Modules on Sites

Introduction

In this document we work out basic notions of sheaves of modules on ringed topoi or ringed sites. We first work out some basic facts on abelian sheaves. After this we introduce ringed sites and ringed topoi. We work through some of the very basic notions on (pre)sheaves of \(\mathcal{O}\)-modules, analogous to the material on (pre)sheaves of \(\mathcal{O}\)-modules in the chapter on sheaves on spaces. Having done this, we duplicate much of the discussion in the chapter on sheaves of modules (see Modules, Section 01AD). Basic references are [FAC], [EGA] and [SGA4].

Abelian presheaves

Let \(\mathcal{C}\) be a category. Abelian presheaves were introduced in Sites, Sections 00V1 and 00VL and discussed a bit more in Sites, Section 00YR. We will follow the convention of this last reference, in that we think of an abelian presheaf as a presheaf of sets endowed with addition rules on all sets of sections compatible with the restriction mappings. Recall that the category of abelian presheaves on \(\mathcal{C}\) is denoted \(\textit{PAb}(\mathcal{C})\).

The category \(\textit{PAb}(\mathcal{C})\) is abelian as defined in Homology, Definition 0109. Given a map of presheaves \(\varphi : \mathcal{G}_1 \to \mathcal{G}_2\) the kernel of \(\varphi\) is the abelian presheaf \(U \mapsto \Ker(\mathcal{G}_1(U) \to \mathcal{G}_2(U))\) and the cokernel of \(\varphi\) is the presheaf \(U \mapsto \Coker(\mathcal{G}_1(U) \to \mathcal{G}_2(U))\). Since the category of abelian groups is abelian it follows that \(\Coim = \Im\) because this holds over each \(U\). A sequence of abelian presheaves \[\mathcal{G}_1 \longrightarrow \mathcal{G}_2 \longrightarrow \mathcal{G}_3\] is exact if and only if \(\mathcal{G}_1(U) \to \mathcal{G}_2(U) \to \mathcal{G}_3(U)\) is an exact sequence of abelian groups for all \(U \in \Ob(\mathcal{C})\). We leave the verifications to the reader.

Lemma

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

1. All limits and colimits exist in \(\textit{PAb}(\mathcal{C})\).

2. All limits and colimits commute with taking sections over objects of \(\mathcal{C}\).

Proof

Let \(\mathcal{I} \to \textit{PAb}(\mathcal{C})\), \(i \mapsto \mathcal{F}_i\) be a diagram. We can simply define abelian presheaves \(L\) and \(C\) by the rules \[L : U \longmapsto \lim_i \mathcal{F}_i(U)\] and \[C : U \longmapsto \colim_i \mathcal{F}_i(U).\] It is clear that there are maps of abelian presheaves \(L \to \mathcal{F}_i\) and \(\mathcal{F}_i \to C\), by using the corresponding maps on groups of sections over each \(U\). It is straightforward to check that \(L\) and \(C\) endowed with these maps are the limit and colimit of the diagram in \(\textit{PAb}(\mathcal{C})\). This proves (1) and (2). Details omitted.

Abelian sheaves

Let \(\mathcal{C}\) be a site. The category of abelian sheaves on \(\mathcal{C}\) is denoted \(\textit{Ab}(\mathcal{C})\). It is the full subcategory of \(\textit{PAb}(\mathcal{C})\) consisting of those abelian presheaves whose underlying presheaves of sets are sheaves. Properties (\(\alpha\)) – (\(\zeta\)) of Sites, Section 00YR hold, see Sites, Proposition 00YV. In particular the inclusion functor \(\textit{Ab}(\mathcal{C}) \to \textit{PAb}(\mathcal{C})\) has a left adjoint, namely the sheafification functor \(\mathcal{G} \mapsto \mathcal{G}^\#\).

We suggest the reader prove the lemma on a piece of scratch paper rather than reading the proof.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be a morphism of abelian sheaves on \(\mathcal{C}\).

1. The category \(\textit{Ab}(\mathcal{C})\) is an abelian category.

2. The kernel \(\Ker(\varphi)\) of \(\varphi\) is the same as the kernel of \(\varphi\) as a morphism of presheaves.

3. The morphism \(\varphi\) is injective (Homology, Definition 010B) if and only if \(\varphi\) is injective as a map of presheaves (Sites, Definition 00V6), if and only if \(\varphi\) is injective as a map of sheaves (Sites, Definition 00WM).

4. The cokernel \(\Coker(\varphi)\) of \(\varphi\) is the sheafification of the cokernel of \(\varphi\) as a morphism of presheaves.

5. The morphism \(\varphi\) is surjective (Homology, Definition 010B) if and only if \(\varphi\) is surjective as a map of sheaves (Sites, Definition 00WM).

6. A complex of abelian sheaves \[\mathcal{F} \to \mathcal{G} \to \mathcal{H}\] is exact at \(\mathcal{G}\) if and only if for all \(U \in \Ob(\mathcal{C})\) and all \(s \in \mathcal{G}(U)\) mapping to zero in \(\mathcal{H}(U)\) there exists a covering \(\{U_i \to U\}_{i \in I}\) in \(\mathcal{C}\) such that each \(s|_{U_i}\) is in the image of \(\mathcal{F}(U_i) \to \mathcal{G}(U_i)\).

Proof

We claim that Homology, Lemma 03A3 applies to the categories \(\mathcal{A} = \textit{Ab}(\mathcal{C})\) and \(\mathcal{B} = \textit{PAb}(\mathcal{C})\), and the functors \(a : \mathcal{A} \to \mathcal{B}\) (inclusion), and \(b : \mathcal{B} \to \mathcal{A}\) (sheafification). Let us check the assumptions of Homology, Lemma 03A3. Assumption (1) is that \(\mathcal{A}\), \(\mathcal{B}\) are additive categories, \(a\), \(b\) are additive functors, and \(a\) is right adjoint to \(b\). The first two statements are clear and adjointness is Sites, Section 00YR (\(\epsilon\)). Assumption (2) says that \(\textit{PAb}(\mathcal{C})\) is abelian which we saw in Section 03A6 and that sheafification is left exact, which is Sites, Section 00YR (\(\zeta\)). The final assumption is that \(ba \cong \text{id}_\mathcal{A}\) which is Sites, Section 00YR (\(\delta\)). Hence Homology, Lemma 03A3 applies and we conclude that \(\textit{Ab}(\mathcal{C})\) is abelian.

In the proof of Homology, Lemma 03A3 it is shown that \(\Ker(\varphi)\) and \(\Coker(\varphi)\) are equal to the sheafification of the kernel and cokernel of \(\varphi\) as a morphism of abelian presheaves. This proves (4). Since the kernel is a equalizer (i.e., a limit) and since sheafification commutes with finite limits, we conclude that (2) holds.

Statement (2) implies (3). Statement (4) implies (5) by our description of sheafification. The characterization of exactness in (6) follows from (2) and (5), and the fact that the sequence is exact if and only if \(\Im(\mathcal{F} \to \mathcal{G}) = \Ker(\mathcal{G} \to \mathcal{H})\).

Another way to say part (6) of the lemma is that a sequence of abelian sheaves \[\mathcal{F}_1 \longrightarrow \mathcal{F}_2 \longrightarrow \mathcal{F}_3\] is exact if and only if the sheafification of \(U \mapsto \Im(\mathcal{F}_1(U) \to \mathcal{F}_2(U))\) is equal to the kernel of \(\mathcal{F}_2 \to \mathcal{F}_3\).

Lemma

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

1. All limits and colimits exist in \(\textit{Ab}(\mathcal{C})\).

2. Limits are the same as the corresponding limits of abelian presheaves over \(\mathcal{C}\) (i.e., commute with taking sections over objects of \(\mathcal{C}\)).

3. Finite direct sums are the same as the corresponding finite direct sums in the category of abelian pre-sheaves over \(\mathcal{C}\).

4. A colimit is the sheafification of the corresponding colimit in the category of abelian presheaves.

5. Filtered colimits are exact.

Proof

By Lemma 03CL limits and colimits of abelian presheaves exist, and are described by taking limits and colimits on the level of sections over objects.

Let \(\mathcal{I} \to \textit{Ab}(\mathcal{C})\), \(i \mapsto \mathcal{F}_i\) be a diagram. Let \(\lim_i \mathcal{F}_i\) be the limit of the diagram as an abelian presheaf. By Sites, Lemma 00W2 this is an abelian sheaf. Then it is quite easy to see that \(\lim_i \mathcal{F}_i\) is the limit of the diagram in \(\textit{Ab}(\mathcal{C})\). This proves limits exist and (2) holds.

By Categories, Lemma 0038, and because sheafification is left adjoint to the inclusion functor we see that \(\colim_i \mathcal{F}\) exists and is the sheafification of the colimit in \(\textit{PAb}(\mathcal{C})\). This proves colimits exist and (4) holds.

Finite direct sums are the same thing as finite products in any abelian category. Hence (3) follows from (2).

Proof of (5). The statement means that given a system \(0 \to \mathcal{F}_i \to \mathcal{G}_i \to \mathcal{H}_i \to 0\) of exact sequences of abelian sheaves over a directed set \(I\) the sequence \(0 \to \colim \mathcal{F}_i \to \colim \mathcal{G}_i \to \colim \mathcal{H}_i \to 0\) is exact as well. A formal argument using Homology, Lemma 05AA and the definition of colimits shows that the sequence \(\colim \mathcal{F}_i \to \colim \mathcal{G}_i \to \colim \mathcal{H}_i \to 0\) is exact. Note that \(\colim \mathcal{F}_i \to \colim \mathcal{G}_i\) is the sheafification of the map of presheaf colimits which is injective as each of the maps \(\mathcal{F}_i \to \mathcal{G}_i\) is injective. Since sheafification is exact we conclude.

Free abelian presheaves

In order to prepare notation for the following definition, let us agree to denote the free abelian group on a set \(S\) as¹ \(\mathbf{Z}[S] = \bigoplus_{s \in S} \mathbf{Z}\). It is characterized by the property \[\Mor_{\textit{Ab}}(\mathbf{Z}[S], A) = \Mor_{\textit{Sets}}(S, A)\] In other words the construction \(S \mapsto \mathbf{Z}[S]\) is a left adjoint to the forgetful functor \(\textit{Ab} \to \textit{Sets}\).

Definition

Let \(\mathcal{C}\) be a category. Let \(\mathcal{G}\) be a presheaf of sets. The free abelian presheaf \(\mathbf{Z}_\mathcal{G}\) on \(\mathcal{G}\) is the abelian presheaf defined by the rule \[U \longmapsto \mathbf{Z}[\mathcal{G}(U)].\] In the special case \(\mathcal{G} = h_X\) of a representable presheaf associated to an object \(X\) of \(\mathcal{C}\) we use the notation \(\mathbf{Z}_X = \mathbf{Z}_{h_X}\). In other words \[\mathbf{Z}_X(U) = \mathbf{Z}[\Mor_\mathcal{C}(U, X)].\]

This construction is clearly functorial in the presheaf \(\mathcal{G}\). In fact it is adjoint to the forgetful functor \(\textit{PAb}(\mathcal{C}) \to \textit{PSh}(\mathcal{C})\). Here is the precise statement.

Lemma

Let \(\mathcal{C}\) be a category. Let \(\mathcal{G}\), \(\mathcal{F}\) be a presheaves of sets. Let \(\mathcal{A}\) be an abelian presheaf. Let \(U\) be an object of \(\mathcal{C}\). Then we have \[\begin{align*} \Mor_{\textit{PSh}(\mathcal{C})}(h_U, \mathcal{F}) & = \mathcal{F}(U), \\ \Mor_{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_\mathcal{G}, \mathcal{A}) & = \Mor_{\textit{PSh}(\mathcal{C})}(\mathcal{G}, \mathcal{A}), \\ \Mor_{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_U, \mathcal{A}) & = \mathcal{A}(U). \end{align*}\] All of these equalities are functorial.

Proof

Omitted.

Lemma

Let \(\mathcal{C}\) be a category. Let \(I\) be a set. For each \(i \in I\) let \(\mathcal{G}_i\) be a presheaf of sets. Then \[\mathbf{Z}_{\coprod_i \mathcal{G}_i} = \bigoplus\nolimits_{i \in I} \mathbf{Z}_{\mathcal{G}_i}\] in \(\textit{PAb}(\mathcal{C})\).

Proof

Omitted.

Free abelian sheaves

Here is the notion of a free abelian sheaf on a sheaf of sets.

Definition

Let \(\mathcal{C}\) be a site. Let \(\mathcal{G}\) be a presheaf of sets. The free abelian sheaf \(\mathbf{Z}_\mathcal{G}^\#\) on \(\mathcal{G}\) is the abelian sheaf \(\mathbf{Z}_\mathcal{G}^\#\) which is the sheafification of the free abelian presheaf on \(\mathcal{G}\). In the special case \(\mathcal{G} = h_X\) of a representable presheaf associated to an object \(X\) of \(\mathcal{C}\) we use the notation \(\mathbf{Z}_X^\#\).

This construction is clearly functorial in the presheaf \(\mathcal{G}\). In fact it provides an adjoint to the forgetful functor \(\textit{Ab}(\mathcal{C}) \to \Sh(\mathcal{C})\). Here is the precise statement.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{G}\), \(\mathcal{F}\) be sheaves of sets. Let \(\mathcal{A}\) be an abelian sheaf. Let \(U\) be an object of \(\mathcal{C}\). Then we have \[\begin{align*} \Mor_{\Sh(\mathcal{C})}(h_U^\#, \mathcal{F}) & = \mathcal{F}(U), \\ \Mor_{\textit{Ab}(\mathcal{C})}(\mathbf{Z}_\mathcal{G}^\#, \mathcal{A}) & = \Mor_{\Sh(\mathcal{C})}(\mathcal{G}, \mathcal{A}), \\ \Mor_{\textit{Ab}(\mathcal{C})}(\mathbf{Z}_U^\#, \mathcal{A}) & = \mathcal{A}(U). \end{align*}\] All of these equalities are functorial.

Proof

Omitted.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{G}\) be a presheaf of sets. Then \(\mathbf{Z}_\mathcal{G}^\# = (\mathbf{Z}_{\mathcal{G}^\#})^\#\).

Proof

Omitted.

Ringed sites

In this chapter we mainly work with sheaves of modules on a ringed site. Hence we need to define this notion.

Definition

Ringed sites.

1. A ringed site is a pair \((\mathcal{C}, \mathcal{O})\) where \(\mathcal{C}\) is a site and \(\mathcal{O}\) is a sheaf of rings on \(\mathcal{C}\). The sheaf \(\mathcal{O}\) is called the structure sheaf of the ringed site.

2. Let \((\mathcal{C}, \mathcal{O})\), \((\mathcal{C}', \mathcal{O}')\) be ringed sites. A morphism of ringed sites \[(f, f^\sharp) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{C}', \mathcal{O}')\] is given by a morphism of sites \(f : \mathcal{C} \to \mathcal{C}'\) (see Sites, Definition 00X1) together with a map of sheaves of rings \(f^\sharp : f^{-1}\mathcal{O}' \to \mathcal{O}\), which by adjunction is the same thing as a map of sheaves of rings \(f^\sharp : \mathcal{O}' \to f_*\mathcal{O}\).

3. Let \((f, f^\sharp) : (\mathcal{C}_1, \mathcal{O}_1) \to (\mathcal{C}_2, \mathcal{O}_2)\) and \((g, g^\sharp) : (\mathcal{C}_2, \mathcal{O}_2) \to (\mathcal{C}_3, \mathcal{O}_3)\) be morphisms of ringed sites. Then we define the composition of morphisms of ringed sites by the rule \[(g, g^\sharp) \circ (f, f^\sharp) = (g \circ f, f^\sharp \circ g^\sharp).\] Here we use composition of morphisms of sites defined in Sites, Definition 03CC and \(f^\sharp \circ g^\sharp\) indicates the morphism of sheaves of rings \[\mathcal{O}_3 \xrightarrow{g^\sharp} g_*\mathcal{O}_2 \xrightarrow{g_*f^\sharp} g_*f_*\mathcal{O}_1 = (g \circ f)_*\mathcal{O}_1\]

Ringed topoi

A ringed topos is just a ringed site, except that the notion of a morphism of ringed topoi is different from the notion of a morphism of ringed sites.

Definition

Ringed topoi.

1. A ringed topos is a pair \((\Sh(\mathcal{C}), \mathcal{O})\) where \(\mathcal{C}\) is a site and \(\mathcal{O}\) is a sheaf of rings on \(\mathcal{C}\). The sheaf \(\mathcal{O}\) is called the structure sheaf of the ringed topos.

2. Let \((\Sh(\mathcal{C}), \mathcal{O})\), \((\Sh(\mathcal{C}'), \mathcal{O}')\) be ringed topoi. A morphism of ringed topoi \[(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}) \longrightarrow (\Sh(\mathcal{C}'), \mathcal{O}')\] is given by a morphism of topoi \(f : \Sh(\mathcal{C}) \to \Sh(\mathcal{C}')\) (see Sites, Definition 00XA) together with a map of sheaves of rings \(f^\sharp : f^{-1}\mathcal{O}' \to \mathcal{O}\), which by adjunction is the same thing as a map of sheaves of rings \(f^\sharp : \mathcal{O}' \to f_*\mathcal{O}\).

3. Let \((f, f^\sharp) : (\Sh(\mathcal{C}_1), \mathcal{O}_1) \to (\Sh(\mathcal{C}_2), \mathcal{O}_2)\) and \((g, g^\sharp) : (\Sh(\mathcal{C}_2), \mathcal{O}_2) \to (\Sh(\mathcal{C}_3), \mathcal{O}_3)\) be morphisms of ringed topoi. Then we define the composition of morphisms of ringed topoi by the rule \[(g, g^\sharp) \circ (f, f^\sharp) = (g \circ f, f^\sharp \circ g^\sharp).\] Here we use composition of morphisms of topoi defined in Sites, Definition 00XA and \(f^\sharp \circ g^\sharp\) indicates the morphism of sheaves of rings \[\mathcal{O}_3 \xrightarrow{g^\sharp} g_*\mathcal{O}_2 \xrightarrow{g_*f^\sharp} g_*f_*\mathcal{O}_1 = (g \circ f)_*\mathcal{O}_1\]

Every morphism of ringed topoi is the composition of an equivalence of ringed topoi with a morphism of ringed topoi associated to a morphism of ringed sites. Here is the precise statement.

Lemma

Let \((f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi. There exists a factorization \[\xymatrix{ (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[rr]_{(f, f^\sharp)} \ar[d]_{(g, g^\sharp)} & & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) \ar[d]^{(e, e^\sharp)} \\ (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[rr]^{(h, h^\sharp)} & & (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) }\] where

1. \(g : \Sh(\mathcal{C}) \to \Sh(\mathcal{C}')\) is an equivalence of topoi induced by a special cocontinuous functor \(\mathcal{C} \to \mathcal{C}'\) (see Sites, Definition 03CG),

2. \(e : \Sh(\mathcal{D}) \to \Sh(\mathcal{D}')\) is an equivalence of topoi induced by a special cocontinuous functor \(\mathcal{D} \to \mathcal{D}'\) (see Sites, Definition 03CG),

3. \(\mathcal{O}_{\mathcal{C}'} = g_*\mathcal{O}_\mathcal{C}\) and \(g^\sharp\) is the obvious map,

4. \(\mathcal{O}_{\mathcal{D}'} = e_*\mathcal{O}_\mathcal{D}\) and \(e^\sharp\) is the obvious map,

5. the sites \(\mathcal{C}'\) and \(\mathcal{D}'\) have final objects and fibre products (i.e., all finite limits),

6. \(h\) is a morphism of sites induced by a continuous functor \(u : \mathcal{D}' \to \mathcal{C}'\) which commutes with all finite limits (i.e., it satisfies the assumptions of Sites, Proposition 00X6), and

7. given any set of sheaves \(\mathcal{F}_i\) (resp. \(\mathcal{G}_j\)) on \(\mathcal{C}\) (resp. \(\mathcal{D}\)) we may assume each of these is a representable sheaf on \(\mathcal{C}'\) (resp. \(\mathcal{D}'\)).

Moreover, if \((f, f^\sharp)\) is an equivalence of ringed topoi, then we can choose the diagram such that \(\mathcal{C}' = \mathcal{D}'\), \(\mathcal{O}_{\mathcal{C}'} = \mathcal{O}_{\mathcal{D}'}\) and \((h, h^\sharp)\) is the identity.

Proof

This follows from Sites, Lemma 03A2, and Sites, Remarks 03CJ and 03CK. You just have to carry along the sheaves of rings. Some details omitted.

2-morphisms of ringed topoi

This is a brief section concerning the notion of a \(2\)-morphism of ringed topoi.

Definition

Let \(f, g : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be two morphisms of ringed topoi. A 2-morphism from \(f\) to \(g\) is given by a transformation of functors \(t : f_* \to g_*\) such that \[\xymatrix{ & \mathcal{O}_\mathcal{D} \ar[ld]_{f^\sharp} \ar[rd]^{g^\sharp} \\ f_*\mathcal{O}_\mathcal{C} \ar[rr]^t & & g_*\mathcal{O}_\mathcal{C} }\] is commutative.

Pictorially we sometimes represent \(t\) as follows: \[\xymatrix{ (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \rrtwocell^f_g{t} & & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) }\] As in Sites, Section 04I9 giving a 2-morphism \(t : f_* \to g_*\) is equivalent to giving \(t : g^{-1} \to f^{-1}\) (usually denoted by the same symbol) such that the diagram \[\xymatrix{ f^{-1}\mathcal{O}_\mathcal{D} \ar[rd]_{f^\sharp} & & g^{-1}\mathcal{O}_\mathcal{D} \ar[ll]^t \ar[ld]^{g^\sharp} \\ & \mathcal{O}_\mathcal{C} }\] is commutative. As in Sites, Section 04I9 the axioms of a strict 2-category hold with horizontal and vertical compositions defined as explained in loc. cit.

Presheaves of modules

Let \(\mathcal{C}\) be a category. Let \(\mathcal{O}\) be a presheaf of rings on \(\mathcal{C}\). At this point we have not yet defined a presheaf of \(\mathcal{O}\)-modules. Thus we do so right now.

Definition

Let \(\mathcal{C}\) be a category, and let \(\mathcal{O}\) be a presheaf of rings on \(\mathcal{C}\).

1. A presheaf of \(\mathcal{O}\)-modules is given by an abelian presheaf \(\mathcal{F}\) together with a map of presheaves of sets \[\mathcal{O} \times \mathcal{F} \longrightarrow \mathcal{F}\] such that for every object \(U\) of \(\mathcal{C}\) the map \(\mathcal{O}(U) \times \mathcal{F}(U) \to \mathcal{F}(U)\) defines the structure of an \(\mathcal{O}(U)\)-module structure on the abelian group \(\mathcal{F}(U)\).

2. A morphism \(\varphi : \mathcal{F} \to \mathcal{G}\) of presheaves of \(\mathcal{O}\)-modules is a morphism of abelian presheaves \(\varphi : \mathcal{F} \to \mathcal{G}\) such that the diagram \[\xymatrix{ \mathcal{O} \times \mathcal{F} \ar[r] \ar[d]_{\text{id} \times \varphi} & \mathcal{F} \ar[d]^{\varphi} \\ \mathcal{O} \times \mathcal{G} \ar[r] & \mathcal{G} }\] commutes.

3. The set of \(\mathcal{O}\)-module morphisms as above is denoted \(\Hom_\mathcal{O}(\mathcal{F}, \mathcal{G})\).

4. The category of presheaves of \(\mathcal{O}\)-modules is denoted \(\textit{PMod}(\mathcal{O})\).

Suppose that \(\mathcal{O}_1 \to \mathcal{O}_2\) is a morphism of presheaves of rings on the category \(\mathcal{C}\). In this case, if \(\mathcal{F}\) is a presheaf of \(\mathcal{O}_2\)-modules then we can think of \(\mathcal{F}\) as a presheaf of \(\mathcal{O}_1\)-modules by using the composition \[\mathcal{O}_1 \times \mathcal{F} \to \mathcal{O}_2 \times \mathcal{F} \to \mathcal{F}.\] We sometimes denote this by \(\mathcal{F}_{\mathcal{O}_1}\) to indicate the restriction of rings. We call this the restriction of \(\mathcal{F}\). We obtain the restriction functor \[\textit{PMod}(\mathcal{O}_2) \longrightarrow \textit{PMod}(\mathcal{O}_1)\]

On the other hand, given a presheaf of \(\mathcal{O}_1\)-modules \(\mathcal{G}\) we can construct a presheaf of \(\mathcal{O}_2\)-modules \(\mathcal{O}_2 \otimes_{p, \mathcal{O}_1} \mathcal{G}\) by the rule \[U \longmapsto \left(\mathcal{O}_2 \otimes_{p, \mathcal{O}_1} \mathcal{G}\right)(U) = \mathcal{O}_2(U) \otimes_{\mathcal{O}_1(U)} \mathcal{G}(U)\] where \(U \in \Ob(\mathcal{C})\), with obvious restriction mappings. The index \(p\) stands for “presheaf” and not “point”. This presheaf is called the tensor product presheaf. We obtain the change of rings functor \[\textit{PMod}(\mathcal{O}_1) \longrightarrow \textit{PMod}(\mathcal{O}_2)\]

Lemma

With \(\mathcal{C}\), \(\mathcal{O}_1 \to \mathcal{O}_2\), \(\mathcal{F}\) and \(\mathcal{G}\) as above there exists a canonical bijection \[\Hom_{\mathcal{O}_1}(\mathcal{G}, \mathcal{F}_{\mathcal{O}_1}) = \Hom_{\mathcal{O}_2}( \mathcal{O}_2 \otimes_{p, \mathcal{O}_1} \mathcal{G}, \mathcal{F} )\] In other words, the restriction and change of rings functors defined above are adjoint to each other.

Proof

This follows from the fact that for a ring map \(A \to B\) the restriction functor and the change of ring functor are adjoint to each other.

Sheaves of modules

Definition

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}\).

1. A sheaf of \(\mathcal{O}\)-modules is a presheaf of \(\mathcal{O}\)-modules \(\mathcal{F}\), see Definition 03CT, such that the underlying presheaf of abelian groups \(\mathcal{F}\) is a sheaf.

2. A morphism of sheaves of \(\mathcal{O}\)-modules is a morphism of presheaves of \(\mathcal{O}\)-modules.

3. Given sheaves of \(\mathcal{O}\)-modules \(\mathcal{F}\) and \(\mathcal{G}\) we denote \(\Hom_\mathcal{O}(\mathcal{F}, \mathcal{G})\) the set of morphism of sheaves of \(\mathcal{O}\)-modules.

4. The category of sheaves of \(\mathcal{O}\)-modules is denoted \(\textit{Mod}(\mathcal{O})\).

This definition kind of makes sense even if \(\mathcal{O}\) is just a presheaf of rings, although we do not know any examples where this is useful, and we will avoid using the terminology “sheaves of \(\mathcal{O}\)-modules” in case \(\mathcal{O}\) is not a sheaf of rings.

Sheafification of presheaves of modules

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O}\) be a presheaf of rings on \(\mathcal{C}\). Let \(\mathcal{F}\) be a presheaf of \(\mathcal{O}\)-modules. Let \(\mathcal{O}^\#\) be the sheafification of \(\mathcal{O}\) as a presheaf of rings, see Sites, Section 00YR. Let \(\mathcal{F}^\#\) be the sheafification of \(\mathcal{F}\) as a presheaf of abelian groups. There exists a unique map of sheaves of sets \[\mathcal{O}^\# \times \mathcal{F}^\# \longrightarrow \mathcal{F}^\#\] which makes the diagram \[\xymatrix{ \mathcal{O} \times \mathcal{F} \ar[r] \ar[d] & \mathcal{F} \ar[d] \\ \mathcal{O}^\# \times \mathcal{F}^\# \ar[r] & \mathcal{F}^\# }\] commute and which makes \(\mathcal{F}^\#\) into a sheaf of \(\mathcal{O}^\#\)-modules. In addition, if \(\mathcal{G}\) is a sheaf of \(\mathcal{O}^\#\)-modules, then any morphism of presheaves of \(\mathcal{O}\)-modules \(\mathcal{F} \to \mathcal{G}\) (into the restriction of \(\mathcal{G}\) to a \(\mathcal{O}\)-module) factors uniquely as \(\mathcal{F} \to \mathcal{F}^\# \to \mathcal{G}\) where \(\mathcal{F}^\# \to \mathcal{G}\) is a morphism of \(\mathcal{O}^\#\)-modules.

Proof

Omitted.

This actually means that the functor \(i : \textit{Mod}(\mathcal{O}^\#) \to \textit{PMod}(\mathcal{O})\) (combining restriction and including sheaves into presheaves) and the sheafification functor of the lemma \({}^\# : \textit{PMod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}^\#)\) are adjoint. In a formula \[\Mor_{\textit{PMod}(\mathcal{O})}(\mathcal{F}, i\mathcal{G}) = \Mor_{\textit{Mod}(\mathcal{O}^\#)}(\mathcal{F}^\#, \mathcal{G})\] An important case happens when \(\mathcal{O}\) is already a sheaf of rings. In this case the formula reads \[\Mor_{\textit{PMod}(\mathcal{O})}(\mathcal{F}, i\mathcal{G}) = \Mor_{\textit{Mod}(\mathcal{O})}(\mathcal{F}^\#, \mathcal{G})\] because \(\mathcal{O} = \mathcal{O}^\#\) in this case.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O}\) be a presheaf of rings on \(\mathcal{C}\) The sheafification functor \[\textit{PMod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}^\#), \quad \mathcal{F} \longmapsto \mathcal{F}^\#\] is exact.

Proof

This is true because it holds for sheafification \(\textit{PAb}(\mathcal{C}) \to \textit{Ab}(\mathcal{C})\). See the discussion in Section 03CM.

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O}_1 \to \mathcal{O}_2\) be a morphism of sheaves of rings on \(\mathcal{C}\). In Section 03CS we defined a restriction functor and a change of rings functor on presheaves of modules associated to this situation.

If \(\mathcal{F}\) is a sheaf of \(\mathcal{O}_2\)-modules then the restriction \(\mathcal{F}_{\mathcal{O}_1}\) of \(\mathcal{F}\) is clearly a sheaf of \(\mathcal{O}_1\)-modules. We obtain the restriction functor \[\textit{Mod}(\mathcal{O}_2) \longrightarrow \textit{Mod}(\mathcal{O}_1)\]

On the other hand, given a sheaf of \(\mathcal{O}_1\)-modules \(\mathcal{G}\) the presheaf of \(\mathcal{O}_2\)-modules \(\mathcal{O}_2 \otimes_{p, \mathcal{O}_1} \mathcal{G}\) is in general not a sheaf. Hence we define the tensor product sheaf \(\mathcal{O}_2 \otimes_{\mathcal{O}_1} \mathcal{G}\) by the formula \[\mathcal{O}_2 \otimes_{\mathcal{O}_1} \mathcal{G} = (\mathcal{O}_2 \otimes_{p, \mathcal{O}_1} \mathcal{G})^\#\] as the sheafification of our construction for presheaves. We obtain the change of rings functor \[\textit{Mod}(\mathcal{O}_1) \longrightarrow \textit{Mod}(\mathcal{O}_2)\]

Lemma

With \(X\), \(\mathcal{O}_1\), \(\mathcal{O}_2\), \(\mathcal{F}\) and \(\mathcal{G}\) as above there exists a canonical bijection \[\Hom_{\mathcal{O}_1}(\mathcal{G}, \mathcal{F}_{\mathcal{O}_1}) = \Hom_{\mathcal{O}_2}( \mathcal{O}_2 \otimes_{\mathcal{O}_1} \mathcal{G}, \mathcal{F} )\] In other words, the restriction and change of rings functors are adjoint to each other.

Proof

This follows from Lemma 03CU and the fact that \(\Hom_{\mathcal{O}_2}( \mathcal{O}_2 \otimes_{\mathcal{O}_1} \mathcal{G}, \mathcal{F} ) = \Hom_{\mathcal{O}_2}( \mathcal{O}_2 \otimes_{p, \mathcal{O}_1} \mathcal{G}, \mathcal{F} )\) because \(\mathcal{F}\) is a sheaf.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O} \to \mathcal{O}'\) be an epimorphism of sheaves of rings. Let \(\mathcal{G}_1, \mathcal{G}_2\) be \(\mathcal{O}'\)-modules. Then \[\Hom_{\mathcal{O}'}(\mathcal{G}_1, \mathcal{G}_2) = \Hom_\mathcal{O}(\mathcal{G}_1, \mathcal{G}_2).\] In other words, the restriction functor \(\textit{Mod}(\mathcal{O}') \to \textit{Mod}(\mathcal{O})\) is fully faithful.

Proof

This is the sheaf version of Algebra, Lemma 08YS and is proved in exactly the same way.

Morphisms of topoi and sheaves of modules

All of this material is completely straightforward. We formulate everything in the case of morphisms of topoi, but of course the results also hold in the case of morphisms of sites.

Lemma

Let \(\mathcal{C}\), \(\mathcal{D}\) be sites. Let \(f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\) be a morphism of topoi. Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}\). Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}\)-modules. There is a natural map of sheaves of sets \[f_*\mathcal{O} \times f_*\mathcal{F} \longrightarrow f_*\mathcal{F}\] which turns \(f_*\mathcal{F}\) into a sheaf of \(f_*\mathcal{O}\)-modules. This construction is functorial in \(\mathcal{F}\).

Proof

Denote \(\mu : \mathcal{O} \times \mathcal{F} \to \mathcal{F}\) the multiplication map. Recall that \(f_*\) (on sheaves of sets) is left exact and hence commutes with products. Hence \(f_*\mu\) is a map as indicated. This proves the lemma.

Lemma

Let \(\mathcal{C}\), \(\mathcal{D}\) be sites. Let \(f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\) be a morphism of topoi. Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{D}\). Let \(\mathcal{G}\) be a sheaf of \(\mathcal{O}\)-modules. There is a natural map of sheaves of sets \[f^{-1}\mathcal{O} \times f^{-1}\mathcal{G} \longrightarrow f^{-1}\mathcal{G}\] which turns \(f^{-1}\mathcal{G}\) into a sheaf of \(f^{-1}\mathcal{O}\)-modules. This construction is functorial in \(\mathcal{G}\).

Proof

Denote \(\mu : \mathcal{O} \times \mathcal{G} \to \mathcal{G}\) the multiplication map. Recall that \(f^{-1}\) (on sheaves of sets) is exact and hence commutes with products. Hence \(f^{-1}\mu\) is a map as indicated. This proves the lemma.

Lemma

Let \(\mathcal{C}\), \(\mathcal{D}\) be sites. Let \(f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\) be a morphism of topoi. Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{D}\). Let \(\mathcal{G}\) be a sheaf of \(\mathcal{O}\)-modules. Let \(\mathcal{F}\) be a sheaf of \(f^{-1}\mathcal{O}\)-modules. Then \[\Mor_{\textit{Mod}(f^{-1}\mathcal{O})}(f^{-1}\mathcal{G}, \mathcal{F}) = \Mor_{\textit{Mod}(\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}).\] Here we use Lemmas 03D2 and 03D1, and we think of \(f_*\mathcal{F}\) as an \(\mathcal{O}\)-module by restriction via \(\mathcal{O} \to f_*f^{-1}\mathcal{O}\).

Proof

First we note that we have \[\Mor_{\textit{Ab}(\mathcal{C})}(f^{-1}\mathcal{G}, \mathcal{F}) = \Mor_{\textit{Ab}(\mathcal{D})}(\mathcal{G}, f_*\mathcal{F}).\] by Sites, Proposition 00YV. Suppose that \(\alpha : f^{-1}\mathcal{G} \to \mathcal{F}\) and \(\beta : \mathcal{G} \to f_*\mathcal{F}\) are morphisms of abelian sheaves which correspond via the formula above. We have to show that \(\alpha\) is \(f^{-1}\mathcal{O}\)-linear if and only if \(\beta\) is \(\mathcal{O}\)-linear. For example, suppose \(\alpha\) is \(f^{-1}\mathcal{O}\)-linear, then clearly \(f_*\alpha\) is \(f_*f^{-1}\mathcal{O}\)-linear, and hence (as restriction is a functor) is \(\mathcal{O}\)-linear. Hence it suffices to prove that the adjunction map \(\mathcal{G} \to f_*f^{-1}\mathcal{G}\) is \(\mathcal{O}\)-linear. Using that both \(f_*\) and \(f^{-1}\) commute with products (on sheaves of sets) this comes down to showing that \[\xymatrix{ \mathcal{O} \times \mathcal{G} \ar[r] \ar[d] & f_*f^{-1}(\mathcal{O} \times \mathcal{G}) \ar[d] \\ \mathcal{G} \ar[r] & f_*f^{-1}\mathcal{G} }\] is commutative. This holds because the adjunction mapping \(\text{id}_{\Sh(\mathcal{D})} \to f_*f^{-1}\) is a transformation of functors. We omit the proof of the implication \(\beta\) linear \(\Rightarrow\) \(\alpha\) linear.

Lemma

Let \(\mathcal{C}\), \(\mathcal{D}\) be sites. Let \(f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\) be a morphism of topoi. Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}\). Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}\)-modules. Let \(\mathcal{G}\) be a sheaf of \(f_*\mathcal{O}\)-modules. Then \[\Mor_{\textit{Mod}(\mathcal{O})}( \mathcal{O} \otimes_{f^{-1}f_*\mathcal{O}} f^{-1}\mathcal{G}, \mathcal{F}) = \Mor_{\textit{Mod}(f_*\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}).\] Here we use Lemmas 03D2 and 03D1, and we use the canonical map \(f^{-1}f_*\mathcal{O} \to \mathcal{O}\) in the definition of the tensor product.

Proof

Note that we have \[\Mor_{\textit{Mod}(\mathcal{O})}( \mathcal{O} \otimes_{f^{-1}f_*\mathcal{O}} f^{-1}\mathcal{G}, \mathcal{F}) = \Mor_{\textit{Mod}(f^{-1}f_*\mathcal{O})}( f^{-1}\mathcal{G}, \mathcal{F}_{f^{-1}f_*\mathcal{O}})\] by Lemma 03CZ. Hence the result follows from Lemma 03D3.

Morphisms of ringed topoi and modules

We have now introduced enough notation so that we are able to define the pullback and pushforward of modules along a morphism of ringed topoi.

Definition

Let \((f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi or ringed sites.

1. Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}_\mathcal{C}\)-modules. We define the pushforward of \(\mathcal{F}\) as the sheaf of \(\mathcal{O}_\mathcal{D}\)-modules which as a sheaf of abelian groups equals \(f_*\mathcal{F}\) and with module structure given by the restriction via \(f^\sharp : \mathcal{O}_\mathcal{D} \to f_*\mathcal{O}_\mathcal{C}\) of the module structure \[f_*\mathcal{O}_\mathcal{C} \times f_*\mathcal{F} \longrightarrow f_*\mathcal{F}\] from Lemma 03D1.

2. Let \(\mathcal{G}\) be a sheaf of \(\mathcal{O}_\mathcal{D}\)-modules. We define the pullback \(f^*\mathcal{G}\) to be the sheaf of \(\mathcal{O}_\mathcal{C}\)-modules defined by the formula \[f^*\mathcal{G} = \mathcal{O}_\mathcal{C} \otimes_{f^{-1}\mathcal{O}_\mathcal{D}} f^{-1}\mathcal{G}\] where the ring map \(f^{-1}\mathcal{O}_\mathcal{D} \to \mathcal{O}_\mathcal{C}\) is \(f^\sharp\), and where the module structure is given by Lemma 03D2.

Thus we have defined functors \[\begin{eqnarray*} f_* : \textit{Mod}(\mathcal{O}_\mathcal{C}) & \longrightarrow & \textit{Mod}(\mathcal{O}_\mathcal{D}) \\ f^* : \textit{Mod}(\mathcal{O}_\mathcal{D}) & \longrightarrow & \textit{Mod}(\mathcal{O}_\mathcal{C}) \end{eqnarray*}\] The final result on these functors is that they are indeed adjoint as expected.

Lemma

Let \((f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi or ringed sites. Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}_\mathcal{C}\)-modules. Let \(\mathcal{G}\) be a sheaf of \(\mathcal{O}_\mathcal{D}\)-modules. There is a canonical bijection \[\Hom_{\mathcal{O}_\mathcal{C}}(f^*\mathcal{G}, \mathcal{F}) = \Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G}, f_*\mathcal{F}).\] In other words: the functor \(f^*\) is the left adjoint to \(f_*\).

Proof

This follows from the work we did before: \[\begin{eqnarray*} \Hom_{\mathcal{O}_\mathcal{C}}(f^*\mathcal{G}, \mathcal{F}) & = & \Mor_{\textit{Mod}(\mathcal{O}_\mathcal{C})}( \mathcal{O}_\mathcal{C} \otimes_{f^{-1}\mathcal{O}_\mathcal{D}} f^{-1}\mathcal{G}, \mathcal{F}) \\ & = & \Mor_{\textit{Mod}(f^{-1}\mathcal{O}_\mathcal{D})}( f^{-1}\mathcal{G}, \mathcal{F}_{f^{-1}\mathcal{O}_\mathcal{D}}) \\ & = & \Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G}, f_*\mathcal{F}). \end{eqnarray*}\] Here we use Lemmas 03CZ and 03D3.

Lemma

\((f, f^\sharp) : (\Sh(\mathcal{C}_1), \mathcal{O}_1) \to (\Sh(\mathcal{C}_2), \mathcal{O}_2)\) and \((g, g^\sharp) : (\Sh(\mathcal{C}_2), \mathcal{O}_2) \to (\Sh(\mathcal{C}_3), \mathcal{O}_3)\) be morphisms of ringed topoi. There are canonical isomorphisms of functors \((g \circ f)_* \cong g_* \circ f_*\) and \((g \circ f)^* \cong f^* \circ g^*\).

Proof

This is clear from the definitions.

The abelian category of sheaves of modules

Let \((\Sh(\mathcal{C}), \mathcal{O})\) be a ringed topos. Let \(\mathcal{F}\), \(\mathcal{G}\) be sheaves of \(\mathcal{O}\)-modules, see Sheaves, Definition 0077. Let \(\varphi, \psi : \mathcal{F} \to \mathcal{G}\) be morphisms of sheaves of \(\mathcal{O}\)-modules. We define \(\varphi + \psi : \mathcal{F} \to \mathcal{G}\) to be the sum of \(\varphi\) and \(\psi\) as morphisms of abelian sheaves. This is clearly again a map of \(\mathcal{O}\)-modules. It is also clear that composition of maps of \(\mathcal{O}\)-modules is bilinear with respect to this addition. Thus \(\textit{Mod}(\mathcal{O})\) is a pre-additive category, see Homology, Definition 00ZY.

We will denote \(0\) the sheaf of \(\mathcal{O}\)-modules which has constant value \(\{0\}\) for all objects \(U\) of \(\mathcal{C}\). Clearly this is both a final and an initial object of \(\textit{Mod}(\mathcal{O})\). Given a morphism of \(\mathcal{O}\)-modules \(\varphi : \mathcal{F} \to \mathcal{G}\) the following are equivalent: (a) \(\varphi\) is zero, (b) \(\varphi\) factors through \(0\), (c) \(\varphi\) is zero on sections over each object \(U\).

Moreover, given a pair \(\mathcal{F}\), \(\mathcal{G}\) of sheaves of \(\mathcal{O}\)-modules we may define the direct sum as \[\mathcal{F} \oplus \mathcal{G} = \mathcal{F} \times \mathcal{G}\] with obvious maps \((i, j, p, q)\) as in Homology, Definition 0102. Thus \(\textit{Mod}(\mathcal{O})\) is an additive category, see Homology, Definition 0104.

Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be a morphism of \(\mathcal{O}\)-modules. We may define \(\Ker(\varphi)\) to be the kernel of \(\varphi\) as a map of abelian sheaves. By Section 03CM this is the subsheaf of \(\mathcal{F}\) with sections \[\Ker(\varphi)(U) = \{ s \in \mathcal{F}(U) \mid \varphi(s) = 0 \text{ in } \mathcal{G}(U)\}\] for all objects \(U\) of \(\mathcal{C}\). It is easy to see that this is indeed a kernel in the category of \(\mathcal{O}\)-modules. In other words, a morphism \(\alpha : \mathcal{H} \to \mathcal{F}\) factors through \(\Ker(\varphi)\) if and only if \(\varphi \circ \alpha = 0\).

Similarly, we define \(\Coker(\varphi)\) as the cokernel of \(\varphi\) as a map of abelian sheaves. There is a unique multiplication map \[\mathcal{O} \times \Coker(\varphi) \longrightarrow \Coker(\varphi)\] such that the map \(\mathcal{G} \to \Coker(\varphi)\) becomes a morphism of \(\mathcal{O}\)-modules (verification omitted). The map \(\mathcal{G} \to \Coker(\varphi)\) is surjective (as a map of sheaves of sets, see Section 03CM). To show that \(\Coker(\varphi)\) is a cokernel in \(\textit{Mod}(\mathcal{O})\), note that if \(\beta : \mathcal{G} \to \mathcal{H}\) is a morphism of \(\mathcal{O}\)-modules such that \(\beta \circ \varphi\) is zero, then you get for every object \(U\) of \(\mathcal{C}\) a map induced by \(\beta\) from \(\mathcal{G}(U)/\varphi(\mathcal{F}(U))\) into \(\mathcal{H}(U)\). By the universal property of sheafification (see Sheaves, Lemma 0089) we obtain a canonical map \(\Coker(\varphi) \to \mathcal{H}\) such that the original \(\beta\) is equal to the composition \(\mathcal{G} \to \Coker(\varphi) \to \mathcal{H}\). The morphism \(\Coker(\varphi) \to \mathcal{H}\) is unique because of the surjectivity mentioned above.

Lemma

Let \((\Sh(\mathcal{C}), \mathcal{O})\) be a ringed topos. The category \(\textit{Mod}(\mathcal{O})\) is an abelian category. The forgetful functor \(\textit{Mod}(\mathcal{O}) \to \textit{Ab}(\mathcal{C})\) is exact, hence kernels, cokernels and exactness of \(\mathcal{O}\)-modules, correspond to the corresponding notions for abelian sheaves.

Proof

Above we have seen that \(\textit{Mod}(\mathcal{O})\) is an additive category, with kernels and cokernels and that \(\textit{Mod}(\mathcal{O}) \to \textit{Ab}(\mathcal{C})\) preserves kernels and cokernels. By Homology, Definition 0109 we have to show that image and coimage agree. This is clear because it is true in \(\textit{Ab}(\mathcal{C})\). The lemma follows.

Lemma

Let \((\Sh(\mathcal{C}), \mathcal{O})\) be a ringed topos. All limits and colimits exist in \(\textit{Mod}(\mathcal{O})\) and the forgetful functor \(\textit{Mod}(\mathcal{O}) \to \textit{Ab}(\mathcal{C})\) commutes with them. Moreover, filtered colimits are exact.

Proof

The final statement follows from the first as filtered colimits are exact in \(\textit{Ab}(\mathcal{C})\) by Lemma 03CO. Let \(\mathcal{I} \to \textit{Mod}(\mathcal{C})\), \(i \mapsto \mathcal{F}_i\) be a diagram. Let \(\lim_i \mathcal{F}_i\) be the limit of the diagram in \(\textit{Ab}(\mathcal{C})\). By the description of this limit in Lemma 03CO we see immediately that there exists a multiplication \[\mathcal{O} \times \lim_i \mathcal{F}_i \longrightarrow \lim_i \mathcal{F}_i\] which turns \(\lim_i \mathcal{F}_i\) into a sheaf of \(\mathcal{O}\)-modules. It is easy to see that this is the limit of the diagram in \(\textit{Mod}(\mathcal{C})\). Let \(\colim_i \mathcal{F}_i\) be the colimit of the diagram in \(\textit{PAb}(\mathcal{C})\). By the description of this colimit in the proof of Lemma 03CL we see immediately that there exists a multiplication \[\mathcal{O} \times \colim_i \mathcal{F}_i \longrightarrow \colim_i \mathcal{F}_i\] which turns \(\colim_i \mathcal{F}_i\) into a presheaf of \(\mathcal{O}\)-modules. Applying sheafification we get a sheaf of \(\mathcal{O}\)-modules \((\colim_i \mathcal{F}_i)^\#\), see Lemma 03CY. It is easy to see that \((\colim_i \mathcal{F}_i)^\#\) is the colimit of the diagram in \(\textit{Mod}(\mathcal{O})\), and by Lemma 03CO forgetting the \(\mathcal{O}\)-module structure is the colimit in \(\textit{Ab}(\mathcal{C})\).

The existence of limits and colimits allows us to consider exactness properties of functors defined on the category of \(\mathcal{O}\)-modules in terms of limits and colimits, as in Categories, Section 0033. See Homology, Lemma 010N for a description of exactness properties in terms of short exact sequences.

Lemma

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

1. The functor \(f_*\) is left exact. In fact it commutes with all limits.

2. The functor \(f^*\) is right exact. In fact it commutes with all colimits.

Proof

This is true because \((f^*, f_*)\) is an adjoint pair of functors, see Lemma 03D7. See Categories, Section 0036.

Lemma

Let \(\mathcal{C}\) be a site. If \(\{p_i\}_{i \in I}\) is a conservative family of points, then we may check exactness of a sequence of abelian sheaves on the stalks at the points \(p_i\), \(i \in I\). If \(\mathcal{C}\) has enough points, then exactness of a sequence of abelian sheaves may be checked on stalks.

Proof

This is immediate from Sites, Lemma 00YL.

Exactness of pushforward

Some technical lemmas concerning exactness properties of pushforward.

Lemma

Let \(f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\) be a morphism of topoi. The following are equivalent:

1. \(f^{-1}f_*\mathcal{F} \to \mathcal{F}\) is surjective for all \(\mathcal{F}\) in \(\textit{Ab}(\mathcal{C})\), and

2. \(f_* : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})\) reflects surjections.

In this case the functor \(f_* : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})\) is faithful.

Proof

Assume (1). Suppose that \(a : \mathcal{F} \to \mathcal{F}'\) is a map of abelian sheaves on \(\mathcal{C}\) such that \(f_*a\) is surjective. As \(f^{-1}\) is exact this implies that \(f^{-1}f_*a : f^{-1}f_*\mathcal{F} \to f^{-1}f_*\mathcal{F}'\) is surjective. Combined with (1) this implies that \(a\) is surjective. This means that (2) holds.

Assume (2). Let \(\mathcal{F}\) be an abelian sheaf on \(\mathcal{C}\). We have to show that the map \(f^{-1}f_*\mathcal{F} \to \mathcal{F}\) is surjective. By (2) it suffices to show that \(f_*f^{-1}f_*\mathcal{F} \to f_*\mathcal{F}\) is surjective. And this is true because there is a canonical map \(f_*\mathcal{F} \to f_*f^{-1}f_*\mathcal{F}\) which is a one-sided inverse.

We omit the proof of the final assertion.

Lemma

Let \(f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\) be a morphism of topoi. Assume at least one of the following properties holds

1. \(f_*\) transforms surjections of sheaves of sets into surjections,

2. \(f_*\) transforms surjections of abelian sheaves into surjections,

3. \(f_*\) commutes with coequalizers on sheaves of sets,

4. \(f_*\) commutes with pushouts on sheaves of sets,

Then \(f_* : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})\) is exact.

Proof

Since \(f_* : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})\) is a right adjoint we already know that it transforms a short exact sequence \(0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\) of abelian sheaves on \(\mathcal{C}\) into an exact sequence \[0 \to f_*\mathcal{F}_1 \to f_*\mathcal{F}_2 \to f_*\mathcal{F}_3\] see Categories, Sections 0033 and 0036 and Homology, Section 010M. Hence it suffices to prove that the map \(f_*\mathcal{F}_2 \to f_*\mathcal{F}_3\) is surjective. If (1), (2) holds, then this is clear from the definitions. By Sites, Lemma 04D6 we see that either (3) or (4) formally implies (1), hence in these cases we are done also.

Lemma

Let \(f : \mathcal{D} \to \mathcal{C}\) be a morphism of sites associated to the continuous functor \(u : \mathcal{C} \to \mathcal{D}\). Assume \(u\) is almost cocontinuous. Then

1. \(f_* : \textit{Ab}(\mathcal{D}) \to \textit{Ab}(\mathcal{C})\) is exact.

2. if \(f^\sharp : f^{-1}\mathcal{O}_\mathcal{C} \to \mathcal{O}_\mathcal{D}\) is given so that \(f\) becomes a morphism of ringed sites, then \(f_* : \textit{Mod}(\mathcal{O}_\mathcal{D}) \to \textit{Mod}(\mathcal{O}_\mathcal{C})\) is exact.

Proof

Part (2) follows from part (1) by Lemma 03DB. Part (1) follows from Sites, Lemmas 04BA and 04D6.

Exactness of lower shriek

Let \(u : \mathcal{C} \to \mathcal{D}\) be a functor between sites. Assume that

1. \(u\) is cocontinuous, and

2. \(u\) is continuous.

Let \(g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\) be the morphism of topoi associated with \(u\), see Sites, Lemma 00XO. Recall that \(g^{-1} = u^p\), i.e., \(g^{-1}\) is given by the simple formula \((g^{-1}\mathcal{G})(U) = \mathcal{G}(u(U))\), see Sites, Lemma 00XR. We would like to show that \(g^{-1} : \textit{Ab}(\mathcal{D}) \to \textit{Ab}(\mathcal{C})\) has a left adjoint \(g_!\). By Sites, Lemma 00XR the functor \(g^{Sh}_! = (u_p\ )^\#\) is a left adjoint on sheaves of sets. Moreover, we know that \(g^{Sh}_!\mathcal{F}\) is the sheaf associated to the presheaf \[V \longmapsto \colim_{V \to u(U)} \mathcal{F}(U)\] where the colimit is over \((\mathcal{I}_V^u)^{opp}\) and is taken in the category of sets. Hence the following definition is natural.

Definition

With \(u : \mathcal{C} \to \mathcal{D}\) satisfying (a), (b) above. For \(\mathcal{F} \in \textit{PAb}(\mathcal{C})\) we define \(g_{p!}\mathcal{F}\) as the presheaf \[V \longmapsto \colim_{V \to u(U)} \mathcal{F}(U)\] with colimits over \((\mathcal{I}_V^u)^{opp}\) taken in \(\textit{Ab}\). For \(\mathcal{F} \in \textit{PAb}(\mathcal{C})\) we set \(g_!\mathcal{F} = (g_{p!}\mathcal{F})^\#\).

The reason for being so explicit with this is that the functors \(g^{Sh}_!\) and \(g_!\) are different. Whenever we use both we have to be careful to make the distinction clear.

Lemma

The functor \(g_{p!}\) is a left adjoint to the functor \(u^p\). The functor \(g_!\) is a left adjoint to the functor \(g^{-1}\). In other words the formulas \[\begin{align*} \Mor_{\textit{PAb}(\mathcal{C})}(\mathcal{F}, u^p\mathcal{G}) & = \Mor_{\textit{PAb}(\mathcal{D})}(g_{p!}\mathcal{F}, \mathcal{G}), \\ \Mor_{\textit{Ab}(\mathcal{C})}(\mathcal{F}, g^{-1}\mathcal{G}) & = \Mor_{\textit{Ab}(\mathcal{D})}(g_!\mathcal{F}, \mathcal{G}) \end{align*}\] hold bifunctorially in \(\mathcal{F}\) and \(\mathcal{G}\).

Proof

The second formula follows formally from the first, since if \(\mathcal{F}\) and \(\mathcal{G}\) are abelian sheaves then \[\begin{align*} \Mor_{\textit{Ab}(\mathcal{C})}(\mathcal{F}, g^{-1}\mathcal{G}) & = \Mor_{\textit{PAb}(\mathcal{D})}(g_{p!}\mathcal{F}, \mathcal{G}) \\ & = \Mor_{\textit{Ab}(\mathcal{D})}(g_!\mathcal{F}, \mathcal{G}) \end{align*}\] by the universal property of sheafification.

To prove the first formula, let \(\mathcal{F}\), \(\mathcal{G}\) be abelian presheaves. To prove the lemma we will construct maps from the group on the left to the group on the right and omit the verification that these are mutually inverse.

Note that there is a canonical map of abelian presheaves \(\mathcal{F} \to u^pg_{p!}\mathcal{F}\) which on sections over \(U\) is the natural map \(\mathcal{F}(U) \to \colim_{u(U) \to u(U')} \mathcal{F}(U')\), see Sites, Lemma 00VD. Given a map \(\alpha : g_{p!}\mathcal{F} \to \mathcal{G}\) we get \(u^p\alpha : u^pg_{p!}\mathcal{F} \to u^p\mathcal{G}\). which we can precompose by the map \(\mathcal{F} \to u^pg_{p!}\mathcal{F}\).

Note that there is a canonical map of abelian presheaves \(g_{p!}u^p\mathcal{G} \to \mathcal{G}\) which on sections over \(V\) is the natural map \(\colim_{V \to u(U)} \mathcal{G}(u(U)) \to \mathcal{G}(V)\). It maps a section \(s \in \mathcal{G}(u(U))\) in the summand corresponding to \(t : V \to u(U)\) to \(t^*s \in \mathcal{G}(V)\). Hence, given a map \(\beta : \mathcal{F} \to u^p\mathcal{G}\) we get a map \(g_{p!}\beta : g_{p!}\mathcal{F} \to g_{p!}u^p\mathcal{G}\) which we can postcompose with the map \(g_{p!}u^p\mathcal{G} \to \mathcal{G}\) above.

Lemma

Let \(\mathcal{C}\) and \(\mathcal{D}\) be sites. Let \(u : \mathcal{C} \to \mathcal{D}\) be a functor. Assume that

1. \(u\) is cocontinuous,

2. \(u\) is continuous, and

3. fibre products and equalizers exist in \(\mathcal{C}\) and \(u\) commutes with them.

In this case the functor \(g_! : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})\) is exact.

Proof

Compare with Sites, Lemma 00XS. Assume (a), (b), and (c). We already know that \(g_!\) is right exact as it is a left adjoint, see Categories, Lemma 0039 and Homology, Section 010M. We have \(g_! = (g_{p!}\ )^\#\). We have to show that \(g_!\) transforms injective maps of abelian sheaves into injective maps of abelian presheaves. Recall that sheafification of abelian presheaves is exact, see Lemma 03CO. Thus it suffices to show that \(g_{p!}\) transforms injective maps of abelian presheaves into injective maps of abelian presheaves. To do this it suffices that colimits over the categories \((\mathcal{I}_V^u)^{opp}\) of Sites, Section 00VC transform injective maps between diagrams into injections. This follows from Sites, Lemma 00X4 and Algebra, Lemma 04B0.

Lemma

Let \(\mathcal{C}\) and \(\mathcal{D}\) be sites. Let \(u : \mathcal{C} \to \mathcal{D}\) be a functor. Assume that

1. \(u\) is cocontinuous,

2. \(u\) is continuous, and

3. \(u\) is fully faithful.

For \(g_!, g^{-1}, g_*\) as above the canonical maps \(\mathcal{F} \to g^{-1}g_!\mathcal{F}\) and \(g^{-1}g_*\mathcal{F} \to \mathcal{F}\) are isomorphisms for all abelian sheaves \(\mathcal{F}\) on \(\mathcal{C}\).

Proof

The map \(g^{-1}g_*\mathcal{F} \to \mathcal{F}\) is an isomorphism by Sites, Lemma 00XT and the fact that pullback and pushforward of abelian sheaves agrees with pullback and pushforward on the underlying sheaves of sets.

Pick \(U \in \Ob(\mathcal{C})\). We will show that \(g^{-1}g_!\mathcal{F}(U) = \mathcal{F}(U)\). First, note that \(g^{-1}g_!\mathcal{F}(U) = g_!\mathcal{F}(u(U))\). Hence it suffices to show that \(g_!\mathcal{F}(u(U)) = \mathcal{F}(U)\). We know that \(g_!\mathcal{F}\) is the (abelian) sheaf associated to the presheaf \(g_{p!}\mathcal{F}\) which is defined by the rule \[V \longmapsto \colim_{V \to u(U')} \mathcal{F}(U')\] with colimit taken in \(\textit{Ab}\). If \(V = u(U)\), then, as \(u\) is fully faithful this colimit is over \(U \to U'\). Hence we conclude that \(g_{p!}\mathcal{F}(u(U)) = \mathcal{F}(U)\). Since \(u\) is cocontinuous and continuous any covering of \(u(U)\) in \(\mathcal{D}\) can be refined by a covering (!) \(\{u(U_i) \to u(U)\}\) of \(\mathcal{D}\) where \(\{U_i \to U\}\) is a covering in \(\mathcal{C}\). This implies that \((g_{p!}\mathcal{F})^+(u(U)) = \mathcal{F}(U)\) also, since in the colimit defining the value of \((g_{p!}\mathcal{F})^+\) on \(u(U)\) we may restrict to the cofinal system of coverings \(\{u(U_i) \to u(U)\}\) as above. Hence we see that \((g_{p!}\mathcal{F})^+(u(U)) = \mathcal{F}(U)\) for all objects \(U\) of \(\mathcal{C}\) as well. Repeating this argument one more time gives the equality \((g_{p!}\mathcal{F})^\#(u(U)) = \mathcal{F}(U)\) for all objects \(U\) of \(\mathcal{C}\). This produces the desired equality \(g^{-1}g_!\mathcal{F} = \mathcal{F}\).

Remark

In general the functor \(g_!\) cannot be extended to categories of modules in case \(g\) is (part of) a morphism of ringed topoi. Namely, given any ring map \(A \to B\) the functor \(M \mapsto B \otimes_A M\) has a right adjoint (restriction) but not in general a left adjoint (because its existence would imply that \(A \to B\) is flat). We will see in Section 03DH below that it is possible to define \(j_!\) on sheaves of modules in the case of a localization of sites. We will discuss this in greater generality in Section 0796 below.

Lemma

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

1. If \(u\) has a left adjoint \(w\), then \(g_!\) agrees with \(g_!^{\Sh}\) on underlying sheaves of sets and \(g_!\) is exact.

2. If in addition \(w\) is cocontinuous, then \(g_! = h^{-1}\) and \(g^{-1} = h_*\) where \(h : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})\) is the morphism of topoi associated to \(w\).

Proof

This Lemma is the analogue of Sites, Lemma 08NH. From Sites, Lemma 09VQ we see that the categories \(\mathcal{I}_V^u\) have an initial object. Thus the underlying set of a colimit of a system of abelian groups over \((\mathcal{I}_V^u)^{opp}\) is the colimit of the underlying sets. Whence the agreement of \(g_!^{\Sh}\) and \(g_!\) by our construction of \(g_!\) in Definition 04BF. The exactness and (2) follow immediately from the corresponding statements of Sites, Lemma 08NH.

Global types of modules

Definition

Let \((\Sh(\mathcal{C}), \mathcal{O})\) be a ringed topos. Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}\)-modules.

1. We say \(\mathcal{F}\) is a free \(\mathcal{O}\)-module if \(\mathcal{F}\) is isomorphic as an \(\mathcal{O}\)-module to a sheaf of the form \(\bigoplus_{i \in I} \mathcal{O}\).

2. We say \(\mathcal{F}\) is finite free if \(\mathcal{F}\) is isomorphic as an \(\mathcal{O}\)-module to a sheaf of the form \(\bigoplus_{i \in I} \mathcal{O}\) with a finite index set \(I\).

3. We say \(\mathcal{F}\) is generated by global sections if there exists a surjection \[\bigoplus\nolimits_{i \in I} \mathcal{O} \longrightarrow \mathcal{F}\] from a free \(\mathcal{O}\)-module onto \(\mathcal{F}\).

4. Given \(r \geq 0\) we say \(\mathcal{F}\) is generated by \(r\) global sections if there exists a surjection \(\mathcal{O}^{\oplus r} \to \mathcal{F}\).

5. We say \(\mathcal{F}\) is generated by finitely many global sections if it is generated by \(r\) global sections for some \(r \geq 0\).

6. We say \(\mathcal{F}\) has a global presentation if there exists an exact sequence \[\bigoplus\nolimits_{j \in J} \mathcal{O} \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O} \longrightarrow \mathcal{F} \longrightarrow 0\] of \(\mathcal{O}\)-modules.

7. We say \(\mathcal{F}\) has a global finite presentation if there exists an exact sequence \[\bigoplus\nolimits_{j \in J} \mathcal{O} \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O} \longrightarrow \mathcal{F} \longrightarrow 0\] of \(\mathcal{O}\)-modules with \(I\) and \(J\) finite sets.

Note that for any set \(I\) the direct sum \(\bigoplus_{i \in I} \mathcal{O}\) exists (Lemma 03DB) and is the sheafification of the presheaf \(U \mapsto \bigoplus_{i \in I} \mathcal{O}(U)\). This module is called the free \(\mathcal{O}\)-module on the set \(I\).

Lemma

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

1. If \(\mathcal{F}\) is free then \(f^*\mathcal{F}\) is free.

2. If \(\mathcal{F}\) is finite free then \(f^*\mathcal{F}\) is finite free.

3. If \(\mathcal{F}\) is generated by global sections then \(f^*\mathcal{F}\) is generated by global sections.

4. Given \(r \geq 0\) if \(\mathcal{F}\) is generated by \(r\) global sections, then \(f^*\mathcal{F}\) is generated by \(r\) global sections.

5. If \(\mathcal{F}\) is generated by finitely many global sections then \(f^*\mathcal{F}\) is generated by finitely many global sections.

6. If \(\mathcal{F}\) has a global presentation then \(f^*\mathcal{F}\) has a global presentation.

7. If \(\mathcal{F}\) has a finite global presentation then \(f^*\mathcal{F}\) has a finite global presentation.

Proof

This is true because \(f^*\) commutes with arbitrary colimits (Lemma 03DC) and \(f^*\mathcal{O}_\mathcal{D} = \mathcal{O}_\mathcal{C}\).

Intrinsic properties of modules

Let \(\mathcal{P}\) be a property of sheaves of modules on ringed topoi. We say \(\mathcal{P}\) is an intrinsic property if we have \(\mathcal{P}(\mathcal{F}) \Leftrightarrow \mathcal{P}(f^*\mathcal{F})\) whenever \((f, f^\sharp) : (\Sh(\mathcal{C}'), \mathcal{O}') \to (\Sh(\mathcal{C}), \mathcal{O})\) is an equivalence of ringed topoi. For example, the property of being free is intrinsic. Indeed, the free \(\mathcal{O}\)-module on the set \(I\) is characterized by the property that \[\Mor_{\textit{Mod}(\mathcal{O})}( \bigoplus\nolimits_{i \in I} \mathcal{O}, \mathcal{F}) = \prod\nolimits_{i \in I} \Mor_{\Sh(\mathcal{C})}(\{*\}, \mathcal{F})\] for a variable \(\mathcal{F}\) in \(\textit{Mod}(\mathcal{O})\). Alternatively, we can also use Lemma 03DF to see that being free is intrinsic. In fact, each of the properties defined in Definition 03DE is intrinsic for the same reason. How will we go about defining other intrinsic properties of \(\mathcal{O}\)-modules?

The upshot of Lemma 03CR is the following: Suppose you want to define an intrinsic property \(\mathcal{P}\) of an \(\mathcal{O}\)-module on a topos. Then you can proceed as follows:

1. Given any site \(\mathcal{C}\), any sheaf of rings \(\mathcal{O}\) on \(\mathcal{C}\) and any \(\mathcal{O}\)-module \(\mathcal{F}\) define the corresponding property \(\mathcal{P}(\mathcal{C}, \mathcal{O}, \mathcal{F})\).

2. For any pair of sites \(\mathcal{C}\), \(\mathcal{C}'\), any special cocontinuous functor \(u : \mathcal{C} \to \mathcal{C}'\), any sheaf of rings \(\mathcal{O}\) on \(\mathcal{C}\) any \(\mathcal{O}\)-module \(\mathcal{F}\), show that \[\mathcal{P}(\mathcal{C}, \mathcal{O}, \mathcal{F}) \Leftrightarrow \mathcal{P}(\mathcal{C}', g_*\mathcal{O}, g_*\mathcal{F})\] where \(g : \Sh(\mathcal{C}) \to \Sh(\mathcal{C}')\) is the equivalence of topoi associated to \(u\).

In this case, given any ringed topos \((\Sh(\mathcal{C}), \mathcal{O})\) and any sheaf of \(\mathcal{O}\)-modules \(\mathcal{F}\) we simply say that \(\mathcal{F}\) has property \(\mathcal{P}\) if \(\mathcal{P}(\mathcal{C}, \mathcal{O}, \mathcal{F})\) is true. And Lemma 03CR combined with (2) above guarantees that this is well defined.

Moreover, the same Lemma 03CR also guarantees that if in addition

1. For any morphism of ringed sites \((f, f^\sharp) : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D})\) such that \(f\) is given by a functor \(u : \mathcal{D} \to \mathcal{C}\) satisfying the assumptions of Sites, Proposition 00X6, and any \(\mathcal{O}_\mathcal{D}\)-module \(\mathcal{G}\) we have \[\mathcal{P}(\mathcal{D}, \mathcal{O}_\mathcal{D}, \mathcal{G}) \Rightarrow \mathcal{P}(\mathcal{C}, \mathcal{O}_\mathcal{C}, f^*\mathcal{G})\]

then it is true that \(\mathcal{P}\) is preserved under pullback of modules w.r.t. arbitrary morphisms of ringed topoi.

We will use this method in the following sections to see that: locally free, locally generated by sections, locally generated by \(r\) sections, finite type, finite presentation, quasi-coherent, and coherent are intrinsic properties of modules.

Perhaps a more satisfying method would be to find an intrinsic definition of these notions, rather than the laborious process sketched here. On the other hand, in many geometric situations where we want to apply these definitions we are given a definite ringed site, and a definite sheaf of modules, and it is nice to have a definition already adapted to this language.

Localization of ringed sites

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(U \in \Ob(\mathcal{C})\). We explain the counterparts of the results in Sites, Section 00XZ in this setting.

Denote \(\mathcal{O}_U = j_U^{-1}\mathcal{O}\) the restriction of \(\mathcal{O}\) to the site \(\mathcal{C}/U\). It is described by the simple rule \(\mathcal{O}_U(V/U) = \mathcal{O}(V)\). With this notation the localization morphism \(j_U\) becomes a morphism of ringed topoi \[(j_U, j_U^\sharp) : (\Sh(\mathcal{C}/U), \mathcal{O}_U) \longrightarrow (\Sh(\mathcal{C}), \mathcal{O})\] namely, we take \(j_U^\sharp : j_U^{-1}\mathcal{O} \to \mathcal{O}_U\) the identity map. Moreover, we obtain the following descriptions for pushforward and pullback of modules.

Definition

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(U \in \Ob(\mathcal{C})\).

1. The ringed site \((\mathcal{C}/U, \mathcal{O}_U)\) is called the localization of the ringed site \((\mathcal{C}, \mathcal{O})\) at the object \(U\).

2. The morphism of ringed topoi \((j_U, j_U^\sharp) : (\Sh(\mathcal{C}/U), \mathcal{O}_U) \to (\Sh(\mathcal{C}), \mathcal{O})\) is called the localization morphism.

3. The functor \(j_{U*} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})\) is called the direct image functor.

4. For a sheaf of \(\mathcal{O}\)-modules \(\mathcal{F}\) on \(\mathcal{C}\) the sheaf \(j_U^*\mathcal{F}\) is called the restriction of \(\mathcal{F}\) to \(\mathcal{C}/U\). We will sometimes denote it by \(\mathcal{F}|_{\mathcal{C}/U}\) or even \(\mathcal{F}|_U\). It is described by the simple rule \(j_U^*(\mathcal{F})(X/U) = \mathcal{F}(X)\).

5. The left adjoint \(j_{U!} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})\) of restriction is called extension by zero. It exists and is exact by Lemmas 03DI and 03DJ.

As in the topological case, see Sheaves, Section 009Z, the extension by zero \(j_{U!}\) functor is different from extension by the empty set \(j_{U!}\) defined on sheaves of sets. Here is the lemma defining extension by zero.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(U \in \Ob(\mathcal{C})\). The restriction functor \(j_U^* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_U)\) has a left adjoint \(j_{U!} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})\). So \[\Mor_{\textit{Mod}(\mathcal{O}_U)}(\mathcal{G}, j_U^*\mathcal{F}) = \Mor_{\textit{Mod}(\mathcal{O})}(j_{U!}\mathcal{G}, \mathcal{F})\] for \(\mathcal{F} \in \Ob(\textit{Mod}(\mathcal{O}))\) and \(\mathcal{G} \in \Ob(\textit{Mod}(\mathcal{O}_U))\). Moreover, the extension by zero \(j_{U!}\mathcal{G}\) of \(\mathcal{G}\) is the sheaf associated to the presheaf \[V \longmapsto \bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U)\] with obvious restriction mappings and an obvious \(\mathcal{O}\)-module structure.

Proof

The \(\mathcal{O}\)-module structure on the presheaf is defined as follows. If \(f \in \mathcal{O}(V)\) and \(s \in \mathcal{G}(V \xrightarrow{\varphi} U)\), then we define \(f \cdot s = fs\) where \(f \in \mathcal{O}_U(\varphi : V \to U) = \mathcal{O}(V)\) (because \(\mathcal{O}_U\) is the restriction of \(\mathcal{O}\) to \(\mathcal{C}/U\)).

Similarly, let \(\alpha : \mathcal{G} \to \mathcal{F}|_U\) be a morphism of \(\mathcal{O}_U\)-modules. In this case we can define a map from the presheaf of the lemma into \(\mathcal{F}\) by mapping \[\bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U) \longrightarrow \mathcal{F}(V)\] by the rule that \(s \in \mathcal{G}(V \xrightarrow{\varphi} U)\) maps to \(\alpha(s) \in \mathcal{F}(V)\). It is clear that this is \(\mathcal{O}\)-linear, and hence induces a morphism of \(\mathcal{O}\)-modules \(\alpha' : j_{U!}\mathcal{G} \to \mathcal{F}\) by the properties of sheafification of modules (Lemma 03CY).

Conversely, let \(\beta : j_{U!}\mathcal{G} \to \mathcal{F}\) by a map of \(\mathcal{O}\)-modules. Recall from Sites, Section 00XZ that there exists an extension by the empty set \(j^{Sh}_{U!} : \Sh(\mathcal{C}/U) \to \Sh(\mathcal{C})\) on sheaves of sets which is left adjoint to \(j_U^{-1}\). Moreover, \(j^{Sh}_{U!}\mathcal{G}\) is the sheaf associated to the presheaf \[V \longmapsto \coprod\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U)\] Hence there is a natural map \(j^{Sh}_{U!}\mathcal{G} \to j_{U!}\mathcal{G}\) of sheaves of sets. Hence precomposing \(\beta\) by this map we get a map of sheaves of sets \(j^{Sh}_{U!}\mathcal{G} \to \mathcal{F}\) which by adjunction corresponds to a map of sheaves of sets \(\beta' : \mathcal{G} \to \mathcal{F}|_U\). We claim that \(\beta'\) is \(\mathcal{O}_U\)-linear. Namely, suppose that \(\varphi : V \to U\) is an object of \(\mathcal{C}/U\) and that \(s, s' \in \mathcal{G}(\varphi : V \to U)\), and \(f \in \mathcal{O}(V) = \mathcal{O}_U(\varphi : V \to U)\). Then by the discussion above we see that \(\beta'(s + s')\), resp. \(\beta'(fs)\) in \(\mathcal{F}|_U(\varphi : V \to U)\) correspond to \(\beta(s + s')\), resp. \(\beta(fs)\) in \(\mathcal{F}(V)\). Since \(\beta\) is a homomorphism we conclude.

To conclude the proof of the lemma we have to show that the constructions \(\alpha \mapsto \alpha'\) and \(\beta \mapsto \beta'\) are mutually inverse. We omit the verifications.

Note that we have in the situation of Definition 04IX we have [0G1V]\[\begin{equation} \Hom_\mathcal{O}(j_{U!}\mathcal{O}_U, \mathcal{F}) = \Hom_{\mathcal{O}_U}(\mathcal{O}_U, j_U^*\mathcal{F}) = \mathcal{F}(U) \end{equation}\] for every \(\mathcal{O}\)-module \(\mathcal{F}\). Namely, the first equality holds by the adjointness of \(j_{U!}\) and \(j_U^*\) and the second because \(\Hom_{\mathcal{O}_U}(\mathcal{O}_U, j_U^*\mathcal{F}) = j_U^*\mathcal{F}(U/U) = \mathcal{F}|_U(U/U) = \mathcal{F}(U)\).

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(U \in \Ob(\mathcal{C})\). The functor \(j_{U!} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})\) is exact.

Proof

Since \(j_{U!}\) is a left adjoint to \(j_U^*\) we see that it is right exact (see Categories, Lemma 0039 and Homology, Section 010M). Hence it suffices to show that if \(\mathcal{G}_1 \to \mathcal{G}_2\) is an injective map of \(\mathcal{O}_U\)-modules, then \(j_{U!}\mathcal{G}_1 \to j_{U!}\mathcal{G}_2\) is injective. The map on sections of presheaves over an object \(V\) (as in Lemma 03DI) is the map \[\bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{G}_1(V \xrightarrow{\varphi} U) \longrightarrow \bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{G}_2(V \xrightarrow{\varphi} U)\] which is injective by assumption. Since sheafification is exact by Lemma 03EI we conclude \(j_{U!}\mathcal{G}_1 \to j_{U!}\mathcal{G}_2\) is injective and we win.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(U \in \Ob(\mathcal{C})\). A complex of \(\mathcal{O}_U\)-modules \(\mathcal{G}_1 \to \mathcal{G}_2 \to \mathcal{G}_3\) is exact if and only if \(j_{U!}\mathcal{G}_1 \to j_{U!}\mathcal{G}_2 \to j_{U!}\mathcal{G}_3\) is exact as a sequence of \(\mathcal{O}\)-modules.

Proof

We already know that \(j_{U!}\) is exact, see Lemma 03DJ. Thus it suffices to show that \(j_{U!} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})\) reflects injections and surjections.

For every \(\mathcal{G}\) in \(\textit{Mod}(\mathcal{O}_U)\) we have the unit \(\mathcal{G} \to j_U^*j_{U!}\mathcal{G}\) of the adjunction. We claim this map is an injection of sheaves. Namely, looking at the construction of Lemma 03DI we see that this map is the sheafification of the rule sending the object \(V/U\) of \(\mathcal{C}/U\) to the injective map \[\mathcal{G}(V/U) \longrightarrow \bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U)\] given by the inclusion of the summand corresponding to the structure morphism \(V \to U\). Since sheafification is exact the claim follows. Some details omitted.

If \(\mathcal{G} \to \mathcal{G}'\) is a map of \(\mathcal{O}_U\)-modules with \(j_{U!}\mathcal{G} \to j_{U!}\mathcal{G}'\) injective, then \(j_U^*j_{U!}\mathcal{G} \to j_U^*j_{U!}\mathcal{G}'\) is injective (restriction is exact), hence \(\mathcal{G} \to j_U^*j_{U!}\mathcal{G}'\) is injective, hence \(\mathcal{G} \to \mathcal{G}'\) is injective. We conclude that \(j_{U!}\) reflects injections.

Let \(a : \mathcal{G} \to \mathcal{G}'\) be a map of \(\mathcal{O}_U\)-modules such that \(j_{U!}\mathcal{G} \to j_{U!}\mathcal{G}'\) is surjective. Let \(\mathcal{H}\) be the cokernel of \(a\). Then \(j_{U!}\mathcal{H} = 0\) as \(j_{U!}\) is exact. By the above the map \(\mathcal{H} \to j^*_U j_{U!}\mathcal{H}\) is injective. Hence \(\mathcal{H} = 0\) as desired.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(f : V \to U\) be a morphism of \(\mathcal{C}\). Then there exists a commutative diagram \[\xymatrix{ (\Sh(\mathcal{C}/V), \mathcal{O}_V) \ar[rd]_{(j_V, j_V^\sharp)} \ar[rr]_{(j, j^\sharp)} & & (\Sh(\mathcal{C}/U), \mathcal{O}_U) \ar[ld]^{(j_U, j_U^\sharp)} \\ & (\Sh(\mathcal{C}), \mathcal{O}) & }\] of ringed topoi. Here \((j, j^\sharp)\) is the localization morphism associated to the object \(V/U\) of the ringed site \((\mathcal{C}/U, \mathcal{O}_U)\).

Proof

The only thing to check is that \(j_V^\sharp = j^\sharp \circ j^{-1}(j_U^\sharp)\), since everything else follows directly from Sites, Lemma 03EH and Sites, Equation (04IK). We omit the verification of the equality.

Remark

In the situation of Lemma 03DI the diagram \[\xymatrix{ \textit{Mod}(\mathcal{O}_U) \ar[r]_{j_{U!}} \ar[d]_{forget} & \textit{Mod}(\mathcal{O}_\mathcal{C}) \ar[d]^{forget} \\ \textit{Ab}(\mathcal{C}/U) \ar[r]^{j^{Ab}_{U!}} & \textit{Ab}(\mathcal{C}) }\] commutes. This is clear from the explicit description of the functor \(j_{U!}\) in the lemma.

Remark

Localization and presheaves of modules; see Sites, Remark 0494. Let \(\mathcal{C}\) be a category. Let \(\mathcal{O}\) be a presheaf of rings. Let \(U\) be an object of \(\mathcal{C}\). Strictly speaking the functors \(j_U^*\), \(j_{U*}\) and \(j_{U!}\) have not been defined for presheaves of \(\mathcal{O}\)-modules. But of course, we can think of a presheaf as a sheaf for the chaotic topology on \(\mathcal{C}\) (see Sites, Examples 07GE). Hence we also obtain a functor \[j_U^* : \textit{PMod}(\mathcal{O}) \longrightarrow \textit{PMod}(\mathcal{O}_U)\] and functors \[j_{U*}, j_{U!} : \textit{PMod}(\mathcal{O}_U) \longrightarrow \textit{PMod}(\mathcal{O})\] which are right, left adjoint to \(j_U^*\). Inspecting the proof of Lemma 03DI we see that \(j_{U!}\mathcal{G}\) is the presheaf \[V \longmapsto \bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U)\] In addition the functor \(j_{U!}\) is exact (by Lemma 03DJ in the case of the discrete topologies). Moreover, if \(\mathcal{C}\) is actually a site, and \(\mathcal{O}\) is actually a sheaf of rings, then the diagram \[\xymatrix{ \textit{Mod}(\mathcal{O}_U) \ar[r]_{j_{U!}} \ar[d]_{forget} & \textit{Mod}(\mathcal{O}) \\ \textit{PMod}(\mathcal{O}_U) \ar[r]^{j_{U!}} & \textit{PMod}(\mathcal{O}) \ar[u]_{(\ )^\#} }\] commutes.

Lemma

Let \(\mathcal{C}\) be a site. Let \(U \in \Ob(\mathcal{C})\). Assume that every \(X\) in \(\mathcal{C}\) has at most one morphism to \(U\). Let \(\mathcal{F}\) be an abelian sheaf on \(\mathcal{C}/U\). The canonical maps \(\mathcal{F} \to j_U^{-1}j_{U!}\mathcal{F}\) and \(j_U^{-1}j_{U*}\mathcal{F} \to \mathcal{F}\) are isomorphisms.

Proof

This is a special case of Lemma 077I because the assumption on \(U\) is equivalent to the fully faithfulness of the localization functor \(\mathcal{C}/U \to \mathcal{C}\).

Localization of morphisms of ringed sites

This section is the analogue of Sites, Section 04I8.

Lemma

Let \((f, f^\sharp) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{D}, \mathcal{O}')\) be a morphism of ringed sites where \(f\) is given by the continuous functor \(u : \mathcal{D} \to \mathcal{C}\). Let \(V\) be an object of \(\mathcal{D}\) and set \(U = u(V)\). Then there is a canonical map of sheaves of rings \((f')^\sharp\) such that the diagram of Sites, Lemma 03CF is turned into a commutative diagram of ringed topoi \[\xymatrix{ (\Sh(\mathcal{C}/U), \mathcal{O}_U) \ar[rr]_{(j_U, j_U^\sharp)} \ar[d]_{(f', (f')^\sharp)} & & (\Sh(\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp)} \\ (\Sh(\mathcal{D}/V), \mathcal{O}'_V) \ar[rr]^{(j_V, j_V^\sharp)} & & (\Sh(\mathcal{D}), \mathcal{O}'). }\] Moreover, in this situation we have \(f'_*j_U^{-1} = j_V^{-1}f_*\) and \(f'_*j_U^* = j_V^*f_*\).

Proof

Just take \((f')^\sharp\) to be \[(f')^{-1}\mathcal{O}'_V = (f')^{-1}j_V^{-1}\mathcal{O}' = j_U^{-1}f^{-1}\mathcal{O}' \xrightarrow{j_U^{-1}f^\sharp} j_U^{-1}\mathcal{O} = \mathcal{O}_U\] and everything else follows from Sites, Lemma 03CF. (Note that \(j^{-1} = j^*\) on sheaves of modules if \(j\) is a localization morphism, hence the first equality of functors implies the second.)

Lemma

Let \((f, f^\sharp) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{D}, \mathcal{O}')\) be a morphism of ringed sites where \(f\) is given by the continuous functor \(u : \mathcal{D} \to \mathcal{C}\). Let \(V \in \Ob(\mathcal{D})\), \(U \in \Ob(\mathcal{C})\) and \(c : U \to u(V)\) a morphism of \(\mathcal{C}\). There exists a commutative diagram of ringed topoi \[\xymatrix{ (\Sh(\mathcal{C}/U), \mathcal{O}_U) \ar[rr]_{(j_U, j_U^\sharp)} \ar[d]_{(f_c, f_c^\sharp)} & & (\Sh(\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp)} \\ (\Sh(\mathcal{D}/V), \mathcal{O}'_V) \ar[rr]^{(j_V, j_V^\sharp)} & & (\Sh(\mathcal{D}), \mathcal{O}'). }\] The morphism \((f_c, f_c^\sharp)\) is equal to the composition of the morphism \[(f', (f')^\sharp) : (\Sh(\mathcal{C}/u(V)), \mathcal{O}_{u(V)}) \longrightarrow (\Sh(\mathcal{D}/V), \mathcal{O}'_V)\] of Lemma 04J0 and the morphism \[(j, j^\sharp) : (\Sh(\mathcal{C}/U), \mathcal{O}_U) \to (\Sh(\mathcal{C}/u(V)), \mathcal{O}_{u(V)})\] of Lemma 04IY. Given any morphisms \(b : V' \to V\), \(a : U' \to U\) and \(c' : U' \to u(V')\) such that \[\xymatrix{ U' \ar[r]_-{c'} \ar[d]_a & u(V') \ar[d]^{u(b)} \\ U \ar[r]^-c & u(V) }\] commutes, then the following diagram of ringed topoi \[\xymatrix{ (\Sh(\mathcal{C}/U'), \mathcal{O}_{U'}) \ar[rr]_{(j_{U'/U}, j_{U'/U}^\sharp)} \ar[d]_{(f_{c'}, f_{c'}^\sharp)} & & (\Sh(\mathcal{C}/U), \mathcal{O}_U) \ar[d]^{(f_c, f_c^\sharp)} \\ (\Sh(\mathcal{D}/V'), \mathcal{O}'_{V'}) \ar[rr]^{(j_{V'/V}, j_{V'/V}^\sharp)} & & (\Sh(\mathcal{D}/V), \mathcal{O}'_V) }\] commutes.

Proof

On the level of morphisms of topoi this is Sites, Lemma 04IN. To check that the diagrams commute as morphisms of ringed topoi use Lemmas 04IY and 04J0 exactly as in the proof of Sites, Lemma 04IN.

Localization of ringed topoi

This section is the analogue of Sites, Section 04GY in the setting of ringed topoi.

Lemma

Let \((\Sh(\mathcal{C}), \mathcal{O})\) be a ringed topos. Let \(\mathcal{F} \in \Sh(\mathcal{C})\) be a sheaf. For a sheaf \(\mathcal{H}\) on \(\mathcal{C}\) denote \(\mathcal{H}_\mathcal{F}\) the sheaf \(\mathcal{H} \times \mathcal{F}\) seen as an object of the category \(\Sh(\mathcal{C})/\mathcal{F}\). The pair \((\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F})\) is a ringed topos and there is a canonical morphism of ringed topoi \[(j_\mathcal{F}, j_\mathcal{F}^\sharp) : (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) \longrightarrow (\Sh(\mathcal{C}), \mathcal{O})\] which is a localization as in Section 03DH such that

1. the functor \(j_\mathcal{F}^{-1}\) is the functor \(\mathcal{H} \mapsto \mathcal{H}_\mathcal{F}\),

2. the functor \(j_\mathcal{F}^*\) is the functor \(\mathcal{H} \mapsto \mathcal{H}_\mathcal{F}\),

3. the functor \(j_{\mathcal{F}!}\) on sheaves of sets is the forgetful functor \(\mathcal{G}/\mathcal{F} \mapsto \mathcal{G}\),

4. the functor \(j_{\mathcal{F}!}\) on sheaves of modules associates to the \(\mathcal{O}_\mathcal{F}\)-module \(\varphi : \mathcal{G} \to \mathcal{F}\) the \(\mathcal{O}\)-module which is the sheafification of the presheaf \[V \longmapsto \bigoplus\nolimits_{s \in \mathcal{F}(V)} \{\sigma \in \mathcal{G}(V) \mid \varphi(\sigma) = s \}\] for \(V \in \Ob(\mathcal{C})\).

Proof

By Sites, Lemma 04GZ we see that \(\Sh(\mathcal{C})/\mathcal{F}\) is a topos and that (1) and (3) are true. In particular this shows that \(j_\mathcal{F}^{-1}\mathcal{O} = \mathcal{O}_\mathcal{F}\) and shows that \(\mathcal{O}_\mathcal{F}\) is a sheaf of rings. Thus we may choose the map \(j_\mathcal{F}^\sharp\) to be the identity, in particular we see that (2) is true. Moreover, the proof of Sites, Lemma 04GZ shows that we may assume \(\mathcal{C}\) is a site with all finite limits and a subcanonical topology and that \(\mathcal{F} = h_U\) for some object \(U\) of \(\mathcal{C}\). Then (4) follows from the description of \(j_{\mathcal{F}!}\) in Lemma 03DI. Alternatively one could show directly that the functor described in (4) is a left adjoint to \(j_\mathcal{F}^*\).

Definition

Let \((\Sh(\mathcal{C}), \mathcal{O})\) be a ringed topos. Let \(\mathcal{F} \in \Sh(\mathcal{C})\).

1. The ringed topos \((\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F})\) is called the localization of the ringed topos \((\Sh(\mathcal{C}), \mathcal{O})\) at \(\mathcal{F}\).

2. The morphism of ringed topoi \((j_\mathcal{F}, j_\mathcal{F}^\sharp) : (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) \to (\Sh(\mathcal{C}), \mathcal{O})\) of Lemma 04IE is called the localization morphism.

We continue the tradition, established in the chapter on sites, that we check the localization constructions on topoi are compatible with the constructions of localization on sites, whenever this makes sense.

Lemma

With \((\Sh(\mathcal{C}), \mathcal{O})\) and \(\mathcal{F} \in \Sh(\mathcal{C})\) as in Lemma 04IE. If \(\mathcal{F} = h_U^\#\) for some object \(U\) of \(\mathcal{C}\) then via the identification \(\Sh(\mathcal{C}/U) = \Sh(\mathcal{C})/h_U^\#\) of Sites, Lemma 00Y1 we have

1. canonically \(\mathcal{O}_U = \mathcal{O}_\mathcal{F}\), and

2. with these identifications we have \((j_\mathcal{F}, j_\mathcal{F}^\sharp) = (j_U, j_U^\sharp)\).

Proof

The assertion for underlying topoi is Sites, Lemma 04IQ. Note that \(\mathcal{O}_U\) is the restriction of \(\mathcal{O}\) which by Sites, Lemma 03EE corresponds to \(\mathcal{O} \times h_U^\#\) under the equivalence of Sites, Lemma 00Y1. By definition of \(\mathcal{O}_\mathcal{F}\) we get (1). What’s left is to prove that \(j_\mathcal{F}^\sharp = j_U^\sharp\) under this identification. We omit the verification.

Localization is functorial in the following two ways: We can “relocalize” a localization (see Lemma 04J4) or we can given a morphism of ringed topoi, localize upstairs at the inverse image of a sheaf downstairs and get a commutative diagram of ringed topoi (see Lemma 04IF).

Lemma

Let \((\Sh(\mathcal{C}), \mathcal{O})\) be a ringed topos. If \(s : \mathcal{G} \to \mathcal{F}\) is a morphism of sheaves on \(\mathcal{C}\) then there exists a natural commutative diagram of morphisms of ringed topoi \[\xymatrix{ (\Sh(\mathcal{C})/\mathcal{G}, \mathcal{O}_\mathcal{G}) \ar[rd]_{(j_\mathcal{G}, j_\mathcal{G}^\sharp)} \ar[rr]_{(j, j^\sharp)} & & (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) \ar[ld]^{(j_\mathcal{F}, j_\mathcal{F}^\sharp)} \\ & (\Sh(\mathcal{C}), \mathcal{O}) & }\] where \((j, j^\sharp)\) is the localization morphism of the ringed topos \((\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F})\) at the object \(\mathcal{G}/\mathcal{F}\).

Proof

All assertions follow from Sites, Lemma 04IR except the assertion that \(j_\mathcal{G}^\sharp = j^\sharp \circ j^{-1}(j_\mathcal{F}^\sharp)\). We omit the verification.

Lemma

With \((\Sh(\mathcal{C}), \mathcal{O})\), \(s : \mathcal{G} \to \mathcal{F}\) as in Lemma 04J4. If there exist a morphism \(f : V \to U\) of \(\mathcal{C}\) such that \(\mathcal{G} = h_V^\#\) and \(\mathcal{F} = h_U^\#\) and \(s\) is induced by \(f\), then the diagrams of Lemma 04IY and Lemma 04J4 agree via the identifications \((j_\mathcal{F}, j_\mathcal{F}^\sharp) = (j_U, j_U^\sharp)\) and \((j_\mathcal{G}, j_\mathcal{G}^\sharp) = (j_V, j_V^\sharp)\) of Lemma 04J3.

Proof

All assertions follow from Sites, Lemma 04IS except for the assertion that the two maps \(j^\sharp\) agree. This holds since in both cases the map \(j^\sharp\) is simply the identity. Some details omitted.

Localization of morphisms of ringed topoi

This section is the analogue of Sites, Section 04IT.

Lemma

Let \[f : (\Sh(\mathcal{C}), \mathcal{O}) \longrightarrow (\Sh(\mathcal{D}), \mathcal{O}')\] be a morphism of ringed topoi. Let \(\mathcal{G}\) be a sheaf on \(\mathcal{D}\). Set \(\mathcal{F} = f^{-1}\mathcal{G}\). Then there exists a commutative diagram of ringed topoi \[\xymatrix{ (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) \ar[rr]_{(j_\mathcal{F}, j_\mathcal{F}^\sharp)} \ar[d]_{(f', (f')^\sharp)} & & (\Sh(\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp)} \\ (\Sh(\mathcal{D})/\mathcal{G}, \mathcal{O}'_\mathcal{G}) \ar[rr]^{(j_\mathcal{G}, j_\mathcal{G}^\sharp)} & & (\Sh(\mathcal{D}), \mathcal{O}') }\] We have \(f'_*j_\mathcal{F}^{-1} = j_\mathcal{G}^{-1}f_*\) and \(f'_*j_\mathcal{F}^* = j_\mathcal{G}^*f_*\). Moreover, the morphism \(f'\) is characterized by the rule \[(f')^{-1}(\mathcal{H} \xrightarrow{\varphi} \mathcal{G}) = (f^{-1}\mathcal{H} \xrightarrow{f^{-1}\varphi} \mathcal{F}).\]

Proof

By Sites, Lemma 04H1 we have the diagram of underlying topoi, the equality \(f'_*j_\mathcal{F}^{-1} = j_\mathcal{G}^{-1}f_*\), and the description of \((f')^{-1}\). To define \((f')^\sharp\) we use the map \[(f')^\sharp : \mathcal{O}'_\mathcal{G} = j_\mathcal{G}^{-1} \mathcal{O}' \xrightarrow{j_\mathcal{G}^{-1}f^\sharp} j_\mathcal{G}^{-1} f_*\mathcal{O} = f'_* j_\mathcal{F}^{-1}\mathcal{O} = f'_* \mathcal{O}_\mathcal{F}\] or equivalently the map \[(f')^\sharp : (f')^{-1}\mathcal{O}'_\mathcal{G} = (f')^{-1}j_\mathcal{G}^{-1} \mathcal{O}' = j_\mathcal{F}^{-1}f^{-1}\mathcal{O}' \xrightarrow{j_\mathcal{F}^{-1}f^\sharp} j_\mathcal{F}^{-1} \mathcal{O} = \mathcal{O}_\mathcal{F}.\] We omit the verification that these two maps are indeed adjoint to each other. The second construction of \((f')^\sharp\) shows that the diagram commutes in the \(2\)-category of ringed topoi (as the maps \(j_\mathcal{F}^\sharp\) and \(j_\mathcal{G}^\sharp\) are identities). Finally, the equality \(f'_*j_\mathcal{F}^* = j_\mathcal{G}^*f_*\) follows from the equality \(f'_*j_\mathcal{F}^{-1} = j_\mathcal{G}^{-1}f_*\) and the fact that pullbacks of sheaves of modules and sheaves of sets agree, see Lemma 04IE.

Lemma

Let \[f : (\Sh(\mathcal{C}), \mathcal{O}) \longrightarrow (\Sh(\mathcal{D}), \mathcal{O}')\] be a morphism of ringed topoi. Let \(\mathcal{G}\) be a sheaf on \(\mathcal{D}\). Set \(\mathcal{F} = f^{-1}\mathcal{G}\). If \(f\) is given by a continuous functor \(u : \mathcal{D} \to \mathcal{C}\) and \(\mathcal{G} = h_V^\#\), then the commutative diagrams of Lemma 04J0 and Lemma 04IF agree via the identifications of Lemma 04J3.

Proof

At the level of morphisms of topoi this is Sites, Lemma 04IU. This works also on the level of morphisms of ringed topoi since the formulas defining \((f')^\sharp\) in the proofs of Lemma 04J0 and Lemma 04IF agree.

Lemma

Let \((f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) be a morphism of ringed topoi. Let \(\mathcal{G}\) be a sheaf on \(\mathcal{D}\), let \(\mathcal{F}\) be a sheaf on \(\mathcal{C}\), and let \(s : \mathcal{F} \to f^{-1}\mathcal{G}\) a morphism of sheaves. There exists a commutative diagram of ringed topoi \[\xymatrix{ (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) \ar[rr]_{(j_\mathcal{F}, j_\mathcal{F}^\sharp)} \ar[d]_{(f_s, f_s^\sharp)} & & (\Sh(\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp)} \\ (\Sh(\mathcal{D})/\mathcal{G}, \mathcal{O}'_\mathcal{G}) \ar[rr]^{(j_\mathcal{G}, j_\mathcal{G}^\sharp)} & & (\Sh(\mathcal{D}), \mathcal{O}'). }\] The morphism \((f_s, f_s^\sharp)\) is equal to the composition of the morphism \[(f', (f')^\sharp) : (\Sh(\mathcal{C})/f^{-1}\mathcal{G}, \mathcal{O}_{f^{-1}\mathcal{G}}) \longrightarrow (\Sh(\mathcal{D})/{\mathcal{G}}, \mathcal{O}'_\mathcal{G})\] of Lemma 04IF and the morphism \[(j, j^\sharp) : (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) \to (\Sh(\mathcal{C})/f^{-1}\mathcal{G}, \mathcal{O}_{f^{-1}\mathcal{G}})\] of Lemma 04J4. Given any morphisms \(b : \mathcal{G}' \to \mathcal{G}\), \(a : \mathcal{F}' \to \mathcal{F}\), and \(s' : \mathcal{F}' \to f^{-1}\mathcal{G}'\) such that \[\xymatrix{ \mathcal{F}' \ar[r]_-{s'} \ar[d]_a & f^{-1}\mathcal{G}' \ar[d]^{f^{-1}b} \\ \mathcal{F} \ar[r]^-s & f^{-1}\mathcal{G} }\] commutes, then the following diagram of ringed topoi \[\xymatrix{ (\Sh(\mathcal{C})/\mathcal{F}', \mathcal{O}_{\mathcal{F}'}) \ar[rr]_{(j_{\mathcal{F}'/\mathcal{F}}, j_{\mathcal{F}'/\mathcal{F}}^\sharp)} \ar[d]_{(f_{s'}, f_{s'}^\sharp)} & & (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) \ar[d]^{(f_s, f_s^\sharp)} \\ (\Sh(\mathcal{D})/\mathcal{G}', \mathcal{O}'_{\mathcal{G}'}) \ar[rr]^{(j_{\mathcal{G}'/\mathcal{G}}, j_{\mathcal{G}'/\mathcal{G}}^\sharp)} & & (\Sh(\mathcal{D})/\mathcal{G}, \mathcal{O}'_\mathcal{G}) }\] commutes.

Proof

On the level of morphisms of topoi this is Sites, Lemma 04IV. To check that the diagrams commute as morphisms of ringed topoi use the commutative diagrams of Lemmas 04J4 and 04IF.

Lemma

Let \((f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\), \(s : \mathcal{F} \to f^{-1}\mathcal{G}\) be as in Lemma 04J8. If \(f\) is given by a continuous functor \(u : \mathcal{D} \to \mathcal{C}\) and \(\mathcal{G} = h_V^\#\), \(\mathcal{F} = h_U^\#\) and \(s\) comes from a morphism \(c : U \to u(V)\), then the commutative diagrams of Lemma 04J1 and Lemma 04J8 agree via the identifications of Lemma 04J3.

Proof

This is formal using Lemmas 04J5 and 04J7.

Local types of modules

According to our general strategy explained in Section 03DG we first define the local types for sheaves of modules on a ringed site, and then we immediately show that these types are intrinsic, hence make sense for sheaves of modules on ringed topoi.

Definition

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}\)-modules. We will freely use the notions defined in Definition 03DE.

1. We say \(\mathcal{F}\) is locally free if for every object \(U\) of \(\mathcal{C}\) there exists a covering \(\{U_i \to U\}_{i \in I}\) of \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/U_i}\) is a free \(\mathcal{O}_{U_i}\)-module.

2. We say \(\mathcal{F}\) is finite locally free if for every object \(U\) of \(\mathcal{C}\) there exists a covering \(\{U_i \to U\}_{i \in I}\) of \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/U_i}\) is a finite free \(\mathcal{O}_{U_i}\)-module.

3. We say \(\mathcal{F}\) is locally generated by sections if for every object \(U\) of \(\mathcal{C}\) there exists a covering \(\{U_i \to U\}_{i \in I}\) of \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/U_i}\) is an \(\mathcal{O}_{U_i}\)-module generated by global sections.

4. Given \(r \geq 0\) we say \(\mathcal{F}\) is locally generated by \(r\) sections if for every object \(U\) of \(\mathcal{C}\) there exists a covering \(\{U_i \to U\}_{i \in I}\) of \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/U_i}\) is an \(\mathcal{O}_{U_i}\)-module generated by \(r\) global sections.

5. We say \(\mathcal{F}\) is of finite type if for every object \(U\) of \(\mathcal{C}\) there exists a covering \(\{U_i \to U\}_{i \in I}\) of \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/U_i}\) is an \(\mathcal{O}_{U_i}\)-module generated by finitely many global sections.

6. We say \(\mathcal{F}\) is quasi-coherent if for every object \(U\) of \(\mathcal{C}\) there exists a covering \(\{U_i \to U\}_{i \in I}\) of \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/U_i}\) is an \(\mathcal{O}_{U_i}\)-module which has a global presentation.

7. We say \(\mathcal{F}\) is of finite presentation if for every object \(U\) of \(\mathcal{C}\) there exists a covering \(\{U_i \to U\}_{i \in I}\) of \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/U_i}\) is an \(\mathcal{O}_{U_i}\)-module which has a finite global presentation.

8. We say \(\mathcal{F}\) is coherent if and only if \(\mathcal{F}\) is of finite type, and for every object \(U\) of \(\mathcal{C}\) and any \(s_1, \ldots, s_n \in \mathcal{F}(U)\) the kernel of the map \(\bigoplus_{i = 1, \ldots, n} \mathcal{O}_U \to \mathcal{F}|_U\) is of finite type on \((\mathcal{C}/U, \mathcal{O}_U)\).

Lemma

Any of the properties (1) – (8) of Definition 03DL is intrinsic (see discussion in Section 03DG).

Proof

Let \(\mathcal{C}\), \(\mathcal{D}\) be sites. Let \(u : \mathcal{C} \to \mathcal{D}\) be a special cocontinuous functor. Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}\). Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}\)-modules on \(\mathcal{C}\). Let \(g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\) be the equivalence of topoi associated to \(u\). Set \(\mathcal{O}' = g_*\mathcal{O}\), and let \(g^\sharp : \mathcal{O}' \to g_*\mathcal{O}\) be the identity. Finally, set \(\mathcal{F}' = g_*\mathcal{F}\). Let \(\mathcal{P}_l\) be one of the properties (1) – (7) listed in Definition 03DL. (We will discuss the coherent case at the end of the proof.) Let \(\mathcal{P}_g\) denote the corresponding property listed in Definition 03DE. We have already seen that \(\mathcal{P}_g\) is intrinsic. We have to show that \(\mathcal{P}_l(\mathcal{C}, \mathcal{O}, \mathcal{F})\) holds if and only if \(\mathcal{P}_l(\mathcal{D}, \mathcal{O}', \mathcal{F}')\) holds.

Assume that \(\mathcal{F}\) has \(\mathcal{P}_l\). Let \(V\) be an object of \(\mathcal{D}\). One of the properties of a special cocontinuous functor is that there exists a covering \(\{u(U_i) \to V\}_{i \in I}\) in the site \(\mathcal{D}\). By assumption, for each \(i\) there exists a covering \(\{U_{ij} \to U_i\}_{j \in J_i}\) in \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{U_{ij}}\) is \(\mathcal{P}_g\). By Sites, Lemma 03CH we have commutative diagrams of ringed topoi \[\xymatrix{ (\Sh(\mathcal{C}/U_{ij}), \mathcal{O}_{U_{ij}}) \ar[r] \ar[d] & (\Sh(\mathcal{C}), \mathcal{O}) \ar[d] \\ (\Sh(\mathcal{D}/u(U_{ij})), \mathcal{O}'_{u(U_{ij})}) \ar[r] & (\Sh(\mathcal{D}), \mathcal{O}') }\] where the vertical arrows are equivalences. Hence we conclude that \(\mathcal{F}'|_{u(U_{ij})}\) has property \(\mathcal{P}_g\) also. And moreover, \(\{u(U_{ij}) \to V\}_{i \in I, j \in J_i}\) is a covering of the site \(\mathcal{D}\). Hence \(\mathcal{F}'\) has property \(\mathcal{P}_l\).

Assume that \(\mathcal{F}'\) has \(\mathcal{P}_l\). Let \(U\) be an object of \(\mathcal{C}\). By assumption, there exists a covering \(\{V_i \to u(U)\}_{i \in I}\) such that \(\mathcal{F}'|_{V_i}\) has property \(\mathcal{P}_g\). Because \(u\) is cocontinuous we can refine this covering by a family \(\{u(U_j) \to u(U)\}_{j \in J}\) where \(\{U_j \to U\}_{j \in J}\) is a covering in \(\mathcal{C}\). Say the refinement is given by \(\alpha : J \to I\) and \(u(U_j) \to V_{\alpha(j)}\). Restricting is transitive, i.e., \((\mathcal{F}'|_{V_{\alpha(j)}})|_{u(U_j)} = \mathcal{F}'|_{u(U_j)}\). Hence by Lemma 03DF we see that \(\mathcal{F}'|_{u(U_j)}\) has property \(\mathcal{P}_g\). Hence the diagram \[\xymatrix{ (\Sh(\mathcal{C}/U_j), \mathcal{O}_{U_j}) \ar[r] \ar[d] & (\Sh(\mathcal{C}), \mathcal{O}) \ar[d] \\ (\Sh(\mathcal{D}/u(U_j)), \mathcal{O}'_{u(U_j)}) \ar[r] & (\Sh(\mathcal{D}), \mathcal{O}') }\] where the vertical arrows are equivalences shows that \(\mathcal{F}|_{U_j}\) has property \(\mathcal{P}_g\) also. Thus \(\mathcal{F}\) has property \(\mathcal{P}_l\) as desired.

Finally, we prove the lemma in case \(\mathcal{P}_l = coherent\)². Assume \(\mathcal{F}\) is coherent. This implies that \(\mathcal{F}\) is of finite type and hence \(\mathcal{F}'\) is of finite type also by the first part of the proof. Let \(V\) be an object of \(\mathcal{D}\) and let \(s_1, \ldots, s_n \in \mathcal{F}'(V)\). We have to show that the kernel \(\mathcal{K}'\) of \(\bigoplus_{j = 1, \ldots, n} \mathcal{O}_V \to \mathcal{F}'|_V\) is of finite type on \(\mathcal{D}/V\). This means we have to show that for any \(V'/V\) there exists a covering \(\{V'_i \to V'\}\) such that \(\mathcal{F}'|_{V'_i}\) is generated by finitely many sections. Replacing \(V\) by \(V'\) (and restricting the sections \(s_j\) to \(V'\)) we reduce to the case where \(V' = V\). Since \(u\) is a special cocontinuous functor, there exists a covering \(\{u(U_i) \to V\}_{i \in I}\) in the site \(\mathcal{D}\). Using the isomorphism of topoi \(\Sh(\mathcal{C}/U_i) = \Sh(\mathcal{D}/u(U_i))\) we see that \(\mathcal{K}'|_{u(U_i)}\) corresponds to the kernel \(\mathcal{K}_i\) of a map \(\bigoplus_{j = 1, \ldots, n} \mathcal{O}_{U_i} \to \mathcal{F}|_{U_i}\). Since \(\mathcal{F}\) is coherent we see that \(\mathcal{K}_i\) is of finite type. Hence we conclude (by the first part of the proof again) that \(\mathcal{K}|_{u(U_i)}\) is of finite type. Thus there exist coverings \(\{V_{il} \to u(U_i)\}\) such that \(\mathcal{K}|_{V_{il}}\) is generated by finitely many global sections. Since \(\{V_{il} \to V\}\) is a covering of \(\mathcal{D}\) we conclude that \(\mathcal{K}\) is of finite type as desired.

Assume \(\mathcal{F}'\) is coherent. This implies that \(\mathcal{F}'\) is of finite type and hence \(\mathcal{F}\) is of finite type also by the first part of the proof. Let \(U\) be an object of \(\mathcal{C}\), and let \(s_1, \ldots, s_n \in \mathcal{F}(U)\). We have to show that the kernel \(\mathcal{K}\) of \(\bigoplus_{j = 1, \ldots, n} \mathcal{O}_U \to \mathcal{F}|_U\) is of finite type on \(\mathcal{C}/U\). Using the isomorphism of topoi \(\Sh(\mathcal{C}/U) = \Sh(\mathcal{D}/u(U))\) we see that \(\mathcal{K}|_U\) corresponds to the kernel \(\mathcal{K}'\) of a map \(\bigoplus_{j = 1, \ldots, n} \mathcal{O}_{u(U)} \to \mathcal{F}'|_{u(U)}\). As \(\mathcal{F}'\) is coherent, we see that \(\mathcal{K}'\) is of finite type. Hence, by the first part of the proof again, we conclude that \(\mathcal{K}\) is of finite type.

Hence from now on we may refer to the properties of \(\mathcal{O}\)-modules defined in Definition 03DL without specifying a site.

Lemma

Let \((\Sh(\mathcal{C}), \mathcal{O})\) be a ringed topos. Let \(\mathcal{F}\) be an \(\mathcal{O}\)-module. Assume that the site \(\mathcal{C}\) has a final object \(X\). Then

1. The following are equivalent

   1. \(\mathcal{F}\) is locally free,

   2. there exists a covering \(\{X_i \to X\}\) in \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/X_i}\) is a locally free \(\mathcal{O}_{X_i}\)-module, and

   3. there exists a covering \(\{X_i \to X\}\) in \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/X_i}\) is a free \(\mathcal{O}_{X_i}\)-module.

2. The following are equivalent

   1. \(\mathcal{F}\) is finite locally free,

   2. there exists a covering \(\{X_i \to X\}\) in \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/X_i}\) is a finite locally free \(\mathcal{O}_{X_i}\)-module, and

   3. there exists a covering \(\{X_i \to X\}\) in \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/X_i}\) is a finite free \(\mathcal{O}_{X_i}\)-module.

3. The following are equivalent

   1. \(\mathcal{F}\) is locally generated by sections,

   2. there exists a covering \(\{X_i \to X\}\) in \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/X_i}\) is an \(\mathcal{O}_{X_i}\)-module locally generated by sections, and

   3. there exists a covering \(\{X_i \to X\}\) in \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/X_i}\) is an \(\mathcal{O}_{X_i}\)-module globally generated by sections.

4. Given \(r \geq 0\), the following are equivalent

   1. \(\mathcal{F}\) is locally generated by \(r\) sections,

   2. there exists a covering \(\{X_i \to X\}\) in \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/X_i}\) is an \(\mathcal{O}_{X_i}\)-module locally generated by \(r\) sections, and

   3. there exists a covering \(\{X_i \to X\}\) in \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/X_i}\) is an \(\mathcal{O}_{X_i}\)-module globally generated by \(r\) sections.

5. The following are equivalent

   1. \(\mathcal{F}\) is of finite type,

   2. there exists a covering \(\{X_i \to X\}\) in \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/X_i}\) is an \(\mathcal{O}_{X_i}\)-module of finite type, and

   3. there exists a covering \(\{X_i \to X\}\) in \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/X_i}\) is an \(\mathcal{O}_{X_i}\)-module globally generated by finitely many sections.

6. The following are equivalent

   1. \(\mathcal{F}\) is quasi-coherent,

   2. there exists a covering \(\{X_i \to X\}\) in \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/X_i}\) is a quasi-coherent \(\mathcal{O}_{X_i}\)-module, and

   3. there exists a covering \(\{X_i \to X\}\) in \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/X_i}\) is an \(\mathcal{O}_{X_i}\)-module which has a global presentation.

7. The following are equivalent

   1. \(\mathcal{F}\) is of finite presentation,

   2. there exists a covering \(\{X_i \to X\}\) in \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/X_i}\) is an \(\mathcal{O}_{X_i}\)-module of finite presentation, and

   3. there exists a covering \(\{X_i \to X\}\) in \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/X_i}\) is an \(\mathcal{O}_{X_i}\)-module has a finite global presentation.

8. The following are equivalent

   1. \(\mathcal{F}\) is coherent, and

   2. there exists a covering \(\{X_i \to X\}\) in \(\mathcal{C}\) such that each restriction \(\mathcal{F}|_{\mathcal{C}/X_i}\) is a coherent \(\mathcal{O}_{X_i}\)-module.

Proof

In each case we have (a) \(\Rightarrow (b)\). In each of the cases (1) - (6) condition (b) implies condition (c) by axiom (2) of a site (see Sites, Definition 00VH) and the definition of the local types of modules. Suppose \(\{X_i \to X\}\) is a covering. Then for every object \(U\) of \(\mathcal{C}\) we get an induced covering \(\{X_i \times_X U \to U\}\). Moreover, the global property for \(\mathcal{F}|_{\mathcal{C}/X_i}\) in part (c) implies the corresponding global property for \(\mathcal{F}|_{\mathcal{C}/X_i \times_X U}\) by Lemma 03DF, hence the sheaf has property (a) by definition. We omit the proof of (b) \(\Rightarrow\) (a) in case (7).

Lemma

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

1. If \(\mathcal{F}\) is locally free then \(f^*\mathcal{F}\) is locally free.

2. If \(\mathcal{F}\) is finite locally free then \(f^*\mathcal{F}\) is finite locally free.

3. If \(\mathcal{F}\) is locally generated by sections then \(f^*\mathcal{F}\) is locally generated by sections.

4. If \(\mathcal{F}\) is locally generated by \(r\) sections then \(f^*\mathcal{F}\) is locally generated by \(r\) sections.

5. If \(\mathcal{F}\) is of finite type then \(f^*\mathcal{F}\) is of finite type.

6. If \(\mathcal{F}\) is quasi-coherent then \(f^*\mathcal{F}\) is quasi-coherent.

7. If \(\mathcal{F}\) is of finite presentation then \(f^*\mathcal{F}\) is of finite presentation.

Proof

According to the discussion in Section 03DG we need only check preservation under pullback for a morphism of ringed sites \((f, f^\sharp) : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D})\) such that \(f\) is given by a left exact, continuous functor \(u : \mathcal{D} \to \mathcal{C}\) between sites which have all finite limits. Let \(\mathcal{G}\) be a sheaf of \(\mathcal{O}_\mathcal{D}\)-modules which has one of the properties (1) – (6) of Definition 03DL. We know \(\mathcal{D}\) has a final object \(Y\) and \(X = u(Y)\) is a final object for \(\mathcal{C}\). By assumption we have a covering \(\{Y_i \to Y\}\) such that \(\mathcal{G}|_{\mathcal{D}/Y_i}\) has the corresponding global property. Set \(X_i = u(Y_i)\) so that \(\{X_i \to X\}\) is a covering in \(\mathcal{C}\). We get a commutative diagram of morphisms ringed sites \[\xymatrix{ (\mathcal{C}/X_i, \mathcal{O}_\mathcal{C}|_{X_i}) \ar[r] \ar[d] & (\mathcal{C}, \mathcal{O}_\mathcal{C}) \ar[d] \\ (\mathcal{D}/Y_i, \mathcal{O}_\mathcal{D}|_{Y_i}) \ar[r] & (\mathcal{D}, \mathcal{O}_\mathcal{D}) }\] by Sites, Lemma 03EF. Hence by Lemma 03DF that \(f^*\mathcal{G}|_{X_i}\) has the corresponding global property. Hence we conclude that \(\mathcal{G}\) has the local property we started out with by Lemma 03DN.

Basic results on local types of modules

Basic lemmas related to the definitions made above.

Lemma

Let \((\mathcal{C},\mathcal{O})\) be a ringed site. Let \(\mathcal{F}\) be a finitely presented \(\mathcal{O}\)-module. Let \(\varphi : \mathcal{G} \to \mathcal{F}\) be a morphism of \(\mathcal{O}\)-modules. If \(\mathcal{G}\) is finite type, then \(\Coker(\varphi)\) is finitely presented.

Proof

Omitted. Hint: See Modules, Lemma 0H96.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\theta : \mathcal{G} \to \mathcal{F}\) be a surjective \(\mathcal{O}\)-module map with \(\mathcal{F}\) of finite presentation and \(\mathcal{G}\) of finite type. Then \(\Ker(\theta)\) is of finite type.

Proof

Omitted. Hint: See Modules, Lemma 01BP.

Lemma

Let \(\mathcal{C}\) be a category viewed as a site with the chaotic topology, see Sites, Example 07GE. Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}\) and let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}\)-modules. Then \(\mathcal{F}\) is quasi-coherent if and only if for all \(U \to V\) in \(\mathcal{C}\) the canonical map \[\mathcal{F}(V) \otimes_{\mathcal{O}(V)} \mathcal{O}(U) \longrightarrow \mathcal{F}(U)\] is an isomorphism.

Proof

Assume \(\mathcal{F}\) is quasi-coherent and let \(U \to V\) be a morphism of \(\mathcal{C}\). Since every covering of \(V\) is given by an isomorphism we conclude from Definition 03DL that there exists a presentation \[\bigoplus\nolimits_{j \in J} \mathcal{O}_V \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O}_V \longrightarrow \mathcal{F}|_{\mathcal{C}/V} \longrightarrow 0\] Since the topology on \(\mathcal{C}\) is chaotic, taking sections over any object of \(\mathcal{C}\) is exact. We conclude that we obtain a presentation \[\bigoplus\nolimits_{j \in J} \mathcal{O}(V) \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O}(V) \longrightarrow \mathcal{F}(V) \longrightarrow 0\] of \(\mathcal{F}(V)\) as an \(\mathcal{O}(V)\)-module and similarly for \(\mathcal{F}(U)\). This easily shows that the displayed map in the statement of the lemma is an isomorphism.

Assume the displayed map in the statement of the lemma is an isomorphism for every morphism \(U \to V\) in \(\mathcal{C}\). Fix \(V\) and choose a presentation \[\bigoplus\nolimits_{j \in J} \mathcal{O}(V) \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O}(V) \longrightarrow \mathcal{F}(V) \longrightarrow 0\] of \(\mathcal{F}(V)\) as an \(\mathcal{O}(V)\)-module. Then the assumption on \(\mathcal{F}\) exactly means that the corresponding sequence \[\bigoplus\nolimits_{j \in J} \mathcal{O}_V \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O}_V \longrightarrow \mathcal{F}|_{\mathcal{C}/V} \longrightarrow 0\] is exact and we conclude that \(\mathcal{F}\) is quasi-coherent.

Lemma

Let \(\mathcal{C}\) be a category viewed as a site with the chaotic topology, see Sites, Example 07GE. Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}\). Assume for all \(U \to V\) in \(\mathcal{C}\) the restriction map \(\mathcal{O}(V) \to \mathcal{O}(U)\) is a flat ring map. Then the category of quasi-coherent \(\mathcal{O}\)-modules is a weak Serre subcategory of \(\textit{Mod}(\mathcal{O})\).

Proof

We will check the definition of a weak Serre subcategory, see Homology, Definition 02MO. To do this we will use the characterization of quasi-coherent modules given in Lemma 0GZN. Consider an exact sequence \[\mathcal{F}_0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to \mathcal{F}_4\] in \(\textit{Mod}(\mathcal{O})\) with \(\mathcal{F}_0\), \(\mathcal{F}_1\), \(\mathcal{F}_3\), and \(\mathcal{F}_4\) quasi-coherent. Let \(U \to V\) be a morphism of \(\mathcal{C}\) and consider the commutative diagram \[\xymatrix{ \mathcal{F}_0(V) \otimes_{\mathcal{O}(V)} \mathcal{O}(U) \ar[r] \ar[d] & \mathcal{F}_1(V) \otimes_{\mathcal{O}(V)} \mathcal{O}(U) \ar[r] \ar[d] & \mathcal{F}_2(V) \otimes_{\mathcal{O}(V)} \mathcal{O}(U) \ar[r] \ar[d] & \mathcal{F}_3(V) \otimes_{\mathcal{O}(V)} \mathcal{O}(U) \ar[r] \ar[d] & \mathcal{F}_4(V) \otimes_{\mathcal{O}(V)} \mathcal{O}(U) \ar[d] \\ \mathcal{F}_0(U) \ar[r] & \mathcal{F}_1(U) \ar[r] & \mathcal{F}_2(U) \ar[r] & \mathcal{F}_3(U) \ar[r] & \mathcal{F}_4(U) }\] By assumption the vertical arrows with indices \(0\), \(1\), \(3\), \(4\) are isomorphisms. Since the topology on \(\mathcal{C}\) is chaotic taking sections over an object of \(\mathcal{C}\) is exact and hence the lower row is exact. Since \(\mathcal{O}(V) \to \mathcal{O}(U)\) is flat also the upper row is exact. Thus we conclude that the middle arrow is an isomorphism by the \(5\) lemma (Homology, Lemma 05QB).

Closed immersions of ringed topoi

When do we declare a morphism of ringed topoi \(i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) to be a closed immersion? By analogy with the discussion in Modules, Section 01C1 it seems natural to assume at least:

1. The functor \(i\) is a closed immersion of topoi (Sites, Definition 08M0).

2. The associated map \(\mathcal{O}' \to i_*\mathcal{O}\) is surjective.

These conditions already imply a number of pleasing results which we discuss in this section. However, it seems prudent to not actually define the notion of a closed immersion of ringed topoi as there are many different definitions we could use.

Lemma

Let \(i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) be a morphism of ringed topoi. Assume \(i\) is a closed immersion of topoi and \(i^\sharp : \mathcal{O}' \to i_*\mathcal{O}\) is surjective. Denote \(\mathcal{I} \subset \mathcal{O}'\) the kernel of \(i^\sharp\). The functor \[i_* : \textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}')\] is exact, fully faithful, with essential image those \(\mathcal{O}'\)-modules \(\mathcal{G}\) such that \(\mathcal{I}\mathcal{G} = 0\).

Proof

By Lemma 04DB and Sites, Lemma 08M1 we see that \(i_*\) is exact. From the fact that \(i_*\) is fully faithful on sheaves of sets, and the fact that \(i^\sharp\) is surjective it follows that \(i_*\) is fully faithful as a functor \(\textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}')\). Namely, suppose that \(\alpha : i_*\mathcal{F}_1 \to i_*\mathcal{F}_2\) is an \(\mathcal{O}'\)-module map. By the fully faithfulness of \(i_*\) we obtain a map \(\beta : \mathcal{F}_1 \to \mathcal{F}_2\) of sheaves of sets. To prove \(\beta\) is a map of modules we have to show that \[\xymatrix{ \mathcal{O} \times \mathcal{F}_1 \ar[r] \ar[d] & \mathcal{F}_1 \ar[d] \\ \mathcal{O} \times \mathcal{F}_2 \ar[r] & \mathcal{F}_2 }\] commutes. It suffices to prove commutativity after applying \(i_*\). Consider \[\xymatrix{ \mathcal{O}' \times i_*\mathcal{F}_1 \ar[r] \ar[d] & i_*\mathcal{O} \times i_*\mathcal{F}_1 \ar[r] \ar[d] & i_*\mathcal{F}_1 \ar[d] \\ \mathcal{O}' \times i_*\mathcal{F}_2 \ar[r] & i_*\mathcal{O} \times i_*\mathcal{F}_2 \ar[r] & i_*\mathcal{F}_2 }\] We know the outer rectangle commutes. Since \(i^\sharp\) is surjective we conclude.

To finish the proof we have to prove the statement on the essential image of \(i_*\). It is clear that \(i_*\mathcal{F}\) is annihilated by \(\mathcal{I}\) for any \(\mathcal{O}\)-module \(\mathcal{F}\). Conversely, let \(\mathcal{G}\) be a \(\mathcal{O}'\)-module with \(\mathcal{I}\mathcal{G} = 0\). By definition of a closed subtopos there exists a subsheaf \(\mathcal{U}\) of the final object of \(\mathcal{D}\) such that the essential image of \(i_*\) on sheaves of sets is the class of sheaves of sets \(\mathcal{H}\) such that \(\mathcal{H} \times \mathcal{U} \to \mathcal{U}\) is an isomorphism. In particular, \(i_*\mathcal{O} \times \mathcal{U} = \mathcal{U}\). This implies that \(\mathcal{I} \times \mathcal{U} = \mathcal{O} \times \mathcal{U}\). Hence our module \(\mathcal{G}\) satisfies \(\mathcal{G} \times \mathcal{U} = \{0\} \times \mathcal{U} = \mathcal{U}\) (because the zero module is isomorphic to the final object of sheaves of sets). Thus there exists a sheaf of sets \(\mathcal{F}\) on \(\mathcal{C}\) with \(i_*\mathcal{F} = \mathcal{G}\). Since \(i_*\) is fully faithful on sheaves of sets, we see that in order to define the addition \(\mathcal{F} \times \mathcal{F} \to \mathcal{F}\) and the multiplication \(\mathcal{O} \times \mathcal{F} \to \mathcal{F}\) it suffices to use the addition \[\mathcal{G} \times \mathcal{G} \longrightarrow \mathcal{G}\] (given to us as \(\mathcal{G}\) is a \(\mathcal{O}'\)-module) and the multiplication \[i_*\mathcal{O} \times \mathcal{G} \to \mathcal{G}\] which is given to us as we have the multiplication by \(\mathcal{O}'\) which annihilates \(\mathcal{I}\) by assumption and \(i_*\mathcal{O} = \mathcal{O}'/\mathcal{I}\). By construction \(\mathcal{G}\) is isomorphic to the pushforward of the \(\mathcal{O}\)-module \(\mathcal{F}\) so constructed.

Tensor product

In Sections 03CS and 03CX we defined the change of rings functor by a tensor product construction. To be sure this construction makes sense also to define the tensor product of presheaves of \(\mathcal{O}\)-modules. To be precise, suppose \(\mathcal{C}\) is a category, \(\mathcal{O}\) is a presheaf of rings, and \(\mathcal{F}\), \(\mathcal{G}\) are presheaves of \(\mathcal{O}\)-modules. In this case we define \(\mathcal{F} \otimes_{p, \mathcal{O}} \mathcal{G}\) to be the presheaf \[U \longmapsto (\mathcal{F} \otimes_{p, \mathcal{O}} \mathcal{G})(U) = \mathcal{F}(U) \otimes_{\mathcal{O}(U)} \mathcal{G}(U)\] If \(\mathcal{C}\) is a site, \(\mathcal{O}\) is a sheaf of rings and \(\mathcal{F}\), \(\mathcal{G}\) are sheaves of \(\mathcal{O}\)-modules then we define \[\mathcal{F} \otimes_\mathcal{O} \mathcal{G} = (\mathcal{F} \otimes_{p, \mathcal{O}} \mathcal{G})^\#\] to be the sheaf of \(\mathcal{O}\)-modules associated to the presheaf \(\mathcal{F} \otimes_{p, \mathcal{O}} \mathcal{G}\).

Here are some formulas which we will use below without further mention: \[(\mathcal{F} \otimes_{p, \mathcal{O}} \mathcal{G}) \otimes_{p, \mathcal{O}} \mathcal{H} = \mathcal{F} \otimes_{p, \mathcal{O}} (\mathcal{G} \otimes_{p, \mathcal{O}} \mathcal{H}),\] and similarly for sheaves. If \(\mathcal{O}_1 \to \mathcal{O}_2\) is a map of presheaves of rings, then \[(\mathcal{F} \otimes_{p, \mathcal{O}_1} \mathcal{G}) \otimes_{p, \mathcal{O}_1} \mathcal{O}_2 = (\mathcal{F} \otimes_{p, \mathcal{O}_1} \mathcal{O}_2) \otimes_{p, \mathcal{O}_2} (\mathcal{G} \otimes_{p, \mathcal{O}_1} \mathcal{O}_2),\] and similarly for sheaves. These follow from their algebraic counterparts and sheafification.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O}\) be a presheaf of rings. Let \(\mathcal{F}\), \(\mathcal{G}\) be presheaves of \(\mathcal{O}\)-modules. Then \(\mathcal{F}^\# \otimes_{\mathcal{O}^\#} \mathcal{G}^\#\) is equal to \((\mathcal{F} \otimes_{p, \mathcal{O}} \mathcal{G})^\#\).

Proof

Omitted. Hint: use the characterization of tensor product in terms of bilinear maps below and use the universal property of sheafification.

Let \(\mathcal{C}\) be a site, let \(\mathcal{O}\) be a sheaf of rings and let \(\mathcal{F}\), \(\mathcal{G}\), \(\mathcal{H}\) be sheaves of \(\mathcal{O}\)-modules. In this case we define \[\text{Bilin}_\mathcal{O}(\mathcal{F} \times \mathcal{G}, \mathcal{H}) = \{\varphi \in \Mor_{\Sh(\mathcal{C})}( \mathcal{F} \times \mathcal{G}, \mathcal{H}) \mid \varphi \text{ is }\mathcal{O}\text{-bilinear}\}.\] With this definition we have \[\Hom_\mathcal{O} (\mathcal{F} \otimes_\mathcal{O} \mathcal{G}, \mathcal{H}) = \text{Bilin}_\mathcal{O}(\mathcal{F} \times \mathcal{G}, \mathcal{H}).\] In other words \(\mathcal{F} \otimes_\mathcal{O} \mathcal{G}\) represents the functor which associates to \(\mathcal{H}\) the set of bilinear maps \(\mathcal{F} \times \mathcal{G} \to \mathcal{H}\). In particular, since the notion of a bilinear map makes sense for a pair of modules on a ringed topos, we see that the tensor product of sheaves of modules is intrinsic to the topos (compare the discussion in Section 03DG). In fact we have the following.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi. Let \(\mathcal{F}\), \(\mathcal{G}\) be \(\mathcal{O}_\mathcal{D}\)-modules. Then \(f^*(\mathcal{F} \otimes_{\mathcal{O}_\mathcal{D}} \mathcal{G}) = f^*\mathcal{F} \otimes_{\mathcal{O}_\mathcal{C}} f^*\mathcal{G}\) functorially in \(\mathcal{F}\), \(\mathcal{G}\).

Proof

For a sheaf \(\mathcal{H}\) of \(\mathcal{O}_\mathcal{C}\) modules we have \[\begin{align*} \Hom_{\mathcal{O}_\mathcal{C}}( f^*(\mathcal{F} \otimes_\mathcal{O} \mathcal{G}), \mathcal{H}) & = \Hom_{\mathcal{O}_\mathcal{D}}( \mathcal{F} \otimes_\mathcal{O} \mathcal{G}, f_*\mathcal{H}) \\ & = \text{Bilin}_{\mathcal{O}_\mathcal{D}}( \mathcal{F} \times \mathcal{G}, f_*\mathcal{H}) \\ & = \text{Bilin}_{f^{-1}\mathcal{O}_\mathcal{D}}( f^{-1}\mathcal{F} \times f^{-1}\mathcal{G}, \mathcal{H}) \\ & = \Hom_{f^{-1}\mathcal{O}_\mathcal{D}}( f^{-1}\mathcal{F} \otimes_{f^{-1}\mathcal{O}_\mathcal{D}} f^{-1}\mathcal{G}, \mathcal{H}) \\ & = \Hom_{\mathcal{O}_\mathcal{C}}( f^*\mathcal{F} \otimes_{f^*\mathcal{O}_\mathcal{D}} f^*\mathcal{G}, \mathcal{H}) \end{align*}\] The interesting “\(=\)” in this sequence of equalities is the third equality. It follows from the definition and adjointness of \(f_*\) and \(f^{-1}\) (as discussed in previous sections) in a straightforward manner.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{F}\), \(\mathcal{G}\) be sheaves of \(\mathcal{O}\)-modules.

1. If \(\mathcal{F}\), \(\mathcal{G}\) are locally free, so is \(\mathcal{F} \otimes_\mathcal{O} \mathcal{G}\).

2. If \(\mathcal{F}\), \(\mathcal{G}\) are finite locally free, so is \(\mathcal{F} \otimes_\mathcal{O} \mathcal{G}\).

3. If \(\mathcal{F}\), \(\mathcal{G}\) are locally generated by sections, so is \(\mathcal{F} \otimes_\mathcal{O} \mathcal{G}\).

4. If \(\mathcal{F}\), \(\mathcal{G}\) are of finite type, so is \(\mathcal{F} \otimes_\mathcal{O} \mathcal{G}\).

5. If \(\mathcal{F}\), \(\mathcal{G}\) are quasi-coherent, so is \(\mathcal{F} \otimes_\mathcal{O} \mathcal{G}\).

6. If \(\mathcal{F}\), \(\mathcal{G}\) are of finite presentation, so is \(\mathcal{F} \otimes_\mathcal{O} \mathcal{G}\).

7. If \(\mathcal{F}\) is of finite presentation and \(\mathcal{G}\) is coherent, then \(\mathcal{F} \otimes_\mathcal{O} \mathcal{G}\) is coherent.

8. If \(\mathcal{F}\), \(\mathcal{G}\) are coherent, so is \(\mathcal{F} \otimes_\mathcal{O} \mathcal{G}\).

Proof

Omitted. Hint: Compare with Sheaves of Modules, Lemma 01CE.

Internal Hom

Let \(\mathcal{C}\) be a category and let \(\mathcal{O}\) be a presheaf of rings. Let \(\mathcal{F}\), \(\mathcal{G}\) be presheaves of \(\mathcal{O}\)-modules. Consider the rule \[U \longmapsto \Hom_{\mathcal{O}_U}(\mathcal{F}|_U, \mathcal{G}|_U).\] For \(\varphi : V \to U\) in \(\mathcal{C}\) we define a restriction mapping \[\Hom_{\mathcal{O}_U}(\mathcal{F}|_U, \mathcal{G}|_U) \longrightarrow \Hom_{\mathcal{O}_V}(\mathcal{F}|_V, \mathcal{G}|_V)\] by restricting via the relocalization morphism \(j : \mathcal{C}/V \to \mathcal{C}/U\), see Sites, Lemma 03EH. Hence this defines a presheaf \(\SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G})\). In addition, given an element \(\varphi \in \Hom_{\mathcal{O}|_U}(\mathcal{F}|_U, \mathcal{G}|_U)\) and a section \(f \in \mathcal{O}(U)\) then we can define \(f\varphi \in \Hom_{\mathcal{O}|_U}(\mathcal{F}|_U, \mathcal{G}|_U)\) by either precomposing with multiplication by \(f\) on \(\mathcal{F}|_U\) or postcomposing with multiplication by \(f\) on \(\mathcal{G}|_U\) (it gives the same result). Hence we in fact get a presheaf of \(\mathcal{O}\)-modules. There is a canonical “evaluation” morphism \[\mathcal{F} \otimes_{p, \mathcal{O}} \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G}) \longrightarrow \mathcal{G}.\]

Lemma

If \(\mathcal{C}\) is a site, \(\mathcal{O}\) is a sheaf of rings, \(\mathcal{F}\) is a presheaf of \(\mathcal{O}\)-modules, and \(\mathcal{G}\) is a sheaf of \(\mathcal{O}\)-modules, then \(\SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G})\) is a sheaf of \(\mathcal{O}\)-modules.

Proof

Omitted. Hints: Note first that \(\SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G}) = \SheafHom_\mathcal{O}(\mathcal{F}^\#, \mathcal{G})\), which reduces the question to the case where both \(\mathcal{F}\) and \(\mathcal{G}\) are sheaves. The result for sheaves of sets is Sites, Lemma 04TQ.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{F}, \mathcal{G}\) be sheaves of \(\mathcal{O}\)-modules. Then formation of \(\SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G})\) commutes with restriction to \(U\) for \(U \in \Ob(\mathcal{C})\).

Proof

Immediate from the definition.

Remark

Let \(f : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D})\) be a morphism of ringed sites. Let \(\mathcal{F}, \mathcal{G}\) be sheaves of \(\mathcal{O}_\mathcal{D}\)-modules. There is a canonical map \[f^*\SheafHom_{\mathcal{O}_\mathcal{D}}(\mathcal{F}, \mathcal{G}) \longrightarrow \SheafHom_{\mathcal{O}_\mathcal{C}}(f^*\mathcal{F}, f^*\mathcal{G})\] Namely, this map is adjoint to the map \[\SheafHom_{\mathcal{O}_\mathcal{D}}(\mathcal{F}, \mathcal{G}) \longrightarrow f_*\SheafHom_{\mathcal{O}_\mathcal{C}}(f^*\mathcal{F}, f^*\mathcal{G})\] defined as follows. Say \(f\) is given by the continuous functor \(u : \mathcal{D} \to \mathcal{C}\). For sections over \(V \in \Ob(\mathcal{D})\) we use the map \[\begin{align*} \Gamma(V, \SheafHom_{\mathcal{O}_\mathcal{D}}(\mathcal{F}, \mathcal{G})) & = \Hom_{\mathcal{O}_V}(\mathcal{F}|_V, \mathcal{G}|_V) \\ & \longrightarrow \Hom_{\mathcal{O}_{u(V)}}(f^*\mathcal{F}|_{u(V)}, f^*\mathcal{G}|_{u(V)}) \\ & = \Gamma(u(V), \SheafHom_{\mathcal{O}_\mathcal{C}}(f^*\mathcal{F}, f^*\mathcal{G})) \\ & = \Gamma(V, f_*\SheafHom_{\mathcal{O}_\mathcal{C}}(f^*\mathcal{F}, f^*\mathcal{G})) \end{align*}\] where for the arrow we use pullback by the morphism \((\mathcal{C}/u(V), \mathcal{O}_{u(V)}) \to (\mathcal{D}/V, \mathcal{O}_V)\) induced by \(f\).

In the situation of Lemma 03EM the “evaluation” morphism factors through the tensor product of sheaves of modules \[\mathcal{F} \otimes_\mathcal{O} \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G}) \longrightarrow \mathcal{G}.\]

Lemma

Internal hom and (co)limits. Let \(\mathcal{C}\) be a category and let \(\mathcal{O}\) be a presheaf of rings.

1. For any presheaf of \(\mathcal{O}\)-modules \(\mathcal{F}\) the functor \[\textit{PMod}(\mathcal{O}) \longrightarrow \textit{PMod}(\mathcal{O}) , \quad \mathcal{G} \longmapsto \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G})\] commutes with arbitrary limits.

2. For any presheaf of \(\mathcal{O}\)-modules \(\mathcal{G}\) the functor \[\textit{PMod}(\mathcal{O}) \longrightarrow \textit{PMod}(\mathcal{O})^{opp} , \quad \mathcal{F} \longmapsto \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G})\] commutes with arbitrary colimits, in a formula \[\SheafHom_\mathcal{O}(\colim_i \mathcal{F}_i, \mathcal{G}) = \lim_i \SheafHom_\mathcal{O}(\mathcal{F}_i, \mathcal{G}).\]

Suppose that \(\mathcal{C}\) is a site, and \(\mathcal{O}\) is a sheaf of rings.

1. For any sheaf of \(\mathcal{O}\)-modules \(\mathcal{F}\) the functor \[\textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}) , \quad \mathcal{G} \longmapsto \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G})\] commutes with arbitrary limits.

2. For any sheaf of \(\mathcal{O}\)-modules \(\mathcal{G}\) the functor \[\textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O})^{opp} , \quad \mathcal{F} \longmapsto \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G})\] commutes with arbitrary colimits, in a formula \[\SheafHom_\mathcal{O}(\colim_i \mathcal{F}_i, \mathcal{G}) = \lim_i \SheafHom_\mathcal{O}(\mathcal{F}_i, \mathcal{G}).\]

Proof

Let \(\mathcal{I} \to \textit{PMod}(\mathcal{O})\), \(i \mapsto \mathcal{G}_i\) be a diagram. Let \(U\) be an object of the category \(\mathcal{C}\). As \(j_U^*\) is both a left and a right adjoint we see that \(\lim_i j_U^*\mathcal{G}_i = j_U^* \lim_i \mathcal{G}_i\). Hence we have \[\begin{align*} \SheafHom_\mathcal{O}(\mathcal{F}, \lim_i \mathcal{G}_i)(U) & = \Hom_{\mathcal{O}_U}(\mathcal{F}|_U, \lim_i \mathcal{G}_i|_U) \\ & = \lim_i \Hom_{\mathcal{O}_U}(\mathcal{F}|_U, \mathcal{G}_i|_U) \\ & = \lim_i \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G}_i)(U) \end{align*}\] by definition of a limit. This proves (1). Part (2) is proved in exactly the same way. Part (3) follows from (1) because the limit of a diagram of sheaves is the same as the limit in the category of presheaves. Finally, (4) follow because, in the formula we have \[\Mor_{\textit{Mod}(\mathcal{O})}( \colim_i \mathcal{F}_i, \mathcal{G}) = \Mor_{\textit{PMod}(\mathcal{O})}( \colim^{PSh}_i \mathcal{F}_i, \mathcal{G})\] as the colimit \(\colim_i \mathcal{F}_i\) is the sheafification of the colimit \(\colim^{PSh}_i \mathcal{F}_i\) in \(\textit{PMod}(\mathcal{O})\). Hence (4) follows from (2) (by the remark on limits above again).

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{F}\), \(\mathcal{G}\) be \(\mathcal{O}\)-modules.

1. If \(\mathcal{F}_2 \to \mathcal{F}_1 \to \mathcal{F} \to 0\) is an exact sequence of \(\mathcal{O}\)-modules, then \[0 \to \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G}) \to \SheafHom_\mathcal{O}(\mathcal{F}_1, \mathcal{G}) \to \SheafHom_\mathcal{O}(\mathcal{F}_2, \mathcal{G})\] is exact.

2. If \(0 \to \mathcal{G} \to \mathcal{G}_1 \to \mathcal{G}_2\) is an exact sequence of \(\mathcal{O}\)-modules, then \[0 \to \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G}) \to \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G}_1) \to \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G}_2)\] is exact.

Proof

Follows from Lemma 03EN and Homology, Lemma 010N.

Lemma

Let \(\mathcal{C}\) be a category. Let \(\mathcal{O}\) be a presheaf of rings.

1. Let \(\mathcal{F}\), \(\mathcal{G}\), \(\mathcal{H}\) be presheaves of \(\mathcal{O}\)-modules. There is a canonical isomorphism \[\SheafHom_\mathcal{O} (\mathcal{F} \otimes_{p, \mathcal{O}} \mathcal{G}, \mathcal{H}) \longrightarrow \SheafHom_\mathcal{O} (\mathcal{F}, \SheafHom_\mathcal{O}(\mathcal{G}, \mathcal{H}))\] which is functorial in all three entries (sheaf Hom in all three spots). In particular, \[\Mor_{\textit{PMod}(\mathcal{O})}( \mathcal{F} \otimes_{p, \mathcal{O}} \mathcal{G}, \mathcal{H}) = \Mor_{\textit{PMod}(\mathcal{O})}( \mathcal{F}, \SheafHom_\mathcal{O}(\mathcal{G}, \mathcal{H}))\]

2. Suppose that \(\mathcal{C}\) is a site, \(\mathcal{O}\) is a sheaf of rings, and \(\mathcal{F}\), \(\mathcal{G}\), \(\mathcal{H}\) are sheaves of \(\mathcal{O}\)-modules. There is a canonical isomorphism \[\SheafHom_\mathcal{O} (\mathcal{F} \otimes_\mathcal{O} \mathcal{G}, \mathcal{H}) \longrightarrow \SheafHom_\mathcal{O} (\mathcal{F}, \SheafHom_\mathcal{O}(\mathcal{G}, \mathcal{H}))\] which is functorial in all three entries (sheaf Hom in all three spots). In particular, \[\Mor_{\textit{Mod}(\mathcal{O})}( \mathcal{F} \otimes_\mathcal{O} \mathcal{G}, \mathcal{H}) = \Mor_{\textit{Mod}(\mathcal{O})}( \mathcal{F}, \SheafHom_\mathcal{O}(\mathcal{G}, \mathcal{H}))\]

Proof

This is the analogue of Algebra, Lemma 00DE. The proof is the same, and is omitted.

Lemma

Tensor product and colimits. Let \(\mathcal{C}\) be a category and let \(\mathcal{O}\) be a presheaf of rings.

1. For any presheaf of \(\mathcal{O}\)-modules \(\mathcal{F}\) the functor \[\textit{PMod}(\mathcal{O}) \longrightarrow \textit{PMod}(\mathcal{O}) , \quad \mathcal{G} \longmapsto \mathcal{F} \otimes_{p, \mathcal{O}} \mathcal{G}\] commutes with arbitrary colimits.

2. Suppose that \(\mathcal{C}\) is a site, and \(\mathcal{O}\) is a sheaf of rings. For any sheaf of \(\mathcal{O}\)-modules \(\mathcal{F}\) the functor \[\textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}) , \quad \mathcal{G} \longmapsto \mathcal{F} \otimes_\mathcal{O} \mathcal{G}\] commutes with arbitrary colimits.

Proof

This is because tensor product is adjoint to internal hom according to Lemma 03EO. See Categories, Lemma 0038.

Lemma

Let \(\mathcal{C}\) be a category, resp. a site Let \(\mathcal{O} \to \mathcal{O}'\) be a map of presheaves, resp. sheaves of rings. Then \[\Hom_\mathcal{O}(\mathcal{G}, \mathcal{F}) = \Hom_{\mathcal{O}'}(\mathcal{G}, \SheafHom_\mathcal{O}(\mathcal{O}', \mathcal{F}))\] for any \(\mathcal{O}'\)-module \(\mathcal{G}\) and \(\mathcal{O}\)-module \(\mathcal{F}\).

Proof

This is the analogue of Algebra, Lemma 08YP. The proof is the same, and is omitted.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(U \in \Ob(\mathcal{C})\). For \(\mathcal{G}\) in \(\textit{Mod}(\mathcal{O}_U)\) and \(\mathcal{F}\) in \(\textit{Mod}(\mathcal{O})\) we have \(j_{U!}\mathcal{G} \otimes_\mathcal{O} \mathcal{F} = j_{U!}(\mathcal{G} \otimes_{\mathcal{O}_U} \mathcal{F}|_U)\).

Proof

Let \(\mathcal{H}\) be an object of \(\textit{Mod}(\mathcal{O})\). Then \[\begin{align*} \Hom_\mathcal{O}(j_{U!}(\mathcal{G} \otimes_{\mathcal{O}_U} \mathcal{F}|_U), \mathcal{H}) & = \Hom_{\mathcal{O}_U}(\mathcal{G} \otimes_{\mathcal{O}_U} \mathcal{F}|_U, \mathcal{H}|_U) \\ & = \Hom_{\mathcal{O}_U}(\mathcal{G}, \SheafHom_{\mathcal{O}_U}(\mathcal{F}|_U, \mathcal{H}|_U)) \\ & = \Hom_{\mathcal{O}_U}(\mathcal{G}, \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{H})|_U) \\ & = \Hom_\mathcal{O}(j_{U!}\mathcal{G}, \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{H})) \\ & = \Hom_\mathcal{O}(j_{U!}\mathcal{G} \otimes_\mathcal{O} \mathcal{F}, \mathcal{H}) \end{align*}\] The first equality because \(j_{U!}\) is a left adjoint to restriction of modules. The second by Lemma 03EO. The third by Lemma 0E8H. The fourth because \(j_{U!}\) is a left adjoint to restriction of modules. The fifth by Lemma 03EO. The lemma follows from this and the Yoneda lemma.

Remark

Let \(\mathcal{C}\) be a site. Let \(\mathcal{F}\) be a sheaf of sets on \(\mathcal{C}\) and consider the localization morphism \(j : \Sh(\mathcal{C})/\mathcal{F} \to \Sh(\mathcal{C})\). See Sites, Definition 04IP. We claim that (a) \(j_!\mathbf{Z} = \mathbf{Z}_\mathcal{F}^\#\) and (b) \(j_!(j^{-1}\mathcal{H}) = j_!\mathbf{Z} \otimes_\mathbf{Z} \mathcal{H}\) for any abelian sheaf \(\mathcal{H}\) on \(\mathcal{C}\). Let \(\mathcal{G}\) be an abelian sheaf on \(\mathcal{C}\). Part (a) follows from the Yoneda lemma because \[\Hom(j_!\mathbf{Z}, \mathcal{G}) = \Hom(\mathbf{Z}, j^{-1}\mathcal{G}) = \Hom(\mathbf{Z}_\mathcal{F}^\#, \mathcal{G})\] where the second equality holds because both sides of the equality evaluate to the set of maps from \(\mathcal{F} \to \mathcal{G}\) viewed as an abelian group. For (b) we use the Yoneda lemma and \[\begin{align*} \Hom(j_!(j^{-1}\mathcal{H}), \mathcal{G}) & = \Hom(j^{-1}\mathcal{H}, j^{-1}\mathcal{G}) \\ & = \Hom(\mathbf{Z}, \SheafHom(j^{-1}\mathcal{H}, j^{-1}\mathcal{G})) \\ & = \Hom(\mathbf{Z}, j^{-1}\SheafHom(\mathcal{H}, \mathcal{G})) \\ & = \Hom(j_!\mathbf{Z}, \SheafHom(\mathcal{H}, \mathcal{G})) \\ & = \Hom(j_!\mathbf{Z} \otimes_\mathbf{Z} \mathcal{H}, \mathcal{G}) \end{align*}\] Here we use adjunction, the fact that taking \(\SheafHom\) commutes with localization, and Lemma 03EO.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{F}\) be an \(\mathcal{O}\)-module of finite presentation. Let \(\mathcal{G} = \colim_{\lambda \in \Lambda} \mathcal{G}_\lambda\) be a filtered colimit of \(\mathcal{O}\)-modules. Then the canonical map \[\colim_\lambda \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G}_\lambda) \longrightarrow \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G})\] is an isomorphism.

Proof

It suffices to show the arrow is an isomorphism after restriction to \(U\) for all \(U\) in \(\mathcal{C}\). Both taking colimits of sheaves of modules and taking internal hom commute with restriction to \(U\). See for example Lemmas 03DC and 0E8H. Fix \(U\). Given a covering \(\{U_i \to U\}_{i \in I}\), then it suffices to prove the restriction to each \(U_i\) is an isomorphism. Hence we may assume \(\mathcal{F}\) has a global presentation \[\bigoplus\nolimits_{j = 1, \ldots, m} \mathcal{O} \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} \mathcal{O} \to \mathcal{F} \to 0\] The functor \(\SheafHom_\mathcal{O}(-, -)\) commutes with finite direct sums in either variable and \(\SheafHom_\mathcal{O}(\mathcal{O}, -)\) is the identity functor. By this and by Lemma 0GMY we obtain an exact sequence \[0 \to \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G}) \to \bigoplus\nolimits_{i = 1, \ldots, n} \mathcal{G} \to \bigoplus\nolimits_{j = 1, \ldots, m} \mathcal{G}\] Since filtered colimits are exact in \(\textit{Mod}(\mathcal{O})\) by Lemma 03DB also the top row in the following commutative diagram is exact \[\xymatrix{ 0 \ar[r] & \colim_\lambda \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G}_\lambda) \ar[r] \ar[d] & \colim_\lambda \bigoplus\nolimits_{i = 1, \ldots, n} \mathcal{G}_\lambda \ar[r] \ar[d] & \colim_\lambda \bigoplus\nolimits_{j = 1, \ldots, m} \mathcal{G}_\lambda \ar[d] \\ 0 \ar[r] & \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G}) \ar[r] & \bigoplus\nolimits_{i = 1, \ldots, n} \mathcal{G} \ar[r] & \bigoplus\nolimits_{j = 1, \ldots, m} \mathcal{G} }\] Since the right two vertical arrows are isomorphisms we conclude.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{G} = \colim_{\lambda \in \Lambda} \mathcal{G}_\lambda\) be a filtered colimit of \(\mathcal{O}\)-modules. Let \(\mathcal{F}\) be an \(\mathcal{O}\)-module of finite presentation. Then we have \[\colim_\lambda \Hom_\mathcal{O}(\mathcal{F}, \mathcal{G}_\lambda) = \Hom_\mathcal{O}(\mathcal{F}, \mathcal{G}).\] if the hypotheses of Sites, Lemma 0GMR part (4) are satisfied for the site \(\mathcal{C}\); please see Sites, Remark 0GMS.

Proof

Set \(\mathcal{H} = \SheafHom_\mathcal{O}(\mathcal{F}, \colim \mathcal{G}_\lambda)\) and \(\mathcal{H}_\lambda = \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G}_\lambda)\). Recall that \[\Hom_\mathcal{O}(\mathcal{F}, \mathcal{G}) = \Gamma(\mathcal{C}, \mathcal{H}) \quad\text{and}\quad \Hom_\mathcal{O}(\mathcal{F}, \mathcal{G}_\lambda) = \Gamma(\mathcal{C}, \mathcal{H}_\lambda)\] by construction. By Lemma 0GMZ we have \(\mathcal{H} = \colim \mathcal{H}_\lambda\). Thus the lemma follows from Sites, Lemma 0GMR.

Flat modules

We can define flat modules exactly as in the case of modules over rings.

Definition

Let \(\mathcal{C}\) be a category. Let \(\mathcal{O}\) be a presheaf of rings.

1. A presheaf \(\mathcal{F}\) of \(\mathcal{O}\)-modules is called flat if the functor \[\textit{PMod}(\mathcal{O}) \longrightarrow \textit{PMod}(\mathcal{O}), \quad \mathcal{G} \mapsto \mathcal{G} \otimes_{p, \mathcal{O}} \mathcal{F}\] is exact.

2. A map \(\mathcal{O} \to \mathcal{O}'\) of presheaves of rings is called flat if \(\mathcal{O}'\) is flat as a presheaf of \(\mathcal{O}\)-modules.

3. If \(\mathcal{C}\) is a site, \(\mathcal{O}\) is a sheaf of rings and \(\mathcal{F}\) is a sheaf of \(\mathcal{O}\)-modules, then we say \(\mathcal{F}\) is flat if the functor \[\textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}), \quad \mathcal{G} \mapsto \mathcal{G} \otimes_\mathcal{O} \mathcal{F}\] is exact.

4. A map \(\mathcal{O} \to \mathcal{O}'\) of sheaves of rings on a site is called flat if \(\mathcal{O}'\) is flat as a sheaf of \(\mathcal{O}\)-modules.

The notion of a flat module or flat ring map is intrinsic (Section 03DG).

Lemma

Let \(\mathcal{C}\) be a category. Let \(\mathcal{O}\) be a presheaf of rings. Let \(\mathcal{F}\) be a presheaf of \(\mathcal{O}\)-modules. If each \(\mathcal{F}(U)\) is a flat \(\mathcal{O}(U)\)-module, then \(\mathcal{F}\) is flat.

Proof

This is immediate from the definitions.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O}\) be a presheaf of rings. Let \(\mathcal{F}\) be a presheaf of \(\mathcal{O}\)-modules. If \(\mathcal{F}\) is a flat \(\mathcal{O}\)-module, then \(\mathcal{F}^\#\) is a flat \(\mathcal{O}^\#\)-module.

Proof

Omitted. (Hint: Sheafification is exact.)

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O}\) be a presheaf of rings. Let \(\mathcal{F}\) be a presheaf of \(\mathcal{O}\)-modules. Assume that every object \(U\) of \(\mathcal{C}\) has a covering \(\{U_i \to U\}_{i \in I}\) such that \(\mathcal{F}(U_i)\) is a flat \(\mathcal{O}(U_i)\)-module. Then \(\mathcal{F}^\#\) is a flat \(\mathcal{O}^\#\)-module.

Proof

Let \(\mathcal{G} \subset \mathcal{G}'\) be an inclusion of \(\mathcal{O}^\#\)-modules. We have to show that \[\mathcal{G} \otimes_{\mathcal{O}^\#} \mathcal{F}^\# \longrightarrow \mathcal{G}' \otimes_{\mathcal{O}^\#} \mathcal{F}^\#\] is injective. By Lemma 0GMW the source of this arrow is the sheafification of the presheaf \(\mathcal{G} \otimes_{p, \mathcal{O}} \mathcal{F}\) and similarly for the target. If \(U\) is an object of \(\mathcal{C}\) such that \(\mathcal{F}(U)\) is a flat \(\mathcal{O}(U)\)-module, then \[(\mathcal{G} \otimes_{p, \mathcal{O}} \mathcal{F})(U) = \mathcal{G}(U) \otimes_{\mathcal{O}(U)} \mathcal{F}(U) \longrightarrow \mathcal{G}'(U) \otimes_{\mathcal{O}(U)} \mathcal{F}(U) = (\mathcal{G}' \otimes_{p, \mathcal{O}} \mathcal{F})(U)\] is injective. Hence we reduce to showing: given a map of presheaves \(f : \mathcal{H} \to \mathcal{H}'\) on \(\mathcal{C}\) such that every \(U\) in \(\mathcal{C}\) has a covering \(\{U_i \to U\}_{i \in I}\) with \(\mathcal{H}(U_i) \to \mathcal{H}'(U_i)\) injective, then \(f^\#\) is injective. This we leave to the reader as an exercise.

Lemma

Colimits and tensor product.

1. A filtered colimit of flat presheaves of modules is flat. A direct sum of flat presheaves of modules is flat.

2. A filtered colimit of flat sheaves of modules is flat. A direct sum of flat sheaves of modules is flat.

Proof

Part (1) follows from Lemma 03EP and Algebra, Lemma 00DB by looking at sections over objects. To see part (2), use Lemma 03EP and the fact that a filtered colimit of exact complexes is an exact complex (this uses that sheafification is exact and commutes with colimits). Some details omitted.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(U\) be an object of \(\mathcal{C}\). If \(\mathcal{F}\) is a flat \(\mathcal{O}\)-module, then \(\mathcal{F}|_U\) is a flat \(\mathcal{O}_U\)-module.

Proof

Let \(\mathcal{G}_1 \to \mathcal{G}_2 \to \mathcal{G}_3\) be an exact complex of \(\mathcal{O}_U\)-modules. Since \(j_{U!}\) is exact (Lemma 03DJ) and \(\mathcal{F}\) is flat as an \(\mathcal{O}\)-modules then we see that the complex made up of the modules \[j_{U!}(\mathcal{G}_i \otimes_{\mathcal{O}_U} \mathcal{F}|_U) = j_{U!}\mathcal{G}_i \otimes_\mathcal{O} \mathcal{F}\] (Lemma 0E8I) is exact. We conclude that \(\mathcal{G}_1 \otimes_{\mathcal{O}_U} \mathcal{F}|_U \to \mathcal{G}_2 \otimes_{\mathcal{O}_U} \mathcal{F}|_U \to \mathcal{G}_3 \otimes_{\mathcal{O}_U} \mathcal{F}|_U\) is exact by Lemma 0E8G.

Lemma

Let \(\mathcal{C}\) be a category. Let \(\mathcal{O}\) be a presheaf of rings. Let \(U\) be an object of \(\mathcal{C}\). Consider the functor \(j_U : \mathcal{C}/U \to \mathcal{C}\).

1. The presheaf of \(\mathcal{O}\)-modules \(j_{U!}\mathcal{O}_U\) (see Remark 03EJ) is flat.

2. If \(\mathcal{C}\) is a site, \(\mathcal{O}\) is a sheaf of rings, \(j_{U!}\mathcal{O}_U\) is a flat sheaf of \(\mathcal{O}\)-modules.

Proof

Proof of (1). By the discussion in Remark 03EJ we see that \[j_{U!}\mathcal{O}_U(V) = \bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{O}(V)\] which is a flat \(\mathcal{O}(V)\)-module. Hence (1) follows from Lemma 03ES. Then (2) follows as \(j_{U!}\mathcal{O}_U = (j_{U!}\mathcal{O}_U)^\#\) (the first \(j_{U!}\) on sheaves, the second on presheaves) and Lemma 03ET.

Lemma

Let \(\mathcal{C}\) be a category. Let \(\mathcal{O}\) be a presheaf of rings.

1. Any presheaf of \(\mathcal{O}\)-modules is a quotient of a direct sum \(\bigoplus j_{U_i!}\mathcal{O}_{U_i}\).

2. Any presheaf of \(\mathcal{O}\)-modules is a quotient of a flat presheaf of \(\mathcal{O}\)-modules.

3. If \(\mathcal{C}\) is a site, \(\mathcal{O}\) is a sheaf of rings, then any sheaf of \(\mathcal{O}\)-modules is a quotient of a direct sum \(\bigoplus j_{U_i!}\mathcal{O}_{U_i}\).

4. If \(\mathcal{C}\) is a site, \(\mathcal{O}\) is a sheaf of rings, then any sheaf of \(\mathcal{O}\)-modules is a quotient of a flat sheaf of \(\mathcal{O}\)-modules.

Proof

Proof of (1). For every object \(U\) of \(\mathcal{C}\) and every \(s \in \mathcal{F}(U)\) we get a morphism \(j_{U!}\mathcal{O}_U \to \mathcal{F}\), namely the adjoint to the morphism \(\mathcal{O}_U \to \mathcal{F}|_U\), \(1 \mapsto s\). Clearly the map \[\bigoplus\nolimits_{(U, s)} j_{U!}\mathcal{O}_U \longrightarrow \mathcal{F}\] is surjective. The source is flat by combining Lemmas 03EU and 03EV which proves (2). The sheaf case follows from this either by sheafifying or repeating the same argument.

Lemma

Let \(\mathcal{C}\) be a category. Let \(\mathcal{O}\) be a presheaf of rings. Let \[0 \to \mathcal{F}'' \to \mathcal{F}' \to \mathcal{F} \to 0\] be a short exact sequence of presheaves of \(\mathcal{O}\)-modules. Let \(\mathcal{G}\) be a presheaf of \(\mathcal{O}\)-modules.

1. If \(\mathcal{F}\) is a flat presheaf of modules, then the sequence \[0 \to \mathcal{F}'' \otimes_{p, \mathcal{O}} \mathcal{G} \to \mathcal{F}' \otimes_{p, \mathcal{O}} \mathcal{G} \to \mathcal{F} \otimes_{p, \mathcal{O}} \mathcal{G} \to 0\] is exact.

2. If \(\mathcal{C}\) is a site, \(\mathcal{O}\), \(\mathcal{F}\), \(\mathcal{F}'\), \(\mathcal{F}''\), and \(\mathcal{G}\) are sheaves, and \(\mathcal{F}\) is flat as a sheaf of modules, then the sequence \[0 \to \mathcal{F}'' \otimes_\mathcal{O} \mathcal{G} \to \mathcal{F}' \otimes_\mathcal{O} \mathcal{G} \to \mathcal{F} \otimes_\mathcal{O} \mathcal{G} \to 0\] is exact.

Proof

Choose a flat presheaf of \(\mathcal{O}\)-modules \(\mathcal{G}'\) which surjects onto \(\mathcal{G}\). This is possible by Lemma 03EW. Let \(\mathcal{G}'' = \Ker(\mathcal{G}' \to \mathcal{G})\). The lemma follows by applying the snake lemma to the following diagram \[\begin{matrix} & & 0 & & 0 & & 0 & & \\ & & \uparrow & & \uparrow & & \uparrow & & \\ & & \mathcal{F}'' \otimes_{p, \mathcal{O}} \mathcal{G} & \to & \mathcal{F}' \otimes_{p, \mathcal{O}} \mathcal{G} & \to & \mathcal{F} \otimes_{p, \mathcal{O}} \mathcal{G} & \to & 0 \\ & & \uparrow & & \uparrow & & \uparrow & & \\ 0 & \to & \mathcal{F}'' \otimes_{p, \mathcal{O}} \mathcal{G}' & \to & \mathcal{F}' \otimes_{p, \mathcal{O}} \mathcal{G}' & \to & \mathcal{F} \otimes_{p, \mathcal{O}} \mathcal{G}' & \to & 0 \\ & & \uparrow & & \uparrow & & \uparrow & & \\ & & \mathcal{F}'' \otimes_{p, \mathcal{O}} \mathcal{G}'' & \to & \mathcal{F}' \otimes_{p, \mathcal{O}} \mathcal{G}'' & \to & \mathcal{F} \otimes_{p, \mathcal{O}} \mathcal{G}'' & \to & 0 \\ & & & & & & \uparrow & & \\ & & & & & & 0 & & \end{matrix}\] with exact rows and columns. The middle row is exact because tensoring with the flat module \(\mathcal{G}'\) is exact. The proof in the case of sheaves is exactly the same.

Lemma

Let \(\mathcal{C}\) be a category. Let \(\mathcal{O}\) be a presheaf of rings. Let \[0 \to \mathcal{F}_2 \to \mathcal{F}_1 \to \mathcal{F}_0 \to 0\] be a short exact sequence of presheaves of \(\mathcal{O}\)-modules.

1. If \(\mathcal{F}_2\) and \(\mathcal{F}_0\) are flat so is \(\mathcal{F}_1\).

2. If \(\mathcal{F}_1\) and \(\mathcal{F}_0\) are flat so is \(\mathcal{F}_2\).

If \(\mathcal{C}\) is a site and \(\mathcal{O}\) is a sheaf of rings then the same result holds in \(\textit{Mod}(\mathcal{O})\).

Proof

Let \(\mathcal{G}^\bullet\) be an arbitrary exact complex of presheaves of \(\mathcal{O}\)-modules. Assume that \(\mathcal{F}_0\) is flat. By Lemma 03EX we see that \[0 \to \mathcal{G}^\bullet \otimes_{p, \mathcal{O}} \mathcal{F}_2 \to \mathcal{G}^\bullet \otimes_{p, \mathcal{O}} \mathcal{F}_1 \to \mathcal{G}^\bullet \otimes_{p, \mathcal{O}} \mathcal{F}_0 \to 0\] is a short exact sequence of complexes of presheaves of \(\mathcal{O}\)-modules. Hence (1) and (2) follow from the snake lemma. The case of sheaves of modules is proved in the same way.

Lemma

Let \(\mathcal{C}\) be a category. Let \(\mathcal{O}\) be a presheaf of rings. Let \[\ldots \to \mathcal{F}_2 \to \mathcal{F}_1 \to \mathcal{F}_0 \to \mathcal{Q} \to 0\] be an exact complex of presheaves of \(\mathcal{O}\)-modules. If \(\mathcal{Q}\) and all \(\mathcal{F}_i\) are flat \(\mathcal{O}\)-modules, then for any presheaf \(\mathcal{G}\) of \(\mathcal{O}\)-modules the complex \[\ldots \to \mathcal{F}_2 \otimes_{p, \mathcal{O}} \mathcal{G} \to \mathcal{F}_1 \otimes_{p, \mathcal{O}} \mathcal{G} \to \mathcal{F}_0 \otimes_{p, \mathcal{O}} \mathcal{G} \to \mathcal{Q} \otimes_{p, \mathcal{O}} \mathcal{G} \to 0\] is exact also. If \(\mathcal{C}\) is a site and \(\mathcal{O}\) is a sheaf of rings then the same result holds \(\textit{Mod}(\mathcal{O})\).

Proof

Follows from Lemma 03EX by splitting the complex into short exact sequences and using Lemma 03EY to prove inductively that \(\Im(\mathcal{F}_{i + 1} \to \mathcal{F}_i)\) is flat.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. If \(\mathcal{G}\) and \(\mathcal{F}\) are flat \(\mathcal{O}\)-modules, then \(\mathcal{G} \otimes_\mathcal{O} \mathcal{F}\) is a flat \(\mathcal{O}\)-module.

Proof

This is true because \[(\mathcal{G} \otimes_\mathcal{O} \mathcal{F}) \otimes_\mathcal{O} \mathcal{H} = \mathcal{G} \otimes_\mathcal{O} (\mathcal{F} \otimes_\mathcal{O} \mathcal{H})\] and a composition of exact functors is exact.

Lemma

Let \(\mathcal{O}_1 \to \mathcal{O}_2\) be a map of sheaves of rings on a site \(\mathcal{C}\). If \(\mathcal{G}\) is a flat \(\mathcal{O}_1\)-module, then \(\mathcal{G} \otimes_{\mathcal{O}_1} \mathcal{O}_2\) is a flat \(\mathcal{O}_2\)-module.

Proof

This is true because \[(\mathcal{G} \otimes_{\mathcal{O}_1} \mathcal{O}_2) \otimes_{\mathcal{O}_2} \mathcal{H} = \mathcal{G} \otimes_{\mathcal{O}_1} \mathcal{H}\] (as sheaves of abelian groups for example).

The following lemma is the analogue of the equational criterion of flatness (Algebra, Lemma 00HK).

Lemma

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

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

2. Let \(U\) be an object of \(\mathcal{C}\) and let \[\mathcal{O}_U \xrightarrow{(f_1, \ldots, f_n)} \mathcal{O}_U^{\oplus n} \xrightarrow{(s_1, \ldots, s_n)} \mathcal{F}|_U\] be a complex of \(\mathcal{O}_U\)-modules. Then there exists a covering \(\{U_i \to U\}\) and for each \(i\) a factorization \[\mathcal{O}_{U_i}^{\oplus n} \xrightarrow{B_i} \mathcal{O}_{U_i}^{\oplus l_i} \xrightarrow{(t_{i1}, \ldots, t_{il_i})} \mathcal{F}|_{U_i}\] of \((s_1, \ldots, s_n)|_{U_i}\) such that \(B_i \circ (f_1, \ldots, f_n)|_{U_i} = 0\).

3. Let \(U\) be an object of \(\mathcal{C}\) and let \[\mathcal{O}_U^{\oplus m} \xrightarrow{A} \mathcal{O}_U^{\oplus n} \xrightarrow{(s_1, \ldots, s_n)} \mathcal{F}|_U\] be a complex of \(\mathcal{O}_U\)-modules. Then there exists a covering \(\{U_i \to U\}\) and for each \(i\) a factorization \[\mathcal{O}_{U_i}^{\oplus n} \xrightarrow{B_i} \mathcal{O}_{U_i}^{\oplus l_i} \xrightarrow{(t_{i1}, \ldots, t_{il_i})} \mathcal{F}|_{U_i}\] of \((s_1, \ldots, s_n)|_{U_i}\) such that \(B_i \circ A|_{U_i} = 0\).

Proof

Assume (1). Let \(\mathcal{I} \subset \mathcal{O}_U\) be the sheaf of ideals generated by \(f_1, \ldots, f_n\). Then \(\sum f_j \otimes s_j\) is a section of \(\mathcal{I} \otimes_{\mathcal{O}_U} \mathcal{F}|_U\) which maps to zero in \(\mathcal{F}|_U\). As \(\mathcal{F}|_U\) is flat (Lemma 0E8J) the map \(\mathcal{I} \otimes_{\mathcal{O}_U} \mathcal{F}|_U \to \mathcal{F}|_U\) is injective. Since \(\mathcal{I} \otimes_{\mathcal{O}_U} \mathcal{F}|_U\) is the sheaf associated to the presheaf tensor product, we see there exists a covering \(\{U_i \to U\}\) such that \(\sum f_j|_{U_i} \otimes s_j|_{U_i}\) is zero in \(\mathcal{I}(U_i) \otimes_{\mathcal{O}(U_i)} \mathcal{F}(U_i)\). Unwinding the definitions using Algebra, Lemma 04VX we find \(t_{i1}, \ldots, t_{i l_i} \in \mathcal{F}(U_i)\) and \(a_{ijk} \in \mathcal{O}(U_i)\) such that \(\sum_j a_{ijk}f_j|_{U_i} = 0\) and \(s_j|_{U_i} = \sum_k a_{ijk}t_{ik}\). Thus (2) holds.

Assume (2). Let \(U\), \(n\), \(m\), \(A\) and \(s_1, \ldots, s_n\) as in (3) be given. Observe that \(A\) has \(m\) columns. We will prove the assertion of (3) is true by induction on \(m\). For the base case \(m = 0\) we can take the covering \(\{U \to U\}\) and for the factorization we use \(\mathcal{O}_U^n \to \mathcal{O}_U^n \to \mathcal{F}|_U\) where the first map is the identity and the second is \((s_1, \ldots, s_n)\). Assume \(m > 0\). Let \((f_1, \ldots, f_n)\) be the last column of \(A\) and apply (2). This gives new diagrams \[\mathcal{O}_{U_i}^{\oplus m} \xrightarrow{B_i \circ A|_{U_i}} \mathcal{O}_{U_i}^{\oplus l_i} \xrightarrow{(t_{i1}, \ldots, t_{il_i})} \mathcal{F}|_{U_i}\] but the last column of \(A_i = B_i \circ A|_{U_i}\) is zero. Hence we can apply the induction hypothesis to \(U_i\), \(l_i\), \(m - 1\), the matrix consisting of the first \(m - 1\) columns of \(A_i\), and \(t_{i1}, \ldots, t_{il_i}\) to get coverings \(\{U_{ij} \to U_i\}\) and factorizations \[\mathcal{O}_{U_{ij}}^{\oplus l_i} \xrightarrow{C_{ij}} \mathcal{O}_{U_{ij}}^{\oplus k_{ij}} \xrightarrow{(v_{ij1}, \ldots, v_{ij k_{ij}})} \mathcal{F}|_{U_{ij}}\] of \((t_{i1}, \ldots, t_{il_i})|_{U_{ij}}\) such that \(C_{ij} \circ B_i|_{U_{ij}} \circ A|_{U_{ij}} = 0\). Then \(\{U_{ij} \to U\}\) is a covering and we get the desired factorizations using \(B_{ij} = C_{ij} \circ B_i|_{U_{ij}}\) and \(v_{ija}\). In this way we see that (2) implies (3).

Assume (3). Let \(\mathcal{G} \to \mathcal{H}\) be an injective homomorphism of \(\mathcal{O}\)-modules. We have to show that \(\mathcal{G} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{H} \otimes_\mathcal{O} \mathcal{F}\) is injective. Let \(U\) be an object of \(\mathcal{C}\) and let \(s \in (\mathcal{G} \otimes_\mathcal{O} \mathcal{F})(U)\) be a section which maps to zero in \(\mathcal{H} \otimes_\mathcal{O} \mathcal{F}\). We have to show that \(s\) is zero. Since \(\mathcal{G} \otimes_\mathcal{O} \mathcal{F}\) is a sheaf, it suffices to find a covering \(\{U_i \to U\}_{i \in I}\) of \(\mathcal{C}\) such that \(s|_{U_i}\) is zero for all \(i \in I\). Hence we may always replace \(U\) by the members of a covering. In particular, since \(\mathcal{G} \otimes_\mathcal{O} \mathcal{F}\) is the sheafification of \(\mathcal{G} \otimes_{p, \mathcal{O}} \mathcal{F}\) we may assume that \(s\) is the image of \(s' \in \mathcal{G}(U) \otimes_{\mathcal{O}(U)} \mathcal{F}(U)\). Arguing similarly for \(\mathcal{H} \otimes_\mathcal{O} \mathcal{F}\) we may assume that \(s'\) maps to zero in \(\mathcal{H}(U) \otimes_{\mathcal{O}(U)} \mathcal{F}(U)\). Write \(\mathcal{F}(U) = \colim M_\alpha\) as a filtered colimit of finitely presented \(\mathcal{O}(U)\)-modules \(M_\alpha\) (Algebra, Lemma 00HA). Since tensor product commutes with filtered colimits (Algebra, Lemma 00DD) we can choose an \(\alpha\) such that \(s'\) comes from some \(s'' \in \mathcal{G}(U) \otimes_{\mathcal{O}(U)} M_\alpha\) and such that \(s''\) maps to zero in \(\mathcal{H}(U) \otimes_{\mathcal{O}(U)} M_\alpha\). Fix \(\alpha\) and \(s''\). Choose a presentation \[\mathcal{O}(U)^{\oplus m} \xrightarrow{A} \mathcal{O}(U)^{\oplus n} \to M_\alpha \to 0\] We apply (3) to the corresponding complex of \(\mathcal{O}_U\)-modules \[\mathcal{O}_U^{\oplus m} \xrightarrow{A} \mathcal{O}_U^{\oplus n} \xrightarrow{(s_1, \ldots, s_n)} \mathcal{F}|_U\] After replacing \(U\) by the members of the covering \(U_i\) we find that the map \[M_\alpha \to \mathcal{F}(U)\] factors through a free module \(\mathcal{O}(U)^{\oplus l}\) for some \(l\). Since \(\mathcal{G}(U) \to \mathcal{H}(U)\) is injective we conclude that \[\mathcal{G}(U) \otimes_{\mathcal{O}(U)} \mathcal{O}(U)^{\oplus l} \to \mathcal{H}(U) \otimes_{\mathcal{O}(U)} \mathcal{O}(U)^{\oplus l}\] is injective too. Hence as \(s''\) maps to zero in the module on the right, it also maps to zero in the module on the left, i.e., \(s\) is zero as desired.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O}' \to \mathcal{O}\) be a surjection of sheaves of rings whose kernel \(\mathcal{I}\) is an ideal of square zero. Let \(\mathcal{F}'\) be an \(\mathcal{O}'\)-module and set \(\mathcal{F} = \mathcal{F}'/\mathcal{I}\mathcal{F}'\). The following are equivalent

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

2. \(\mathcal{F}\) is a flat \(\mathcal{O}\)-module and \(\mathcal{I} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{F}'\) is injective.

Proof

If (1) holds, then \(\mathcal{F} = \mathcal{F}' \otimes_{\mathcal{O}'} \mathcal{O}\) is flat over \(\mathcal{O}\) by Lemma 05V4 and we see the map \(\mathcal{I} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{F}'\) is injective by applying \(- \otimes_{\mathcal{O}'} \mathcal{F}'\) to the exact sequence \(0 \to \mathcal{I} \to \mathcal{O}' \to \mathcal{O} \to 0\), see Lemma 03EX. Assume (2). In the rest of the proof we will use without further mention that \(\mathcal{K} \otimes_{\mathcal{O}'} \mathcal{F}' = \mathcal{K} \otimes_\mathcal{O} \mathcal{F}\) for any \(\mathcal{O}'\)-module \(\mathcal{K}\) annihilated by \(\mathcal{I}\). Let \(\alpha : \mathcal{G}' \to \mathcal{H}'\) be an injective map of \(\mathcal{O}'\)-modules. Let \(\mathcal{G} \subset \mathcal{G}'\), resp. \(\mathcal{H} \subset \mathcal{H}'\) be the subsheaf of sections annihilated by \(\mathcal{I}\). Consider the diagram \[\xymatrix{ \mathcal{G} \otimes_{\mathcal{O}'} \mathcal{F}' \ar[r] \ar[d] & \mathcal{G}' \otimes_{\mathcal{O}'} \mathcal{F}' \ar[r] \ar[d] & \mathcal{G}'/\mathcal{G} \otimes_{\mathcal{O}'} \mathcal{F}' \ar[r] \ar[d] & 0 \\ \mathcal{H} \otimes_{\mathcal{O}'} \mathcal{F}' \ar[r] & \mathcal{H}' \otimes_{\mathcal{O}'} \mathcal{F}' \ar[r] & \mathcal{H}'/\mathcal{H} \otimes_{\mathcal{O}'} \mathcal{F}' \ar[r] & 0 }\] Note that \(\mathcal{G}'/\mathcal{G}\) and \(\mathcal{H}'/\mathcal{H}\) are annihilated by \(\mathcal{I}\) and that \(\mathcal{G}'/\mathcal{G} \to \mathcal{H}'/\mathcal{H}\) is injective. Thus the right vertical arrow is injective as \(\mathcal{F}\) is flat over \(\mathcal{O}\). The same is true for the left vertical arrow. Hence the middle vertical arrow is injective and \(\mathcal{F}'\) is flat.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O} \to \mathcal{O}'\) be a flat homomorphism of sheaves of rings. Let \(\mathcal{I} \subset \mathcal{O}\) be a sheaf of ideals such that the induced map \(\mathcal{O}/\mathcal{I} \to \mathcal{O}'/\mathcal{I}\mathcal{O}'\) is an isomorphism. For any \(\mathcal{O}\)-module \(\mathcal{F}\) annihilated by \(\mathcal{I}^n\) for some \(n \geq 0\) the map \(\text{id} \otimes 1 : \mathcal{F} \to \mathcal{F} \otimes_\mathcal{O} \mathcal{O}'\) is an isomorphism.

Proof

Omitted. Hint: See More on Algebra, Lemma 05E9.

Duals

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. The category of \(\mathcal{O}\)-modules endowed with the tensor product constructed in Section 03EK is a symmetric monoidal category. For an \(\mathcal{O}\)-module \(\mathcal{F}\) the following are equivalent

1. \(\mathcal{F}\) has a left dual in the monoidal category of \(\mathcal{O}\)-modules,

2. for every object \(U\) of \(\mathcal{C}\) there exists a covering \(\{U_i \to U\}\) such that \(\mathcal{F}|_{U_i}\) is a direct summand of a finite free \(\mathcal{O}|_{U_i}\)-module, and

3. \(\mathcal{F}\) is of finite presentation and flat as an \(\mathcal{O}\)-module.

This is proved in Example 0FNY and Lemmas 0FNZ and 08FD of this section.

Example

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{F}\) be an \(\mathcal{O}\)-module such that for every object \(U\) of \(\mathcal{C}\) there exists a covering \(\{U_i \to U\}\) such that \(\mathcal{F}|_{U_i}\) is a direct summand of a finite free \(\mathcal{O}|_{U_i}\)-module. Then the map \[\mathcal{F} \otimes_\mathcal{O} \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{O}) \longrightarrow \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{F})\] is an isomorphism. Namely, this is a local question, it is true if \(\mathcal{F}\) is finite free, and it holds for any summand of a module for which it is true (details omitted). Denote \[\eta : \mathcal{O} \longrightarrow \mathcal{F} \otimes_\mathcal{O} \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{O})\] the map sending \(1\) to the section corresponding to \(\text{id}_\mathcal{F}\) under the isomorphism above. Denote \[\epsilon : \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{O}) \otimes_\mathcal{O} \mathcal{F} \longrightarrow \mathcal{O}\] the evaluation map. Then we see that \(\SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{O}), \eta, \epsilon\) is a left dual for \(\mathcal{F}\) as in Categories, Definition 0FFP. We omit the verification that \((1 \otimes \epsilon) \circ (\eta \otimes 1) = \text{id}_\mathcal{F}\) and \((\epsilon \otimes 1) \circ (1 \otimes \eta) = \text{id}_{\SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{O})}\).

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{F}\) be a \(\mathcal{O}\)-module. Let \(\mathcal{G}, \eta, \epsilon\) be a left dual of \(\mathcal{F}\) in the monoidal category of \(\mathcal{O}\)-modules, see Categories, Definition 0FFP. Then

1. for every object \(U\) of \(\mathcal{C}\) there exists a covering \(\{U_i \to U\}\) such that \(\mathcal{F}|_{U_i}\) is a direct summand of a finite free \(\mathcal{O}|_{U_i}\)-module,

2. the map \(e : \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{O}) \to \mathcal{G}\) sending a local section \(\lambda\) to \((\lambda \otimes 1)(\eta)\) is an isomorphism,

3. we have \(\epsilon(f, g) = e^{-1}(g)(f)\) for local sections \(f\) and \(g\) of \(\mathcal{F}\) and \(\mathcal{G}\).

Proof

The assumptions mean that \[\mathcal{F} \xrightarrow{\eta \otimes 1} \mathcal{F} \otimes_\mathcal{O} \mathcal{G} \otimes_\mathcal{O} \mathcal{F} \xrightarrow{1 \otimes \epsilon} \mathcal{F} \quad\text{and}\quad \mathcal{G} \xrightarrow{1 \otimes \eta} \mathcal{G} \otimes_\mathcal{O} \mathcal{F} \otimes_\mathcal{O} \mathcal{G} \xrightarrow{\epsilon \otimes 1} \mathcal{G}\] are the identity map. Let \(U\) be an object of \(\mathcal{C}\). After replacing \(U\) by the members of a covering of \(U\), we can find a finite number of sections \(f_1, \ldots, f_n\) and \(g_1, \ldots, g_n\) of \(\mathcal{F}\) and \(\mathcal{G}\) over \(U\) such that \(\eta(1) = \sum f_i g_i\). Denote \[\mathcal{O}_U^{\oplus n} \to \mathcal{F}|_U\] the map sending the \(i\)th basis vector to \(f_i\). Then we can factor the map \(\eta|_U\) over a map \(\tilde \eta : \mathcal{O}_U \to \mathcal{O}_U^{\oplus n} \otimes_{\mathcal{O}_U} \mathcal{G}|_U\). We obtain a commutative diagram \[\xymatrix{ \mathcal{F}|_U \ar[rr]_-{\eta \otimes 1} \ar[rrd]_-{\tilde \eta \otimes 1} & & \mathcal{F}|_U \otimes \mathcal{G}|_U \otimes \mathcal{F}|_U \ar[r]_-{1 \otimes \epsilon} & \mathcal{F}|_U \\ & & \mathcal{O}_U^{\oplus n} \otimes \mathcal{G}|_U \otimes \mathcal{F}|_U \ar[u] \ar[r]^-{1 \otimes \epsilon} & \mathcal{O}_U^{\oplus n} \ar[u] }\] This shows that the identity on \(\mathcal{F}|_U\) factors through a finite free \(\mathcal{O}_U\)-module. This proves (1). Part (2) follows from Categories, Lemma 0FFQ and its proof. Part (3) follows from the first equality of the proof. You can also deduce (2) and (3) from the uniqueness of left duals (Categories, Remark 0FFR) and the construction of the left dual in Example 0FNY.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{F}\) be locally of finite presentation and flat. Then given an object \(U\) of \(\mathcal{C}\) there exists a covering \(\{U_i \to U\}\) such that \(\mathcal{F}|_{U_i}\) is a direct summand of a finite free \(\mathcal{O}_{U_i}\)-module.

Proof

Choose an object \(U\) of \(\mathcal{C}\). After replacing \(U\) by the members of a covering, we may assume there exists a presentation \[\mathcal{O}_U^{\oplus r} \to \mathcal{O}_U^{\oplus n} \to \mathcal{F}|_U \to 0\] By Lemma 08FC we may, after replacing \(U\) by the members of a covering, assume there exists a factorization \[\mathcal{O}_U^{\oplus n} \to \mathcal{O}_U^{\oplus n_1} \to \mathcal{F}|_U\] such that the composition \(\mathcal{O}_U^{\oplus r} \to \mathcal{O}_U^{\oplus n} \to \mathcal{O}_U^{\oplus n_r}\) is zero. This means that the surjection \(\mathcal{O}_U^{\oplus n_r} \to \mathcal{F}|_U\) has a section and we win.

Towards constructible modules

Recall that a quasi-compact object of a site is roughly an object such that every covering of it can be refined by a finite covering (the actual definition is slightly more involved, see Sites, Section 090G). It turns out that if every object of a site has a covering by quasi-compact objects, then the modules \(j_!\mathcal{O}_U\) with \(U\) quasi-compact form a particularly nice set of generators for the category of all modules.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\{U_i \to U\}\) be a covering of \(\mathcal{C}\). Then the sequence \[\bigoplus j_{U_i \times_U U_j!}\mathcal{O}_{U_i \times_U U_j} \to \bigoplus j_{U_i!}\mathcal{O}_{U_i} \to j_!\mathcal{O}_U \to 0\] is exact.

Proof

For any \(\mathcal{O}\)-module \(\mathcal{F}\) the functor \(\Hom_\mathcal{O}(-, \mathcal{F})\) turns our sequence into the exact sequence \(0 \to \mathcal{F}(U) \to \prod \mathcal{F}(U_i) \to \prod \mathcal{F}(U_i \times_U U_j)\), see (0G1V). The lemma follows from this and Homology, Lemma 05AA.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) be covering of \(\mathcal{C}\). If \(U\) is quasi-compact, then there exist a finite subset \(I' \subset I\) such that the sequence \[\bigoplus\nolimits_{i, i' \in I'} j_{U_i \times_U U_{i'}!}\mathcal{O}_{U_i \times_U U_{i'}} \to \bigoplus\nolimits_{i \in I'} j_{U_i!}\mathcal{O}_{U_i} \to j_!\mathcal{O}_U \to 0\] is exact.

Proof

This lemma is immediate from Lemma 0934 if \(U\) satisfies condition (3) of Sites, Lemma 0D05. We urge the reader to skip the proof in the general case. By definition there exists a covering \(\mathcal{V} = \{V_j \to U\}_{j \in J}\) and a morphism \(\mathcal{V} \to \mathcal{U}\) of families of maps with fixed target given by \(\text{id} : U \to U\), \(\alpha : J \to I\), and \(f_j : V_j \to U_{\alpha(j)}\) (see Sites, Definition 00VT) such that the image \(I' \subset I\) of \(\alpha\) is finite. By Homology, Lemma 05AA it suffices to show that for any sheaf of \(\mathcal{O}\)-modules \(\mathcal{F}\) the functor \(\Hom_\mathcal{O}(-, \mathcal{F})\) turns the sequence of the lemma into an exact sequence. By (0G1V) we obtain the usual sequence \[0 \to \mathcal{F}(U) \to \prod\nolimits_{i \in I'} \mathcal{F}(U_i) \to \prod\nolimits_{i, i' \in I'} \mathcal{F}(U_i \times_U U_{i'})\] This is an exact sequence by Sites, Lemma 0G1L applied to the family of maps \(\{U_i \to U\}_{i \in I'}\) which is refined by the covering \(\mathcal{V}\).

Lemma

Let \(\mathcal{C}\) be a site. Let \(W\) be a quasi-compact object of \(\mathcal{C}\).

1. The functor \(\Sh(\mathcal{C}) \to \textit{Sets}\), \(\mathcal{F} \mapsto \mathcal{F}(W)\) commutes with coproducts.

2. Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{C}\). The functor \(\textit{Mod}(\mathcal{O}) \to \textit{Ab}\), \(\mathcal{F} \mapsto \mathcal{F}(W)\) commutes with direct sums.

Proof

Proof of (1). Taking sections over \(W\) commutes with filtered colimits with injective transition maps by Sites, Lemma 0738. If \(\mathcal{F}_i\) is a family of sheaves of sets indexed by a set \(I\). Then \(\coprod \mathcal{F}_i\) is the filtered colimit over the partially ordered set of finite subsets \(E \subset I\) of the coproducts \(\mathcal{F}_E = \coprod_{i \in E} \mathcal{F}_i\). Since the transition maps are injective we conclude.

Proof of (2). Let \(\mathcal{F}_i\) be a family of sheaves of \(\mathcal{O}\)-modules indexed by a set \(I\). Then \(\bigoplus \mathcal{F}_i\) is the filtered colimit over the partially ordered set of finite subsets \(E \subset I\) of the direct sums \(\mathcal{F}_E = \bigoplus_{i \in E} \mathcal{F}_i\). A filtered colimit of abelian sheaves can be computed in the category of sheaves of sets. Moreover, for \(E \subset E'\) the transition map \(\mathcal{F}_E \to \mathcal{F}_{E'}\) is injective (as sheafification is exact and the injectivity is clear on underlying presheaves). Hence it suffices to show the result for a finite index set by Sites, Lemma 0738. The finite case is dealt with in Lemma 03CO (it holds over any object of \(\mathcal{C}\)).

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(U\) be a quasi-compact object of \(\mathcal{C}\). Then the functor \(\Hom_\mathcal{O}(j_!\mathcal{O}_U, -)\) commutes with direct sums.

Proof

This is true because \(\Hom_\mathcal{O}(j_!\mathcal{O}_U, \mathcal{F}) = \mathcal{F}(U)\) by (0G1V) and because the functor \(\mathcal{F} \mapsto \mathcal{F}(U)\) commutes with direct sums by Lemma 0935.

In order to state the sharpest possible results in the following we introduce some notation.

Situation

Let \(\mathcal{C}\) be a site. Let \(\mathcal{B} \subset \text{Ob}(\mathcal{C})\) be a set of objects. We consider the following conditions

1. Every object of \(\mathcal{C}\) has a covering by elements of \(\mathcal{B}\).

2. Every \(U \in \mathcal{B}\) is quasi-compact (Sites, Section 090G).

3. For a covering \(\{U_i \to U\}\) with \(U_i, U \in \mathcal{B}\) the fibre products \(U_i \times_U U_j\) are quasi-compact.

Lemma

In Situation 0937 assume (0938) holds.

1. Every sheaf of sets is the target of a surjective map whose source is a coproduct \(\coprod h_{U_i}^\#\) with \(U_i\) in \(\mathcal{B}\).

2. If \(\mathcal{O}\) is a sheaf of rings, then every \(\mathcal{O}\)-module is a quotient of a direct sum \(\bigoplus\nolimits j_{U_i!}\mathcal{O}_{U_i}\) with \(U_i\) in \(\mathcal{B}\).

Proof

Part (1) follows from Sites, Lemmas 00WS and 00WT. Part (2) follows from Lemmas 03EW and 0934.

Lemma

In Situation 0937 assume (0938) and (0939) hold.

1. Every sheaf of sets is a filtered colimit of sheaves of the form [09Y7]\[\begin{equation} \text{Coequalizer}\left( \xymatrix{ \coprod\nolimits_{j = 1, \ldots, m} h_{V_j}^\# \ar@<1ex>[r] \ar@<-1ex>[r] & \coprod\nolimits_{i = 1, \ldots, n} h_{U_i}^\# } \right) \end{equation}\] with \(U_i\) and \(V_j\) in \(\mathcal{B}\).

2. If \(\mathcal{O}\) is a sheaf of rings, then every \(\mathcal{O}\)-module is a filtered colimit of sheaves of the form [093D]\[\begin{equation} \Coker\left( \bigoplus\nolimits_{j = 1, \ldots, m} j_{V_j!}\mathcal{O}_{V_j} \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} j_{U_i!}\mathcal{O}_{U_i} \right) \end{equation}\] with \(U_i\) and \(V_j\) in \(\mathcal{B}\).

Proof

Proof of (1). By Lemma 093B every sheaf of sets \(\mathcal{F}\) is the target of a surjection whose source is a coprod \(\mathcal{F}_0\) of sheaves the form \(h_{U}^\#\) with \(U \in \mathcal{B}\). Applying this to \(\mathcal{F}_0 \times_\mathcal{F} \mathcal{F}_0\) we find that \(\mathcal{F}\) is a coequalizer of a pair of maps \[\xymatrix{ \coprod\nolimits_{j \in J} h_{V_j}^\# \ar@<1ex>[r] \ar@<-1ex>[r] & \coprod\nolimits_{i \in I} h_{U_i}^\# }\] for some index sets \(I\), \(J\) and \(V_j\) and \(U_i\) in \(\mathcal{B}\). For every finite subset \(J' \subset J\) there is a finite subset \(I' \subset I\) such that the coproduct over \(j \in J'\) maps into the coprod over \(i \in I'\) via both maps, see Sites, Lemma 0738. (Details omitted; hint: an infinite coproduct is the filtered colimit of the finite sub-coproducts.) Thus our sheaf is the colimit of the cokernels of these maps between finite coproducts.

Proof of (2). By Lemma 093B every module is a quotient of a direct sum of modules of the form \(j_{U!}\mathcal{O}_U\) with \(U \in \mathcal{B}\). Thus every module is a cokernel \[\Coker\left( \bigoplus\nolimits_{j \in J} j_{V_j!}\mathcal{O}_{V_j} \longrightarrow \bigoplus\nolimits_{i \in I} j_{U_i!}\mathcal{O}_{U_i} \right)\] for some index sets \(I\), \(J\) and \(V_j\) and \(U_i\) in \(\mathcal{B}\). For every finite subset \(J' \subset J\) there is a finite subset \(I' \subset I\) such that the direct sum over \(j \in J'\) maps into the direct sum over \(i \in I'\), see Lemma 0936. Thus our module is the colimit of the cokernels of these maps between finite direct sums.

Lemma

In Situation 0937 assume (0938) and (0939) hold. Let \(\mathcal{O}\) be a sheaf of rings. Then a cokernel of a map between modules as in (093D) is another module as in (093D).

Proof

Let \(\mathcal{F} = \Coker(\bigoplus j_{V_j!}\mathcal{O}_{V_j} \to \bigoplus j_{U_i!}\mathcal{O}_{U_i})\) as in (093D). It suffices to show that the cokernel of a map \(\varphi : j_{W!}\mathcal{O}_W \to \mathcal{F}\) with \(W \in \mathcal{B}\) is another module of the same type. The map \(\varphi\) corresponds to \(s \in \mathcal{F}(W)\). Since \(\bigoplus j_{U_i!}\mathcal{O}_{U_i} \to \mathcal{F}\) is surjective, by (0938) we may choose a covering \(\{W_k \to W\}_{k \in K}\) with \(W_k \in \mathcal{B}\) such that \(s|_{W_k}\) is the image of some section \(s_k\) of \(\bigoplus j_{U_i!}\mathcal{O}_{U_i})\). By (0939) the object \(W\) is quasi-compact. By Lemma 0G1W there is a finite subset \(K' \subset K\) such that \(\bigoplus_{k \in K'} j_{W_k!}\mathcal{O}_{W_k} \to j_{W!}\mathcal{O}_W\) is surjective. We conclude that \(\Coker(\varphi)\) is equal to \[\Coker\left( \bigoplus\nolimits_{k \in K'} j_{W_k!}\mathcal{O}_{W_k} \oplus \bigoplus j_{V_j!}\mathcal{O}_{V_j} \longrightarrow \bigoplus j_{U_i!}\mathcal{O}_{U_i} \right)\] where the map \(\bigoplus_{k \in K'} j_{W_k!}\mathcal{O}_{W_k} \to \bigoplus j_{U_i!}\mathcal{O}_{U_i}\) corresponds to \(\sum_{k \in K'} s_k\). This finishes the proof.

Lemma

In Situation 0937 assume (0938), (0939), and (093A) hold. Let \(\mathcal{O}\) be a sheaf of rings. Assume given a map \[\bigoplus\nolimits_{j = 1, \ldots, m} j_{V_j!}\mathcal{O}_{V_j} \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} j_{U_i!}\mathcal{O}_{U_i}\] with \(U_i\) and \(V_j\) in \(\mathcal{B}\), and coverings \(\{U_{ik} \to U_i\}_{k \in K_i}\) with \(U_{ik} \in \mathcal{B}\). Then there exist finite subsets \(K'_i \subset K_i\) and a finite set \(L\) of \(W_l \in \mathcal{B}\) and a commutative diagram \[\xymatrix{ \bigoplus_{l \in L} j_{W_l!}\mathcal{O}_{W_l} \ar[d] \ar[r] & \bigoplus_{i = 1, \ldots, n} \bigoplus_{k \in K'_i} j_{U_{ik}!}\mathcal{O}_{U_{ik}} \ar[d] \\ \bigoplus_{j = 1, \ldots, m} j_{V_j!}\mathcal{O}_{V_j} \ar[r] & \bigoplus_{i = 1, \ldots, n} j_{U_i!}\mathcal{O}_{U_i} }\] inducing an isomorphism on cokernels of the horizontal maps.

Proof

Since \(U_i\) is quasi-compact, we may choose finite subsets \(K'_i \subset K_i\) as in Lemma 0G1W. Then since \(\bigoplus_{i = 1, \ldots, n} \bigoplus_{k \in K'_i} j_{U_{ik}!}\mathcal{O}_{U_{ik}} \to \bigoplus_{i = 1, \ldots, n} j_{U_i!}\mathcal{O}_{U_i}\) is surjective, we can find coverings \(\{V_{jm} \to V_j\}_{m \in M_j}\) with \(V_{jm} \in \mathcal{B}\) such that we can find a commutative diagram \[\xymatrix{ \bigoplus_{j = 1, \ldots, m} \bigoplus_{m \in M_j} j_{V_{jm}!}\mathcal{O}_{V_{jm}} \ar[d] \ar[r] & \bigoplus_{i = 1, \ldots n} \bigoplus_{k \in K'_i} j_{U_{ik}!}\mathcal{O}_{U_{ik}} \ar[d] \\ \bigoplus_{j = 1, \ldots, m} j_{V_j!}\mathcal{O}_{V_j} \ar[r] & \bigoplus_{i = 1, \ldots, n} j_{U_i!}\mathcal{O}_{U_i} }\] Since \(V_j\) is quasi-compact, we can choose finite subsets \(M'_j \subset M_j\) as in Lemma 0G1W. Set \[L = \left(\coprod\nolimits_{i = 1, \ldots, n} K'_i \times K'_i \right) \coprod \left(\coprod\nolimits_{j = 1, \ldots, m} M'_j\right)\] and for \(l = (k, k') \in K'_i \times K'_i \subset L\) set \(W_l = U_{ik} \times_{U_i} U_{ik'}\) and for \(l = m \in M'_j \subset L\) set \(W_l = V_{jm}\). Since we have the exact sequences of Lemma 0G1W for the families \(\{U_{ik} \to U_i\}_{k \in K'_i}\) we conclude that we get a diagram as in the statement of the lemma (details omitted), except that it is not yet clear that \(W_l \in \mathcal{B}\). However, since \(W_l\) is quasi-compact for all \(l \in L\) we do another application of Lemma 0G1W and find finite families of maps \(\{W_{lt} \to W_l\}_{t \in T_l}\) with \(W_{lt} \in \mathcal{B}\) such that \(\bigoplus j_{W_{lt}!}\mathcal{O}_{W_{lt}} \to j_{W_l!}\mathcal{O}_{W_l}\) is surjective. Then we replace \(L\) by \(\coprod_{l \in L} T_l\) and everything is clear.

Lemma

In Situation 0937 assume (0938), (0939), and (093A) hold. Let \(\mathcal{O}\) be a sheaf of rings. Then an extension of modules as in (093D) is another module as in (093D).

Proof

Let \(0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\) be a short exact sequence of \(\mathcal{O}\)-modules with \(\mathcal{F}_1\) and \(\mathcal{F}_3\) as in (093D). Choose presentations \[\bigoplus A_{V_j} \to \bigoplus A_{U_i} \to \mathcal{F}_1 \to 0 \quad\text{and}\quad \bigoplus A_{T_j} \to \bigoplus A_{W_i} \to \mathcal{F}_3 \to 0\] In this proof the direct sums are always finite, and we write \(A_U = j_{U!}\mathcal{O}_U\) for \(U \in \mathcal{B}\). Since \(\mathcal{F}_2 \to \mathcal{F}_3\) is surjective, we can choose coverings \(\{W_{ik} \to W_i\}\) with \(W_{ik} \in \mathcal{B}\) such that \(A_{W_{ik}} \to \mathcal{F}_3\) lifts to a map \(A_{W_{ik}} \to \mathcal{F}_2\). By Lemma 093F we may replace our collection \(\{W_i\}\) by a finite subcollection of the collection \(\{W_{ik}\}\) and assume the map \(\bigoplus A_{W_i} \to \mathcal{F}_3\) lifts to a map into \(\mathcal{F}_2\). Consider the kernel \[\mathcal{K}_2 = \Ker(\bigoplus A_{U_i} \oplus \bigoplus A_{W_i} \longrightarrow \mathcal{F}_2)\] By the snake lemma this kernel surjects onto \(\mathcal{K}_3 = \Ker(\bigoplus A_{W_i} \to \mathcal{F}_3)\). Thus, arguing as above, after replacing each \(T_j\) by a finite family of elements of \(\mathcal{B}\) (permissible by Lemma 0G1W) we may assume there is a map \(\bigoplus A_{T_j} \to \mathcal{K}_2\) lifting the given map \(\bigoplus A_{T_j} \to \mathcal{K}_3\). Then \(\bigoplus A_{V_j} \oplus \bigoplus A_{T_j} \to \mathcal{K}_2\) is surjective which finishes the proof.

Lemma

In Situation 0937 assume (0938), (0939), and (093A) hold. Let \(\mathcal{O}\) be a sheaf of rings. Let \(\mathcal{A} \subset \textit{Mod}(\mathcal{O})\) be the full subcategory of modules isomorphic to a cokernel as in (093D). If the kernel of every map of \(\mathcal{O}\)-modules of the form \[\bigoplus\nolimits_{j = 1, \ldots, m} j_{V_j!}\mathcal{O}_{V_j} \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} j_{U_i!}\mathcal{O}_{U_i}\] with \(U_i\) and \(V_j\) in \(\mathcal{B}\), is in \(\mathcal{A}\), then \(\mathcal{A}\) is weak Serre subcategory of \(\textit{Mod}(\mathcal{O})\).

Proof

We will use the criterion of Homology, Lemma 0754. By the results of Lemmas 093E and 093G it suffices to see that the kernel of a map \(\mathcal{F} \to \mathcal{G}\) between objects of \(\mathcal{A}\) is in \(\mathcal{A}\). To prove this choose presentations \[\bigoplus A_{V_j} \to \bigoplus A_{U_i} \to \mathcal{F} \to 0 \quad\text{and}\quad \bigoplus A_{T_j} \to \bigoplus A_{W_i} \to \mathcal{G} \to 0\] In this proof the direct sums are always finite, and we write \(A_U = j_{U!}\mathcal{O}_U\) for \(U \in \mathcal{B}\). Using Lemmas 0934 and 093F and arguing as in the proof of Lemma 093G we may assume that the map \(\mathcal{F} \to \mathcal{G}\) lifts to a map of presentations \[\xymatrix{ \bigoplus A_{V_j} \ar[r] \ar[d] & \bigoplus A_{U_i} \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & 0 \\ \bigoplus A_{T_j} \ar[r] & \bigoplus A_{W_i} \ar[r] & \mathcal{G} \ar[r] & 0 }\] Then we see that \[\Ker(\mathcal{F} \to \mathcal{G}) = \Coker\left(\bigoplus A_{V_j} \to \Ker\left( \bigoplus A_{T_j} \oplus \bigoplus A_{U_i} \to \bigoplus A_{W_i}\right)\right)\] and the lemma follows from the assumption and Lemma 093E.

Flat morphisms

Definition

Let \((f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}) \longrightarrow (\Sh(\mathcal{C}'), \mathcal{O}')\) be a morphism of ringed topoi. We say \((f, f^\sharp)\) is flat if the ring map \(f^\sharp : f^{-1}\mathcal{O}' \to \mathcal{O}\) is flat. We say a morphism of ringed sites is flat if the associated morphism of ringed topoi is flat.

Lemma

Let \(f : \Sh(\mathcal{C}) \to \Sh(\mathcal{C}')\) be a morphism of ringed topoi. Then \[f^{-1} : \textit{Ab}(\mathcal{C}') \longrightarrow \textit{Ab}(\mathcal{C}), \quad \mathcal{F} \longmapsto f^{-1}\mathcal{F}\] is exact. If \((f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')\) is a flat morphism of ringed topoi then \[f^* : \textit{Mod}(\mathcal{O}') \longrightarrow \textit{Mod}(\mathcal{O}), \quad \mathcal{F} \longmapsto f^*\mathcal{F}\] is exact.

Proof

Given an abelian sheaf \(\mathcal{G}\) on \(\mathcal{C}'\) the underlying sheaf of sets of \(f^{-1}\mathcal{G}\) is the same as \(f^{-1}\) of the underlying sheaf of sets of \(\mathcal{G}\), see Sites, Section 00YR. Hence the exactness of \(f^{-1}\) for sheaves of sets (required in the definition of a morphism of topoi, see Sites, Definition 00XA) implies the exactness of \(f^{-1}\) as a functor on abelian sheaves.

To see the statement on modules recall that \(f^*\mathcal{F}\) is defined as the tensor product \(f^{-1}\mathcal{F} \otimes_{f^{-1}\mathcal{O}', f^\sharp} \mathcal{O}\). Hence \(f^*\) is a composition of functors both of which are exact.

Definition

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) be a morphism of ringed topoi. Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}\)-modules. We say that \(\mathcal{F}\) is flat over \((\Sh(\mathcal{D}), \mathcal{O}')\) if \(\mathcal{F}\) is flat as an \(f^{-1}\mathcal{O}'\)-module.

This is compatible with the notion as defined for morphisms of ringed spaces, see Modules, Definition 08KT and the discussion following.

Lemma

Let \(f : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D})\) be a morphism of ringed sites. Let \(\mathcal{F}\), \(\mathcal{G}\) be \(\mathcal{O}_\mathcal{D}\)-modules. If \(\mathcal{F}\) is finitely presented and \(f\) is flat, then the canonical map \[f^*\SheafHom_{\mathcal{O}_\mathcal{D}}(\mathcal{F}, \mathcal{G}) \longrightarrow \SheafHom_{\mathcal{O}_\mathcal{C}}(f^*\mathcal{F}, f^*\mathcal{G})\] of Remark 0GMX is an isomorphism.

Proof

Say \(f\) is given by the continuous functor \(u : \mathcal{D} \to \mathcal{C}\). We have to show that the restriction of the map to \(\mathcal{C}/U\) for any \(U \in \Ob(\mathcal{C})\) is an isomorphism. We may replace \(U\) by the members of a covering of \(U\). Hence by Sites, Lemma 08H2 we may assume there exists a morphism \(U \to u(V)\) for some \(V \in \Ob(\mathcal{D})\). Of course, then we may replace \(U\) by \(u(V)\). Then since \(u\) is continuous, we may replace \(V\) by a covering and assume there is a presentation \(\mathcal{O}_V^{\oplus m} \to \mathcal{O}_V^{\oplus n} \to \mathcal{F}|_V \to 0\) over \(\mathcal{D}/V\). Since formation of \(\SheafHom\) commutes with localization (Lemma 0E8H) we may replace \(f\) by the morphism \((\mathcal{C}/u(V), \mathcal{O}_{u(V)}) \to (\mathcal{D}/V, \mathcal{O}_V)\) induced by \(f\). Hence we reduce to the case where \(\mathcal{F}\) has a global presentation \(\mathcal{O}_\mathcal{D}^{\oplus m} \to \mathcal{O}_\mathcal{D}^{\oplus n} \to \mathcal{F} \to 0\). Since \(f\) is flat and \(f^*\mathcal{O}_\mathcal{D} = \mathcal{O}_\mathcal{C}\) we obtain a corresponding presentation \(\mathcal{O}_\mathcal{C}^{\oplus m} \to \mathcal{O}_\mathcal{C}^{\oplus n} \to f^*\mathcal{F} \to 0\), see Lemma 04JC. Using that \(\SheafHom\) commutes with finite direct sums in the first variable, using that both \(\SheafHom_{\mathcal{O}_\mathcal{C}}(\mathcal{O}_\mathcal{C}, -)\) and \(\SheafHom_{\mathcal{O}_\mathcal{D}}(\mathcal{O}_\mathcal{D}, -)\) are the identity functor, and using the functoriality of the construction of Remark 0GMX we obtain a commutative diagram \[\xymatrix{ 0 \ar[r] & f^*\SheafHom_{\mathcal{O}_\mathcal{D}}(\mathcal{F}, \mathcal{G}) \ar[d] \ar[r] & f^*\mathcal{G}^{\oplus n} \ar[d] \ar[r] & f^*\mathcal{G}^{\oplus m} \ar[d] \\ 0 \ar[r] & \SheafHom_{\mathcal{O}_\mathcal{C}}(f^*\mathcal{F}, f^*\mathcal{G}) \ar[r] & f^*\mathcal{G}^{\oplus n} \ar[r] & f^*\mathcal{G}^{\oplus m} }\] where the right two vertical arrows are isomorphisms. By Lemma 0GMY the rows are exact. We conclude by the 5 lemma.

Invertible modules

Here is the definition.

Definition

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

1. A finite locally free \(\mathcal{O}\)-module \(\mathcal{F}\) is said to have rank \(r\) if for every object \(U\) of \(\mathcal{C}\) there exists a covering \(\{U_i \to U\}\) of \(U\) such that \(\mathcal{F}|_{U_i}\) is isomorphic to \(\mathcal{O}_{U_i}^{\oplus r}\) as an \(\mathcal{O}_{U_i}\)-module.

2. An \(\mathcal{O}\)-module \(\mathcal{L}\) is invertible if the functor \[\textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}),\quad \mathcal{F} \longmapsto \mathcal{F} \otimes_\mathcal{O} \mathcal{L}\] is an equivalence.

3. The sheaf \(\mathcal{O}^*\) is the subsheaf of \(\mathcal{O}\) defined by the rule \[U \longmapsto \mathcal{O}^*(U) = \{f \in \mathcal{O}(U) \mid \exists g \in \mathcal{O}(U)\text{ such that }fg = 1\}\] It is a sheaf of abelian groups with multiplication as the group law.

Lemma 0B8Q below explains the relationship with locally free modules of rank \(1\).

Lemma

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

1. \(\mathcal{L}\) is invertible, and

2. there exists an \(\mathcal{O}\)-module \(\mathcal{N}\) such that \(\mathcal{L} \otimes_\mathcal{O} \mathcal{N} \cong \mathcal{O}\).

In this case we have

1. \(\mathcal{L}\) is a flat \(\mathcal{O}\)-module of finite presentation,

2. for every object \(U\) of \(\mathcal{C}\) there exists a covering \(\{U_i \to U\}\) such that \(\mathcal{L}|_{U_i}\) is a direct summand of a finite free module, and

3. the module \(\mathcal{N}\) in (2) is isomorphic to \(\SheafHom_\mathcal{O}(\mathcal{L}, \mathcal{O})\).

Proof

Assume (1). Then the functor \(- \otimes_\mathcal{O} \mathcal{L}\) is essentially surjective, hence there exists an \(\mathcal{O}\)-module \(\mathcal{N}\) as in (2). If (2) holds, then the functor \(- \otimes_\mathcal{O} \mathcal{N}\) is a quasi-inverse to the functor \(- \otimes_\mathcal{O} \mathcal{L}\) and we see that (1) holds.

Assume (1) and (2) hold. Since \(- \otimes_\mathcal{O} \mathcal{L}\) is an equivalence, it is exact, and hence \(\mathcal{L}\) is flat. Denote \(\psi : \mathcal{L} \otimes_\mathcal{O} \mathcal{N} \to \mathcal{O}\) the given isomorphism. Let \(U\) be an object of \(\mathcal{C}\). We will show that the restriction \(\mathcal{L}\) to the members of a covering of \(U\) is a direct summand of a free module, which will certainly imply that \(\mathcal{L}\) is of finite presentation. By construction of \(\otimes\) we may assume (after replacing \(U\) by the members of a covering) that there exists an integer \(n \geq 1\) and sections \(x_i \in \mathcal{L}(U)\), \(y_i \in \mathcal{N}(U)\) such that \(\psi(\sum x_i \otimes y_i) = 1\). Consider the isomorphisms \[\mathcal{L}|_U \to \mathcal{L}|_U \otimes_{\mathcal{O}_U} \mathcal{L}|_U \otimes_{\mathcal{O}_U} \mathcal{N}|_U \to \mathcal{L}|_U\] where the first arrow sends \(x\) to \(\sum x_i \otimes x \otimes y_i\) and the second arrow sends \(x \otimes x' \otimes y\) to \(\psi(x' \otimes y)x\). We conclude that \(x \mapsto \sum \psi(x \otimes y_i)x_i\) is an automorphism of \(\mathcal{L}|_U\). This automorphism factors as \[\mathcal{L}|_U \to \mathcal{O}_U^{\oplus n} \to \mathcal{L}|_U\] where the first arrow is given by \(x \mapsto (\psi(x \otimes y_1), \ldots, \psi(x \otimes y_n))\) and the second arrow by \((a_1, \ldots, a_n) \mapsto \sum a_i x_i\). In this way we conclude that \(\mathcal{L}|_U\) is a direct summand of a finite free \(\mathcal{O}_U\)-module.

Assume (1) and (2) hold. Consider the evaluation map \[\mathcal{L} \otimes_\mathcal{O} \SheafHom_\mathcal{O}(\mathcal{L}, \mathcal{O}_X) \longrightarrow \mathcal{O}_X\] To finish the proof of the lemma we will show this is an isomorphism. By Lemma 03EO we have \[\Hom_\mathcal{O}(\mathcal{O}, \mathcal{O}) = \Hom_\mathcal{O} (\mathcal{N} \otimes_\mathcal{O} \mathcal{L}, \mathcal{O}) \longrightarrow \Hom_\mathcal{O} (\mathcal{N}, \SheafHom_\mathcal{O}(\mathcal{L}, \mathcal{O}))\] The image of \(1\) gives a morphism \(\mathcal{N} \to \SheafHom_\mathcal{O}(\mathcal{L}, \mathcal{O})\). Tensoring with \(\mathcal{L}\) we obtain \[\mathcal{O} = \mathcal{L} \otimes_\mathcal{O} \mathcal{N} \longrightarrow \mathcal{L} \otimes_\mathcal{O} \SheafHom_\mathcal{O}(\mathcal{L}, \mathcal{O})\] This map is the inverse to the evaluation map; computation omitted.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi. The pullback \(f^*\mathcal{L}\) of an invertible \(\mathcal{O}_\mathcal{D}\)-module is invertible.

Proof

By Lemma 0B8N there exists an \(\mathcal{O}_\mathcal{D}\)-module \(\mathcal{N}\) such that \(\mathcal{L} \otimes_{\mathcal{O}_\mathcal{D}} \mathcal{N} \cong \mathcal{O}_\mathcal{D}\). Pulling back we get \(f^*\mathcal{L} \otimes_{\mathcal{O}_\mathcal{C}} f^*\mathcal{N} \cong \mathcal{O}_\mathcal{C}\) by Lemma 03EL. Thus \(f^*\mathcal{L}\) is invertible by Lemma 0B8N.

Lemma

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

1. If \(\mathcal{L}\), \(\mathcal{N}\) are invertible \(\mathcal{O}\)-modules, then so is \(\mathcal{L} \otimes_\mathcal{O} \mathcal{N}\).

2. If \(\mathcal{L}\) is an invertible \(\mathcal{O}\)-module, then so is \(\SheafHom_\mathcal{O}(\mathcal{L}, \mathcal{O})\) and the evaluation map \(\mathcal{L} \otimes_\mathcal{O} \SheafHom_\mathcal{O}(\mathcal{L}, \mathcal{O}) \to \mathcal{O}\) is an isomorphism.

Proof

Part (1) is clear from the definition and part (2) follows from Lemma 0B8N and its proof.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. There exists a set of invertible modules \(\{\mathcal{L}_i\}_{i \in I}\) such that each invertible module on \((\mathcal{C}, \mathcal{O})\) is isomorphic to exactly one of the \(\mathcal{L}_i\).

Proof

Omitted, but see Sheaves of Modules, Lemma 01CW.

Lemma 040B says that the collection of isomorphism classes of invertible sheaves forms a set. Lemma 040A says that tensor product defines the structure of an abelian group on this set with inverse of \(\mathcal{L}\) given by \(\SheafHom_\mathcal{O}(\mathcal{L}, \mathcal{O})\).

In fact, given an invertible \(\mathcal{O}\)-module \(\mathcal{L}\) and \(n \in \mathbf{Z}\) we define the \(n\)th tensor power \(\mathcal{L}^{\otimes n}\) of \(\mathcal{L}\) as the image of \(\mathcal{O}\) under applying the equivalence \(\mathcal{F} \mapsto \mathcal{F} \otimes_\mathcal{O} \mathcal{L}\) exactly \(n\) times. This makes sense also for negative \(n\) as we’ve defined an invertible \(\mathcal{O}\)-module as one for which tensoring is an equivalence. More explicitly, we have \[\mathcal{L}^{\otimes n} = \left\{ \begin{matrix} \mathcal{O} & \text{if} & n = 0 \\ \SheafHom_\mathcal{O}(\mathcal{L}, \mathcal{O}) & \text{if} & n = -1\\ \mathcal{L} \otimes_\mathcal{O} \ldots \otimes_\mathcal{O} \mathcal{L} & \text{if} & n > 0 \\ \mathcal{L}^{\otimes -1} \otimes_\mathcal{O} \ldots \otimes_\mathcal{O} \mathcal{L}^{\otimes -1} & \text{if} & n < -1 \end{matrix} \right.\] see Lemma 040A. With this definition we have canonical isomorphisms \(\mathcal{L}^{\otimes n} \otimes_\mathcal{O} \mathcal{L}^{\otimes m} \to \mathcal{L}^{\otimes n + m}\), and these isomorphisms satisfy a commutativity and an associativity constraint (formulation omitted).

Definition

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. The Picard group \(\Pic(\mathcal{O})\) of the ringed site is the abelian group whose elements are isomorphism classes of invertible \(\mathcal{O}\)-modules, with addition corresponding to tensor product.

Modules of differentials

In this section we briefly explain how to define the module of relative differentials for a morphism of ringed topoi. We suggest the reader take a look at the corresponding section in the chapter on commutative algebra (Algebra, Section 00RM).

Definition

Let \(\mathcal{C}\) be a site. Let \(\varphi : \mathcal{O}_1 \to \mathcal{O}_2\) be a homomorphism of sheaves of rings. Let \(\mathcal{F}\) be an \(\mathcal{O}_2\)-module. A \(\mathcal{O}_1\)-derivation or more precisely a \(\varphi\)-derivation into \(\mathcal{F}\) is a map \(D : \mathcal{O}_2 \to \mathcal{F}\) which is additive, annihilates the image of \(\mathcal{O}_1 \to \mathcal{O}_2\), and satisfies the Leibniz rule \[D(ab) = aD(b) + D(a)b\] for all \(a, b\) local sections of \(\mathcal{O}_2\) (wherever they are both defined). We denote \(\text{Der}_{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{F})\) the set of \(\varphi\)-derivations into \(\mathcal{F}\).

This is the sheaf theoretic analogue of Algebra, Definition 04BK. Given a derivation \(D : \mathcal{O}_2 \to \mathcal{F}\) as in the definition the map on global sections \[D : \Gamma(\mathcal{O}_2) \longrightarrow \Gamma(\mathcal{F})\] clearly is a \(\Gamma(\mathcal{O}_1)\)-derivation as in the algebra definition. Note that if \(\alpha : \mathcal{F} \to \mathcal{G}\) is a map of \(\mathcal{O}_2\)-modules, then there is an induced map \[\text{Der}_{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{F}) \longrightarrow \text{Der}_{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{G})\] given by the rule \(D \mapsto \alpha \circ D\). In other words we obtain a functor.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\varphi : \mathcal{O}_1 \to \mathcal{O}_2\) be a homomorphism of sheaves of rings. The functor \[\textit{Mod}(\mathcal{O}_2) \longrightarrow \textit{Ab}, \quad \mathcal{F} \longmapsto \text{Der}_{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{F})\] is representable.

Proof

This is proved in exactly the same way as the analogous statement in algebra. During this proof, for any sheaf of sets \(\mathcal{F}\) on \(\mathcal{C}\), let us denote \(\mathcal{O}_2[\mathcal{F}]\) the sheafification of the presheaf \(U \mapsto \mathcal{O}_2(U)[\mathcal{F}(U)]\) where this denotes the free \(\mathcal{O}_2(U)\)-module on the set \(\mathcal{F}(U)\). For \(s \in \mathcal{F}(U)\) we denote \([s]\) the corresponding section of \(\mathcal{O}_2[\mathcal{F}]\) over \(U\). If \(\mathcal{F}\) is a sheaf of \(\mathcal{O}_2\)-modules, then there is a canonical map \[c : \mathcal{O}_2[\mathcal{F}] \longrightarrow \mathcal{F}\] which on the presheaf level is given by the rule \(\sum f_s[s] \mapsto \sum f_s s\). We will employ the short hand \([s] \mapsto s\) to describe this map and similarly for other maps below. Consider the map of \(\mathcal{O}_2\)-modules [04BM]\[\begin{equation} \begin{matrix} \mathcal{O}_2[\mathcal{O}_2 \times \mathcal{O}_2] \oplus \mathcal{O}_2[\mathcal{O}_2 \times \mathcal{O}_2] \oplus \mathcal{O}_2[\mathcal{O}_1] & \longrightarrow & \mathcal{O}_2[\mathcal{O}_2] \\ [(a, b)] \oplus [(f, g)] \oplus [h] & \longmapsto & [a + b] - [a] - [b] + \\ & & [fg] - g[f] - f[g] + \\ & & [\varphi(h)] \end{matrix} \end{equation}\] with short hand notation as above. Set \(\Omega_{\mathcal{O}_2/\mathcal{O}_1}\) equal to the cokernel of this map. Then it is clear that there exists a map of sheaves of sets \[\text{d} : \mathcal{O}_2 \longrightarrow \Omega_{\mathcal{O}_2/\mathcal{O}_1}\] mapping a local section \(f\) to the image of \([f]\) in \(\Omega_{\mathcal{O}_2/\mathcal{O}_1}\). By construction \(\text{d}\) is a \(\mathcal{O}_1\)-derivation. Next, let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}_2\)-modules and let \(D : \mathcal{O}_2 \to \mathcal{F}\) be a \(\mathcal{O}_1\)-derivation. Then we can consider the \(\mathcal{O}_2\)-linear map \(\mathcal{O}_2[\mathcal{O}_2] \to \mathcal{F}\) which sends \([g]\) to \(D(g)\). It follows from the definition of a derivation that this map annihilates sections in the image of the map (04BM) and hence defines a map \[\alpha_D : \Omega_{\mathcal{O}_2/\mathcal{O}_1} \longrightarrow \mathcal{F}\] Since it is clear that \(D = \alpha_D \circ \text{d}\) the lemma is proved.

Definition

Let \(\mathcal{C}\) be a site. Let \(\varphi : \mathcal{O}_1 \to \mathcal{O}_2\) be a homomorphism of sheaves of rings. The module of differentials of the ring map \(\varphi\) is the object representing the functor \(\mathcal{F} \mapsto \text{Der}_{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{F})\) which exists by Lemma 04BL. It is denoted \(\Omega_{\mathcal{O}_2/\mathcal{O}_1}\), and the universal \(\varphi\)-derivation is denoted \(\text{d} : \mathcal{O}_2 \to \Omega_{\mathcal{O}_2/\mathcal{O}_1}\).

Since this module and the derivation form the universal object representing a functor, this notion is clearly intrinsic (i.e., does not depend on the choice of the site underlying the ringed topos, see Section 03DG). Note that \(\Omega_{\mathcal{O}_2/\mathcal{O}_1}\) is the cokernel of the map (04BM) of \(\mathcal{O}_2\)-modules. Moreover the map \(\text{d}\) is described by the rule that \(\text{d}f\) is the image of the local section \([f]\).

Lemma

Let \(\mathcal{C}\) be a site. Let \(\varphi : \mathcal{O}_1 \to \mathcal{O}_2\) be a homomorphism of presheaves of rings. Then \(\Omega_{\mathcal{O}_2^\#/\mathcal{O}_1^\#}\) is the sheaf associated to the presheaf \(U \mapsto \Omega_{\mathcal{O}_2(U)/\mathcal{O}_1(U)}\).

Proof

Consider the map (04BM). There is a similar map of presheaves whose value on \(U \in \Ob(\mathcal{C})\) is \[\mathcal{O}_2(U)[\mathcal{O}_2(U) \times \mathcal{O}_2(U)] \oplus \mathcal{O}_2(U)[\mathcal{O}_2(U) \times \mathcal{O}_2(U)] \oplus \mathcal{O}_2(U)[\mathcal{O}_1(U)] \longrightarrow \mathcal{O}_2(U)[\mathcal{O}_2(U)]\] The cokernel of this map has value \(\Omega_{\mathcal{O}_2(U)/\mathcal{O}_1(U)}\) over \(U\) by the construction of the module of differentials in Algebra, Definition 07BK. On the other hand, the sheaves in (04BM) are the sheafifications of the presheaves above. Thus the result follows as sheafification is exact.

Lemma

Let \(f : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})\) be a morphism of topoi. Let \(\varphi : \mathcal{O}_1 \to \mathcal{O}_2\) be a homomorphism of sheaves of rings on \(\mathcal{C}\). Then there is a canonical identification \(f^{-1}\Omega_{\mathcal{O}_2/\mathcal{O}_1} = \Omega_{f^{-1}\mathcal{O}_2/f^{-1}\mathcal{O}_1}\) compatible with universal derivations.

Proof

This holds because the sheaf \(\Omega_{\mathcal{O}_2/\mathcal{O}_1}\) is the cokernel of the map (04BM) and a similar statement holds for \(\Omega_{f^{-1}\mathcal{O}_2/f^{-1}\mathcal{O}_1}\), because the functor \(f^{-1}\) is exact, and because \(f^{-1}(\mathcal{O}_2[\mathcal{O}_2]) = f^{-1}\mathcal{O}_2[f^{-1}\mathcal{O}_2]\), \(f^{-1}(\mathcal{O}_2[\mathcal{O}_2 \times \mathcal{O}_2]) = f^{-1}\mathcal{O}_2[f^{-1}\mathcal{O}_2 \times f^{-1}\mathcal{O}_2]\), and \(f^{-1}(\mathcal{O}_2[\mathcal{O}_1]) = f^{-1}\mathcal{O}_2[f^{-1}\mathcal{O}_1]\).

Lemma

Let \(\mathcal{C}\) be a site. Let \(\varphi : \mathcal{O}_1 \to \mathcal{O}_2\) be a homomorphism of sheaves of rings. For any object \(U\) of \(\mathcal{C}\) there is a canonical isomorphism \[\Omega_{\mathcal{O}_2/\mathcal{O}_1}|_U = \Omega_{(\mathcal{O}_2|_U)/(\mathcal{O}_1|_U)}\] compatible with universal derivations.

Proof

This is a special case of Lemma 08TQ.

Lemma

Let \(\mathcal{C}\) be a site. Let \[\xymatrix{ \mathcal{O}_2 \ar[r]_\varphi & \mathcal{O}_2' \\ \mathcal{O}_1 \ar[r] \ar[u] & \mathcal{O}'_1 \ar[u] }\] be a commutative diagram of sheaves of rings on \(\mathcal{C}\). The map \(\mathcal{O}_2 \to \mathcal{O}'_2\) composed with the map \(\text{d} : \mathcal{O}'_2 \to \Omega_{\mathcal{O}'_2/\mathcal{O}'_1}\) is a \(\mathcal{O}_1\)-derivation. Hence we obtain a canonical map of \(\mathcal{O}_2\)-modules \(\Omega_{\mathcal{O}_2/\mathcal{O}_1} \to \Omega_{\mathcal{O}'_2/\mathcal{O}'_1}\). It is uniquely characterized by the property that \(\text{d}(f)\) maps to \(\text{d}(\varphi(f))\) for any local section \(f\) of \(\mathcal{O}_2\). In this way \(\Omega_{-/-}\) becomes a functor on the category of arrows of sheaves of rings.

Proof

This lemma proves itself.

Lemma

In Lemma 08TR suppose that \(\mathcal{O}_2 \to \mathcal{O}'_2\) is surjective with kernel \(\mathcal{I} \subset \mathcal{O}_2\) and assume that \(\mathcal{O}_1 = \mathcal{O}'_1\). Then there is a canonical exact sequence of \(\mathcal{O}'_2\)-modules \[\mathcal{I}/\mathcal{I}^2 \longrightarrow \Omega_{\mathcal{O}_2/\mathcal{O}_1} \otimes_{\mathcal{O}_2} \mathcal{O}'_2 \longrightarrow \Omega_{\mathcal{O}'_2/\mathcal{O}_1} \longrightarrow 0\] The leftmost map is characterized by the rule that a local section \(f\) of \(\mathcal{I}\) maps to \(\text{d}f \otimes 1\).

Proof

For a local section \(f\) of \(\mathcal{I}\) denote \(\overline{f}\) the image of \(f\) in \(\mathcal{I}/\mathcal{I}^2\). To show that the map \(\overline{f} \mapsto \text{d}f \otimes 1\) is well defined we just have to check that \(\text{d} f_1f_2 \otimes 1 = 0\) if \(f_1, f_2\) are local sections of \(\mathcal{I}\). And this is clear from the Leibniz rule \(\text{d} f_1f_2 \otimes 1 = (f_1 \text{d}f_2 + f_2 \text{d} f_1 )\otimes 1 = \text{d}f_2 \otimes f_1 + \text{d}f_1 \otimes f_2 = 0\). A similar computation show this map is \(\mathcal{O}'_2 = \mathcal{O}_2/\mathcal{I}\)-linear. The map on the right is the one from Lemma 08TR.

To see that the sequence is exact, we argue as follows. Let \(\mathcal{O}''_2 \subset \mathcal{O}'_2\) be the presheaf of \(\mathcal{O}_1\)-algebras whose value on \(U\) is the image of \(\mathcal{O}_2(U) \to \mathcal{O}'_2(U)\). By Algebra, Lemma 00RU the sequences \[\mathcal{I}(U)/\mathcal{I}(U)^2 \longrightarrow \Omega_{\mathcal{O}_2(U)/\mathcal{O}_1(U)} \otimes_{\mathcal{O}_2(U)} \mathcal{O}''_2(U) \longrightarrow \Omega_{\mathcal{O}''_2(U)/\mathcal{O}_1(U)} \longrightarrow 0\] are exact for all objects \(U\) of \(\mathcal{C}\). Since sheafification is exact this gives an exact sequence of sheaves of \((\mathcal{O}'_2)^\#\)-modules. By Lemma 08TP and the fact that \((\mathcal{O}''_2)^\# = \mathcal{O}'_2\) we conclude.

Here is a particular situation where derivations come up naturally.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\varphi : \mathcal{O}_1 \to \mathcal{O}_2\) be a homomorphism of sheaves of rings. Consider a short exact sequence \[0 \to \mathcal{F} \to \mathcal{A} \to \mathcal{O}_2 \to 0\] Here \(\mathcal{A}\) is a sheaf of \(\mathcal{O}_1\)-algebras, \(\pi : \mathcal{A} \to \mathcal{O}_2\) is a surjection of sheaves of \(\mathcal{O}_1\)-algebras, and \(\mathcal{F} = \Ker(\pi)\) is its kernel. Assume \(\mathcal{F}\) an ideal sheaf with square zero in \(\mathcal{A}\). So \(\mathcal{F}\) has a natural structure of an \(\mathcal{O}_2\)-module. A section \(s : \mathcal{O}_2 \to \mathcal{A}\) of \(\pi\) is a \(\mathcal{O}_1\)-algebra map such that \(\pi \circ s = \text{id}\). Given any section \(s : \mathcal{O}_2 \to \mathcal{A}\) of \(\pi\) and any \(\varphi\)-derivation \(D : \mathcal{O}_2 \to \mathcal{F}\) the map \[s + D : \mathcal{O}_2 \to \mathcal{A}\] is a section of \(\pi\) and every section \(s'\) is of the form \(s + D\) for a unique \(\varphi\)-derivation \(D\).

Proof

Recall that the \(\mathcal{O}_2\)-module structure on \(\mathcal{F}\) is given by \(h \tau = \tilde h \tau\) (multiplication in \(\mathcal{A}\)) where \(h\) is a local section of \(\mathcal{O}_2\), and \(\tilde h\) is a local lift of \(h\) to a local section of \(\mathcal{A}\), and \(\tau\) is a local section of \(\mathcal{F}\). In particular, given \(s\), we may use \(\tilde h = s(h)\). To verify that \(s + D\) is a homomorphism of sheaves of rings we compute \[\begin{eqnarray*} (s + D)(ab) & = & s(ab) + D(ab) \\ & = & s(a)s(b) + aD(b) + D(a)b \\ & = & s(a) s(b) + s(a)D(b) + D(a)s(b) \\ & = & (s(a) + D(a))(s(b) + D(b)) \end{eqnarray*}\] by the Leibniz rule. In the same manner one shows \(s + D\) is a \(\mathcal{O}_1\)-algebra map because \(D\) is an \(\mathcal{O}_1\)-derivation. Conversely, given \(s'\) we set \(D = s' - s\). Details omitted.

Definition

Let \(X = (\Sh(\mathcal{C}), \mathcal{O})\) and \(Y = (\Sh(\mathcal{C}'), \mathcal{O}')\) be ringed topoi. Let \((f, f^\sharp) : X \to Y\) be a morphism of ringed topoi. In this situation

1. for a sheaf \(\mathcal{F}\) of \(\mathcal{O}\)-modules a \(Y\)-derivation \(D : \mathcal{O} \to \mathcal{F}\) is just a \(f^\sharp\)-derivation, and

2. the sheaf of differentials \(\Omega_{X/Y}\) of \(X\) over \(Y\) is the module of differentials of \(f^\sharp : f^{-1}\mathcal{O}' \to \mathcal{O}\), see Definition 04BN.

Thus \(\Omega_{X/Y}\) comes equipped with a universal \(Y\)-derivation \(\text{d}_{X/Y} : \mathcal{O} \longrightarrow \Omega_{X/Y}\). We sometimes write \(\Omega_{X/Y} = \Omega_f\).

Recall that \(f^\sharp : f^{-1}\mathcal{O}' \to \mathcal{O}\) so that this definition makes sense.

Lemma

Let \(X = (\Sh(\mathcal{C}_X), \mathcal{O}_X)\), \(Y = (\Sh(\mathcal{C}_Y), \mathcal{O}_Y)\), \(X' = (\Sh(\mathcal{C}_{X'}), \mathcal{O}_{X'})\), and \(Y' = (\Sh(\mathcal{C}_{Y'}), \mathcal{O}_{Y'})\) be ringed topoi. Let \[\xymatrix{ X' \ar[d] \ar[r]_f & X \ar[d] \\ Y' \ar[r] & Y }\] be a commutative diagram of morphisms of ringed topoi. The map \(f^\sharp : \mathcal{O}_X \to f_*\mathcal{O}_{X'}\) composed with the map \(f_*\text{d}_{X'/Y'} : f_*\mathcal{O}_{X'} \to f_*\Omega_{X'/Y'}\) is a \(Y\)-derivation. Hence we obtain a canonical map of \(\mathcal{O}_X\)-modules \(\Omega_{X/Y} \to f_*\Omega_{X'/Y'}\), and by adjointness of \(f_*\) and \(f^*\) a canonical \(\mathcal{O}_{X'}\)-module homomorphism \[c_f : f^*\Omega_{X/Y} \longrightarrow \Omega_{X'/Y'}.\] It is uniquely characterized by the property that \(f^*\text{d}_{X/Y}(t)\) maps to \(\text{d}_{X'/Y'}(f^* t)\) for any local section \(t\) of \(\mathcal{O}_X\).

Proof

This is clear except for the last assertion. Let us explain the meaning of this. Let \(U \in \Ob(\mathcal{C}_X)\) and let \(t \in \mathcal{O}_X(U)\). This is what it means for \(t\) to be a local section of \(\mathcal{O}_X\). Now, we may think of \(t\) as a map of sheaves of sets \(t : h_U^\# \to \mathcal{O}_X\). Then \(f^{-1}t : f^{-1}h_U^\# \to f^{-1}\mathcal{O}_X\). By \(f^*t\) we mean the composition \[\xymatrix{ f^{-1}h_U^\# \ar[rr]^{f^{-1}t} \ar@/^4ex/[rrrr]^{f^*t} & & f^{-1}\mathcal{O}_X \ar[rr]^{f^\sharp} & & \mathcal{O}_{X'} }\] Note that \(\text{d}_{X/Y}(t) \in \Omega_{X/Y}(U)\). Hence we may think of \(\text{d}_{X/Y}(t)\) as a map \(\text{d}_{X/Y}(t) : h_U^\# \to \Omega_{X/Y}\). Then \(f^{-1}\text{d}_{X/Y}(t) : f^{-1}h_U^\# \to f^{-1}\Omega_{X/Y}\). By \(f^*\text{d}_{X/Y}(t)\) we mean the composition \[\xymatrix{ f^{-1}h_U^\# \ar[rr]^{f^{-1}\text{d}_{X/Y}(t)} \ar@/^4ex/[rrrr]^{f^*\text{d}_{X/Y}(t)} & & f^{-1}\Omega_{X/Y} \ar[rr]^{1 \otimes \text{id}} & & f^*\Omega_{X/Y} }\] OK, and now the statement of the lemma means that we have \[c_f \circ f^*\text{d}_{X/Y}(t) = \text{d}_{X'/Y'}(f^*t)\] as maps from \(f^{-1}h_U^\#\) to \(\Omega_{X'/Y'}\). We omit the verification that this property holds for \(c_f\) as defined in the lemma. (Hint: The first map \(c'_f : \Omega_{X/Y} \to f_*\Omega_{X'/Y'}\) satisfies \(c'_f(\text{d}_{X/Y}(t)) = f_*\text{d}_{X'/Y'}(f^\sharp(t))\) as sections of \(f_*\Omega_{X'/Y'}\) over \(U\), and you have to turn this into the equality above by using adjunction.) The reason that this uniquely characterizes \(c_f\) is that the images of \(f^*\text{d}_{X/Y}(t)\) generate the \(\mathcal{O}_{X'}\)-module \(f^*\Omega_{X/Y}\) simply because the local sections \(\text{d}_{X/Y}(t)\) generate the \(\mathcal{O}_X\)-module \(\Omega_{X/Y}\).

Finite order differential operators

In this section we introduce differential operators of finite order. We suggest the reader take a look at the corresponding section in the chapter on commutative algebra (Algebra, Section 09CH).

Definition

Let \(\mathcal{C}\) be a site. Let \(\varphi : \mathcal{O}_1 \to \mathcal{O}_2\) be a homomorphism of sheaves of rings. Let \(k \geq 0\) be an integer. Let \(\mathcal{F}\), \(\mathcal{G}\) be sheaves of \(\mathcal{O}_2\)-modules. A differential operator \(D : \mathcal{F} \to \mathcal{G}\) of order \(k\) is an \(\mathcal{O}_1\)-linear map such that for all local sections \(g\) of \(\mathcal{O}_2\) the map \(s \mapsto D(gs) - gD(s)\) is a differential operator of order \(k - 1\). For the base case \(k = 0\) we define a differential operator of order \(0\) to be an \(\mathcal{O}_2\)-linear map.

If \(D : \mathcal{F} \to \mathcal{G}\) is a differential operator of order \(k\), then for all local sections \(g\) of \(\mathcal{O}_2\) the map \(gD\) is a differential operator of order \(k\). The sum of two differential operators of order \(k\) is another. Hence the set of all these \[\text{Diff}^k(\mathcal{F}, \mathcal{G}) = \text{Diff}^k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}, \mathcal{G})\] is a \(\Gamma(\mathcal{C}, \mathcal{O}_2)\)-module. We have \[\text{Diff}^0(\mathcal{F}, \mathcal{G}) \subset \text{Diff}^1(\mathcal{F}, \mathcal{G}) \subset \text{Diff}^2(\mathcal{F}, \mathcal{G}) \subset \ldots\] The rule which maps \(U \in \Ob(\mathcal{C})\) to the module of differential operators \(D : \mathcal{F}|_U \to \mathcal{G}|_U\) of order \(k\) is a sheaf of \(\mathcal{O}_2\)-modules on the site \(\mathcal{C}\). Thus we obtain a sheaf of differential operators (if we ever need this we will add a definition here).

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O}_1 \to \mathcal{O}_2\) be a map of sheaves of rings. Let \(\mathcal{E}, \mathcal{F}, \mathcal{G}\) be sheaves of \(\mathcal{O}_2\)-modules. If \(D : \mathcal{E} \to \mathcal{F}\) and \(D' : \mathcal{F} \to \mathcal{G}\) are differential operators of order \(k\) and \(k'\), then \(D' \circ D\) is a differential operator of order \(k + k'\).

Proof

Let \(g\) be a local section of \(\mathcal{O}_2\). Then the map which sends a local section \(x\) of \(\mathcal{E}\) to \[D'(D(gx)) - gD'(D(x)) = D'(D(gx)) - D'(gD(x)) + D'(gD(x)) - gD'(D(x))\] is a sum of two compositions of differential operators of lower order. Hence the lemma follows by induction on \(k + k'\).

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O}_1 \to \mathcal{O}_2\) be a map of sheaves of rings. Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}_2\)-modules. Let \(k \geq 0\). There exists a sheaf of \(\mathcal{O}_2\)-modules \(\mathcal{P}^k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F})\) and a canonical isomorphism \[\text{Diff}^k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}, \mathcal{G}) = \Hom_{\mathcal{O}_2}( \mathcal{P}^k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}), \mathcal{G})\] functorial in the \(\mathcal{O}_2\)-module \(\mathcal{G}\).

Proof

The existence follows from general category theoretic arguments (insert future reference here), but we will also give a direct construction as this construction will be useful in the future proofs. We will freely use the notation introduced in the proof of Lemma 04BL. Given any differential operator \(D : \mathcal{F} \to \mathcal{G}\) we obtain an \(\mathcal{O}_2\)-linear map \(L_D : \mathcal{O}_2[\mathcal{F}] \to \mathcal{G}\) sending \([m]\) to \(D(m)\). If \(D\) has order \(0\) then \(L_D\) annihilates the local sections \[[m + m'] - [m] - [m'],\quad g_0[m] - [g_0m]\] where \(g_0\) is a local section of \(\mathcal{O}_2\) and \(m, m'\) are local sections of \(\mathcal{F}\). If \(D\) has order \(1\), then \(L_D\) annihilates the local sections \[[m + m'] - [m] - [m'],\quad f[m] - [fm], \quad g_0g_1[m] - g_0[g_1m] - g_1[g_0m] + [g_1g_0m]\] where \(f\) is a local section of \(\mathcal{O}_1\), \(g_0, g_1\) are local sections of \(\mathcal{O}_2\), and \(m, m'\) are local sections of \(\mathcal{F}\). If \(D\) has order \(k\), then \(L_D\) annihilates the local sections \([m + m'] - [m] - [m']\), \(f[m] - [fm]\), and the local sections \[g_0g_1\ldots g_k[m] - \sum g_0 \ldots \hat g_i \ldots g_k[g_im] + \ldots +(-1)^{k + 1}[g_0\ldots g_km]\] Conversely, if \(L : \mathcal{O}_2[\mathcal{F}] \to \mathcal{G}\) is an \(\mathcal{O}_2\)-linear map annihilating all the local sections listed in the previous sentence, then \(m \mapsto L([m])\) is a differential operator of order \(k\). Thus we see that \(\mathcal{P}^k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F})\) is the quotient of \(\mathcal{O}_2[\mathcal{F}]\) by the \(\mathcal{O}_2\)-submodule generated by these local sections.

Definition

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O}_1 \to \mathcal{O}_2\) be a map of sheaves of rings. Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}_2\)-modules. The module \(\mathcal{P}^k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F})\) constructed in Lemma 09CT is called the module of principal parts of order \(k\) of \(\mathcal{F}\).

Note that the inclusions \[\text{Diff}^0(\mathcal{F}, \mathcal{G}) \subset \text{Diff}^1(\mathcal{F}, \mathcal{G}) \subset \text{Diff}^2(\mathcal{F}, \mathcal{G}) \subset \ldots\] correspond via Yoneda’s lemma (Categories, Lemma 001P) to surjections \[\ldots \to \mathcal{P}^2_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}) \to \mathcal{P}^1_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}) \to \mathcal{P}^0_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}) = \mathcal{F}\]

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O}_1 \to \mathcal{O}_2\) be a homomorphism of presheaves of rings. Let \(\mathcal{F}\) be a presheaf of \(\mathcal{O}_2\)-modules. Then \(\mathcal{P}^k_{\mathcal{O}_2^\#/\mathcal{O}_1^\#}(\mathcal{F}^\#)\) is the sheaf associated to the presheaf \(U \mapsto P^k_{\mathcal{O}_2(U)/\mathcal{O}_1(U)}(\mathcal{F}(U))\).

Proof

This can be proved in exactly the same way as is done for the sheaf of differentials in Lemma 08TP. Perhaps a more pleasing approach is to use the universal property of Lemma 09CT directly to see the equality. We omit the details.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O}_1 \to \mathcal{O}_2\) be a homomorphism of sheaves of rings. Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}_2\)-modules. There is a canonical short exact sequence \[0 \to \Omega_{\mathcal{O}_2/\mathcal{O}_1} \otimes_{\mathcal{O}_2} \mathcal{F} \to \mathcal{P}^1_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}) \to \mathcal{F} \to 0\] functorial in \(\mathcal{F}\) called the sequence of principal parts.

Proof

Follows from the commutative algebra version (Algebra, Lemma 09CN) and Lemmas 08TP and 09CV.

Remark

Let \(\mathcal{C}\) be a site. Suppose given a commutative diagram of sheaves of rings \[\xymatrix{ \mathcal{B} \ar[r] & \mathcal{B}' \\ \mathcal{A} \ar[u] \ar[r] & \mathcal{A}' \ar[u] }\] a \(\mathcal{B}\)-module \(\mathcal{F}\), a \(\mathcal{B}'\)-module \(\mathcal{F}'\), and a \(\mathcal{B}\)-linear map \(\mathcal{F} \to \mathcal{F}'\). Then we get a compatible system of module maps \[\xymatrix{ \ldots \ar[r] & \mathcal{P}^2_{\mathcal{B}'/\mathcal{A}'}(\mathcal{F}') \ar[r] & \mathcal{P}^1_{\mathcal{B}'/\mathcal{A}'}(\mathcal{F}') \ar[r] & \mathcal{P}^0_{\mathcal{B}'/\mathcal{A}'}(\mathcal{F}') \\ \ldots \ar[r] & \mathcal{P}^2_{\mathcal{B}/\mathcal{A}}(\mathcal{F}) \ar[r] \ar[u] & \mathcal{P}^1_{\mathcal{B}/\mathcal{A}}(\mathcal{F}) \ar[r] \ar[u] & \mathcal{P}^0_{\mathcal{B}/\mathcal{A}}(\mathcal{F}) \ar[u] }\] These maps are compatible with further composition of maps of this type. The easiest way to see this is to use the description of the modules \(\mathcal{P}^k_{\mathcal{B}/\mathcal{A}}(\mathcal{M})\) in terms of (local) generators and relations in the proof of Lemma 09CT but it can also be seen directly from the universal property of these modules. Moreover, these maps are compatible with the short exact sequences of Lemma 09CW.

The naive cotangent complex

This section is the analogue of Algebra, Section 00S0 and Modules, Section 08TG. We advise the reader to read those sections first.

Let \(\mathcal{C}\) be a site. Let \(\mathcal{A} \to \mathcal{B}\) be a homomorphism of sheaves of rings on \(\mathcal{C}\). In this section, for any sheaf of sets \(\mathcal{E}\) on \(\mathcal{C}\) we denote \(\mathcal{A}[\mathcal{E}]\) the sheafification of the presheaf \(U \mapsto \mathcal{A}(U)[\mathcal{E}(U)]\). Here \(\mathcal{A}(U)[\mathcal{E}(U)]\) denotes the polynomial algebra over \(\mathcal{A}(U)\) whose variables correspond to the elements of \(\mathcal{E}(U)\). We denote \([e] \in \mathcal{A}(U)[\mathcal{E}(U)]\) the variable corresponding to \(e \in \mathcal{E}(U)\). There is a canonical surjection of \(\mathcal{A}\)-algebras [08TU]\[\begin{equation} \mathcal{A}[\mathcal{B}] \longrightarrow \mathcal{B},\quad [b] \longmapsto b \end{equation}\] whose kernel we denote \(\mathcal{I} \subset \mathcal{A}[\mathcal{B}]\). It is a simple observation that \(\mathcal{I}\) is generated by the local sections \([b][b'] - [bb']\) and \([a] - a\). According to Lemma 08TS there is a canonical map [08TV]\[\begin{equation} \mathcal{I}/\mathcal{I}^2 \longrightarrow \Omega_{\mathcal{A}[\mathcal{B}]/\mathcal{A}} \otimes_{\mathcal{A}[\mathcal{B}]} \mathcal{B} \end{equation}\] whose cokernel is canonically isomorphic to \(\Omega_{\mathcal{B}/\mathcal{A}}\).

Definition

Let \(\mathcal{C}\) be a site. Let \(\mathcal{A} \to \mathcal{B}\) be a homomorphism of sheaves of rings on \(\mathcal{C}\). The naive cotangent complex \(\NL_{\mathcal{B}/\mathcal{A}}\) is the chain complex (08TV) \[\NL_{\mathcal{B}/\mathcal{A}} = \left(\mathcal{I}/\mathcal{I}^2 \longrightarrow \Omega_{\mathcal{A}[\mathcal{B}]/\mathcal{A}} \otimes_{\mathcal{A}[\mathcal{B}]} \mathcal{B}\right)\] with \(\mathcal{I}/\mathcal{I}^2\) placed in degree \(-1\) and \(\Omega_{\mathcal{A}[\mathcal{B}]/\mathcal{A}} \otimes_{\mathcal{A}[\mathcal{B}]} \mathcal{B}\) placed in degree \(0\).

This construction satisfies a functoriality similar to that discussed in Lemma 08TR for modules of differentials. Namely, given a commutative diagram [08TX]\[\begin{equation} \vcenter{ \xymatrix{ \mathcal{B} \ar[r] & \mathcal{B}' \\ \mathcal{A} \ar[u] \ar[r] & \mathcal{A}' \ar[u] } } \end{equation}\] of sheaves of rings on \(\mathcal{C}\) there is a canonical \(\mathcal{B}\)-linear map of complexes \[\NL_{\mathcal{B}/\mathcal{A}} \longrightarrow \NL_{\mathcal{B}'/\mathcal{A}'}\] Namely, the maps in the commutative diagram give rise to a canonical map \(\mathcal{A}[\mathcal{B}] \to \mathcal{A}'[\mathcal{B}']\) which maps \(\mathcal{I}\) into \(\mathcal{I}' = \Ker(\mathcal{A}'[\mathcal{B}'] \to \mathcal{B}')\). Thus a map \(\mathcal{I}/\mathcal{I}^2 \to \mathcal{I}'/(\mathcal{I}')^2\) and a map between modules of differentials, which together give the desired map between the naive cotangent complexes.

We can choose a different presentation of \(\mathcal{B}\) as a quotient of a polynomial algebra over \(\mathcal{A}\) and still obtain the same object of \(D(\mathcal{B})\). To explain this, suppose that \(\mathcal{E}\) is a sheaf of sets on \(\mathcal{C}\) and \(\alpha : \mathcal{E} \to \mathcal{B}\) a map of sheaves of sets. Then we obtain an \(\mathcal{A}\)-algebra homomorphism \(\mathcal{A}[\mathcal{E}] \to \mathcal{B}\). Assume this map is surjective, and let \(\mathcal{J} \subset \mathcal{A}[\mathcal{E}]\) be the kernel. Set \[\NL(\alpha) = \left( \mathcal{J}/\mathcal{J}^2 \longrightarrow \Omega_{\mathcal{A}[\mathcal{E}]/\mathcal{A}} \otimes_{\mathcal{A}[\mathcal{E}]} \mathcal{B}\right)\] Here is the result.

Lemma

In the situation above there is a canonical isomorphism \(\NL(\alpha) = \NL_{\mathcal{B}/\mathcal{A}}\) in \(D(\mathcal{B})\).

Proof

Observe that \(\NL_{\mathcal{B}/\mathcal{A}} = \NL(\text{id}_\mathcal{B})\). Thus it suffices to show that given two maps \(\alpha_i : \mathcal{E}_i \to \mathcal{B}\) as above, there is a canonical quasi-isomorphism \(\NL(\alpha_1) = \NL(\alpha_2)\) in \(D(\mathcal{B})\). To see this set \(\mathcal{E} = \mathcal{E}_1 \amalg \mathcal{E}_2\) and \(\alpha = \alpha_1 \amalg \alpha_2 : \mathcal{E} \to \mathcal{B}\). Set \(\mathcal{J}_i = \Ker(\mathcal{A}[\mathcal{E}_i] \to \mathcal{B})\) and \(\mathcal{J} = \Ker(\mathcal{A}[\mathcal{E}] \to \mathcal{B})\). We obtain maps \(\mathcal{A}[\mathcal{E}_i] \to \mathcal{A}[\mathcal{E}]\) which send \(\mathcal{J}_i\) into \(\mathcal{J}\). Thus we obtain canonical maps of complexes \[\NL(\alpha_i) \longrightarrow \NL(\alpha)\] and it suffices to show these maps are quasi-isomorphism. To see this we argue as follows. First, observe that \(H^0(\NL(\alpha_i)) = \Omega_{\mathcal{B}/\mathcal{A}}\) and \(H^0(\NL(\alpha)) = \Omega_{\mathcal{B}/\mathcal{A}}\) by Lemma 08TS hence the map is an isomorphism on cohomology sheaves in degree \(0\). Similarly, we claim that \(H^{-1}(\NL(\alpha_i))\) and \(H^{-1}(\NL(\alpha))\) are the sheaves associated to the presheaf \(U \mapsto H_1(L_{\mathcal{B}(U)/\mathcal{A}(U)})\) where \(H_1(L_{-/-})\) is as in Algebra, Definition 07BN. If the claim holds, then the proof is finished.

Proof of the claim. Let \(\alpha : \mathcal{E} \to \mathcal{B}\) be as above. Let \(\mathcal{B}' \subset \mathcal{B}\) be the subpresheaf of \(\mathcal{A}\)-algebras whose value on \(U\) is the image of \(\mathcal{A}(U)[\mathcal{E}(U)] \to \mathcal{B}(U)\). Let \(\mathcal{I}'\) be the presheaf whose value on \(U\) is the kernel of \(\mathcal{A}(U)[\mathcal{E}(U)] \to \mathcal{B}(U)\). Then \(\mathcal{I}\) is the sheafification of \(\mathcal{I}'\) and \(\mathcal{B}\) is the sheafification of \(\mathcal{B}'\). Similarly, \(H^{-1}(\NL(\alpha))\) is the sheafification of the presheaf \[U \longmapsto \Ker(\mathcal{I}'(U)/\mathcal{I}'(U)^2 \to \Omega_{\mathcal{A}(U)[\mathcal{E}(U)]/\mathcal{A}(U)} \otimes_{\mathcal{A}(U)[\mathcal{E}(U)]} \mathcal{B}'(U))\] by Lemma 08TP. By Algebra, Lemma 00S1 we conclude \(H^{-1}(\NL(\alpha))\) is the sheaf associated to the presheaf \(U \mapsto H_1(L_{\mathcal{B}'(U)/\mathcal{A}(U)})\). Thus we have to show that the maps \(H_1(L_{\mathcal{B}'(U)/\mathcal{A}(U)}) \to H_1(L_{\mathcal{B}(U)/\mathcal{A}(U)})\) induce an isomorphism \(\mathcal{H}'_1 \to \mathcal{H}_1\) of sheafifications.

Injectivity of \(\mathcal{H}'_1 \to \mathcal{H}_1\). Let \(f \in H_1(L_{\mathcal{B}'(U)/\mathcal{A}(U)})\) map to zero in \(\mathcal{H}_1(U)\). To show: \(f\) maps to zero in \(\mathcal{H}'_1(U)\). The assumption means there is a covering \(\{U_i \to U\}\) such that \(f\) maps to zero in \(H_1(L_{\mathcal{B}(U_i)/\mathcal{A}(U_i)})\) for all \(i\). Replace \(U\) by \(U_i\) to get to the point where \(f\) maps to zero in \(H_1(L_{\mathcal{B}(U)/\mathcal{A}(U)})\). By Algebra, Lemma 07BQ we can find a finitely generated subalgebra \(\mathcal{B}'(U) \subset B \subset \mathcal{B}(U)\) such that \(f\) maps to zero in \(H_1(L_{B/\mathcal{A}(U)})\). Since \(\mathcal{B} = (\mathcal{B}')^\#\) we can find a covering \(\{U_i \to U\}\) such that \(B \to \mathcal{B}(U_i)\) factors through \(\mathcal{B}'(U_i)\). Hence \(f\) maps to zero in \(H_1(L_{\mathcal{B}'(U_i)/\mathcal{A}(U_i)})\) as desired.

The surjectivity of \(\mathcal{H}'_1 \to \mathcal{H}_1\) is proved in exactly the same way.

Lemma

Let \(f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\) be a morphism of topoi. Let \(\mathcal{A} \to \mathcal{B}\) be a homomorphism of sheaves of rings on \(\mathcal{D}\). Then \(f^{-1}\NL_{\mathcal{B}/\mathcal{A}} = \NL_{f^{-1}\mathcal{B}/f^{-1}\mathcal{A}}\).

Proof

Omitted. Hint: Use Lemma 08TQ.

The cotangent complex of a morphism of ringed topoi is defined in terms of the cotangent complex we defined above.

Definition

Let \(X = (\Sh(\mathcal{C}), \mathcal{O})\) and \(Y = (\Sh(\mathcal{C}'), \mathcal{O}')\) be ringed topoi. Let \((f, f^\sharp) : X \to Y\) be a morphism of ringed topoi. The naive cotangent complex \(\NL_f = \NL_{X/Y}\) of the given morphism of ringed topoi is \(\NL_{\mathcal{O}/f^{-1}\mathcal{O}'}\). We sometimes write \(\NL_{X/Y} = \NL_{\mathcal{O}/\mathcal{O}'}\).

Stalks of modules

We have to be a bit careful when taking stalks at points, since the colimit defining a stalk (see Sites, Equation 04EH) may not be filtered³. On the other hand, by definition of a point of a site the stalk functor is exact and commutes with arbitrary colimits. In other words, it behaves exactly as if the colimit were filtered.

Lemma

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

1. We have \((\mathcal{F}^\#)_p = \mathcal{F}_p\) for any presheaf of sets on \(\mathcal{C}\).

2. The stalk functor \(\Sh(\mathcal{C}) \to \textit{Sets}\), \(\mathcal{F} \mapsto \mathcal{F}_p\) is exact (see Categories, Definition 0034) and commutes with arbitrary colimits.

3. The stalk functor \(\textit{PSh}(\mathcal{C}) \to \textit{Sets}\), \(\mathcal{F} \mapsto \mathcal{F}_p\) is exact (see Categories, Definition 0034) and commutes with arbitrary colimits.

Proof

By Sites, Lemma 00Y8 we have (1). By Sites, Lemmas 00Y7 we see that \(\textit{PSh}(\mathcal{C}) \to \textit{Sets}\), \(\mathcal{F} \mapsto \mathcal{F}_p\) is a left adjoint, and by Sites, Lemma 00Y8 we see the same thing for \(\Sh(\mathcal{C}) \to \textit{Sets}\), \(\mathcal{F} \mapsto \mathcal{F}_p\). Hence the stalk functor commutes with arbitrary colimits (see Categories, Lemma 0038). It follows from the definition of a point of a site, see Sites, Definition 00Y5 that \(\Sh(\mathcal{C}) \to \textit{Sets}\), \(\mathcal{F} \mapsto \mathcal{F}_p\) is exact. Since sheafification is exact (Sites, Lemma 00WJ) it follows that \(\textit{PSh}(\mathcal{C}) \to \textit{Sets}\), \(\mathcal{F} \mapsto \mathcal{F}_p\) is exact.

In particular, since the stalk functor \(\mathcal{F} \mapsto \mathcal{F}_p\) on presheaves commutes with all finite limits and colimits we may apply the reasoning of the proof of Sites, Proposition 00YV. The result of such an argument is that if \(\mathcal{F}\) is a (pre)sheaf of algebraic structures listed in Sites, Proposition 00YV then the stalk \(\mathcal{F}_p\) is naturally an algebraic structure of the same kind. Let us explain this in detail when \(\mathcal{F}\) is an abelian presheaf. In this case the addition map \(+ : \mathcal{F} \times \mathcal{F} \to \mathcal{F}\) induces a map \[+ : \mathcal{F}_p \times \mathcal{F}_p = (\mathcal{F} \times \mathcal{F})_p \longrightarrow \mathcal{F}_p\] where the equal sign uses that stalk functor on presheaves of sets commutes with finite limits. This defines a group structure on the stalk \(\mathcal{F}_p\). In this way we obtain our stalk functor \[\textit{PAb}(\mathcal{C}) \longrightarrow \textit{Ab}, \quad \mathcal{F} \longmapsto \mathcal{F}_p\] By construction the underlying set of \(\mathcal{F}_p\) is the stalk of the underlying presheaf of sets. This also defines our stalk functor for sheaves of abelian groups by precomposing with the inclusion \(\textit{Ab}(\mathcal{C}) \subset \textit{PAb}(\mathcal{C})\).

Lemma

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

1. The functor \(\textit{Ab}(\mathcal{C}) \to \textit{Ab}\), \(\mathcal{F} \mapsto \mathcal{F}_p\) is exact.

2. The stalk functor \(\textit{PAb}(\mathcal{C}) \to \textit{Ab}\), \(\mathcal{F} \mapsto \mathcal{F}_p\) is exact.

3. For \(\mathcal{F} \in \Ob(\textit{PAb}(\mathcal{C}))\) we have \(\mathcal{F}_p = \mathcal{F}^\#_p\).

Proof

This is formal from the results of Lemma 04EN and the construction of the stalk functor above.

Next, we turn to the case of sheaves of modules. Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. (It suffices for the discussion that \(\mathcal{O}\) be a presheaf of rings.) Let \(\mathcal{F}\) be a presheaf of \(\mathcal{O}\)-modules. Let \(p\) be a point of \(\mathcal{C}\). In this case we get a map \[\cdot : \mathcal{O}_p \times \mathcal{O}_p = (\mathcal{O} \times \mathcal{O})_p \longrightarrow \mathcal{O}_p\] which is the stalk of the multiplication map and \[\cdot : \mathcal{O}_p \times \mathcal{F}_p = (\mathcal{O} \times \mathcal{F})_p \longrightarrow \mathcal{F}_p\] which is the stalk of the multiplication map. We omit the verification that this defines a ring structure on \(\mathcal{O}_p\) and an \(\mathcal{O}_p\)-module structure on \(\mathcal{F}_p\). In this way we obtain a functor \[\textit{PMod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}_p), \quad \mathcal{F} \longmapsto \mathcal{F}_p\] By construction the underlying set of \(\mathcal{F}_p\) is the stalk of the underlying presheaf of sets. This also defines our stalk functor for sheaves of \(\mathcal{O}\)-modules by precomposing with the inclusion \(\textit{Mod}(\mathcal{O}) \subset \textit{PMod}(\mathcal{O})\).

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(p\) be a point of \(\mathcal{C}\).

1. The functor \(\textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_p)\), \(\mathcal{F} \mapsto \mathcal{F}_p\) is exact.

2. The stalk functor \(\textit{PMod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_p)\), \(\mathcal{F} \mapsto \mathcal{F}_p\) is exact.

3. For \(\mathcal{F} \in \Ob(\textit{PMod}(\mathcal{O}))\) we have \(\mathcal{F}_p = \mathcal{F}^\#_p\).

Proof

This is formal from the results of Lemma 04EP, the construction of the stalk functor above, and Lemma 03DA.

Lemma

Let \((f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi or ringed sites. Let \(p\) be a point of \(\mathcal{C}\) or \(\Sh(\mathcal{C})\) and set \(q = f \circ p\). Then \[(f^*\mathcal{F})_p = \mathcal{F}_q \otimes_{\mathcal{O}_{\mathcal{D}, q}} \mathcal{O}_{\mathcal{C}, p}\] for any \(\mathcal{O}_\mathcal{D}\)-module \(\mathcal{F}\).

Proof

We have \[f^*\mathcal{F} = f^{-1}\mathcal{F} \otimes_{f^{-1}\mathcal{O}_\mathcal{D}} \mathcal{O}_\mathcal{C}\] by definition. Since taking stalks at \(p\) (i.e., applying \(p^{-1}\)) commutes with \(\otimes\) by Lemma 03EL we win by the relation between the stalk of pullbacks at \(p\) and stalks at \(q\) explained in Sites, Lemma 05V1 or Sites, Lemma 05V2.

Skyscraper sheaves

Let \(p\) be a point of a site \(\mathcal{C}\) or a topos \(\Sh(\mathcal{C})\). In this section we study the exactness properties of the functor which associates to an abelian group \(A\) the skyscraper sheaf \(p_*A\). First, recall that \(p_* : \textit{Sets} \to \Sh(\mathcal{C})\) has a lot of exactness properties, see Sites, Lemmas 05UX and 05UY.

Lemma

Let \(\mathcal{C}\) be a site. Let \(p\) be a point of \(\mathcal{C}\) or of its associated topos.

1. The functor \(p_* : \textit{Ab} \to \textit{Ab}(\mathcal{C})\), \(A \mapsto p_*A\) is exact.

2. There is a functorial direct sum decomposition \[p^{-1}p_*A = A \oplus I(A)\] for \(A \in \Ob(\textit{Ab})\).

Proof

By Sites, Lemma 05UX there are functorial maps \(A \to p^{-1}p_*A \to A\) whose composition equals \(\text{id}_A\). Hence a functorial direct sum decomposition as in (2) with \(I(A)\) the kernel of the adjunction map \(p^{-1}p_*A \to A\). The functor \(p_*\) is left exact by Lemma 03DC. The functor \(p_*\) transforms surjections into surjections by Sites, Lemma 05UY. Hence (1) holds.

To do the same thing for sheaves of modules, suppose given a point \(p\) of a ringed topos \((\Sh(\mathcal{C}), \mathcal{O})\). Recall that \(p^{-1}\) is just the stalk functor. Hence we can think of \(p\) as a morphism of ringed topoi \[(p, \text{id}_{\mathcal{O}_p}) : (\Sh(pt), \mathcal{O}_p) \longrightarrow (\Sh(\mathcal{C}), \mathcal{O}).\] Thus we get a pullback functor \(p^* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_p)\) which equals the stalk functor, and which we discussed in Lemma 04EQ. In this section we consider the functor \(p_* : \textit{Mod}(\mathcal{O}_p) \to \textit{Mod}(\mathcal{O})\).

Lemma

Let \((\Sh(\mathcal{C}), \mathcal{O})\) be a ringed topos. Let \(p\) be a point of the topos \(\Sh(\mathcal{C})\).

1. The functor \(p_* : \textit{Mod}(\mathcal{O}_p) \to \textit{Mod}(\mathcal{O})\), \(M \mapsto p_*M\) is exact.

2. The canonical surjection \(p^{-1}p_*M \to M\) is \(\mathcal{O}_p\)-linear.

3. The functorial direct sum decomposition \(p^{-1}p_*M = M \oplus I(M)\) of Lemma 05V7 is not \(\mathcal{O}_p\)-linear in general.

Proof

Part (1) and surjectivity in (2) follow immediately from the corresponding result for abelian sheaves in Lemma 05V7. Since \(p^{-1}\mathcal{O} = \mathcal{O}_p\) we have \(p^{-1} = p^*\) and hence \(p^{-1}p_*M \to M\) is the same as the counit \(p^*p_*M \to M\) of the adjunction for modules, whence linear.

Proof of (3). Suppose that \(G\) is a group. Consider the topos \(G\textit{-Sets} = \Sh(\mathcal{T}_G)\) and the point \(p : \textit{Sets} \to G\textit{-Sets}\). See Sites, Section 00VZ and Example 00YF. Here \(p^{-1}\) is the functor forgetting about the \(G\)-action. And \(p_*\) is the right adjoint of the forgetful functor, sending \(M\) to \(\text{Map}(G, M)\). The maps in the direct sum decomposition are the maps \[M \to \text{Map}(G, M) \to M\] where the first sends \(m \in M\) to the constant map with value \(m\) and where the second map is evaluation at the identity element \(1\) of \(G\). Next, suppose that \(R\) is a ring endowed with an action of \(G\). This determines a sheaf of rings \(\mathcal{O}\) on \(\mathcal{T}_G\). The category of \(\mathcal{O}\)-modules is the category of \(R\)-modules \(M\) endowed with an action of \(G\) compatible with the action on \(R\). The \(R\)-module structure on \(\text{Map}(G, M)\) is given by \[( r f ) (\sigma) = \sigma(r) f(\sigma)\] for \(r \in R\) and \(f \in \text{Map}(G, M)\). This is true because it is the unique \(G\)-invariant \(R\)-module structure compatible with evaluation at \(1\). The reader observes that in general the image of \(M \to \text{Map}(G, M)\) is not an \(R\)-submodule (for example take \(M = R\) and assume the \(G\)-action is nontrivial), which concludes the proof.

Example

Let \(G\) be a group. Consider the site \(\mathcal{T}_G\) and its point \(p\), see Sites, Example 00YF. Let \(R\) be a ring with a \(G\)-action which corresponds to a sheaf of rings \(\mathcal{O}\) on \(\mathcal{T}_G\). Then \(\mathcal{O}_p = R\) where we forget the \(G\)-action. In this case \(p^{-1}p_*M = \text{Map}(G, M)\) and \(I(M) = \{f : G \to M \mid f(1_G) = 0\}\) and \(M \to \text{Map}(G, M)\) assigns to \(m \in M\) the constant function with value \(m\).

Localization and points

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(p\) be a point of \(\mathcal{C}\). Let \(U\) be an object of \(\mathcal{C}\). For \(\mathcal{G}\) in \(\textit{Mod}(\mathcal{O}_U)\) we have \[(j_{U!}\mathcal{G})_p = \bigoplus\nolimits_q \mathcal{G}_q\] where the coproduct is over the points \(q\) of \(\mathcal{C}/U\) lying over \(p\), see Sites, Lemma 04H3.

Proof

We use the description of \(j_{U!}\mathcal{G}\) as the sheaf associated to the presheaf \(V \mapsto \bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{G}(V/_\varphi U)\) of Lemma 03DI. The stalk of \(j_{U!}\mathcal{G}\) at \(p\) is equal to the stalk of this presheaf, see Lemma 04EQ. Let \(u : \mathcal{C} \to \textit{Sets}\) be the functor corresponding to \(p\) (see Sites, Section 00Y3). Hence we see that \[(j_{U!}\mathcal{G})_p = \colim_{(V, y)} \bigoplus\nolimits_{\varphi : V \to U} \mathcal{G}(V/_\varphi U)\] where the colimit is taken in the category of abelian groups. To a quadruple \((V, y, \varphi, s)\) occurring in this colimit, we can assign \(x = u(\varphi)(y) \in u(U)\). Hence we obtain \[(j_{U!}\mathcal{G})_p = \bigoplus\nolimits_{x \in u(U)} \colim_{(\varphi : V \to U, y), \ u(\varphi)(y) = x} \mathcal{G}(V/_\varphi U).\] This is equal to the expression of the lemma by the description of the points \(q\) lying over \(x\) in Sites, Lemma 04H3.

Remark

Warning: The result of Lemma 0710 has no analogue for \(j_{U, *}\).

Pullbacks of flat modules

The pullback of a flat module along a morphism of ringed topoi is flat. This is a bit tricky to prove.

Lemma

Let \((f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi or ringed sites. Then \(f^*\mathcal{F}\) is a flat \(\mathcal{O}_\mathcal{C}\)-module whenever \(\mathcal{F}\) is a flat \(\mathcal{O}_\mathcal{D}\)-module.

Proof

Choose a diagram as in Lemma 03CR. Recall that being a flat module is intrinsic (see Section 03DG and Definition 03ER). Hence it suffices to prove the lemma for the morphism \((h, h^\sharp) : (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \to (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'})\). In other words, we may assume that our sites \(\mathcal{C}\) and \(\mathcal{D}\) have all finite limits and that \(f\) is a morphism of sites induced by a continuous functor \(u : \mathcal{D} \to \mathcal{C}\) which commutes with finite limits.

Recall that \(f^*\mathcal{F} = \mathcal{O}_\mathcal{C} \otimes_{f^{-1}\mathcal{O}_\mathcal{D}} f^{-1}\mathcal{F}\) (Definition 03D6). By Lemma 05V4 it suffices to prove that \(f^{-1}\mathcal{F}\) is a flat \(f^{-1}\mathcal{O}_\mathcal{D}\)-module. Combined with the previous paragraph this reduces us to the situation of the next paragraph.

Assume \(\mathcal{C}\) and \(\mathcal{D}\) are sites which have all finite limits and that \(u : \mathcal{D} \to \mathcal{C}\) is a continuous functor which commutes with finite limits. Let \(\mathcal{O}\) be a sheaf of rings on \(\mathcal{D}\) and let \(\mathcal{F}\) be a flat \(\mathcal{O}\)-module. Then \(u\) defines a morphism of sites \(f : \mathcal{C} \to \mathcal{D}\) (Sites, Proposition 00X6). To show: \(f^{-1}\mathcal{F}\) is a flat \(f^{-1}\mathcal{O}\)-module. Let \(U\) be an object of \(\mathcal{C}\) and let \[f^{-1}\mathcal{O}|_U \xrightarrow{(f_1, \ldots, f_n)} f^{-1}\mathcal{O}|_U^{\oplus n} \xrightarrow{(s_1, \ldots, s_n)} f^{-1}\mathcal{F}|_U\] be a complex of \(f^{-1}\mathcal{O}|_U\)-modules. Our goal is to construct a factorization of \((s_1, \ldots, s_n)\) on the members of a covering of \(U\) as in Lemma 08FC part (2). Consider the elements \(s_a \in f^{-1}\mathcal{F}(U)\) and \(f_a \in f^{-1}\mathcal{O}(U)\). Since \(f^{-1}\mathcal{F}\), resp. \(f^{-1}\mathcal{O}\) is the sheafification of \(u_p\mathcal{F}\) we may, after replacing \(U\) by the members of a covering, assume that \(s_a\) is the image of an element \(s'_a \in u_p\mathcal{F}(U)\) and \(f_a\) is the image of an element \(f'_a \in u_p\mathcal{O}(U)\). Then after another replacement of \(U\) by the members of a covering we may assume that \(\sum f'_as'_a\) is zero in \(u_p\mathcal{F}(U)\). Recall that the category \((\mathcal{I}_U^u)^{opp}\) is directed (Sites, Lemma 00X3) and that \(u_p\mathcal{F}(U) = \colim_{(\mathcal{I}_U^u)^{opp}} \mathcal{F}(V)\) and \(u_p\mathcal{O}(U) = \colim_{(\mathcal{I}_U^u)^{opp}} \mathcal{O}(V)\). Hence we may assume there is a pair \((V, \phi) \in \Ob(\mathcal{I}_U^u)\) where \(V\) is an object of \(\mathcal{D}\) and \(\phi\) is a morphism \(\phi : U \to u(V)\) of \(\mathcal{C}\) and elements \(s''_a \in \mathcal{F}(V)\) and \(f''_a \in \mathcal{O}(V)\) whose images in \(u_p\mathcal{F}(U)\) and \(u_p\mathcal{O}(U)\) are equal to \(s'_a\) and \(f'_a\) and such that \(\sum f''_a s''_a = 0\) in \(\mathcal{F}(V)\). Then we obtain a complex \[\mathcal{O}|_V \xrightarrow{(f''_1, \ldots, f''_n)} \mathcal{O}|_V^{\oplus n} \xrightarrow{(s''_1, \ldots, s''_n)} \mathcal{F}|_V\] and we can apply the other direction of Lemma 08FC to see there exists a covering \(\{V_i \to V\}\) of \(\mathcal{D}\) and for each \(i\) a factorization \[\mathcal{O}|_{V_i}^{\oplus n} \xrightarrow{B''_i} \mathcal{O}|_{V_i}^{\oplus l_i} \xrightarrow{(t''_{i1}, \ldots, t''_{il_i})} \mathcal{F}|_{V_i}\] of \((s''_1, \ldots, s''_n)|_{V_i}\) such that \(B''_i \circ (f''_1, \ldots, f''_n)|_{V_i} = 0\). Set \(U_i = U \times_{\phi, u(V)} u(V_i)\), denote \(B_i \in \text{Mat}(l_i \times n, f^{-1}\mathcal{O}(U_i))\) the image of \(B''_i\), and denote \(t_{ij} \in f^{-1}\mathcal{F}(U_i)\) the image of \(t''_{ij}\). Then we get a factorization \[f^{-1}\mathcal{O}|_{U_i}^{\oplus n} \xrightarrow{B_i} f^{-1}\mathcal{O}|_{U_i}^{\oplus l_i} \xrightarrow{(t_{i1}, \ldots, t_{il_i})} \mathcal{F}|_{U_i}\] of \((s_1, \ldots, s_n)|_{U_i}\) such that \(B_i \circ (f_1, \ldots, f_n)|_{U_i} = 0\). This finishes the proof.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(p\) be a point of \(\mathcal{C}\). If \(\mathcal{F}\) is a flat \(\mathcal{O}\)-module, then \(\mathcal{F}_p\) is a flat \(\mathcal{O}_p\)-module.

Proof

In Section 05V6 we have seen that we can think of \(p\) as a morphism of ringed topoi \[(p, \text{id}_{\mathcal{O}_p}) : (\Sh(pt), \mathcal{O}_p) \longrightarrow (\Sh(\mathcal{C}), \mathcal{O}).\] such that the pullback functor \(p^* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_p)\) equals the stalk functor. Thus the lemma follows from Lemma 05VD.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}\)-modules. Let \(\{p_i\}_{i \in I}\) be a conservative family of points of \(\mathcal{C}\). Then \(\mathcal{F}\) is flat if and only if \(\mathcal{F}_{p_i}\) is a flat \(\mathcal{O}_{p_i}\)-module for all \(i \in I\).

Proof

By Lemma 05VB we see one of the implications. For the converse, use that \((\mathcal{F} \otimes_\mathcal{O} \mathcal{G})_p = \mathcal{F}_p \otimes_{\mathcal{O}_p} \mathcal{G}_p\) by Lemma 03EL (as taking stalks at \(p\) is given by \(p^{-1}\)) and Lemma 05V3.

Lemma

Let \(f : (\Sh(\mathcal{C}'), \mathcal{O}') \to (\Sh(\mathcal{C}), \mathcal{O})\) be a morphism of ringed topoi. Let \(0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0\) be a short exact sequence of \(\mathcal{O}\)-modules with \(\mathcal{H}\) a flat \(\mathcal{O}\)-module. Then the sequence \(0 \to f^*\mathcal{F} \to f^*\mathcal{G} \to f^*\mathcal{H} \to 0\) is exact as well.

Proof

Since \(f^{-1}\) is exact we have the short exact sequence \(0 \to f^{-1}\mathcal{F} \to f^{-1}\mathcal{G} \to f^{-1}\mathcal{H} \to 0\) of \(f^{-1}\mathcal{O}\)-modules. By Lemma 05VD the \(f^{-1}\mathcal{O}\)-module \(f^{-1}\mathcal{H}\) is flat. By Lemma 03EX this implies that tensoring the sequence over \(f^{-1}\mathcal{O}\) with \(\mathcal{O}'\) the sequence remains exact. Since \(f^*\mathcal{F} = f^{-1}\mathcal{F} \otimes_{f^{-1}\mathcal{O}} \mathcal{O}'\) and similarly for \(\mathcal{G}\) and \(\mathcal{H}\) we conclude.

Locally ringed topoi

A reference for this section is [SGA4, Exposé IV, Exercice 13.9].

Lemma

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

1. For every object \(U\) of \(\mathcal{C}\) and \(f \in \mathcal{O}(U)\) there exists a covering \(\{U_j \to U\}\) such that for each \(j\) either \(f|_{U_j}\) is invertible or \((1 - f)|_{U_j}\) is invertible.

2. For \(U \in \Ob(\mathcal{C})\), \(n \geq 1\), and \(f_1, \ldots, f_n \in \mathcal{O}(U)\) which generate the unit ideal in \(\mathcal{O}(U)\) there exists a covering \(\{U_j \to U\}\) such that for each \(j\) there exists an \(i\) such that \(f_i|_{U_j}\) is invertible.

3. The map of sheaves of sets \[(\mathcal{O} \times \mathcal{O}) \amalg (\mathcal{O} \times \mathcal{O}) \longrightarrow \mathcal{O} \times \mathcal{O}\] which maps \((f, a)\) in the first component to \((f, af)\) and \((f, b)\) in the second component to \((f, b(1 - f))\) is surjective.

Proof

It is clear that (2) implies (1). To show that (1) implies (2) we argue by induction on \(n\). The first case is \(n = 2\) (since \(n = 1\) is trivial). In this case we have \(a_1f_1 + a_2f_2 = 1\) for some \(a_1, a_2 \in \mathcal{O}(U)\). By assumption we can find a covering \(\{U_j \to U\}\) such that for each \(j\) either \(a_1f_1|_{U_j}\) is invertible or \(a_2f_2|_{U_j}\) is invertible. Hence either \(f_1|_{U_j}\) is invertible or \(f_2|_{U_j}\) is invertible as desired. For \(n > 2\) we have \(a_1f_1 + \ldots + a_nf_n = 1\) for some \(a_1, \ldots, a_n \in \mathcal{O}(U)\). By the case \(n = 2\) we see that we have some covering \(\{U_j \to U\}_{j \in J}\) such that for each \(j\) either \(f_n|_{U_j}\) is invertible or \(a_1f_1 + \ldots + a_{n - 1}f_{n - 1}|_{U_j}\) is invertible. Say the first case happens for \(j \in J_n\). Set \(J' = J \setminus J_n\). By induction hypothesis, for each \(j \in J'\) we can find a covering \(\{U_{jk} \to U_j\}_{k \in K_j}\) such that for each \(k \in K_j\) there exists an \(i \in \{1, \ldots, n - 1\}\) such that \(f_i|_{U_{jk}}\) is invertible. By the axioms of a site the family of morphisms \(\{U_j \to U\}_{j \in J_n} \cup \{U_{jk} \to U\}_{j \in J', k \in K_j}\) is a covering which has the desired property.

Assume (1). To see that the map in (3) is surjective, let \((f, c)\) be a section of \(\mathcal{O} \times \mathcal{O}\) over \(U\). By assumption there exists a covering \(\{U_j \to U\}\) such that for each \(j\) either \(f\) or \(1 - f\) restricts to an invertible section. In the first case we can take \(a = c|_{U_j} (f|_{U_j})^{-1}\), and in the second case we can take \(b = c|_{U_j} (1 - f|_{U_j})^{-1}\). Hence \((f, c)\) is in the image of the map on each of the members. Conversely, assume (3) holds. For any \(U\) and \(f \in \mathcal{O}(U)\) there exists a covering \(\{U_j \to U\}\) of \(U\) such that the section \((f, 1)|_{U_j}\) is in the image of the map in (3) on sections over \(U_j\). This means precisely that either \(f\) or \(1 - f\) restricts to an invertible section over \(U_j\), and we see that (1) holds.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Consider the following conditions

1. For every object \(U\) of \(\mathcal{C}\) and \(f \in \mathcal{O}(U)\) there exists a covering \(\{U_j \to U\}\) such that for each \(j\) either \(f|_{U_j}\) is invertible or \((1 - f)|_{U_j}\) is invertible.

2. For every point \(p\) of \(\mathcal{C}\) the stalk \(\mathcal{O}_p\) is either the zero ring or a local ring.

We always have (1) \(\Rightarrow\) (2). If \(\mathcal{C}\) has enough points then (1) and (2) are equivalent.

Proof

Assume (1). Let \(p\) be a point of \(\mathcal{C}\) given by a functor \(u : \mathcal{C} \to \textit{Sets}\). Let \(f_p \in \mathcal{O}_p\). Since \(\mathcal{O}_p\) is computed by Sites, Equation (04EH) we may represent \(f_p\) by a triple \((U, x, f)\) where \(x \in U(U)\) and \(f \in \mathcal{O}(U)\). By assumption there exists a covering \(\{U_i \to U\}\) such that for each \(i\) either \(f\) or \(1 - f\) is invertible on \(U_i\). Because \(u\) defines a point of the site we see that for some \(i\) there exists an \(x_i \in u(U_i)\) which maps to \(x \in u(U)\). By the discussion surrounding Sites, Equation (04EH) we see that \((U, x, f)\) and \((U_i, x_i, f|_{U_i})\) define the same element of \(\mathcal{O}_p\). Hence we conclude that either \(f_p\) or \(1 - f_p\) is invertible. Thus \(\mathcal{O}_p\) is a ring such that for every element \(a\) either \(a\) or \(1 - a\) is invertible. This means that \(\mathcal{O}_p\) is either zero or a local ring, see Algebra, Lemma 00E9.

Assume (2) and assume that \(\mathcal{C}\) has enough points. Consider the map of sheaves of sets \[\mathcal{O} \times \mathcal{O} \amalg \mathcal{O} \times \mathcal{O} \longrightarrow \mathcal{O} \times \mathcal{O}\] of Lemma 04ES part (3). For any local ring \(R\) the corresponding map \((R \times R) \amalg (R \times R) \to R \times R\) is surjective, see for example Algebra, Lemma 00E9. Since each \(\mathcal{O}_p\) is a local ring or zero the map is surjective on stalks. Hence, by our assumption that \(\mathcal{C}\) has enough points it is surjective and we win.

In Modules, Section 01AE we pointed out how in a ringed space \((X, \mathcal{O}_X)\) there can be an open subspace over which the structure sheaf is zero. To prevent this we can require the sections \(1\) and \(0\) to have different values in every stalk of the space \(X\). In the setting of ringed topoi and ringed sites the condition is that [05D7]\[\begin{equation} \emptyset^\# \longrightarrow \text{Equalizer}(0, 1 : * \longrightarrow \mathcal{O}) \end{equation}\] is an isomorphism of sheaves. Here \(*\) is the singleton sheaf, resp. \(\emptyset^\#\) is the “empty sheaf”, i.e., the final, resp. initial object in the category of sheaves, see Sites, Example 00W3, resp. Section 04B4. In other words, the condition is that whenever \(U \in \Ob(\mathcal{C})\) is not sheaf theoretically empty, then \(1, 0 \in \mathcal{O}(U)\) are not equal. Let us state the obligatory lemma.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Consider the statements

1. (05D7) is an isomorphism, and

2. for every point \(p\) of \(\mathcal{C}\) the stalk \(\mathcal{O}_p\) is not the zero ring.

We always have (1) \(\Rightarrow\) (2) and if \(\mathcal{C}\) has enough points then (1) \(\Leftrightarrow\) (2).

Proof

Omitted.

Lemmas 04ES, 04ET, and 05D8 motivate the following definition.

Definition

A ringed site \((\mathcal{C}, \mathcal{O})\) is said to be locally ringed site if (05D7) is an isomorphism, and the equivalent properties of Lemma 04ES are satisfied.

In [SGA4, Exposé IV, Exercice 13.9] the condition that (05D7) be an isomorphism is missing leading to a slightly different notion of a locally ringed site and locally ringed topos. As we are motivated by the notion of a locally ringed space we decided to add this condition (see explanation above).

Lemma

Being a locally ringed site is an intrinsic property. More precisely,

1. if \(f : \Sh(\mathcal{C}') \to \Sh(\mathcal{C})\) is a morphism of topoi and \((\mathcal{C}, \mathcal{O})\) is a locally ringed site, then \((\mathcal{C}', f^{-1}\mathcal{O})\) is a locally ringed site, and

2. if \((f, f^\sharp) : (\Sh(\mathcal{C}'), \mathcal{O}') \to (\Sh(\mathcal{C}), \mathcal{O})\) is an equivalence of ringed topoi, then \((\mathcal{C}, \mathcal{O})\) is locally ringed if and only if \((\mathcal{C}', \mathcal{O}')\) is locally ringed.

Proof

It is clear that (2) follows from (1). To prove (1) note that as \(f^{-1}\) is exact we have \(f^{-1}* = *\), \(f^{-1}\emptyset^\# = \emptyset^\#\), and \(f^{-1}\) commutes with products, equalizers and transforms isomorphisms and surjections into isomorphisms and surjections. Thus \(f^{-1}\) transforms the isomorphism (05D7) into its analogue for \(f^{-1}\mathcal{O}\) and transforms the surjection of Lemma 04ES part (3) into the corresponding surjection for \(f^{-1}\mathcal{O}\).

In fact Lemma 04H7 part (2) is the analogue of Schemes, Lemma 01HC. It assures us that the following definition makes sense.

Definition

A ringed topos \((\Sh(\mathcal{C}), \mathcal{O})\) is said to be locally ringed if the underlying ringed site \((\mathcal{C}, \mathcal{O})\) is locally ringed.

Here is an example of a consequence of being locally ringed.

Lemma

Let \((\Sh(\mathcal{C}), \mathcal{O})\) be a ringed topos. Any locally free \(\mathcal{O}\)-module of rank \(1\) is invertible. If \((\mathcal{C}, \mathcal{O})\) is locally ringed, then the converse holds as well (but in general this is not the case).

Proof

Assume \(\mathcal{L}\) is locally free of rank \(1\) and consider the evaluation map \[\mathcal{L} \otimes_\mathcal{O} \SheafHom_\mathcal{O}(\mathcal{L}, \mathcal{O}) \longrightarrow \mathcal{O}\] Given any object \(U\) of \(\mathcal{C}\) and restricting to the members of a covering trivializing \(\mathcal{L}\), we see that this map is an isomorphism (details omitted). Hence \(\mathcal{L}\) is invertible by Lemma 0B8N.

Assume \((\Sh(\mathcal{C}), \mathcal{O})\) is locally ringed. Let \(U\) be an object of \(\mathcal{C}\). In the proof of Lemma 0B8N we have seen that there exists a covering \(\{U_i \to U\}\) such that \(\mathcal{L}|_{\mathcal{C}/U_i}\) is a direct summand of a finite free \(\mathcal{O}_{U_i}\)-module. After replacing \(U\) by \(U_i\), let \(p : \mathcal{O}_U^{\oplus r} \to \mathcal{O}_U^{\oplus r}\) be a projector whose image is isomorphic to \(\mathcal{L}|_{\mathcal{C}/U}\). Then \(p\) corresponds to a matrix \[P = (p_{ij}) \in \text{Mat}(r \times r, \mathcal{O}(U))\] which is a projector: \(P^2 = P\). Set \(A = \mathcal{O}(U)\) so that \(P \in \text{Mat}(r \times r, A)\). By Algebra, Lemma 00NX the image of \(P\) is a finite locally free module \(M\) over \(A\). Hence there are \(f_1, \ldots, f_t \in A\) generating the unit ideal, such that \(M_{f_i}\) is finite free. By Lemma 04ES after replacing \(U\) by the members of an open covering, we may assume that \(M\) is free. This means that \(\mathcal{L}|_U\) is free (details omitted). Of course, since \(\mathcal{L}\) is invertible, this is only possible if the rank of \(\mathcal{L}|_U\) is \(1\) and the proof is complete.

Next, we want to work out what it means to have a morphism of locally ringed spaces. In order to do this we have the following lemma.

Lemma

Let \((f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi. Consider the following conditions

1. The diagram of sheaves \[\xymatrix{ f^{-1}(\mathcal{O}^*_\mathcal{D}) \ar[r]_-{f^\sharp} \ar[d] & \mathcal{O}^*_\mathcal{C} \ar[d] \\ f^{-1}(\mathcal{O}_\mathcal{D}) \ar[r]^-{f^\sharp} & \mathcal{O}_\mathcal{C} }\] is cartesian.

2. For any point \(p\) of \(\mathcal{C}\), setting \(q = f \circ p\), the diagram \[\xymatrix{ \mathcal{O}^*_{\mathcal{D}, q} \ar[r] \ar[d] & \mathcal{O}^*_{\mathcal{C}, p} \ar[d] \\ \mathcal{O}_{\mathcal{D}, q} \ar[r] & \mathcal{O}_{\mathcal{C}, p} }\] of sets is cartesian.

We always have (1) \(\Rightarrow\) (2). If \(\mathcal{C}\) has enough points then (1) and (2) are equivalent. If \((\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})\) and \((\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) are locally ringed topoi then (2) is equivalent to

1. For any point \(p\) of \(\mathcal{C}\), setting \(q = f \circ p\), the ring map \(\mathcal{O}_{\mathcal{D}, q} \to \mathcal{O}_{\mathcal{C}, p}\) is a local ring map.

In fact, properties (2), or (3) for a conservative family of points implies (1).

Proof

This lemma proves itself, in other words, it follows by unwinding the definitions.

Definition

Let \((f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) be a morphism of ringed topoi. Assume \((\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})\) and \((\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})\) are locally ringed topoi. We say that \((f, f^\sharp)\) is a morphism of locally ringed topoi if and only if the diagram of sheaves \[\xymatrix{ f^{-1}(\mathcal{O}^*_\mathcal{D}) \ar[r]_-{f^\sharp} \ar[d] & \mathcal{O}^*_\mathcal{C} \ar[d] \\ f^{-1}(\mathcal{O}_\mathcal{D}) \ar[r]^-{f^\sharp} & \mathcal{O}_\mathcal{C} }\] (see Lemma 04H9) is cartesian. If \((f, f^\sharp)\) is a morphism of ringed sites, then we say that it is a morphism of locally ringed sites if the associated morphism of ringed topoi is a morphism of locally ringed topoi.

It is clear that an isomorphism of ringed topoi between locally ringed topoi is automatically an isomorphism of locally ringed topoi.

Lemma

Let \((f, f^\sharp) : (\Sh(\mathcal{C}_1), \mathcal{O}_1) \to (\Sh(\mathcal{C}_2), \mathcal{O}_2)\) and \((g, g^\sharp) : (\Sh(\mathcal{C}_2), \mathcal{O}_2) \to (\Sh(\mathcal{C}_3), \mathcal{O}_3)\) be morphisms of locally ringed topoi. Then the composition \((g, g^\sharp) \circ (f, f^\sharp)\) (see Definition 01D3) is also a morphism of locally ringed topoi.

Proof

Omitted.

Lemma

Let \(f : \Sh(\mathcal{C}') \to \Sh(\mathcal{C})\) be a morphism of topoi. If \(\mathcal{O}\) is a sheaf of rings on \(\mathcal{C}\), then \[f^{-1}(\mathcal{O}^*) = (f^{-1}\mathcal{O})^*.\] In particular, if \(\mathcal{O}\) turns \(\mathcal{C}\) into a locally ringed site, then setting \(f^\sharp = \text{id}\) the morphism of ringed topoi \[(f, f^\sharp) : (\Sh(\mathcal{C}'), f^{-1}\mathcal{O}) \to (\Sh(\mathcal{C}), \mathcal{O})\] is a morphism of locally ringed topoi.

Proof

Note that the diagram \[\xymatrix{ \mathcal{O}^* \ar[rr] \ar[d]_{u \mapsto (u, u^{-1})} & & {*} \ar[d]^{1} \\ \mathcal{O} \times \mathcal{O} \ar[rr]^-{(a, b) \mapsto ab} & & \mathcal{O} }\] is cartesian. Since \(f^{-1}\) is exact we conclude that \[\xymatrix{ f^{-1}(\mathcal{O}^*) \ar[d]_{u \mapsto (u, u^{-1})} \ar[rr] & & {*} \ar[d]^{1} \\ f^{-1}\mathcal{O} \times f^{-1}\mathcal{O} \ar[rr]^-{(a, b) \mapsto ab} & & f^{-1}\mathcal{O} }\] is cartesian which implies the first assertion. For the second, note that \((\mathcal{C}', f^{-1}\mathcal{O})\) is a locally ringed site by Lemma 04H7 so that the assertion makes sense. Now the first part implies that the morphism is a morphism of locally ringed topoi.

Lemma

Localization of locally ringed sites and topoi.

1. Let \((\mathcal{C}, \mathcal{O})\) be a locally ringed site. Let \(U\) be an object of \(\mathcal{C}\). Then the localization \((\mathcal{C}/U, \mathcal{O}_U)\) is a locally ringed site, and the localization morphism \[(j_U, j_U^\sharp) : (\Sh(\mathcal{C}/U), \mathcal{O}_U) \to (\Sh(\mathcal{C}), \mathcal{O})\] is a morphism of locally ringed topoi.

2. Let \((\mathcal{C}, \mathcal{O})\) be a locally ringed site. Let \(f : V \to U\) be a morphism of \(\mathcal{C}\). Then the morphism \[(j, j^\sharp) : (\Sh(\mathcal{C}/V), \mathcal{O}_V) \to (\Sh(\mathcal{C}/U), \mathcal{O}_U)\] of Lemma 04IY is a morphism of locally ringed topoi.

3. Let \((f, f^\sharp) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{D}, \mathcal{O}')\) be a morphism of locally ringed sites where \(f\) is given by the continuous functor \(u : \mathcal{D} \to \mathcal{C}\). Let \(V\) be an object of \(\mathcal{D}\) and let \(U = u(V)\). Then the morphism \[(f', (f')^\sharp) : (\Sh(\mathcal{C}/U), \mathcal{O}_U) \to (\Sh(\mathcal{D}/V), \mathcal{O}'_V)\] of Lemma 04J0 is a morphism of locally ringed sites.

4. Let \((f, f^\sharp) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{D}, \mathcal{O}')\) be a morphism of locally ringed sites where \(f\) is given by the continuous functor \(u : \mathcal{D} \to \mathcal{C}\). Let \(V \in \Ob(\mathcal{D})\), \(U \in \Ob(\mathcal{C})\), and \(c : U \to u(V)\). Then the morphism \[(f_c, (f_c)^\sharp) : (\Sh(\mathcal{C}/U), \mathcal{O}_U) \to (\Sh(\mathcal{D}/V), \mathcal{O}'_V)\] of Lemma 04J1 is a morphism of locally ringed topoi.

5. Let \((\Sh(\mathcal{C}), \mathcal{O})\) be a locally ringed topos. Let \(\mathcal{F}\) be a sheaf on \(\mathcal{C}\). Then the localization \((\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F})\) is a locally ringed topos and the localization morphism \[(j_\mathcal{F}, j_\mathcal{F}^\sharp) : (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) \to (\Sh(\mathcal{C}), \mathcal{O})\] is a morphism of locally ringed topoi.

6. Let \((\Sh(\mathcal{C}), \mathcal{O})\) be a locally ringed topos. Let \(s : \mathcal{G} \to \mathcal{F}\) be a map of sheaves on \(\mathcal{C}\). Then the morphism \[(j, j^\sharp) : (\Sh(\mathcal{C})/\mathcal{G}, \mathcal{O}_\mathcal{G}) \longrightarrow (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F})\] of Lemma 04J4 is a morphism of locally ringed topoi.

7. Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \longrightarrow (\Sh(\mathcal{D}), \mathcal{O}')\) be a morphism of locally ringed topoi. Let \(\mathcal{G}\) be a sheaf on \(\mathcal{D}\). Set \(\mathcal{F} = f^{-1}\mathcal{G}\). Then the morphism \[(f', (f')^\sharp) : (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) \longrightarrow (\Sh(\mathcal{D})/\mathcal{G}, \mathcal{O}'_\mathcal{G})\] of Lemma 04IF is a morphism of locally ringed topoi.

8. Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \longrightarrow (\Sh(\mathcal{D}), \mathcal{O}')\) be a morphism of locally ringed topoi. Let \(\mathcal{G}\) be a sheaf on \(\mathcal{D}\), let \(\mathcal{F}\) be a sheaf on \(\mathcal{C}\), and let \(s : \mathcal{F} \to f^{-1}\mathcal{G}\) be a morphism of sheaves. Then the morphism \[(f_s, (f_s)^\sharp) : (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) \longrightarrow (\Sh(\mathcal{D})/\mathcal{G}, \mathcal{O}'_\mathcal{G})\] of Lemma 04J8 is a morphism of locally ringed topoi.

Proof

Part (1) is clear since \(\mathcal{O}_U\) is just the restriction of \(\mathcal{O}\), so Lemmas 04H7 and 04KR apply. Part (2) is clear as the morphism \((j, j^\sharp)\) is actually a localization of a locally ringed site so (1) applies. Part (3) is clear also since \((f')^\sharp\) is just the restriction of \(f^\sharp\) to the topos \(\Sh(\mathcal{C}/U)\), see proof of Lemma 04IF (hence the diagram of Definition 04HA for the morphism \(f'\) is just the restriction of the corresponding diagram for \(f\), and restriction is an exact functor). Part (4) follows formally on combining (2) and (3). Parts (5), (6), (7), and (8) follow from their counterparts (1), (2), (3), and (4) by enlarging the sites as in Lemma 03CR and translating everything in terms of sites and morphisms of sites using the comparisons of Lemmas 04J3, 04J5, 04J7, and 04J9. (Alternatively one could use the same arguments as in the proofs of (1), (2), (3), and (4) to prove (5), (6), (7), and (8) directly.)

Lower shriek for modules

In this section we extend the construction of \(g_!\) discussed in Section 04BE to the case of sheaves of modules.

Lemma

Let \(u : \mathcal{C} \to \mathcal{D}\) be a continuous and cocontinuous functor between sites. Denote \(g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\) the associated morphism of topoi. Let \(\mathcal{O}_\mathcal{D}\) be a sheaf of rings on \(\mathcal{D}\). Set \(\mathcal{O}_\mathcal{C} = g^{-1}\mathcal{O}_\mathcal{D}\). Hence \(g\) becomes a morphism of ringed topoi with \(g^* = g^{-1}\). In this case there exists a functor \[g_! : \textit{Mod}(\mathcal{O}_\mathcal{C}) \longrightarrow \textit{Mod}(\mathcal{O}_\mathcal{D})\] which is left adjoint to \(g^*\).

Proof

Let \(U\) be an object of \(\mathcal{C}\). For any \(\mathcal{O}_\mathcal{D}\)-module \(\mathcal{G}\) we have \[\begin{align*} \Hom_{\mathcal{O}_\mathcal{C}}(j_{U!}\mathcal{O}_U, g^{-1}\mathcal{G}) & = g^{-1}\mathcal{G}(U) \\ & = \mathcal{G}(u(U)) \\ & = \Hom_{\mathcal{O}_\mathcal{D}}(j_{u(U)!}\mathcal{O}_{u(U)}, \mathcal{G}) \end{align*}\] because \(g^{-1}\) is described by restriction, see Sites, Lemma 00XR. Of course a similar formula holds for a direct sum of modules of the form \(j_{U!}\mathcal{O}_U\). By Homology, Lemma 0793 and Lemma 03EW we see that \(g_!\) exists.

Remark

Warning! Let \(u : \mathcal{C} \to \mathcal{D}\), \(g\), \(\mathcal{O}_\mathcal{D}\), and \(\mathcal{O}_\mathcal{C}\) be as in Lemma 0797. In general it is not the case that the diagram \[\xymatrix{ \textit{Mod}(\mathcal{O}_\mathcal{C}) \ar[r]_{g_!} \ar[d]_{forget} & \textit{Mod}(\mathcal{O}_\mathcal{D}) \ar[d]^{forget} \\ \textit{Ab}(\mathcal{C}) \ar[r]^{g^{Ab}_!} & \textit{Ab}(\mathcal{D}) }\] commutes (here \(g^{Ab}_!\) is the one from Lemma 04BG). There is a transformation of functors \[g_!^{Ab} \circ forget \longrightarrow forget \circ g_!\] From the proof of Lemma 0797 we see that this is an isomorphism if and only if \(g^{Ab}_!j_{U!}\mathcal{O}_U \to g_!j_{U!}\mathcal{O}_U\) is an isomorphism for all objects \(U\) of \(\mathcal{C}\). Since we have \(g_!j_{U!}\mathcal{O}_U = j_{u(U)!}\mathcal{O}_{u(U)}\) this holds if and only if \[g^{Ab}_!j_{U!}\mathcal{O}_U \longrightarrow j_{u(U)!}\mathcal{O}_{u(U)}\] is an isomorphism for all objects \(U\) of \(\mathcal{C}\). Note that for such a \(U\) we obtain a commutative diagram \[\xymatrix{ \mathcal{C}/U \ar[r]_-{u'} \ar[d]_{j_U} & \mathcal{D}/u(U) \ar[d]^{j_{u(U)}} \\ \mathcal{C} \ar[r]^u & \mathcal{D} }\] of cocontinuous functors of sites, see Sites, Lemma 03EG and therefore \(g^{Ab}_!j_{U!} = j_{u(U)!}(g')^{Ab}_!\) where \(g' : \Sh(\mathcal{C}/U) \to \Sh(\mathcal{D}/u(U))\) is the morphism of topoi induced by the cocontinuous functor \(u'\). Hence we see that \(g_! = g^{Ab}_!\) if the canonical map [0799]\[\begin{equation} (g')^{Ab}_!\mathcal{O}_U \longrightarrow \mathcal{O}_{u(U)} \end{equation}\] is an isomorphism for all objects \(U\) of \(\mathcal{C}\).

The following two results are of a slightly different nature.

Lemma

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

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

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

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

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

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

Then we have \(f'_* \circ (g')^* = g^* \circ f_*\) and \(g'_! \circ (f')^{-1} = f^{-1} \circ g_!\) on modules.

Proof

We have \((g')^*\mathcal{F} = (g')^{-1}\mathcal{F}\) and \(g^*\mathcal{G} = g^{-1}\mathcal{G}\) because of condition (5). Thus the first equality follows immediately from the corresponding equality in Sites, Lemma 0FN1. Since the left adjoint functors \(g_!\) and \(g'_!\) to \(g^*\) and \((g')^*\) exist by Lemma 0797 we see that the second equality follows by uniqueness of adjoint functors.

Lemma

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

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

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

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

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

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

Then \(f'_* \circ (g')^* = g^* \circ f_*\) and \(g'_! \circ (f')^{-1} = f^{-1} \circ g_!\) on modules.

Proof

We have \((g')^*\mathcal{F} = (g')^{-1}\mathcal{F}\) and \(g^*\mathcal{G} = g^{-1}\mathcal{G}\) because of condition (5). Thus the first equality follows immediately from the corresponding equality in Sites, Lemma 0FN2. Since the left adjoint functors \(g_!\) and \(g'_!\) to \(g^*\) and \((g')^*\) exist by Lemma 0797 we see that the second equality follows by uniqueness of adjoint functors.

Constant sheaves

Let \(E\) be a set and let \(\mathcal{C}\) be a site. The constant sheaf with value \(E\) is denoted \(\underline{E}\), see Sites, Example 0H7J. If \(E\) is an abelian group, ring, module, etc, then \(\underline{E}\) is a sheaf of abelian groups, rings, modules, etc.

Lemma

Let \(\mathcal{C}\) be a site. If \(0 \to A \to B \to C \to 0\) is a short exact sequence of abelian groups, then \(0 \to \underline{A} \to \underline{B} \to \underline{C} \to 0\) is an exact sequence of abelian sheaves and in fact it is even exact as a sequence of abelian presheaves.

Proof

Since sheafification is exact it is clear that \(0 \to \underline{A} \to \underline{B} \to \underline{C} \to 0\) is an exact sequence of abelian sheaves. Thus \(0 \to \underline{A} \to \underline{B} \to \underline{C}\) is an exact sequence of abelian presheaves. To see that \(\underline{B} \to \underline{C}\) is surjective, pick a set theoretical section \(s : C \to B\). This induces a section \(\underline{s} : \underline{C} \to \underline{B}\) of sheaves of sets left inverse to the surjection \(\underline{B} \to \underline{C}\).

Lemma

Let \(\mathcal{C}\) be a site. Let \(\Lambda\) be a ring and let \(M\) and \(Q\) be \(\Lambda\)-modules. If \(Q\) is a finitely presented \(\Lambda\)-module, then we have \(\underline{M \otimes_\Lambda Q}(U) = \underline{M}(U) \otimes_\Lambda Q\) for all \(U \in \Ob(\mathcal{C})\).

Proof

Choose a presentation \(\Lambda^{\oplus m} \to \Lambda^{\oplus n} \to Q \to 0\). This gives an exact sequence \(M^{\oplus m} \to M^{\oplus n} \to M \otimes Q \to 0\). By Lemma 093J we obtain an exact sequence \[\underline{M}(U)^{\oplus m} \to \underline{M}(U)^{\oplus n} \to \underline{M \otimes Q}(U) \to 0\] which proves the lemma. (Note that taking sections over \(U\) always commutes with finite direct sums, but not arbitrary direct sums.)

Lemma

Let \(\mathcal{C}\) be a site. Let \(\Lambda\) be a coherent ring. Let \(M\) be a flat \(\Lambda\)-module. For \(U \in \Ob(\mathcal{C})\) the module \(\underline{M}(U)\) is a flat \(\Lambda\)-module.

Proof

Let \(I \subset \Lambda\) be a finitely generated ideal. By Algebra, Lemma 00HD it suffices to show that \(\underline{M}(U) \otimes_\Lambda I \to \underline{M}(U)\) is injective. As \(\Lambda\) is coherent \(I\) is finitely presented as a \(\Lambda\)-module. By Lemma 093K we see that \(\underline{M}(U) \otimes I = \underline{M \otimes I}\). Since \(M\) is flat the map \(M \otimes I \to M\) is injective, whence \(\underline{M \otimes I} \to \underline{M}\) is injective.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\Lambda\) be a Noetherian ring. Let \(I \subset \Lambda\) be an ideal. The sheaf \(\underline{\Lambda}^\wedge = \lim \underline{\Lambda/I^n}\) is a flat \(\underline{\Lambda}\)-algebra. Moreover we have canonical identifications \[\underline{\Lambda}/I\underline{\Lambda} = \underline{\Lambda}/\underline{I} = \underline{\Lambda}^\wedge/I\underline{\Lambda}^\wedge = \underline{\Lambda}^\wedge/\underline{I} \cdot \underline{\Lambda}^\wedge = \underline{\Lambda}^\wedge/\underline{I}^\wedge = \underline{\Lambda/I}\] where \(\underline{I}^\wedge = \lim \underline{I/I^n}\).

Proof

To prove \(\underline{\Lambda}^\wedge\) is flat, it suffices to show that \(\underline{\Lambda}^\wedge(U)\) is flat as a \(\Lambda\)-module for each \(U \in \Ob(\mathcal{C})\), see Lemmas 03ES and 03ET. By Lemma 093L we see that \[\underline{\Lambda}^\wedge(U) = \lim \underline{\Lambda/I^n}(U)\] is a limit of a system of flat \(\Lambda/I^n\)-modules. By Lemma 093J we see that the transition maps are surjective. We conclude by More on Algebra, Lemma 0912.

To see the equalities, note that \(\underline{\Lambda}(U)/I\underline{\Lambda}(U) = \underline{\Lambda/I}(U)\) by Lemma 093K. It follows that \(\underline{\Lambda}/I\underline{\Lambda} = \underline{\Lambda}/\underline{I} = \underline{\Lambda/I}\). The system of short exact sequences \[0 \to \underline{I/I^n}(U) \to \underline{\Lambda/I^n}(U) \to \underline{\Lambda/I}(U) \to 0\] has surjective transition maps, hence gives a short exact sequence \[0 \to \lim \underline{I/I^n}(U) \to \lim \underline{\Lambda/I^n}(U) \to \lim \underline{\Lambda/I}(U) \to 0\] see Homology, Lemma 02N1. Thus we see that \(\underline{\Lambda}^\wedge/\underline{I}^\wedge = \underline{\Lambda/I}\). Since \[I \underline{\Lambda}^\wedge \subset \underline{I} \cdot \underline{\Lambda}^\wedge \subset \underline{I}^\wedge\] it suffices to show that \(I \underline{\Lambda}^\wedge(U) = \underline{I}^\wedge(U)\) for all \(U\). Choose generators \(I = (f_1, \ldots, f_r)\). For every \(n\) we obtain a short exact sequence \[0 \to K_n/(I^n)^{\oplus r} \to (\Lambda/I^n)^{\oplus r} \xrightarrow{(f_1, \ldots, f_r)} I/I^{n + 1} \to 0\] where \(K_n = \{(x_1, \ldots, x_r) \in \Lambda^{\oplus r} \mid \sum x_i f_i \in I^{n + 1}\}\). We obtain short exact sequences \[0 \to \underline{K_n/(I^n)^{\oplus r}}(U) \to \underline{(\Lambda/I^n)^{\oplus r}}(U) \to \underline{I/I^{n + 1}}(U) \to 0\] A calculation shows \(K_n = K + (I^n)^{\oplus r}\), hence the transition maps \(K_{n + 1}/(I^{n + 1})^{\oplus r} \to K_n/(I^n)^{\oplus r}\) are surjective. Hence the system of modules on the left hand side has surjective transition maps and a fortiori has ML. Thus we see that \((f_1, \ldots, f_r) : (\underline{\Lambda}^\wedge)^{\oplus r}(U) \to \underline{I}^\wedge(U)\) is surjective by Homology, Lemma 02N1 which is what we wanted to show.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\Lambda\) be a ring and let \(M\) be a \(\Lambda\)-module. Assume \(\Sh(\mathcal{C})\) is not the empty topos. Then

1. \(\underline{M}\) is a finite type sheaf of \(\underline{\Lambda}\)-modules if and only if \(M\) is a finite \(\Lambda\)-module, and

2. \(\underline{M}\) is a finitely presented sheaf of \(\underline{\Lambda}\)-modules if and only if \(M\) is a finitely presented \(\Lambda\)-module.

Proof

Proof of (1). If \(M\) is generated by \(x_1, \ldots, x_r\) then \(x_1, \ldots, x_r\) define global sections of \(\underline{M}\) which generate it, hence \(\underline{M}\) is of finite type. Conversely, assume \(\underline{M}\) is of finite type. Let \(U \in \mathcal{C}\) be an object which is not sheaf theoretically empty (Sites, Definition 04B5). Such an object exists as we assumed \(\Sh(\mathcal{C})\) is not the empty topos. Then there exists a covering \(\{U_i \to U\}\) and finitely many sections \(s_{ij} \in \underline{M}(U_i)\) generating \(\underline{M}|_{U_i}\). After refining the covering we may assume that \(s_{ij}\) come from elements \(x_{ij}\) of \(M\). Then \(x_{ij}\) define global sections of \(\underline{M}\) whose restriction to \(U\) generate \(\underline{M}\).

Assume there exist elements \(x_1, \ldots, x_r\) of \(M\) which define global sections of \(\underline{M}\) generating \(\underline{M}\) as a sheaf of \(\underline{\Lambda}\)-modules. We will show that \(x_1, \ldots, x_r\) generate \(M\) as a \(\Lambda\)-module. Let \(x \in M\). We can find a covering \(\{U_i \to U\}_{i \in I}\) and \(f_{i, j} \in \underline{\Lambda}(U_i)\) such that \(x|_{U_i} = \sum f_{i, j} x_j|_{U_i}\). After refining the covering we may assume \(f_{i, j} \in \Lambda\). Since \(U\) is not sheaf theoretically empty, there is at least one \(i \in I\) such that \(U_i\) is not sheaf theoretically empty. Then the map \(M \to \underline{M}(U_i)\) is injective (details omitted). We conclude that \(x = \sum f_{i, j}x_j\) in \(M\) as desired.

Proof of (2). Assume \(\underline{M}\) is a \(\underline{\Lambda}\)-module of finite presentation. By (1) we see that \(M\) is of finite type. Choose generators \(x_1, \ldots, x_r\) of \(M\) as a \(\Lambda\)-module. This determines a short exact sequence \(0 \to K \to \Lambda^{\oplus r} \to M \to 0\) which turns into a short exact sequence \[0 \to \underline{K} \to \underline{\Lambda}^{\oplus r} \to \underline{M} \to 0\] by Lemma 093J. By Lemma 082T we see that \(\underline{K}\) is of finite type. Hence \(K\) is a finite \(\Lambda\)-module by (1). Thus \(M\) is a \(\Lambda\)-module of finite presentation.

Locally constant sheaves

Here is the general definition.

Definition

Let \(\mathcal{C}\) be a site. Let \(\mathcal{F}\) be a sheaf of sets, groups, abelian groups, rings, modules over a fixed ring \(\Lambda\), etc.

1. We say \(\mathcal{F}\) is a constant sheaf of sets, groups, abelian groups, rings, modules over a fixed ring \(\Lambda\), etc if it is isomorphic as a sheaf of sets, groups, abelian groups, rings, modules over a fixed ring \(\Lambda\), etc to a constant sheaf \(\underline{E}\) as in Section 093I.

2. We say \(\mathcal{F}\) is locally constant if for every object \(U\) of \(\mathcal{C}\) there exists a covering \(\{U_i \to U\}\) such that \(\mathcal{F}|_{U_i}\) is a constant sheaf.

3. If \(\mathcal{F}\) is a sheaf of sets or groups, then we say \(\mathcal{F}\) is finite locally constant if the constant values are finite sets or finite groups.

Lemma

Let \(f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\) be a morphism of topoi. If \(\mathcal{G}\) is a locally constant sheaf of sets, groups, abelian groups, rings, modules over a fixed ring \(\Lambda\), etc on \(\mathcal{D}\), the same is true for \(f^{-1}\mathcal{G}\) on \(\mathcal{C}\).

Proof

Omitted.

Lemma

Let \(\mathcal{C}\) be a site with a final object \(X\).

1. Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be a map of locally constant sheaves of sets on \(\mathcal{C}\). If \(\mathcal{F}\) is finite locally constant, there exists a covering \(\{U_i \to X\}\) such that \(\varphi|_{U_i}\) is the map of constant sheaves associated to a map of sets.

2. Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be a map of locally constant sheaves of abelian groups on \(\mathcal{C}\). If \(\mathcal{F}\) is finite locally constant, there exists a covering \(\{U_i \to X\}\) such that \(\varphi|_{U_i}\) is the map of constant abelian sheaves associated to a map of abelian groups.

3. Let \(\Lambda\) be a ring. Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be a map of locally constant sheaves of \(\Lambda\)-modules on \(\mathcal{C}\). If \(\mathcal{F}\) is of finite type, then there exists a covering \(\{U_i \to X\}\) such that \(\varphi|_{U_i}\) is the map of constant sheaves of \(\Lambda\)-modules associated to a map of \(\Lambda\)-modules.

Proof

Proof omitted.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\Lambda\) be a ring. Let \(M\), \(N\) be \(\Lambda\)-modules. Let \(\mathcal{F}, \mathcal{G}\) be a locally constant sheaves of \(\Lambda\)-modules.

1. If \(M\) is of finite presentation, then \[\underline{\Hom_\Lambda(M, N)} = \SheafHom_{\underline{\Lambda}}(\underline{M}, \underline{N})\]

2. If \(M\) and \(N\) are both of finite presentation, then \[\underline{\text{Isom}_\Lambda(M, N)} = \mathit{Isom}_{\underline{\Lambda}}(\underline{M}, \underline{N})\]

3. If \(\mathcal{F}\) is of finite presentation, then \(\SheafHom_{\underline{\Lambda}}(\mathcal{F}, \mathcal{G})\) is a locally constant sheaf of \(\Lambda\)-modules.

4. If \(\mathcal{F}\) and \(\mathcal{G}\) are both of finite presentation, then \(\mathit{Isom}_{\underline{\Lambda}}(\mathcal{F}, \mathcal{G})\) is a locally constant sheaf of sets.

Proof

Proof of (1). Set \(E = \Hom_\Lambda(M, N)\). We want to show the canonical map \[\underline{E} \longrightarrow \SheafHom_{\underline{\Lambda}}(\underline{M}, \underline{N})\] is an isomorphism. The module \(M\) has a presentation \(\Lambda^{\oplus s} \to \Lambda^{\oplus t} \to M \to 0\). Then \(E\) sits in an exact sequence \[0 \to E \to \Hom_\Lambda(\Lambda^{\oplus t}, N) \to \Hom_\Lambda(\Lambda^{\oplus s}, N)\] and we have similarly \[0 \to \SheafHom_{\underline{\Lambda}}(\underline{M}, \underline{N}) \to \SheafHom_{\underline{\Lambda}}(\underline{\Lambda^{\oplus t}}, \underline{N}) \to \SheafHom_{\underline{\Lambda}}(\underline{\Lambda^{\oplus s}}, \underline{N})\] This reduces the question to the case where \(M\) is a finite free module where the result is clear.

Proof of (3). The question is local on \(\mathcal{C}\), hence we may assume \(\mathcal{F} = \underline{M}\) and \(\mathcal{G} = \underline{N}\) for some \(\Lambda\)-modules \(M\) and \(N\). By Lemma 093N the module \(M\) is of finite presentation. Thus the result follows from (1).

Parts (2) and (4) follow from parts (1) and (3) and the fact that \(\mathit{Isom}\) can be viewed as the subsheaf of sections of \(\SheafHom_{\underline{\Lambda}}(\mathcal{F}, \mathcal{G})\) which have an inverse in \(\SheafHom_{\underline{\Lambda}}(\mathcal{G}, \mathcal{F})\).

Lemma

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

1. The category of finite locally constant sheaves of sets is closed under finite limits and colimits inside \(\Sh(\mathcal{C})\).

2. The category of finite locally constant abelian sheaves is a weak Serre subcategory of \(\textit{Ab}(\mathcal{C})\).

3. Let \(\Lambda\) be a Noetherian ring. The category of finite type, locally constant sheaves of \(\Lambda\)-modules on \(\mathcal{C}\) is a weak Serre subcategory of \(\textit{Mod}(\mathcal{C}, \Lambda)\).

Proof

Proof of (1). We may work locally on \(\mathcal{C}\). Hence by Lemma 093S we may assume we are given a finite diagram of finite sets such that our diagram of sheaves is the associated diagram of constant sheaves. Then we just take the limit or colimit in the category of sets and take the associated constant sheaf. Some details omitted.

To prove (2) and (3) we use the criterion of Homology, Lemma 0754. Existence of kernels and cokernels is argued in the same way as above. Of course, the reason for using a Noetherian ring in (3) is to assure us that the kernel of a map of finite \(\Lambda\)-modules is a finite \(\Lambda\)-module. To see that the category is closed under extensions (in the case of sheaves \(\Lambda\)-modules), assume given an extension of sheaves of \(\Lambda\)-modules \[0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{G} \to 0\] on \(\mathcal{C}\) with \(\mathcal{F}\), \(\mathcal{G}\) finite type and locally constant. Localizing on \(\mathcal{C}\) we may assume \(\mathcal{F}\) and \(\mathcal{G}\) are constant, i.e., we get \[0 \to \underline{M} \to \mathcal{E} \to \underline{N} \to 0\] for some \(\Lambda\)-modules \(M, N\). Choose generators \(y_1, \ldots, y_m\) of \(N\), so that we get a short exact sequence \(0 \to K \to \Lambda^{\oplus m} \to N \to 0\) of \(\Lambda\)-modules. Localizing further we may assume \(y_j\) lifts to a section \(s_j\) of \(\mathcal{E}\). Thus we see that \(\mathcal{E}\) is a pushout as in the following diagram \[\xymatrix{ 0 \ar[r] & \underline{K} \ar[d] \ar[r] & \underline{\Lambda^{\oplus m}} \ar[d] \ar[r] & \underline{N} \ar[d] \ar[r] & 0 \\ 0 \ar[r] & \underline{M} \ar[r] & \mathcal{E} \ar[r] & \underline{N} \ar[r] & 0 }\] By Lemma 093S again (and the fact that \(K\) is a finite \(\Lambda\)-module as \(\Lambda\) is Noetherian) we see that the map \(\underline{K} \to \underline{M}\) is locally constant, hence we conclude.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\Lambda\) be a ring. The tensor product of two locally constant sheaves of \(\Lambda\)-modules on \(\mathcal{C}\) is a locally constant sheaf of \(\Lambda\)-modules.

Proof

Omitted.

Localizing sheaves of rings

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{S} \subset \mathcal{O}\) be a sub-presheaf of sets such that for all \(U \in \Ob(\mathcal{C})\) the set \(\mathcal{S}(U) \subset \mathcal{O}(U)\) is a multiplicative subset, see Algebra, Definition 00CN. In this case we can consider the presheaf of rings \[\mathcal{S}^{-1}\mathcal{O} : U \longmapsto \mathcal{S}(U)^{-1}\mathcal{O}(U).\] The restriction mapping sends the section \(f/s\), \(f \in \mathcal{O}(U)\), \(s \in \mathcal{S}(U)\) to \((f|_V)/(s|_V)\) for \(V \to U\) in \(\mathcal{C}\).

Lemma

In the situation above the map to the sheafification \[\mathcal{O} \longrightarrow (\mathcal{S}^{-1}\mathcal{O})^\#\] is a homomorphism of sheaves of rings with the following universal property: for any homomorphism of sheaves of rings \(\mathcal{O} \to \mathcal{A}\) such that each local section of \(\mathcal{S}\) maps to an invertible section of \(\mathcal{A}\) there exists a unique factorization \((\mathcal{S}^{-1}\mathcal{O})^\# \to \mathcal{A}\).

Proof

Omitted.

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{S} \subset \mathcal{O}\) be a sub-presheaf of sets such that for all \(U \in \mathcal{C}\) the set \(\mathcal{S}(U) \subset \mathcal{O}(U)\) is a multiplicative subset. Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}\)-modules. In this case we can consider the presheaf of \(\mathcal{S}^{-1}\mathcal{O}\)-modules \[\mathcal{S}^{-1}\mathcal{F} : U \longmapsto \mathcal{S}(U)^{-1}\mathcal{F}(U).\] The restriction mapping sends the section \(t/s\), \(t \in \mathcal{F}(U)\), \(s \in \mathcal{S}(U)\) to \((t|_V)/(s|_V)\) if \(V \to U\) is a morphism of \(\mathcal{C}\). Then \(\mathcal{S}^{-1}\mathcal{F}\) is a presheaf of \(\mathcal{S}^{-1}\mathcal{O}\)-modules.

Lemma

In the situation above the map to the sheafification \[\mathcal{F} \longrightarrow (\mathcal{S}^{-1}\mathcal{F})^\#\] has the following universal property: for any homomorphism of \(\mathcal{O}\)-modules \(\mathcal{F} \to \mathcal{G}\) such that each local section of \(\mathcal{S}\) acts invertibly on \(\mathcal{G}\) there exists a unique factorization \((\mathcal{S}^{-1}\mathcal{F})^\# \to \mathcal{G}\). Moreover we have \[(\mathcal{S}^{-1}\mathcal{F})^\# = (\mathcal{S}^{-1}\mathcal{O})^\# \otimes_\mathcal{O} \mathcal{F}\] as sheaves of \((\mathcal{S}^{-1}\mathcal{O})^\#\)-modules.

Proof

Omitted.

Sheaves of pointed sets

In this section we collect some facts about sheaves of pointed sets which we’ve previously mentioned only for abelian sheaves.

A pointed set is a pair \((S, 0)\) where \(S\) is a set and \(0 \in S\) is an element of \(S\). A morphism \((S, 0) \to (S', 0')\) of pointed sets is simply a map of sets \(S \to S'\) sending \(0\) to \(0'\). We’ll abuse notation and say “let \(S\) be a pointed set” to mean \(S\) is endowed with a marked element \(0 \in S\). A sheaf of pointed sets is the same thing as a sheaf of sets \(\mathcal{F}\) endowed with a “marking” \(0 : * \to \mathcal{F}\) where \(*\) is the final sheaf (Sites, Example 00W3).

Given a morphism of sites or of topoi, there are pushforward and pullback functors on the categories of sheaves of pointed sets, see Sites, Section 00YR. These are constructed by taking the pushforward, resp. pullback of the underlying sheaf of sets and suitably marking it (using that the pullback of the final sheaf is the final sheaf).

Let \(u : \mathcal{C} \to \mathcal{D}\) be a continuous and cocontinuous functor between sites. Let \(g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})\) be the morphism of topoi associated with \(u\), see Sites, Lemma 00XO. Then \(g^{-1}\) on sheaves of pointed sets has a left adjoint \(g_!\) as well. The construction of this functor is entirely analogous to the construction of \(g_!\) on abelian sheaves in Section 04BE.

Similarly, if \(j : \mathcal{C}/U \to \mathcal{C}\) is as in Section 03DH then there is a left adjoint \(j_!\) to the functor \(j^{-1}\) on sheaves of pointed sets

If we ever need these facts and constructions we will precisely state and prove here the corresponding lemmas.