullet, \mathcal{I}[n]) \\ & = H^n(\Hom_{\mathcal{O}_\mathcal{D}}( g_!\mathcal{K}^\bullet, \mathcal{I})) \\ & = H^n(\Hom_{\mathcal{O}_\mathcal{C}}( \mathcal{K}^\bullet, g^{-1}\mathcal{I})) \end{align*}\] the last equality by the adjointness of \(g_!\) and \(g^{-1}\).

The vanishing of this group would be clear if \(g^{-1}\mathcal{I}\) were an injective \(\mathcal{O}_\mathcal{C}\)-module. But \(g^{-1}\mathcal{I}\) isn’t necessarily an injective \(\mathcal{O}_\mathcal{C}\)-module as \(g_!\) isn’t exact in general. We do know that \[\Ext^p_{\mathcal{O}_\mathcal{C}}( j_{U!}\mathcal{O}_U, g^{-1}\mathcal{I}) = H^p(U, g^{-1}\mathcal{I}) = 0 \text{ for }p \geq 1\] Here the first equality follows from \(\Hom_{\mathcal{O}_\mathcal{C}}(j_{U!}\mathcal{O}_U, \mathcal{H}) = \mathcal{H}(U)\) and taking derived functors and the vanishing of \(H^p(U, g^{-1}\mathcal{I})\) for \(p > 0\) and \(U \in \Ob(\mathcal{C})\) follows from Lemma 0D6X. Since each \(\mathcal{K}^{-q}\) is a direct sum of modules of the form \(j_{U!}\mathcal{O}_U\) we see that \[\Ext^p_{\mathcal{O}_\mathcal{C}}(\mathcal{K}^{-q}, g^{-1}\mathcal{I}) = 0 \text{ for }p \geq 1\text{ and all }q\] Let us use the spectral sequence (see Example 07AA) \[E_1^{p, q} = \Ext^q_{\mathcal{O}_\mathcal{C}}( \mathcal{K}^{-p}, g^{-1}\mathcal{I}) \Rightarrow \Ext^{p + q}_{\mathcal{O}_\mathcal{C}}( \mathcal{K}^\bullet, g^{-1}\mathcal{I}) = 0.\] Note that the spectral sequence abuts to zero as \(\mathcal{K}^\bullet\) is acyclic (hence vanishes in the derived category, hence produces vanishing ext groups). By the vanishing of higher exts proved above the only nonzero terms on the \(E_1\) page are the terms \(E_1^{p, 0} = \Hom_{\mathcal{O}_\mathcal{C}}( \mathcal{K}^{-p}, g^{-1}\mathcal{I})\). We conclude that the complex \(\Hom_{\mathcal{O}_\mathcal{C}}( \mathcal{K}^\bullet, g^{-1}\mathcal{I})\) is acyclic as desired.

Thus the left derived functor \(Lg_!\) exists. It is left adjoint to \(g^{-1} = g^* = Rg^* = Lg^*\), i.e., we have [07AD]\[\begin{equation} \Hom_{D(\mathcal{O}_\mathcal{C})}(K, g^*L) = \Hom_{D(\mathcal{O}_\mathcal{D})}(Lg_!K, L) \end{equation}\] by Derived Categories, Lemma 09T5. This finishes the proof.

Remark

Warning! Let \(u : \mathcal{C} \to \mathcal{D}\), \(g\), \(\mathcal{O}_\mathcal{D}\), and \(\mathcal{O}_\mathcal{C}\) be as in Lemma 07AC. In general it is not the case that the diagram \[\xymatrix{ D(\mathcal{O}_\mathcal{C}) \ar[r]_{Lg_!} \ar[d]_{forget} & D(\mathcal{O}_\mathcal{D}) \ar[d]^{forget} \\ D(\mathcal{C}) \ar[r]^{Lg^{Ab}_!} & D(\mathcal{D}) }\] commutes where the functor \(Lg_!^{Ab}\) is the one constructed in Lemma 07AC but using the constant sheaf \(\mathbf{Z}\) as the structure sheaf on both \(\mathcal{C}\) and \(\mathcal{D}\). In general it isn’t even the case that \(g_! = g_!^{Ab}\) (see Modules on Sites, Remark 0798), but this phenomenon can occur even if \(g_! = g_!^{Ab}\)! Namely, the construction of \(Lg_!\) in the proof of Lemma 07AC shows that \(Lg_!\) agrees with \(Lg_!^{\textit{Ab}}\) if and only if the canonical maps \[Lg^{Ab}_!j_{U!}\mathcal{O}_U \longrightarrow j_{u(U)!}\mathcal{O}_{u(U)}\] are isomorphisms in \(D(\mathcal{D})\) for all objects \(U\) in \(\mathcal{C}\). In general all we can say is that there exists a natural transformation \[Lg_!^{Ab} \circ forget \longrightarrow forget \circ Lg_!\]

Lemma

Let \(u : \mathcal{C} \to \mathcal{D}\) be a continuous and cocontinuous functor of sites. Let \(g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\) be the corresponding morphism of topoi. Let \(\mathcal{O}_\mathcal{D}\) be a sheaf of rings and let \(\mathcal{I}\) be an injective \(\mathcal{O}_\mathcal{D}\)-module. If \(g_!^{Sh} : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\) commutes with fibre products⁷, then \(g^{-1}\mathcal{I}\) is totally acyclic.

Proof

We will use the criterion of Lemma 07A1. Condition (1) holds by Lemma 0D6X. Let \(K' \to K\) be a surjective map of sheaves of sets on \(\mathcal{C}\). Since \(g_!^{Sh}\) is a left adjoint, we see that \(g_!^{Sh}K' \to g_!^{Sh}K\) is surjective. Observe that \[\begin{align*} H^0(K' \times_K \ldots \times_K K', g^{-1}\mathcal{I}) & = H^0(g_!^{Sh}(K' \times_K \ldots \times_K K'), \mathcal{I}) \\ & = H^0(g_!^{Sh}K' \times_{g_!^{Sh}K} \ldots \times_{g_!^{Sh}K} g_!^{Sh}K', \mathcal{I}) \end{align*}\] by our assumption on \(g_!^{Sh}\). Since \(\mathcal{I}\) is an injective module it is totally acyclic by Lemma 072Z (applied to the identity). Hence we can use the converse of Lemma 07A1 to see that the complex \[0 \to H^0(K, g^{-1}\mathcal{I}) \to H^0(K', g^{-1}\mathcal{I}) \to H^0(K' \times_K K', g^{-1}\mathcal{I}) \to \ldots\] is exact as desired.

Lemma

Let \(u : \mathcal{C} \to \mathcal{D}\) be a continuous and cocontinuous functor of sites. Let \(g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\) be the corresponding morphism of topoi. Let \(U \in \Ob(\mathcal{C})\).

1. For \(M\) in \(D(\mathcal{D})\) we have \(R\Gamma(U, g^{-1}M) = R\Gamma(u(U), M)\).

2. If \(\mathcal{O}_\mathcal{D}\) is a sheaf of rings and \(\mathcal{O}_\mathcal{C} = g^{-1}\mathcal{O}_\mathcal{D}\), then for \(M\) in \(D(\mathcal{O}_\mathcal{D})\) we have \(R\Gamma(U, g^*M) = R\Gamma(u(U), M)\).

Proof

In the bounded below case (1) and (2) can be seen by representing \(K\) by a bounded below complex of injectives and using Lemma 0D6X as well as Leray’s acyclicity lemma. In the unbounded case, first note that (1) is a special case of (2). For (2) we can use \[R\Gamma(U, g^*M) = R\Hom_{\mathcal{O}_\mathcal{C}}(j_{U!}\mathcal{O}_U, g^*M) = R\Hom_{\mathcal{O}_\mathcal{D}}(j_{u(U)!}\mathcal{O}_{u(U)}, M) = R\Gamma(u(U), M)\] where the middle equality is a consequence of Lemma 07AC.

Lemma

Assume given a commutative diagram \[\xymatrix{ (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_{(g', (g')^\sharp)} \ar[d]_{(f', (f')^\sharp)} & (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^{(f, f^\sharp)} \\ (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^{(g, g^\sharp)} & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) }\] of ringed topoi. Assume

1. \(f\), \(f'\), \(g\), and \(g'\) correspond to cocontinuous functors \(u\), \(u'\), \(v\), and \(v'\) as in Sites, Lemma 00XO,

2. \(v \circ u' = u \circ v'\),

3. \(v\) and \(v'\) are continuous as well as cocontinuous,

4. for any object \(V'\) of \(\mathcal{D}'\) the functor \({}^{u'}_{V'}\mathcal{I} \to {}^{\ \ \ u}_{v(V')}\mathcal{I}\) given by \(v\) is cofinal,

5. \(g^{-1}\mathcal{O}_{\mathcal{D}} = \mathcal{O}_{\mathcal{D}'}\) and \((g')^{-1}\mathcal{O}_{\mathcal{C}} = \mathcal{O}_{\mathcal{C}'}\), and

6. \(g'_! : \textit{Ab}(\mathcal{C}') \to \textit{Ab}(\mathcal{C})\) is exact⁸.

Then we have \(Rf'_* \circ (g')^* = g^* \circ Rf_*\) as functors \(D(\mathcal{O}_\mathcal{C}) \to D(\mathcal{O}_{\mathcal{D}'})\).

Proof

We have \(g^* = Lg^* = g^{-1}\) and \((g')^* = L(g')^* = (g')^{-1}\) by condition (5). By Lemma 0D6J it suffices to prove the result on the derived category \(D(\mathcal{C})\) of abelian sheaves. Choose an object \(K \in D(\mathcal{C})\). Let \(\mathcal{I}^\bullet\) be a K-injective complex of abelian sheaves on \(\mathcal{C}\) representing \(K\). By Derived Categories, Lemma 08BJ and assumption (6) we find that \((g')^{-1}\mathcal{I}^\bullet\) is a K-injective complex of abelian sheaves on \(\mathcal{C}'\). By Modules on Sites, Lemma 0FN3 we find that \(f'_*(g')^{-1}\mathcal{I}^\bullet = g^{-1}f_*\mathcal{I}^\bullet\). Since \(f_*\mathcal{I}^\bullet\) represents \(Rf_*K\) and since \(f'_*(g')^{-1}\mathcal{I}^\bullet\) represents \(Rf'_*(g')^{-1}K\) we conclude.

Lemma

Consider a commutative diagram \[\xymatrix{ (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_{(g', (g')^\sharp)} \ar[d]_{(f', (f')^\sharp)} & (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^{(f, f^\sharp)} \\ (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^{(g, g^\sharp)} & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) }\] of ringed topoi and suppose we have functors \[\xymatrix{ \mathcal{C}' \ar[r]_{v'} & \mathcal{C} \\ \mathcal{D}' \ar[r]^v \ar[u]^{u'} & \mathcal{D} \ar[u]_u }\] such that (with notation as in Sites, Sections 00X0 and 00XN) we have

1. \(u\) and \(u'\) are continuous and give rise to the morphisms \(f\) and \(f'\),

2. \(v\) and \(v'\) are cocontinuous giving rise to the morphisms \(g\) and \(g'\),

3. \(u \circ v = v' \circ u'\),

4. \(v\) and \(v'\) are continuous as well as cocontinuous, and

5. \(g^{-1}\mathcal{O}_{\mathcal{D}} = \mathcal{O}_{\mathcal{D}'}\) and \((g')^{-1}\mathcal{O}_{\mathcal{C}} = \mathcal{O}_{\mathcal{C}'}\).

Then \(Rf'_* \circ (g')^* = g^* \circ Rf_*\) as functors \(D^+(\mathcal{O}_\mathcal{C}) \to D^+(\mathcal{O}_{\mathcal{D}'})\). If in addition

1. \(g'_! : \textit{Ab}(\mathcal{C}') \to \textit{Ab}(\mathcal{C})\) is exact⁹,

then \(Rf'_* \circ (g')^* = g^* \circ Rf_*\) as functors \(D(\mathcal{O}_\mathcal{C}) \to D(\mathcal{O}_{\mathcal{D}'})\).

Proof

We have \(g^* = Lg^* = g^{-1}\) and \((g')^* = L(g')^* = (g')^{-1}\) by condition (5). By Lemma 0D6J it suffices to prove the result on the derived category \(D^+(\mathcal{C})\) or \(D(\mathcal{C})\) of abelian sheaves.

Choose an object \(K \in D^+(\mathcal{C})\). Let \(\mathcal{I}^\bullet\) be a bounded below complex of injective abelian sheaves on \(\mathcal{C}\) representing \(K\). By Lemma 0D6X we see that \(H^p(U', (g')^{-1}\mathcal{I}^q) = 0\) for all \(p > 0\) and any \(q\) and any \(U' \in \Ob(\mathcal{C}')\). Recall that \(R^pf'_*(g')^{-1}\mathcal{I}^q\) is the sheaf associated to the presheaf \(V' \mapsto H^p(u'(V'), (g')^{-1}\mathcal{I}^q)\), see Lemma 072W. Thus we see that \((g')^{-1}\mathcal{I}^q\) is right acyclic for the functor \(f'_*\). By Leray’s acyclicity lemma (Derived Categories, Lemma 015E) we find that \(f'_*(g')^*\mathcal{I}^\bullet\) represents \(Rf'_*(g')^{-1}K\). By Modules on Sites, Lemma 0FN4 we find that \(f'_*(g')^{-1}\mathcal{I}^\bullet = g^{-1}f_*\mathcal{I}^\bullet\). Since \(g^{-1}f_*\mathcal{I}^\bullet\) represents \(g^{-1}Rf_*K\) we conclude.

Choose an object \(K \in D(\mathcal{C})\). Let \(\mathcal{I}^\bullet\) be a K-injective complex of abelian sheaves on \(\mathcal{C}\) representing \(K\). By Derived Categories, Lemma 08BJ and assumption (6) we find that \((g')^{-1}\mathcal{I}^\bullet\) is a K-injective complex of abelian sheaves on \(\mathcal{C}'\). By Modules on Sites, Lemma 0FN4 we find that \(f'_*(g')^{-1}\mathcal{I}^\bullet = g^{-1}f_*\mathcal{I}^\bullet\). Since \(f_*\mathcal{I}^\bullet\) represents \(Rf_*K\) and since \(f'_*(g')^{-1}\mathcal{I}^\bullet\) represents \(Rf'_*(g')^{-1}K\) we conclude.

Derived lower shriek for fibred categories

In this section we work out some special cases of the situation discussed in Section 07AB. We make sure that we have equality between lower shriek on modules and sheaves of abelian groups. We encourage the reader to skip this section on a first reading.

Situation

Here \((\mathcal{D}, \mathcal{O}_\mathcal{D})\) be a ringed site and \(p : \mathcal{C} \to \mathcal{D}\) is a fibred category. We endow \(\mathcal{C}\) with the topology inherited from \(\mathcal{D}\) (Stacks, Section 06NT). We denote \(\pi : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\) the morphism of topoi associated to \(p\) (Stacks, Lemma 06NW). We set \(\mathcal{O}_\mathcal{C} = \pi^{-1}\mathcal{O}_\mathcal{D}\) so that we obtain a morphism of ringed topoi \[\pi : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \longrightarrow (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\]

Lemma

Assumptions and notation as in Situation 08P8. For \(U \in \Ob(\mathcal{C})\) consider the induced morphism of topoi \[\pi_U : \Sh(\mathcal{C}/U) \longrightarrow \Sh(\mathcal{D}/p(U))\] Then there exists a morphism of topoi \[\sigma : \Sh(\mathcal{D}/p(U)) \to \Sh(\mathcal{C}/U)\] such that \(\pi_U \circ \sigma = \text{id}\) and \(\sigma^{-1} = \pi_{U, *}\).

Proof

Observe that \(\pi_U\) is the restriction of \(\pi\) to the localizations, see Sites, Lemma 03EG. For an object \(V \to p(U)\) of \(\mathcal{D}/p(U)\) denote \(V \times_{p(U)} U \to U\) the strongly cartesian morphism of \(\mathcal{C}\) over \(\mathcal{D}\) which exists as \(p\) is a fibred category. The functor \[v : \mathcal{D}/p(U) \to \mathcal{C}/U,\quad V/p(U) \mapsto V \times_{p(U)} U/U\] is continuous by the definition of the topology on \(\mathcal{C}\). Moreover, it is a right adjoint to \(p\) by the definition of strongly cartesian morphisms. Hence we are in the situation discussed in Sites, Section 00XW and we see that the sheaf \(\pi_{U, *}\mathcal{F}\) is equal to \(V \mapsto \mathcal{F}(V \times_{p(U)} U)\) (see especially Sites, Lemma 00XY).

But here we have more. Namely, the functor \(v\) is also cocontinuous (as all morphisms in coverings of \(\mathcal{C}\) are strongly cartesian). Hence \(v\) defines a morphism \(\sigma\) as indicated in the lemma. The equality \(\sigma^{-1} = \pi_{U, *}\) is immediate from the definition. Since \(\pi_U^{-1}\mathcal{G}\) is given by the rule \(U'/U \mapsto \mathcal{G}(p(U')/p(U))\) it follows that \(\sigma^{-1} \circ \pi_U^{-1} = \text{id}\) which proves the equality \(\pi_U \circ \sigma = \text{id}\).

Situation

Let \((\mathcal{D}, \mathcal{O}_\mathcal{D})\) be a ringed site. Let \(u : \mathcal{C}' \to \mathcal{C}\) be a \(1\)-morphism of fibred categories over \(\mathcal{D}\) (Categories, Definition 02XP). Endow \(\mathcal{C}\) and \(\mathcal{C}'\) with their inherited topologies (Stacks, Definition 06NV) and let \(\pi : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\), \(\pi' : \Sh(\mathcal{C}') \to \Sh(\mathcal{D})\), and \(g : \Sh(\mathcal{C}') \to \Sh(\mathcal{C})\) be the corresponding morphisms of topoi (Stacks, Lemma 06NW). Set \(\mathcal{O}_\mathcal{C} = \pi^{-1}\mathcal{O}_\mathcal{D}\) and \(\mathcal{O}_{\mathcal{C}'} = (\pi')^{-1}\mathcal{O}_\mathcal{D}\). Observe that \(g^{-1}\mathcal{O}_\mathcal{C} = \mathcal{O}_{\mathcal{C}'}\) so that \[\xymatrix{ (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[rd]_{\pi'} \ar[rr]_g & & (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[ld]^\pi \\ & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) }\] is a commutative diagram of morphisms of ringed topoi.

Lemma

Assumptions and notation as in Situation 08PA. For \(U' \in \Ob(\mathcal{C}')\) set \(U = u(U')\) and \(V = p'(U')\) and consider the induced morphisms of ringed topoi \[\xymatrix{ (\Sh(\mathcal{C}'/U'), \mathcal{O}_{U'}) \ar[rd]_{\pi'_{U'}} \ar[rr]_{g'} & & (\Sh(\mathcal{C}/U), \mathcal{O}_U) \ar[ld]^{\pi_U} \\ & (\Sh(\mathcal{D}/V), \mathcal{O}_V) }\] Then there exists a morphism of topoi \[\sigma' : \Sh(\mathcal{D}/V) \to \Sh(\mathcal{C}'/U'),\] such that setting \(\sigma = g' \circ \sigma'\) we have \(\pi'_{U'} \circ \sigma' = \text{id}\), \(\pi_U \circ \sigma = \text{id}\), \((\sigma')^{-1} = \pi'_{U', *}\), and \(\sigma^{-1} = \pi_{U, *}\).

Proof

Let \(v' : \mathcal{D}/V \to \mathcal{C}'/U'\) be the functor constructed in the proof of Lemma 08P9 starting with \(p' : \mathcal{C}' \to \mathcal{D}\) and the object \(U'\). Since \(u\) is a \(1\)-morphism of fibred categories over \(\mathcal{D}\) it transforms strongly cartesian morphisms into strongly cartesian morphisms, hence the functor \(v = u \circ v'\) is the functor of the proof of Lemma 08P9 relative to \(p : \mathcal{C} \to \mathcal{D}\) and \(U\). Thus our lemma follows from that lemma.

Lemma

Assumption and notation as in Situation 08PA.

1. There are left adjoints \(g_! : \textit{Mod}(\mathcal{O}_{\mathcal{C}'}) \to \textit{Mod}(\mathcal{O}_\mathcal{C})\) and \(g_!^{\textit{Ab}} : \textit{Ab}(\mathcal{C}') \to \textit{Ab}(\mathcal{C})\) to \(g^* = g^{-1}\) on modules and on abelian sheaves.

2. The diagram \[\xymatrix{ \textit{Mod}(\mathcal{O}_{\mathcal{C}'}) \ar[d] \ar[r]_{g_!} & \textit{Mod}(\mathcal{O}_\mathcal{C}) \ar[d] \\ \textit{Ab}(\mathcal{C}') \ar[r]^{g_!^{\textit{Ab}}} & \textit{Ab}(\mathcal{C}) }\] commutes.

3. There are left adjoints \(Lg_! : D(\mathcal{O}_{\mathcal{C}'}) \to D(\mathcal{O}_\mathcal{C})\) and \(Lg_!^{\textit{Ab}} : D(\mathcal{C}') \to D(\mathcal{C})\) to \(g^* = g^{-1}\) on derived categories of modules and abelian sheaves.

4. The diagram \[\xymatrix{ D(\mathcal{O}_{\mathcal{C}'}) \ar[d] \ar[r]_{Lg_!} & D(\mathcal{O}_\mathcal{C}) \ar[d] \\ D(\mathcal{C}') \ar[r]^{Lg_!^{\textit{Ab}}} & D(\mathcal{C}) }\] commutes.

Proof

The functor \(u\) is continuous and cocontinuous Stacks, Lemma 06NW. Hence the existence of the functors \(g_!\), \(g_!^{\textit{Ab}}\), \(Lg_!\), and \(Lg_!^{\textit{Ab}}\) can be found in Modules on Sites, Sections 04BE and 0796 and Section 07AB.

To prove (2) it suffices to show that the canonical map \[g_!^{\textit{Ab}}j_{U'!}\mathcal{O}_{U'} \to j_{u(U')!}\mathcal{O}_{u(U')}\] is an isomorphism for all objects \(U'\) of \(\mathcal{C}'\), see Modules on Sites, Remark 0798. Similarly, to prove (4) it suffices to show that the canonical map \[Lg_!^{\textit{Ab}}j_{U'!}\mathcal{O}_{U'} \to j_{u(U')!}\mathcal{O}_{u(U')}\] is an isomorphism in \(D(\mathcal{C})\) for all objects \(U'\) of \(\mathcal{C}'\), see Remark 07AE. This will also imply the previous formula hence this is what we will show.

We will use that for a localization morphism \(j\) the functors \(j_!\) and \(j_!^{\textit{Ab}}\) agree (see Modules on Sites, Remark 08P4) and that \(j_!\) is exact (Modules on Sites, Lemma 03DJ). Let us adopt the notation of Lemma 08PB. Since \(Lg_!^{\textit{Ab}} \circ j_{U'!} = j_{U!} \circ L(g')^{\textit{Ab}}_!\) (by commutativity of Sites, Lemma 03EG and uniqueness of adjoint functors) it suffices to prove that \(L(g')^{\textit{Ab}}_!\mathcal{O}_{U'} = \mathcal{O}_U\). Using the results of Lemma 08PB we have for any object \(E\) of \(D(\mathcal{C}/u(U'))\) the following sequence of equalities \[\begin{align*} \Hom_{D(\mathcal{C}/U)}(L(g')_!^{\textit{Ab}}\mathcal{O}_{U'}, E) & = \Hom_{D(\mathcal{C}'/U')}(\mathcal{O}_{U'}, (g')^{-1}E) \\ & = \Hom_{D(\mathcal{C}'/U')}((\pi'_{U'})^{-1}\mathcal{O}_V, (g')^{-1}E) \\ & = \Hom_{D(\mathcal{D}/V)}(\mathcal{O}_V, R\pi'_{U', *}(g')^{-1}E) \\ & = \Hom_{D(\mathcal{D}/V)}(\mathcal{O}_V, (\sigma')^{-1}(g')^{-1}E) \\ & = \Hom_{D(\mathcal{D}/V)}(\mathcal{O}_V, \sigma^{-1}E) \\ & = \Hom_{D(\mathcal{D}/V)}(\mathcal{O}_V, \pi_{U, *}E) \\ & = \Hom_{D(\mathcal{C}/U)}(\pi_U^{-1}\mathcal{O}_V, E) \\ & = \Hom_{D(\mathcal{C}/U)}(\mathcal{O}_U, E) \end{align*}\] By Yoneda’s lemma we conclude.