Simplicial Spaces

Introduction

This chapter develops some theory concerning simplicial topological spaces, simplicial ringed spaces, simplicial schemes, and simplicial algebraic spaces. The theory of simplicial spaces sometimes allows one to prove local to global principles which appear difficult to prove in other ways. Some example applications can be found in the papers [faltings_finiteness], [Kiehl], and [HodgeIII].

We assume throughout that the reader is familiar with the basic concepts and results of the chapter Simplicial Methods, see Simplicial, Section 0163. In particular, we continue to write \(X\) and not \(X_\bullet\) for a simplicial object.

Simplicial topological spaces

A simplicial space is a simplicial object in the category of topological spaces where morphisms are continuous maps of topological spaces. (We will use “simplicial algebraic space” to refer to simplicial objects in the category of algebraic spaces.) We may picture a simplicial space \(X\) as follows \[\xymatrix{ X_2 \ar@<2ex>[r] \ar@<0ex>[r] \ar@<-2ex>[r] & X_1 \ar@<1ex>[r] \ar@<-1ex>[r] \ar@<1ex>[l] \ar@<-1ex>[l] & X_0 \ar@<0ex>[l] }\] Here there are two morphisms \(d^1_0, d^1_1 : X_1 \to X_0\) and a single morphism \(s^0_0 : X_0 \to X_1\), etc. It is important to keep in mind that \(d^n_i : X_n \to X_{n - 1}\) should be thought of as a “projection forgetting the \(i\)th coordinate” and \(s^n_j : X_n \to X_{n + 1}\) as the diagonal map repeating the \(j\)th coordinate.

Let \(X\) be a simplicial space. We associate a site \(X_{Zar}\)¹ to \(X\) as follows.

1. An object of \(X_{Zar}\) is an open \(U\) of \(X_n\) for some \(n\),

2. a morphism \(U \to V\) of \(X_{Zar}\) is given by a \(\varphi : [m] \to [n]\) where \(n, m\) are such that \(U \subset X_n\), \(V \subset X_m\) and \(\varphi\) is such that \(X(\varphi)(U) \subset V\), and

3. a covering \(\{U_i \to U\}\) in \(X_{Zar}\) means that \(U, U_i \subset X_n\) are open, the maps \(U_i \to U\) are given by \(\text{id} : [n] \to [n]\), and \(U = \bigcup U_i\).

Note that in particular, if \(U \to V\) is a morphism of \(X_{Zar}\) given by \(\varphi\), then \(X(\varphi) : X_n \to X_m\) does in fact induce a continuous map \(U \to V\) of topological spaces.

It is clear that the above is a special case of a construction that associates to any diagram of topological spaces a site. We formulate the obligatory lemma.

Lemma

Let \(X\) be a simplicial space. Then \(X_{Zar}\) as defined above is a site.

Proof

Omitted.

Let \(X\) be a simplicial space. Let \(\mathcal{F}\) be a sheaf on \(X_{Zar}\). It is clear from the definition of coverings, that the restriction of \(\mathcal{F}\) to the opens of \(X_n\) defines a sheaf \(\mathcal{F}_n\) on the topological space \(X_n\). For every \(\varphi : [m] \to [n]\) the restriction maps of \(\mathcal{F}\) for pairs \(U \subset X_n\), \(V \subset X_m\) with \(X(\varphi)(U) \subset V\), define an \(X(\varphi)\)-map \(\mathcal{F}(\varphi) : \mathcal{F}_m \to \mathcal{F}_n\), see Sheaves, Definition 008J. Moreover, given \(\varphi : [m] \to [n]\) and \(\psi : [l] \to [m]\) we have \[\mathcal{F}(\varphi) \circ \mathcal{F}(\psi) = \mathcal{F}(\varphi \circ \psi)\] (LHS uses composition of \(f\)-maps, see Sheaves, Definition 008L). Clearly, the converse is true as well: if we have a system \((\{\mathcal{F}_n\}_{n \geq 0}, \{\mathcal{F}(\varphi)\}_{\varphi \in \text{Arrows}(\Delta)})\) as above, satisfying the displayed equalities, then we obtain a sheaf on \(X_{Zar}\).

Lemma

Let \(X\) be a simplicial space. There is an equivalence of categories between

1. \(\Sh(X_{Zar})\), and

2. category of systems \((\mathcal{F}_n, \mathcal{F}(\varphi))\) described above.

Proof

See discussion above.

Lemma

Let \(f : Y \to X\) be a morphism of simplicial spaces. Then the functor \(u : X_{Zar} \to Y_{Zar}\) which associates to the open \(U \subset X_n\) the open \(f_n^{-1}(U) \subset Y_n\) defines a morphism of sites \(f_{Zar} : Y_{Zar} \to X_{Zar}\).

Proof

It is clear that \(u\) is a continuous functor. Hence we obtain functors \(f_{Zar, *} = u^s\) and \(f_{Zar}^{-1} = u_s\), see Sites, Section 00X0. To see that we obtain a morphism of sites we have to show that \(u_s\) is exact. We will use Sites, Lemma 00X5 to see this. Let \(V \subset Y_n\) be an open subset. The category \(\mathcal{I}_V^u\) (see Sites, Section 00VC) consists of pairs \((U, \varphi)\) where \(\varphi : [m] \to [n]\) and \(U \subset X_m\) open such that \(Y(\varphi)(V) \subset f_m^{-1}(U)\). Moreover, a morphism \((U, \varphi) \to (U', \varphi')\) is given by a \(\psi : [m'] \to [m]\) such that \(X(\psi)(U) \subset U'\) and \(\varphi \circ \psi = \varphi'\). It is our task to show that \(\mathcal{I}_V^u\) is cofiltered.

We verify the conditions of Categories, Definition 04AZ. Condition (1) holds because \((X_n, \text{id}_{[n]})\) is an object. Let \((U, \varphi)\) be an object. The condition \(Y(\varphi)(V) \subset f_m^{-1}(U)\) is equivalent to \(V \subset f_n^{-1}(X(\varphi)^{-1}(U))\). Hence we obtain a morphism \((X(\varphi)^{-1}(U), \text{id}_{[n]}) \to (U, \varphi)\) given by setting \(\psi = \varphi\). Moreover, given a pair of objects of the form \((U, \text{id}_{[n]})\) and \((U', \text{id}_{[n]})\) we see there exists an object, namely \((U \cap U', \text{id}_{[n]})\), which maps to both of them. Thus condition (2) holds. To verify condition (3) suppose given two morphisms \(a, a': (U, \varphi) \to (U', \varphi')\) given by \(\psi, \psi' : [m'] \to [m]\). Then precomposing with the morphism \((X(\varphi)^{-1}(U), \text{id}_{[n]}) \to (U, \varphi)\) given by \(\varphi\) equalizes \(a, a'\) because \(\varphi \circ \psi = \varphi' = \varphi \circ \psi'\). This finishes the proof.

Lemma

Let \(f : Y \to X\) be a morphism of simplicial spaces. In terms of the description of sheaves in Lemma 09VM the morphism \(f_{Zar}\) of Lemma 09VN can be described as follows.

1. If \(\mathcal{G}\) is a sheaf on \(Y\), then \((f_{Zar, *}\mathcal{G})_n = f_{n, *}\mathcal{G}_n\).

2. If \(\mathcal{F}\) is a sheaf on \(X\), then \((f_{Zar}^{-1}\mathcal{F})_n = f_n^{-1}\mathcal{F}_n\).

Proof

The first part is immediate from the definitions. For the second part, note that in the proof of Lemma 09VN we have shown that for a \(V \subset Y_n\) open the category \((\mathcal{I}_V^u)^{opp}\) contains as a cofinal subcategory the category of opens \(U \subset X_n\) with \(f_n^{-1}(U) \supset V\) and morphisms given by inclusions. Hence we see that the restriction of \(u_p\mathcal{F}\) to opens of \(Y_n\) is the presheaf \(f_{n, p}\mathcal{F}_n\) as defined in Sheaves, Lemma 008F. Since \(f_{Zar}^{-1}\mathcal{F} = u_s\mathcal{F}\) is the sheafification of \(u_p\mathcal{F}\) and since sheafification uses only coverings and since coverings in \(Y_{Zar}\) use only inclusions between opens on the same \(Y_n\), the result follows from the fact that \(f_n^{-1}\mathcal{F}_n\) is (correspondingly) the sheafification of \(f_{n, p}\mathcal{F}_n\), see Sheaves, Section 008C.

Let \(X\) be a topological space. In Sites, Example 00VJ we denoted \(X_{Zar}\) the site consisting of opens of \(X\) with inclusions as morphisms and coverings given by open coverings. We identify the topos \(\Sh(X_{Zar})\) with the category of sheaves on \(X\).

Lemma

Let \(X\) be a simplicial space. The functor \(X_{n, Zar} \to X_{Zar}\), \(U \mapsto U\) is continuous and cocontinuous. The associated morphism of topoi \(g_n : \Sh(X_n) \to \Sh(X_{Zar})\) satisfies

1. \(g_n^{-1}\) associates to the sheaf \(\mathcal{F}\) on \(X\) the sheaf \(\mathcal{F}_n\) on \(X_n\),

2. \(g_n^{-1} : \Sh(X_{Zar}) \to \Sh(X_n)\) has a left adjoint \(g^{Sh}_{n!}\),

3. \(g^{Sh}_{n!}\) commutes with finite connected limits,

4. \(g_n^{-1} : \textit{Ab}(X_{Zar}) \to \textit{Ab}(X_n)\) has a left adjoint \(g_{n!}\), and

5. \(g_{n!}\) is exact.

Proof

Besides the properties of our functor mentioned in the statement, the category \(X_{n, Zar}\) has fibre products and equalizers and the functor commutes with them (beware that \(X_{Zar}\) does not have all fibre products). Hence the lemma follows from the discussion in Sites, Sections 00XI and 00XN and Modules on Sites, Section 04BE. More precisely, Sites, Lemmas 00XO, 00XR, and 00XS and Modules on Sites, Lemmas 04BG and 04BH.

Lemma

Let \(X\) be a simplicial space. If \(\mathcal{I}\) is an injective abelian sheaf on \(X_{Zar}\), then \(\mathcal{I}_n\) is an injective abelian sheaf on \(X_n\).

Proof

This follows from Homology, Lemma 015Z and Lemma 09W0.

Lemma

Let \(f : Y \to X\) be a morphism of simplicial spaces. Then \[\xymatrix{ \Sh(Y_n) \ar[d] \ar[r]_{f_n} & \Sh(X_n) \ar[d] \\ \Sh(Y_{Zar}) \ar[r]^{f_{Zar}} & \Sh(X_{Zar}) }\] is a commutative diagram of topoi.

Proof

Direct from the description of pullback functors in Lemmas 09VP and 09W0.

Lemma

Let \(Y\) be a simplicial space and let \(a : Y \to X\) be an augmentation (Simplicial, Definition 018G). Let \(a_n : Y_n \to X\) be the corresponding morphisms of topological spaces. There is a canonical morphism of topoi \[a : \Sh(Y_{Zar}) \to \Sh(X)\] with the following properties:

1. \(a^{-1}\mathcal{F}\) is the sheaf restricting to \(a_n^{-1}\mathcal{F}\) on \(Y_n\),

2. \(a_m \circ Y(\varphi) = a_n\) for all \(\varphi : [m] \to [n]\),

3. \(a \circ g_n = a_n\) as morphisms of topoi with \(g_n\) as in Lemma 09W0,

4. \(a_*\mathcal{G}\) for \(\mathcal{G} \in \Sh(Y_{Zar})\) is the equalizer of the two maps \(a_{0, *}\mathcal{G}_0 \to a_{1, *}\mathcal{G}_1\).

Proof

Part (2) holds for augmentations of simplicial objects in any category. Thus \(Y(\varphi)^{-1} a_m^{-1} \mathcal{F} = a_n^{-1}\mathcal{F}\) which defines an \(Y(\varphi)\)-map from \(a_m^{-1}\mathcal{F}\) to \(a_n^{-1}\mathcal{F}\). Thus we can use (1) as the definition of \(a^{-1}\mathcal{F}\) (using Lemma 09VM) and (4) as the definition of \(a_*\). If this defines a morphism of topoi then part (3) follows because we’ll have \(g_n^{-1} \circ a^{-1} = a_n^{-1}\) by construction. To check \(a\) is a morphism of topoi we have to show that \(a^{-1}\) is left adjoint to \(a_*\) and we have to show that \(a^{-1}\) is exact. The last fact is immediate from the exactness of the functors \(a_n^{-1}\).

Let \(\mathcal{F}\) be an object of \(\Sh(X)\) and let \(\mathcal{G}\) be an object of \(\Sh(Y_{Zar})\). Given \(\beta : a^{-1}\mathcal{F} \to \mathcal{G}\) we can look at the components \(\beta_n : a_n^{-1}\mathcal{F} \to \mathcal{G}_n\). These maps are adjoint to maps \(\beta_n : \mathcal{F} \to a_{n, *}\mathcal{G}_n\). Compatibility with the simplicial structure shows that \(\beta_0\) maps into \(a_*\mathcal{G}\). Conversely, suppose given a map \(\alpha : \mathcal{F} \to a_*\mathcal{G}\). For any \(n\) choose a \(\varphi : [0] \to [n]\). Then we can look at the composition \[\mathcal{F} \xrightarrow{\alpha} a_*\mathcal{G} \to a_{0, *}\mathcal{G}_0 \xrightarrow{\mathcal{G}(\varphi)} a_{n, *}\mathcal{G}_n\] These are adjoint to maps \(a_n^{-1}\mathcal{F} \to \mathcal{G}_n\) which define a morphism of sheaves \(a^{-1}\mathcal{F} \to \mathcal{G}\). We omit the proof that the constructions given above define mutually inverse bijections \[\Mor_{\Sh(Y_{Zar})}(a^{-1}\mathcal{F}, \mathcal{G}) = \Mor_{\Sh(X)}(\mathcal{F}, a_*\mathcal{G})\] This finishes the proof. An interesting observation is here that this morphism of topoi does not correspond to any obvious geometric functor between the sites defining the topoi.

Lemma

Let \(X\) be a simplicial topological space. The complex of abelian presheaves on \(X_{Zar}\) \[\ldots \to \mathbf{Z}_{X_2} \to \mathbf{Z}_{X_1} \to \mathbf{Z}_{X_0}\] with boundary \(\sum (-1)^i d^n_i\) is a resolution of the constant presheaf \(\mathbf{Z}\).

Proof

Let \(U \subset X_m\) be an object of \(X_{Zar}\). Then the value of the complex above on \(U\) is the complex of abelian groups \[\ldots \to \mathbf{Z}[\Mor_\Delta([2], [m])] \to \mathbf{Z}[\Mor_\Delta([1], [m])] \to \mathbf{Z}[\Mor_\Delta([0], [m])]\] In other words, this is the complex associated to the free abelian group on the simplicial set \(\Delta[m]\), see Simplicial, Example 0176. Since \(\Delta[m]\) is homotopy equivalent to \(\Delta[0]\), see Simplicial, Example 08Q3, and since “taking free abelian groups” is a functor, we see that the complex above is homotopy equivalent to the free abelian group on \(\Delta[0]\) (Simplicial, Remark 019M and Lemma 019T). This complex is acyclic in positive degrees and equal to \(\mathbf{Z}\) in degree \(0\).

Lemma

Let \(X\) be a simplicial topological space. Let \(\mathcal{F}\) be an abelian sheaf on \(X\). There is a spectral sequence \((E_r, d_r)_{r \geq 0}\) with \[E_1^{p, q} = H^q(X_p, \mathcal{F}_p)\] converging to \(H^{p + q}(X_{Zar}, \mathcal{F})\). This spectral sequence is functorial in \(\mathcal{F}\).

Proof

Let \(\mathcal{F} \to \mathcal{I}^\bullet\) be an injective resolution. Consider the double complex with terms \[A^{p, q} = \mathcal{I}^q(X_p)\] and first differential given by the alternating sum along the maps \(d^{p + 1}_i\)-maps \(\mathcal{I}_p^q \to \mathcal{I}_{p + 1}^q\), see Lemma 09VM. Note that \[A^{p, q} = \Gamma(X_p, \mathcal{I}_p^q) = \Mor_{\textit{PSh}}(h_{X_p}, \mathcal{I}^q) = \Mor_{\textit{PAb}}(\mathbf{Z}_{X_p}, \mathcal{I}^q)\] Hence it follows from Lemma 09W5 and Cohomology on Sites, Lemma 03F6 that the rows of the double complex are exact in positive degrees and evaluate to \(\Gamma(X_{Zar}, \mathcal{I}^q)\) in degree \(0\). On the other hand, since restriction is exact (Lemma 09W0) the map \[\mathcal{F}_p \to \mathcal{I}_p^\bullet\] is a resolution. The sheaves \(\mathcal{I}_p^q\) are injective abelian sheaves on \(X_p\) (Lemma 09W1). Hence the cohomology of the columns computes the groups \(H^q(X_p, \mathcal{F}_p)\). We conclude by applying Homology, Lemmas 0132 and 0133.

Lemma

Let \(X\) be a simplicial space and let \(a : X \to Y\) be an augmentation. Let \(\mathcal{F}\) be an abelian sheaf on \(X_{Zar}\). Then \(R^na_*\mathcal{F}\) is the sheaf associated to the presheaf \[V \longmapsto H^n((X \times_Y V)_{Zar}, \mathcal{F}|_{(X \times_Y V)_{Zar}})\]

Proof

This is the analogue of Cohomology, Lemma 01E4 or of Cohomology on Sites, Lemma 072W and we strongly encourage the reader to skip the proof. Choosing an injective resolution of \(\mathcal{F}\) on \(X_{Zar}\) and using the definitions we see that it suffices to show: (1) the restriction of an injective abelian sheaf on \(X_{Zar}\) to \((X \times_Y V)_{Zar}\) is an injective abelian sheaf and (2) \(a_*\mathcal{F}\) is equal to the rule \[V \longmapsto H^0((X \times_Y V)_{Zar}, \mathcal{F}|_{(X \times_Y V)_{Zar}})\] Part (2) follows from the following facts

1. \(a_*\mathcal{F}\) is the equalizer of the two maps \(a_{0, *}\mathcal{F}_0 \to a_{1, *}\mathcal{F}_1\) by Lemma 09W4,

2. \(a_{0, *}\mathcal{F}_0(V) = H^0(a_0^{-1}(V), \mathcal{F}_0)\) and \(a_{1, *}\mathcal{F}_1(V) = H^0(a_1^{-1}(V), \mathcal{F}_1)\),

3. \(X_0 \times_Y V = a_0^{-1}(V)\) and \(X_1 \times_Y V = a_1^{-1}(V)\),

4. \(H^0((X \times_Y V)_{Zar}, \mathcal{F}|_{(X \times_Y V)_{Zar}})\) is the equalizer of the two maps \(H^0(X_0 \times_Y V, \mathcal{F}_0) \to H^0(X_1 \times_Y V, \mathcal{F}_1)\) for example by Lemma 09W6.

Part (1) follows after one defines an exact left adjoint \(j_! : \textit{Ab}((X \times_Y V)_{Zar}) \to \textit{Ab}(X_{Zar})\) (extension by zero) to restriction \(\textit{Ab}(X_{Zar}) \to \textit{Ab}((X \times_Y V)_{Zar})\) and using Homology, Lemma 015Z.

Let \(X\) be a topological space. Denote \(X_\bullet\) the constant simplicial topological space with value \(X\). By Lemma 09VM a sheaf on \(X_{\bullet, Zar}\) is the same thing as a cosimplicial object in the category of sheaves on \(X\).

Lemma

Let \(X\) be a topological space. Let \(X_\bullet\) be the constant simplicial topological space with value \(X\). The functor \[X_{\bullet, Zar} \longrightarrow X_{Zar},\quad U \longmapsto U\] is continuous and cocontinuous and defines a morphism of topoi \(g : \Sh(X_{\bullet, Zar}) \to \Sh(X)\) as well as a left adjoint \(g_!\) to \(g^{-1}\). We have

1. \(g^{-1}\) associates to a sheaf on \(X\) the constant cosimplicial sheaf on \(X\),

2. \(g_!\) associates to a sheaf \(\mathcal{F}\) on \(X_{\bullet, Zar}\) the sheaf \(\mathcal{F}_0\), and

3. \(g_*\) associates to a sheaf \(\mathcal{F}\) on \(X_{\bullet, Zar}\) the equalizer of the two maps \(\mathcal{F}_0 \to \mathcal{F}_1\).

Proof

The statements about the functor are straightforward to verify. The existence of \(g\) and \(g_!\) follow from Sites, Lemmas 00XO and 00XR. The description of \(g^{-1}\) is immediate from Sites, Lemma 00XR. The description of \(g_*\) and \(g_!\) follows as the functors given are right and left adjoint to \(g^{-1}\).

Simplicial sites and topoi

It seems natural to define a simplicial site as a simplicial object in the (big) category whose objects are sites and whose morphisms are morphisms of sites. See Sites, Definitions 00VH and 00X1 with composition of morphisms as in Sites, Lemma 03CB. But here are some variants one might want to consider: (a) we could work with cocontinuous functors (see Sites, Sections 00XI and 00XN) between sites instead, (b) we could work in a suitable \(2\)-category of sites where one introduces the notion of a \(2\)-morphism between morphisms of sites, (c) we could work in a \(2\)-category constructed out of cocontinuous functors. Instead of picking one of these variants as a definition we will simply develop theory as needed.

Certainly a simplicial topos should probably be defined as a pseudo-functor from \(\Delta^{opp}\) into the \(2\)-category of topoi. See Categories, Definition 003N and Sites, Section 00X9 and 04I9. We will try to avoid working with such a beast if possible.

Case A. Let \(\mathcal{C}\) be a simplicial object in the category whose objects are sites and whose morphisms are morphisms of sites. This means that for every morphism \(\varphi : [m] \to [n]\) of \(\Delta\) we have a morphism of sites \(f_\varphi : \mathcal{C}_n \to \mathcal{C}_m\). This morphism is given by a continuous functor in the opposite direction which we will denote \(u_\varphi : \mathcal{C}_m \to \mathcal{C}_n\).

Lemma

Let \(\mathcal{C}\) be a simplicial object in the category of sites. With notation as above we construct a site \(\mathcal{C}_{total}\) as follows.

1. An object of \(\mathcal{C}_{total}\) is an object \(U\) of \(\mathcal{C}_n\) for some \(n\),

2. a morphism \((\varphi, f) : U \to V\) of \(\mathcal{C}_{total}\) is given by a map \(\varphi : [m] \to [n]\) with \(U \in \Ob(\mathcal{C}_n)\), \(V \in \Ob(\mathcal{C}_m)\) and a morphism \(f : U \to u_\varphi(V)\) of \(\mathcal{C}_n\), and

3. a covering \(\{(\text{id}, f_i) : U_i \to U\}\) in \(\mathcal{C}_{total}\) is given by an \(n\) and a covering \(\{f_i : U_i \to U\}\) of \(\mathcal{C}_n\).

Proof

Composition of \((\varphi, f) : U \to V\) with \((\psi, g) : V \to W\) is given by \((\varphi \circ \psi, u_\varphi(g) \circ f)\). This uses that \(u_\varphi \circ u_\psi = u_{\varphi \circ \psi}\).

Let \(\{(\text{id}, f_i) : U_i \to U\}\) be a covering as in (3) and let \((\varphi, g) : W \to U\) be a morphism with \(W \in \Ob(\mathcal{C}_m)\). We claim that \[W \times_{(\varphi, g), U, (\text{id}, f_i)} U_i = W \times_{g, u_\varphi(U), u_\varphi(f_i)} u_\varphi(U_i)\] in the category \(\mathcal{C}_{total}\). This makes sense as by our definition of morphisms of sites, the required fibre products in \(\mathcal{C}_m\) exist since \(u_\varphi\) transforms coverings into coverings. The same reasoning implies the claim (details omitted). Thus we see that the collection of coverings is stable under base change. The other axioms of a site are immediate.

Case B. Let \(\mathcal{C}\) be a simplicial object in the category whose objects are sites and whose morphisms are cocontinuous functors. This means that for every morphism \(\varphi : [m] \to [n]\) of \(\Delta\) we have a cocontinuous functor denoted \(u_\varphi : \mathcal{C}_n \to \mathcal{C}_m\). The associated morphism of topoi is denoted \(f_\varphi : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_m)\).

Lemma

Let \(\mathcal{C}\) be a simplicial object in the category whose objects are sites and whose morphisms are cocontinuous functors. With notation as above, assume the functors \(u_\varphi : \mathcal{C}_n \to \mathcal{C}_m\) have property \(P\) of Sites, Remark 09W7. Then we can construct a site \(\mathcal{C}_{total}\) as follows.

1. An object of \(\mathcal{C}_{total}\) is an object \(U\) of \(\mathcal{C}_n\) for some \(n\),

2. a morphism \((\varphi, f) : U \to V\) of \(\mathcal{C}_{total}\) is given by a map \(\varphi : [m] \to [n]\) with \(U \in \Ob(\mathcal{C}_n)\), \(V \in \Ob(\mathcal{C}_m)\) and a morphism \(f : u_\varphi(U) \to V\) of \(\mathcal{C}_m\), and

3. a covering \(\{(\text{id}, f_i) : U_i \to U\}\) in \(\mathcal{C}_{total}\) is given by an \(n\) and a covering \(\{f_i : U_i \to U\}\) of \(\mathcal{C}_n\).

Proof

Composition of \((\varphi, f) : U \to V\) with \((\psi, g) : V \to W\) is given by \((\varphi \circ \psi, g \circ u_\psi(f))\). This uses that \(u_\psi \circ u_\varphi = u_{\varphi \circ \psi}\).

Let \(\{(\text{id}, f_i) : U_i \to U\}\) be a covering as in (3) and let \((\varphi, g) : W \to U\) be a morphism with \(W \in \Ob(\mathcal{C}_m)\). We claim that \[W \times_{(\varphi, g), U, (\text{id}, f_i)} U_i = W \times_{g, U, f_i} U_i\] in the category \(\mathcal{C}_{total}\) where the right hand side is the object of \(\mathcal{C}_m\) defined in Sites, Remark 09W7 which exists by property \(P\). Compatibility of this type of fibre product with compositions of functors implies the claim (details omitted). Since the family \(\{W \times_{g, U, f_i} U_i \to W\}\) is a covering of \(\mathcal{C}_m\) by property \(P\) we see that the collection of coverings is stable under base change. The other axioms of a site are immediate.

Situation

Here we have one of the following two cases:

1. \(\mathcal{C}\) is a simplicial object in the category whose objects are sites and whose morphisms are morphisms of sites. For every morphism \(\varphi : [m] \to [n]\) of \(\Delta\) we have a morphism of sites \(f_\varphi : \mathcal{C}_n \to \mathcal{C}_m\) given by a continuous functor \(u_\varphi : \mathcal{C}_m \to \mathcal{C}_n\).

2. \(\mathcal{C}\) is a simplicial object in the category whose objects are sites and whose morphisms are cocontinuous functors having property \(P\) of Sites, Remark 09W7. For every morphism \(\varphi : [m] \to [n]\) of \(\Delta\) we have a cocontinuous functor \(u_\varphi : \mathcal{C}_n \to \mathcal{C}_m\) which induces a morphism of topoi \(f_\varphi : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_m)\).

As usual we will denote \(f_\varphi^{-1}\) and \(f_{\varphi, *}\) the pullback and pushforward. We let \(\mathcal{C}_{total}\) denote the site defined in Lemma 09WC (case A) or Lemma 09WD (case B).

Let \(\mathcal{C}\) be as in Situation 09WE. Let \(\mathcal{F}\) be a sheaf on \(\mathcal{C}_{total}\). It is clear from the definition of coverings, that the restriction of \(\mathcal{F}\) to the objects of \(\mathcal{C}_n\) defines a sheaf \(\mathcal{F}_n\) on the site \(\mathcal{C}_n\). For every \(\varphi : [m] \to [n]\) the restriction maps of \(\mathcal{F}\) along the morphisms \((\varphi, f) : U \to V\) with \(U \in \Ob(\mathcal{C}_n)\) and \(V \in \Ob(\mathcal{C}_m)\) define an element \(\mathcal{F}(\varphi)\) of \[\Mor_{\Sh(\mathcal{C}_m)}(\mathcal{F}_m, f_{\varphi, *}\mathcal{F}_n) = \Mor_{\Sh(\mathcal{C}_n)}(f_\varphi^{-1}\mathcal{F}_m, \mathcal{F}_n)\] Moreover, given \(\varphi : [m] \to [n]\) and \(\psi : [l] \to [m]\) the diagrams \[\vcenter{ \xymatrix{ \mathcal{F}_l \ar[rr]_{\mathcal{F}(\varphi \circ \psi)} \ar[rd]_{\mathcal{F}(\psi)} & & f_{\varphi \circ \psi, *} \mathcal{F}_n \\ & f_{\psi, *}\mathcal{F}_m \ar[ur]_{f_{\psi, *}\mathcal{F}(\varphi)} } } \quad\text{and}\quad \vcenter{ \xymatrix{ f_{\varphi \circ \psi}^{-1}\mathcal{F}_l \ar[rr]_{\mathcal{F}(\varphi \circ \psi)} \ar[rd]_{f_\varphi^{-1}\mathcal{F}(\psi)} & & \mathcal{F}_n \\ & f_\varphi^{-1}\mathcal{F}_m \ar[ur]_{\mathcal{F}(\varphi)} } }\] commute. Clearly, the converse statement is true as well: if we have a system \((\{\mathcal{F}_n\}_{n \geq 0}, \{\mathcal{F}(\varphi)\}_{\varphi \in \text{Arrows}(\Delta)})\) satisfying the commutativity constraints above, then we obtain a sheaf on \(\mathcal{C}_{total}\).

Lemma

In Situation 09WE there is an equivalence of categories between

1. \(\Sh(\mathcal{C}_{total})\), and

2. the category of systems \((\mathcal{F}_n, \mathcal{F}(\varphi))\) described above.

In particular, the topos \(\Sh(\mathcal{C}_{total})\) only depends on the topoi \(\Sh(\mathcal{C}_n)\) and the morphisms of topoi \(f_\varphi\).

Proof

See discussion above.

Lemma

In Situation 09WE the functor \(\mathcal{C}_n \to \mathcal{C}_{total}\), \(U \mapsto U\) is continuous and cocontinuous. The associated morphism of topoi \(g_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_{total})\) satisfies

1. \(g_n^{-1}\) associates to the sheaf \(\mathcal{F}\) on \(\mathcal{C}_{total}\) the sheaf \(\mathcal{F}_n\) on \(\mathcal{C}_n\),

2. \(g_n^{-1} : \Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{C}_n)\) has a left adjoint \(g^{Sh}_{n!}\),

3. for \(\mathcal{G}\) in \(\Sh(\mathcal{C}_n)\) the restriction of \(g_{n!}^{Sh}\mathcal{G}\) to \(\mathcal{C}_m\) is \(\coprod\nolimits_{\varphi : [n] \to [m]} f_\varphi^{-1}\mathcal{G}\),

4. \(g_{n!}^{Sh}\) commutes with finite connected limits,

5. \(g_n^{-1} : \textit{Ab}(\mathcal{C}_{total}) \to \textit{Ab}(\mathcal{C}_n)\) has a left adjoint \(g_{n!}\),

6. for \(\mathcal{G}\) in \(\textit{Ab}(\mathcal{C}_n)\) the restriction of \(g_{n!}\mathcal{G}\) to \(\mathcal{C}_m\) is \(\bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^{-1}\mathcal{G}\), and

7. \(g_{n!}\) is exact.

Proof

Case A. If \(\{U_i \to U\}_{i \in I}\) is a covering in \(\mathcal{C}_n\) then the image \(\{U_i \to U\}_{i \in I}\) is a covering in \(\mathcal{C}_{total}\) by definition (Lemma 09WC). For a morphism \(V \to U\) of \(\mathcal{C}_n\), the fibre product \(V \times_U U_i\) in \(\mathcal{C}_n\) is also the fibre product in \(\mathcal{C}_{total}\) (by the claim in the proof of Lemma 09WC). Therefore our functor is continuous. On the other hand, our functor defines a bijection between coverings of \(U\) in \(\mathcal{C}_n\) and coverings of \(U\) in \(\mathcal{C}_{total}\). Therefore it is certainly the case that our functor is cocontinuous.

Case B. If \(\{U_i \to U\}_{i \in I}\) is a covering in \(\mathcal{C}_n\) then the image \(\{U_i \to U\}_{i \in I}\) is a covering in \(\mathcal{C}_{total}\) by definition (Lemma 09WD). For a morphism \(V \to U\) of \(\mathcal{C}_n\), the fibre product \(V \times_U U_i\) in \(\mathcal{C}_n\) is also the fibre product in \(\mathcal{C}_{total}\) (by the claim in the proof of Lemma 09WD). Therefore our functor is continuous. On the other hand, our functor defines a bijection between coverings of \(U\) in \(\mathcal{C}_n\) and coverings of \(U\) in \(\mathcal{C}_{total}\). Therefore it is certainly the case that our functor is cocontinuous.

At this point part (1) and the existence of \(g^{Sh}_{n!}\) and \(g_{n!}\) in cases A and B follows from Sites, Lemmas 00XO and 00XR and Modules on Sites, Lemma 04BG.

Proof of (3). Let \(\mathcal{G}\) be a sheaf on \(\mathcal{C}_n\). Consider the sheaf \(\mathcal{H}\) on \(\mathcal{C}_{total}\) whose degree \(m\) part is the sheaf \[\mathcal{H}_m = \coprod\nolimits_{\varphi : [n] \to [m]} f_\varphi^{-1}\mathcal{G}\] given in part (3) of the statement of the lemma. Given a map \(\psi : [m] \to [m']\) the map \(\mathcal{H}(\psi) : f_\psi^{-1}\mathcal{H}_m \to \mathcal{H}_{m'}\) is given on components by the identifications \[f_\psi^{-1} f_\varphi^{-1} \mathcal{G} \to f_{\psi \circ \varphi}^{-1}\mathcal{G}\] Observe that given a map \(\alpha : \mathcal{H} \to \mathcal{F}\) of sheaves on \(\mathcal{C}_{total}\) we obtain a map \(\mathcal{G} \to \mathcal{F}_n\) corresponding to the restriction of \(\alpha_n\) to the component \(\mathcal{G}\) in \(\mathcal{H}_n\). Conversely, given a map \(\beta : \mathcal{G} \to \mathcal{F}_n\) of sheaves on \(\mathcal{C}_n\) we can define \(\alpha : \mathcal{H} \to \mathcal{F}\) by letting \(\alpha_m\) be the map which on components \[f_\varphi^{-1}\mathcal{G} \to \mathcal{F}_m\] uses the maps adjoint to \(\mathcal{F}(\varphi) \circ f_\varphi^{-1}\beta\). We omit the arguments showing these two constructions give mutually inverse maps \[\Mor_{\Sh(\mathcal{C}_n)}(\mathcal{G}, \mathcal{F}_n) = \Mor_{\Sh(\mathcal{C}_{total})}(\mathcal{H}, \mathcal{F})\] Thus \(\mathcal{H} = g^{Sh}_{n!}\mathcal{G}\) as desired.

Proof of (4). If \(\mathcal{G}\) is an abelian sheaf on \(\mathcal{C}_n\), then we proceed in exactly the same ammner as above, except that we define \(\mathcal{H}\) is the abelian sheaf on \(\mathcal{C}_{total}\) whose degree \(m\) part is the sheaf \[\bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^{-1}\mathcal{G}\] with transition maps defined exactly as above. The bijection \[\Mor_{\textit{Ab}(\mathcal{C}_n)}(\mathcal{G}, \mathcal{F}_n) = \Mor_{\textit{Ab}(\mathcal{C}_{total})}(\mathcal{H}, \mathcal{F})\] is proved exactly as above. Thus \(\mathcal{H} = g_{n!}\mathcal{G}\) as desired.

The exactness properties of \(g^{Sh}_{n!}\) and \(g_{n!}\) follow from formulas given for these functors.

Lemma

In Situation 09WE. If \(\mathcal{I}\) is injective in \(\textit{Ab}(\mathcal{C}_{total})\), then \(\mathcal{I}_n\) is injective in \(\textit{Ab}(\mathcal{C}_n)\). If \(\mathcal{I}^\bullet\) is a K-injective complex in \(\textit{Ab}(\mathcal{C}_{total})\), then \(\mathcal{I}_n^\bullet\) is K-injective in \(\textit{Ab}(\mathcal{C}_n)\).

Proof

The first statement follows from Homology, Lemma 015Z and Lemma 09WG. The second statement from Derived Categories, Lemma 08BJ and Lemma 09WG.

Augmentations of simplicial sites

We continue in the fashion described in Section 09WB working out the meaning of augmentations in cases A and B treated in that section.

Remark

In Situation 09WE an augmentation \(a_0\) towards a site \(\mathcal{D}\) will mean

1. \(a_0 : \mathcal{C}_0 \to \mathcal{D}\) is a morphism of sites given by a continuous functor \(u_0 : \mathcal{D} \to \mathcal{C}_0\) such that for all \(\varphi, \psi : [0] \to [n]\) we have \(u_\varphi \circ u_0 = u_\psi \circ u_0\).

2. \(a_0 : \Sh(\mathcal{C}_0) \to \Sh(\mathcal{D})\) is a morphism of topoi given by a cocontinuous functor \(u_0 : \mathcal{C}_0 \to \mathcal{D}\) such that for all \(\varphi, \psi : [0] \to [n]\) we have \(u_0 \circ u_\varphi = u_0 \circ u_\psi\).

Lemma

In Situation 09WE let \(a_0\) be an augmentation towards a site \(\mathcal{D}\) as in Remark 0D6Z. Then \(a_0\) induces

1. a morphism of topoi \(a_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{D})\) for all \(n \geq 0\),

2. a morphism of topoi \(a : \Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{D})\)

such that

1. for all \(\varphi : [m] \to [n]\) we have \(a_m \circ f_\varphi = a_n\),

2. if \(g_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_{total})\) is as in Lemma 09WG, then \(a \circ g_n = a_n\), and

3. \(a_*\mathcal{F}\) for \(\mathcal{F} \in \Sh(\mathcal{C}_{total})\) is the equalizer of the two maps \(a_{0, *}\mathcal{F}_0 \to a_{1, *}\mathcal{F}_1\).

Proof

Case A. Let \(u_n : \mathcal{D} \to \mathcal{C}_n\) be the common value of the functors \(u_\varphi \circ u_0\) for \(\varphi : [0] \to [n]\). Then \(u_n\) corresponds to a morphism of sites \(a_n : \mathcal{C}_n \to \mathcal{D}\), see Sites, Lemma 03CB. The same lemma shows that for all \(\varphi : [m] \to [n]\) we have \(a_m \circ f_\varphi = a_n\).

Case B. Let \(u_n : \mathcal{C}_n \to \mathcal{D}\) be the common value of the functors \(u_0 \circ u_\varphi\) for \(\varphi : [0] \to [n]\). Then \(u_n\) is cocontinuous and hence defines a morphism of topoi \(a_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{D)}\), see Sites, Lemma 03L5. The same lemma shows that for all \(\varphi : [m] \to [n]\) we have \(a_m \circ f_\varphi = a_n\).

Consider the functor \(a^{-1} : \Sh(\mathcal{D}) \to \Sh(\mathcal{C}_{total})\) which to a sheaf of sets \(\mathcal{G}\) associates the sheaf \(\mathcal{F} = a^{-1}\mathcal{G}\) whose components are \(a_n^{-1}\mathcal{G}\) and whose transition maps \(\mathcal{F}(\varphi)\) are the identifications \[f_\varphi^{-1}\mathcal{F}_m = f_\varphi^{-1} a_m^{-1}\mathcal{G} = a_n^{-1}\mathcal{G} = \mathcal{F}_n\] for \(\varphi : [m] \to [n]\), see the description of \(\Sh(\mathcal{C}_{total})\) in Lemma 09WF. Since the functors \(a_n^{-1}\) are exact, \(a^{-1}\) is an exact functor. Finally, for \(a_* : \Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{D})\) we take the functor which to a sheaf \(\mathcal{F}\) on \(\Sh(\mathcal{D})\) associates \[\xymatrix{ a_*\mathcal{F} \ar@{=}[r] & \text{Equalizer}(a_{0, *}\mathcal{F}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & a_{1, *}\mathcal{F}_1) }\] Here the two maps come from the two maps \(\varphi : [0] \to [1]\) via \[a_{0, *}\mathcal{F}_0 \to a_{0, *}f_{\varphi, *} f_\varphi^{-1}\mathcal{F}_0 \xrightarrow{\mathcal{F}(\varphi)} a_{0, *}f_{\varphi, *} \mathcal{F}_1 = a_{1, *}\mathcal{F}_1\] where the first arrow comes from \(1 \to f_{\varphi, *} f_\varphi^{-1}\). Let \(\mathcal{G}_\bullet\) denote the constant coosimplicial sheaf with value \(\mathcal{G}\) and let \(a_{\bullet, *}\mathcal{F}\) denote the cosimplicial sheaf having \(a_{n, *}\mathcal{F}_n\) in degree \(n\). By the usual adjuntion for the morphisms of topoi \(a_n\) we see that a map \(a^{-1}\mathcal{G} \to \mathcal{F}\) is the same thing as a map \[\mathcal{G}_\bullet \longrightarrow a_{\bullet, *}\mathcal{F}\] of cosimplicial sheaves. By the dual to Simplicial, Lemma 018H this is the same thing as a map \(\mathcal{G} \to a_*\mathcal{F}\). Thus \(a^{-1}\) and \(a_*\) are adjoint functors and we obtain our morphism of topoi \(a\)². The equalities \(a \circ g_n = f_n\) follow immediately from the definitions.

Morphisms of simplicial sites

We continue in the fashion described in Section 09WB working out the meaning of morphisms of simplicial sites in cases A and B treated in that section.

Remark

Let \(\mathcal{C}_n, f_\varphi, u_\varphi\) and \(\mathcal{C}'_n, f'_\varphi, u'_\varphi\) be as in Situation 09WE. A morphism \(h\) between simplicial sites will mean

1. Morphisms of sites \(h_n : \mathcal{C}_n \to \mathcal{C}'_n\) such that \(f'_\varphi \circ h_n = h_m \circ f_\varphi\) as morphisms of sites for all \(\varphi : [m] \to [n]\).

2. Cocontinuous functors \(v_n : \mathcal{C}_n \to \mathcal{C}'_n\) inducing morphisms of topoi \(h_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}'_n)\) such that \(u'_\varphi \circ v_n = v_m \circ u_\varphi\) as functors for all \(\varphi : [m] \to [n]\).

In both cases we have \(f'_\varphi \circ h_n = h_m \circ f_\varphi\) as morphisms of topoi, see Sites, Lemma 03L5 for case B and Sites, Definition 03CC for case A.

Lemma

Let \(\mathcal{C}_n, f_\varphi, u_\varphi\) and \(\mathcal{C}'_n, f'_\varphi, u'_\varphi\) be as in Situation 09WE. Let \(h\) be a morphism between simplicial sites as in Remark 0D95. Then we obtain a morphism of topoi \[h_{total} : \Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{C}'_{total})\] and commutative diagrams \[\xymatrix{ \Sh(\mathcal{C}_n) \ar[d]_{g_n} \ar[r]_{h_n} & \Sh(\mathcal{C}'_n) \ar[d]^{g'_n} \\ \Sh(\mathcal{C}_{total}) \ar[r]^{h_{total}} & \Sh(\mathcal{C}'_{total}) }\] Moreover, we have \((g'_n)^{-1} \circ h_{total, *} = h_{n, *} \circ g_n^{-1}\).

Proof

Case A. Say \(h_n\) corresponds to the continuous functor \(v_n : \mathcal{C}'_n \to \mathcal{C}_n\). Then we can define a functor \(v_{total} : \mathcal{C}'_{total} \to \mathcal{C}_{total}\) by using \(v_n\) in degree \(n\). This is clearly a continuous functor (see definition of coverings in Lemma 09WC). Let \(h_{total}^{-1} = v_{total, s} : \Sh(\mathcal{C}'_{total}) \to \Sh(\mathcal{C}_{total})\) and \(h_{total, *} = v_{total}^s = v_{total}^p : \Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{C}'_{total})\) be the adjoint pair of functors constructed and studied in Sites, Sections 00WU and 00X0. To see that \(h_{total}\) is a morphism of topoi we still have to verify that \(h_{total}^{-1}\) is exact. We first observe that \((g'_n)^{-1} \circ h_{total, *} = h_{n, *} \circ g_n^{-1}\); this is immediate by computing sections over an object \(U\) of \(\mathcal{C}'_n\). Thus, if we think of a sheaf \(\mathcal{F}\) on \(\mathcal{C}_{total}\) as a system \((\mathcal{F}_n, \mathcal{F}(\varphi))\) as in Lemma 09WF, then \(h_{total, *}\mathcal{F}\) corresponds to the system \((h_{n, *}\mathcal{F}_n, h_{n, *}\mathcal{F}(\varphi))\). Clearly, the functor \((\mathcal{F}'_n, \mathcal{F}'(\varphi)) \to (h_n^{-1}\mathcal{F}'_n, h_n^{-1}\mathcal{F}'(\varphi))\) is its left adjoint. By uniqueness of adjoints, we conclude that \(h_{total}^{-1}\) is given by this rule on systems. In particular, \(h_{total}^{-1}\) is exact (by the description of sheaves on \(\mathcal{C}_{total}\) given in the lemma and the exactness of the functors \(h_n^{-1}\)) and we have our morphism of topoi. Finally, we obtain \(g_n^{-1} \circ h_{total}^{-1} = h_n^{-1} \circ (g'_n)^{-1}\) as well, which proves that the displayed diagram of the lemma commutes.

Case B. Here we have a functor \(v_{total} : \mathcal{C}_{total} \to \mathcal{C}'_{total}\) by using \(v_n\) in degree \(n\). This is clearly a cocontinuous functor (see definition of coverings in Lemma 09WD). Let \(h_{total}\) be the morphism of topoi associated to \(v_{total}\). The commutativity of the displayed diagram of the lemma follows immediately from Sites, Lemma 03L5. Taking left adjoints the final equality of the lemma becomes \[h_{total}^{-1} \circ (g'_n)^{Sh}_! = g^{Sh}_{n!} \circ h_n^{-1}\] This follows immediately from the explicit description of the functors \((g'_n)^{Sh}_!\) and \(g^{Sh}_{n!}\) in Lemma 09WG, the fact that \(h_n^{-1} \circ (f'_\varphi)^{-1} = f_\varphi^{-1} \circ h_m^{-1}\) for \(\varphi : [m] \to [n]\), and the fact that we already know \(h_{total}^{-1}\) commutes with restrictions to the degree \(n\) parts of the simplicial sites.

Lemma

With notation and hypotheses as in Lemma 0D96. For \(K \in D(\mathcal{C}_{total})\) we have \((g'_n)^{-1}Rh_{total, *}K = Rh_{n, *}g_n^{-1}K\).

Proof

Let \(\mathcal{I}^\bullet\) be a K-injective complex on \(\mathcal{C}_{total}\) representing \(K\). Then \(g_n^{-1}K\) is represented by \(g_n^{-1}\mathcal{I}^\bullet = \mathcal{I}_n^\bullet\) which is K-injective by Lemma 09WH. We have \((g'_n)^{-1}h_{total, *}\mathcal{I}^\bullet = h_{n, *}g_n^{-1}\mathcal{I}_n^\bullet\) by Lemma 0D96 which gives the desired equality.

Remark

Let \(\mathcal{C}_n, f_\varphi, u_\varphi\) and \(\mathcal{C}'_n, f'_\varphi, u'_\varphi\) be as in Situation 09WE. Let \(a_0\), resp. \(a'_0\) be an augmentation towards a site \(\mathcal{D}\), resp. \(\mathcal{D}'\) as in Remark 0D6Z. Let \(h\) be a morphism between simplicial sites as in Remark 0D95. We say a morphism of topoi \(h_{-1} : \Sh(\mathcal{D}) \to \Sh(\mathcal{D}')\) is compatible with \(h\), \(a_0\), \(a'_0\) if

1. \(h_{-1}\) comes from a morphism of sites \(h_{-1} : \mathcal{D} \to \mathcal{D}'\) such that \(a'_0 \circ h_0 = h_{-1} \circ a_0\) as morphisms of sites.

2. \(h_{-1}\) comes from a cocontinuous functor \(v_{-1} : \mathcal{D} \to \mathcal{D}'\) such that \(u'_0 \circ v_0 = v_{-1} \circ u_0\) as functors.

In both cases we have \(a'_0 \circ h_0 = h_{-1} \circ a_0\) as morphisms of topoi, see Sites, Lemma 03L5 for case B and Sites, Definition 03CC for case A.

Lemma

Let \(\mathcal{C}_n, f_\varphi, u_\varphi, \mathcal{D}, a_0\), \(\mathcal{C}'_n, f'_\varphi, u'_\varphi, \mathcal{D}', a'_0\), and \(h_n\), \(n \geq -1\) be as in Remark 0D98. Then we obtain a commutative diagram \[\xymatrix{ \Sh(\mathcal{C}_{total}) \ar[d]_a \ar[r]_{h_{total}} & \Sh(\mathcal{C}'_{total}) \ar[d]^{a'} \\ \Sh(\mathcal{D}) \ar[r]^{h_{-1}} & \Sh(\mathcal{D}') }\]

Proof

The morphism \(h\) is defined in Lemma 0D96. The morphisms \(a\) and \(a'\) are defined in Lemma 0D70. Thus the only thing is to prove the commutativity of the diagram. To do this, we prove that \(a^{-1} \circ h_{-1}^{-1} = h_{total}^{-1} \circ (a')^{-1}\). By the commutative diagrams of Lemma 0D96 and the description of \(\Sh(\mathcal{C}_{total})\) and \(\Sh(\mathcal{C}'_{total})\) in terms of components in Lemma 09WF, it suffices to show that \[\xymatrix{ \Sh(\mathcal{C}_n) \ar[d]_{a_n} \ar[r]_{h_n} & \Sh(\mathcal{C}'_n) \ar[d]^{a'_n} \\ \Sh(\mathcal{D}) \ar[r]^{h_{-1}} & \Sh(\mathcal{D}') }\] commutes for all \(n\). This follows from the case for \(n = 0\) (which is an assumption in Remark 0D98) and for \(n > 0\) we pick \(\varphi : [0] \to [n]\) and then the required commutativity follows from the case \(n = 0\) and the relations \(a_n = a_0 \circ f_\varphi\) and \(a'_n = a'_0 \circ f'_\varphi\) as well as the commutation relations \(f'_\varphi \circ h_n = h_0 \circ f_\varphi\).

Ringed simplicial sites

Let us endow our simplicial topos with a sheaf of rings.

Lemma

In Situation 09WE. Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}_{total}\). There is a canonical morphism of ringed topoi \(g_n : (\Sh(\mathcal{C}_n), \mathcal{O}_n) \to (\Sh(\mathcal{C}_{total}), \mathcal{O})\) agreeing with the morphism \(g_n\) of Lemma 09WG on underlying topoi. The functor \(g_n^* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_n)\) has a left adjoint \(g_{n!}\). For \(\mathcal{G}\) in \(\textit{Mod}(\mathcal{O}_n)\)-modules the restriction of \(g_{n!}\mathcal{G}\) to \(\mathcal{C}_m\) is \[\bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^*\mathcal{G}\] where \(f_\varphi : (\Sh(\mathcal{C}_m), \mathcal{O}_m) \to (\Sh(\mathcal{C}_n), \mathcal{O}_n)\) is the morphism of ringed topoi agreeing with the previously defined \(f_\varphi\) on topoi and using the map \(\mathcal{O}(\varphi) : f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m\) on sheaves of rings.

Proof

By Lemma 09WG we have \(g_n^{-1}\mathcal{O} = \mathcal{O}_n\) and hence we obtain our morphism of ringed topoi. By Modules on Sites, Lemma 0797 we obtain the adjoint \(g_{n!}\). To prove the formula for \(g_{n!}\) we first define a sheaf of \(\mathcal{O}\)-modules \(\mathcal{H}\) on \(\mathcal{C}_{total}\) with degree \(m\) component the \(\mathcal{O}_m\)-module \[\mathcal{H}_m = \bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^*\mathcal{G}\] Given a map \(\psi : [m] \to [m']\) the map \(\mathcal{H}(\psi) : f_\psi^{-1}\mathcal{H}_m \to \mathcal{H}_{m'}\) is given on components by \[f_\psi^{-1} f_\varphi^*\mathcal{G} \to f_\psi^* f_\varphi^*\mathcal{G} \to f_{\psi \circ \varphi}^*\mathcal{G}\] Since this map \(f_\psi^{-1}\mathcal{H}_m \to \mathcal{H}_{m'}\) is \(\mathcal{O}(\psi) : f_\psi^{-1}\mathcal{O}_m \to \mathcal{O}_{m'}\)-semi-linear, this indeed does define an \(\mathcal{O}\)-module (use Lemma 09WF). Then one proves directly that \[\Mor_{\mathcal{O}_n}(\mathcal{G}, \mathcal{F}_n) = \Mor_{\mathcal{O}}(\mathcal{H}, \mathcal{F})\] proceeding as in the proof of Lemma 09WG. Thus \(\mathcal{H} = g_{n!}\mathcal{G}\) as desired.

Lemma

In Situation 09WE. Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}_{total}\). If \(\mathcal{I}\) is injective in \(\textit{Mod}(\mathcal{O})\), then \(\mathcal{I}_n\) is a totally acyclic sheaf on \(\mathcal{C}_n\).

Proof

This follows from Cohomology on Sites, Lemma 0D6Y applied to the inclusion functor \(\mathcal{C}_n \to \mathcal{C}_{total}\) and its properties proven in Lemma 09WG.

Lemma

With assumptions as in Lemma 0D72 the functor \(g_{n!} : \textit{Mod}(\mathcal{O}_n) \to \textit{Mod}(\mathcal{O})\) is exact if the maps \(f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m\) are flat for all \(\varphi : [n] \to [m]\).

Proof

Recall that \(g_{n!}\mathcal{G}\) is the \(\mathcal{O}\)-module whose degree \(m\) part is the \(\mathcal{O}_m\)-module \[\bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^*\mathcal{G}\] Here the morphism of ringed topoi \(f_\varphi : (\Sh(\mathcal{C}_m), \mathcal{O}_m) \to (\Sh(\mathcal{C}_n), \mathcal{O}_n)\) uses the map \(f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m\) of the statement of the lemma. If these maps are flat, then \(f_\varphi^*\) is exact (Modules on Sites, Lemma 04JC). By definition of the site \(\mathcal{C}_{total}\) we see that these functors have the desired exactness properties and we conclude.

Lemma

In Situation 09WE. Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}_{total}\) such that \(f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m\) is flat for all \(\varphi : [n] \to [m]\). If \(\mathcal{I}\) is injective in \(\textit{Mod}(\mathcal{O})\), then \(\mathcal{I}_n\) is injective in \(\textit{Mod}(\mathcal{O}_n)\).

Proof

This follows from Homology, Lemma 015Z and Lemma 0D74.

Morphisms of ringed simplicial sites

We continue the discussion of Section 0D94.

Remark

Let \(\mathcal{C}_n, f_\varphi, u_\varphi\) and \(\mathcal{C}'_n, f'_\varphi, u'_\varphi\) be as in Situation 09WE. Let \(\mathcal{O}\) and \(\mathcal{O}'\) be a sheaf of rings on \(\mathcal{C}_{total}\) and \(\mathcal{C}'_{total}\). We will say that \((h, h^\sharp)\) is a morphism between ringed simplicial sites if \(h\) is a morphism between simplicial sites as in Remark 0D95 and \(h^\sharp : h_{total}^{-1}\mathcal{O}' \to \mathcal{O}\) or equivalently \(h^\sharp : \mathcal{O}' \to h_{total, *}\mathcal{O}\) is a homomorphism of sheaves of rings.

Lemma

Let \(\mathcal{C}_n, f_\varphi, u_\varphi\) and \(\mathcal{C}'_n, f'_\varphi, u'_\varphi\) be as in Situation 09WE. Let \(\mathcal{O}\) and \(\mathcal{O}'\) be a sheaf of rings on \(\mathcal{C}_{total}\) and \(\mathcal{C}'_{total}\). Let \((h, h^\sharp)\) be a morphism between simplicial sites as in Remark 0DGZ. Then we obtain a morphism of ringed topoi \[h_{total} : (\Sh(\mathcal{C}_{total}), \mathcal{O}) \to (\Sh(\mathcal{C}'_{total}), \mathcal{O}')\] and commutative diagrams \[\xymatrix{ (\Sh(\mathcal{C}_n), \mathcal{O}_n) \ar[d]_{g_n} \ar[r]_{h_n} & (\Sh(\mathcal{C}'_n), \mathcal{O}'_n) \ar[d]^{g'_n} \\ (\Sh(\mathcal{C}_{total}), \mathcal{O}) \ar[r]^{h_{total}} & (\Sh(\mathcal{C}'_{total}), \mathcal{O}') }\] of ringed topoi where \(g_n\) and \(g'_n\) are as in Lemma 0D72. Moreover, we have \((g'_n)^* \circ h_{total, *} = h_{n, *} \circ g_n^*\) as functor \(\textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}'_n)\).

Proof

Follows from Lemma 0D96 and 0D72 by keeping track of the sheaves of rings. A small point is that in order to define \(h_n\) as a morphism of ringed topoi we set \(h_n^\sharp = g_n^{-1}h^\sharp : g_n^{-1}h_{total}^{-1}\mathcal{O}' \to g_n^{-1}\mathcal{O}\) which makes sense because \(g_n^{-1}h_{total}^{-1}\mathcal{O}' = h_n^{-1}(g'_n)^{-1}\mathcal{O}' = h_n^{-1}\mathcal{O}'_n\) and \(g_n^{-1}\mathcal{O} = \mathcal{O}_n\). Note that \(g_n^*\mathcal{F} = g_n^{-1}\mathcal{F}\) for a sheaf of \(\mathcal{O}\)-modules \(\mathcal{F}\) and similarly for \(g'_n\) and this helps explain why \((g'_n)^* \circ h_{total, *} = h_{n, *} \circ g_n^*\) follows from the corresponding statement of Lemma 0D96.

Lemma

With notation and hypotheses as in Lemma 0DH0. For \(K \in D(\mathcal{O})\) we have \((g'_n)^*Rh_{total, *}K = Rh_{n, *}g_n^*K\).

Proof

Recall that \(g_n^* = g_n^{-1}\) because \(g_n^{-1}\mathcal{O} = \mathcal{O}_n\) by the construction in Lemma 0D72. In particular \(g_n^*\) is exact and \(Lg_n^*\) is given by applying \(g_n^*\) to any representative complex of modules. Similarly for \(g'_n\). There is a canonical base change map \((g'_n)^*Rh_{total, *}K \to Rh_{n, *}g_n^*K\), see Cohomology on Sites, Remark 07A7. By Cohomology on Sites, Lemma 0D6J the image of this in \(D(\mathcal{C}'_n)\) is the map \((g'_n)^{-1}Rh_{total, *}K_{ab} \to Rh_{n, *}g_n^{-1}K_{ab}\) where \(K_{ab}\) is the image of \(K\) in \(D(\mathcal{C}_{total})\). This we proved to be an isomorphism in Lemma 0D97 and the result follows.

Cohomology on simplicial sites

Let \(\mathcal{C}\) be as in Situation 09WE. In statement of the following lemmas we will let \(g_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_{total})\) be the morphism of topoi of Lemma 09WG. If \(\varphi : [m] \to [n]\) is a morphism of \(\Delta\), then the diagram of topoi \[\xymatrix{ \Sh(\mathcal{C}_n) \ar[rd]_{g_n} \ar[rr]_{f_\varphi} & & \Sh(\mathcal{C}_m) \ar[ld]^{g_m} \\ & \Sh(\mathcal{C}_{total}) }\] is not commutative, but there is a \(2\)-morphism \(g_n \to g_m \circ f_\varphi\) coming from the maps \(\mathcal{F}(\varphi) : f_\varphi^{-1}\mathcal{F}_m \to \mathcal{F}_n\). See Sites, Section 04I9.

Lemma

In Situation 09WE and with notation as above there is a complex \[\ldots \to g_{2!}\mathbf{Z} \to g_{1!}\mathbf{Z} \to g_{0!}\mathbf{Z}\] of abelian sheaves on \(\mathcal{C}_{total}\) which forms a resolution of the constant sheaf with value \(\mathbf{Z}\) on \(\mathcal{C}_{total}\).

Proof

We will use the description of the functors \(g_{n!}\) in Lemma 09WG without further mention. As maps of the complex we take \(\sum (-1)^i d^n_i\) where \(d^n_i : g_{n!}\mathbf{Z} \to g_{n - 1!}\mathbf{Z}\) is the adjoint to the map \(\mathbf{Z} \to \bigoplus_{[n - 1] \to [n]} \mathbf{Z} = g_n^{-1}g_{n - 1!}\mathbf{Z}\) corresponding to the factor labeled with \(\delta^n_i : [n - 1] \to [n]\). Then \(g_m^{-1}\) applied to the complex gives the complex \[\ldots \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([2], [m])]} \mathbf{Z} \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([1], [m])]} \mathbf{Z} \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([0], [m])]} \mathbf{Z}\] on \(\mathcal{C}_m\). In other words, this is the complex associated to the free abelian sheaf on the simplicial set \(\Delta[m]\), see Simplicial, Example 0176. Since \(\Delta[m]\) is homotopy equivalent to \(\Delta[0]\), see Simplicial, Example 08Q3, and since “taking free abelian sheaf on” is a functor, we see that the complex above is homotopy equivalent to the free abelian sheaf on \(\Delta[0]\) (Simplicial, Remark 019M and Lemma 019T). This complex is acyclic in positive degrees and equal to \(\mathbf{Z}\) in degree \(0\).

Lemma

In Situation 09WE. Let \(\mathcal{F}\) be an abelian sheaf on \(\mathcal{C}_{total}\) there is a canonical complex \[0 \to \Gamma(\mathcal{C}_{total}, \mathcal{F}) \to \Gamma(\mathcal{C}_0, \mathcal{F}_0) \to \Gamma(\mathcal{C}_1, \mathcal{F}_1) \to \Gamma(\mathcal{C}_2, \mathcal{F}_2) \to \ldots\] which is exact in degrees \(-1, 0\) and exact everywhere if \(\mathcal{F}\) is injective.

Proof

Observe that \(\Hom(\mathbf{Z}, \mathcal{F}) = \Gamma(\mathcal{C}_{total}, \mathcal{F})\) and \(\Hom(g_{n!}\mathbf{Z}, \mathcal{F}) = \Gamma(\mathcal{C}_n, \mathcal{F}_n)\). Hence this lemma is an immediate consequence of Lemma 09WI and the fact that \(\Hom(-, \mathcal{F})\) is exact if \(\mathcal{F}\) is injective.

Lemma

In Situation 09WE. For \(K\) in \(D^+(\mathcal{C}_{total})\) there is a spectral sequence \((E_r, d_r)_{r \geq 0}\) with \[E_1^{p, q} = H^q(\mathcal{C}_p, K_p),\quad d_1^{p, q} : E_1^{p, q} \to E_1^{p + 1, q}\] converging to \(H^{p + q}(\mathcal{C}_{total}, K)\). This spectral sequence is functorial in \(K\).

Proof

Let \(\mathcal{I}^\bullet\) be a bounded below complex of injectives representing \(K\). Consider the double complex with terms \[A^{p, q} = \Gamma(\mathcal{C}_p, \mathcal{I}^q_p)\] where the horizontal arrows come from Lemma 0D77 and the vertical arrows from the differentials of the complex \(\mathcal{I}^\bullet\). The rows of the double complex are exact in positive degrees and evaluate to \(\Gamma(\mathcal{C}_{total}, \mathcal{I}^q)\) in degree \(0\). On the other hand, since restriction to \(\mathcal{C}_p\) is exact (Lemma 09WG) the complex \(\mathcal{I}_p^\bullet\) represents \(K_p\) in \(D(\mathcal{C}_p)\). The sheaves \(\mathcal{I}_p^q\) are injective abelian sheaves on \(\mathcal{C}_p\) (Lemma 09WH). Hence the cohomology of the columns computes the groups \(H^q(\mathcal{C}_p, K_p)\). We conclude by applying Homology, Lemmas 0132 and 0133.

Remark

Assumptions and notation as in Lemma 09WJ except we do not require \(K\) in \(D(\mathcal{C}_{total})\) to be bounded below. We claim there is a natural spectral sequence in this case also. Namely, suppose that \(\mathcal{I}^\bullet\) is a K-injective complex of sheaves on \(\mathcal{C}_{total}\) with injective terms representing \(K\). We have \[\begin{align*} R\Gamma(\mathcal{C}_{total}, K) & = R\Hom(\mathbf{Z}, K) \\ & = R\Hom( \ldots \to g_{2!}\mathbf{Z} \to g_{1!}\mathbf{Z} \to g_{0!}\mathbf{Z}, K) \\ & = \Gamma(\mathcal{C}_{total}, \SheafHom^\bullet( \ldots \to g_{2!}\mathbf{Z} \to g_{1!}\mathbf{Z} \to g_{0!}\mathbf{Z}, \mathcal{I}^\bullet)) \\ & = \text{Tot}_\pi(A^{\bullet, \bullet}) \end{align*}\] where \(A^{\bullet, \bullet}\) is the double complex with terms \(A^{p, q} = \Gamma(\mathcal{C}_p, \mathcal{I}^q_p)\) and \(\text{Tot}_\pi\) denotes the product totalization of this double complex. Namely, the first equality holds in any site. The second equality holds by Lemma 09WI. The third equality holds because \(\mathcal{I}^\bullet\) is K-injective, see Cohomology on Sites, Sections 0A8X and 08J7. The final equality holds by the construction of \(\SheafHom^\bullet\) and the fact that \(\Hom(g_{p!}\mathbf{Z}, \mathcal{I}^q) = \Gamma(\mathcal{C}_p, \mathcal{I}^q_p)\). Then we get our spectral sequence by viewing \(\text{Tot}_\pi(A^{\bullet, \bullet})\) as a filtered complex with \(F^i\text{Tot}^n_\pi(A^{\bullet, \bullet}) = \prod_{p + q = n,\ p \geq i} A^{p, q}\). The spectral sequence we obtain behaves like the spectral sequence \(({}'E_r, {}'d_r)_{r \geq 0}\) in Homology, Section 012X (where the case of the direct sum totalization is discussed) except for regularity, boundedness, convergence, and abutment issues. In particular we obtain \(E_1^{p, q} = H^q(\mathcal{C}_p, K_p)\) as in Lemma 09WJ.

Lemma

In Situation 09WE. Let \(K\) be an object of \(D(\mathcal{C}_{total})\).

1. If \(H^{-p}(\mathcal{C}_p, K_p) = 0\) for all \(p \geq 0\), then \(H^0(\mathcal{C}_{total}, K) = 0\).

2. If \(R\Gamma(\mathcal{C}_p, K_p) = 0\) for all \(p \geq 0\), then \(R\Gamma(\mathcal{C}_{total}, K) = 0\).

Proof

With notation as in Remark 0H0V we see that \(R\Gamma(\mathcal{C}_{total}, K)\) is represented by \(\text{Tot}_\pi(A^{\bullet, \bullet})\). The assumption in (1) tells us that \(H^{-p}(A^{p, \bullet}) = 0\). Thus the vanishing in (1) follows from More on Algebra, Lemma 0H0Q. Part (2) follows from part (1) and taking shifts.

Lemma

Let \(\mathcal{C}\) be as in Situation 09WE. Let \(U \in \Ob(\mathcal{C}_n)\). Let \(\mathcal{F} \in \textit{Ab}(\mathcal{C}_{total})\). Then \(H^p(U, \mathcal{F}) = H^p(U, g_n^{-1}\mathcal{F})\) where on the left hand side \(U\) is viewed as an object of \(\mathcal{C}_{total}\).

Proof

Observe that “\(U\) viewed as object of \(\mathcal{C}_{total}\)” is explained by the construction of \(\mathcal{C}_{total}\) in Lemma 09WC in case (A) and Lemma 09WD in case (B). The equality then follows from Lemma 09WH and the definition of cohomology.

Cohomology and augmentations of simplicial sites

Consider a simplicial site \(\mathcal{C}\) as in Situation 09WE. Let \(a_0\) be an augmentation towards a site \(\mathcal{D}\) as in Remark 0D6Z. By Lemma 0D70 we obtain a morphism of topoi \[a : \Sh(\mathcal{C}_{total}) \longrightarrow \Sh(\mathcal{D})\] and morphisms of topoi \(g_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_{total})\) as in Lemma 09WG. The compositions \(a \circ g_n\) are denoted \(a_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{D})\). Furthermore, the simplicial structure gives morphisms of topoi \(f_\varphi : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_m)\) such that \(a_n \circ f_\varphi = a_m\) for all \(\varphi : [m] \to [n]\).

Lemma

In Situation 09WE let \(a_0\) be an augmentation towards a site \(\mathcal{D}\) as in Remark 0D6Z. For any abelian sheaf \(\mathcal{G}\) on \(\mathcal{D}\) there is an exact complex \[\ldots \to g_{2!}(a_2^{-1}\mathcal{G}) \to g_{1!}(a_1^{-1}\mathcal{G}) \to g_{0!}(a_0^{-1}\mathcal{G}) \to a^{-1}\mathcal{G} \to 0\] of abelian sheaves on \(\mathcal{C}_{total}\).

Proof

We encourage the reader to read the proof of Lemma 09WI first. We will use Lemma 0D70 and the description of the functors \(g_{n!}\) in Lemma 09WG without further mention. In particular \(g_{n!}(a_n^{-1}\mathcal{G})\) is the sheaf on \(\mathcal{C}_{total}\) whose restriction to \(\mathcal{C}_m\) is the sheaf \[\bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^{-1}a_n^{-1}\mathcal{G} = \bigoplus\nolimits_{\varphi : [n] \to [m]} a_m^{-1}\mathcal{G}\] As maps of the complex we take \(\sum (-1)^i d^n_i\) where \(d^n_i : g_{n!}(a_n^{-1}\mathcal{G}) \to g_{n - 1!}(a_{n - 1}^{-1}\mathcal{G})\) is the adjoint to the map \(a_n^{-1}\mathcal{G} \to \bigoplus_{[n - 1] \to [n]} a_n^{-1}\mathcal{G} = g_n^{-1}g_{n - 1!}(a_{n - 1}^{-1}\mathcal{G})\) corresponding to the factor labeled with \(\delta^n_i : [n - 1] \to [n]\). The map \(g_{0!}(a_0^{-1}\mathcal{G}) \to a^{-1}\mathcal{G}\) is adjoint to the identity map of \(a_0^{-1}\mathcal{G}\). Then \(g_m^{-1}\) applied to the chain complex in degrees \(\ldots, 2, 1, 0\) gives the complex \[\ldots \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([2], [m])]} a_m^{-1}\mathcal{G} \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([1], [m])]} a_m^{-1}\mathcal{G} \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([0], [m])]} a_m^{-1}\mathcal{G}\] on \(\mathcal{C}_m\). This is equal to \(a_m^{-1}\mathcal{G}\) tensored over the constant sheaf \(\mathbf{Z}\) with the complex \[\ldots \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([2], [m])]} \mathbf{Z} \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([1], [m])]} \mathbf{Z} \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([0], [m])]} \mathbf{Z}\] discussed in the proof of Lemma 09WI. There we have seen that this complex is homotopy equivalent to \(\mathbf{Z}\) placed in degree \(0\) which finishes the proof.

Lemma

In Situation 09WE let \(a_0\) be an augmentation towards a site \(\mathcal{D}\) as in Remark 0D6Z. For an abelian sheaf \(\mathcal{F}\) on \(\mathcal{C}_{total}\) there is a canonical complex \[0 \to a_*\mathcal{F} \to a_{0, *}\mathcal{F}_0 \to a_{1, *}\mathcal{F}_1 \to a_{2, *}\mathcal{F}_2 \to \ldots\] on \(\mathcal{D}\) which is exact in degrees \(-1, 0\) and exact everywhere if \(\mathcal{F}\) is injective.

Proof

To construct the complex, by the Yoneda lemma, it suffices for any abelian sheaf \(\mathcal{G}\) on \(\mathcal{D}\) to construct a complex \[0 \to \Hom(\mathcal{G}, a_*\mathcal{F}) \to \Hom(\mathcal{G}, a_{0, *}\mathcal{F}_0) \to \Hom(\mathcal{G}, a_{1, *}\mathcal{F}_1) \to \ldots\] functorially in \(\mathcal{G}\). To do this apply \(\Hom(-, \mathcal{F})\) to the exact complex of Lemma 0D78 and use adjointness of pullback and pushforward. The exactness properties in degrees \(-1, 0\) follow from the construction as \(\Hom(-, \mathcal{F})\) is left exact. If \(\mathcal{F}\) is an injective abelian sheaf, then the complex is exact because \(\Hom(-, \mathcal{F})\) is exact.

Lemma

In Situation 09WE let \(a_0\) be an augmentation towards a site \(\mathcal{D}\) as in Remark 0D6Z. For any \(K\) in \(D^+(\mathcal{C}_{total})\) there is a spectral sequence \((E_r, d_r)_{r \geq 0}\) with \[E_1^{p, q} = R^qa_{p, *} K_p,\quad d_1^{p, q} : E_1^{p, q} \to E_1^{p + 1, q}\] converging to \(R^{p + q}a_*K\). This spectral sequence is functorial in \(K\).

Proof

Let \(\mathcal{I}^\bullet\) be a bounded below complex of injectives representing \(K\). Consider the double complex with terms \[A^{p, q} = a_{p, *}\mathcal{I}^q_p\] where the horizontal arrows come from Lemma 0D79 and the vertical arrows from the differentials of the complex \(\mathcal{I}^\bullet\). The rows of the double complex are exact in positive degrees and evaluate to \(a_*\mathcal{I}^q\) in degree \(0\). On the other hand, since restriction to \(\mathcal{C}_p\) is exact (Lemma 09WG) the complex \(\mathcal{I}_p^\bullet\) represents \(K_p\) in \(D(\mathcal{C}_p)\). The sheaves \(\mathcal{I}_p^q\) are injective abelian sheaves on \(\mathcal{C}_p\) (Lemma 09WH). Hence the cohomology of the columns computes \(R^qa_{p, *}K_p\). We conclude by applying Homology, Lemmas 0132 and 0133.

Cohomology on ringed simplicial sites

This section is the analogue of Section 0D76 for sheaves of modules.

In Situation 09WE let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}_{total}\). In statement of the following lemmas we will let \(g_n : (\Sh(\mathcal{C}_n), \mathcal{O}_n) \to (\Sh(\mathcal{C}_{total}), \mathcal{O})\) be the morphism of ringed topoi of Lemma 0D72. If \(\varphi : [m] \to [n]\) is a morphism of \(\Delta\), then the diagram of ringed topoi \[\xymatrix{ (\Sh(\mathcal{C}_n), \mathcal{O}_n) \ar[rd]_{g_n} \ar[rr]_{f_\varphi} & & (\Sh(\mathcal{C}_m), \mathcal{O}_m) \ar[ld]^{g_m} \\ & (\Sh(\mathcal{C}_{total}), \mathcal{O}) }\] is not commutative, but there is a \(2\)-morphism \(g_n \to g_m \circ f_\varphi\) coming from the maps \(\mathcal{F}(\varphi) : f_\varphi^{-1}\mathcal{F}_m \to \mathcal{F}_n\). See Sites, Section 04I9.

Lemma

In Situation 09WE let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}_{total}\). There is a complex \[\ldots \to g_{2!}\mathcal{O}_2 \to g_{1!}\mathcal{O}_1 \to g_{0!}\mathcal{O}_0\] of \(\mathcal{O}\)-modules which forms a resolution of \(\mathcal{O}\). Here \(g_{n!}\) is as in Lemma 0D72.

Proof

We will use the description of \(g_{n!}\) given in Lemma 09WG. As maps of the complex we take \(\sum (-1)^i d^n_i\) where \(d^n_i : g_{n!}\mathcal{O}_n \to g_{n - 1!}\mathcal{O}_{n - 1}\) is the adjoint to the map \(\mathcal{O}_n \to \bigoplus_{[n - 1] \to [n]} \mathcal{O}_n = g_n^*g_{n - 1!}\mathcal{O}_{n - 1}\) corresponding to the factor labeled with \(\delta^n_i : [n - 1] \to [n]\). Then \(g_m^{-1}\) applied to the complex gives the complex \[\ldots \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([2], [m])]} \mathcal{O}_m \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([1], [m])]} \mathcal{O}_m \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([0], [m])]} \mathcal{O}_m\] on \(\mathcal{C}_m\). In other words, this is the complex associated to the free \(\mathcal{O}_m\)-module on the simplicial set \(\Delta[m]\), see Simplicial, Example 0176. Since \(\Delta[m]\) is homotopy equivalent to \(\Delta[0]\), see Simplicial, Example 08Q3, and since “taking free abelian sheaf on” is a functor, we see that the complex above is homotopy equivalent to the free abelian sheaf on \(\Delta[0]\) (Simplicial, Remark 019M and Lemma 019T). This complex is acyclic in positive degrees and equal to \(\mathcal{O}_m\) in degree \(0\).

Lemma

In Situation 09WE let \(\mathcal{O}\) be a sheaf of rings. Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}\)-modules. There is a canonical complex \[0 \to \Gamma(\mathcal{C}_{total}, \mathcal{F}) \to \Gamma(\mathcal{C}_0, \mathcal{F}_0) \to \Gamma(\mathcal{C}_1, \mathcal{F}_1) \to \Gamma(\mathcal{C}_2, \mathcal{F}_2) \to \ldots\] which is exact in degrees \(-1, 0\) and exact everywhere if \(\mathcal{F}\) is an injective \(\mathcal{O}\)-module.

Proof

Observe that \(\Hom(\mathcal{O}, \mathcal{F}) = \Gamma(\mathcal{C}_{total}, \mathcal{F})\) and \(\Hom(g_{n!}\mathcal{O}_n, \mathcal{F}) = \Gamma(\mathcal{C}_n, \mathcal{F}_n)\). Hence this lemma is an immediate consequence of Lemma 0D9B and the fact that \(\Hom(-, \mathcal{F})\) is exact if \(\mathcal{F}\) is injective.

Lemma

In Situation 09WE let \(\mathcal{O}\) be a sheaf of rings. For \(K\) in \(D^+(\mathcal{O})\) there is a spectral sequence \((E_r, d_r)_{r \geq 0}\) with \[E_1^{p, q} = H^q(\mathcal{C}_p, K_p),\quad d_1^{p, q} : E_1^{p, q} \to E_1^{p + 1, q}\] converging to \(H^{p + q}(\mathcal{C}_{total}, K)\). This spectral sequence is functorial in \(K\).

Proof

Let \(\mathcal{I}^\bullet\) be a bounded below complex of injective \(\mathcal{O}\)-modules representing \(K\). Consider the double complex with terms \[A^{p, q} = \Gamma(\mathcal{C}_p, \mathcal{I}^q_p)\] where the horizontal arrows come from Lemma 0D9C and the vertical arrows from the differentials of the complex \(\mathcal{I}^\bullet\). Observe that \(\Gamma(\mathcal{D}, -) = \Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{O}_\mathcal{D}, -)\) on \(\textit{Mod}(\mathcal{O}_\mathcal{D})\). Hence the lemma says rows of the double complex are exact in positive degrees and evaluate to \(\Gamma(\mathcal{C}_{total}, \mathcal{I}^q)\) in degree \(0\). Thus the total complex associated to the double complex computes \(R\Gamma(\mathcal{C}_{total}, K)\) by Homology, Lemma 0133. On the other hand, since restriction to \(\mathcal{C}_p\) is exact (Lemma 09WG) the complex \(\mathcal{I}_p^\bullet\) represents \(K_p\) in \(D(\mathcal{C}_p)\). The sheaves \(\mathcal{I}_p^q\) are totally acyclic on \(\mathcal{C}_p\) (Lemma 0D73). Hence the cohomology of the columns computes the groups \(H^q(\mathcal{C}_p, K_p)\) by Leray’s acyclicity lemma (Derived Categories, Lemma 015E) and Cohomology on Sites, Lemma 0731. We conclude by applying Homology, Lemma 0132.

Lemma

In Situation 09WE let \(\mathcal{O}\) be a sheaf of rings. Let \(U \in \Ob(\mathcal{C}_n)\). Let \(\mathcal{F} \in \textit{Mod}(\mathcal{O})\). Then \(H^p(U, \mathcal{F}) = H^p(U, g_n^*\mathcal{F})\) where on the left hand side \(U\) is viewed as an object of \(\mathcal{C}_{total}\).

Proof

Observe that “\(U\) viewed as object of \(\mathcal{C}_{total}\)” is explained by the construction of \(\mathcal{C}_{total}\) in Lemma 09WC in case (A) and Lemma 09WD in case (B). In both cases the functor \(\mathcal{C}_n \to \mathcal{C}\) is continuous and cocontinuous, see Lemma 09WG, and \(g_n^{-1}\mathcal{O} = \mathcal{O}_n\) by definition. Hence the result is a special case of Cohomology on Sites, Lemma 0DD8.

Cohomology and augmentations of ringed simplicial sites

This section is the analogue of Section 0D9A for sheaves of modules.

Consider a simplicial site \(\mathcal{C}\) as in Situation 09WE. Let \(a_0\) be an augmentation towards a site \(\mathcal{D}\) as in Remark 0D6Z. Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}_{total}\). Let \(\mathcal{O}_\mathcal{D}\) be a sheaf of rings on \(\mathcal{D}\). Suppose we are given a morphism \[a^\sharp : \mathcal{O}_\mathcal{D} \longrightarrow a_*\mathcal{O}\] where \(a\) is as in Lemma 0D70. Consequently, we obtain a morphism of ringed topoi \[a : (\Sh(\mathcal{C}_{total}), \mathcal{O}) \longrightarrow (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\] We will think of \(g_n : (\Sh(\mathcal{C}_n), \mathcal{O}_n) \to (\Sh(\mathcal{C}_{total}), \mathcal{O})\) as a morphism of ringed topoi as in Lemma 0D72, then taking the composition \(a_n = a \circ g_n\) (Lemma 0D70) as morphisms of ringed topoi we obtain \[a_n : (\Sh(\mathcal{C}_n), \mathcal{O}_n) \longrightarrow (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\] Using the transition maps \(f_\varphi^{-1}\mathcal{O}_m \to \mathcal{O}_n\) we obtain morphisms of ringed topoi \[f_\varphi : (\Sh(\mathcal{C}_n), \mathcal{O}_n) \to (\Sh(\mathcal{C}_m), \mathcal{O}_m)\] such that \(a_m \circ f_\varphi = a_n\) as morphisms of ringed topoi for all \(\varphi : [m] \to [n]\).

Lemma

With notation as above. The morphism \(a : (\Sh(\mathcal{C}_{total}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) is flat if and only if \(a_n : (\Sh(\mathcal{C}_n), \mathcal{O}_n) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) is flat for \(n \geq 0\).

Proof

Since \(g_n : (\Sh(\mathcal{C}_n), \mathcal{O}_n) \to (\Sh(\mathcal{C}_{total}), \mathcal{O})\) is flat, we see that if \(a\) is flat, then \(a_n = a \circ g_n\) is flat as a composition. Conversely, suppose that \(a_n\) is flat for all \(n\). We have to check that \(\mathcal{O}\) is flat as a sheaf of \(a^{-1}\mathcal{O}_\mathcal{D}\)-modules. Let \(\mathcal{F} \to \mathcal{G}\) be an injective map of \(a^{-1}\mathcal{O}_\mathcal{D}\)-modules. We have to show that \[\mathcal{F} \otimes_{a^{-1}\mathcal{O}_\mathcal{D}} \mathcal{O} \to \mathcal{G} \otimes_{a^{-1}\mathcal{O}_\mathcal{D}} \mathcal{O}\] is injective. We can check this on \(\mathcal{C}_n\), i.e., after applying \(g_n^{-1}\). Since \(g_n^* = g_n^{-1}\) because \(g_n^{-1}\mathcal{O} = \mathcal{O}_n\) we obtain \[g_n^{-1}\mathcal{F} \otimes_{g_n^{-1}a^{-1}\mathcal{O}_\mathcal{D}} \mathcal{O}_n \to g_n^{-1}\mathcal{G} \otimes_{g_n^{-1}a^{-1}\mathcal{O}_\mathcal{D}} \mathcal{O}_n\] which is injective because \(g_n^{-1}a^{-1}\mathcal{O}_\mathcal{D} = a_n^{-1}\mathcal{O}_\mathcal{D}\) and we assume \(a_n\) was flat.

Lemma

With notation as above. For a \(\mathcal{O}_\mathcal{D}\)-module \(\mathcal{G}\) there is an exact complex \[\ldots \to g_{2!}(a_2^*\mathcal{G}) \to g_{1!}(a_1^*\mathcal{G}) \to g_{0!}(a_0^*\mathcal{G}) \to a^*\mathcal{G} \to 0\] of sheaves of \(\mathcal{O}\)-modules on \(\mathcal{C}_{total}\). Here \(g_{n!}\) is as in Lemma 0D72.

Proof

Observe that \(a^*\mathcal{G}\) is the \(\mathcal{O}\)-module on \(\mathcal{C}_{total}\) whose restriction to \(\mathcal{C}_m\) is the \(\mathcal{O}_m\)-module \(a_m^*\mathcal{G}\). The description of the functors \(g_{n!}\) on modules in Lemma 0D72 shows that \(g_{n!}(a_n^*\mathcal{G})\) is the \(\mathcal{O}\)-module on \(\mathcal{C}_{total}\) whose restriction to \(\mathcal{C}_m\) is the \(\mathcal{O}_m\)-module \[\bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^*a_n^*\mathcal{G} = \bigoplus\nolimits_{\varphi : [n] \to [m]} a_m^*\mathcal{G}\] The rest of the proof is exactly the same as the proof of Lemma 0D78, replacing \(a_m^{-1}\mathcal{G}\) by \(a_m^*\mathcal{G}\).

Lemma

With notation as above. For an \(\mathcal{O}\)-module \(\mathcal{F}\) on \(\mathcal{C}_{total}\) there is a canonical complex \[0 \to a_*\mathcal{F} \to a_{0, *}\mathcal{F}_0 \to a_{1, *}\mathcal{F}_1 \to a_{2, *}\mathcal{F}_2 \to \ldots\] of \(\mathcal{O}_\mathcal{D}\)-modules which is exact in degrees \(-1, 0\). If \(\mathcal{F}\) is an injective \(\mathcal{O}\)-module, then the complex is exact in all degrees and remains exact on applying the functor \(\Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G}, -)\) for any \(\mathcal{O}_\mathcal{D}\)-module \(\mathcal{G}\).

Proof

To construct the complex, by the Yoneda lemma, it suffices for any \(\mathcal{O}_\mathcal{D}\)-modules \(\mathcal{G}\) on \(\mathcal{D}\) to construct a complex \[0 \to \Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G}, a_*\mathcal{F}) \to \Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G}, a_{0, *}\mathcal{F}_0) \to \Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G}, a_{1, *}\mathcal{F}_1) \to \ldots\] functorially in \(\mathcal{G}\). To do this apply \(\Hom_\mathcal{O}(-, \mathcal{F})\) to the exact complex of Lemma 0D7C and use adjointness of pullback and pushforward. The exactness properties in degrees \(-1, 0\) follow from the construction as \(\Hom_\mathcal{O}(-, \mathcal{F})\) is left exact. If \(\mathcal{F}\) is an injective \(\mathcal{O}\)-module, then the complex is exact because \(\Hom_\mathcal{O}(-, \mathcal{F})\) is exact.

Lemma

With notation as above for any \(K\) in \(D^+(\mathcal{O})\) there is a spectral sequence \((E_r, d_r)_{r \geq 0}\) in \(\textit{Mod}(\mathcal{O}_\mathcal{D})\) with \[E_1^{p, q} = R^qa_{p, *} K_p\] converging to \(R^{p + q}a_*K\). This spectral sequence is functorial in \(K\).

Proof

Let \(\mathcal{I}^\bullet\) be a bounded below complex of injective \(\mathcal{O}\)-modules representing \(K\). Consider the double complex with terms \[A^{p, q} = a_{p, *}\mathcal{I}^q_p\] where the horizontal arrows come from Lemma 0D7D and the vertical arrows from the differentials of the complex \(\mathcal{I}^\bullet\). The lemma says rows of the double complex are exact in positive degrees and evaluate to \(a_*\mathcal{I}^q\) in degree \(0\). Thus the total complex associated to the double complex computes \(Ra_*K\) by Homology, Lemma 0133. On the other hand, since restriction to \(\mathcal{C}_p\) is exact (Lemma 09WG) the complex \(\mathcal{I}_p^\bullet\) represents \(K_p\) in \(D(\mathcal{C}_p)\). The sheaves \(\mathcal{I}_p^q\) are totally acyclic on \(\mathcal{C}_p\) (Lemma 0D73). Hence the cohomology of the columns are the sheaves \(R^qa_{p, *}K_p\) by Leray’s acyclicity lemma (Derived Categories, Lemma 015E) and Cohomology on Sites, Lemma 0731. We conclude by applying Homology, Lemma 0132.

Cartesian sheaves and modules

Here is the definition.

Definition

In Situation 09WE.

1. A sheaf \(\mathcal{F}\) of sets or of abelian groups on \(\mathcal{C}_{total}\) is cartesian if the maps \(\mathcal{F}(\varphi) : f_\varphi^{-1}\mathcal{F}_m \to \mathcal{F}_n\) are isomorphisms for all \(\varphi : [m] \to [n]\).

2. If \(\mathcal{O}\) is a sheaf of rings on \(\mathcal{C}_{total}\), then a sheaf \(\mathcal{F}\) of \(\mathcal{O}\)-modules is cartesian if the maps \(f_\varphi^*\mathcal{F}_m \to \mathcal{F}_n\) are isomorphisms for all \(\varphi : [m] \to [n]\).

3. An object \(K\) of \(D(\mathcal{C}_{total})\) is cartesian if the maps \(f_\varphi^{-1}K_m \to K_n\) are isomorphisms for all \(\varphi : [m] \to [n]\).

4. If \(\mathcal{O}\) is a sheaf of rings on \(\mathcal{C}_{total}\), then an object \(K\) of \(D(\mathcal{O})\) is cartesian if the maps \(Lf_\varphi^*K_m \to K_n\) are isomorphisms for all \(\varphi : [m] \to [n]\).

Of course there is a general notion of a cartesian section of a fibred category and the above are merely examples of this. The property on pullbacks needs only be checked for the degeneracies.

Lemma

In Situation 09WE.

1. A sheaf \(\mathcal{F}\) of sets or abelian groups is cartesian if and only if the maps \((f_{\delta^n_j})^{-1}\mathcal{F}_{n - 1} \to \mathcal{F}_n\) are isomorphisms.

2. An object \(K\) of \(D(\mathcal{C}_{total})\) is cartesian if and only if the maps \((f_{\delta^n_j})^{-1}K_{n - 1} \to K_n\) are isomorphisms.

3. If \(\mathcal{O}\) is a sheaf of rings on \(\mathcal{C}_{total}\) a sheaf \(\mathcal{F}\) of \(\mathcal{O}\)-modules is cartesian if and only if the maps \((f_{\delta^n_j})^*\mathcal{F}_{n - 1} \to \mathcal{F}_n\) are isomorphisms.

4. If \(\mathcal{O}\) is a sheaf of rings on \(\mathcal{C}_{total}\) an object \(K\) of \(D(\mathcal{O})\) is cartesian if and only if the maps \(L(f_{\delta^n_j})^*K_{n - 1} \to K_n\) are isomorphisms.

5. Add more here.

Proof

In each case the key is that the pullback functors compose to pullback functor; for part (4) see Cohomology on Sites, Lemma 0D6D. We show how the argument works in case (1) and omit the proof in the other cases. The category \(\Delta\) is generated by the morphisms the morphisms \(\delta^n_j\) and \(\sigma^n_j\), see Simplicial, Lemma 0166. Hence we only need to check the maps \((f_{\delta^n_j})^{-1}\mathcal{F}_{n - 1} \to \mathcal{F}_n\) and \((f_{\sigma^n_j})^{-1}\mathcal{F}_{n + 1} \to \mathcal{F}_n\) are isomorphisms, see Simplicial, Lemma 016B for notation. Since \(\sigma^n_j \circ \delta_j^{n + 1} = \text{id}_{[n]}\) the composition \[\mathcal{F}_n = (f_{\sigma^n_j})^{-1} (f_{\delta_j^{n + 1}})^{-1} \mathcal{F}_n \to (f_{\sigma^n_j})^{-1} \mathcal{F}_{n + 1} \to \mathcal{F}_n\] is the identity. Thus the result for \(\delta^{n + 1}_j\) implies the result for \(\sigma^n_j\).

Lemma

In Situation 09WE let \(a_0\) be an augmentation towards a site \(\mathcal{D}\) as in Remark 0D6Z.

1. The pullback \(a^{-1}\mathcal{G}\) of a sheaf of sets or abelian groups on \(\mathcal{D}\) is cartesian.

2. The pullback \(a^{-1}K\) of an object \(K\) of \(D(\mathcal{D})\) is cartesian.

Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}_{total}\) and \(\mathcal{O}_\mathcal{D}\) a sheaf of rings on \(\mathcal{D}\) and \(a^\sharp : \mathcal{O}_\mathcal{D} \to a_*\mathcal{O}\) a morphism as in Section 0D9D.

1. The pullback \(a^*\mathcal{F}\) of a sheaf of \(\mathcal{O}_\mathcal{D}\)-modules is cartesian.

2. The derived pullback \(La^*K\) of an object \(K\) of \(D(\mathcal{O}_\mathcal{D})\) is cartesian.

Proof

This follows immediately from the identities \(a_m \circ f_\varphi = a_n\) for all \(\varphi : [m] \to [n]\). See Lemma 0D70 and the discussion in Section 0D9D.

Lemma

In Situation 09WE. The category of cartesian sheaves of sets (resp. abelian groups) is equivalent to the category of pairs \((\mathcal{F}, \alpha)\) where \(\mathcal{F}\) is a sheaf of sets (resp. abelian groups) on \(\mathcal{C}_0\) and \[\alpha : (f_{\delta_1^1})^{-1}\mathcal{F} \longrightarrow (f_{\delta_0^1})^{-1}\mathcal{F}\] is an isomorphism of sheaves of sets (resp. abelian groups) on \(\mathcal{C}_1\) such that \((f_{\delta^2_1})^{-1}\alpha = (f_{\delta^2_0})^{-1}\alpha \circ (f_{\delta^2_2})^{-1}\alpha\) as maps of sheaves on \(\mathcal{C}_2\).

Proof

We abbreviate \(d^n_j = f_{\delta^n_j} : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_{n - 1})\). The condition on \(\alpha\) in the statement of the lemma makes sense because \[d^1_1 \circ d^2_2 = d^1_1 \circ d^2_1, \quad d^1_1 \circ d^2_0 = d^1_0 \circ d^2_2, \quad d^1_0 \circ d^2_0 = d^1_0 \circ d^2_1\] as morphisms of topoi \(\Sh(\mathcal{C}_2) \to \Sh(\mathcal{C}_0)\), see Simplicial, Remark 016C. Hence we can picture these maps as follows \[\xymatrix{ & (d^2_0)^{-1}(d^1_1)^{-1}\mathcal{F} \ar[r]_-{(d^2_0)^{-1}\alpha} & (d^2_0)^{-1}(d^1_0)^{-1}\mathcal{F} \ar@{=}[rd] & \\ (d^2_2)^{-1}(d^1_0)^{-1}\mathcal{F} \ar@{=}[ru] & & & (d^2_1)^{-1}(d^1_0)^{-1}\mathcal{F} \\ & (d^2_2)^{-1}(d^1_1)^{-1}\mathcal{F} \ar[lu]^{(d^2_2)^{-1}\alpha} \ar@{=}[r] & (d^2_1)^{-1}(d^1_1)^{-1}\mathcal{F} \ar[ru]_{(d^2_1)^{-1}\alpha} }\] and the condition signifies the diagram is commutative. It is clear that given a cartesian sheaf \(\mathcal{G}\) of sets (resp. abelian groups) on \(\mathcal{C}_{total}\) we can set \(\mathcal{F} = \mathcal{G}_0\) and \(\alpha\) equal to the composition \[(d_1^1)^{-1}\mathcal{G}_0 \to \mathcal{G}_1 \leftarrow (d_1^0)^{-1}\mathcal{G}_0\] where the arrows are invertible as \(\mathcal{G}\) is cartesian. To prove this functor is an equivalence we construct a quasi-inverse. The construction of the quasi-inverse is analogous to the construction discussed in Descent, Section 023F from which we borrow the notation \(\tau^n_i : [0] \to [n]\), \(0 \mapsto i\) and \(\tau^n_{ij} : [1] \to [n]\), \(0 \mapsto i\), \(1 \mapsto j\). Namely, given a pair \((\mathcal{F}, \alpha)\) as in the lemma we set \(\mathcal{G}_n = (f_{\tau^n_n})^{-1}\mathcal{F}\). Given \(\varphi : [n] \to [m]\) we define \(\mathcal{G}(\varphi) : (f_\varphi)^{-1}\mathcal{G}_n \to \mathcal{G}_m\) using \[\xymatrix{ (f_\varphi)^{-1}\mathcal{G}_n \ar@{=}[r] & (f_\varphi)^{-1}(f_{\tau^n_n})^{-1}\mathcal{F} \ar@{=}[r] & (f_{\tau^m_{\varphi(n)}})^{-1}\mathcal{F} \ar@{=}[r] & (f_{\tau^m_{\varphi(n)m}})^{-1}(d^1_1)^{-1}\mathcal{F} \ar[d]^{(f_{\tau^m_{\varphi(n)m}})^{-1}\alpha} \\ & \mathcal{G}_m \ar@{=}[r] & (f_{\tau^m_m})^{-1}\mathcal{F} \ar@{=}[r] & (f_{\tau^m_{\varphi(n)m}})^{-1}(d^1_0)^{-1}\mathcal{F} }\] We omit the verification that the commutativity of the displayed diagram above implies the maps compose correctly and hence give rise to a sheaf on \(\mathcal{C}_{total}\), see Lemma 09WF. We also omit the verification that the two functors are quasi-inverse to each other.

Lemma

In Situation 09WE let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}_{total}\). The category of cartesian \(\mathcal{O}\)-modules is equivalent to the category of pairs \((\mathcal{F}, \alpha)\) where \(\mathcal{F}\) is a \(\mathcal{O}_0\)-module and \[\alpha : (f_{\delta_1^1})^*\mathcal{F} \longrightarrow (f_{\delta_0^1})^*\mathcal{F}\] is an isomorphism of \(\mathcal{O}_1\)-modules such that \((f_{\delta^2_1})^*\alpha = (f_{\delta^2_0})^*\alpha \circ (f_{\delta^2_2})^*\alpha\) as \(\mathcal{O}_2\)-module maps.

Proof

The proof is identical to the proof of Lemma 0D7I with pullback of sheaves of abelian groups replaced by pullback of modules.

Lemma

In Situation 09WE.

1. The full subcategory of cartesian abelian sheaves forms a weak Serre subcategory of \(\textit{Ab}(\mathcal{C}_{total})\). Colimits of systems of cartesian abelian sheaves are cartesian.

2. Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}_{total}\) such that the morphisms \[f_{\delta^n_j} : (\Sh(\mathcal{C}_n), \mathcal{O}_n) \to (\Sh(\mathcal{C}_{n - 1}), \mathcal{O}_{n - 1})\] are flat. The full subcategory of cartesian \(\mathcal{O}\)-modules forms a weak Serre subcategory of \(\textit{Mod}(\mathcal{O})\). Colimits of systems of cartesian \(\mathcal{O}\)-modules are cartesian.

Proof

To see we obtain a weak Serre subcategory in (1) we check the conditions listed in Homology, Lemma 0754. First, if \(\varphi : \mathcal{F} \to \mathcal{G}\) is a map between cartesian abelian sheaves, then \(\Ker(\varphi)\) and \(\Coker(\varphi)\) are cartesian too because the restriction functors \(\Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{C}_n)\) and the functors \(f_\varphi^{-1}\) are exact. Similarly, if \[0 \to \mathcal{F} \to \mathcal{H} \to \mathcal{G} \to 0\] is a short exact sequence of abelian sheaves on \(\mathcal{C}_{total}\) with \(\mathcal{F}\) and \(\mathcal{G}\) cartesian, then it follows that \(\mathcal{H}\) is cartesian from the 5-lemma. To see the property of colimits, use that colimits commute with pullback as pullback is a left adjoint. In the case of modules we argue in the same manner, using the exactness of flat pullback (Modules on Sites, Lemma 04JC) and the fact that it suffices to check the condition for \(f_{\delta^n_j}\), see Lemma 07TG.

Remark

Lemma 0D7J notwithstanding, it can happen that the category of cartesian \(\mathcal{O}\)-modules is abelian without being a Serre subcategory of \(\textit{Mod}(\mathcal{O})\). Namely, suppose that we only know that \(f_{\delta_1^1}\) and \(f_{\delta_0^1}\) are flat. Then it follows easily from Lemma 07TH that the category of cartesian \(\mathcal{O}\)-modules is abelian. But if \(f_{\delta_0^2}\) is not flat (for example), there is no reason for the inclusion functor from the category of cartesian \(\mathcal{O}\)-modules to all \(\mathcal{O}\)-modules to be exact.

Lemma

In Situation 09WE.

1. An object \(K\) of \(D(\mathcal{C}_{total})\) is cartesian if and only if \(H^q(K)\) is a cartesian abelian sheaf for all \(q\).

2. Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}_{total}\) such that the morphisms \(f_{\delta^n_j} : (\Sh(\mathcal{C}_n), \mathcal{O}_n) \to (\Sh(\mathcal{C}_{n - 1}), \mathcal{O}_{n - 1})\) are flat. Then an object \(K\) of \(D(\mathcal{O})\) is cartesian if and only if \(H^q(K)\) is a cartesian \(\mathcal{O}\)-module for all \(q\).

Proof

Part (1) is true because the pullback functors \((f_\varphi)^{-1}\) are exact. Part (2) follows from the characterization in Lemma 07TG and the fact that \(L(f_{\delta^n_j})^* = (f_{\delta^n_j})^*\) by flatness.

Lemma

In Situation 09WE.

1. An object \(K\) of \(D(\mathcal{C}_{total})\) is cartesian if and only the canonical map \[g_{n!}K_n \longrightarrow g_{n!}\mathbf{Z} \otimes^\mathbf{L}_\mathbf{Z} K\] is an isomorphism for all \(n\).

2. Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}_{total}\) such that the morphisms \(f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m\) are flat for all \(\varphi : [n] \to [m]\). Then an object \(K\) of \(D(\mathcal{O})\) is cartesian if and only if the canonical map \[g_{n!}K_n \longrightarrow g_{n!}\mathcal{O}_n \otimes^\mathbf{L}_\mathcal{O} K\] is an isomorphism for all \(n\).

Proof

Proof of (1). Since \(g_{n!}\) is exact, it induces a functor on derived categories adjoint to \(g_n^{-1}\). The map is the adjoint of the map \(K_n \to (g_n^{-1}g_{n!}\mathbf{Z}) \otimes^\mathbf{L}_\mathbf{Z} K_n\) corresponding to \(\mathbf{Z} \to g_n^{-1}g_{n!}\mathbf{Z}\) which in turn is adjoint to \(\text{id} : g_{n!}\mathbf{Z} \to g_{n!}\mathbf{Z}\). Using the description of \(g_{n!}\) given in Lemma 09WG we see that the restriction to \(\mathcal{C}_m\) of this map is \[\bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^{-1}K_n \longrightarrow \bigoplus\nolimits_{\varphi : [n] \to [m]} K_m\] Thus the statement is clear.

Proof of (2). Since \(g_{n!}\) is exact (Lemma 0D74), it induces a functor on derived categories adjoint to \(g_n^*\) (also exact). The map is the adjoint of the map \(K_n \to (g_n^*g_{n!}\mathcal{O}_n) \otimes^\mathbf{L}_{\mathcal{O}_n} K_n\) corresponding to \(\mathcal{O}_n \to g_n^*g_{n!}\mathcal{O}_n\) which in turn is adjoint to \(\text{id} : g_{n!}\mathcal{O}_n \to g_{n!}\mathcal{O}_n\). Using the description of \(g_{n!}\) given in Lemma 0D72 we see that the restriction to \(\mathcal{C}_m\) of this map is \[\bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^*K_n \longrightarrow \bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^*\mathcal{O}_n \otimes_{\mathcal{O}_m} K_m = \bigoplus\nolimits_{\varphi : [n] \to [m]} K_m\] Thus the statement is clear.

Lemma

In Situation 09WE let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}_{total}\). Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}\)-modules. Then \(\mathcal{F}\) is quasi-coherent in the sense of Modules on Sites, Definition 03DL if and only if \(\mathcal{F}\) is cartesian and \(\mathcal{F}_n\) is a quasi-coherent \(\mathcal{O}_n\)-module for all \(n\).

Proof

Assume \(\mathcal{F}\) is quasi-coherent. Since pullbacks of quasi-coherent modules are quasi-coherent (Modules on Sites, Lemma 03DO) we see that \(\mathcal{F}_n\) is a quasi-coherent \(\mathcal{O}_n\)-module for all \(n\). To show that \(\mathcal{F}\) is cartesian, let \(U\) be an object of \(\mathcal{C}_n\) for some \(n\). Let us view \(U\) as an object of \(\mathcal{C}_{total}\). Because \(\mathcal{F}\) is quasi-coherent there exists a covering \(\{U_i \to U\}\) and for each \(i\) a presentation \[\bigoplus\nolimits_{j \in J_i} \mathcal{O}_{\mathcal{C}_{total}/U_i} \to \bigoplus\nolimits_{k \in K_i} \mathcal{O}_{\mathcal{C}_{total}/U_i} \to \mathcal{F}|_{\mathcal{C}_{total}/U_i} \to 0\] Observe that \(\{U_i \to U\}\) is a covering of \(\mathcal{C}_n\) by the construction of the site \(\mathcal{C}_{total}\). Next, let \(V\) be an object of \(\mathcal{C}_m\) for some \(m\) and let \(V \to U\) be a morphism of \(\mathcal{C}_{total}\) lying over \(\varphi : [n] \to [m]\). The fibre products \(V_i = V \times_U U_i\) exist and we get an induced covering \(\{V_i \to V\}\) in \(\mathcal{C}_m\). Restricting the presentation above to the sites \(\mathcal{C}_n/U_i\) and \(\mathcal{C}_m/V_i\) we obtain presentations \[\bigoplus\nolimits_{j \in J_i} \mathcal{O}_{\mathcal{C}_m/U_i} \to \bigoplus\nolimits_{k \in K_i} \mathcal{O}_{\mathcal{C}_m/U_i} \to \mathcal{F}_n|_{\mathcal{C}_n/U_i} \to 0\] and \[\bigoplus\nolimits_{j \in J_i} \mathcal{O}_{\mathcal{C}_m/V_i} \to \bigoplus\nolimits_{k \in K_i} \mathcal{O}_{\mathcal{C}_m/V_i} \to \mathcal{F}_m|_{\mathcal{C}_m/V_i} \to 0\] These presentations are compatible with the map \(\mathcal{F}(\varphi) : f_\varphi^*\mathcal{F}_n \to \mathcal{F}_m\) (as this map is defined using the restriction maps of \(\mathcal{F}\) along morphisms of \(\mathcal{C}_{total}\) lying over \(\varphi\)). We conclude that \(\mathcal{F}(\varphi)|_{\mathcal{C}_m/V_i}\) is an isomorphism. As \(\{V_i \to V\}\) is a covering we conclude \(\mathcal{F}(\varphi)|_{\mathcal{C}_m/V}\) is an isomorphism. Since \(V\) and \(U\) were arbitrary this proves that \(\mathcal{F}\) is cartesian. (In case A use Sites, Lemma 08H2.)

Conversely, assume \(\mathcal{F}_n\) is quasi-coherent for all \(n\) and that \(\mathcal{F}\) is cartesian. Then for any \(n\) and object \(U\) of \(\mathcal{C}_n\) we can choose a covering \(\{U_i \to U\}\) of \(\mathcal{C}_n\) and for each \(i\) a presentation \[\bigoplus\nolimits_{j \in J_i} \mathcal{O}_{\mathcal{C}_m/U_i} \to \bigoplus\nolimits_{k \in K_i} \mathcal{O}_{\mathcal{C}_m/U_i} \to \mathcal{F}_n|_{\mathcal{C}_n/U_i} \to 0\] Pulling back to \(\mathcal{C}_{total}/U_i\) we obtain complexes \[\bigoplus\nolimits_{j \in J_i} \mathcal{O}_{\mathcal{C}_{total}/U_i} \to \bigoplus\nolimits_{k \in K_i} \mathcal{O}_{\mathcal{C}_{total}/U_i} \to \mathcal{F}|_{\mathcal{C}_{total}/U_i} \to 0\] of modules on \(\mathcal{C}_{total}/U_i\). Then the property that \(\mathcal{F}\) is cartesian implies that this is exact. We omit the details.

Simplicial systems of the derived category

In this section we are going to prove a special case of [BBD, Proposition 3.2.9] in the setting of derived categories of abelian sheaves. The case of modules is discussed in Section 0D9M.

Definition

In Situation 09WE. A simplicial system of the derived category consists of the following data

1. for every \(n\) an object \(K_n\) of \(D(\mathcal{C}_n)\),

2. for every \(\varphi : [m] \to [n]\) a map \(K_\varphi : f_\varphi^{-1}K_m \to K_n\) in \(D(\mathcal{C}_n)\)

subject to the condition that \[K_{\varphi \circ \psi} = K_\varphi \circ f_\varphi^{-1}K_\psi : f_{\varphi \circ \psi}^{-1}K_l = f_\varphi^{-1} f_\psi^{-1}K_l \longrightarrow K_n\] for any morphisms \(\varphi : [m] \to [n]\) and \(\psi : [l] \to [m]\) of \(\Delta\). We say the simplicial system is cartesian if the maps \(K_\varphi\) are isomorphisms for all \(\varphi\). Given two simplicial systems of the derived category there is an obvious notion of a morphism of simplicial systems of the derived category.

We have given this notion a ridiculously long name intentionally. The goal is to show that a simplicial system of the derived category comes from an object of \(D(\mathcal{C}_{total})\) under certain hypotheses.

Lemma

In Situation 09WE. If \(K \in D(\mathcal{C}_{total})\) is an object, then \((K_n, K(\varphi))\) is a simplicial system of the derived category. If \(K\) is cartesian, so is the system.

Proof

This is obvious.

Lemma

In Situation 09WE suppose given \(K_0 \in D(\mathcal{C}_0)\) and an isomorphism \[\alpha : f_{\delta_1^1}^{-1}K_0 \longrightarrow f_{\delta_0^1}^{-1}K_0\] satisfying the cocycle condition. Set \(\tau^n_i : [0] \to [n]\), \(0 \mapsto i\) and set \(K_n = f_{\tau^n_n}^{-1}K_0\). Then the \(K_n\) form a cartesian simplicial system of the derived category.

Proof

Please compare with Lemma 0D7I and its proof (also to see the cocycle condition spelled out). The construction is analogous to the construction discussed in Descent, Section 023F from which we borrow the notation \(\tau^n_i : [0] \to [n]\), \(0 \mapsto i\) and \(\tau^n_{ij} : [1] \to [n]\), \(0 \mapsto i\), \(1 \mapsto j\). Given \(\varphi : [n] \to [m]\) we define \(K_\varphi : f_\varphi^{-1}K_n \to K_m\) using \[\xymatrix{ f_\varphi^{-1}K_n \ar@{=}[r] & f_\varphi^{-1} f_{\tau^n_n}^{-1}K_0 \ar@{=}[r] & f_{\tau^m_{\varphi(n)}}^{-1}K_0 \ar@{=}[r] & f_{\tau^m_{\varphi(n)m}}^{-1}f_{\delta^1_1}^{-1}K_0 \ar[d]_{f_{\tau^m_{\varphi(n)m}}^{-1}\alpha} \\ & K_m \ar@{=}[r] & f_{\tau^m_m}^{-1}K_0 \ar@{=}[r] & f_{\tau^m_{\varphi(n)m}}^{-1}f_{\delta^1_0}^{-1}K_0 }\] We omit the verification that the cocycle condition implies the maps compose correctly (in their respective derived categories) and hence give rise to a simplicial system in the derived category.

Lemma

In Situation 09WE. Let \(K\) be an object of \(D(\mathcal{C}_{total})\). Set \[X_n = (g_{n!}\mathbf{Z}) \otimes^\mathbf{L}_\mathbf{Z} K \quad\text{and}\quad Y_n = (g_{n!}\mathbf{Z} \to \ldots \to g_{0!}\mathbf{Z})[-n] \otimes^\mathbf{L}_\mathbf{Z} K\] as objects of \(D(\mathcal{C}_{total})\) where the maps are as in Lemma 09WI. With the evident canonical maps \(Y_n \to X_n\) and \(Y_0 \to Y_1[1] \to Y_2[2] \to \ldots\) we have

1. the distinguished triangles \(Y_n \to X_n \to Y_{n - 1} \to Y_n[1]\) define a Postnikov system (Derived Categories, Definition 0D7Z) for \(\ldots \to X_2 \to X_1 \to X_0\),

2. \(K = \text{hocolim} Y_n[n]\) in \(D(\mathcal{C}_{total})\).

Proof

First, if \(K = \mathbf{Z}\), then this is the construction of Derived Categories, Example 0D8Z applied to the complex \[\ldots \to g_{2!}\mathbf{Z} \to g_{1!}\mathbf{Z} \to g_{0!}\mathbf{Z}\] in \(\textit{Ab}(\mathcal{C}_{total})\) combined with the fact that this complex represents \(K = \mathbf{Z}\) in \(D(\mathcal{C}_{total})\) by Lemma 09WI. The general case follows from this, the fact that the exact functor \(- \otimes^\mathbf{L}_\mathbf{Z} K\) sends Postnikov systems to Postnikov systems, and that \(- \otimes^\mathbf{L}_\mathbf{Z} K\) commutes with homotopy colimits.

Lemma

In Situation 09WE. If \(K, K' \in D(\mathcal{C}_{total})\). Assume

1. \(K\) is cartesian,

2. \(\Hom(K_i[i], K'_i) = 0\) for \(i > 0\), and

3. \(\Hom(K_i[i + 1], K'_i) = 0\) for \(i \geq 0\).

Then any map \(K \to K'\) which induces the zero map \(K_0 \to K'_0\) is zero.

Proof

Consider the objects \(X_n\) and the Postnikov system \(Y_n\) associated to \(K\) in Lemma 0D9I. As \(K = \text{hocolim} Y_n[n]\) the map \(K \to K'\) induces a compatible family of morphisms \(Y_n[n] \to K'\). By (1) and Lemma 0D9E we have \(X_n = g_{n!}K_n\). Since \(Y_0 = X_0\) we find that \(K_0 \to K'_0\) being zero implies \(Y_0 \to K'\) is zero. Suppose we’ve shown that the map \(Y_n[n] \to K'\) is zero for some \(n \geq 0\). From the distinguished triangle \[Y_n[n] \to Y_{n + 1}[n + 1] \to X_{n + 1}[n + 1] \to Y_n[n + 1]\] we get an exact sequence \[\Hom(X_{n + 1}[n + 1], K') \to \Hom(Y_{n + 1}[n + 1], K') \to \Hom(Y_n[n], K')\] As \(X_{n + 1}[n + 1] = g_{n + 1!}K_{n + 1}[n + 1]\) the first group is equal to \[\Hom(K_{n + 1}[n + 1], K'_{n + 1})\] which is zero by assumption (2). By induction we conclude all the maps \(Y_n[n] \to K'\) are zero. Consider the defining distinguished triangle \[\bigoplus Y_n[n] \to \bigoplus Y_n[n] \to K \to (\bigoplus Y_n[n])[1]\] for the homotopy colimit. Arguing as above, we find that it suffices to show that \[\Hom((\bigoplus Y_n[n])[1], K') = \prod \Hom(Y_n[n + 1], K')\] is zero for all \(n \geq 0\). To see this, arguing as above, it suffices to show that \[\Hom(K_n[n + 1], K'_n) = 0\] for all \(n \geq 0\) which follows from condition (3).

Lemma

In Situation 09WE. If \(K, K' \in D(\mathcal{C}_{total})\). Assume

1. \(K\) is cartesian,

2. \(\Hom(K_i[i - 1], K'_i) = 0\) for \(i > 1\).

Then any map \(\{K_n \to K'_n\}\) between the associated simplicial systems of \(K\) and \(K'\) comes from a map \(K \to K'\) in \(D(\mathcal{C}_{total})\).

Proof

Let \(\{K_n \to K'_n\}_{n \geq 0}\) be a morphism of simplicial systems of the derived category. Consider the objects \(X_n\) and Postnikov system \(Y_n\) associated to \(K\) of Lemma 0D9I. By (1) and Lemma 0D9E we have \(X_n = g_{n!}K_n\). In particular, the map \(K_0 \to K'_0\) induces a morphism \(X_0 \to K'\). Since \(\{K_n \to K'_n\}\) is a morphism of systems, a computation (omitted) shows that the composition \[X_1 \to X_0 \to K'\] is zero. As \(Y_0 = X_0\) and as \(Y_1\) fits into a distinguished triangle \[Y_1 \to X_1 \to Y_0 \to Y_1[1]\] we conclude that there exists a morphism \(Y_1[1] \to K'\) whose composition with \(X_0 = Y_0 \to Y_1[1]\) is the morphism \(X_0 \to K'\) given above. Suppose given a map \(Y_n[n] \to K'\) for \(n \geq 1\). From the distinguished triangle \[X_{n + 1}[n] \to Y_n[n] \to Y_{n + 1}[n + 1] \to X_{n + 1}[n + 1]\] we get an exact sequence \[\Hom(Y_{n + 1}[n + 1], K') \to \Hom(Y_n[n], K') \to \Hom(X_{n + 1}[n], K')\] As \(X_{n + 1}[n] = g_{n + 1!}K_{n + 1}[n]\) the last group is equal to \[\Hom(K_{n + 1}[n], K'_{n + 1})\] which is zero by assumption (2). By induction we get a system of maps \(Y_n[n] \to K'\) compatible with transition maps and reducing to the given map on \(Y_0\). This produces a map \[\gamma : K = \text{hocolim} Y_n[n] \longrightarrow K'\] This map in any case has the property that the diagram \[\xymatrix{ X_0 \ar[rd] \ar[r] & K \ar[d]^\gamma \\ & K' }\] is commutative. Restricting to \(\mathcal{C}_0\) we deduce that the map \(\gamma_0 : K_0 \to K'_0\) is the same as the first map \(K_0 \to K'_0\) of the morphism of simplicial systems. Since \(K\) is cartesian, this easily gives that \(\{\gamma_n\}\) is the map of simplicial systems we started out with.

Lemma

In Situation 09WE. Let \((K_n, K_\varphi)\) be a simplicial system of the derived category. Assume

1. \((K_n, K_\varphi)\) is cartesian,

2. \(\Hom(K_i[t], K_i) = 0\) for \(i \geq 0\) and \(t > 0\).

Then there exists a cartesian object \(K\) of \(D(\mathcal{C}_{total})\) whose associated simplicial system is isomorphic to \((K_n, K_\varphi)\).

Proof

Set \(X_n = g_{n!}K_n\) in \(D(\mathcal{C}_{total})\). For each \(n \geq 1\) we have \[\Hom(X_n, X_{n - 1}) = \Hom(K_n, g_n^{-1}g_{n - 1!}K_{n - 1}) = \bigoplus\nolimits_{\varphi : [n - 1] \to [n]} \Hom(K_n, f_\varphi^{-1}K_{n - 1})\] Thus we get a map \(X_n \to X_{n - 1}\) corresponding to the alternating sum of the maps \(K_\varphi^{-1} : K_n \to f_\varphi^{-1}K_{n - 1}\) where \(\varphi\) runs over \(\delta^n_0, \ldots, \delta^n_n\). We can do this because \(K_\varphi\) is invertible by assumption (1). Please observe the similarity with the definition of the maps in the proof of Lemma 09WI. We obtain a complex \[\ldots \to X_2 \to X_1 \to X_0\] in \(D(\mathcal{C}_{total})\). We omit the computation which shows that the compositions are zero. By Derived Categories, Lemma 0D83 if we have \[\Hom(X_i[i - j - 2], X_j) = 0\text{ for }i > j + 2\] then we can extend this complex to a Postnikov system. The group is equal to \[\Hom(K_i[i - j - 2], g_i^{-1}g_{j!}K_j)\] Again using that \((K_n, K_\varphi)\) is cartesian we see that \(g_i^{-1}g_{j!}K_j\) is isomorphic to a finite direct sum of copies of \(K_i\). Hence the group vanishes by assumption (2). Let the Postnikov system be given by \(Y_0 = X_0\) and distinguished sequences \(Y_n \to X_n \to Y_{n - 1} \to Y_n[1]\) for \(n \geq 1\). We set \[K = \text{hocolim} Y_n[n]\] To finish the proof we have to show that \(g_m^{-1}K\) is isomorphic to \(K_m\) for all \(m\) compatible with the maps \(K_\varphi\). Observe that \[g_m^{-1} K = \text{hocolim} g_m^{-1}Y_n[n]\] and that \(g_m^{-1}Y_n[n]\) is a Postnikov system for \(g_m^{-1}X_n\). Consider the isomorphisms \[g_m^{-1}X_n = \bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^{-1}K_n \xrightarrow{\bigoplus K_\varphi} \bigoplus\nolimits_{\varphi : [n] \to [m]} K_m\] These maps define an isomorphism of complexes \[\xymatrix{ \ldots \ar[r] & g_m^{-1}X_2 \ar[r] \ar[d] & g_m^{-1}X_1 \ar[r] \ar[d] & g_m^{-1}X_0 \ar[d] \\ \ldots \ar[r] & \bigoplus\nolimits_{\varphi : [2] \to [m]} K_m \ar[r] & \bigoplus\nolimits_{\varphi : [1] \to [m]} K_m \ar[r] & \bigoplus\nolimits_{\varphi : [0] \to [m]} K_m }\] in \(D(\mathcal{C}_m)\) where the arrows in the bottom row are as in the proof of Lemma 09WI. The squares commute by our choice of the arrows of the complex \(\ldots \to X_2 \to X_1 \to X_0\); we omit the computation. The bottom row complex has a postnikov tower given by \[Y'_{m, n} = \left(\bigoplus\nolimits_{\varphi : [n] \to [m]} \mathbf{Z} \to \ldots \to \bigoplus\nolimits_{\varphi : [0] \to [m]} \mathbf{Z}\right)[-n] \otimes^\mathbf{L}_\mathbf{Z} K_m\] and \(\text{hocolim} Y'_{m, n} = K_m\) (please compare with the proof of Lemma 0D9I and Derived Categories, Example 0D8Z). Applying the second part of Derived Categories, Lemma 0D83 the vertical maps in the big diagram extend to an isomorphism of Postnikov systems provided we have \[\Hom(g_m^{-1}X_i[i - j - 1], \bigoplus\nolimits_{\varphi : [j] \to [m]} K_m) = 0\text{ for }i > j + 1\] The is true if \(\Hom(K_m[i - j - 1], K_m) = 0\) for \(i > j + 1\) which holds by assumption (2). Choose an isomorphism given by \(\gamma_{m, n} : g_m^{-1}Y_n \to Y'_{m, n}\) of Postnikov systems in \(D(\mathcal{C}_m)\). By uniqueness of homotopy colimits, we can find an isomorphism \[g_m^{-1} K = \text{hocolim} g_m^{-1}Y_n[n] \xrightarrow{\gamma_m} \text{hocolim} Y'_{m, n} = K_m\] compatible with \(\gamma_{m, n}\).

We still have to prove that the maps \(\gamma_m\) fit into commutative diagrams \[\xymatrix{ f_\varphi^{-1}g_m^{-1}K \ar[d]_{f_\varphi^{-1}\gamma_m} \ar[r]_{K(\varphi)} & g_n^{-1}K \ar[d]^{\gamma_n} \\ f_\varphi^{-1}K_m \ar[r]^{K_\varphi} & K_n }\] for every \(\varphi : [m] \to [n]\). Consider the diagram \[\xymatrix{ f_\varphi^{-1}(\bigoplus_{\psi : [0] \to [m]} f_\psi^{-1}K_0) \ar@{=}[r] \ar[d]_{f_\varphi^{-1}(\bigoplus K_\psi)} & f_\varphi^{-1}g_m^{-1}X_0 \ar[d] \ar[r]_{X_0(\varphi)} & g_n^{-1}X_0 \ar[d] & \bigoplus_{\chi : [0] \to [n]} f_\chi^{-1}K_0 \ar@{=}[l] \ar[d]^{\bigoplus K_\chi} \\ f_\varphi^{-1}(\bigoplus_{\psi : [0] \to [m]} K_m) \ar@{=}[d] & f_\varphi^{-1}g_m^{-1}K \ar[d]_{f_\varphi^{-1}\gamma_m} \ar[r]_{K(\varphi)} & g_n^{-1}K \ar[d]^{\gamma_n} & \bigoplus_{\chi : [0] \to [n]} K_n \ar@{=}[d] \\ f_\varphi^{-1}Y'_{0, m} \ar[r] & f_\varphi^{-1}K_m \ar[r]^{K_\varphi} & K_n & Y'_{0, n} \ar[l] }\] The top middle square is commutative as \(X_0 \to K\) is a morphism of simplicial objects. The left, resp. the right rectangles are commutative as \(\gamma_m\), resp. \(\gamma_n\) is compatible with \(\gamma_{0, m}\), resp. \(\gamma_{0, n}\) which are the arrows \(\bigoplus K_\psi\) and \(\bigoplus K_\chi\) in the diagram. Going around the outer rectangle of the diagram is commutative as \((K_n, K_\varphi)\) is a simplical system and the map \(X_0(\varphi)\) is given by the obvious identifications \(f_\varphi^{-1}f_\psi^{-1}K_0 = f_{\varphi \circ \psi}^{-1}K_0\). Note that the arrow \(\bigoplus_\psi K_m \to Y'_{0, m} \to K_m\) induces an isomorphism on any of the direct summands (because of our explicit construction of the Postnikov systems \(Y'_{i, j}\) above). Hence, if we take a direct summand of the upper left and corner, then this maps isomorphically to \(f_\varphi^{-1}g_m^{-1}K\) as \(\gamma_m\) is an isomorphism. Working out what the above says, but looking only at this direct summand we conclude the lower middle square commutes as we well. This concludes the proof.

Simplicial systems of the derived category: modules

In this section we are going to prove a special case of [BBD, Proposition 3.2.9] in the setting of derived categories of \(\mathcal{O}\)-modules. The (slightly) easier case of abelian sheaves is discussed in Section 0D9F.

Definition

In Situation 09WE. Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}_{total}\). A simplicial system of the derived category of modules consists of the following data

1. for every \(n\) an object \(K_n\) of \(D(\mathcal{O}_n)\),

2. for every \(\varphi : [m] \to [n]\) a map \(K_\varphi : Lf_\varphi^*K_m \to K_n\) in \(D(\mathcal{O}_n)\)

subject to the condition that \[K_{\varphi \circ \psi} = K_\varphi \circ Lf_\varphi^*K_\psi : Lf_{\varphi \circ \psi}^*K_l = Lf_\varphi^* Lf_\psi^*K_l \longrightarrow K_n\] for any morphisms \(\varphi : [m] \to [n]\) and \(\psi : [l] \to [m]\) of \(\Delta\). We say the simplicial system is cartesian if the maps \(K_\varphi\) are isomorphisms for all \(\varphi\). Given two simplicial systems of the derived category there is an obvious notion of a morphism of simplicial systems of the derived category of modules.

We have given this notion a ridiculously long name intentionally. The goal is to show that a simplicial system of the derived category of modules comes from an object of \(D(\mathcal{O})\) under certain hypotheses.

Lemma

In Situation 09WE let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}_{total}\). If \(K \in D(\mathcal{O})\) is an object, then \((K_n, K(\varphi))\) is a simplicial system of the derived category of modules. If \(K\) is cartesian, so is the system.

Proof

This is immediate from the definitions.

Lemma

In Situation 09WE let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}_{total}\). Suppose given \(K_0 \in D(\mathcal{O}_0)\) and an isomorphism \[\alpha : L(f_{\delta_1^1})^*K_0 \longrightarrow L(f_{\delta_0^1})^*K_0\] satisfying the cocycle condition. Set \(\tau^n_i : [0] \to [n]\), \(0 \mapsto i\) and set \(K_n = Lf_{\tau^n_n}^*K_0\). The objects \(K_n\) form the members of a cartesian simplicial system of the derived category of modules.

Proof

Please compare with Lemmas 0GME and 0D7I and its proof (also to see the cocycle condition spelled out). The construction is analogous to the construction discussed in Descent, Section 023F from which we borrow the notation \(\tau^n_i : [0] \to [n]\), \(0 \mapsto i\) and \(\tau^n_{ij} : [1] \to [n]\), \(0 \mapsto i\), \(1 \mapsto j\). Given \(\varphi : [n] \to [m]\) we define \(K_\varphi : L(f_\varphi)^*K_n \to K_m\) using \[\xymatrix{ L(f_\varphi)^*K_n \ar@{=}[r] & L(f_\varphi)^* L(f_{\tau^n_n})^*K_0 \ar@{=}[r] & L(f_{\tau^m_{\varphi(n)}})^*K_0 \ar@{=}[r] & L(f_{\tau^m_{\varphi(n)m}})^* L(f_{\delta^1_1})^*K_0 \ar[d]_{L(f_{\tau^m_{\varphi(n)m}})^*\alpha} \\ & K_m \ar@{=}[r] & L(f_{\tau^m_m})^*K_0 \ar@{=}[r] & L(f_{\tau^m_{\varphi(n)m}})^* L(f_{\delta^1_0})^*K_0 }\] We omit the verification that the cocycle condition implies the maps compose correctly (in their respective derived categories) and hence give rise to a simplicial systems of the derived category of modules.

Lemma

In Situation 09WE let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}_{total}\). Let \(K\) be an object of \(D(\mathcal{C}_{total})\). Set \[X_n = (g_{n!}\mathcal{O}_n) \otimes^\mathbf{L}_\mathcal{O} K \quad\text{and}\quad Y_n = (g_{n!}\mathcal{O}_n \to \ldots \to g_{0!}\mathcal{O}_0)[-n] \otimes^\mathbf{L}_\mathcal{O} K\] as objects of \(D(\mathcal{O})\) where the maps are as in Lemma 09WI. With the evident canonical maps \(Y_n \to X_n\) and \(Y_0 \to Y_1[1] \to Y_2[2] \to \ldots\) we have

1. the distinguished triangles \(Y_n \to X_n \to Y_{n - 1} \to Y_n[1]\) define a Postnikov system (Derived Categories, Definition 0D7Z) for \(\ldots \to X_2 \to X_1 \to X_0\),

2. \(K = \text{hocolim} Y_n[n]\) in \(D(\mathcal{O})\).

Proof

First, if \(K = \mathcal{O}\), then this is the construction of Derived Categories, Example 0D8Z applied to the complex \[\ldots \to g_{2!}\mathcal{O}_2 \to g_{1!}\mathcal{O}_1 \to g_{0!}\mathcal{O}_0\] in \(\textit{Ab}(\mathcal{C}_{total})\) combined with the fact that this complex represents \(K = \mathcal{O}\) in \(D(\mathcal{C}_{total})\) by Lemma 0D9B. The general case follows from this, the fact that the exact functor \(- \otimes^\mathbf{L}_\mathcal{O} K\) sends Postnikov systems to Postnikov systems, and that \(- \otimes^\mathbf{L}_\mathcal{O} K\) commutes with homotopy colimits.

Lemma

In Situation 09WE let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}_{total}\). If \(K, K' \in D(\mathcal{O})\). Assume

1. \(f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m\) is flat for \(\varphi : [m] \to [n]\),

2. \(K\) is cartesian,

3. \(\Hom(K_i[i], K'_i) = 0\) for \(i > 0\), and

4. \(\Hom(K_i[i + 1], K'_i) = 0\) for \(i \geq 0\).

Then any map \(K \to K'\) which induces the zero map \(K_0 \to K'_0\) is zero.

Proof

The proof is exactly the same as the proof of Lemma 0D9J except using Lemma 0D9Q instead of Lemma 0D9I.

Lemma

In Situation 09WE let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}_{total}\). If \(K, K' \in D(\mathcal{O})\). Assume

1. \(f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m\) is flat for \(\varphi : [m] \to [n]\),

2. \(K\) is cartesian,

3. \(\Hom(K_i[i - 1], K'_i) = 0\) for \(i > 1\).

Then any map \(\{K_n \to K'_n\}\) between the associated simplicial systems of \(K\) and \(K'\) comes from a map \(K \to K'\) in \(D(\mathcal{O})\).

Proof

The proof is exactly the same as the proof of Lemma 0D9K except using Lemma 0D9Q instead of Lemma 0D9I.

Lemma

In Situation 09WE let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}_{total}\). Let \((K_n, K_\varphi)\) be a simplicial system of the derived category of modules. Assume

1. \(f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m\) is flat for \(\varphi : [m] \to [n]\),

2. \((K_n, K_\varphi)\) is cartesian,

3. \(\Hom(K_i[t], K_i) = 0\) for \(i \geq 0\) and \(t > 0\).

Then there exists a cartesian object \(K\) of \(D(\mathcal{O})\) whose associated simplicial system is isomorphic to \((K_n, K_\varphi)\).

Proof

The proof is exactly the same as the proof of Lemma 0D9L with the following changes

1. use \(g_n^* = Lg_n^*\) everywhere instead of \(g_n^{-1}\),

2. use \(f_\varphi^* = Lf_\varphi^*\) everywhere instead of \(f_\varphi^{-1}\),

3. refer to Lemma 0D9B instead of Lemma 09WI,

4. in the construction of \(Y'_{m, n}\) use \(\mathcal{O}_m\) instead of \(\mathbf{Z}\),

5. compare with the proof of Lemma 0D9Q rather than the proof of Lemma 0D9I.

This ends the proof.

The site associated to a semi-representable object

Let \(\mathcal{C}\) be a site. Recall that a semi-representable object of \(\mathcal{C}\) is simply a family \(\{U_i\}_{i \in I}\) of objects of \(\mathcal{C}\). A morphism \(\{U_i\}_{i \in I} \to \{V_j\}_{j \in J}\) of semi-representable objects is given by a map \(\alpha : I \to J\) and for every \(i \in I\) a morphism \(f_i : U_i \to V_{\alpha(i)}\) of \(\mathcal{C}\). The category of semi-representable objects of \(\mathcal{C}\) is denoted \(\text{SR}(\mathcal{C})\). See Hypercoverings, Definition 01G0 and the enclosing section for more information.

For a semi-representable object \(K = \{U_i\}_{i \in I}\) of \(\mathcal{C}\) we let \[\mathcal{C}/K = \coprod\nolimits_{i \in I} \mathcal{C}/U_i\] be the disjoint union of the localizations of \(\mathcal{C}\) at \(U_i\). There is a natural structure of a site on this category, with coverings inherited from the localizations \(\mathcal{C}/U_i\). The site \(\mathcal{C}/K\) is called the localization of \(\mathcal{C}\) at \(K\). Observe that a sheaf on \(\mathcal{C}/K\) is the same thing as a family of sheaves \(\mathcal{F}_i\) on \(\mathcal{C}/U_i\), i.e., \[\Sh(\mathcal{C}/K) = \prod\nolimits_{i \in I} \Sh(\mathcal{C}/U_i)\] This is occasionally useful to understand what is going on.

Let \(\mathcal{C}\) be a site. Let \(K = \{U_i\}_{i \in I}\) be an object of \(\text{SR}(\mathcal{C})\). There is a continuous and cocontinuous localization functor \(j : \mathcal{C}/K \to \mathcal{C}\) which is the product of the localization functors \(j_i : \mathcal{C}/V_i \to \mathcal{C}\). We obtain functors \(j_!\), \(j^{-1}\), \(j_*\) exactly as in Sites, Section 00XZ. In terms of the product decomposition \(\Sh(\mathcal{C}/K) = \prod\nolimits_{i \in I} \Sh(\mathcal{C}/U_i)\) we have \[\begin{matrix} j_! & : & (\mathcal{F}_i)_{i \in I} & \longmapsto & \coprod j_{i, !}\mathcal{F}_i \\ j^{-1} & : & \mathcal{G} & \longmapsto & (j_i^{-1}\mathcal{G})_{i \in I} \\ j_* & : & (\mathcal{F}_i)_{i \in I} & \longmapsto & \prod j_{i, *}\mathcal{F}_i \end{matrix}\] as the reader easily verifies.

Let \(f : K \to L\) be a morphism of \(\text{SR}(\mathcal{C})\). Then we obtain a continuous and cocontinuous functor \[v : \mathcal{C}/K \longrightarrow \mathcal{C}/L\] by applying the construction of Sites, Lemma 03EH to the components. More precisely, suppose \(f = (\alpha, f_i)\) where \(K = \{U_i\}_{i \in I}\), \(L = \{V_j\}_{j \in J}\), \(\alpha : I \to J\), and \(f_i : U_i \to V_{\alpha(i)}\). Then the functor \(v\) maps the component \(\mathcal{C}/U_i\) into the component \(\mathcal{C}/V_{\alpha(i)}\) via the construction of the aforementioned lemma. In particular we obtain a morphism \[f : \Sh(\mathcal{C}/K) \to \Sh(\mathcal{C}/L)\] of topoi. In terms of the product decompositions \(\Sh(\mathcal{C}/K) = \prod\nolimits_{i \in I} \Sh(\mathcal{C}/U_i)\) and \(\Sh(\mathcal{C}/L) = \prod\nolimits_{j \in J} \Sh(\mathcal{C}/V_j)\) the reader verifies that \[\begin{matrix} f_! & : & (\mathcal{F}_i)_{i \in I} & \longmapsto & (\coprod\nolimits_{i \in I, \alpha(i) = j} f_{i, !}\mathcal{F}_i)_{j \in J} \\ f^{-1} & : & (\mathcal{G}_j)_{j \in J} & \longmapsto & (f_i^{-1}\mathcal{G}_{\alpha(i)})_{i \in I} \\ f_* & : & (\mathcal{F}_i)_{i \in I} & \longmapsto & (\prod\nolimits_{i \in I, \alpha(i) = j} f_{i, *}\mathcal{F}_i)_{j \in J} \end{matrix}\] where \(f_i : \Sh(\mathcal{C}/U_i) \to \Sh(\mathcal{C}/V_{\alpha(i)})\) is the morphism associated to the localization functor \(\mathcal{C}/U_i \to \mathcal{C}/V_{\alpha(i)}\) corresponding to \(f_i : U_i \to V_{\alpha(i)}\).

Lemma

Let \(\mathcal{C}\) be a site.

1. For \(K\) in \(\text{SR}(\mathcal{C})\) the functor \(j : \mathcal{C}/K \to \mathcal{C}\) is continuous, cocontinuous, and has property P of Sites, Remark 09W7.

2. For \(f : K \to L\) in \(\text{SR}(\mathcal{C})\) the functor \(v : \mathcal{C}/K \to \mathcal{C}/L\) (see above) is continuous, cocontinuous, and has property P of Sites, Remark 09W7.

Proof

Proof of (2). In the notation of the discussion preceding the lemma, the localization functors \(\mathcal{C}/U_i \to \mathcal{C}/V_{\alpha(i)}\) are continuous and cocontinuous by Sites, Section 00XZ and satisfy \(P\) by Sites, Remark 09W8. It is formal to deduce \(v\) is continuous and cocontinuous and has \(P\). We omit the details. We also omit the proof of (1).

Lemma

Let \(\mathcal{C}\) be a site and \(K\) in \(\text{SR}(\mathcal{C})\). For \(\mathcal{F}\) in \(\Sh(\mathcal{C})\) we have \[j_*j^{-1}\mathcal{F} = \SheafHom(F(K)^\#, \mathcal{F})\] where \(F\) is as in Hypercoverings, Definition 01G1.

Proof

Say \(K = \{U_i\}_{i \in I}\). Using the description of the functors \(j^{-1}\) and \(j_*\) given above we see that \[j_*j^{-1}\mathcal{F} = \prod\nolimits_{i \in I} j_{i, *}(\mathcal{F}|_{\mathcal{C}/U_i}) = \prod\nolimits_{i \in I} \SheafHom(h_{U_i}^\#, \mathcal{F})\] The second equality by Sites, Lemma 0D7X. Since \(F(K) = \coprod h_{U_i}\) in \(\textit{PSh}(\mathcal{C}\), we have \(F(K)^\# = \coprod h_{U_i}^\#\) in \(\Sh(\mathcal{C})\) and since \(\SheafHom(-, \mathcal{F})\) turns coproducts into products (immediate from the construction in Sites, Section 04TP), we conclude.

Lemma

Let \(\mathcal{C}\) be a site.

1. For \(K\) in \(\text{SR}(\mathcal{C})\) the functor \(j_!\) gives an equivalence \(\Sh(\mathcal{C}/K) \to \Sh(\mathcal{C})/F(K)^\#\) where \(F\) is as in Hypercoverings, Definition 01G1.

2. The functor \(j^{-1} : \Sh(\mathcal{C}) \to \Sh(\mathcal{C}/K)\) corresponds via the identification of (1) with \(\mathcal{F} \mapsto (\mathcal{F} \times F(K)^\# \to F(K)^\#)\).

3. For \(f : K \to L\) in \(\text{SR}(\mathcal{C})\) the functor \(f^{-1}\) corresponds via the identifications of (1) to the functor \(\Sh(\mathcal{C})/F(L)^\# \to \Sh(\mathcal{C})/F(K)^\#\), \((\mathcal{G} \to F(L)^\#) \mapsto (\mathcal{G} \times_{F(L)^\#} F(K)^\# \to F(K)^\#)\).

Proof

Observe that if \(K = \{U_i\}_{i \in I}\) then the category \(\Sh(\mathcal{C}/K)\) decomposes as the product of the categories \(\Sh(\mathcal{C}/U_i)\). Observe that \(F(K)^\# = \coprod_{i \in I} h_{U_i}^\#\) (coproduct in sheaves). Hence \(\Sh(\mathcal{C})/F(K)^\#\) is the product of the categories \(\Sh(\mathcal{C})/h_{U_i}^\#\). Thus (1) and (2) follow from the corresponding statements for each \(i\), see Sites, Lemmas 00Y1 and 03EE. Similarly, if \(L = \{V_j\}_{j \in J}\) and \(f\) is given by \(\alpha : I \to J\) and \(f_i : U_i \to V_{\alpha(i)}\), then we can apply Sites, Lemma 04IL to each of the re-localization morphisms \(\mathcal{C}/U_i \to \mathcal{C}/V_{\alpha(i)}\) to get (3).

Lemma

Let \(\mathcal{C}\) be a site. For \(K\) in \(\text{SR}(\mathcal{C})\) the functor \(j^{-1}\) sends injective abelian sheaves to injective abelian sheaves. Similarly, the functor \(j^{-1}\) sends K-injective complexes of abelian sheaves to K-injective complexes of abelian sheaves.

Proof

The first statement is the natural generalization of Cohomology on Sites, Lemma 03F3 to semi-representable objects. In fact, it follows from this lemma by the product decomposition of \(\Sh(\mathcal{C}/K)\) and the description of the functor \(j^{-1}\) given above. The second statement is the natural generalization of Cohomology on Sites, Lemma 08FI and follows from it by the product decomposition of the topos.

Alternative: since \(j\) induces a localization of topoi by Lemma 0D87 part (1) it also follows immediately from Cohomology on Sites, Lemmas 03F3 and 08FI by enlarging the site; compare with the proof of Cohomology on Sites, Lemma 07A0 in the case of injective sheaves.

Remark

Let \(\mathcal{C}\) be a site. Let \(X \in \Ob(\mathcal{C})\). The category \(\text{SR}(\mathcal{C}, X)\) of semi-representable objects over \(X\) is defined by the formula \(\text{SR}(\mathcal{C}, X) = \text{SR}(\mathcal{C}/X)\). See Hypercoverings, Definition 01G0. Thus we may apply the above discussion to the site \(\mathcal{C}/X\). Briefly, the constructions above give

1. a site \(\mathcal{C}/K\) for \(K\) in \(\text{SR}(\mathcal{C}, X)\),

2. a decomposition \(\Sh(\mathcal{C}/K) = \prod \Sh(\mathcal{C}/U_i)\) if \(K = \{U_i/X\}\),

3. a localization functor \(j : \mathcal{C}/K \to \mathcal{C}/X\),

4. a morphism \(f : \Sh(\mathcal{C}/K) \to \Sh(\mathcal{C}/L)\) for \(f : K \to L\) in \(\text{SR}(\mathcal{C}, X)\).

All results of this section hold in this situation by replacing \(\mathcal{C}\) everywhere by \(\mathcal{C}/X\).

Remark

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O}_\mathcal{C}\) be a sheaf of rings on \(\mathcal{C}\). In this case, for any semi-representable object \(K\) of \(\mathcal{C}\) the site \(\mathcal{C}/K\) is a ringed site with sheaf of rings \(\mathcal{O}_K = j^{-1}\mathcal{O}_\mathcal{C}\). The constructions above give

1. a ringed site \((\mathcal{C}/K, \mathcal{O}_K)\) for \(K\) in \(\text{SR}(\mathcal{C})\),

2. a decomposition \(\textit{Mod}(\mathcal{O}_K) = \prod \textit{Mod}(\mathcal{O}_{U_i})\) if \(K = \{U_i\}\),

3. a localization morphism \(j : (\Sh(\mathcal{C}/K), \mathcal{O}_K) \to (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})\) of ringed topoi,

4. a morphism \(f : (\Sh(\mathcal{C}/K), \mathcal{O}_K) \to (\Sh(\mathcal{C}/L), \mathcal{O}_L)\) of ringed topoi for \(f : K \to L\) in \(\text{SR}(\mathcal{C})\).

Many of the results above hold in this setting. For example, the functor \(j^*\) has an exact left adjoint \[j_! : \textit{Mod}(\mathcal{O}_K) \to \textit{Mod}(\mathcal{O}_\mathcal{C}),\] which in terms of the product decomposition given in (2) sends \((\mathcal{F}_i)_{i \in I}\) to \(\bigoplus j_{i, !}\mathcal{F}_i\). Similarly, given \(f : K \to L\) as above, the functor \(f^*\) has an exact left adjoint \(f_! : \textit{Mod}(\mathcal{O}_K) \to \textit{Mod}(\mathcal{O}_L)\). Thus the functors \(j^*\) and \(f^*\) are exact, i.e., \(j\) and \(f\) are flat morphisms of ringed topoi (also follows from the equalities \(\mathcal{O}_K = j^{-1}\mathcal{O}_\mathcal{C}\) and \(\mathcal{O}_K = f^{-1}\mathcal{O}_L\)).

Remark

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O}_\mathcal{C}\) be a sheaf of rings on \(\mathcal{C}\). Let \(X \in \Ob(\mathcal{C})\) and denote \(\mathcal{O}_X = \mathcal{O}_\mathcal{C}|_{\mathcal{C}/U}\). Then we can combine the constructions given in Remarks 0D89 and 0D9U to get

1. a ringed site \((\mathcal{C}/K, \mathcal{O}_K)\) for \(K\) in \(\text{SR}(\mathcal{C}, X)\),

2. a decomposition \(\textit{Mod}(\mathcal{O}_K) = \prod \textit{Mod}(\mathcal{O}_{U_i})\) if \(K = \{U_i\}\),

3. a localization morphism \(j : (\Sh(\mathcal{C}/K), \mathcal{O}_K) \to (\Sh(\mathcal{C}/X), \mathcal{O}_X)\) of ringed topoi,

4. a morphism \(f : (\Sh(\mathcal{C}/K), \mathcal{O}_K) \to (\Sh(\mathcal{C}/L), \mathcal{O}_L)\) of ringed topoi for \(f : K \to L\) in \(\text{SR}(\mathcal{C}, X)\).

Of course all of the results mentioned in Remark 0D9U hold in this setting as well.

The site associate to a simplicial semi-representable object

Let \(\mathcal{C}\) be a site. Let \(K\) be a simplicial object of \(\text{SR}(\mathcal{C})\). As usual, set \(K_n = K([n])\) and denote \(K(\varphi) : K_n \to K_m\) the morphism associated to \(\varphi : [m] \to [n]\). By the construction in Section 09WK we obtain a simplicial object \(n \mapsto \mathcal{C}/K_n\) in the category whose objects are sites and whose morphisms are cocontinuous functors. In other words, we get a gadget as in Case B of Section 09WB. The functors satisfy property P by Lemma 0D85. Hence we may apply Lemma 09WD to obtain a site \((\mathcal{C}/K)_{total}\).

We can describe the site \((\mathcal{C}/K)_{total}\) explicitly as follows. Say \(K_n = \{U_{n, i}\}_{i \in I_n}\). For \(\varphi : [m] \to [n]\) the morphism \(K(\varphi) : K_n \to K_m\) is given by a map \(\alpha(\varphi) : I_n \to I_m\) and morphisms \(f_{\varphi, i} : U_{n, i} \to U_{m, \alpha(\varphi)(i)}\) for \(i \in I_n\). Then we have

1. an object of \((\mathcal{C}/K)_{total}\) corresponds to an object \((U/U_{n, i})\) of \(\mathcal{C}/U_{n, i}\) for some \(n\) and some \(i \in I_n\),

2. a morphism between \(U/U_{n, i}\) and \(V/U_{m, j}\) is a pair \((\varphi, f)\) where \(\varphi : [m] \to [n]\), \(j = \alpha(\varphi)(i)\), and \(f : U \to V\) is a morphism of \(\mathcal{C}\) such that \[\vcenter{ \xymatrix{ U \ar[r]_f \ar[d] & V \ar[d] \\ U_{n, i} \ar[r]^-{f_{\varphi, i}} & U_{m, j} } }\] is commutative, and

3. coverings of the object \(U/U_{n, i}\) are constructed by starting with a covering \(\{f_j : U_j \to U\}\) in \(\mathcal{C}\) and letting \(\{(\text{id}, f_j) : U_j/U_{n, i} \to U/U_{n, i}\}\) be a covering in \((\mathcal{C}/K)_{total}\).

All of our general theory developed for simplicial sites applies to \((\mathcal{C}/K)_{total}\). Observe that the obvious forgetful functor \[j_{total} : (\mathcal{C}/K)_{total} \longrightarrow \mathcal{C}\] is continuous and cocontinuous. It turns out that the associated morphism of topoi comes from an (obvious) augmentation.

Lemma

Let \(\mathcal{C}\) be a site. Let \(K\) be a simplicial object of \(\text{SR}(\mathcal{C})\). The localization functor \(j_0 : \mathcal{C}/K_0 \to \mathcal{C}\) defines an augmentation \(a_0 : \Sh(\mathcal{C}/K_0) \to \Sh(\mathcal{C})\), as in case (B) of Remark 0D6Z. The corresponding morphisms of topoi \[a_n : \Sh(\mathcal{C}/K_n) \longrightarrow \Sh(\mathcal{C}),\quad a : \Sh((\mathcal{C}/K)_{total}) \longrightarrow \Sh(\mathcal{C})\] of Lemma 0D70 are equal to the morphisms of topoi associated to the continuous and cocontinuous localization functors \(j_n : \mathcal{C}/K_n \to \mathcal{C}\) and \(j_{total} : (\mathcal{C}/K)_{total} \to \mathcal{C}\).

Proof

This is immediate from working through the definitions. See in particular the footnote in the proof of Lemma 0D70 for the relationship between \(a\) and \(j_{total}\).

Lemma

With assumption and notation as in Lemma 0D8B we have the following properties:

1. there is a functor \(a^{Sh}_! : \Sh((\mathcal{C}/K)_{total}) \to \Sh(\mathcal{C})\) left adjoint to \(a^{-1} : \Sh(\mathcal{C}) \to \Sh((\mathcal{C}/K)_{total})\),

2. there is a functor \(a_! : \textit{Ab}((\mathcal{C}/K)_{total}) \to \textit{Ab}(\mathcal{C})\) left adjoint to \(a^{-1} : \textit{Ab}(\mathcal{C}) \to \textit{Ab}((\mathcal{C}/K)_{total})\),

3. the functor \(a^{-1}\) associates to \(\mathcal{F}\) in \(\Sh(\mathcal{C})\) the sheaf on \((\mathcal{C}/K)_{total}\) which in degree \(n\) is equal to \(a_n^{-1}\mathcal{F}\),

4. the functor \(a_*\) associates to \(\mathcal{G}\) in \(\textit{Ab}((\mathcal{C}/K)_{total})\) the equalizer of the two maps \(j_{0, *}\mathcal{G}_0 \to j_{1, *}\mathcal{G}_1\),

Proof

Parts (3) and (4) hold for any augmentation of a simplicial site, see Lemma 0D70. Parts (1) and (2) follow as \(j_{total}\) is continuous and cocontinuous. The functor \(a^{Sh}_!\) is constructed in Sites, Lemma 00XR and the functor \(a_!\) is constructed in Modules on Sites, Lemma 04BG.

Lemma

Let \(\mathcal{C}\) be a site. Let \(K\) be a simplicial object of \(\text{SR}(\mathcal{C})\). Let \(U/U_{n, i}\) be an object of \(\mathcal{C}/K_n\). Let \(\mathcal{F} \in \textit{Ab}((\mathcal{C}/K)_{total})\). Then \[H^p(U, \mathcal{F}) = H^p(U, \mathcal{F}_{n, i})\] where

1. on the left hand side \(U\) is viewed as an object of \(\mathcal{C}_{total}\), and

2. on the right hand side \(\mathcal{F}_{n, i}\) is the \(i\)th component of the sheaf \(\mathcal{F}_n\) on \(\mathcal{C}/K_n\) in the decomposition \(\Sh(\mathcal{C}/K_n) = \prod \Sh(\mathcal{C}/U_{n, i})\) of Section 09WK.

Proof

This follows immediately from Lemma 0DBZ and the product decompositions of Section 09WK.

Remark

Let \(\mathcal{C}\) be a site. Let \(X \in \Ob(\mathcal{C})\). Recall that we have a category \(\text{SR}(\mathcal{C}, X) = \text{SR}(\mathcal{C}/X)\) of semi-representable objects over \(X\), see Remark 0D89. We may apply the above discussion to the site \(\mathcal{C}/X\). Briefly, the constructions above give

1. a site \((\mathcal{C}/K)_{total}\) for a simplicial \(K\) object of \(\text{SR}(\mathcal{C}, X)\),

2. a localization functor \(j_{total} : (\mathcal{C}/K)_{total} \to \mathcal{C}/X\),

3. localization functors \(j_n : \mathcal{C}/K_n \to \mathcal{C}/X\),

4. a morphism of topoi \(a : \Sh((\mathcal{C}/K)_{total}) \to \Sh(\mathcal{C}/X)\),

5. morphisms of topoi \(a_n : \Sh(\mathcal{C}/K_n) \to \Sh(\mathcal{C}/X)\),

6. a functor \(a^{Sh}_! : \Sh((\mathcal{C}/K)_{total}) \to \Sh(\mathcal{C}/X)\) left adjoint to \(a^{-1}\), and

7. a functor \(a_! : \textit{Ab}((\mathcal{C}/K)_{total}) \to \textit{Ab}(\mathcal{C}/X)\) left adjoint to \(a^{-1}\).

All of the results of this section hold in this setting. To prove this one replaces the site \(\mathcal{C}\) everywhere by \(\mathcal{C}/X\).

Remark

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O}_\mathcal{C}\) be a sheaf of rings. Given a simplicial semi-representable object \(K\) of \(\mathcal{C}\) we set \(\mathcal{O} = a^{-1}\mathcal{O}_\mathcal{C}\), where \(a\) is as in Lemmas 0D8B and 09WM. The constructions above, keeping track of the sheaves of rings as in Remark 0D9U, give

1. a ringed site \(((\mathcal{C}/K)_{total}, \mathcal{O})\) for a simplicial \(K\) object of \(\text{SR}(\mathcal{C})\),

2. a morphism of ringed topoi \(a : (\Sh((\mathcal{C}/K)_{total}), \mathcal{O}) \to (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})\),

3. morphisms of ringed topoi \(a_n : (\Sh(\mathcal{C}/K_n), \mathcal{O}_n) \to (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})\),

4. a functor \(a_! : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_\mathcal{C})\) left adjoint to \(a^*\).

The functor \(a_!\) exists (but in general is not exact) because \(a^{-1}\mathcal{O}_\mathcal{C} = \mathcal{O}\) and we can replace the use of Modules on Sites, Lemma 04BG in the proof of Lemma 09WM by Modules on Sites, Lemma 0797. As discussed in Remark 0D9U there are exact functors \(a_{n!} : \textit{Mod}(\mathcal{O}_n) \to \textit{Mod}(\mathcal{O}_\mathcal{C})\) left adjoint to \(a_n^*\). Consequently, the morphisms \(a\) and \(a_n\) are flat. Remark 0D9U implies the morphism of ringed topoi \(f_\varphi : (\Sh(\mathcal{C}/K_n), \mathcal{O}_n) \to (\Sh(\mathcal{C}/K_m), \mathcal{O}_m)\) for \(\varphi : [m] \to [n]\) is flat and there exists an exact functor \(f_{\varphi !} : \textit{Mod}(\mathcal{O}_n) \to \textit{Mod}(\mathcal{O}_m)\) left adjoint to \(f_\varphi^*\). This in turn implies that for the flat morphism of ringed topoi \(g_n : (\Sh(\mathcal{C}/K_n), \mathcal{O}_n) \to (\Sh((\mathcal{C}/K)_{total}), \mathcal{O})\) the functor \(g_{n!} : \textit{Mod}(\mathcal{O}_n) \to \textit{Mod}(\mathcal{O})\) left adjoint to \(g_n^*\) is exact, see Lemma 0D74.

Remark

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O}_\mathcal{C}\) be a sheaf of rings. Let \(X \in \Ob(\mathcal{C})\) and denote \(\mathcal{O}_X = \mathcal{O}_\mathcal{C}|_{\mathcal{C}/X}\). Then we can combine the constructions given in Remarks 0D8C and 0D9W to get

1. a ringed site \(((\mathcal{C}/K)_{total}, \mathcal{O})\) for a simplicial \(K\) object of \(\text{SR}(\mathcal{C}, X)\),

2. a morphism of ringed topoi \(a : (\Sh((\mathcal{C}/K)_{total}), \mathcal{O}) \to (\Sh(\mathcal{C}/X), \mathcal{O}_X)\),

3. morphisms of ringed topoi \(a_n : (\Sh(\mathcal{C}/K_n), \mathcal{O}_n) \to (\Sh(\mathcal{C}/X), \mathcal{O}_X)\),

4. a functor \(a_! : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_X)\) left adjoint to \(a^*\).

Of course, all the results mentioned in Remark 0D9W hold in this setting as well.

Cohomological descent for hypercoverings

Let \(\mathcal{C}\) be a site. In this section we assume \(\mathcal{C}\) has equalizers and fibre products. We let \(K\) be a hypercovering as defined in Hypercoverings, Definition 09VU. We will study the augmentation \[a : \Sh((\mathcal{C}/K)_{total}) \longrightarrow \Sh(\mathcal{C})\] of Section 0D8A.

Lemma

Let \(\mathcal{C}\) be a site with equalizers and fibre products. Let \(K\) be a hypercovering. Then

1. \(a^{-1} : \Sh(\mathcal{C}) \to \Sh((\mathcal{C}/K)_{total})\) is fully faithful with essential image the cartesian sheaves of sets,

2. \(a^{-1} : \textit{Ab}(\mathcal{C}) \to \textit{Ab}((\mathcal{C}/K)_{total})\) is fully faithful with essential image the cartesian sheaves of abelian groups.

In both cases \(a_*\) provides the quasi-inverse functor.

Proof

The case of abelian sheaves follows immediately from the case of sheaves of sets as the functor \(a^{-1}\) commutes with products. In the rest of the proof we work with sheaves of sets. Observe that \(a^{-1}\mathcal{F}\) is cartesian for \(\mathcal{F}\) in \(\Sh(\mathcal{C})\) by Lemma 0D7H. It suffices to show that the adjunction map \(\mathcal{F} \to a_*a^{-1}\mathcal{F}\) is an isomorphism \(\mathcal{F}\) in \(\Sh(\mathcal{C})\) and that for a cartesian sheaf \(\mathcal{G}\) on \((\mathcal{C}/K)_{total}\) the adjunction map \(a^{-1}a_*\mathcal{G} \to \mathcal{G}\) is an isomorphism.

Let \(\mathcal{F}\) be a sheaf on \(\mathcal{C}\). Recall that \(a_*a^{-1}\mathcal{F}\) is the equalizer of the two maps \(a_{0, *}a_0^{-1}\mathcal{F} \to a_{1, *}a_1^{-1}\mathcal{F}\), see Lemma 09WM. By Lemma 0D86 \[a_{0, *}a_0^{-1}\mathcal{F} = \SheafHom(F(K_0)^\#, \mathcal{F}) \quad\text{and}\quad a_{1, *}a_1^{-1}\mathcal{F} = \SheafHom(F(K_1)^\#, \mathcal{F})\] On the other hand, we know that \[\xymatrix{ F(K_1)^\# \ar@<1ex>[r] \ar@<-1ex>[r] & F(K_0)^\# \ar[r] & \text{final object }*\text{ of }\Sh(\mathcal{C}) }\] is a coequalizer diagram in sheaves of sets by definition of a hypercovering. Thus it suffices to prove that \(\SheafHom(-, \mathcal{F})\) transforms coequalizers into equalizers which is immediate from the construction in Sites, Section 04TP.

Let \(\mathcal{G}\) be a cartesian sheaf on \((\mathcal{C}/K)_{total}\). We will show that \(\mathcal{G} = a^{-1}\mathcal{F}\) for some sheaf \(\mathcal{F}\) on \(\mathcal{C}\). This will finish the proof because then \(a^{-1}a_*\mathcal{G} = a^{-1}a_*a^{-1}\mathcal{F} = a^{-1}\mathcal{F} = \mathcal{G}\) by the result of the previous paragraph. Set \(\mathcal{K}_n = F(K_n)^\#\) for \(n \geq 0\). Then we have maps of sheaves \[\xymatrix{ \mathcal{K}_2 \ar@<1ex>[r] \ar@<0ex>[r] \ar@<-1ex>[r] & \mathcal{K}_1 \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & \mathcal{K}_0 }\] coming from the fact that \(K\) is a simplicial semi-representable object. The fact that \(K\) is a hypercovering means that \[\mathcal{K}_1 \to \mathcal{K}_0 \times \mathcal{K}_0 \quad\text{and}\quad \mathcal{K}_2 \to \left(\text{cosk}_1( \xymatrix{ \mathcal{K}_1 \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & \mathcal{K}_0 \ar[l] })\right)_2\] are surjective maps of sheaves. Using the description of cartesian sheaves on \((\mathcal{C}/K)_{total}\) given in Lemma 0D7I and using the description of \(\Sh(\mathcal{C}/K_n)\) in Lemma 0D87 we find that our problem can be entirely formulated³ in terms of

1. the topos \(\Sh(\mathcal{C})\), and

2. the simplicial object \(\mathcal{K}\) in \(\Sh(\mathcal{C})\) whose terms are \(\mathcal{K}_n\).

Thus, after replacing \(\mathcal{C}\) by a different site \(\mathcal{C}'\) as in Sites, Lemma 03CI, we may assume \(\mathcal{C}\) has all finite limits, the topology on \(\mathcal{C}\) is subcanonical, a family \(\{V_j \to V\}\) of morphisms of \(\mathcal{C}\) is a covering if and only if \(\coprod h_{V_j} \to V\) is surjective, and there exists a simplicial object \(U\) of \(\mathcal{C}\) such that \(\mathcal{K}_n = h_{U_n}\) as simplicial sheaves. Working backwards through the equivalences we may assume \(K_n = \{U_n\}\) for all \(n\).

Let \(X\) be the final object of \(\mathcal{C}\). Then \(\{U_0 \to X\}\) is a covering, \(\{U_1 \to U_0 \times U_0\}\) is a covering, and \(\{U_2 \to (\text{cosk}_1 \text{sk}_1 U)_2\}\) is a covering. Let us use \(d^n_i : U_n \to U_{n - 1}\) and \(s^n_j : U_n \to U_{n + 1}\) the morphisms corresponding to \(\delta^n_i\) and \(\sigma^n_j\) as in Simplicial, Definition 0165. By abuse of notation, given a morphism \(c : V \to W\) of \(\mathcal{C}\) we denote the morphism of topoi \(c : \Sh(\mathcal{C}/V) \to \Sh(\mathcal{C}/W)\) by the same letter. Now \(\mathcal{G}\) is given by a sheaf \(\mathcal{G}_0\) on \(\mathcal{C}/U_0\) and an isomorphism \(\alpha : (d^1_1)^{-1}\mathcal{G}_0 \to (d^1_0)^{-1}\mathcal{G}_0\) satisfying the cocycle condition on \(\mathcal{C}/U_2\) formulated in Lemma 0D7I. Since \(\{U_2 \to (\text{cosk}_1 \text{sk}_1 U)_2\}\) is a covering, the corresponding pullback functor on sheaves is faithful (small detail omitted). Hence we may replace \(U\) by \(\text{cosk}_1 \text{sk}_1 U\), because this replaces \(U_2\) by \((\text{cosk}_1 \text{sk}_1 U)_2\) and leaves \(U_1\) and \(U_0\) unchanged. Then \[(d^2_0, d^2_1, d^2_2) : U_2 \to U_1 \times U_1 \times U_1\] is a monomorphism whose its image on \(T\)-valued points is described in Simplicial, Lemma 0187. In particular, there is a morphism \(c\) fitting into a commutative diagram \[\xymatrix{ U_1 \times_{(d^1_1, d^1_0), U_0 \times U_0, (d^1_1, d^1_0)} U_1 \ar[d] \ar[rr]_c & & U_2 \ar[d] \\ U_1 \times U_1 \ar[rr]^{(\text{pr}_1, \text{pr}_2, s^0_0 \circ d^1_1 \circ \text{pr}_1)} & & U_1 \times U_1 \times U_1 }\] as going around the other way defines a point of \(U_2\). Pulling back the cocycle condition for \(\alpha\) on \(U_2\) translates into the condition that the pullbacks of \(\alpha\) via the projections to \(U_1 \times_{(d^1_1, d^1_0), U_0 \times U_0, (d^1_1, d^1_0)} U_1\) are the same as the pullback of \(\alpha\) via \(s^0_0 \circ d^1_1 \circ \text{pr}_1\) is the identity map (namely, the pullback of \(\alpha\) by \(s^0_0\) is the identity). By Sites, Lemma 04TQ this means that \(\alpha\) comes from an isomorphism \[\alpha' : \text{pr}_1^{-1}\mathcal{G}_0 \to \text{pr}_2^{-1}\mathcal{G}_0\] of sheaves on \(\mathcal{C}/U_0 \times U_0\). Then finally, the morphism \(U_2 \to U_0 \times U_0 \times U_0\) is surjective on associated sheaves as is easily seen using the surjectivity of \(U_1 \to U_0 \times U_0\) and the description of \(U_2\) given above. Therefore \(\alpha'\) satisfies the cocycle condition on \(U_0 \times U_0 \times U_0\). The proof is finished by an application of Sites, Lemma 04TS to the covering \(\{U_0 \to X\}\).

Lemma

Let \(\mathcal{C}\) be a site with equalizers and fibre products. Let \(K\) be a hypercovering. The Čech complex of Lemma 0D79 associated to \(a^{-1}\mathcal{F}\) \[a_{0, *}a_0^{-1}\mathcal{F} \to a_{1, *}a_1^{-1}\mathcal{F} \to a_{2, *}a_2^{-1}\mathcal{F} \to \ldots\] is equal to the complex \(\SheafHom(s(\mathbf{Z}_{F(K)}^\#), \mathcal{F})\). Here \(s(\mathbf{Z}_{F(K)}^\#)\) is as in Hypercoverings, Definition 01GB.

Proof

By Lemma 0D86 we have \[a_{n, *}a_n^{-1}\mathcal{F} = \SheafHom'(F(K_n)^\#, \mathcal{F})\] where \(\SheafHom'\) is as in Sites, Section 04TP. The boundary maps in the complex of Lemma 0D79 come from the simplicial structure. Thus the equality of complexes comes from the canonical identifications \(\SheafHom'(\mathcal{G}, \mathcal{F}) = \SheafHom(\mathbf{Z}_\mathcal{G}, \mathcal{F})\) for \(\mathcal{G}\) in \(\Sh(\mathcal{C})\).

Lemma

Let \(\mathcal{C}\) be a site with equalizers and fibre products. Let \(K\) be a hypercovering. For \(E \in D(\mathcal{C})\) the map \[E \longrightarrow Ra_*a^{-1}E\] is an isomorphism.

Proof

First, let \(\mathcal{I}\) be an injective abelian sheaf on \(\mathcal{C}\). Then the spectral sequence of Lemma 0D7A for the sheaf \(a^{-1}\mathcal{I}\) degenerates as \((a^{-1}\mathcal{I})_p = a_p^{-1}\mathcal{I}\) is injective by Lemma 0D88. Thus the complex \[a_{0, *}a_0^{-1}\mathcal{I} \to a_{1, *}a_1^{-1}\mathcal{I} \to a_{2, *}a_2^{-1}\mathcal{I} \to\ldots\] computes \(Ra_*a^{-1}\mathcal{I}\). By Lemma 0D8F this is equal to the complex \(\SheafHom(s(\mathbf{Z}_{F(K)}^\#), \mathcal{I})\). Because \(K\) is a hypercovering, we see that \(s(\mathbf{Z}_{F(K)}^\#)\) is exact in degrees \(> 0\) by Hypercoverings, Lemma 01GE applied to the simplicial presheaf \(F(K)\). Since \(\mathcal{I}\) is injective, the functor \(\SheafHom(-, \mathcal{I})\) is exact and we conclude that \(\SheafHom(s(\mathbf{Z}_{F(K)}^\#), \mathcal{I})\) is exact in positive degrees. We conclude that \(R^pa_*a^{-1}\mathcal{I} = 0\) for \(p > 0\). On the other hand, we have \(\mathcal{I} = a_*a^{-1}\mathcal{I}\) by Lemma 0D8E.

Bounded case. Let \(E \in D^+(\mathcal{C})\). Choose a bounded below complex \(\mathcal{I}^\bullet\) of injectives representing \(E\). By the result of the first paragraph and Leray’s acyclicity lemma (Derived Categories, Lemma 015E) \(Ra_*a^{-1}\mathcal{I}^\bullet\) is computed by the complex \(a_*a^{-1}\mathcal{I}^\bullet = \mathcal{I}^\bullet\) and we conclude the lemma is true in this case.

Unbounded case. We urge the reader to skip this, since the argument is the same as above, except that we use explicit representation by double complexes to get around convergence issues. Let \(E \in D(\mathcal{C})\). To show the map \(E \to Ra_*a^{-1}E\) is an isomorphism, it suffices to show for every object \(U\) of \(\mathcal{C}\) that \[R\Gamma(U, E) = R\Gamma(U, Ra_*a^{-1}E)\] We will compute both sides and show the map \(E \to Ra_*a^{-1}E\) induces an isomorphism. Choose a K-injective complex \(\mathcal{I}^\bullet\) representing \(E\). Choose a quasi-isomorphism \(a^{-1}\mathcal{I}^\bullet \to \mathcal{J}^\bullet\) for some K-injective complex \(\mathcal{J}^\bullet\) on \((\mathcal{C}/K)_{total}\). We have \[R\Gamma(U, E) = R\Hom(\mathbf{Z}_U^\#, E)\] and \[R\Gamma(U, Ra_*a^{-1}E) = R\Hom(\mathbf{Z}_U^\#, Ra_*a^{-1}E) = R\Hom(a^{-1}\mathbf{Z}_U^\#, a^{-1}E)\] By Lemma 0D78 we have a quasi-isomorphism \[\Big(\ldots \to g_{2!}(a_2^{-1}\mathbf{Z}_U^\#) \to g_{1!}(a_1^{-1}\mathbf{Z}_U^\#) \to g_{0!}(a_0^{-1}\mathbf{Z}_U^\#)\Big) \longrightarrow a^{-1}\mathbf{Z}_U^\#\] Hence \(R\Hom(a^{-1}\mathbf{Z}_U^\#, a^{-1}E)\) is equal to \[R\Gamma((\mathcal{C}/K)_{total}, R\SheafHom( \ldots \to g_{2!}(a_2^{-1}\mathbf{Z}_U^\#) \to g_{1!}(a_1^{-1}\mathbf{Z}_U^\#) \to g_{0!}(a_0^{-1}\mathbf{Z}_U^\#), \mathcal{J}^\bullet))\] By the construction in Cohomology on Sites, Section 08J7 and since \(\mathcal{J}^\bullet\) is K-injective, we see that this is represented by the complex of abelian groups with terms \[\prod\nolimits_{p + q = n} \Hom(g_{p!}(a_p^{-1}\mathbf{Z}_U^\#), \mathcal{J}^q) = \prod\nolimits_{p + q = n} \Hom(a_p^{-1}\mathbf{Z}_U^\#, g_p^{-1}\mathcal{J}^q)\] See Cohomology on Sites, Lemmas 0A94 and 08JA for more information. Thus we find that \(R\Gamma(U, Ra_*a^{-1}E)\) is computed by the product total complex \(\text{Tot}_\pi(B^{\bullet, \bullet})\) with \(B^{p, q} = \Hom(a_p^{-1}\mathbf{Z}_U^\#, g_p^{-1}\mathcal{J}^q)\). For the other side we argue similarly. First we note that \[s(\mathbf{Z}_{F(K)}^\#) \longrightarrow \mathbf{Z}\] is a quasi-isomorphism of complexes on \(\mathcal{C}\) by Hypercoverings, Lemma 01GE. Since \(\mathbf{Z}_U^\#\) is a flat sheaf of \(\mathbf{Z}\)-modules we see that \[s(\mathbf{Z}_{F(K)}^\#) \otimes_\mathbf{Z} \mathbf{Z}_U^\# \longrightarrow \mathbf{Z}_U^\#\] is a quasi-isomorphism. Therefore \(R\Hom(\mathbf{Z}_U^\#, E)\) is equal to \[R\Gamma(\mathcal{C}, R\SheafHom( s(\mathbf{Z}_{F(K)}^\#) \otimes_\mathbf{Z} \mathbf{Z}_U^\#, \mathcal{I}^\bullet))\] By the construction of \(R\SheafHom\) and since \(\mathcal{I}^\bullet\) is K-injective, this is represented by the complex of abelian groups with terms \[\prod\nolimits_{p + q = n} \Hom(\mathbf{Z}^\#_{K_p} \otimes_\mathbf{Z} \mathbf{Z}_U^\#, \mathcal{I}^q) = \prod\nolimits_{p + q = n} \Hom(a_p^{-1}\mathbf{Z}_U^\#, a_p^{-1}\mathcal{I}^q)\] The equality of terms follows from the fact that \(\mathbf{Z}^\#_{K_p} \otimes_\mathbf{Z} \mathbf{Z}_U^\# = a_{p!}a_p^{-1}\mathbf{Z}_U^\#\) by Modules on Sites, Remark 0EYY. Thus we find that \(R\Gamma(U, E)\) is computed by the product total complex \(\text{Tot}_\pi(A^{\bullet, \bullet})\) with \(A^{p, q} = \Hom(a_p^{-1}\mathbf{Z}_U^\#, a_p^{-1}\mathcal{I}^q)\).

Since \(\mathcal{I}^\bullet\) is K-injective we see that \(a_p^{-1}\mathcal{I}^\bullet\) is K-injective, see Lemma 0D88. Since \(\mathcal{J}^\bullet\) is K-injective we see that \(g_p^{-1}\mathcal{J}^\bullet\) is K-injective, see Lemma 09WH. Both represent the object \(a_p^{-1}E\). Hence for every \(p \geq 0\) the map of complexes \[A^{p, \bullet} = \Hom(a_p^{-1}\mathbf{Z}_U^\#, a_p^{-1}\mathcal{I}^\bullet) \longrightarrow \Hom(a_p^{-1}\mathbf{Z}_U^\#, g_p^{-1}\mathcal{J}^\bullet) = B^{p, \bullet}\] induced by \(g_p^{-1}\) applied to the given map \(a^{-1}\mathcal{I}^\bullet \to \mathcal{J}^\bullet\) is a quasi-isomorphisms as these complexes both compute \[R\Hom(a_p^{-1}\mathbf{Z}_U^\#, a_p^{-1}E)\] By More on Algebra, Lemma 0EYX we conclude that the right vertical arrow in the commutative diagram \[\xymatrix{ R\Gamma(U, E) \ar[r] \ar[d] & \text{Tot}_\pi(A^{\bullet, \bullet}) \ar[d] \\ R\Gamma(U, Ra_*a^{-1}E) \ar[r] & \text{Tot}_\pi(B^{\bullet, \bullet}) }\] is a quasi-isomorphism. Since we saw above that the horizontal arrows are quasi-isomorphisms, so is the left vertical arrow.

Lemma

Let \(\mathcal{C}\) be a site with equalizers and fibre products. Let \(K\) be a hypercovering. Then we have a canonical isomorphism \[R\Gamma(\mathcal{C}, E) = R\Gamma((\mathcal{C}/K)_{total}, a^{-1}E)\] for \(E \in D(\mathcal{C})\).

Proof

This follows from Lemma 0D8G because \(R\Gamma((\mathcal{C}/K)_{total}, -) = R\Gamma(\mathcal{C}, -) \circ Ra_*\) by Cohomology on Sites, Remark 08J6.

Lemma

Let \(\mathcal{C}\) be a site with equalizers and fibre products. Let \(K\) be a hypercovering. Let \(\mathcal{A} \subset \textit{Ab}((\mathcal{C}/K)_{total})\) denote the weak Serre subcategory of cartesian abelian sheaves. Then the functor \(a^{-1}\) defines an equivalence \[D^+(\mathcal{C}) \longrightarrow D_\mathcal{A}^+((\mathcal{C}/K)_{total})\] with quasi-inverse \(Ra_*\).

Proof

Observe that \(\mathcal{A}\) is a weak Serre subcategory by Lemma 0D7J. The equivalence is a formal consequence of the results obtained so far. Use Lemmas 0D8E and 0D8G and Cohomology on Sites, Lemma 0D7U

We urge the reader to skip the following remark.

Remark

Let \(\mathcal{C}\) be a site. Let \(\mathcal{G}\) be a presheaf of sets on \(\mathcal{C}\). If \(\mathcal{C}\) has equalizers and fibre products, then we’ve defined the notion of a hypercovering of \(\mathcal{G}\) in Hypercoverings, Definition 09VU. We claim that all the results in this section have a valid counterpart in this setting. To see this, define the localization \(\mathcal{C}/\mathcal{G}\) of \(\mathcal{C}\) at \(\mathcal{G}\) exactly as in Sites, Lemma 0791 (which is stated only for sheaves; the topos \(\Sh(\mathcal{C}/\mathcal{G})\) is equal to the localization of the topos \(\Sh(\mathcal{C})\) at the sheaf \(\mathcal{G}^\#\)). Then the reader easily shows that the site \(\mathcal{C}/\mathcal{G}\) has fibre products and equalizers and that a hypercovering of \(\mathcal{G}\) in \(\mathcal{C}\) is the same thing as a hypercovering for the site \(\mathcal{C}/\mathcal{G}\). Hence replacing the site \(\mathcal{C}\) by \(\mathcal{C}/\mathcal{G}\) in the lemmas on hypercoverings above we obtain proofs of the corresponding results for hypercoverings of \(\mathcal{G}\). Example: for a hypercovering \(K\) of \(\mathcal{G}\) we have \[R\Gamma(\mathcal{C}/\mathcal{G}, E) = R\Gamma((\mathcal{C}/K)_{total}, a^{-1}E)\] for \(E \in D^+(\mathcal{C}/\mathcal{G})\) where \(a : \Sh((\mathcal{C}/K)_{total}) \to \Sh(\mathcal{C}/\mathcal{G})\) is the canonical augmentation. This is Lemma 0D8H. Let \(R\Gamma(\mathcal{G}, -) : D(\mathcal{C}) \to D(\textit{Ab})\) be defined as the derived functor of the functor \(H^0(\mathcal{G}, -) = H^0(\mathcal{G}^\#, -)\) discussed in Hypercoverings, Section 09VT and Cohomology on Sites, Section 079X. We have \[R\Gamma(\mathcal{G}, E) = R\Gamma(\mathcal{C}/\mathcal{G}, j^{-1}E)\] by the analogue of Cohomology on Sites, Lemma 03F3 for the localization fuctor \(j : \mathcal{C}/\mathcal{G} \to \mathcal{C}\). Putting everything together we obtain \[R\Gamma(\mathcal{G}, E) = R\Gamma((\mathcal{C}/K)_{total}, a^{-1}j^{-1}E) = R\Gamma((\mathcal{C}/K)_{total}, g^{-1}E)\] for \(E \in D^+(\mathcal{C})\) where \(g : \Sh((\mathcal{C}/K)_{total}) \to \Sh(\mathcal{C})\) is the composition of \(a\) and \(j\).

Cohomological descent for hypercoverings: modules

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O}_\mathcal{C}\) be a sheaf of rings. Assume \(\mathcal{C}\) has equalizers and fibre products and let \(K\) be a hypercovering as defined in Hypercoverings, Definition 09VU. We will study cohomological descent for the augmentation \[a : (\Sh((\mathcal{C}/K)_{total}), \mathcal{O}) \longrightarrow (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})\] of Remark 0D9W.

Lemma

Let \(\mathcal{C}\) be a site with equalizers and fibre products. Let \(\mathcal{O}_\mathcal{C}\) be a sheaf of rings. Let \(K\) be a hypercovering. With notation as above \[a^* : \textit{Mod}(\mathcal{O}_\mathcal{C}) \to \textit{Mod}(\mathcal{O})\] is fully faithful with essential image the cartesian \(\mathcal{O}\)-modules. The functor \(a_*\) provides the quasi-inverse.

Proof

Since \(a^{-1}\mathcal{O}_\mathcal{C} = \mathcal{O}\) we have \(a^* = a^{-1}\). Hence the lemma follows immediately from Lemma 0D8E.

Lemma

Let \(\mathcal{C}\) be a site with equalizers and fibre products. Let \(\mathcal{O}_\mathcal{C}\) be a sheaf of rings. Let \(K\) be a hypercovering. For \(E \in D(\mathcal{O}_\mathcal{C})\) the map \[E \longrightarrow Ra_*La^*E\] is an isomorphism.

Proof

Since \(a^{-1}\mathcal{O}_\mathcal{C} = \mathcal{O}\) we have \(La^* = a^* = a^{-1}\). Moreover \(Ra_*\) agrees with \(Ra_*\) on abelian sheaves, see Cohomology on Sites, Lemma 0D6J. Hence the lemma follows immediately from Lemma 0D8G.

Lemma

Let \(\mathcal{C}\) be a site with equalizers and fibre products. Let \(\mathcal{O}_\mathcal{C}\) be a sheaf of rings. Let \(K\) be a hypercovering. Then we have a canonical isomorphism \[R\Gamma(\mathcal{C}, E) = R\Gamma((\mathcal{C}/K)_{total}, La^*E)\] for \(E \in D(\mathcal{O}_\mathcal{C})\).

Proof

This follows from Lemma 0DA0 because \(R\Gamma((\mathcal{C}/K)_{total}, -) = R\Gamma(\mathcal{C}, -) \circ Ra_*\) by Cohomology on Sites, Remark 08J6 or by Cohomology on Sites, Lemma 0D6H.

Lemma

Let \(\mathcal{C}\) be a site with equalizers and fibre products. Let \(\mathcal{O}_\mathcal{C}\) be a sheaf of rings. Let \(K\) be a hypercovering. Let \(\mathcal{A} \subset \textit{Mod}(\mathcal{O})\) denote the weak Serre subcategory of cartesian \(\mathcal{O}\)-modules. Then the functor \(La^*\) defines an equivalence \[D^+(\mathcal{O}_\mathcal{C}) \longrightarrow D_\mathcal{A}^+(\mathcal{O})\] with quasi-inverse \(Ra_*\).

Proof

Observe that \(\mathcal{A}\) is a weak Serre subcategory by Lemma 0D7J (the required hypotheses hold by the discussion in Remark 0D9W). The equivalence is a formal consequence of the results obtained so far. Use Lemmas 0D9Z and 0DA0 and Cohomology on Sites, Lemma 0D7U.

Cohomological descent for hypercoverings of an object

In this section we assume \(\mathcal{C}\) has fibre products and \(X \in \Ob(\mathcal{C})\). We let \(K\) be a hypercovering of \(X\) as defined in Hypercoverings, Definition 01G5. We will study the augmentation \[a : \Sh((\mathcal{C}/K)_{total}) \longrightarrow \Sh(\mathcal{C}/X)\] of Remark 0D8C. Observe that \(\mathcal{C}/X\) is a site which has equalizers and fibre products and that \(K\) is a hypercovering for the site \(\mathcal{C}/X\)⁴ by Hypercoverings, Lemma 01G9. This means that every single result proved for hypercoverings in Section 0D8D has an immediate analogue in the situation in this section.

Lemma

Let \(\mathcal{C}\) be a site with fibre products and \(X \in \Ob(\mathcal{C})\). Let \(K\) be a hypercovering of \(X\). Then

1. \(a^{-1} : \Sh(\mathcal{C}/X) \to \Sh((\mathcal{C}/K)_{total})\) is fully faithful with essential image the cartesian sheaves of sets,

2. \(a^{-1} : \textit{Ab}(\mathcal{C}/X) \to \textit{Ab}((\mathcal{C}/K)_{total})\) is fully faithful with essential image the cartesian sheaves of abelian groups.

In both cases \(a_*\) provides the quasi-inverse functor.

Proof

Via Remarks 0D89 and 0D8C and the discussion in the introduction to this section this follows from Lemma 0D8E.

Lemma

Let \(\mathcal{C}\) be a site with fibre product and \(X \in \Ob(\mathcal{C})\). Let \(K\) be a hypercovering of \(X\). For \(E \in D(\mathcal{C}/X)\) the map \[E \longrightarrow Ra_*a^{-1}E\] is an isomorphism.

Proof

Via Remarks 0D89 and 0D8C and the discussion in the introduction to this section this follows from Lemma 0D8G.

Lemma

Let \(\mathcal{C}\) be a site with fibre products and \(X \in \Ob(\mathcal{C})\). Let \(K\) be a hypercovering of \(X\). Then we have a canonical isomorphism \[R\Gamma(X, E) = R\Gamma((\mathcal{C}/K)_{total}, a^{-1}E)\] for \(E \in D(\mathcal{C}/X)\).

Proof

Via Remarks 0D89 and 0D8C this follows from Lemma 0D8H.

Lemma

Let \(\mathcal{C}\) be a site with fibre products and \(X \in \Ob(\mathcal{C})\). Let \(K\) be a hypercovering of \(X\). Let \(\mathcal{A} \subset \textit{Ab}((\mathcal{C}/K)_{total})\) denote the weak Serre subcategory of cartesian abelian sheaves. Then the functor \(a^{-1}\) defines an equivalence \[D^+(\mathcal{C}/X) \longrightarrow D_\mathcal{A}^+((\mathcal{C}/K)_{total})\] with quasi-inverse \(Ra_*\).

Proof

Via Remarks 0D89 and 0D8C this follows from Lemma 0D8I.

Cohomological descent for hypercoverings of an object: modules

In this section we assume \(\mathcal{C}\) has fibre products and \(X \in \Ob(\mathcal{C})\). We let \(K\) be a hypercovering of \(X\) as defined in Hypercoverings, Definition 01G5. Let \(\mathcal{O}_\mathcal{C}\) be a sheaf of rings on \(\mathcal{C}\). Set \(\mathcal{O}_X = \mathcal{O}_\mathcal{C}|_{\mathcal{C}/X}\). We will study the augmentation \[a : (\Sh((\mathcal{C}/K)_{total}), \mathcal{O}) \longrightarrow (\Sh(\mathcal{C}/X), \mathcal{O}_X)\] of Remark 0D9X. Observe that \(\mathcal{C}/X\) is a site which has equalizers and fibre products and that \(K\) is a hypercovering for the site \(\mathcal{C}/X\). Therefore the results in this section are immediate consequences of the corresponding results in Section 0D9Y.

Lemma

Let \(\mathcal{C}\) be a site with fibre products and \(X \in \Ob(\mathcal{C})\). Let \(\mathcal{O}_\mathcal{C}\) be a sheaf of rings. Let \(K\) be a hypercovering of \(X\). With notation as above \[a^* : \textit{Mod}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O})\] is fully faithful with essential image the cartesian \(\mathcal{O}\)-modules. The functor \(a_*\) provides the quasi-inverse.

Proof

Via Remarks 0D9V and 0D9X and the discussion in the introduction to this section this follows from Lemma 0D9Z.

Lemma

Let \(\mathcal{C}\) be a site with fibre products and \(X \in \Ob(\mathcal{C})\). Let \(\mathcal{O}_\mathcal{C}\) be a sheaf of rings. Let \(K\) be a hypercovering of \(X\). For \(E \in D(\mathcal{O}_X)\) the map \[E \longrightarrow Ra_*La^*E\] is an isomorphism.

Proof

Via Remarks 0D9V and 0D9X and the discussion in the introduction to this section this follows from Lemma 0DA0.

Lemma

Let \(\mathcal{C}\) be a site with fibre products and \(X \in \Ob(\mathcal{C})\). Let \(\mathcal{O}_\mathcal{C}\) be a sheaf of rings. Let \(K\) be a hypercovering of \(X\). Then we have a canonical isomorphism \[R\Gamma(X, E) = R\Gamma((\mathcal{C}/K)_{total}, La^*E)\] for \(E \in D(\mathcal{O}_\mathcal{C})\).

Proof

Via Remarks 0D9V and 0D9X and the discussion in the introduction to this section this follows from Lemma 0DA1.

Lemma

Let \(\mathcal{C}\) be a site with fibre products and \(X \in \Ob(\mathcal{C})\). Let \(\mathcal{O}_\mathcal{C}\) be a sheaf of rings. Let \(K\) be a hypercovering of \(X\). Let \(\mathcal{A} \subset \textit{Mod}(\mathcal{O})\) denote the weak Serre subcategory of cartesian \(\mathcal{O}\)-modules. Then the functor \(La^*\) defines an equivalence \[D^+(\mathcal{O}_X) \longrightarrow D_\mathcal{A}^+(\mathcal{O})\] with quasi-inverse \(Ra_*\).

Proof

Via Remarks 0D9V and 0D9X and the discussion in the introduction to this section this follows from Lemma 0DA2.

Hypercovering by a simplicial object of the site

Let \(\mathcal{C}\) be a site with fibre products and let \(X \in \Ob(\mathcal{C})\). In this section we elucidate the results of Section 0D8J in the case that our hypercovering is given by a simplicial object of the site. Let \(U\) be a simplicial object of \(\mathcal{C}\). As usual we denote \(U_n = U([n])\) and \(f_\varphi : U_n \to U_m\) the morphism \(f_\varphi = U(\varphi)\) corresponding to \(\varphi : [m] \to [n]\). Assume we have an augmentation \[a : U \to X\] From this we obtain a simplicial site \((\mathcal{C}/U)_{total}\) and an augmentation morphism \[a : \Sh((\mathcal{C}/U)_{total}) \longrightarrow \Sh(\mathcal{C}/X)\] Namely, from \(U\) we obtain a simiplical object \(K\) of \(\text{SR}(\mathcal{C}, X)\) with degree \(n\) part \(K_n = \{U_n \to X\}\) and we can apply the constructions in Remark 0D8C. More precisely, an object of the site \((\mathcal{C}/U)_{total}\) is given by a \(V/U_n\) and a morphism \((\varphi, f) : V/U_n \to W/U_m\) is given by a morphism \(\varphi : [m] \to [n]\) in \(\Delta\) and a morphism \(f : V \to W\) such that the diagram \[\xymatrix{ V \ar[r]_f \ar[d] & W \ar[d] \\ U_n \ar[r]^{f_\varphi} & U_m }\] is commutative. The morphism of topoi \(a\) is given by the cocontinuous functor \(V/U_n \mapsto V/X\). That’s all folks!

In this section we will say the augmentation \(a : U \to X\) is a hypercovering of \(X\) in \(\mathcal{C}\) if the following hold

1. \(\{U_0 \to X\}\) is a covering of \(\mathcal{C}\),

2. \(\{U_1 \to U_0 \times_X U_0\}\) is a covering of \(\mathcal{C}\),

3. \(\{U_{n + 1} \to (\text{cosk}_n\text{sk}_n U)_{n + 1}\}\) is a covering of \(\mathcal{C}\) for \(n \geq 1\).

This is equivalent to the condition that \(K\) (as above) is a hypercovering of \(X\), see Hypercoverings, Example 0GM9.

Lemma

Let \(\mathcal{C}\) be a site with fibre product and \(X \in \Ob(\mathcal{C})\). Let \(a : U \to X\) be a hypercovering of \(X\) in \(\mathcal{C}\) as defined above. Then

1. \(a^{-1} : \Sh(\mathcal{C}/X) \to \Sh((\mathcal{C}/U)_{total})\) is fully faithful with essential image the cartesian sheaves of sets,

2. \(a^{-1} : \textit{Ab}(\mathcal{C}/X) \to \textit{Ab}((\mathcal{C}/U)_{total})\) is fully faithful with essential image the cartesian sheaves of abelian groups.

In both cases \(a_*\) provides the quasi-inverse functor.

Proof

This is a special case of Lemma 0D8K.

Lemma

Let \(\mathcal{C}\) be a site with fibre product and \(X \in \Ob(\mathcal{C})\). Let \(a : U \to X\) be a hypercovering of \(X\) in \(\mathcal{C}\) as defined above. For \(E \in D(\mathcal{C}/X)\) the map \[E \longrightarrow Ra_*a^{-1}E\] is an isomorphism.

Proof

This is a special case of Lemma 0D8L.

Lemma

Let \(\mathcal{C}\) be a site with fibre products and \(X \in \Ob(\mathcal{C})\). Let \(a : U \to X\) be a hypercovering of \(X\) in \(\mathcal{C}\) as defined above. Then we have a canonical isomorphism \[R\Gamma(X, E) = R\Gamma((\mathcal{C}/U)_{total}, a^{-1}E)\] for \(E \in D(\mathcal{C}/X)\).

Proof

This is a special case of Lemma 09X7.

Lemma

Let \(\mathcal{C}\) be a site with fibre product and \(X \in \Ob(\mathcal{C})\). Let \(a : U \to X\) be a hypercovering of \(X\) in \(\mathcal{C}\) as defined above. Let \(\mathcal{A} \subset \textit{Ab}((\mathcal{C}/U)_{total})\) denote the weak Serre subcategory of cartesian abelian sheaves. Then the functor \(a^{-1}\) defines an equivalence \[D^+(\mathcal{C}/X) \longrightarrow D_\mathcal{A}^+((\mathcal{C}/U)_{total})\] with quasi-inverse \(Ra_*\).

Proof

This is a special case of Lemma 0D8M

Lemma

Let \(U\) be a simplicial object of a site \(\mathcal{C}\) with fibre products.

1. \(\mathcal{C}/U\) has the structure of a simplicial object in the category whose objects are sites and whose morphisms are morphisms of sites,

2. the construction of Lemma 09WC applied to the structure in (1) reproduces the site \((\mathcal{C}/U)_{total}\) above,

3. if \(a : U \to X\) is an augmentation, then \(a_0 : \mathcal{C}/U_0 \to \mathcal{C}/X\) is an augmentation as in Remark 0D6Z part (A) and gives the same morphism of topoi \(a : \Sh((\mathcal{C}/U)_{total}) \to \Sh(\mathcal{C}/X)\) as the one above.

Proof

Given a morphism of objects \(V \to W\) of \(\mathcal{C}\) the localization morphism \(j : \mathcal{C}/V \to \mathcal{C}/W\) is a left adjoint to the base change functor \(\mathcal{C}/W \to \mathcal{C}/V\). The base change functor is continuous and induces the same morphism of topoi as \(j\). See Sites, Lemma 09W9. This proves (1).

Part (2) holds because a morphism \(V/U_n \to W/U_m\) of the category constructed in Lemma 09WC is a morphism \(V \to W \times_{U_m, f_\varphi} U_n\) over \(U_n\) which is the same thing as a morphism \(f : V \to W\) over the morphism \(f_\varphi : U_n \to U_m\), i.e., the same thing as a morphism in the category \((\mathcal{C}/U)_{total}\) defined above. Equality of sets of coverings is immediate from the definition.

We omit the proof of (3).

Hypercovering by a simplicial object of the site: modules

Let \(\mathcal{C}\) be a site with fibre products and \(X \in \Ob(\mathcal{C})\). Let \(\mathcal{O}_\mathcal{C}\) be a sheaf of rings on \(\mathcal{C}\). Let \(U \to X\) be a hypercovering of \(X\) in \(\mathcal{C}\) as defined in Section 09X8. In this section we study the augmentation \[a : (\Sh((\mathcal{C}/U)_{total}), \mathcal{O}) \longrightarrow (\Sh(\mathcal{C}/X), \mathcal{O}_X)\] we obtain by thinking of \(U\) as a simiplical semi-representable object of \(\mathcal{C}/X\) whose degree \(n\) part is the singleton element \(\{U_n/X\}\) and applying the constructions in Remark 0D9X. Thus all the results in this section are immediate consequences of the corresponding results in Section 0DA3.

Lemma

Let \(\mathcal{C}\) be a site with fibre products and \(X \in \Ob(\mathcal{C})\). Let \(\mathcal{O}_\mathcal{C}\) be a sheaf of rings. Let \(U\) be a hypercovering of \(X\) in \(\mathcal{C}\). With notation as above \[a^* : \textit{Mod}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O})\] is fully faithful with essential image the cartesian \(\mathcal{O}\)-modules. The functor \(a_*\) provides the quasi-inverse.

Proof

This is a special case of Lemma 0DA4.

Lemma

Let \(\mathcal{C}\) be a site with fibre products and \(X \in \Ob(\mathcal{C})\). Let \(\mathcal{O}_\mathcal{C}\) be a sheaf of rings. Let \(U\) be a hypercovering of \(X\) in \(\mathcal{C}\). For \(E \in D(\mathcal{O}_X)\) the map \[E \longrightarrow Ra_*La^*E\] is an isomorphism.

Proof

This is a special case of Lemma 0DA5.

Lemma

Let \(\mathcal{C}\) be a site with fibre products and \(X \in \Ob(\mathcal{C})\). Let \(\mathcal{O}_\mathcal{C}\) be a sheaf of rings. Let \(U\) be a hypercovering of \(X\) in \(\mathcal{C}\). Then we have a canonical isomorphism \[R\Gamma(X, E) = R\Gamma((\mathcal{C}/U)_{total}, La^*E)\] for \(E \in D(\mathcal{O}_\mathcal{C})\).

Proof

This is a special case of Lemma 0DA6.

Lemma

Let \(\mathcal{C}\) be a site with fibre products and \(X \in \Ob(\mathcal{C})\). Let \(\mathcal{O}_\mathcal{C}\) be a sheaf of rings. Let \(U\) be a hypercovering of \(X\) in \(\mathcal{C}\). Let \(\mathcal{A} \subset \textit{Mod}(\mathcal{O})\) denote the weak Serre subcategory of cartesian \(\mathcal{O}\)-modules. Then the functor \(La^*\) defines an equivalence \[D^+(\mathcal{O}_X) \longrightarrow D_\mathcal{A}^+(\mathcal{O})\] with quasi-inverse \(Ra_*\).

Proof

This is a special case of Lemma 0DA7.

Unbounded cohomological descent for hypercoverings

In this section we discuss unbounded cohomological descent. The results themselves will be immediate consequences of our results on bounded cohomological descent in the previous sections and Cohomology on Sites, Lemmas 0D7V and/or 0D7W; the real work lies in setting up notation and choosing appropriate assumptions. Our discussion is motivated by the discussion in [six-I] although the details are a good bit different.

Let \((\mathcal{C}, \mathcal{O}_\mathcal{C})\) be a ringed site. Assume given for every object \(U\) of \(\mathcal{C}\) a weak Serre subcategory \(\mathcal{A}_U \subset \textit{Mod}(\mathcal{O}_U)\) satisfying the following properties

1. given a morphism \(U \to V\) of \(\mathcal{C}\) the restriction functor \(\textit{Mod}(\mathcal{O}_V) \to \textit{Mod}(\mathcal{O}_U)\) sends \(\mathcal{A}_V\) into \(\mathcal{A}_U\),

2. given a covering \(\{U_i \to U\}_{i \in I}\) of \(\mathcal{C}\) an object \(\mathcal{F}\) of \(\textit{Mod}(\mathcal{O}_U)\) is in \(\mathcal{A}_U\) if and only if the restriction of \(\mathcal{F}\) to \(\mathcal{C}/U_i\) is in \(\mathcal{A}_{U_i}\) for all \(i \in I\).

3. 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 \[H^p(V_i, \mathcal{F}) = 0 \text{ for } \{V_i \to V\} \in \text{Cov}_V,\ p > d_V, \text{ and } \mathcal{F} \in \Ob(\mathcal{A}_V)\]

Note that we require this to be true for \(\mathcal{F}\) in \(\mathcal{A}_V\) and not just for “global” objects (and thus it is stronger than the condition imposed in Cohomology on Sites, Situation 0D6R). In this situation, there is a weak Serre subcategory \(\mathcal{A} \subset \textit{Mod}(\mathcal{O}_\mathcal{C})\) consisting of objects whose restriction to \(\mathcal{C}/U\) is in \(\mathcal{A}_U\) for all \(U \in \Ob(\mathcal{C})\). Moreover, there are derived categories \(D_\mathcal{A}(\mathcal{O}_\mathcal{C})\) and \(D_{\mathcal{A}_U}(\mathcal{O}_U)\) and the restriction functors send these into each other.

Example

Let \(S\) be a scheme and let \(X\) be an algebraic space over \(S\). Let \(\mathcal{C} = X_{spaces, \etale}\) be the étale site on the category of algebraic spaces étale over \(X\), see Properties of Spaces, Definition 03G0. Denote \(\mathcal{O}_\mathcal{C}\) the structure sheaf, i.e., the sheaf given by the rule \(U \mapsto \Gamma(U, \mathcal{O}_U)\). Denote \(\mathcal{A}_U\) the category of quasi-coherent \(\mathcal{O}_U\)-modules. Let \(\mathcal{B} = \Ob(\mathcal{C})\) and for \(V \in \mathcal{B}\) set \(d_V = 0\) and let \(\text{Cov}_V\) denote the coverings \(\{V_i \to V\}\) with \(V_i\) affine for all \(i\). Then the assumptions (1), (2), (3) are satisfied. See Properties of Spaces, Lemmas 03GA and 03M1 for properties (1) and (2) and the vanishing in (3) follows from Cohomology of Schemes, Lemma 01XB and the discussion in Cohomology of Spaces, Section 071Y.

Example

Let \(S\) be one of the following types of schemes

1. the spectrum of a finite field,

2. the spectrum of a separably closed field,

3. the spectrum of a strictly henselian Noetherian local ring,

4. the spectrum of a henselian Noetherian local ring with finite residue field,

5. add more here.

Let \(\Lambda\) be a finite ring whose order is invertible on \(S\). Let \(\mathcal{C} \subset (\Sch/S)_\etale\) be the full subcategory consisting of schemes locally of finite type over \(S\) endowed with the étale topology. Let \(\mathcal{O}_\mathcal{C} = \underline{\Lambda}\) be the constant sheaf. Set \(\mathcal{A}_U = \textit{Mod}(\mathcal{O}_U)\), in other words, we consider all étale sheaves of \(\Lambda\)-modules. Let \(\mathcal{B} \subset \Ob(\mathcal{C})\) be the set of quasi-compact objects. For \(V \in \mathcal{B}\) set \[d_V = 1 + 2\dim(S) + \sup\nolimits_{v \in V}(\text{trdeg}_{\kappa(s)}(\kappa(v)) + 2 \dim \mathcal{O}_{V, v})\] and let \(\text{Cov}_V\) denote the étale coverings \(\{V_i \to V\}\) with \(V_i\) quasi-compact for all \(i\). Our choice of bound \(d_V\) comes from Gabber’s theorem on cohomological dimension. To see that condition (3) holds with this choice, use [Traveaux, Exposé VIII-A, Corollary 1.2 and Lemma 2.2] plus elementary arguments on cohomological dimensions of fields. We add \(1\) to the formula because our list contains cases where we allow \(S\) to have finite residue field. We will come back to this example later (insert future reference).

Let \((\mathcal{C}, \mathcal{O}_\mathcal{C})\) be a ringed site. Assume given weak Serre subcategories \(\mathcal{A}_U \subset \textit{Mod}(\mathcal{O}_U)\) satisfying condition (0DC2). Then

1. given a semi-representable object \(K = \{U_i\}_{i \in I}\) we get a weak Serre subcategory \(\mathcal{A}_K \subset \textit{Mod}(\mathcal{O}_K)\) by taking \(\prod \mathcal{A}_{U_i} \subset \prod \textit{Mod}(\mathcal{O}_{U_i}) = \textit{Mod}(\mathcal{O}_K)\), and

2. given a morphism of semi-representable objects \(f : K \to L\) the pullback map \(f^* : \textit{Mod}(\mathcal{O}_L) \to \textit{Mod}(\mathcal{O}_L)\) sends \(\mathcal{A}_L\) into \(\mathcal{A}_K\).

See Remark 0D9U for notation and explanation. In particular, given a simplicial semi-representable object \(K\) it is unambiguous to say what it means for an object \(\mathcal{F}\) of \(\textit{Mod}(\mathcal{O})\) as in Remark 0D9W to have restrictions \(\mathcal{F}_n\) in \(\mathcal{A}_{K_n}\) for all \(n\).

Lemma

Let \((\mathcal{C}, \mathcal{O}_\mathcal{C})\) be a ringed site. Assume given weak Serre subcategories \(\mathcal{A}_U \subset \textit{Mod}(\mathcal{O}_U)\) satisfying conditions (0DC2), (0DC3), and (0DC4) above. Assume \(\mathcal{C}\) has equalizers and fibre products and let \(K\) be a hypercovering. Let \(((\mathcal{C}/K)_{total}, \mathcal{O})\) be as in Remark 0D9W. Let \(\mathcal{A}_{total} \subset \textit{Mod}(\mathcal{O})\) denote the weak Serre subcategory of cartesian \(\mathcal{O}\)-modules \(\mathcal{F}\) whose restriction \(\mathcal{F}_n\) is in \(\mathcal{A}_{K_n}\) for all \(n\) (as defined above). Then the functor \(La^*\) defines an equivalence \[D_\mathcal{A}(\mathcal{O}_\mathcal{C}) \longrightarrow D_{\mathcal{A}_{total}}(\mathcal{O})\] with quasi-inverse \(Ra_*\).

Proof

The cartesian \(\mathcal{O}\)-modules form a weak Serre subcategory by Lemma 0D7J (the required hypotheses hold by the discussion in Remark 0D9W). Since the restriction functor \(g_n^* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_n)\) are exact, it follows that \(\mathcal{A}_{total}\) is a weak Serre subcategory.

Let us show that \(a^* : \mathcal{A} \to \mathcal{A}_{total}\) is an equivalence of categories with inverse given by \(La_*\). We already know that \(La_*a^*\mathcal{F} = \mathcal{F}\) by the bounded version (Lemma 0DA2). It is clear that \(a^*\mathcal{F}\) is in \(\mathcal{A}_{total}\) for \(\mathcal{F}\) in \(\mathcal{A}\). Conversely, assume that \(\mathcal{G} \in \mathcal{A}_{total}\). Because \(\mathcal{G}\) is cartesian we see that \(\mathcal{G} = a^*\mathcal{F}\) for some \(\mathcal{O}_\mathcal{C}\)-module \(\mathcal{F}\) by Lemma 0D9Z. We want to show that \(\mathcal{F}\) is in \(\mathcal{A}\). Take \(U \in \Ob(\mathcal{C})\). We have to show that the restriction of \(\mathcal{F}\) to \(\mathcal{C}/U\) is in \(\mathcal{A}_U\). As usual, write \(K_0 = \{U_{0, i}\}_{i \in I_0}\). Since \(K\) is a hypercovering, the map \(\coprod_{i \in I_0} h_{U_{0, i}} \to *\) becomes surjective after sheafification. This implies there is a covering \(\{U_j \to U\}_{j \in J}\) and a map \(\tau : J \to I_0\) and for each \(j \in J\) a morphism \(\varphi_j : U_j \to U_{0, \tau(j)}\). Since \(\mathcal{G}_0 = a_0^*\mathcal{F}\) we find that the restriction of \(\mathcal{F}\) to \(\mathcal{C}/U_j\) is equal to the restriction of the \(\tau(j)\)th component of \(\mathcal{G}_0\) to \(\mathcal{C}/U_j\) via the morphism \(\varphi_j : U_j \to U_{0, \tau(i)}\). Hence by (0DC2) we find that \(\mathcal{F}|_{\mathcal{C}/U_j}\) is in \(\mathcal{A}_{U_j}\) and in turn by (0DC3) we find that \(\mathcal{F}|_{\mathcal{C}/U}\) is in \(\mathcal{A}_U\).

In particular the statement of the lemma makes sense. The lemma now follows from Cohomology on Sites, Lemma 0D7V. Assumption (1) is clear (see Remark 0D9W). Assumptions (2) and (3) we proved in the preceding paragraph. Assumption (4) is immediate from (0DC4). For assumption (5) let \(\mathcal{B}_{total}\) be the set of objects \(U/U_{n, i}\) of the site \((\mathcal{C}/K)_{total}\) such that \(U \in \mathcal{B}\) where \(\mathcal{B}\) is as in (0DC4). Here we use the description of the site \((\mathcal{C}/K)_{total}\) given in Section 0D8A. Moreover, we set \(\text{Cov}_{U/U_{n, i}}\) equal to \(\text{Cov}_U\) and \(d_{U/U_{n, i}}\) equal \(d_U\) where \(\text{Cov}_U\) and \(d_U\) are given to us by (0DC4). Then we claim that condition (5) holds with these choices. This follows immediately from Lemma 0DC0 and the fact that \(\mathcal{F} \in \mathcal{A}_{total}\) implies \(\mathcal{F}_n \in \mathcal{A}_{K_n}\) and hence \(\mathcal{F}_{n, i} \in \mathcal{A}_{U_{n, i}}\). (The reader who worries about the difference between cohomology of abelian sheaves versus cohomology of sheaves of modules may consult Cohomology on Sites, Lemma 03FD.)

Glueing complexes

This section is the continuation of Cohomology, Section 0D65. The goal is to prove a slight generalization of [BBD, Theorem 3.2.4]. Our method will be a tiny bit different in that we use the material from Sections 0D9F and 0D9M. We will also reprove the unbounded version as it is proved in [six-I].

Advice to the reader: We suggest the reader first look at the statement of Lemma 0GMG as well as the second proof of this lemma.

Here is the situation we are interested in.

Situation

Let \((\mathcal{C}, \mathcal{O}_\mathcal{C})\) be a ringed site. We are given

1. a category \(\mathcal{B}\) and a functor \(u : \mathcal{B} \to \mathcal{C}\),

2. an object \(E_U\) in \(D(\mathcal{O}_{u(U)})\) for \(U \in \Ob(\mathcal{B})\),

3. an isomorphism \(\rho_a : E_U|_{\mathcal{C}/u(V)} \to E_V\) in \(D(\mathcal{O}_{u(V)})\) for \(a : V \to U\) in \(\mathcal{B}\)

such that whenever we have composable arrows \(b : W \to V\) and \(a : V \to U\) of \(\mathcal{B}\), then \(\rho_{a \circ b} = \rho_b \circ \rho_a|_{\mathcal{C}/u(W)}\).

We won’t be able to prove anything about this without making more assumptions. An interesting case is where \(\mathcal{B}\) is a full subcategory such that every object of \(\mathcal{C}\) has a covering whose members are objects of \(\mathcal{B}\) (this is the case considered in [BBD]). For us it is important to allow cases where this is not the case; the main alternative case is where we have a morphism of sites \(f : \mathcal{C} \to \mathcal{D}\) and \(\mathcal{B}\) is a full subcategory of \(\mathcal{D}\) such that every object of \(\mathcal{D}\) has a covering whose members are objects of \(\mathcal{B}\).

In Situation 0DC9 a solution will be a pair \((E, \rho_U)\) where \(E\) is an object of \(D(\mathcal{O}_\mathcal{C})\) and \(\rho_U : E|_{\mathcal{C}/u(U)} \to E_U\) for \(U \in \Ob(\mathcal{B})\) are isomorphisms such that we have \(\rho_a \circ \rho_U|_{\mathcal{C}/u(V)} = \rho_V\) for \(a : V \to U\) in \(\mathcal{B}\).

Lemma

In Situation 0DC9. Assume negative self-exts of \(E_U\) in \(D(\mathcal{O}_{u(U)})\) are zero. Let \(L\) be a simplicial object of \(\text{SR}(\mathcal{B})\). Consider the simplicial object \(K = u(L)\) of \(\text{SR}(\mathcal{C})\) and let \(((\mathcal{C}/K)_{total}, \mathcal{O})\) be as in Remark 0D9W. There exists a cartesian object \(E\) of \(D(\mathcal{O})\) such that writing \(L_n = \{U_{n, i}\}_{i \in I_n}\) the restriction of \(E\) to \(D(\mathcal{O}_{\mathcal{C}/u(U_{n, i})})\) is \(E_{U_{n, i}}\) compatibly (see proof for details). Moreover, \(E\) is unique up to unique isomorphism.

Proof

Recall that \(\Sh(\mathcal{C}/K_n) = \prod_{i \in I_n} \Sh(\mathcal{C}/u(U_{n, i}))\) and similarly for the categories of modules. This product decomposition is also inherited by the derived categories of sheaves of modules. Moreover, this product decomposition is compatible with the morphisms in the simplicial semi-representable object \(K\). See Section 09WK. Hence we can set \(E_n = \prod_{i \in I_n} E_{U_{n, i}}\) (“formal” product) in \(D(\mathcal{O}_n)\). Taking (formal) products of the maps \(\rho_a\) of Situation 0DC9 we obtain isomorphisms \(E_\varphi : f_\varphi^*E_n \to E_m\). The assumption about compostions of the maps \(\rho_a\) immediately implies that \((E_n, E_\varphi)\) defines a simplicial system of the derived category of modules as in Definition 0D9N. The vanishing of negative exts assumed in the lemma implies that \(\Hom(E_n[t], E_n) = 0\) for \(n \geq 0\) and \(t > 0\). Thus by Lemma 0D9T we obtain \(E\). Uniqueness up to unique isomorphism follows from Lemmas 0D9R and 0D9S.

Lemma

In Situation 0DC9. Assume

1. \(\mathcal{C}\) has equalizers and fibre products,

2. there is a morphism of sites \(f : \mathcal{C} \to \mathcal{D}\) given by a continuous functor \(u : \mathcal{D} \to \mathcal{C}\) such that

   1. \(\mathcal{D}\) has equalizers and fibre products and \(u\) commutes with them,

   2. \(\mathcal{B}\) is a full subcategory of \(\mathcal{D}\) and \(u : \mathcal{B} \to \mathcal{C}\) is the restriction of \(u\),

   3. every object of \(\mathcal{D}\) has a covering whose members are objects of \(\mathcal{B}\),

3. for all \(U\) in \(\mathcal{B}\) all negative self-exts of \(E_U\) in \(D(\mathcal{O}_{u(U)})\) are zero, and

4. there exists a \(t \in \mathbf{Z}\) such that \(H^i(E_U) = 0\) for \(i < t\) and \(U \in \Ob(\mathcal{B})\).

Then there exists a solution unique up to unique isomorphism.

Proof

By Hypercoverings, Lemma 0DAX there exists a hypercovering \(L\) for the site \(\mathcal{D}\) such that \(L_n = \{U_{n, i}\}_{i \in I_n}\) with \(U_{i, n} \in \Ob(\mathcal{B})\). Set \(K = u(L)\). Apply Lemma 0DCA to get a cartesian object \(E\) of \(D(\mathcal{O})\) on the site \((\mathcal{C}/K)_{total}\) restricting to \(E_{U_{n, i}}\) on \(\mathcal{C}/u(U_{n, i})\) compatibly. The assumption on \(t\) implies that \(E \in D^+(\mathcal{O})\). By Hypercoverings, Lemma 0DAY we see that \(K\) is a hypercovering too. By Lemma 0DA2 we find that \(E = a^*F\) for some \(F\) in \(D^+(\mathcal{O}_\mathcal{C})\).

To prove that \(F\) is a solution we will use the construction of \(L_0\) and \(L_1\) given in the proof of Hypercoverings, Lemma 0DAX. (This is a bit inelegant but there does not seem to be a completely straightforward way around it.)

Namely, we have \(I_0 = \Ob(\mathcal{B})\) and so \(L_0 = \{U\}_{U \in \Ob(\mathcal{B})}\). Hence the isomorphism \(a^*F \to E\) restricted to the components \(\mathcal{C}/u(U)\) of \(\mathcal{C}/K_0\) defines isomorphisms \(\rho_U : F|_{\mathcal{C}/u(U)} \to E_U\) for \(U \in \Ob(\mathcal{B})\) by our choice of \(E\).

To prove that \(\rho_U\) satisfy the requirement of compatibility with the maps \(\rho_a\) of Situation 0DC9 we use that \(I_1\) contains the set \[\Omega = \{(U, V, W, a, b) \mid U, V, W \in \mathcal{B}, a : U \to V, b : U \to W\}\] and that for \(i = (U, V, W, a, b)\) in \(\Omega\) we have \(U_{1, i} = U\). Moreover, the component maps \(f_{\delta^1_0, i}\) and \(f_{\delta^1_1, i}\) of the two morphisms \(K_1 \to K_0\) are the morphisms \[a : U \to V \quad\text{and}\quad b : U \to V\] Hence the compatibility mentioned in Lemma 0DCA gives that \[\rho_a \circ \rho_V|_{\mathcal{C}/u(U)} = \rho_U \quad\text{and}\quad \rho_b \circ \rho_W|_{\mathcal{C}/u(U)} = \rho_U\] Taking \(i = (U, V, U, a, \text{id}_U) \in \Omega\) for example, we find that we have the desired compatibility. The uniqueness of \(F\) follows from the uniqueness of \(E\) in the previous lemma (small detail omitted).

Lemma

In Situation 0DC9. Assume

1. \(\mathcal{C}\) has equalizers and fibre products,

2. there is a morphism of sites \(f : \mathcal{C} \to \mathcal{D}\) given by a continuous functor \(u : \mathcal{D} \to \mathcal{C}\) such that

   1. \(\mathcal{D}\) has equalizers and fibre products and \(u\) commutes with them,

   2. \(\mathcal{B}\) is a full subcategory of \(\mathcal{D}\) and \(u : \mathcal{B} \to \mathcal{C}\) is the restriction of \(u\),

   3. every object of \(\mathcal{D}\) has a covering whose members are objects of \(\mathcal{B}\),

3. all negative self-exts of \(E_U\) in \(D(\mathcal{O}_{u(U)})\) are zero, and

4. there exist weak Serre subcategories \(\mathcal{A}_U \subset \textit{Mod}(\mathcal{O}_U)\) for all \(U \in \Ob(\mathcal{C})\) satisfying conditions (0DC2), (0DC3), and (0DC4),

5. \(E_U \in D_{\mathcal{A}_U}(\mathcal{O}_U)\).

Then there exists a solution unique up to unique isomorphism.

Proof

The proof is exactly the same as the proof of Lemma 0DCB. The only change is that \(E\) is an object of \(D_{\mathcal{A}_{total}}(\mathcal{O})\) and hence we use Lemma 0DC7 to obtain \(F\) with \(E = a^*F\) instead of Lemma 0DA2.

Here is an example application of the general theory above.

Lemma

Let \((\mathcal{C}, \mathcal{O}_\mathcal{C})\) be a ringed site. Assume \(\mathcal{C}\) has fibre products. Let \(\{U_i \to X\}_{i \in I}\) be a covering in \(\mathcal{C}\). For \(i \in I\) let \(E_i\) be an object of \(D(\mathcal{O}_{U_i})\) and for \(i, j \in I\) let \[\rho_{ij} : E_i|_{\mathcal{C}/U_{ij}} \longrightarrow E_j|_{\mathcal{C}/U_{ij}}\] be an isomorphism in \(D(\mathcal{O}_{U_{ij}})\) where \(U_{ij} = U_i \times_X U_j\). Assume

1. the \(\rho_{ij}\) satisfy the cocycle condition on \(U_i \times_X U_j \times_X U_k\) for all \(i, j, k \in I\),

2. \(\SheafExt^p_{\mathcal{O}_{U_i}}(E_i, E_i) = 0\) for all \(p < 0\) and \(i \in I\), and

3. there exists a \(t \in \mathbf{Z}\) such that \(H^p(E_i) = 0\) for \(p < t\) and all \(i \in I\).

Then there exists a unique pair \((E, \rho_i)\) where \(E\) is an object of \(D(\mathcal{O}_X)\) and \(\rho_i : E|_{U_i} \to E_i\) are isomorphisms in \(D(\mathcal{O}_{U_i})\) compatible with the \(\rho_{ij}\).

Proof

In this proof we deduce the lemma from the very general Lemma 0DCB. We urge the reader to look at the second proof in stead.

We may replace \(\mathcal{C}\) with \(\mathcal{C}/X\). Thus we may and do assume \(X\) is the final object of \(\mathcal{C}\) and that \(\mathcal{C}\) has all finite limits.

Let \(\mathcal{B}\) be the full subcategory of \(\mathcal{C}\) consisting of \(U \in \Ob(\mathcal{C})\) such that there exists an \(i(U) \in I\) and a morphism \(a_U : U \to U_{i(U)}\). We denote \(E_U = a_U^*E_{i(U)}\) in \(D(\mathcal{O}_U)\) the pullback (restriction) of \(E_i\) via \(a_U\). Given a morphism \(a : U \to U'\) of \(\mathcal{B}\) we obtain a morphism \((a_{U'} \circ a, a_U) : U \to U_{i(U')} \times_X U_{i(U)} = U_{i(U')i(U)}\) and hence an isomorphism \[\rho_a : a^*E_{U'} = a^*a_{U'}^*E_{i(U')} \xrightarrow{(a_{U'} \circ a, a_U)^*\rho_{i(U')i(U)}} a_{U}^*E_{i(U)} = E_{U}\] in \(D(\mathcal{O}_U)\). The data \(\mathcal{B}, E_U, \rho_a\) are as in Situation 0DC9; the isomorphisms \(\rho_a\) satisfy the cocycle condition exactly because of condition (1) in the statement of the lemma (details omitted).

We are going to apply Lemma 0DCB with \(\mathcal{B}\), \(E_U\), \(\rho_a\) as above and with \(\mathcal{D} = \mathcal{C}\) and \(f : \mathcal{C} \to \mathcal{D}\) the identity morphism. Assumptions (1) and (2)(a) of Lemma 0DCB we have seen above. Assumption (2)(b) of Lemma 0DCB is clear. Assumption (2)(c) of Lemma 0DCB holds because \(\{U_i \to X\}\) is a covering⁵. Assumption (3) of Lemma 0DCB holds because we have assumed the vanishing of all negative Ext sheaves of \(E_i\) which certainly implies that for any object \(U\) lying over \(U_i\) the negative self-Exts of \(E_i|_U\) are zero. Assumption (4) of Lemma 0DCB holds because we have assumed the cohomology sheaves of each \(E_i\) are zero to the left of \(t\).

We obtain a unique solution \((E, \rho_U)\). Setting \(\rho_i = \rho_{U_i}\) the lemma follows.

Proof

We sketch a more direct proof. Denote \(K\) the Čech hypercovering of \(X\) associated to the covering \(\{U_i \to X\}_{i \in I}\), see Hypercoverings, Example 01G6. Thus for example \(K_0 = \{U_i \to X\}_{i \in I}\) and \(K_1 = \{U_i \times_X U_j \to X\}_{i, j \in I}\) and so on. Let \(((\mathcal{C}/K)_{total}, \mathcal{O})\), \(a\), \(a_n\) be as in Remark 0D9X. The objects \(E_i\) determine an object \(M_0\) in \(D(\mathcal{O}_0) = \prod D(\mathcal{O}_{U_i})\). Similarly, the isomorphisms \(\rho_{ij}\) determine an isomorphism \[\alpha : L(f_{\delta_1^1})^*M_0 \longrightarrow L(f_{\delta_0^1})^*M_0\] in \(D(\mathcal{O}_1)\) satisfying the cocycle condition. By Lemma 0GMF we obtain a cartesian simplicial system \((M_n)\) of the derived category. By the assumed vanishing of the negative Ext sheaves we see that the objects \(M_n\) have vanishing negative self-exts. Thus we find a cartesian object \(M\) of \(D(\mathcal{O})\) whose associated simplicial system is isomorphic to \((M_n)\) by Lemma 0D9T. Since the cohomology sheaves of \(M\) are zero in degrees \(< t\) we see that by Lemma 0DA7 we have \(M = La^*E\) for some \(E\) in \(D(\mathcal{O}_X)\). The isomorphism \(La^*E \to M\) restricted to \(\mathcal{C}/U_i\) produces the isomorphisms \(\rho_i\). We omit the verification of the compatibility with the isomorphisms \(\rho_{ij}\).

Proper hypercoverings in topology

Let’s work in the category \(\textit{LC}\) of Hausdorff and locally quasi-compact topological spaces and continuous maps, see Cohomology on Sites, Section 09WY. Let \(X\) be an object of \(\textit{LC}\) and let \(U\) be a simplicial object of \(\textit{LC}\). Assume we have an augmentation \[a : U \to X\] We say that \(U\) is a proper hypercovering of \(X\) if

1. \(U_0 \to X\) is a proper surjective map,

2. \(U_1 \to U_0 \times_X U_0\) is a proper surjective map,

3. \(U_{n + 1} \to (\text{cosk}_n\text{sk}_n U)_{n + 1}\) is a proper surjective map for \(n \geq 1\).

The category \(\textit{LC}\) has all finite limits, hence the coskeleta used in the formulation above exist. \[\fbox{Principle: Proper hypercoverings can be used to compute cohomology.}\] A key idea behind the proof of the principle is to find a topology on \(\textit{LC}\) which is stronger than the usual one such that (a) a surjective proper map defines a covering, and (b) cohomology of usual sheaves with respect to this stronger topology agrees with the usual cohomology. Properties (a) and (b) hold for the qc topology, see Cohomology on Sites, Section 09WY. Once we have (a) and (b) we deduce the principle via the earlier work done in this chapter.

Lemma

Let \(U\) be a simplicial object of \(\textit{LC}\) and let \(a : U \to X\) be an augmentation. There is a commutative diagram \[\xymatrix{ \Sh((\textit{LC}_{qc}/U)_{total}) \ar[r]_-h \ar[d]_{a_{qc}} & \Sh(U_{Zar}) \ar[d]^a \\ \Sh(\textit{LC}_{qc}/X) \ar[r]^-{h_{-1}} & \Sh(X) }\] where the left vertical arrow is defined in Section 09X8 and the right vertical arrow is defined in Lemma 09W4.

Proof

Write \(\Sh(X) = \Sh(X_{Zar})\). Observe that both \((\textit{LC}_{qc}/U)_{total}\) and \(U_{Zar}\) fall into case A of Situation 09WE. This is immediate from the construction of \(U_{Zar}\) in Section 09VK and it follows from Lemma 09WL for \((\textit{LC}_{qc}/U)_{total}\). Next, consider the functors \(U_{n, Zar} \to \textit{LC}_{qc}/U_n\), \(U \mapsto U/U_n\) and \(X_{Zar} \to \textit{LC}_{qc}/X\), \(U \mapsto U/X\). We have seen that these define morphisms of sites in Cohomology on Sites, Section 09WY. Thus we obtain a morphism of simplicial sites compatible with augmentations as in Remark 0D98 and we may apply Lemma 0D99 to conclude.

Lemma

Let \(U\) be a simplicial object of \(\textit{LC}\) and let \(a : U \to X\) be an augmentation. If \(a : U \to X\) gives a proper hypercovering of \(X\), then \[a^{-1} : \Sh(X) \to \Sh(U_{Zar}) \quad\text{and}\quad a^{-1} : \textit{Ab}(X) \to \textit{Ab}(U_{Zar})\] are fully faithful with essential image the cartesian sheaves and quasi-inverse given by \(a_*\). Here \(a : \Sh(U_{Zar}) \to \Sh(X)\) is as in Lemma 09W4.

Proof

We will prove the statement for sheaves of sets. It will be an almost formal consequence of results already established. Consider the diagram of Lemma 0DAF. By Cohomology on Sites, Lemma 09X3 the functor \((h_{-1})^{-1}\) is fully faithful with quasi-inverse \(h_{-1, *}\). The same holds true for the components \(h_n\) of \(h\). By the description of the functors \(h^{-1}\) and \(h_*\) of Lemma 0D96 we conclude that \(h^{-1}\) is fully faithful with quasi-inverse \(h_*\). Observe that \(U\) is a hypercovering of \(X\) in \(\textit{LC}_{qc}\) (as defined in Section 09X8) by Cohomology on Sites, Lemma 09X5. By Lemma 0DA8 we see that \(a_{qc}^{-1}\) is fully faithful with quasi-inverse \(a_{qc, *}\) and with essential image the cartesian sheaves on \((\textit{LC}_{qc}/U)_{total}\). A formal argument (chasing around the diagram) now shows that \(a^{-1}\) is fully faithful.

Finally, suppose that \(\mathcal{G}\) is a cartesian sheaf on \(U_{Zar}\). Then \(h^{-1}\mathcal{G}\) is a cartesian sheaf on \(\textit{LC}_{qc}/U\). Hence \(h^{-1}\mathcal{G} = a_{qc}^{-1}\mathcal{H}\) for some sheaf \(\mathcal{H}\) on \(\textit{LC}_{qc}/X\). We compute \[\begin{align*} (h_{-1})^{-1}(a_*\mathcal{G}) & = (h_{-1})^{-1} \text{Eq}( \xymatrix{ a_{0, *}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & a_{1, *}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ (h_{-1})^{-1}a_{0, *}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & (h_{-1})^{-1}a_{1, *}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ a_{qc, 0, *}h_0^{-1}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & a_{qc, 1, *}h_1^{-1}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ a_{qc, 0, *}a_{qc, 0}^{-1}\mathcal{H} \ar@<1ex>[r] \ar@<-1ex>[r] & a_{qc, 1, *}a_{qc, 1}^{-1}\mathcal{H} } ) \\ & = a_{qc, *}a_{qc}^{-1}\mathcal{H} \\ & = \mathcal{H} \end{align*}\] Here the first equality follows from Lemma 09W4, the second equality follows as \((h_{-1})^{-1}\) is an exact functor, the third equality follows from Cohomology on Sites, Lemma 0D92 (here we use that \(a_0 : U_0 \to X\) and \(a_1: U_1 \to X\) are proper), the fourth follows from \(a_{qc}^{-1}\mathcal{H} = h^{-1}\mathcal{G}\), the fifth from Lemma 0D70, and the sixth we’ve seen above. Since \(a_{qc}^{-1}\mathcal{H} = h^{-1}\mathcal{G}\) we deduce that \(h^{-1}\mathcal{G} \cong h^{-1}a^{-1}a_*\mathcal{G}\) which ends the proof by fully faithfulness of \(h^{-1}\).

Lemma

Let \(U\) be a simplicial object of \(\textit{LC}\) and let \(a : U \to X\) be an augmentation. If \(a : U \to X\) gives a proper hypercovering of \(X\), then for \(K \in D^+(X)\) \[K \to Ra_*(a^{-1}K)\] is an isomorphism where \(a : \Sh(U_{Zar}) \to \Sh(X)\) is as in Lemma 09W4.

Proof

Consider the diagram of Lemma 0DAF. Observe that \(Rh_{n, *}h_n^{-1}\) is the identity functor on \(D^+(U_n)\) by Cohomology on Sites, Lemma 0D91. Hence \(Rh_*h^{-1}\) is the identity functor on \(D^+(U_{Zar})\) by Lemma 0D97. We have \[\begin{align*} Ra_*(a^{-1}K) & = Ra_*Rh_*h^{-1}a^{-1}K \\ & = Rh_{-1, *}Ra_{qc, *}a_{qc}^{-1}(h_{-1})^{-1}K \\ & = Rh_{-1, *}(h_{-1})^{-1}K \\ & = K \end{align*}\] The first equality by the discussion above, the second equality because of the commutativity of the diagram in Lemma 0DAF, the third equality by Lemma 0D8N (\(U\) is a hypercovering of \(X\) in \(\textit{LC}_{qc}\) by Cohomology on Sites, Lemma 09X5), and the last equality by the already used Cohomology on Sites, Lemma 0D91.

Lemma

Let \(U\) be a simplicial object of \(\textit{LC}\) and let \(a : U \to X\) be an augmentation. If \(U\) is a proper hypercovering of \(X\), then \[R\Gamma(X, K) = R\Gamma(U_{Zar}, a^{-1}K)\] for \(K \in D^+(X)\) where \(a : \Sh(U_{Zar}) \to \Sh(X)\) is as in Lemma 09W4.

Proof

This follows from Lemma 09XS because \(R\Gamma(U_{Zar}, -) = R\Gamma(X, -) \circ Ra_*\) by Cohomology on Sites, Remark 08J6.

Lemma

Let \(U\) be a simplicial object of \(\textit{LC}\) and let \(a : U \to X\) be an augmentation. Let \(\mathcal{A} \subset \textit{Ab}(U_{Zar})\) denote the weak Serre subcategory of cartesian abelian sheaves. If \(U\) is a proper hypercovering of \(X\), then the functor \(a^{-1}\) defines an equivalence \[D^+(X) \longrightarrow D_\mathcal{A}^+(U_{Zar})\] with quasi-inverse \(Ra_*\) where \(a : \Sh(U_{Zar}) \to \Sh(X)\) is as in Lemma 09W4.

Proof

Observe that \(\mathcal{A}\) is a weak Serre subcategory by Lemma 0D7J. The equivalence is a formal consequence of the results obtained so far. Use Lemmas 0DAG and 09XS and Cohomology on Sites, Lemma 0D7U.

Lemma

Let \(U\) be a simplicial object of \(\textit{LC}\) and let \(a : U \to X\) be an augmentation. Let \(\mathcal{F}\) be an abelian sheaf on \(X\). Let \(\mathcal{F}_n\) be the pullback to \(U_n\). If \(U\) is a proper hypercovering of \(X\), then there exists a canonical spectral sequence \[E_1^{p, q} = H^q(U_p, \mathcal{F}_p)\] converging to \(H^{p + q}(X, \mathcal{F})\).

Proof

Immediate consequence of Lemmas 09XC and 09W6.

Simplicial schemes

A simplicial scheme is a simplicial object in the category of schemes, see Simplicial, Definition 016A. Recall that a simplicial scheme looks like \[\xymatrix{ X_2 \ar@<2ex>[r] \ar@<0ex>[r] \ar@<-2ex>[r] & X_1 \ar@<1ex>[r] \ar@<-1ex>[r] \ar@<1ex>[l] \ar@<-1ex>[l] & X_0 \ar@<0ex>[l] }\] Here there are two morphisms \(d^1_0, d^1_1 : X_1 \to X_0\) and a single morphism \(s^0_0 : X_0 \to X_1\), etc. These morphisms satisfy some required relations such as \(d^1_0 \circ s^0_0 = \text{id}_{X_0} = d^1_1 \circ s^0_0\), see Simplicial, Lemma 016B. It is useful to think of \(d^n_i : X_n \to X_{n - 1}\) as the “projection forgetting the \(i\)th coordinate” and to think of \(s^n_j : X_n \to X_{n + 1}\) as the “diagonal map repeating the \(j\)th coordinate”.

A morphism of simplicial schemes \(h : X \to Y\) is the same thing as a morphism of simplicial objects in the category of schemes, see Simplicial, Definition 016A. Thus \(h\) consists of morphisms of schemes \(h_n : X_n \to Y_n\) such that \(h_{n - 1} \circ d^n_j = d^n_j \circ h_n\) and \(h_{n + 1} \circ s^n_j = s^n_j \circ h_n\) whenever this makes sense.

An augmentation of a simplicial scheme \(X\) is a morphism of schemes \(a_0 : X_0 \to S\) such that \(a_0 \circ d^1_0 = a_0 \circ d^1_1\). See Simplicial, Section 018F.

Let \(X\) be a simplicial scheme. The construction of Section 09VK applied to the underlying simplicial topological space gives a site \(X_{Zar}\). On the other hand, for every \(n\) we have the small Zariski site \(X_{n, Zar}\) (Topologies, Definition 020T) and for every morphism \(\varphi : [m] \to [n]\) we have a morphism of sites \(f_\varphi = X(\varphi)_{small} : X_{n, Zar} \to X_{m, Zar}\), associated to the morphism of schemes \(X(\varphi) : X_n \to X_m\) (Topologies, Lemma 0211). This gives a simplicial object \(\mathcal{C}\) in the category of sites. In Lemma 09WC we constructed an associated site \(\mathcal{C}_{total}\). Assigning to an open immersion its image defines an equivalence \(\mathcal{C}_{total} \to X_{Zar}\) which identifies sheaves, i.e., \(\Sh(\mathcal{C}_{total}) = \Sh(X_{Zar})\). The difference between \(\mathcal{C}_{total}\) and \(X_{Zar}\) is similar to the difference between the small Zariski site \(S_{Zar}\) and the underlying topological space of \(S\). We will silently identify these sites in what follows.

Let \(X_{Zar}\) be the site associated to a simplicial scheme \(X\). There is a sheaf of rings \(\mathcal{O}\) on \(X_{Zar}\) whose restriction to \(X_n\) is the structure sheaf \(\mathcal{O}_{X_n}\). This follows from Lemma 09VM or from Lemma 09WF. We will say \(\mathcal{O}\) is the structure sheaf of the simplicial scheme \(X\). At this point all the material developed for simplicial (ringed) sites applies, see Sections 09WB, 0D93, 0D94, 0D71, 0D76, 0D9A, 0D7B, 0D9D, 0D7G, 0D9F, and 0D9M.

Let \(X\) be a simplicial scheme with structure sheaf \(\mathcal{O}\). As on any ringed topos, there is a notion of a quasi-coherent \(\mathcal{O}\)-module on \(X_{Zar}\), see Modules on Sites, Definition 03DL. However, a quasi-coherent \(\mathcal{O}\)-module on \(X_{Zar}\) is just a cartesian \(\mathcal{O}\)-module \(\mathcal{F}\) whose restrictions \(\mathcal{F}_n\) are quasi-coherent on \(X_n\), see Lemma 0D7M.

Let \(h : X \to Y\) be a morphism of simplicial schemes. Either by Lemma 09VN or by (the proof of) Lemma 0D96 we obtain a morphism of sites \(h_{Zar} : X_{Zar} \to Y_{Zar}\). Recall that \(h_{Zar}^{-1}\) and \(h_{Zar, *}\) have a simple description in terms of the components, see Lemma 09VP or Lemma 0D96. Let \(\mathcal{O}_X\), resp. \(\mathcal{O}_Y\) denote the structure sheaf of \(X\), resp. \(Y\). We define \(h_{Zar}^\sharp : h_{Zar, *}\mathcal{O}_X \to \mathcal{O}_Y\) to be the map of sheaves of rings on \(Y_{Zar}\) given by \(h_n^\sharp : h_{n, *}\mathcal{O}_{X_n} \to \mathcal{O}_{Y_n}\) on \(Y_n\). We obtain a morphism of ringed sites \[h_{Zar} : (X_{Zar}, \mathcal{O}_X) \longrightarrow (Y_{Zar}, \mathcal{O}_Y)\]

Let \(X\) be a simplicial scheme with structure sheaf \(\mathcal{O}\). Let \(S\) be a scheme and let \(a_0 : X_0 \to S\) be an augmentation of \(X\). Either by Lemma 09W4 or by Lemma 0D70 we obtain a corresponding morphism of topoi \(a : \Sh(X_{Zar}) \to \Sh(S)\). Observe that \(a^{-1}\mathcal{G}\) is the sheaf on \(X_{Zar}\) with components \(a_n^{-1}\mathcal{G}\). Hence we can use the maps \(a_n^\sharp : a_n^{-1}\mathcal{O}_S \to \mathcal{O}_{X_n}\) to define a map \(a^\sharp : a^{-1}\mathcal{O}_S \to \mathcal{O}\), or equivalently by adjunction a map \(a^\sharp : \mathcal{O}_S \to a_*\mathcal{O}\) (which as usual has the same name). This puts us in the situation discussed in Section 0D9D. Therefore we obtain a morphism of ringed topoi \[a : (\Sh(X_{Zar}), \mathcal{O}) \longrightarrow (\Sh(S), \mathcal{O}_S)\]

A final observation is the following. Suppose we are given a morphism \(h : X \to Y\) of simplicial schemes \(X\) and \(Y\) with structure sheaves \(\mathcal{O}_X\), \(\mathcal{O}_Y\), augmentations \(a_0 : X_0 \to X_{-1}\), \(b_0 : Y_0 \to Y_{-1}\) and a morphism \(h_{-1} : X_{-1} \to Y_{-1}\) such that \[\xymatrix{ X_0 \ar[r]_{h_0} \ar[d]_{a_0} & Y_0 \ar[d]^{b_0} \\ X_{-1} \ar[r]^{h_{-1}} & Y_{-1} }\] commutes. Then from the constructions elucidated above we obtain a commutative diagram of morphisms of ringed topoi as follows \[\xymatrix{ (\Sh(X_{Zar}), \mathcal{O}_X) \ar[r]_{h_{Zar}} \ar[d]_a & (\Sh(Y_{Zar}), \mathcal{O}_Y) \ar[d]^b \\ (\Sh(X_{-1}), \mathcal{O}_{X_{-1}}) \ar[r]^{h_{-1}} & (\Sh(Y_{-1}), \mathcal{O}_{Y_{-1}}) }\]

Descent in terms of simplicial schemes

Cartesian morphisms are defined as follows.

Definition

Let \(a : Y \to X\) be a morphism of simplicial schemes. We say \(a\) is cartesian, or that \(Y\) is cartesian over \(X\), if for every morphism \(\varphi : [n] \to [m]\) of \(\Delta\) the corresponding diagram \[\xymatrix{ Y_m \ar[r]_a \ar[d]_{Y(\varphi)} & X_m \ar[d]^{X(\varphi)}\\ Y_n \ar[r]^{a} & X_n }\] is a fibre square in the category of schemes.

Cartesian morphisms are related to descent data. First we prove a general lemma describing the category of cartesian simplicial schemes over a fixed simplicial scheme. In this lemma we denote \(f^* : \Sch/X \to \Sch/Y\) the base change functor associated to a morphism of schemes \(f :Y \to X\).

Lemma

Let \(X\) be a simplicial scheme. The category of simplicial schemes cartesian over \(X\) is equivalent to the category of pairs \((V, \varphi)\) where \(V\) is a scheme over \(X_0\) and \[\varphi : V \times_{X_0, d^1_1} X_1 \longrightarrow X_1 \times_{d^1_0, X_0} V\] is an isomorphism over \(X_1\) such that \((s_0^0)^*\varphi = \text{id}_V\) and such that \[(d^2_1)^*\varphi = (d^2_0)^*\varphi \circ (d^2_2)^*\varphi\] as morphisms of schemes over \(X_2\).

Proof

The statement of the displayed equality makes sense because \(d^1_1 \circ d^2_2 = d^1_1 \circ d^2_1\), \(d^1_1 \circ d^2_0 = d^1_0 \circ d^2_2\), and \(d^1_0 \circ d^2_0 = d^1_0 \circ d^2_1\) as morphisms \(X_2 \to X_0\), see Simplicial, Remark 016C hence we can picture these maps as follows \[\xymatrix{ & X_2 \times_{d^1_1 \circ d^2_0, X_0} V \ar[r]_-{(d^2_0)^*\varphi} & X_2 \times_{d^1_0 \circ d^2_0, X_0} V \ar@{=}[rd] & \\ X_2 \times_{d^1_0 \circ d^2_2, X_0} V \ar@{=}[ru] & & & X_2 \times_{d^1_0 \circ d^2_1, X_0} V \\ & X_2 \times_{d^1_1 \circ d^2_2, X_0} V \ar[lu]^{(d^2_2)^*\varphi} \ar@{=}[r] & X_2 \times_{d^1_1 \circ d^2_1, X_0} V \ar[ru]_{(d^2_1)^*\varphi} }\] and the condition signifies the diagram is commutative. It is clear that given a simplicial scheme \(Y\) cartesian over \(X\) we can set \(V = Y_0\) and \(\varphi\) equal to the composition \[V \times_{X_0, d^1_1} X_1 = Y_0 \times_{X_0, d^1_1} X_1 = Y_1 = X_1 \times_{X_0, d^1_0} Y_0 = X_1 \times_{X_0, d^1_0} V\] of identifications given by the cartesian structure. To prove this functor is an equivalence we construct a quasi-inverse. The construction of the quasi-inverse is analogous to the construction discussed in Descent, Section 023F from which we borrow the notation \(\tau^n_i : [0] \to [n]\), \(0 \mapsto i\) and \(\tau^n_{ij} : [1] \to [n]\), \(0 \mapsto i\), \(1 \mapsto j\). Namely, given a pair \((V, \varphi)\) as in the lemma we set \(Y_n = X_n \times_{X(\tau^n_n), X_0} V\). Then given \(\beta : [n] \to [m]\) we define \(V(\beta) : Y_m \to Y_n\) as the pullback by \(X(\tau^m_{\beta(n)m})\) of the map \(\varphi\) postcomposed by the projection \(X_m \times_{X(\beta), X_n} Y_n \to Y_n\). This makes sense because \[X_m \times_{X(\tau^m_{\beta(n)m}), X_1} X_1 \times_{d^1_1, X_0} V = X_m \times_{X(\tau^m_m), X_0} V = Y_m\] and \[X_m \times_{X(\tau^m_{\beta(n)m}), X_1} X_1 \times_{d^1_0, X_0} V = X_m \times_{X(\tau^m_{\beta(n)}), X_0} V = X_m \times_{X(\beta), X_n} Y_n.\] We omit the verification that the commutativity of the displayed diagram above implies the maps compose correctly. We also omit the verification that the two functors are quasi-inverse to each other.

Definition

Let \(f : X \to S\) be a morphism of schemes. The simplicial scheme associated to \(f\), denoted \((X/S)_\bullet\), is the functor \(\Delta^{opp} \to \Sch\), \([n] \mapsto X \times_S \ldots \times_S X\) described in Simplicial, Example 016E.

Thus \((X/S)_n\) is the \((n + 1)\)-fold fibre product of \(X\) over \(S\). The morphism \(d^1_0 : X \times_S X \to X\) is the map \((x_0, x_1) \mapsto x_1\) and the morphism \(d^1_1\) is the other projection. The morphism \(s^0_0\) is the diagonal morphism \(X \to X \times_S X\).

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(\pi : Y \to (X/S)_\bullet\) be a cartesian morphism of simplicial schemes. Set \(V = Y_0\) considered as a scheme over \(X\). The morphisms \(d^1_0, d^1_1 : Y_1 \to Y_0\) and the morphism \(\pi_1 : Y_1 \to X \times_S X\) induce isomorphisms \[\xymatrix{ V \times_S X & & Y_1 \ar[ll]_-{(d^1_1, \text{pr}_1 \circ \pi_1)} \ar[rr]^-{(\text{pr}_0 \circ \pi_1, d^1_0)} & & X \times_S V. }\] Denote \(\varphi : V \times_S X \to X \times_S V\) the resulting isomorphism. Then the pair \((V, \varphi)\) is a descent datum relative to \(X \to S\).

Proof

This is a special case of (part of) Lemma 07TC as the displayed equation of that lemma is equivalent to the cocycle condition of Descent, Definition 023V.

Lemma

Let \(f : X \to S\) be a morphism of schemes. The construction \[\begin{matrix} \text{category of cartesian } \\ \text{schemes over } (X/S)_\bullet \end{matrix} \longrightarrow \begin{matrix} \text{ category of descent data} \\ \text{ relative to } X/S \end{matrix}\] of Lemma 024B is an equivalence of categories.

Proof

The functor from left to right is given in Lemma 024B. Hence this is a special case of Lemma 07TC.

We may reinterpret the pullback of Descent, Lemma 023Y as follows. Suppose given a morphism of simplicial schemes \(f : X' \to X\) and a cartesian morphism of simplicial schemes \(Y \to X\). Then the fibre product (viewed as a “pullback”) \[f^*Y = Y \times_X X'\] of simplicial schemes is a simplicial scheme cartesian over \(X'\). Suppose given a commutative diagram of morphisms of schemes \[\xymatrix{ X' \ar[r]_f \ar[d] & X \ar[d] \\ S' \ar[r] & S. }\] This gives rise to a morphism of simplicial schemes \[f_\bullet : (X'/S')_\bullet \longrightarrow (X/S)_\bullet.\] We claim that the “pullback” \(f_\bullet^*\) along the morphism \(f_\bullet : (X'/S')_\bullet \to (X/S)_\bullet\) corresponds via Lemma 024C with the pullback defined in terms of descent data in the aforementioned Descent, Lemma 023Y.

Quasi-coherent modules on simplicial schemes

Lemma

Let \(f : V \to U\) be a morphism of simplicial schemes. Given a quasi-coherent module \(\mathcal{F}\) on \(U_{Zar}\) the pullback \(f^*\mathcal{F}\) is a quasi-coherent module on \(V_{Zar}\).

Proof

Recall that \(\mathcal{F}\) is cartesian with \(\mathcal{F}_n\) quasi-coherent, see Lemma 0D7M. By Lemma 09VP we see that \((f^*\mathcal{F})_n = f_n^*\mathcal{F}_n\) (some details omitted). Hence \((f^*\mathcal{F})_n\) is quasi-coherent. The same fact and the cartesian property for \(\mathcal{F}\) imply the cartesian property for \(f^*\mathcal{F}\). Thus \(\mathcal{F}\) is quasi-coherent by Lemma 0D7M again.

Lemma

Let \(f : V \to U\) be a cartesian morphism of simplicial schemes. Assume the morphisms \(d^n_j : U_n \to U_{n - 1}\) are flat and the morphisms \(V_n \to U_n\) are quasi-compact and quasi-separated. For a quasi-coherent module \(\mathcal{G}\) on \(V_{Zar}\) the pushforward \(f_*\mathcal{G}\) is a quasi-coherent module on \(U_{Zar}\).

Proof

If \(\mathcal{F} = f_* \mathcal{G}\), then \(\mathcal{F}_n = f_{n , *}\mathcal{G}_n\) by Lemma 09VP. The maps \(\mathcal{F}(\varphi)\) are defined using the base change maps, see Cohomology, Section 02N6. The sheaves \(\mathcal{F}_n\) are quasi-coherent by Schemes, Lemma 01LC and the fact that \(\mathcal{G}_n\) is quasi-coherent by Lemma 0D7M. The base change maps along the degeneracies \(d^n_j\) are isomorphisms by Cohomology of Schemes, Lemma 02KH and the fact that \(\mathcal{G}\) is cartesian by Lemma 0D7M. Hence \(\mathcal{F}\) is cartesian by Lemma 07TG. Thus \(\mathcal{F}\) is quasi-coherent by Lemma 0D7M.

Lemma

Let \(f : V \to U\) be a cartesian morphism of simplicial schemes. Assume the morphisms \(d^n_j : U_n \to U_{n - 1}\) are flat and the morphisms \(V_n \to U_n\) are quasi-compact and quasi-separated. Then \(f^*\) and \(f_*\) form an adjoint pair of functors between the categories of quasi-coherent modules on \(U_{Zar}\) and \(V_{Zar}\).

Proof

We have seen in Lemmas 07TI and 07TJ that the statement makes sense. The adjointness property follows immediately from the fact that each \(f_n^*\) is adjoint to \(f_{n, *}\).

Lemma

Let \(f : X \to S\) be a morphism of schemes which has a section⁶. Let \((X/S)_\bullet\) be the simplicial scheme associated to \(X \to S\), see Definition 024A. Then pullback defines an equivalence between the category of quasi-coherent \(\mathcal{O}_S\)-modules and the category of quasi-coherent modules on \(((X/S)_\bullet)_{Zar}\).

Proof

Let \(\sigma : S \to X\) be a section of \(f\). Let \((\mathcal{F}, \alpha)\) be a pair as in Lemma 07TH. Set \(\mathcal{G} = \sigma^*\mathcal{F}\). Consider the diagram \[\xymatrix{ X \ar[r]_-{(\sigma \circ f, 1)} \ar[d]_f & X \times_S X \ar[d]^{\text{pr}_0} \ar[r]_-{\text{pr}_1} & X \\ S \ar[r]^\sigma & X }\] Note that \(\text{pr}_0 = d^1_1\) and \(\text{pr}_1 = d^1_0\). Hence we see that \((\sigma \circ f, 1)^*\alpha\) defines an isomorphism \[f^*\mathcal{G} = (\sigma \circ f, 1)^*\text{pr}_0^*\mathcal{F} \longrightarrow (\sigma \circ f, 1)^*\text{pr}_1^*\mathcal{F} = \mathcal{F}\] We omit the verification that this isomorphism is compatible with \(\alpha\) and the canonical isomorphism \(\text{pr}_0^*f^*\mathcal{G} \to \text{pr}_1^*f^*\mathcal{G}\).

Groupoids and simplicial schemes

Given a groupoid in schemes we can build a simplicial scheme. It will turn out that the category of quasi-coherent sheaves on a groupoid is equivalent to the category of cartesian quasi-coherent sheaves on the associated simplicial scheme.

Lemma

Let \((U, R, s, t, c, e, i)\) be a groupoid scheme over \(S\). There exists a simplicial scheme \(X\) over \(S\) with the following properties

1. \(X_0 = U\), \(X_1 = R\), \(X_2 = R \times_{s, U, t} R\),

2. \(s_0^0 = e : X_0 \to X_1\),

3. \(d^1_0 = s : X_1 \to X_0\), \(d^1_1 = t : X_1 \to X_0\),

4. \(s_0^1 = (e \circ t, 1) : X_1 \to X_2\), \(s_1^1 = (1, e \circ t) : X_1 \to X_2\),

5. \(d^2_0 = \text{pr}_1 : X_2 \to X_1\), \(d^2_1 = c : X_2 \to X_1\), \(d^2_2 = \text{pr}_0\), and

6. \(X = \text{cosk}_2 \text{sk}_2 X\).

For all \(n\) we have \(X_n = R \times_{s, U, t} \ldots \times_{s, U, t} R\) with \(n\) factors. The map \(d^n_j : X_n \to X_{n - 1}\) is given on functors of points by \[(r_1, \ldots, r_n) \longmapsto (r_1, \ldots, c(r_j, r_{j + 1}), \ldots, r_n)\] for \(1 \leq j \leq n - 1\) whereas \(d^n_0(r_1, \ldots, r_n) = (r_2, \ldots, r_n)\) and \(d^n_n(r_1, \ldots, r_n) = (r_1, \ldots, r_{n - 1})\).

Proof

We only have to verify that the rules prescribed in (1), (2), (3), (4), (5) define a \(2\)-truncated simplicial scheme \(U'\) over \(S\), since then (6) allows us to set \(X = \text{cosk}_2 U'\), see Simplicial, Lemma 0183. Using the functor of points approach, all we have to verify is that if \((\text{Ob}, \text{Arrows}, s, t, c, e, i)\) is a groupoid, then \[\xymatrix{ \text{Arrows} \times_{s, \text{Ob}, t} \text{Arrows} \ar@<8ex>[d]^{\text{pr}_0} \ar@<0ex>[d]_c \ar@<-8ex>[d]_{\text{pr}_1} \\ \text{Arrows} \ar@<4ex>[d]^t \ar@<-4ex>[d]_s \ar@<4ex>[u]^{1, e} \ar@<-4ex>[u]_{e, 1} \\ \text{Ob} \ar@<0ex>[u]_e }\] is a \(2\)-truncated simplicial set. We omit the details.

Finally, the description of \(X_n\) for \(n > 2\) follows by induction from the description of \(X_0\), \(X_1\), \(X_2\), and Simplicial, Remark 018A and Lemma 0187. Alternately, one shows that \(\text{cosk}_2\) applied to the \(2\)-truncated simplicial set displayed above gives a simplicial set whose \(n\)th term equals \(\text{Arrows} \times_{s, \text{Ob}, t} \ldots \times_{s, \text{Ob}, t} \text{Arrows}\) with \(n\) factors and degeneracy maps as given in the lemma. Some details omitted.

Lemma

Let \(S\) be a scheme. Let \((U, R, s, t, c)\) be a groupoid scheme over \(S\). Let \(X\) be the simplicial scheme over \(S\) constructed in Lemma 07TN. Then the category of quasi-coherent modules on \((U, R, s, t, c)\) is equivalent to the category of quasi-coherent modules on \(X_{Zar}\).

Proof

This is clear from Lemmas 0D7M and 07TH and Groupoids, Definition 03LI.

In the following lemma we will use the concept of a cartesian morphism \(V \to U\) of simplicial schemes as defined in Definition 0249.

Lemma

Let \((U, R, s, t, c)\) be a groupoid scheme over a scheme \(S\). Let \(X\) be the simplicial scheme over \(S\) constructed in Lemma 07TN. Let \((R/U)_\bullet\) be the simplicial scheme associated to \(s : R \to U\), see Definition 024A. There exists a cartesian morphism \(t_\bullet : (R/U)_\bullet \to X\) of simplicial schemes with low degree morphisms given by \[\xymatrix{ R \times_{s, U, s} R \times_{s, U, s} R \ar@<3ex>[r]_-{\text{pr}_{12}} \ar@<0ex>[r]_-{\text{pr}_{02}} \ar@<-3ex>[r]_-{\text{pr}_{01}} \ar[dd]_{(r_0, r_1, r_2) \mapsto (r_0 \circ r_1^{-1}, r_1 \circ r_2^{-1})} & R \times_{s, U, s} R \ar@<1ex>[r]_-{\text{pr}_1} \ar@<-2ex>[r]_-{\text{pr}_0} \ar[dd]_{(r_0, r_1) \mapsto r_0 \circ r_1^{-1}} & R \ar[dd]^t \\ \\ R \times_{s, U, t} R \ar@<3ex>[r]_{\text{pr}_1} \ar@<0ex>[r]_c \ar@<-3ex>[r]_{\text{pr}_0} & R \ar@<1ex>[r]_s \ar@<-2ex>[r]_t & U }\]

Proof

For arbitrary \(n\) we define \((R/U)_\bullet \to X_n\) by the rule \[(r_0, \ldots, r_n) \longrightarrow (r_0 \circ r_1^{-1}, \ldots, r_{n - 1} \circ r_n^{-1})\] Compatibility with degeneracy maps is clear from the description of the degeneracies in Lemma 07TN. We omit the verification that the maps respect the morphisms \(s^n_j\). Groupoids, Lemma 03C6 (with the roles of \(s\) and \(t\) reversed) shows that the two right squares are cartesian. In exactly the same manner one shows all the other squares are cartesian too. Hence the morphism is cartesian.

Descent data give equivalence relations

In Section 0248 we saw how descent data relative to \(X \to S\) can be formulated in terms of cartesian simplicial schemes over \((X/S)_\bullet\). Here we link this to equivalence relations as follows.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(\pi : Y \to (X/S)_\bullet\) be a cartesian morphism of simplicial schemes, see Definitions 0249 and 024A. Then the morphism \[j = (d^1_1, d^1_0) : Y_1 \to Y_0 \times_S Y_0\] defines an equivalence relation on \(Y_0\) over \(S\), see Groupoids, Definition 022P.

Proof

Note that \(j\) is a monomorphism. Namely the composition \(Y_1 \to Y_0 \times_S Y_0 \to Y_0 \times_S X\) is an isomorphism as \(\pi\) is cartesian.

Consider the morphism \[(d^2_2, d^2_0) : Y_2 \to Y_1 \times_{d^1_0, Y_0, d^1_1} Y_1.\] This works because \(d_0 \circ d_2 = d_1 \circ d_0\), see Simplicial, Remark 016C. Also, it is a morphism over \((X/S)_2\). It is an isomorphism because \(Y \to (X/S)_\bullet\) is cartesian. Note for example that the right hand side is isomorphic to \(Y_0 \times_{\pi_0, X, \text{pr}_1} (X \times_S X \times_S X) = X \times_S Y_0 \times_S X\) because \(\pi\) is cartesian. Details omitted.

As in Groupoids, Definition 022P we denote \(t = \text{pr}_0 \circ j = d^1_1\) and \(s = \text{pr}_1 \circ j = d^1_0\). The isomorphism above, combined with the morphism \(d^2_1 : Y_2 \to Y_1\) give us a composition morphism \[c : Y_1 \times_{s, Y_0, t} Y_1 \longrightarrow Y_1\] over \(Y_0 \times_S Y_0\). This immediately implies that for any scheme \(T/S\) the relation \(Y_1(T) \subset Y_0(T) \times Y_0(T)\) is transitive.

Reflexivity follows from the fact that the restriction of the morphism \(j\) to the diagonal \(\Delta : X \to X \times_S X\) is an isomorphism (again use the cartesian property of \(\pi\)).

To see symmetry we consider the morphism \[(d^2_2, d^2_1) : Y_2 \to Y_1 \times_{d^1_1, Y_0, d^1_1} Y_1.\] This works because \(d_1 \circ d_2 = d_1 \circ d_1\), see Simplicial, Remark 016C. It is an isomorphism because \(Y \to (X/S)_\bullet\) is cartesian. Note for example that the right hand side is isomorphic to \(Y_0 \times_{\pi_0, X, \text{pr}_0} (X \times_S X \times_S X) = Y_0 \times_S X \times_S X\) because \(\pi\) is cartesian. Details omitted.

Let \(T/S\) be a scheme. Let \(a \sim b\) for \(a, b \in Y_0(T)\) be synonymous with \((a, b) \in Y_1(T)\). The isomorphism \((d^2_2, d^2_1)\) above implies that if \(a \sim b\) and \(a \sim c\), then \(b \sim c\). Combined with reflexivity this shows that \(\sim\) is an equivalence relation.

An example case

In this section we show that disjoint unions of spectra of Artinian rings can be descended along a quasi-compact surjective flat morphism of schemes.

Lemma

Let \(X \to S\) be a morphism of schemes. Suppose \(Y \to (X/S)_\bullet\) is a cartesian morphism of simplicial schemes. For \(y \in Y_0\) a point define \[T_y = \{y' \in Y_0 \mid \exists\ y_1 \in Y_1: d^1_1(y_1) = y, d^1_0(y_1) = y'\}\] as a subset of \(Y_0\). Then \(y \in T_y\) and \(T_y \cap T_{y'} \not = \emptyset \Rightarrow T_y = T_{y'}\).

Proof

Combine Lemma 024E and Groupoids, Lemma 022Q.

Lemma

Let \(X \to S\) be a morphism of schemes. Suppose \(Y \to (X/S)_\bullet\) is a cartesian morphism of simplicial schemes. Let \(y \in Y_0\) be a point. If \(X \to S\) is quasi-compact, then \[T_y = \{y' \in Y_0 \mid \exists\ y_1 \in Y_1: d^1_1(y_1) = y, d^1_0(y_1) = y'\}\] is a quasi-compact subset of \(Y_0\).

Proof

Let \(F_y\) be the scheme theoretic fibre of \(d^1_1 : Y_1 \to Y_0\) at \(y\). Then we see that \(T_y\) is the image of the morphism \[\xymatrix{ F_y \ar[r] \ar[d] & Y_1 \ar[r]^{d^1_0} \ar[d]^{d^1_1} & Y_0 \\ y \ar[r] & Y_0 & }\] Note that \(F_y\) is quasi-compact. This proves the lemma.

Lemma

Let \(X \to S\) be a quasi-compact flat surjective morphism. Let \((V, \varphi)\) be a descent datum relative to \(X \to S\). If \(V\) is a disjoint union of spectra of Artinian rings, then \((V, \varphi)\) is effective.

Proof

Let \(Y \to (X/S)_\bullet\) be the cartesian morphism of simplicial schemes corresponding to \((V, \varphi)\) by Lemma 024C. Observe that \(Y_0 = V\). Write \(V = \coprod_{i \in I} \Spec(A_i)\) with each \(A_i\) local Artinian. Moreover, let \(v_i \in V\) be the unique closed point of \(\Spec(A_i)\) for all \(i \in I\). Write \(i \sim j\) if and only if \(v_i \in T_{v_j}\) with notation as in Lemma 024G above. By Lemmas 024G and 024H this is an equivalence relation with finite equivalence classes. Let \(\overline{I} = I/\sim\). Then we can write \(V = \coprod_{\overline{i} \in \overline{I}} V_{\overline{i}}\) with \(V_{\overline{i}} = \coprod_{i \in \overline{i}} \Spec(A_i)\). By construction we see that \(\varphi : V \times_S X \to X \times_S V\) maps the open and closed subspaces \(V_{\overline{i}} \times_S X\) into the open and closed subspaces \(X \times_S V_{\overline{i}}\). In other words, we get descent data \((V_{\overline{i}}, \varphi_{\overline{i}})\), and \((V, \varphi)\) is the coproduct of them in the category of descent data. Since each of the \(V_{\overline{i}}\) is a finite union of spectra of Artinian local rings the morphism \(V_{\overline{i}} \to X\) is affine, see Morphisms, Lemma 01SI. Since \(\{X \to S\}\) is an fpqc covering we see that all the descent data \((V_{\overline{i}}, \varphi_{\overline{i}})\) are effective by Descent, Lemma 0245.

To be sure, the lemma above has very limited applicability!

Simplicial algebraic spaces

Let \(S\) be a scheme. A simplicial algebraic space is a simplicial object in the category of algebraic spaces over \(S\), see Simplicial, Definition 016A. Recall that a simplicial algebraic space looks like \[\xymatrix{ X_2 \ar@<2ex>[r] \ar@<0ex>[r] \ar@<-2ex>[r] & X_1 \ar@<1ex>[r] \ar@<-1ex>[r] \ar@<1ex>[l] \ar@<-1ex>[l] & X_0 \ar@<0ex>[l] }\] Here there are two morphisms \(d^1_0, d^1_1 : X_1 \to X_0\) and a single morphism \(s^0_0 : X_0 \to X_1\), etc. These morphisms satisfy some required relations such as \(d^1_0 \circ s^0_0 = \text{id}_{X_0} = d^1_1 \circ s^0_0\), see Simplicial, Lemma 016B. It is useful to think of \(d^n_i : X_n \to X_{n - 1}\) as the “projection forgetting the \(i\)th coordinate” and to think of \(s^n_j : X_n \to X_{n + 1}\) as the “diagonal map repeating the \(j\)th coordinate”.

A morphism of simplicial algebraic spaces \(h : X \to Y\) is the same thing as a morphism of simplicial objects in the category of algebraic spaces over \(S\), see Simplicial, Definition 016A. Thus \(h\) consists of morphisms of algebraic spaces \(h_n : X_n \to Y_n\) such that \(h_{n - 1} \circ d^n_j = d^n_j \circ h_n\) and \(h_{n + 1} \circ s^n_j = s^n_j \circ h_n\) whenever this makes sense.

An augmentation \(a : X \to X_{-1}\) of a simplicial algebraic space \(X\) is given by a morphism of algebraic spaces \(a_0 : X_0 \to X_{-1}\) such that \(a_0 \circ d^1_0 = a_0 \circ d^1_1\). See Simplicial, Section 018F. In this situation we always indicate \(a_n : X_n \to X_{-1}\) the induced morphisms for \(n \geq 0\).

Let \(X\) be a simplicial algebraic space. For every \(n\) we have the site \(X_{n, spaces, \etale}\) (Properties of Spaces, Definition 03G0) and for every morphism \(\varphi : [m] \to [n]\) we have a morphism of sites \[f_\varphi = X(\varphi)_{spaces, \etale} : X_{n, spaces, \etale} \to X_{m, spaces, \etale},\] associated to the morphism of algebraic spaces \(X(\varphi) : X_n \to X_m\) (Properties of Spaces, Lemma 03G2). This gives a simplicial object in the category of sites. In Lemma 09WC we constructed an associated site which we denote \(X_{spaces, \etale}\). An object of the site \(X_{spaces, \etale}\) is a an algebraic space \(U\) étale over \(X_n\) for some \(n\) and a morphism \((\varphi, f) : U/X_n \to V/X_m\) is given by a morphism \(\varphi : [m] \to [n]\) in \(\Delta\) and a morphism \(f : U \to V\) of algebraic spaces such that the diagram \[\xymatrix{ U \ar[r]_f \ar[d] & V \ar[d] \\ X_n \ar[r]^{f_\varphi} & X_m }\] is commutative. Consider the full subcategories \[X_{affine, \etale} \subset X_\etale \subset X_{spaces, \etale}\] whose objects are \(U/X_n\) with \(U\) affine, respectively a scheme. Endowing these categories with their natural topologies (see Properties of Spaces, Lemma 04JS, Definition 03ED, and Lemma 03G1) these inclusion functors define equivalences of topoi \[\Sh(X_{affine, \etale}) = \Sh(X_\etale) = \Sh(X_{spaces, \etale})\] In the following we will silently identify these topoi. We will say that \(X_\etale\) is the small étale site of \(X\) and its topos is the small étale topos of \(X\).

Let \(X_\etale\) be the small étale site of a simplicial algebraic space \(X\). There is a sheaf of rings \(\mathcal{O}\) on \(X_\etale\) whose restriction to \(X_n\) is the structure sheaf \(\mathcal{O}_{X_n}\). This follows from Lemma 09WF. We will say \(\mathcal{O}\) is the structure sheaf of the simplicial algebraic space \(X\). At this point all the material developed for simplicial (ringed) sites applies, see Sections 09WB, 0D93, 0D94, 0D71, 0D76, 0D9A, 0D7B, 0D9D, 0D7G, 0D9F, and 0D9M.

Let \(X\) be a simplicial algebraic space with structure sheaf \(\mathcal{O}\). As on any ringed topos, there is a notion of a quasi-coherent \(\mathcal{O}\)-module on \(X_\etale\), see Modules on Sites, Definition 03DL. However, a quasi-coherent \(\mathcal{O}\)-module on \(X_\etale\) is just a cartesian \(\mathcal{O}\)-module \(\mathcal{F}\) whose restrictions \(\mathcal{F}_n\) are quasi-coherent on \(X_n\), see Lemma 0D7M.

Let \(h : X \to Y\) be a morphism of simplicial algebraic spaces over \(S\). By Lemma 0D96 applied to the morphisms of sites \((h_n)_{spaces, \etale} : X_{spaces, \etale} \to Y_{spaces, \etale}\) (Properties of Spaces, Lemma 03G2) we obtain a morphism of small étale topoi \(h_\etale : \Sh(X_\etale) \to \Sh(Y_\etale)\). Recall that \(h_\etale^{-1}\) and \(h_{\etale, *}\) have a simple description in terms of the components, see Lemma 0D96. Let \(\mathcal{O}_X\), resp. \(\mathcal{O}_Y\) denote the structure sheaf of \(X\), resp. \(Y\). We define \(h_\etale^\sharp : h_{\etale, *}\mathcal{O}_X \to \mathcal{O}_Y\) to be the map of sheaves of rings on \(Y_\etale\) given by \(h_n^\sharp : h_{n, *}\mathcal{O}_{X_n} \to \mathcal{O}_{Y_n}\) on \(Y_n\). We obtain a morphism of ringed topoi \[h_\etale : (\Sh(X_\etale), \mathcal{O}_X) \longrightarrow (\Sh(Y_\etale), \mathcal{O}_Y)\]

Let \(X\) be a simplicial algebraic space with structure sheaf \(\mathcal{O}\). Let \(X_{-1}\) be an algebraic space over \(S\) and let \(a_0 : X_0 \to X_{-1}\) be an augmentation of \(X\). By Lemma 0D70 applied to the morphism of sites \((a_0)_{spaces, \etale} : X_{0, spaces, \etale} \to X_{-1, spaces, \etale}\) we obtain a corresponding morphism of topoi \(a : \Sh(X_\etale) \to \Sh(X_{-1, \etale})\). Observe that \(a^{-1}\mathcal{G}\) is the sheaf on \(X_\etale\) with components \(a_n^{-1}\mathcal{G}\). Hence we can use the maps \(a_n^\sharp : a_n^{-1}\mathcal{O}_{X_{-1}} \to \mathcal{O}_{X_n}\) to define a map \(a^\sharp : a^{-1}\mathcal{O}_{X_{-1}} \to \mathcal{O}\), or equivalently by adjunction a map \(a^\sharp : \mathcal{O}_{X_{-1}} \to a_*\mathcal{O}\) (which as usual has the same name). This puts us in the situation discussed in Section 0D9D. Therefore we obtain a morphism of ringed topoi \[a : (\Sh(X_\etale), \mathcal{O}) \longrightarrow (\Sh(X_{-1}), \mathcal{O}_{X_{-1}})\]

A final observation is the following. Suppose we are given a morphism \(h : X \to Y\) of simplicial algebraic spaces \(X\) and \(Y\) with structure sheaves \(\mathcal{O}_X\), \(\mathcal{O}_Y\), augmentations \(a_0 : X_0 \to X_{-1}\), \(b_0 : Y_0 \to Y_{-1}\) and a morphism \(h_{-1} : X_{-1} \to Y_{-1}\) such that \[\xymatrix{ X_0 \ar[r]_{h_0} \ar[d]_{a_0} & Y_0 \ar[d]^{b_0} \\ X_{-1} \ar[r]^{h_{-1}} & Y_{-1} }\] commutes. Then from the constructions elucidated above we obtain a commutative diagram of morphisms of ringed topoi as follows \[\xymatrix{ (\Sh(X_\etale), \mathcal{O}_X) \ar[r]_{h_\etale} \ar[d]_a & (\Sh(Y_\etale), \mathcal{O}_Y) \ar[d]^b \\ (\Sh(X_{-1}), \mathcal{O}_{X_{-1}}) \ar[r]^{h_{-1}} & (\Sh(Y_{-1}), \mathcal{O}_{Y_{-1}}) }\]

Fppf hypercoverings of algebraic spaces

This section is the analogue of Section 09XA for the case of algebraic spaces and fppf hypercoverings. The reader who wishes to do so, can replace “algebraic space” everywhere with “scheme” and get equally valid results. This has the advantage of replacing the references to More on Cohomology of Spaces, Section 0DGE with references to Étale Cohomology, Section 0DDK.

We fix a base scheme \(S\). Let \(X\) be an algebraic space over \(S\) and let \(U\) be a simplicial algebraic space over \(S\). Assume we have an augmentation \[a : U \to X\] See Section 0DE7. We say that \(U\) is an fppf hypercovering of \(X\) if

1. \(U_0 \to X\) is flat, locally of finite presentation, and surjective,

2. \(U_1 \to U_0 \times_X U_0\) is flat, locally of finite presentation, and surjective,

3. \(U_{n + 1} \to (\text{cosk}_n\text{sk}_n U)_{n + 1}\) is flat, locally of finite presentation, and surjective for \(n \geq 1\).

The category of algebraic spaces over \(S\) has all finite limits, hence the coskeleta used in the formulation above exist. \[\fbox{Principle: Fppf hypercoverings can be used to compute \'etale cohomology.}\] The key idea behind the proof of the principle is to compare the fppf and étale topologies on the category \(\textit{Spaces}/S\). Namely, the fppf topology is stronger than the étale topology and we have (a) a flat, locally finitely presented, surjective map defines an fppf covering, and (b) fppf cohomology of sheaves pulled back from the small étale site agrees with étale cohomology as we have seen in More on Cohomology of Spaces, Section 0DGE.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(U\) be a simplicial algebraic space over \(S\). Let \(a : U \to X\) be an augmentation. There is a commutative diagram \[\xymatrix{ \Sh((\textit{Spaces}/U)_{fppf, total}) \ar[r]_-h \ar[d]_{a_{fppf}} & \Sh(U_\etale) \ar[d]^a \\ \Sh((\textit{Spaces}/X)_{fppf}) \ar[r]^-{h_{-1}} & \Sh(X_\etale) }\] where the left vertical arrow is defined in Section 09X8 and the right vertical arrow is defined in Section 0DE7.

Proof

The notation \((\textit{Spaces}/U)_{fppf, total}\) indicates that we are using the construction of Section 09X8 for the site \((\textit{Spaces}/S)_{fppf}\) and the simplicial object \(U\) of this site⁷. We will use the sites \(X_{spaces, \etale}\) and \(U_{spaces, \etale}\) for the topoi on the right hand side; this is permissible see discussion in Section 0DE7.

Observe that both \((\textit{Spaces}/U)_{fppf, total}\) and \(U_{spaces, \etale}\) fall into case A of Situation 09WE. This is immediate from the construction of \(U_\etale\) in Section 0DE7 and it follows from Lemma 09WL for \((\textit{Spaces}/U)_{fppf, total}\). Next, consider the functors \(U_{n, spaces, \etale} \to (\textit{Spaces}/U_n)_{fppf}\), \(U \mapsto U/U_n\) and \(X_{spaces, \etale} \to (\textit{Spaces}/X)_{fppf}\), \(U \mapsto U/X\). We have seen that these define morphisms of sites in More on Cohomology of Spaces, Section 0DGE where these were denoted \(a_{U_n} = \epsilon_{U_n} \circ \pi_{u_n}\) and \(a_X = \epsilon_X \circ \pi_X\). Thus we obtain a morphism of simplicial sites compatible with augmentations as in Remark 0D98 and we may apply Lemma 0D99 to conclude.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(U\) be a simplicial algebraic space over \(S\). Let \(a : U \to X\) be an augmentation. If \(a : U \to X\) is an fppf hypercovering of \(X\), then \[a^{-1} : \Sh(X_\etale) \to \Sh(U_\etale) \quad\text{and}\quad a^{-1} : \textit{Ab}(X_\etale) \to \textit{Ab}(U_\etale)\] are fully faithful with essential image the cartesian sheaves and quasi-inverse given by \(a_*\). Here \(a : \Sh(U_\etale) \to \Sh(X_\etale)\) is as in Section 0DE7.

Proof

We will prove the statement for sheaves of sets. It will be an almost formal consequence of results already established. Consider the diagram of Lemma 0DH5. In the proof of this lemma we have seen that \(h_{-1}\) is the morphism \(a_X\) of More on Cohomology of Spaces, Section 0DGE. Thus it follows from More on Cohomology of Spaces, Lemma 0DGF that \((h_{-1})^{-1}\) is fully faithful with quasi-inverse \(h_{-1, *}\). The same holds true for the components \(h_n\) of \(h\). By the description of the functors \(h^{-1}\) and \(h_*\) of Lemma 0D96 we conclude that \(h^{-1}\) is fully faithful with quasi-inverse \(h_*\). Observe that \(U\) is a hypercovering of \(X\) in \((\textit{Spaces}/S)_{fppf}\) as defined in Section 09X8. By Lemma 0DA8 we see that \(a_{fppf}^{-1}\) is fully faithful with quasi-inverse \(a_{fppf, *}\) and with essential image the cartesian sheaves on \((\textit{Spaces}/U)_{fppf, total}\). A formal argument (chasing around the diagram) now shows that \(a^{-1}\) is fully faithful.

Finally, suppose that \(\mathcal{G}\) is a cartesian sheaf on \(U_\etale\). Then \(h^{-1}\mathcal{G}\) is a cartesian sheaf on \((\textit{Spaces}/U)_{fppf, total}\). Hence \(h^{-1}\mathcal{G} = a_{fppf}^{-1}\mathcal{H}\) for some sheaf \(\mathcal{H}\) on \((\textit{Spaces}/X)_{fppf}\). In particular we find that \(h_0^{-1}\mathcal{G}_0 = (a_{0, big, fppf})^{-1}\mathcal{H}\). Recalling that \(h_0 = a_{U_0}\) and that \(U_0 \to X\) is flat, locally of finite presentation, and surjective, we find from More on Cohomology of Spaces, Lemma 0DGL that there exists a sheaf \(\mathcal{F}\) on \(X_\etale\) and isomorphism \(\mathcal{H} = (h_{-1})^{-1}\mathcal{F}\). Since \(a_{fppf}^{-1}\mathcal{H} = h^{-1}\mathcal{G}\) we deduce that \(h^{-1}\mathcal{G} \cong h^{-1}a^{-1}\mathcal{F}\). By fully faithfulness of \(h^{-1}\) we conclude that \(a^{-1}\mathcal{F} \cong \mathcal{G}\).

Fix an isomorphism \(\theta : a^{-1}\mathcal{F} \to \mathcal{G}\). To finish the proof we have to show \(\mathcal{G} = a^{-1}a_*\mathcal{G}\) (in order to show that the quasi-inverse is given by \(a_*\); everything else has been proven above). Because \(a^{-1}\) is fully faithful we have \(\text{id} \cong a_*a^{-1}\) by Categories, Lemma 07RB. Thus \(\mathcal{F} \cong a_*a^{-1}\mathcal{F}\) and \(a_*\theta : a_*a^{-1}\mathcal{F} \to a_*\mathcal{G}\) combine to an isomorphism \(\mathcal{F} \to a_*\mathcal{G}\). Pulling back by \(a\) and precomposing by \(\theta^{-1}\) we find the desired isomorphism.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(U\) be a simplicial algebraic space over \(S\). Let \(a : U \to X\) be an augmentation. If \(a : U \to X\) is an fppf hypercovering of \(X\), then for \(K \in D^+(X_\etale)\) \[K \to Ra_*(a^{-1}K)\] is an isomorphism. Here \(a : \Sh(U_\etale) \to \Sh(X_\etale)\) is as in Section 0DE7.

Proof

Consider the diagram of Lemma 0DH5. Observe that \(Rh_{n, *}h_n^{-1}\) is the identity functor on \(D^+(U_{n, \etale})\) by More on Cohomology of Spaces, Lemma 0DGG. Hence \(Rh_*h^{-1}\) is the identity functor on \(D^+(U_\etale)\) by Lemma 0D97. We have \[\begin{align*} Ra_*(a^{-1}K) & = Ra_*Rh_*h^{-1}a^{-1}K \\ & = Rh_{-1, *}Ra_{fppf, *}a_{fppf}^{-1}(h_{-1})^{-1}K \\ & = Rh_{-1, *}(h_{-1})^{-1}K \\ & = K \end{align*}\] The first equality by the discussion above, the second equality because of the commutativity of the diagram in Lemma 0DAF, the third equality by Lemma 0D8N as \(U\) is a hypercovering of \(X\) in \((\textit{Spaces}/S)_{fppf}\), and the last equality by the already used More on Cohomology of Spaces, Lemma 0DGG.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(U\) be a simplicial algebraic space over \(S\). Let \(a : U \to X\) be an augmentation. If \(a : U \to X\) is an fppf hypercovering of \(X\), then \[R\Gamma(X_\etale, K) = R\Gamma(U_\etale, a^{-1}K)\] for \(K \in D^+(X_\etale)\). Here \(a : \Sh(U_\etale) \to \Sh(X_\etale)\) is as in Section 0DE7.

Proof

This follows from Lemma 0DH7 because \(R\Gamma(U_\etale, -) = R\Gamma(X_\etale, -) \circ Ra_*\) by Cohomology on Sites, Remark 08J6.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(U\) be a simplicial algebraic space over \(S\). Let \(a : U \to X\) be an augmentation. Let \(\mathcal{A} \subset \textit{Ab}(U_\etale)\) denote the weak Serre subcategory of cartesian abelian sheaves. If \(U\) is an fppf hypercovering of \(X\), then the functor \(a^{-1}\) defines an equivalence \[D^+(X_\etale) \longrightarrow D_\mathcal{A}^+(U_\etale)\] with quasi-inverse \(Ra_*\). Here \(a : \Sh(U_\etale) \to \Sh(X_\etale)\) is as in Section 0DE7.

Proof

Observe that \(\mathcal{A}\) is a weak Serre subcategory by Lemma 0D7J. The equivalence is a formal consequence of the results obtained so far. Use Lemmas 0DH6 and 0DH7 and Cohomology on Sites, Lemma 0D7U.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(U\) be a simplicial algebraic space over \(S\). Let \(a : U \to X\) be an augmentation. Let \(\mathcal{F}\) be an abelian sheaf on \(X_\etale\). Let \(\mathcal{F}_n\) be the pullback to \(U_{n, \etale}\). If \(U\) is an fppf hypercovering of \(X\), then there exists a canonical spectral sequence \[E_1^{p, q} = H^q_\etale(U_p, \mathcal{F}_p)\] converging to \(H^{p + q}_\etale(X, \mathcal{F})\).

Proof

Immediate consequence of Lemmas 0DH8 and 09WJ.

Fppf hypercoverings of algebraic spaces: modules

We continue the discussion of (cohomological) descent for fppf hypercoverings started in Section 0DH4 but in this section we discuss what happens for sheaves of modules. We mainly discuss quasi-coherent modules and it turns out that we can do unbounded cohomological descent for those.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(U\) be a simplicial algebraic space over \(S\). Let \(a : U \to X\) be an augmentation. There is a commutative diagram \[\xymatrix{ (\Sh((\textit{Spaces}/U)_{fppf, total}), \mathcal{O}_{big, total}) \ar[r]_-h \ar[d]_{a_{fppf}} & (\Sh(U_\etale), \mathcal{O}_U) \ar[d]^a \\ (\Sh((\textit{Spaces}/X)_{fppf}), \mathcal{O}_{big}) \ar[r]^-{h_{-1}} & (\Sh(X_\etale), \mathcal{O}_X) }\] of ringed topoi where the left vertical arrow is defined in Section 0DAA and the right vertical arrow is defined in Section 0DE7.

Proof

For the underlying diagram of topoi we refer to the discussion in the proof of Lemma 0DH5. The sheaf \(\mathcal{O}_U\) is the structure sheaf of the simplicial algebraic space \(U\) as defined in Section 0DE7. The sheaf \(\mathcal{O}_X\) is the usual structure sheaf of the algebraic space \(X\). The sheaves of rings \(\mathcal{O}_{big, total}\) and \(\mathcal{O}_{big}\) come from the structure sheaf on \((\textit{Spaces}/S)_{fppf}\) in the manner explained in Section 0DAA which also constructs \(a_{fppf}\) as a morphism of ringed topoi. The component morphisms \(h_n = a_{U_n}\) and \(h_{-1} = a_X\) are morphisms of ringed topoi by More on Cohomology of Spaces, Section 0DGM. Finally, since the continuous functor \(u : U_{spaces, \etale} \to (\textit{Spaces}/U)_{fppf, total}\) used to define \(h\)⁸ is given by \(V/U_n \mapsto V/U_n\) we see that \(h_*\mathcal{O}_{big, total} = \mathcal{O}_U\) which is how we endow \(h\) with the structure of a morphism of ringed simplicial sites as in Remark 0DGZ. Then we obtain \(h\) as a morphism of ringed topoi by Lemma 0DH0. Please observe that the morphisms \(h_n\) indeed agree with the morphisms \(a_{U_n}\) described above. We omit the verification that the diagram is commutative (as a diagram of ringed topoi – we already know it is commutative as a diagram of topoi).

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(U\) be a simplicial algebraic space over \(S\). Let \(a : U \to X\) be an augmentation. If \(a : U \to X\) is an fppf hypercovering of \(X\), then \[a^* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_U)\] is an equivalence fully faithful with quasi-inverse given by \(a_*\). Here \(a : \Sh(U_\etale) \to \Sh(X_\etale)\) is as in Section 0DE7.

Proof

Consider the diagram of Lemma 0DHC. In the proof of this lemma we have seen that \(h_{-1}\) is the morphism \(a_X\) of More on Cohomology of Spaces, Section 0DGM. Thus it follows from More on Cohomology of Spaces, Lemma 0DGN that \[(h_{-1})^* : \QCoh(\mathcal{O}_X) \longrightarrow \QCoh(\mathcal{O}_{big})\] is an equivalence with quasi-inverse \(h_{-1, *}\). The same holds true for the components \(h_n\) of \(h\). Recall that \(\QCoh(\mathcal{O}_U)\) and \(\QCoh(\mathcal{O}_{big, total})\) consist of cartesian modules whose components are quasi-coherent, see Lemma 0D7M. Since the functors \(h^*\) and \(h_*\) of Lemma 0DH0 agree with the functors \(h_n^*\) and \(h_{n, *}\) on components we conclude that \[h^* : \QCoh(\mathcal{O}_U) \longrightarrow \QCoh(\mathcal{O}_{big, total})\] is an equivalence with quasi-inverse \(h_*\). Observe that \(U\) is a hypercovering of \(X\) in \((\textit{Spaces}/S)_{fppf}\) as defined in Section 09X8. By Lemma 0DAB we see that \(a_{fppf}^*\) is fully faithful with quasi-inverse \(a_{fppf, *}\) and with essential image the cartesian sheaves of \(\mathcal{O}_{fppf, total}\)-modules. Thus, by the description of \(\QCoh(\mathcal{O}_{big})\) and \(\QCoh(\mathcal{O}_{big, total})\) of Lemma 0D7M, we get an equivalence \[a_{fppf}^* : \QCoh(\mathcal{O}_{big}) \longrightarrow \QCoh(\mathcal{O}_{big, total})\] with quasi-inverse given by \(a_{fppf, *}\). A formal argument (chasing around the diagram) now shows that \(a^*\) is fully faithful on \(\QCoh(\mathcal{O}_X)\) and has image contained in \(\QCoh(\mathcal{O}_U)\).

Finally, suppose that \(\mathcal{G}\) is in \(\QCoh(\mathcal{O}_U)\). Then \(h^*\mathcal{G}\) is in \(\QCoh(\mathcal{O}_{big, total})\). Hence \(h^*\mathcal{G} = a_{fppf}^*\mathcal{H}\) with \(\mathcal{H} = a_{fppf, *}h^*\mathcal{G}\) in \(\QCoh(\mathcal{O}_{big})\) (see above). In turn we see that \(\mathcal{H} = (h_{-1})^*\mathcal{F}\) with \(\mathcal{F} = h_{-1, *}\mathcal{H}\) in \(\QCoh(\mathcal{O}_X)\). Going around the diagram we deduce that \(h^*\mathcal{G} \cong h^*a^*\mathcal{F}\). By fully faithfulness of \(h^*\) we conclude that \(a^*\mathcal{F} \cong \mathcal{G}\). Since \(\mathcal{F} = h_{-1, *}a_{fppf, *}h^*\mathcal{G} = a_*h_*h^*\mathcal{G} = a_*\mathcal{G}\) we also obtain the statement that the quasi-inverse is given by \(a_*\).

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(U\) be a simplicial algebraic space over \(S\). Let \(a : U \to X\) be an augmentation. If \(a : U \to X\) is an fppf hypercovering of \(X\), then for \(\mathcal{F}\) a quasi-coherent \(\mathcal{O}_X\)-module the map \[\mathcal{F} \to Ra_*(a^*\mathcal{F})\] is an isomorphism. Here \(a : \Sh(U_\etale) \to \Sh(X_\etale)\) is as in Section 0DE7.

Proof

Consider the diagram of Lemma 0DH5. Let \(\mathcal{F}_n = a_n^*\mathcal{F}\) be the \(n\)th component of \(a^*\mathcal{F}\). This is a quasi-coherent \(\mathcal{O}_{U_n}\)-module. Then \(\mathcal{F}_n = Rh_{n, *}h_n^*\mathcal{F}_n\) by More on Cohomology of Spaces, Lemma 0DGP. Hence \(a^*\mathcal{F} = Rh_*h^*a^*\mathcal{F}\) by Lemma 0DH1. We have \[\begin{align*} Ra_*(a^*\mathcal{F}) & = Ra_*Rh_*h^*a^*\mathcal{F} \\ & = Rh_{-1, *}Ra_{fppf, *}a_{fppf}^*(h_{-1})^*\mathcal{F} \\ & = Rh_{-1, *}(h_{-1})^*\mathcal{F} \\ & = \mathcal{F} \end{align*}\] The first equality by the discussion above, the second equality because of the commutativity of the diagram in Lemma 0DAF, the third equality by Lemma 0DAC as \(U\) is a hypercovering of \(X\) in \((\textit{Spaces}/S)_{fppf}\) and \(La_{fppf}^* = a_{fppf}^*\) as \(a_{fppf}\) is flat (namely \(a_{fppf}^{-1}\mathcal{O}_{big} = \mathcal{O}_{big, total}\), see Remark 0D9W), and the last equality by the already used More on Cohomology of Spaces, Lemma 0DGP.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(U\) be a simplicial algebraic space over \(S\). Let \(a : U \to X\) be an augmentation. Assume \(a : U \to X\) is an fppf hypercovering of \(X\). Then \(\QCoh(\mathcal{O}_U)\) is a weak Serre subcategory of \(\textit{Mod}(\mathcal{O}_U)\) and \[a^* : D_\QCoh(\mathcal{O}_X) \longrightarrow D_\QCoh(\mathcal{O}_U)\] is an equivalence of categories with quasi-inverse given by \(Ra_*\). Here \(a : \Sh(U_\etale) \to \Sh(X_\etale)\) is as in Section 0DE7.

Proof

First observe that the maps \(a_n : U_n \to X\) and \(d^n_i : U_n \to U_{n - 1}\) are flat, locally of finite presentation, and surjective by Hypercoverings, Remark 0DER.

Recall that an \(\mathcal{O}_U\)-module \(\mathcal{F}\) is quasi-coherent if and only if it is cartesian and \(\mathcal{F}_n\) is quasi-coherent for all \(n\). See Lemma 0D7M. By Lemma 0D7J (and flatness of the maps \(d^n_i : U_n \to U_{n - 1}\) shown above) the cartesian modules for a weak Serre subcategory of \(\textit{Mod}(\mathcal{O}_U)\). On the other hand \(\QCoh(\mathcal{O}_{U_n}) \subset \textit{Mod}(\mathcal{O}_{U_n})\) is a weak Serre subcategory for each \(n\) (Properties of Spaces, Lemma 03M1). Combined we see that \(\QCoh(\mathcal{O}_U) \subset \textit{Mod}(\mathcal{O}_U)\) is a weak Serre subcategory.

To finish the proof we check the conditions (1) – (5) of Cohomology on Sites, Lemma 0D7V one by one.

Ad (1). This holds since \(a_n\) flat (seen above) implies \(a\) is flat by Lemma 0DH3.

Ad (2). This is the content of Lemma 0DHD.

Ad (3). This is the content of Lemma 0DHE.

Ad (4). Recall that we can use either the site \(U_\etale\) or \(U_{spaces, \etale}\) to define the small étale topos \(\Sh(U_\etale)\), see Section 0DE7. The assumption of Cohomology on Sites, Situation 0D6R holds for the triple \((U_{spaces, \etale}, \mathcal{O}_U, \QCoh(\mathcal{O}_U))\) and by the same reasoning for the triple \((U_\etale, \mathcal{O}_U, \QCoh(\mathcal{O}_U))\). Namely, take \[\mathcal{B} \subset \Ob(U_\etale) \subset \Ob(U_{spaces, \etale})\] to be the set of affine objects. For \(V/U_n \in \mathcal{B}\) take \(d_{V/U_n} = 0\) and take \(\text{Cov}_{V/U_n}\) to be the set of étale coverings \(\{V_i \to V\}\) with \(V_i\) affine. Then we get the desired vanishing because for \(\mathcal{F} \in \QCoh(\mathcal{O}_U)\) and any \(V/U_n \in \mathcal{B}\) we have \[H^p(V/U_n, \mathcal{F}) = H^p(V, \mathcal{F}_n)\] by Lemma 0DH2. Here on the right hand side we have the cohomology of the quasi-coherent sheaf \(\mathcal{F}_n\) on \(U_n\) over the affine object \(V\) of \(U_{n, \etale}\). This vanishes for \(p > 0\) by the discussion in Cohomology of Spaces, Section 071Y and Cohomology of Schemes, Lemma 01XB.

Ad (5). Follows by taking \(\mathcal{B} \subset \Ob(X_{spaces, \etale})\) the set of affine objects and the references given above.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(U\) be a simplicial algebraic space over \(S\). Let \(a : U \to X\) be an augmentation. If \(a : U \to X\) is an fppf hypercovering of \(X\), then \[R\Gamma(X_\etale, K) = R\Gamma(U_\etale, a^*K)\] for \(K \in D_\QCoh(\mathcal{O}_X)\). Here \(a : \Sh(U_\etale) \to \Sh(X_\etale)\) is as in Section 0DE7.

Proof

This follows from Lemma 0DHF because \(R\Gamma(U_\etale, -) = R\Gamma(X_\etale, -) \circ Ra_*\) by Cohomology on Sites, Remark 08J6.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(U\) be a simplicial algebraic space over \(S\). Let \(a : U \to X\) be an augmentation. Let \(\mathcal{F}\) be quasi-coherent \(\mathcal{O}_X\)-module. Let \(\mathcal{F}_n\) be the pullback to \(U_{n, \etale}\). If \(U\) is an fppf hypercovering of \(X\), then there exists a canonical spectral sequence \[E_1^{p, q} = H^q_\etale(U_p, \mathcal{F}_p)\] converging to \(H^{p + q}_\etale(X, \mathcal{F})\).

Proof

Immediate consequence of Lemmas 0DHG and 0D7E.

Fppf descent of complexes

In this section we pull some of the previously shown results together for fppf coverings of algebraic spaces and derived categories of quasi-coherent modules.

Lemma

Let \(X\) be an algebraic space over a scheme \(S\). Let \(K, E \in D_\QCoh(\mathcal{O}_X)\). Let \(a : U \to X\) be an fppf hypercovering. Assume that for all \(n \geq 0\) we have \[\Ext_{\mathcal{O}_{U_n}}^i(La_n^*K, La_n^*E) = 0 \text{ for } i < 0\] Then we have

1. \(\Ext_{\mathcal{O}_X}^i(K, E) = 0\) for \(i < 0\), and

2. there is an exact sequence \[0 \to \Hom_{\mathcal{O}_X}(K, E) \to \Hom_{\mathcal{O}_{U_0}}(La_0^*K, La_0^*E) \to \Hom_{\mathcal{O}_{U_1}}(La_1^*K, La_1^*E)\]

Proof

Write \(K_n = La_n^*K\) and \(E_n = La_n^*E\). Then these are the simplicial systems of the derived category of modules (Definition 0D9N) associated to \(La^*K\) and \(La^*E\) (Lemma 0D9P) where \(a : U_\etale \to X_\etale\) is as in Section 0DE7. Let us prove (2) first. By Lemma 0DHF we have \[\Hom_{\mathcal{O}_X}(K, E) = \Hom_{\mathcal{O}_U}(La^*K, La^*E)\] Thus the sequence looks like this: \[0 \to \Hom_{\mathcal{O}_U}(La^*K, La^*E) \to \Hom_{\mathcal{O}_{U_0}}(K_0, E_0) \to \Hom_{\mathcal{O}_{U_1}}(K_1, E_1)\] The first arrow is injective by Lemma 0D9R. The image of this arrow is the kernel of the second by Lemma 0D9S. This finishes the proof of (2). Part (1) follows by applying part (2) with \(K[i]\) and \(E\) for \(i > 0\).

Lemma

Let \(X\) be an algebraic space over a scheme \(S\). Let \(a : U \to X\) be an fppf hypercovering. Suppose given \(K_0 \in D_\QCoh(U_0)\) and an isomorphism \[\alpha : L(f_{\delta_1^1})^*K_0 \longrightarrow L(f_{\delta_0^1})^*K_0\] satisfying the cocycle condition on \(U_1\). Set \(\tau^n_i : [0] \to [n]\), \(0 \mapsto i\) and set \(K_n = Lf_{\tau^n_n}^*K_0\). Assume \(\Ext^i_{\mathcal{O}_{U_n}}(K_n, K_n) = 0\) for \(i < 0\). Then there exists an object \(K \in D_\QCoh(\mathcal{O}_X)\) and an isomorphism \(La_0^*K \to K\) compatible with \(\alpha\).

Proof

The objects \(K_n\) form the members of a simplicial system of the derived category of modules by Lemma 0GMF. Then we obtain an object \(K' \in D_\QCoh(\mathcal{O}_{U_\etale})\) such that \((K_n, K_\varphi)\) is the system deduced from \(K'\), see Lemma 0D9T. Finally, we apply Lemma 0DHF to see that \(K' = La^*K\) for some \(K \in D_\QCoh(\mathcal{O}_X)\) as desired.

Proper hypercoverings of algebraic spaces

This section is the analogue of Section 09XA for the case of algebraic spaces. The reader who wishes to do so, can replace “algebraic space” everywhere with “scheme” and get equally valid results. This has the advantage of replacing the references to More on Cohomology of Spaces, Section 0DGS with references to Étale Cohomology, Section 0DDV.

We fix a base scheme \(S\). Let \(X\) be an algebraic space over \(S\) and let \(U\) be a simplicial algebraic space over \(S\). Assume we have an augmentation \[a : U \to X\] See Section 0DE7. We say that \(U\) is a proper hypercovering of \(X\) if

1. \(U_0 \to X\) is proper and surjective,

2. \(U_1 \to U_0 \times_X U_0\) is proper and surjective,

3. \(U_{n + 1} \to (\text{cosk}_n\text{sk}_n U)_{n + 1}\) is proper and surjective for \(n \geq 1\).

The category of algebraic spaces over \(S\) has all finite limits, hence the coskeleta used in the formulation above exist. \[\fbox{Principle: Proper hypercoverings can be used to compute \'etale cohomology.}\] The key idea behind the proof of the principle is to compare the ph and étale topologies on the category \(\textit{Spaces}/S\). Namely, the ph topology is stronger than the étale topology and we have (a) a proper surjective map defines a ph covering, and (b) ph cohomology of sheaves pulled back from the small étale site agrees with étale cohomology as we have seen in More on Cohomology of Spaces, Section 0DGS.

All results in this section generalize to the case where \(U \to X\) is merely a “ph hypercovering”, meaning a hypercovering of \(X\) in the site \((\textit{Spaces}/S)_{ph}\) as defined in Section 09X8. If we ever need this, we will precisely formulate and prove this here.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(U\) be a simplicial algebraic space over \(S\). Let \(a : U \to X\) be an augmentation. There is a commutative diagram \[\xymatrix{ \Sh((\textit{Spaces}/U)_{ph, total}) \ar[r]_-h \ar[d]_{a_{ph}} & \Sh(U_\etale) \ar[d]^a \\ \Sh((\textit{Spaces}/X)_{ph}) \ar[r]^-{h_{-1}} & \Sh(X_\etale) }\] where the left vertical arrow is defined in Section 09X8 and the right vertical arrow is defined in Section 0DE7.

Proof

The notation \((\textit{Spaces}/U)_{ph, total}\) indicates that we are using the construction of Section 09X8 for the site \((\textit{Spaces}/S)_{ph}\) and the simplicial object \(U\) of this site⁹. We will use the sites \(X_{spaces, \etale}\) and \(U_{spaces, \etale}\) for the topoi on the right hand side; this is permissible see discussion in Section 0DE7.

Observe that both \((\textit{Spaces}/U)_{ph, total}\) and \(U_{spaces, \etale}\) fall into case A of Situation 09WE. This is immediate from the construction of \(U_\etale\) in Section 0DE7 and it follows from Lemma 09WL for \((\textit{Spaces}/U)_{ph, total}\). Next, consider the functors \(U_{n, spaces, \etale} \to (\textit{Spaces}/U_n)_{ph}\), \(U \mapsto U/U_n\) and \(X_{spaces, \etale} \to (\textit{Spaces}/X)_{ph}\), \(U \mapsto U/X\). We have seen that these define morphisms of sites in More on Cohomology of Spaces, Section 0DGS where these were denoted \(a_{U_n} = \epsilon_{U_n} \circ \pi_{u_n}\) and \(a_X = \epsilon_X \circ \pi_X\). Thus we obtain a morphism of simplicial sites compatible with augmentations as in Remark 0D98 and we may apply Lemma 0D99 to conclude.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(U\) be a simplicial algebraic space over \(S\). Let \(a : U \to X\) be an augmentation. If \(a : U \to X\) is a proper hypercovering of \(X\), then \[a^{-1} : \Sh(X_\etale) \to \Sh(U_\etale) \quad\text{and}\quad a^{-1} : \textit{Ab}(X_\etale) \to \textit{Ab}(U_\etale)\] are fully faithful with essential image the cartesian sheaves and quasi-inverse given by \(a_*\). Here \(a : \Sh(U_\etale) \to \Sh(X_\etale)\) is as in Section 0DE7.

Proof

We will prove the statement for sheaves of sets. It will be an almost formal consequence of results already established. Consider the diagram of Lemma 0DHJ. In the proof of this lemma we have seen that \(h_{-1}\) is the morphism \(a_X\) of More on Cohomology of Spaces, Section 0DGS. Thus it follows from More on Cohomology of Spaces, Lemma 0DGT that \((h_{-1})^{-1}\) is fully faithful with quasi-inverse \(h_{-1, *}\). The same holds true for the components \(h_n\) of \(h\). By the description of the functors \(h^{-1}\) and \(h_*\) of Lemma 0D96 we conclude that \(h^{-1}\) is fully faithful with quasi-inverse \(h_*\). Observe that \(U\) is a hypercovering of \(X\) in \((\textit{Spaces}/S)_{ph}\) as defined in Section 09X8 since a surjective proper morphism gives a ph covering by Topologies on Spaces, Lemma 0DFI. By Lemma 0DA8 we see that \(a_{ph}^{-1}\) is fully faithful with quasi-inverse \(a_{ph, *}\) and with essential image the cartesian sheaves on \((\textit{Spaces}/U)_{ph, total}\). A formal argument (chasing around the diagram) now shows that \(a^{-1}\) is fully faithful.

Finally, suppose that \(\mathcal{G}\) is a cartesian sheaf on \(U_\etale\). Then \(h^{-1}\mathcal{G}\) is a cartesian sheaf on \((\textit{Spaces}/U)_{ph, total}\). Hence \(h^{-1}\mathcal{G} = a_{ph}^{-1}\mathcal{H}\) for some sheaf \(\mathcal{H}\) on \((\textit{Spaces}/X)_{ph}\). We compute using somewhat pedantic notation \[\begin{align*} (h_{-1})^{-1}(a_*\mathcal{G}) & = (h_{-1})^{-1} \text{Eq}( \xymatrix{ a_{0, small, *}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & a_{1, small, *}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ (h_{-1})^{-1}a_{0, small, *}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & (h_{-1})^{-1}a_{1, small, *}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ a_{0, big, ph, *}h_0^{-1}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & a_{1, big, ph, *}h_1^{-1}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ a_{0, big, ph, *}(a_{0, big, ph})^{-1}\mathcal{H} \ar@<1ex>[r] \ar@<-1ex>[r] & a_{1, big, ph, *}(a_{1, big, ph})^{-1}\mathcal{H} } ) \\ & = a_{ph, *}a_{ph}^{-1}\mathcal{H} \\ & = \mathcal{H} \end{align*}\] Here the first equality follows from Lemma 0D70, the second equality follows as \((h_{-1})^{-1}\) is an exact functor, the third equality follows from More on Cohomology of Spaces, Lemma 0DGX (here we use that \(a_0 : U_0 \to X\) and \(a_1: U_1 \to X\) are proper), the fourth follows from \(a_{ph}^{-1}\mathcal{H} = h^{-1}\mathcal{G}\), the fifth from Lemma 0D70, and the sixth we’ve seen above. Since \(a_{ph}^{-1}\mathcal{H} = h^{-1}\mathcal{G}\) we deduce that \(h^{-1}\mathcal{G} \cong h^{-1}a^{-1}a_*\mathcal{G}\) which ends the proof by fully faithfulness of \(h^{-1}\).

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(U\) be a simplicial algebraic space over \(S\). Let \(a : U \to X\) be an augmentation. If \(a : U \to X\) is a proper hypercovering of \(X\), then for \(K \in D^+(X_\etale)\) \[K \to Ra_*(a^{-1}K)\] is an isomorphism. Here \(a : \Sh(U_\etale) \to \Sh(X_\etale)\) is as in Section 0DE7.

Proof

Consider the diagram of Lemma 0DHJ. Observe that \(Rh_{n, *}h_n^{-1}\) is the identity functor on \(D^+(U_{n, \etale})\) by More on Cohomology of Spaces, Lemma 0DGU. Hence \(Rh_*h^{-1}\) is the identity functor on \(D^+(U_\etale)\) by Lemma 0D97. We have \[\begin{align*} Ra_*(a^{-1}K) & = Ra_*Rh_*h^{-1}a^{-1}K \\ & = Rh_{-1, *}Ra_{ph, *}a_{ph}^{-1}(h_{-1})^{-1}K \\ & = Rh_{-1, *}(h_{-1})^{-1}K \\ & = K \end{align*}\] The first equality by the discussion above, the second equality because of the commutativity of the diagram in Lemma 0DAF, the third equality by Lemma 0D8N as \(U\) is a hypercovering of \(X\) in \((\textit{Spaces}/S)_{ph}\) by Topologies on Spaces, Lemma 0DFI, and the last equality by the already used More on Cohomology of Spaces, Lemma 0DGU.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(U\) be a simplicial algebraic space over \(S\). Let \(a : U \to X\) be an augmentation. If \(a : U \to X\) is a proper hypercovering of \(X\), then \[R\Gamma(X_\etale, K) = R\Gamma(U_\etale, a^{-1}K)\] for \(K \in D^+(X_\etale)\). Here \(a : \Sh(U_\etale) \to \Sh(X_\etale)\) is as in Section 0DE7.

Proof

This follows from Lemma 0DHL because \(R\Gamma(U_\etale, -) = R\Gamma(X_\etale, -) \circ Ra_*\) by Cohomology on Sites, Remark 08J6.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(U\) be a simplicial algebraic space over \(S\). Let \(a : U \to X\) be an augmentation. Let \(\mathcal{A} \subset \textit{Ab}(U_\etale)\) denote the weak Serre subcategory of cartesian abelian sheaves. If \(U\) is a proper hypercovering of \(X\), then the functor \(a^{-1}\) defines an equivalence \[D^+(X_\etale) \longrightarrow D_\mathcal{A}^+(U_\etale)\] with quasi-inverse \(Ra_*\). Here \(a : \Sh(U_\etale) \to \Sh(X_\etale)\) is as in Section 0DE7.

Proof

Observe that \(\mathcal{A}\) is a weak Serre subcategory by Lemma 0D7J. The equivalence is a formal consequence of the results obtained so far. Use Lemmas 0DHK and 0DHL and Cohomology on Sites, Lemma 0D7U.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(U\) be a simplicial algebraic space over \(S\). Let \(a : U \to X\) be an augmentation. Let \(\mathcal{F}\) be an abelian sheaf on \(X_\etale\). Let \(\mathcal{F}_n\) be the pullback to \(U_{n, \etale}\). If \(U\) is a ph hypercovering of \(X\), then there exists a canonical spectral sequence \[E_1^{p, q} = H^q_\etale(U_p, \mathcal{F}_p)\] converging to \(H^{p + q}_\etale(X, \mathcal{F})\).

Proof

Immediate consequence of Lemmas 0DHM and 09WJ.