Etale Cohomology

Introduction

This chapter is the first in a series of chapter on the étale cohomology of schemes. In this chapter we discuss the very basics of the étale topology and cohomology of abelian sheaves in this topology. Many of the topics discussed may be safely skipped on a first reading; please see the advice in the next section as to how to decide what to skip.

The initial version of this chapter was formed by the notes of the first part of a course on étale cohomology taught by Johan de Jong at Columbia University in the Fall of 2009. The original note takers were Thibaut Pugin, Zachary Maddock and Min Lee. The second part of the course can be found in the chapter on the trace formula, see The Trace Formula, Section 0F5Q.

Which sections to skip on a first reading?

We want to use the material in this chapter for the development of theory related to algebraic spaces, Deligne-Mumford stacks, algebraic stacks, etc. Thus we have added some pretty technical material to the original exposition of étale cohomology for schemes. The reader can recognize this material by the frequency of the word “topos”, or by discussions related to set theory, or by proofs dealing with very general properties of morphisms of schemes. Some of these discussions can be skipped on a first reading.

In particular, we suggest that the reader skip the following sections:

1. Comparing big and small topoi, Section 0757.

2. Recovering morphisms, Section 04JH.

3. Push and pull, Section 04C6.

4. Property (A), Section 04DJ.

5. Property (B), Section 04DN.

6. Property (C), Section 04DS.

7. Topological invariance of the small étale site, Section 04DY.

8. Integral universally injective morphisms, Section 04FY.

9. Big sites and pushforward, Section 04E2.

10. Exactness of big lower shriek, Section 04CB.

Besides these sections there are some sporadic results that may be skipped that the reader can recognize by the keywords given above.

Prologue

These lectures are about another cohomology theory. The first thing to remark is that the Zariski topology is not entirely satisfactory. One of the main reasons that it fails to give the results that we would want is that if \(X\) is a complex variety and \(\mathcal{F}\) is a constant sheaf then \[H^i(X, \mathcal{F}) = 0, \quad \text{ for all } i > 0.\] The reason for that is the following. In an irreducible scheme (a variety in particular), any two nonempty open subsets meet, and so the restriction mappings of a constant sheaf are surjective. We say that the sheaf is flasque. In this case, all higher Čech cohomology groups vanish, and so do all higher Zariski cohomology groups. In other words, there are “not enough” open sets in the Zariski topology to detect this higher cohomology.

On the other hand, if \(X\) is a smooth projective complex variety, then \[H_{Betti}^{2 \dim X}(X (\mathbf{C}), \Lambda) = \Lambda \quad \text{ for } \Lambda = \mathbf{Z}, \ \mathbf{Z}/n\mathbf{Z},\] where \(X(\mathbf{C})\) means the set of complex points of \(X\). This is a feature that would be nice to replicate in algebraic geometry. In positive characteristic in particular.

The étale topology

It is very hard to simply “add” extra open sets to refine the Zariski topology. One efficient way to define a topology is to consider not only open sets, but also some schemes that lie over them. To define the étale topology, one considers all morphisms \(\varphi : U \to X\) which are étale. If \(X\) is a smooth projective variety over \(\mathbf{C}\), then this means

1. \(U\) is a disjoint union of smooth varieties, and

2. \(\varphi\) is (analytically) locally an isomorphism.

The word “analytically” refers to the usual (transcendental) topology over \(\mathbf{C}\). So the second condition means that the derivative of \(\varphi\) has full rank everywhere (and in particular all the components of \(U\) have the same dimension as \(X\)).

A double cover – loosely defined as a finite degree \(2\) map between varieties – for example \[\Spec(\mathbf{C}[t]) \longrightarrow \Spec(\mathbf{C}[t]), \quad t \longmapsto t^2\] will not be an étale morphism if it has a fibre consisting of a single point. In the example this happens when \(t = 0\). For a finite map between varieties over \(\mathbf{C}\) to be étale all the fibers should have the same number of points. Removing the point \(t = 0\) from the source of the map in the example will make the morphism étale. But we can remove other points from the source of the morphism also, and the morphism will still be étale. To consider the étale topology, we have to look at all such morphisms. Unlike the Zariski topology, these need not be merely open subsets of \(X\), even though their images always are.

Definition

A family of morphisms \(\{ \varphi_i : U_i \to X\}_{i \in I}\) is called an étale covering if each \(\varphi_i\) is an étale morphism and their images cover \(X\), i.e., \(X = \bigcup_{i \in I} \varphi_i(U_i)\).

This “defines” the étale topology. In other words, we can now say what the sheaves are. An étale sheaf \(\mathcal{F}\) of sets (resp. abelian groups, vector spaces, etc) on \(X\) is the data:

1. for each étale morphism \(\varphi : U \to X\) a set (resp. abelian group, vector space, etc) \(\mathcal{F}(U)\),

2. for each pair \(U, \ U'\) of étale schemes over \(X\), and each morphism \(U \to U'\) over \(X\) (which is automatically étale) a restriction map \(\rho^{U'}_U : \mathcal{F}(U') \to \mathcal{F}(U)\)

These data have to satisfy the condition that \(\rho^U_U = \text{id}\) in case of the identity morphism \(U \to U\) and that \(\rho^{U'}_U \circ \rho^{U''}_{U'} = \rho^{U''}_U\) when we have morphisms \(U \to U' \to U''\) of schemes étale over \(X\) as well as the following sheaf axiom:

• for every étale covering \(\{ \varphi_i : U_i \to U\}_{i \in I}\), the diagram \[\xymatrix{ \mathcal{F} (U) \ar[r] & \Pi_{i \in I} \mathcal{F} (U_i) \ar@<1ex>[r] \ar@<-1ex>[r] & \Pi_{i, j \in I} \mathcal{F} (U_i \times_U U_j) }\] is an equalizer diagram in the category of sets (resp. abelian groups, vector spaces, etc).

Remark

In the last statement, it is essential not to forget the case where \(i = j\) which is in general a highly nontrivial condition (unlike in the Zariski topology). In fact, frequently important coverings have only one element.

Since the identity is an étale morphism, we can compute the global sections of an étale sheaf, and cohomology will simply be the corresponding right-derived functors. In other words, once more theory has been developed and statements have been made precise, there will be no obstacle to defining cohomology.

Feats of the étale topology

For a natural number \(n \in \mathbf{N} = \{1, 2, 3, 4, \ldots\}\) it is true that \[H_\etale^2 (\mathbf{P}^1_\mathbf{C}, \mathbf{Z}/n\mathbf{Z}) = \mathbf{Z}/n\mathbf{Z}.\] More generally, if \(X\) is a complex variety, then its étale Betti numbers with coefficients in a finite field agree with the usual Betti numbers of \(X(\mathbf{C})\), i.e., \[\dim_{\mathbf{F}_q} H_\etale^{2i} (X, \mathbf{F}_q) = \dim_{\mathbf{F}_q} H_{Betti}^{2i} (X(\mathbf{C}), \mathbf{F}_q).\] This is extremely satisfactory. However, these equalities only hold for torsion coefficients, not in general. For integer coefficients, one has \[H_\etale^2 (\mathbf{P}^1_\mathbf{C}, \mathbf{Z}) = 0.\] By contrast \(H_{Betti}^2(\mathbf{P}^1(\mathbf{C}), \mathbf{Z}) = \mathbf{Z}\) as the topological space \(\mathbf{P}^1(\mathbf{C})\) is homeomorphic to a \(2\)-sphere. There are ways to get back to nontorsion coefficients from torsion ones by a limit procedure which we will come to shortly.

A computation

How do we compute the cohomology of \(\mathbf{P}^1_\mathbf{C}\) with coefficients \(\Lambda = \mathbf{Z}/n\mathbf{Z}\)? We use Čech cohomology. A covering of \(\mathbf{P}^1_\mathbf{C}\) is given by the two standard opens \(U_0, U_1\), which are both isomorphic to \(\mathbf{A}^1_\mathbf{C}\), and whose intersection is isomorphic to \(\mathbf{A}^1_\mathbf{C} \setminus \{0\} = \mathbf{G}_{m, \mathbf{C}}\). It turns out that the Mayer-Vietoris sequence holds in étale cohomology. This gives an exact sequence \[H_\etale^{i-1}(U_0\cap U_1, \Lambda) \to H_\etale^i(\mathbf{P}^1_\mathbf{C}, \Lambda) \to H_\etale^i(U_0, \Lambda) \oplus H_\etale^i(U_1, \Lambda) \to H_\etale^i(U_0\cap U_1, \Lambda).\] To get the answer we expect, we would need to show that the direct sum in the third term vanishes. In fact, it is true that, as for the usual topology, \[H_\etale^q (\mathbf{A}^1_\mathbf{C}, \Lambda) = 0 \quad \text{ for } q \geq 1,\] and \[H_\etale^q (\mathbf{A}^1_\mathbf{C} \setminus \{0\}, \Lambda) = \left\{ \begin{matrix} \Lambda & \text{ if }q = 1\text{, and} \\ 0 & \text{ for }q \geq 2. \end{matrix} \right.\] These results are already quite hard (what is an elementary proof?). Let us explain how we would compute this once the machinery of étale cohomology is at our disposal.

Higher cohomology. This is taken care of by the following general fact: if \(X\) is an affine curve over \(\mathbf{C}\), then \[H_\etale^q (X, \mathbf{Z}/n\mathbf{Z}) = 0 \quad \text{ for } q \geq 2.\] This is proved by considering the generic point of the curve and doing some Galois cohomology. So we only have to worry about the cohomology in degree 1.

Cohomology in degree 1. We use the following identifications: \[\begin{eqnarray*} H_\etale^1 (X, \mathbf{Z}/n\mathbf{Z}) = \left\{ \begin{matrix} \text{sheaves of sets }\mathcal{F}\text{ on the \'etale site }X_\etale \text{ endowed with an} \\ \text{action }\mathbf{Z}/n\mathbf{Z} \times \mathcal{F} \to \mathcal{F} \text{ such that }\mathcal{F}\text{ is a }\mathbf{Z}/n\mathbf{Z}\text{-torsor.} \end{matrix} \right\} \Big/ \cong \\ = \left\{ \begin{matrix} \text{morphisms }Y \to X\text{ which are finite \'etale together} \\ \text{ with a free }\mathbf{Z}/n\mathbf{Z}\text{ action such that } X = Y/(\mathbf{Z}/n\mathbf{Z}). \end{matrix} \right\} \Big/ \cong. \end{eqnarray*}\] The first identification is very general (it is true for any cohomology theory on a site) and has nothing to do with the étale topology. The second identification is a consequence of descent theory. The last set describes a collection of geometric objects on which we can get our hands.

The curve \(\mathbf{A}^1_\mathbf{C}\) has no nontrivial finite étale covering and hence \(H_\etale^1 (\mathbf{A}^1_\mathbf{C}, \mathbf{Z}/n\mathbf{Z}) = 0\). This can be seen either topologically or by using the argument in the next paragraph.

Let us describe the finite étale coverings \(\varphi : Y \to \mathbf{A}^1_\mathbf{C} \setminus \{0\}\). It suffices to consider the case where \(Y\) is connected, which we assume. We are going to find out what \(Y\) can be by applying the Riemann-Hurwitz formula (of course this is a bit silly, and you can go ahead and skip the next section if you like). Say that this morphism is \(n\) to 1, and consider a projective compactification \[\xymatrix{ {Y\ } \ar@{^{(}->}[r] \ar[d]^\varphi & {\bar Y} \ar[d]^{\bar\varphi} \\ {\mathbf{A}^1_\mathbf{C} \setminus \{0\}} \ar@{^{(}->}[r] & {\mathbf{P}^1_\mathbf{C}} }\] Even though \(\varphi\) is étale and does not ramify, \(\bar{\varphi}\) may ramify at 0 and \(\infty\). Say that the preimages of 0 are the points \(y_1, \ldots, y_r\) with indices of ramification \(e_1, \ldots e_r\), and that the preimages of \(\infty\) are the points \(y_1', \ldots, y_s'\) with indices of ramification \(d_1, \ldots d_s\). In particular, \(\sum e_i = n = \sum d_j\). Applying the Riemann-Hurwitz formula, we get \[2 g_Y - 2 = -2n + \sum (e_i - 1) + \sum (d_j - 1)\] and therefore \(g_Y = 0\), \(r = s = 1\) and \(e_1 = d_1 = n\). Hence \(Y \cong {\mathbf{A}^1_\mathbf{C} \setminus \{0\}}\), and it is easy to see that \(\varphi(z) = \lambda z^n\) for some \(\lambda \in \mathbf{C}^*\). After reparametrizing \(Y\) we may assume \(\lambda = 1\). Thus our covering is given by taking the \(n\)th root of the coordinate on \(\mathbf{A}^1_{\mathbf{C}} \setminus \{0\}\).

Remember that we need to classify the coverings of \({\mathbf{A}^1_\mathbf{C} \setminus \{0\}}\) together with free \(\mathbf{Z}/n\mathbf{Z}\)-actions on them. In our case any such action corresponds to an automorphism of \(Y\) sending \(z\) to \(\zeta_n z\), where \(\zeta_n\) is a primitive \(n\)th root of unity. There are \(\phi(n)\) such actions (here \(\phi(n)\) means the Euler function). Thus there are exactly \(\phi(n)\) connected finite étale coverings with a given free \(\mathbf{Z}/n\mathbf{Z}\)-action, each corresponding to a primitive \(n\)th root of unity. We leave it to the reader to see that the disconnected finite étale degree \(n\) coverings of \(\mathbf{A}^1_{\mathbf{C}} \setminus \{0\}\) with a given free \(\mathbf{Z}/n\mathbf{Z}\)-action correspond one-to-one with \(n\)th roots of \(1\) which are not primitive. In other words, this computation shows that \[H_\etale^1 (\mathbf{A}^1_\mathbf{C} \setminus \{0\}, \mathbf{Z}/n\mathbf{Z}) = \Hom(\mu_n(\mathbf{C}), \mathbf{Z}/n\mathbf{Z}) \cong \mathbf{Z}/n\mathbf{Z}.\] The first identification is canonical, the second isn’t, see Remark 0A44. Since the proof of Riemann-Hurwitz does not use the computation of cohomology, the above actually constitutes a proof (provided we fill in the details on vanishing, etc).

Nontorsion coefficients

To study nontorsion coefficients, one makes the following definition: \[H_\etale^i (X, \mathbf{Q}_\ell) := \left( \lim_n H_\etale^i(X, \mathbf{Z}/\ell^n\mathbf{Z}) \right) \otimes_{\mathbf{Z}_\ell} \mathbf{Q}_\ell.\] The symbol \(\lim_n\) denote the limit of the system of cohomology groups \(H_\etale^i(X, \mathbf{Z}/\ell^n\mathbf{Z})\) indexed by \(n\), see Categories, Section 002Z. Thus we will need to study systems of sheaves satisfying some compatibility conditions.

Sheaf theory

At this point we start talking about sites and sheaves in earnest. There is an amazing amount of useful abstract material that could fit in the next few sections. Some of this material is worked out in earlier chapters, such as the chapter on sites, modules on sites, and cohomology on sites. We try to refrain from adding too much material here, just enough so the material later in this chapter makes sense.

Presheaves

A reference for this section is Sites, Section 00V1.

Definition

Let \(\mathcal{C}\) be a category. A presheaf of sets (respectively, an abelian presheaf) on \(\mathcal{C}\) is a functor \(\mathcal{C}^{opp} \to \textit{Sets}\) (resp. \(\textit{Ab}\)).

Terminology. If \(U \in \Ob(\mathcal{C})\), then elements of \(\mathcal{F}(U)\) are called sections of \(\mathcal{F}\) over \(U\). For \(\varphi : V \to U\) in \(\mathcal{C}\), the map \(\mathcal{F}(\varphi) : \mathcal{F}(U) \to \mathcal{F}(V)\) is called the restriction map and is often denoted \(s \mapsto s|_V\) or sometimes \(s \mapsto \varphi^*s\). The notation \(s|_V\) is ambiguous since the restriction map depends on \(\varphi\), but it is a standard abuse of notation. We also use the notation \(\Gamma(U, \mathcal{F}) = \mathcal{F}(U)\).

Saying that \(\mathcal{F}\) is a functor means that if \(W \to V \to U\) are morphisms in \(\mathcal{C}\) and \(s \in \Gamma(U, \mathcal{F})\) then \((s|_V)|_W = s |_W\), with the abuse of notation just seen. Moreover, the restriction mappings corresponding to the identity morphisms \(\text{id}_U : U \to U\) are the identity.

The category of presheaves of sets (respectively of abelian presheaves) on \(\mathcal{C}\) is denoted \(\textit{PSh} (\mathcal{C})\) (resp. \(\textit{PAb} (\mathcal{C})\)). It is the category of functors from \(\mathcal{C}^{opp}\) to \(\textit{Sets}\) (resp. \(\textit{Ab}\)), which is to say that the morphisms of presheaves are natural transformations of functors. We only consider the categories \(\textit{PSh}(\mathcal{C})\) and \(\textit{PAb}(\mathcal{C})\) when the category \(\mathcal{C}\) is small. (Our convention is that a category is small unless otherwise mentioned, and if it isn’t small it should be listed in Categories, Remark 0015.)

Example

Given an object \(X \in \Ob(\mathcal{C})\), we consider the rule \[\begin{matrix} U & \longmapsto & h_X(U) = \Mor_\mathcal{C}(U, X) \\ V \xrightarrow{\varphi} U & \longmapsto & (\psi \mapsto \psi \circ \varphi) : h_X(U) \to h_X(V). \end{matrix}\] This defines a functor \(h_X : \mathcal{C}^{opp} \to \textit{Sets}\) and hence a presheaf. This is called the representable presheaf associated to \(X\). It is not true that representable presheaves are sheaves in every topology on every site.

Lemma

Let \(\mathcal{C}\) be a category, and \(X, Y \in \Ob(\mathcal{C})\). There is a natural bijection \[\begin{matrix} \Mor_\mathcal{C}(X, Y) & \longrightarrow & \Mor_{\textit{PSh}(\mathcal{C})} (h_X, h_Y) \\ \psi & \longmapsto & h_\psi = \psi \circ - : h_X \to h_Y. \end{matrix}\]

Proof

See Categories, Lemma 001P.

Sites

Definition

Let \(\mathcal{C}\) be a category. A family of morphisms with fixed target \(\mathcal{U} = \{\varphi_i : U_i \to U\}_{i\in I}\) is the data of

1. an object \(U \in \mathcal{C}\),

2. a set \(I\) (possibly empty), and

3. for all \(i\in I\), a morphism \(\varphi_i : U_i \to U\) of \(\mathcal{C}\) with target \(U\).

There is a notion of a morphism of families of morphisms with fixed target. A special case of that is the notion of a refinement. A reference for this material is Sites, Section 00VS.

Definition

A site¹ consists of a category \(\mathcal{C}\) and a set \(\text{Cov}(\mathcal{C})\) consisting of families of morphisms with fixed target called coverings, such that

1. (isomorphism) if \(\varphi : V \to U\) is an isomorphism in \(\mathcal{C}\), then \(\{\varphi : V \to U\}\) is a covering,

2. (locality) if \(\{\varphi_i : U_i \to U\}_{i\in I}\) is a covering and for all \(i \in I\) we are given a covering \(\{\psi_{ij} : U_{ij} \to U_i \}_{j\in I_i}\), then \[\{ \varphi_i \circ \psi_{ij} : U_{ij} \to U \}_{(i, j)\in \prod_{i\in I} \{i\} \times I_i}\] is also a covering, and

3. (base change) if \(\{U_i \to U\}_{i\in I}\) is a covering and \(V \to U\) is a morphism in \(\mathcal{C}\), then

   1. for all \(i \in I\) the fibre product \(U_i \times_U V\) exists in \(\mathcal{C}\), and

   2. \(\{U_i \times_U V \to V\}_{i\in I}\) is a covering.

For us the category underlying a site is always “small”, i.e., its collection of objects form a set, and the collection of coverings of a site is a set as well (as in the definition above). We will mostly, in this chapter, leave out the arguments that cut down the collection of objects and coverings to a set. For further discussion, see Sites, Remark 00VI.

Example

If \(X\) is a topological space, then it has an associated site \(X_{Zar}\) defined as follows: the objects of \(X_{Zar}\) are the open subsets of \(X\), the morphisms between these are the inclusion mappings, and the coverings are the usual topological (surjective) coverings. Observe that if \(U, V \subset W \subset X\) are open subsets then \(U \times_W V = U \cap V\) exists: this category has fiber products. All the verifications are trivial and everything works as expected.

Sheaves

Definition

A presheaf \(\mathcal{F}\) of sets (resp. abelian presheaf) on a site \(\mathcal{C}\) is said to be a separated presheaf if for all coverings \(\{\varphi_i : U_i \to U\}_{i\in I} \in \text{Cov} (\mathcal{C})\) the map \[\mathcal{F}(U) \longrightarrow \prod\nolimits_{i\in I} \mathcal{F}(U_i)\] is injective. Here the map is \(s \mapsto (s|_{U_i})_{i\in I}\). The presheaf \(\mathcal{F}\) is a sheaf if for all coverings \(\{\varphi_i : U_i \to U\}_{i\in I} \in \text{Cov} (\mathcal{C})\), the diagram [03NL]\[\begin{equation} \xymatrix{ \mathcal{F}(U) \ar[r] & \prod_{i\in I} \mathcal{F}(U_i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod_{i, j \in I} \mathcal{F}(U_i \times_U U_j), } \end{equation}\] where the first map is \(s \mapsto (s|_{U_i})_{i\in I}\) and the two maps on the right are \((s_i)_{i\in I} \mapsto (s_i |_{U_i \times_U U_j})\) and \((s_i)_{i\in I} \mapsto (s_j |_{U_i \times_U U_j})\), is an equalizer diagram in the category of sets (resp. abelian groups).

Remark

For the empty covering (where \(I = \emptyset\)), this implies that \(\mathcal{F}(\emptyset)\) is an empty product, which is a final object in the corresponding category (a singleton, for both \(\textit{Sets}\) and \(\textit{Ab}\)).

Example

Working this out for the site \(X_{Zar}\) associated to a topological space, see Example 03NI, gives the usual notion of sheaves.

Definition

We denote \(\Sh(\mathcal{C})\) (resp. \(\textit{Ab}(\mathcal{C})\)) the full subcategory of \(\textit{PSh}(\mathcal{C})\) (resp. \(\textit{PAb}(\mathcal{C})\)) whose objects are sheaves. This is the category of sheaves of sets (resp. abelian sheaves) on \(\mathcal{C}\).

The example of G-sets

Let \(G\) be a group and define a site \(\mathcal{T}_G\) as follows: the underlying category is the category of \(G\)-sets, i.e., its objects are sets endowed with a left \(G\)-action and the morphisms are equivariant maps; and the coverings of \(\mathcal{T}_G\) are the families \(\{\varphi_i : U_i \to U\}_{i\in I}\) satisfying \(U = \bigcup_{i\in I} \varphi_i(U_i)\).

There is a special object in the site \(\mathcal{T}_G\), namely the \(G\)-set \(G\) endowed with its natural action by left translations. We denote it \({}_G G\). Observe that there is a natural group isomorphism \[\begin{matrix} \rho : & G^{opp} & \longrightarrow & \text{Aut}_{G\textit{-Sets}}({}_G G) \\ & g & \longmapsto & (h \mapsto hg). \end{matrix}\] In particular, for any presheaf \(\mathcal{F}\), the set \(\mathcal{F}({}_G G)\) inherits a \(G\)-action via \(\rho\). (Note that by contravariance of \(\mathcal{F}\), the set \(\mathcal{F}({}_G G)\) is again a left \(G\)-set.) In fact, the functor \[\begin{matrix} \Sh(\mathcal{T}_G) & \longrightarrow & G\textit{-Sets} \\ \mathcal{F} & \longmapsto & \mathcal{F}({}_G G) \end{matrix}\] is an equivalence of categories. Its quasi-inverse is the functor \(X \mapsto h_X\). Without giving the complete proof (which can be found in Sites, Section 00VZ) let us try to explain why this is true.

1. If \(S\) is a \(G\)-set, we can decompose it into orbits \(S = \coprod_{i\in I} O_i\). The sheaf axiom for the covering \(\{O_i \to S\}_{i\in I}\) says that \[\xymatrix{ \mathcal{F}(S) \ar[r] & \prod_{i\in I} \mathcal{F}(O_i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod_{i, j \in I} \mathcal{F}(O_i \times_S O_j) }\] is an equalizer. Observing that fibered products in \(G\textit{-Sets}\) are induced from fibered products in \(\textit{Sets}\), and using the fact that \(\mathcal{F}(\emptyset)\) is a \(G\)-singleton, we get that \[\prod_{i, j \in I} \mathcal{F}(O_i \times_S O_j) = \prod_{i \in I} \mathcal{F}(O_i)\] and the two maps above are in fact the same. Therefore the sheaf axiom merely says that \(\mathcal{F}(S) = \prod_{i\in I} \mathcal{F}(O_i)\).

2. If \(S\) is the \(G\)-set \(S= G/H\) and \(\mathcal{F}\) is a sheaf on \(\mathcal{T}_G\), then we claim that \[\mathcal{F}(G/H) = \mathcal{F}({}_G G)^H\] and in particular \(\mathcal{F}(\{*\}) = \mathcal{F}({}_G G)^G\). To see this, let’s use the sheaf axiom for the covering \(\{ {}_G G \to G/H \}\) of \(S\). We have \[\begin{eqnarray*} {}_G G \times_{G/H} {}_G G & \cong & G \times H \\ (g_1, g_2) & \longmapsto & (g_1, g_1^{-1} g_2) \end{eqnarray*}\] is a disjoint union of copies of \({}_G G\) (as a \(G\)-set). Hence the sheaf axiom reads \[\xymatrix{ \mathcal{F} (G/H) \ar[r] & \mathcal{F}({}_G G) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod_{h\in H} \mathcal{F}({}_G G) }\] where the two maps on the right are \(s \mapsto (s)_{h \in H}\) and \(s \mapsto (hs)_{h \in H}\). Therefore \(\mathcal{F}(G/H) = \mathcal{F}({}_G G)^H\) as claimed.

This doesn’t quite prove the claimed equivalence of categories, but it shows at least that a sheaf \(\mathcal{F}\) is entirely determined by its sections over \({}_G G\). Details (and set theoretical remarks) can be found in Sites, Section 00VZ.

Sheafification

Definition

Let \(\mathcal{F}\) be a presheaf on the site \(\mathcal{C}\) and \(\mathcal{U} = \{U_i \to U\} \in \text{Cov} (\mathcal{C})\). We define the zeroth Čech cohomology group of \(\mathcal{F}\) with respect to \(\mathcal{U}\) by \[\check H^0 (\mathcal{U}, \mathcal{F}) = \left\{ (s_i)_{i\in I} \in \prod\nolimits_{i\in I }\mathcal{F}(U_i) \text{ such that } s_i|_{U_i \times_U U_j} = s_j |_{U_i \times_U U_j} \right\}.\]

There is a canonical map \(\mathcal{F}(U) \to \check H^0 (\mathcal{U}, \mathcal{F})\), \(s \mapsto (s |_{U_i})_{i\in I}\). We say that a morphism of coverings from a covering \(\mathcal{V} = \{V_j \to V\}_{j \in J}\) to \(\mathcal{U}\) is a triple \((\chi, \alpha, \chi_j)\), where \(\chi : V \to U\) is a morphism, \(\alpha : J \to I\) is a map of sets, and for all \(j \in J\) the morphism \(\chi_j\) fits into a commutative diagram \[\xymatrix{ V_j \ar[rr]_{\chi_j} \ar[d] & & U_{\alpha(j)} \ar[d] \\ V \ar[rr]^\chi & & U. }\] Given the data \(\chi, \alpha, \{\chi_j\}_{j \in J}\) we define \[\begin{eqnarray*} \check H^0(\mathcal{U}, \mathcal{F}) & \longrightarrow & \check H^0(\mathcal{V}, \mathcal{F}) \\ (s_i)_{i\in I} & \longmapsto & \left(\chi_j^*\left(s_{\alpha(j)}\right)\right)_{j\in J}. \end{eqnarray*}\] We then claim that

1. the map is well-defined, and

2. depends only on \(\chi\) and is independent of the choice of \(\alpha, \{\chi_j\}_{j \in J}\).

We omit the proof of the first fact. To see part (2), consider another triple \((\psi, \beta, \psi_j)\) with \(\chi = \psi\). Then we have the commutative diagram \[\xymatrix{ V_j \ar[rrr]_{(\chi_j, \psi_j)} \ar[dd] & & & U_{\alpha(j)} \times_U U_{\beta(j)} \ar[dl] \ar[dr] \\ & & U_{\alpha(j)} \ar[dr] & & U_{\beta(j)} \ar[dl] \\ V \ar[rrr]^{\chi = \psi} & & & U. }\] Given a section \(s \in \mathcal{F}(\mathcal{U})\), its image in \(\mathcal{F}(V_j)\) under the map given by \((\chi, \alpha, \{\chi_j\}_{j \in J})\) is \(\chi_j^*s_{\alpha(j)}\), and its image under the map given by \((\psi, \beta, \{\psi_j\}_{j \in J})\) is \(\psi_j^*s_{\beta(j)}\). These two are equal since by assumption \(s \in \check H^0(\mathcal{U}, \mathcal{F})\) and hence both are equal to the pullback of the common value \[s_{\alpha(j)}|_{U_{\alpha(j)} \times_U U_{\beta(j)}} = s_{\beta(j)}|_{U_{\alpha(j)} \times_U U_{\beta(j)}}\] pulled back by the map \((\chi_j, \psi_j)\) in the diagram.

Theorem

Let \(\mathcal{C}\) be a site and \(\mathcal{F}\) a presheaf on \(\mathcal{C}\).

1. The rule \[U \mapsto \mathcal{F}^+(U) := \colim_{\mathcal{U} \text{ covering of }U} \check H^0(\mathcal{U}, \mathcal{F})\] is a presheaf. And the colimit is a directed one.

2. There is a canonical map of presheaves \(\mathcal{F} \to \mathcal{F}^+\).

3. If \(\mathcal{F}\) is a separated presheaf then \(\mathcal{F}^+\) is a sheaf and the map in (2) is injective.

4. \(\mathcal{F}^+\) is a separated presheaf.

5. \(\mathcal{F}^\# = (\mathcal{F}^+)^+\) is a sheaf, and the canonical map induces a functorial isomorphism \[\Hom_{\textit{PSh}(\mathcal{C})}(\mathcal{F}, \mathcal{G}) = \Hom_{\Sh(\mathcal{C})}(\mathcal{F}^\#, \mathcal{G})\] for any \(\mathcal{G} \in \Sh(\mathcal{C})\).

Proof

See Sites, Theorem 00WB.

In other words, this means that the natural map \(\mathcal{F} \to \mathcal{F}^\#\) is a left adjoint to the forgetful functor \(\Sh(\mathcal{C}) \to \textit{PSh}(\mathcal{C})\).

Cohomology

The following is the basic result that makes it possible to define cohomology for abelian sheaves on sites.

Theorem

The category of abelian sheaves on a site is an abelian category which has enough injectives.

Proof

See Modules on Sites, Lemma 03CN and Injectives, Theorem 01DP.

So we can define cohomology as the right-derived functors of the sections functor: if \(U \in \Ob(\mathcal{C})\) and \(\mathcal{F} \in \textit{Ab}(\mathcal{C})\), \[H^p(U, \mathcal{F}) := R^p\Gamma(U, \mathcal{F}) = H^p(\Gamma(U, \mathcal{I}^\bullet))\] where \(\mathcal{F} \to \mathcal{I}^\bullet\) is an injective resolution. To do this, we should check that the functor \(\Gamma(U, -)\) is left exact. This is true and is part of why the category \(\textit{Ab}(\mathcal{C})\) is abelian, see Modules on Sites, Lemma 03CN. For more general discussion of cohomology on sites (including the global sections functor and its right derived functors), see Cohomology on Sites, Section 01FT.

The fpqc topology

Before doing étale cohomology we study a bit the fpqc topology, since it works well for quasi-coherent sheaves.

Definition

Let \(T\) be a scheme. An fpqc covering of \(T\) is a family \(\{ \varphi_i : T_i \to T\}_{i \in I}\) such that

1. each \(\varphi_i\) is a flat morphism and \(\bigcup_{i\in I} \varphi_i(T_i) = T\), and

2. for each affine open \(U \subset T\) there exists a finite set \(K\), a map \(\mathbf{i} : K \to I\) and affine opens \(U_{\mathbf{i}(k)} \subset T_{\mathbf{i}(k)}\) such that \(U = \bigcup_{k \in K} \varphi_{\mathbf{i}(k)}(U_{\mathbf{i}(k)})\).

Remark

The first condition corresponds to fp, which stands for fidèlement plat, faithfully flat in french, and the second to qc, quasi-compact. The second part of the first condition is unnecessary when the second condition holds.

Example

Examples of fpqc coverings.

1. Any Zariski open covering of \(T\) is an fpqc covering.

2. A family \(\{\Spec(B) \to \Spec(A)\}\) is an fpqc covering if and only if \(A \to B\) is a faithfully flat ring map.

3. If \(f: X \to Y\) is flat, surjective and quasi-compact, then \(\{ f: X\to Y\}\) is an fpqc covering.

4. The morphism \(\varphi : \coprod_{x \in \mathbf{A}^1_k} \Spec(\mathcal{O}_{\mathbf{A}^1_k, x}) \to \mathbf{A}^1_k\), where \(k\) is a field, is flat and surjective. It is not quasi-compact, and in fact the family \(\{\varphi\}\) is not an fpqc covering.

5. Write \(\mathbf{A}^2_k = \Spec(k[x, y])\). Denote \(i_x : D(x) \to \mathbf{A}^2_k\) and \(i_y : D(y) \to \mathbf{A}^2_k\) the standard opens. Then the families \(\{i_x, i_y, \Spec(k[[x, y]]) \to \mathbf{A}^2_k\}\) and \(\{i_x, i_y, \Spec(\mathcal{O}_{\mathbf{A}^2_k, 0}) \to \mathbf{A}^2_k\}\) are fpqc coverings.

Lemma

The collection of fpqc coverings on the category of schemes satisfies the axioms of site.

Proof

See Topologies, Lemma 022D.

It seems that this lemma allows us to define the fpqc site of the category of schemes. However, there is a set theoretical problem that comes up when considering the fpqc topology, see Topologies, Section 022A. It comes from our requirement that sites are “small”, but that no small category of schemes can contain a cofinal system of fpqc coverings of a given nonempty scheme. Although this does not strictly speaking prevent us from defining “partial” fpqc sites, it does not seem prudent to do so. The work-around is to allow the notion of a sheaf for the fpqc topology (see below) but to prohibit considering the category of all fpqc sheaves.

Definition

Let \(S\) be a scheme. The category of schemes over \(S\) is denoted \(\Sch/S\). Consider a functor \(\mathcal{F} : (\Sch/S)^{opp} \to \textit{Sets}\), in other words a presheaf of sets. We say \(\mathcal{F}\) satisfies the sheaf property for the fpqc topology if for every fpqc covering \(\{U_i \to U\}_{i \in I}\) of schemes over \(S\) the diagram (03NL) is an equalizer diagram.

We similarly say that \(\mathcal{F}\) satisfies the sheaf property for the Zariski topology if for every open covering \(U = \bigcup_{i \in I} U_i\) the diagram (03NL) is an equalizer diagram. See Schemes, Definition 01JI. Clearly, this is equivalent to saying that for every scheme \(T\) over \(S\) the restriction of \(\mathcal{F}\) to the opens of \(T\) is a (usual) sheaf.

Lemma

Let \(\mathcal{F}\) be a presheaf on \(\Sch/S\). Then \(\mathcal{F}\) satisfies the sheaf property for the fpqc topology if and only if

1. \(\mathcal{F}\) satisfies the sheaf property with respect to the Zariski topology, and

2. for every faithfully flat morphism \(\Spec(B) \to \Spec(A)\) of affine schemes over \(S\), the sheaf axiom holds for the covering \(\{\Spec(B) \to \Spec(A)\}\). Namely, this means that \[\xymatrix{ \mathcal{F}(\Spec(A)) \ar[r] & \mathcal{F}(\Spec(B)) \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}(\Spec(B \otimes_A B)) }\] is an equalizer diagram.

Proof

See Topologies, Lemma 022H.

An alternative way to think of a presheaf \(\mathcal{F}\) on \(\Sch/S\) which satisfies the sheaf condition for the fpqc topology is as the following data:

1. for each \(T/S\), a usual (i.e., Zariski) sheaf \(\mathcal{F}_T\) on \(T_{Zar}\),

2. for every map \(f : T' \to T\) over \(S\), a restriction mapping \(f^{-1}\mathcal{F}_T \to \mathcal{F}_{T'}\)

such that

1. the restriction mappings are functorial,

2. if \(f : T' \to T\) is an open immersion then the restriction mapping \(f^{-1}\mathcal{F}_T \to \mathcal{F}_{T'}\) is an isomorphism, and

3. for every faithfully flat morphism \(\Spec(B) \to \Spec(A)\) over \(S\), the diagram \[\xymatrix{ \mathcal{F}_{\Spec(A)}(\Spec(A)) \ar[r] & \mathcal{F}_{\Spec(B)}(\Spec(B)) \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}_{\Spec(B \otimes_A B)}(\Spec(B \otimes_A B)) }\] is an equalizer.

Data (1) and (2) and conditions (a), (b) give the data of a presheaf on \(\Sch/S\) satisfying the sheaf condition for the Zariski topology. By Lemma 03O1 condition (c) then suffices to get the sheaf condition for the fpqc topology.

Example

Consider the presheaf \[\begin{matrix} \mathcal{F} : & (\Sch/S)^{opp} & \longrightarrow & \textit{Ab} \\ & T/S & \longmapsto & \Gamma(T, \Omega_{T/S}). \end{matrix}\] The compatibility of differentials with localization implies that \(\mathcal{F}\) is a sheaf on the Zariski site. However, it does not satisfy the sheaf condition for the fpqc topology. Namely, consider the case \(S = \Spec(\mathbf{F}_p)\) and the morphism \[\varphi : V = \Spec(\mathbf{F}_p[v]) \to U = \Spec(\mathbf{F}_p[u])\] given by mapping \(u\) to \(v^p\). The family \(\{\varphi\}\) is an fpqc covering, yet the restriction mapping \(\mathcal{F}(U) \to \mathcal{F}(V)\) sends the generator \(\text{d}u\) to \(\text{d}(v^p) = 0\), so it is the zero map, and the diagram \[\xymatrix{ \mathcal{F}(U) \ar[r]^{0} & \mathcal{F}(V) \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}(V \times_U V) }\] is not an equalizer. We will see later that \(\mathcal{F}\) does in fact give rise to a sheaf on the étale and smooth sites.

Lemma

Any representable presheaf on \(\Sch/S\) satisfies the sheaf condition for the fpqc topology.

Proof

See Descent, Lemma 023Q.

We will return to this later, since the proof of this fact uses descent for quasi-coherent sheaves, which we will discuss in the next section. A fancy way of expressing the lemma is to say that the fpqc topology is weaker than the canonical topology, or that the fpqc topology is subcanonical. In the setting of sites this is discussed in Sites, Section 00WO.

Remark

The fpqc is finer than the Zariski, étale, smooth, syntomic, and fppf topologies. Hence any presheaf satisfying the sheaf condition for the fpqc topology will be a sheaf on the Zariski, étale, smooth, syntomic, and fppf sites. In particular representable presheaves will be sheaves on the étale site of a scheme for example.

Example

Let \(S\) be a scheme. Consider the additive group scheme \(\mathbf{G}_{a, S} = \mathbf{A}^1_S\) over \(S\), see Groupoids, Example 022V. The associated representable presheaf is given by \[h_{\mathbf{G}_{a, S}}(T) = \Mor_S(T, \mathbf{G}_{a, S}) = \Gamma(T, \mathcal{O}_T).\] By the above we now know that this is a presheaf of sets which satisfies the sheaf condition for the fpqc topology. On the other hand, it is clearly a presheaf of rings as well. Hence we can think of this as a functor \[\begin{matrix} \mathcal{O} : & (\Sch/S)^{opp} & \longrightarrow & \textit{Rings} \\ & T/S & \longmapsto & \Gamma(T, \mathcal{O}_T) \end{matrix}\] which satisfies the sheaf condition for the fpqc topology. Correspondingly there is a notion of \(\mathcal{O}\)-module, and so on and so forth.

Faithfully flat descent

In this section we discuss faithfully flat descent for quasi-coherent modules. More precisely, we will prove quasi-coherent modules satisfy effective descent with respect to fpqc coverings.

Definition

Let \(\mathcal{U} = \{ t_i : T_i \to T\}_{i \in I}\) be a family of morphisms of schemes with fixed target. A descent datum for quasi-coherent sheaves with respect to \(\mathcal{U}\) is a collection \(((\mathcal{F}_i)_{i \in I}, (\varphi_{ij})_{i, j \in I})\) where

1. \(\mathcal{F}_i\) is a quasi-coherent sheaf on \(T_i\), and

2. \(\varphi_{ij} : \text{pr}_0^* \mathcal{F}_i \to \text{pr}_1^* \mathcal{F}_j\) is an isomorphism of modules on \(T_i \times_T T_j\),

such that the cocycle condition holds: the diagrams \[\xymatrix{ \text{pr}_0^*\mathcal{F}_i \ar[dr]_{\text{pr}_{02}^*\varphi_{ik}} \ar[rr]^{\text{pr}_{01}^*\varphi_{ij}} & & \text{pr}_1^*\mathcal{F}_j \ar[dl]^{\text{pr}_{12}^*\varphi_{jk}} \\ & \text{pr}_2^*\mathcal{F}_k }\] commute on \(T_i \times_T T_j \times_T T_k\). This descent datum is called effective if there exist a quasi-coherent sheaf \(\mathcal{F}\) over \(T\) and \(\mathcal{O}_{T_i}\)-module isomorphisms \(\varphi_i : t_i^* \mathcal{F} \cong \mathcal{F}_i\) compatible with the maps \(\varphi_{ij}\), namely \[\varphi_{ij} = \text{pr}_1^* (\varphi_j) \circ \text{pr}_0^* (\varphi_i)^{-1}.\]

In this and the next section we discuss some ingredients of the proof of the following theorem, as well as some related material.

Theorem

If \(\mathcal{V} = \{T_i \to T\}_{i\in I}\) is an fpqc covering, then all descent data for quasi-coherent sheaves with respect to \(\mathcal{V}\) are effective.

Proof

See Descent, Proposition 023T.

In other words, the fibered category of quasi-coherent sheaves is a stack on the fpqc site. The proof of the theorem is in two steps. The first one is to realize that for Zariski coverings this is easy (or well-known) using standard glueing of sheaves (see Sheaves, Section 00AK) and the locality of quasi-coherence. The second step is the case of an fpqc covering of the form \(\{\Spec(B) \to \Spec(A)\}\) where \(A \to B\) is a faithfully flat ring map. This is a lemma in algebra, which we now present.

Descent of modules. If \(A \to B\) is a ring map, we consider the complex \[(B/A)_\bullet : B \to B \otimes_A B \to B \otimes_A B \otimes_A B \to \ldots\] where \(B\) is in degree 0, \(B \otimes_A B\) in degree 1, etc, and the maps are given by \[\begin{eqnarray*} b & \mapsto & 1 \otimes b - b \otimes 1, \\ b_0 \otimes b_1 & \mapsto & 1 \otimes b_0 \otimes b_1 - b_0 \otimes 1 \otimes b_1 + b_0 \otimes b_1 \otimes 1, \\ & \text{etc.} \end{eqnarray*}\]

Lemma

If \(A \to B\) is faithfully flat, then the complex \((B/A)_\bullet\) is exact in positive degrees, and \(H^0((B/A)_\bullet) = A\).

Proof

See Descent, Lemma 023M.

Grothendieck proves this in three steps. Firstly, he assumes that the map \(A \to B\) has a section, and constructs an explicit homotopy to the complex where \(A\) is the only nonzero term, in degree 0. Secondly, he observes that to prove the result, it suffices to do so after a faithfully flat base change \(A \to A'\), replacing \(B\) with \(B' = B \otimes_A A'\). Thirdly, he applies the faithfully flat base change \(A \to A' = B\) and remark that the map \(A' = B \to B' = B \otimes_A B\) has a natural section.

The same strategy proves the following lemma.

Lemma

If \(A \to B\) is faithfully flat and \(M\) is an \(A\)-module, then the complex \((B/A)_\bullet \otimes_A M\) is exact in positive degrees, and \(H^0((B/A)_\bullet \otimes_A M) = M\).

Proof

See Descent, Lemma 023M.

Definition

Let \(A \to B\) be a ring map and \(N\) a \(B\)-module. A descent datum for \(N\) with respect to \(A \to B\) is an isomorphism \(\varphi : N \otimes_A B \cong B \otimes_A N\) of \(B \otimes_A B\)-modules such that the diagram of \(B \otimes_A B \otimes_A B\)-modules \[\xymatrix{ {N \otimes_A B \otimes_A B} \ar[dr]_{\varphi_{02}} \ar[rr]^{\varphi_{01}} & & {B \otimes_A N \otimes_A B} \ar[dl]^{\varphi_{12}} \\ & {B \otimes_A B \otimes_A N} }\] commutes where \(\varphi_{01} = \varphi \otimes \text{id}_B\) and similarly for \(\varphi_{12}\) and \(\varphi_{02}\).

If \(N' = B \otimes_A M\) for some \(A\)-module M, then it has a canonical descent datum given by the map \[\begin{matrix} \varphi_\text{can}: & N' \otimes_A B & \to & B \otimes_A N' \\ & b_0 \otimes m \otimes b_1 & \mapsto & b_0 \otimes b_1 \otimes m. \end{matrix}\]

Definition

A descent datum \((N, \varphi)\) is called effective if there exists an \(A\)-module \(M\) such that \((N, \varphi) \cong (B \otimes_A M, \varphi_\text{can})\), with the obvious notion of isomorphism of descent data.

Theorem 03O8 is a consequence the following result.

Theorem

If \(A \to B\) is faithfully flat then descent data with respect to \(A\to B\) are effective.

Proof

See Descent, Proposition 023N. See also Descent, Remark 039Y for an alternative view of the proof.

Remarks

The results on descent of modules have several applications:

1. The exactness of the Čech complex in positive degrees for the covering \(\{\Spec(B) \to \Spec(A)\}\) where \(A \to B\) is faithfully flat. This will give some vanishing of cohomology.

2. If \((N, \varphi)\) is a descent datum with respect to a faithfully flat map \(A \to B\), then the corresponding \(A\)-module is given by \[M = \Ker \left( \begin{matrix} N & \longrightarrow & B \otimes_A N \\ n & \longmapsto & 1 \otimes n - \varphi(n \otimes 1) \end{matrix} \right).\] See Descent, Proposition 023N.

Quasi-coherent sheaves

We can apply the descent of modules to study quasi-coherent sheaves.

Proposition

For any quasi-coherent sheaf \(\mathcal{F}\) on \(S\) the presheaf \[\begin{matrix} \mathcal{F}^a : & \Sch/S & \to & \textit{Ab}\\ & (f: T \to S) & \mapsto & \Gamma(T, f^*\mathcal{F}) \end{matrix}\] is an \(\mathcal{O}\)-module which satisfies the sheaf condition for the fpqc topology.

Proof

This is proved in Descent, Lemma 03DT. We indicate the proof here. As established in Lemma 03O1, it is enough to check the sheaf property on Zariski coverings and faithfully flat morphisms of affine schemes. The sheaf property for Zariski coverings is standard scheme theory, since \(\Gamma(U, i^\ast \mathcal{F}) = \mathcal{F}(U)\) when \(i : U \hookrightarrow S\) is an open immersion.

For \(\left\{\Spec(B)\to \Spec(A)\right\}\) with \(A\to B\) faithfully flat and \(\mathcal{F}|_{\Spec(A)} = \widetilde{M}\) this corresponds to the fact that \(M = H^0\left((B/A)_\bullet \otimes_A M \right)\), i.e., that \[\begin{align*} 0 \to M \to B \otimes_A M \to B \otimes_A B \otimes_A M \end{align*}\] is exact by Lemma 03OA.

There is an abstract notion of a quasi-coherent sheaf on a ringed site. We briefly introduce this here. For more information please consult Modules on Sites, Section 03DK. Let \(\mathcal{C}\) be a category, and let \(U\) be an object of \(\mathcal{C}\). Then \(\mathcal{C}/U\) indicates the category of objects over \(U\), see Categories, Example 001G. If \(\mathcal{C}\) is a site, then \(\mathcal{C}/U\) is a site as well, namely the coverings of \(V/U\) are families \(\{V_i/U \to V/U\}\) of morphisms of \(\mathcal{C}/U\) with fixed target such that \(\{V_i \to V\}\) is a covering of \(\mathcal{C}\). Moreover, given any sheaf \(\mathcal{F}\) on \(\mathcal{C}\) the restriction \(\mathcal{F}|_{\mathcal{C}/U}\) (defined in the obvious manner) is a sheaf as well. See Sites, Section 00XZ for details.

Definition

Let \(\mathcal{C}\) be a ringed site, i.e., a site endowed with a sheaf of rings \(\mathcal{O}\). A sheaf of \(\mathcal{O}\)-modules \(\mathcal{F}\) on \(\mathcal{C}\) is called quasi-coherent if for all \(U \in \Ob(\mathcal{C})\) there exists a covering \(\{U_i \to U\}_{i\in I}\) of \(\mathcal{C}\) such that the restriction \(\mathcal{F}|_{\mathcal{C}/U_i}\) is isomorphic to the cokernel of an \(\mathcal{O}\)-linear map of free \(\mathcal{O}\)-modules \[\bigoplus\nolimits_{k \in K} \mathcal{O}|_{\mathcal{C}/U_i} \longrightarrow \bigoplus\nolimits_{l \in L} \mathcal{O}|_{\mathcal{C}/U_i}.\] The direct sum over \(K\) is the sheaf associated to the presheaf \(V \mapsto \bigoplus_{k \in K} \mathcal{O}(V)\) and similarly for the other.

Although it is useful to be able to give a general definition as above this notion is not well behaved in general.

Remark

In the case where \(\mathcal{C}\) has a final object, e.g. \(S\), it suffices to check the condition of the definition for \(U = S\) in the above statement. See Modules on Sites, Lemma 03DN.

Theorem

Let \(S\) be a scheme. Let \(\mathcal{C}\) be a site. Assume that

1. the underlying category \(\mathcal{C}\) is a full subcategory of \(\Sch/S\),

2. any Zariski covering of \(T \in \Ob(\mathcal{C})\) can be refined by a covering of \(\mathcal{C}\),

3. \(S/S\) is an object of \(\mathcal{C}\),

4. every covering of \(\mathcal{C}\) is an fpqc covering of schemes.

Then the presheaf \(\mathcal{O}\) is a sheaf on \(\mathcal{C}\) and any quasi-coherent \(\mathcal{O}\)-module on \((\mathcal{C}, \mathcal{O})\) is of the form \(\mathcal{F}^a\) for some quasi-coherent sheaf \(\mathcal{F}\) on \(S\).

Proof

After some formal arguments this is exactly Theorem 03O8. Details omitted. In Descent, Proposition 03DX we prove a more precise version of the theorem for the big Zariski, fppf, étale, smooth, and syntomic sites of \(S\), as well as the small Zariski and étale sites of \(S\).

In other words, there is no difference between quasi-coherent modules on the scheme \(S\) and quasi-coherent \(\mathcal{O}\)-modules on sites \(\mathcal{C}\) as in the theorem. More precise statements for the big and small sites \((\Sch/S)_{fppf}\), \(S_\etale\), etc can be found in Descent, Sections 03DR, 0GN8, and 0GN9. In this chapter we will sometimes refer to a “site as in Theorem 03OJ” in order to conveniently state results which hold in any of those situations.

Čech cohomology

Our next goal is to use descent theory to show that \(H^i(\mathcal{C}, \mathcal{F}^a) = H_{Zar}^i(S, \mathcal{F})\) for all quasi-coherent sheaves \(\mathcal{F}\) on \(S\), and any site \(\mathcal{C}\) as in Theorem 03OJ. To this end, we introduce Čech cohomology on sites. See [ArtinTopologies] and Cohomology on Sites, Sections 03AK, 03AO and 03AV for more details.

Definition

Let \(\mathcal{C}\) be a category. Let \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) be a family of morphisms of \(\mathcal{C}\) with fixed target. Assume that all the fibre products \(U_{i_0} \times_U \ldots \times_U U_{i_p}\) exists in \(\mathcal{C}\). Let \(\mathcal{F} \in \textit{PAb}(\mathcal{C})\) be an abelian presheaf. We define the Čech complex \(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})\) by \[\prod_{i_0 \in I} \mathcal{F}(U_{i_0}) \to \prod_{i_0, i_1 \in I} \mathcal{F}(U_{i_0} \times_U U_{i_1}) \to \prod_{i_0, i_1, i_2 \in I} \mathcal{F}(U_{i_0} \times_U U_{i_1} \times_U U_{i_2}) \to \ldots\] where the first term is in degree 0 and the maps are the usual ones. The Čech cohomology groups are defined by \[\check{H}^p(\mathcal{U}, \mathcal{F}) = H^p(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})).\]

In the above definition, it is essential to allow the indices \((i_0, \ldots, i_p)\) to have repetitions.

Lemma

Notation and assumptions as in Definition 03OL. The functor \(\check{\mathcal{C}}^\bullet(\mathcal{U}, -)\) is exact on the category \(\textit{PAb}(\mathcal{C})\).

In other words, if \(0\to \mathcal{F}_1\to \mathcal{F}_2\to \mathcal{F}_3\to 0\) is a short exact sequence of presheaves of abelian groups, then \[0 \to \check{\mathcal{C}}^\bullet\left(\mathcal{U}, \mathcal{F}_1\right) \to\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}_2) \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}_3)\to 0\] is a short exact sequence of complexes.

Proof

This follows at once from the definition of a short exact sequence of presheaves. Namely, as the category of abelian presheaves is the category of functors on some category with values in \(\textit{Ab}\), it is automatically an abelian category: a sequence \(\mathcal{F}_1\to \mathcal{F}_2\to \mathcal{F}_3\) is exact in \(\textit{PAb}\) if and only if for all \(U \in \Ob(\mathcal{C})\), the sequence \(\mathcal{F}_1(U) \to \mathcal{F}_2(U) \to \mathcal{F}_3(U)\) is exact in \(\textit{Ab}\). So the complex above is merely a product of short exact sequences in each degree. See also Cohomology on Sites, Lemma 03AQ.

This shows that \(\check{H}^\bullet(\mathcal{U}, -)\) is a \(\delta\)-functor. We now proceed to show that it is a universal \(\delta\)-functor. We thus need to show that it is an effaceable functor. We start by recalling the Yoneda lemma.

Lemma

For any presheaf \(\mathcal{F}\) on a category \(\mathcal{C}\) and \(U \in \Ob(\mathcal{C})\) there is a functorial isomorphism \[\Hom_{\textit{PSh}(\mathcal{C})}(h_U, \mathcal{F}) = \mathcal{F}(U).\]

Proof

See Categories, Lemma 001P.

Given a set \(E\) we denote (in this section) \(\mathbf{Z}[E]\) the free abelian group on \(E\). In a formula \(\mathbf{Z}[E] = \bigoplus_{e \in E} \mathbf{Z}\), i.e., \(\mathbf{Z}[E]\) is a free \(\mathbf{Z}\)-module having a basis consisting of the elements of \(E\). Using this notation we introduce the free abelian presheaf on a presheaf of sets.

Definition

Let \(\mathcal{C}\) be a category. Given a presheaf of sets \(\mathcal{G}\), we define the free abelian presheaf on \(\mathcal{G}\), denoted \(\mathbf{Z}_\mathcal{G}\), by the rule \[\mathbf{Z}_\mathcal{G}(U) = \mathbf{Z}[\mathcal{G}(U)]\] for \(U \in \Ob(\mathcal{C})\) with restriction maps induced by the restriction maps of \(\mathcal{G}\). In the special case \(\mathcal{G} = h_U\) we write simply \(\mathbf{Z}_U = \mathbf{Z}_{h_U}\).

The functor \(\mathcal{G} \mapsto \mathbf{Z}_\mathcal{G}\) is left adjoint to the forgetful functor \(\textit{PAb}(\mathcal{C}) \to \textit{PSh}(\mathcal{C})\). Thus, for any presheaf \(\mathcal{F}\), there is a canonical isomorphism \[\Hom_{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_U, \mathcal{F}) = \Hom_{\textit{PSh}(\mathcal{C})}(h_U, \mathcal{F}) = \mathcal{F}(U)\] the last equality by the Yoneda lemma. In particular, we have the following result.

Lemma

Notation and assumptions as in Definition 03OL. The Čech complex \(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})\) can be described explicitly as follows \[\begin{eqnarray*} \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) & = & \left( \prod_{i_0 \in I} \Hom_{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_{U_{i_0}}, \mathcal{F}) \to \prod_{i_0, i_1 \in I} \Hom_{\textit{PAb}(\mathcal{C})}( \mathbf{Z}_{U_{i_0} \times_U U_{i_1}}, \mathcal{F}) \to \ldots \right) \\ & = & \Hom_{\textit{PAb}(\mathcal{C})}\left( \left( \bigoplus_{i_0 \in I} \mathbf{Z}_{U_{i_0}} \leftarrow \bigoplus_{i_0, i_1 \in I} \mathbf{Z}_{U_{i_0} \times_U U_{i_1}} \leftarrow \ldots \right), \mathcal{F}\right) \end{eqnarray*}\]

Proof

This follows from the formula above. See Cohomology on Sites, Lemma 03AS.

This reduces us to studying only the complex in the first argument of the last \(\Hom\).

Lemma

Notation and assumptions as in Definition 03OL. The complex of abelian presheaves \[\begin{align*} \mathbf{Z}_\mathcal{U}^\bullet \quad : \quad \bigoplus_{i_0 \in I} \mathbf{Z}_{U_{i_0}} \leftarrow \bigoplus_{i_0, i_1 \in I} \mathbf{Z}_{U_{i_0} \times_U U_{i_1}} \leftarrow \bigoplus_{i_0, i_1, i_2 \in I} \mathbf{Z}_{U_{i_0} \times_U U_{i_1} \times_U U_{i_2}} \leftarrow \ldots \end{align*}\] is exact in all degrees except \(0\) in \(\textit{PAb}(\mathcal{C})\).

Proof

For any \(V\in \Ob(\mathcal{C})\) the complex of abelian groups \(\mathbf{Z}_\mathcal{U}^\bullet(V)\) is \[\begin{matrix} \mathbf{Z}\left[ \coprod_{i_0\in I} \Mor_\mathcal{C}(V, U_{i_0})\right] \leftarrow \mathbf{Z}\left[ \coprod_{i_0, i_1 \in I} \Mor_\mathcal{C}(V, U_{i_0} \times_U U_{i_1})\right] \leftarrow \ldots = \\ \bigoplus_{\varphi : V \to U} \left( \mathbf{Z}\left[\coprod_{i_0 \in I} \Mor_\varphi(V, U_{i_0})\right] \leftarrow \mathbf{Z}\left[\coprod_{i_0, i_1\in I} \Mor_\varphi(V, U_{i_0}) \times \Mor_\varphi(V, U_{i_1})\right] \leftarrow \ldots \right) \end{matrix}\] where \[\Mor_{\varphi}(V, U_i) = \{ V \to U_i \text{ such that } V \to U_i \to U \text{ equals } \varphi \}.\] Set \(S_\varphi = \coprod_{i\in I} \Mor_\varphi(V, U_i)\), so that \[\mathbf{Z}_\mathcal{U}^\bullet(V) = \bigoplus_{\varphi : V \to U} \left( \mathbf{Z}[S_\varphi] \leftarrow \mathbf{Z}[S_\varphi \times S_\varphi] \leftarrow \mathbf{Z}[S_\varphi \times S_\varphi \times S_\varphi] \leftarrow \ldots \right).\] Thus it suffices to show that for each \(S = S_\varphi\), the complex \[\begin{align*} \mathbf{Z}[S] \leftarrow \mathbf{Z}[S \times S] \leftarrow \mathbf{Z}[S \times S \times S] \leftarrow \ldots \end{align*}\] is exact in negative degrees. To see this, we can give an explicit homotopy. Fix \(s\in S\) and define \(K: n_{(s_0, \ldots, s_p)} \mapsto n_{(s, s_0, \ldots, s_p)}.\) One easily checks that \(K\) is a nullhomotopy for the operator \[\delta : \eta_{(s_0, \ldots, s_p)} \mapsto \sum\nolimits_{i = 0}^p (-1)^p \eta_{(s_0, \ldots, \hat s_i, \ldots, s_p)}.\] See Cohomology on Sites, Lemma 03AT for more details.

Lemma

Notation and assumptions as in Definition 03OL. If \(\mathcal{I}\) is an injective object of \(\textit{PAb}(\mathcal{C})\), then \(\check H^p(\mathcal{U}, \mathcal{I}) = 0\) for all \(p > 0\).

Proof

The Čech complex is the result of applying the functor \(\Hom_{\textit{PAb}(\mathcal{C})}(-, \mathcal{I})\) to the complex \(\mathbf{Z}^\bullet_\mathcal{U}\), i.e., \[\check H^p(\mathcal{U}, \mathcal{I}) = H^p (\Hom_{\textit{PAb}(\mathcal{C})} (\mathbf{Z}^\bullet_\mathcal{U}, \mathcal{I})).\] But we have just seen that \(\mathbf{Z}^\bullet_\mathcal{U}\) is exact in negative degrees, and the functor \(\Hom_{\textit{PAb}(\mathcal{C})}(-, \mathcal{I})\) is exact, hence \(\Hom_{\textit{PAb}(\mathcal{C})} (\mathbf{Z}^\bullet_\mathcal{U}, \mathcal{I})\) is exact in positive degrees.

Theorem

Notation and assumptions as in Definition 03OL. On \(\textit{PAb}(\mathcal{C})\) the functors \(\check{H}^p(\mathcal{U}, -)\) are the right derived functors of \(\check{H}^0(\mathcal{U}, -)\).

Proof

By the Lemma 03OR, the functors \(\check H^p(\mathcal{U}, -)\) are universal \(\delta\)-functors since they are effaceable. So are the right derived functors of \(\check H^0(\mathcal{U}, -)\). Since they agree in degree \(0\), they agree by the universal property of universal \(\delta\)-functors. For more details see Cohomology on Sites, Lemma 03AU.

Remark

Observe that all of the preceding statements are about presheaves so we haven’t made use of the topology yet.

The Čech-to-cohomology spectral sequence

This spectral sequence is fundamental in proving foundational results on cohomology of sheaves.

Lemma

The forgetful functor \(\textit{Ab}(\mathcal{C})\to \textit{PAb}(\mathcal{C})\) transforms injectives into injectives.

Proof

This is formal using the fact that the forgetful functor has a left adjoint, namely sheafification, which is an exact functor. For more details see Cohomology on Sites, Lemma 03F6.

Theorem

Let \(\mathcal{C}\) be a site. For any covering \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) of \(U \in \Ob(\mathcal{C})\) and any abelian sheaf \(\mathcal{F}\) on \(\mathcal{C}\) there is a spectral sequence \[E_2^{p, q} = \check H^p(\mathcal{U}, \underline{H}^q(\mathcal{F})) \Rightarrow H^{p+q}(U, \mathcal{F}),\] where \(\underline{H}^q(\mathcal{F})\) is the abelian presheaf \(V \mapsto H^q(V, \mathcal{F})\).

Proof

Choose an injective resolution \(\mathcal{F}\to \mathcal{I}^\bullet\) in \(\textit{Ab}(\mathcal{C})\), and consider the double complex \(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet)\) and the maps \[\xymatrix{ \Gamma(U, I^\bullet) \ar[r] & \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet) \\ & \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \ar[u] }\] Here the horizontal map is the natural map \(\Gamma(U, I^\bullet) \to \check{\mathcal{C}}^0(\mathcal{U}, \mathcal{I}^\bullet)\) to the left column, and the vertical map is induced by \(\mathcal{F}\to \mathcal{I}^0\) and lands in the bottom row. By assumption, \(\mathcal{I}^\bullet\) is a complex of injectives in \(\textit{Ab}(\mathcal{C})\), hence by Lemma 03OV, it is a complex of injectives in \(\textit{PAb}(\mathcal{C})\). Thus, the rows of the double complex are exact in positive degrees (Lemma 03OR), and the kernel of \(\check{\mathcal{C}}^0(\mathcal{U}, \mathcal{I}^\bullet) \to \check{\mathcal{C}}^1(\mathcal{U}, \mathcal{I}^\bullet)\) is equal to \(\Gamma(U, \mathcal{I}^\bullet)\), since \(\mathcal{I}^\bullet\) is a complex of sheaves. In particular, the cohomology of the total complex is the standard cohomology of the global sections functor \(H^0(U, \mathcal{F})\).

For the vertical direction, the \(q\)th cohomology group of the \(p\)th column is \[\prod_{i_0, \ldots, i_p} H^q(U_{i_0} \times_U \ldots \times_U U_{i_p}, \mathcal{F}) = \prod_{i_0, \ldots, i_p} \underline{H}^q(\mathcal{F})(U_{i_0} \times_U \ldots \times_U U_{i_p})\] in the entry \(E_1^{p, q}\). So this is a standard double complex spectral sequence, and the \(E_2\)-page is as prescribed. For more details see Cohomology on Sites, Lemma 03AZ.

Remark

This is a Grothendieck spectral sequence for the composition of functors \[\textit{Ab}(\mathcal{C}) \longrightarrow \textit{PAb}(\mathcal{C}) \xrightarrow{\check H^0} \textit{Ab}.\]

Big and small sites of schemes

Let \(S\) be a scheme. Let \(\tau\) be one of the topologies we will be discussing. Thus \(\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}\). Of course if you are only interested in the étale topology, then you can simply assume \(\tau = \etale\) throughout. Moreover, we will discuss étale morphisms, étale coverings, and étale sites in more detail starting in Section 03P9. In order to proceed with the discussion of cohomology of quasi-coherent sheaves it is convenient to introduce the big \(\tau\)-site and in case \(\tau \in \{\etale, Zariski\}\), the small \(\tau\)-site of \(S\). In order to do this we first introduce the notion of a \(\tau\)-covering.

Definition

(See Topologies, Definitions 021M, 0225, 021Z, 0215, and 020O.) Let \(\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}\). A family of morphisms of schemes \(\{f_i : T_i \to T\}_{i \in I}\) with fixed target is called a \(\tau\)-covering if and only if each \(f_i\) is flat of finite presentation, syntomic, smooth, étale, resp. an open immersion, and we have \(\bigcup f_i(T_i) = T\).

The class of all \(\tau\)-coverings satisfies the axioms (1), (2) and (3) of Definition 03NH (our definition of a site), see Topologies, Lemmas 021O, 0227, 0221, 0217, and 020P.

Let us introduce the sites we will be working with. Contrary to what happens in [SGA4], we do not want to choose a universe. Instead we pick a “partial universe” (which is a suitably large set as in Sets, Section 000B), and consider all schemes contained in this set. Of course we make sure that our favorite base scheme \(S\) is contained in the partial universe. Having picked the underlying category we pick a suitably large set of \(\tau\)-coverings which turns this into a site. The details are in the chapter on topologies on schemes; there is a lot of freedom in the choices made, but in the end the actual choices made will not affect the étale (or other) cohomology of \(S\) (just as in [SGA4] the actual choice of universe doesn’t matter at the end). Moreover, the way the material is written the reader who is happy using strongly inaccessible cardinals (i.e., universes) can do so as a substitute.

Definition

Let \(S\) be a scheme. Let \(\tau \in \{fppf, syntomic, smooth, \etale, \linebreak[0] Zariski\}\).

1. A big \(\tau\)-site of \(S\) is any of the sites \((\Sch/S)_\tau\) constructed as explained above and in more detail in Topologies, Definitions 021S, 03X3, 03X0, 021B, and 020T.

2. If \(\tau \in \{\etale, Zariski\}\), then the small \(\tau\)-site of \(S\) is the full subcategory \(S_\tau\) of \((\Sch/S)_\tau\) whose objects are schemes \(T\) over \(S\) whose structure morphism \(T \to S\) is étale, resp. an open immersion. A covering in \(S_\tau\) is a covering \(\{U_i \to U\}\) in \((\Sch/S)_\tau\) such that \(U\) is an object of \(S_\tau\).

The underlying category of the site \((\Sch/S)_\tau\) has reasonable “closure” properties, i.e., given a scheme \(T\) in it any locally closed subscheme of \(T\) is isomorphic to an object of \((\Sch/S)_\tau\). Other such closure properties are: closed under fibre products of schemes, taking countable disjoint unions, taking finite type schemes over a given scheme, given an affine scheme \(\Spec(R)\) one can complete, localize, or take the quotient of \(R\) by an ideal while staying inside the category, etc. On the other hand, for example arbitrary disjoint unions of schemes in \((\Sch/S)_\tau\) will take you outside of it. Also note that, given an object \(T\) of \((\Sch/S)_\tau\) there will exist \(\tau\)-coverings \(\{T_i \to T\}_{i \in I}\) (as in Definition 03X8) which are not coverings in \((\Sch/S)_\tau\) for example because the schemes \(T_i\) are not objects of the category \((\Sch/S)_\tau\). But our choice of the sites \((\Sch/S)_\tau\) is such that there always does exist a covering \(\{U_j \to T\}_{j \in J}\) of \((\Sch/S)_\tau\) which refines the covering \(\{T_i \to T\}_{i \in I}\), see Topologies, Lemmas 03WX, 03X2, 03WZ, 03WW, and 03WV. We will mostly ignore these issues in this chapter.

If \(\mathcal{F}\) is a sheaf on \((\Sch/S)_\tau\) or \(S_\tau\), then we denote \[H^p_\tau(U, \mathcal{F}), \text{ in particular } H^p_\tau(S, \mathcal{F})\] the cohomology groups of \(\mathcal{F}\) over the object \(U\) of the site, see Section 03NT. Thus we have \(H^p_{fppf}(S, \mathcal{F})\), \(H^p_{syntomic}(S, \mathcal{F})\), \(H^p_{smooth}(S, \mathcal{F})\), \(H^p_\etale(S, \mathcal{F})\), and \(H^p_{Zar}(S, \mathcal{F})\). The last two are potentially ambiguous since they might refer to either the big or small étale or Zariski site. However, this ambiguity is harmless by the following lemma.

Lemma

Let \(\tau \in \{\etale, Zariski\}\). If \(\mathcal{F}\) is an abelian sheaf defined on \((\Sch/S)_\tau\), then the cohomology groups of \(\mathcal{F}\) over \(S\) agree with the cohomology groups of \(\mathcal{F}|_{S_\tau}\) over \(S\).

Proof

By Topologies, Lemmas 020Z and 021G the functors \(S_\tau \to (\Sch/S)_\tau\) satisfy the hypotheses of Sites, Lemma 00XU. Hence our lemma follows from Cohomology on Sites, Lemma 03YU.

The category of sheaves on the big or small étale site of \(S\) depends only on the full subcategory of \((\Sch/S)_\etale\) or \(S_\etale\) consisting of affines and one only needs to consider the standard étale coverings between them (as defined below). This gives rise to sites \((\textit{Aff}/S)_\etale\) and \(S_{affine, \etale}\), see Topologies, Definition 021B. The comparison results are proven in Topologies, Lemmas 021E and 04HR. Here is our definition of standard coverings in some of the topologies we will consider in this chapter.

Definition

(See Topologies, Definitions 021Q, 0229, 0223, 0219, and 020R.) Let \(\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}\). Let \(T\) be an affine scheme. A standard \(\tau\)-covering of \(T\) is a family \(\{f_j : U_j \to T\}_{j = 1, \ldots, m}\) with each \(U_j\) is affine, and each \(f_j\) flat and of finite presentation, standard syntomic, standard smooth, étale, resp. the immersion of a standard principal open in \(T\) and \(T = \bigcup f_j(U_j)\).

Lemma

Let \(\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}\). Any \(\tau\)-covering of an affine scheme can be refined by a standard \(\tau\)-covering.

Proof

See Topologies, Lemmas 021P, 0228, 0222, 0218, and 020Q.

For completeness we state and prove the invariance under choice of partial universe of the cohomology groups we are considering. We will prove invariance of the small étale topos in Lemma 0958 below. For notation and terminology used in this lemma we refer to Topologies, Section 022I.

Lemma

Let \(\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}\). Let \(S\) be a scheme. Let \((\Sch/S)_\tau\) and \((\Sch'/S)_\tau\) be two big \(\tau\)-sites of \(S\), and assume that the first is contained in the second. In this case

1. for any abelian sheaf \(\mathcal{F}'\) defined on \((\Sch'/S)_\tau\) and any object \(U\) of \((\Sch/S)_\tau\) we have \[H^p_\tau(U, \mathcal{F}'|_{(\Sch/S)_\tau}) = H^p_\tau(U, \mathcal{F}')\] In words: the cohomology of \(\mathcal{F}'\) over \(U\) computed in the bigger site agrees with the cohomology of \(\mathcal{F}'\) restricted to the smaller site over \(U\).

2. for any abelian sheaf \(\mathcal{F}\) on \((\Sch/S)_\tau\) there is an abelian sheaf \(\mathcal{F}'\) on \((\Sch/S)_\tau'\) whose restriction to \((\Sch/S)_\tau\) is isomorphic to \(\mathcal{F}\).

Proof

By Topologies, Lemma 022K the inclusion functor \((\Sch/S)_\tau \to (\Sch'/S)_\tau\) satisfies the assumptions of Sites, Lemma 00XU. This implies (2) and (1) follows from Cohomology on Sites, Lemma 03YU.

The étale topos

A topos is the category of sheaves of sets on a site, see Sites, Definition 00XA. Hence it is customary to refer to the use the phrase “étale topos of a scheme” to refer to the category of sheaves on the small étale site of a scheme. Here is the formal definition.

Definition

Let \(S\) be a scheme.

1. The étale topos, or the small étale topos of \(S\) is the category \(\Sh(S_\etale)\) of sheaves of sets on the small étale site of \(S\).

2. The Zariski topos, or the small Zariski topos of \(S\) is the category \(\Sh(S_{Zar})\) of sheaves of sets on the small Zariski site of \(S\).

3. For \(\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}\) a big \(\tau\)-topos is the category of sheaves of set on a big \(\tau\)-topos of \(S\).

Note that the small Zariski topos of \(S\) is simply the category of sheaves of sets on the underlying topological space of \(S\), see Topologies, Lemma 020X. Whereas the small étale topos does not depend on the choices made in the construction of the small étale site, in general the big topoi do depend on those choices.

It turns out that the big or small étale topos only depends on the full subcategory of \((\Sch/S)_\etale\) or \(S_\etale\) consisting of affines, see Topologies, Lemmas 021E and 04HR. We will use this for example in the proof of the following lemma.

Lemma

Let \(S\) be a scheme. The étale topos of \(S\) is independent (up to canonical equivalence) of the construction of the small étale site in Definition 03XB.

Proof

We have to show, given two big étale sites \(\Sch_\etale\) and \(\Sch_\etale'\) containing \(S\), then \(\Sh(S_\etale) \cong \Sh(S_\etale')\) with obvious notation. By Topologies, Lemma 022J we may assume \(\Sch_\etale \subset \Sch_\etale'\). By Sets, Lemma 000R any affine scheme étale over \(S\) is isomorphic to an object of both \(\Sch_\etale\) and \(\Sch_\etale'\). Thus the induced functor \(S_{affine, \etale} \to S_{affine, \etale}'\) is an equivalence. Moreover, it is clear that both this functor and a quasi-inverse map transform standard étale coverings into standard étale coverings. Hence the result follows from Topologies, Lemma 04HR.

Cohomology of quasi-coherent sheaves

We start with a simple lemma (which holds in greater generality than stated). It says that the Čech complex of a standard covering is equal to the Čech complex of an fpqc covering of the form \(\{\Spec(B) \to \Spec(A)\}\) with \(A \to B\) faithfully flat.

Lemma

Let \(\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}\). Let \(S\) be a scheme. Let \(\mathcal{F}\) be an abelian sheaf on \((\Sch/S)_\tau\), or on \(S_\tau\) in case \(\tau = \etale\), and let \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) be a standard \(\tau\)-covering of this site. Let \(V = \coprod_{i \in I} U_i\). Then

1. \(V\) is an affine scheme,

2. \(\mathcal{V} = \{V \to U\}\) is an fpqc covering and also a \(\tau\)-covering unless \(\tau = Zariski\),

3. the Čech complexes \(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})\) and \(\check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F})\) agree.

Proof

The definition of a standard \(\tau\)-covering is given in Topologies, Definition 020R, 0219, 0223, 0229, and 021Q. By definition each of the schemes \(U_i\) is affine and \(I\) is a finite set. Hence \(V\) is an affine scheme. It is clear that \(V \to U\) is flat and surjective, hence \(\mathcal{V}\) is an fpqc covering, see Example 03NY. Excepting the Zariski case, the covering \(\mathcal{V}\) is also a \(\tau\)-covering, see Topologies, Definition 0215, 021Z, 0225, and 021M.

Note that \(\mathcal{U}\) is a refinement of \(\mathcal{V}\) and hence there is a map of Čech complexes \(\check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})\), see Cohomology on Sites, Equation (03F4). Next, we observe that if \(T = \coprod_{j \in J} T_j\) is a disjoint union of schemes in the site on which \(\mathcal{F}\) is defined then the family of morphisms with fixed target \(\{T_j \to T\}_{j \in J}\) is a Zariski covering, and so [03XC]\[\begin{equation} \mathcal{F}(T) = \mathcal{F}(\coprod\nolimits_{j \in J} T_j) = \prod\nolimits_{j \in J} \mathcal{F}(T_j) \end{equation}\] by the sheaf condition of \(\mathcal{F}\). This implies the map of Čech complexes above is an isomorphism in each degree because \[V \times_U \ldots \times_U V = \coprod\nolimits_{i_0, \ldots i_p} U_{i_0} \times_U \ldots \times_U U_{i_p}\] as schemes.

Note that Equality (03XC) is false for a general presheaf. Even for sheaves it does not hold on any site, since coproducts may not lead to coverings, and may not be disjoint. But it does for all the usual ones (at least all the ones we will study).

Remark

In the statement of Lemma 03OZ the covering \(\mathcal{U}\) is a refinement of \(\mathcal{V}\) but not the other way around. Coverings of the form \(\{V \to U\}\) do not form an initial subcategory of the category of all coverings of \(U\). Yet it is still true that we can compute Čech cohomology \(\check H^n(U, \mathcal{F})\) (which is defined as the colimit over the opposite of the category of coverings \(\mathcal{U}\) of \(U\) of the Čech cohomology groups of \(\mathcal{F}\) with respect to \(\mathcal{U}\)) in terms of the coverings \(\{V \to U\}\). We will formulate a precise lemma (it only works for sheaves) and add it here if we ever need it.

Lemma

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

Proof

Choose an injective resolution \(\mathcal{F} \to \mathcal{I}^\bullet\). Then \(\xi\) is represented by a cocycle \(\tilde{\xi} \in \mathcal{I}^p(U)\) with \(d^p(\tilde{\xi}) = 0\). By assumption, the sequence \(\mathcal{I}^{p - 1} \to \mathcal{I}^p \to \mathcal{I}^{p + 1}\) in exact in \(\textit{Ab}(\mathcal{C})\), which means that there exists a covering \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) such that \(\tilde{\xi}|_{U_i} = d^{p - 1}(\xi_i)\) for some \(\xi_i \in \mathcal{I}^{p-1}(U_i)\). Since the cohomology class \(\xi|_{U_i}\) is represented by the cocycle \(\tilde{\xi}|_{U_i}\) which is a coboundary, it vanishes. For more details see Cohomology on Sites, Lemma 01FW.

Theorem

Let \(S\) be a scheme and \(\mathcal{F}\) a quasi-coherent \(\mathcal{O}_S\)-module. Let \(\mathcal{C}\) be either \((\Sch/S)_\tau\) for \(\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}\) or \(S_\etale\). Then \[H^p(S, \mathcal{F}) = H^p_\tau(S, \mathcal{F}^a)\] for all \(p \geq 0\) where

1. the left hand side indicates the usual cohomology of the sheaf \(\mathcal{F}\) on the underlying topological space of the scheme \(S\), and

2. the right hand side indicates cohomology of the abelian sheaf \(\mathcal{F}^a\) (see Proposition 03OG) on the site \(\mathcal{C}\).

Proof

We are going to show that \(H^p(U, f^*\mathcal{F}) = H^p_\tau(U, \mathcal{F}^a)\) for any object \(f : U \to S\) of the site \(\mathcal{C}\). The result is true for \(p = 0\) by the sheaf property.

Assume that \(U\) is affine. Then we want to prove that \(H^p_\tau(U, \mathcal{F}^a) = 0\) for all \(p > 0\). We use induction on \(p\).

1. Pick \(\xi \in H^1_\tau(U, \mathcal{F}^a)\). By Lemma 03P1, there exists an fpqc covering \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) such that \(\xi|_{U_i} = 0\) for all \(i \in I\). Up to refining \(\mathcal{U}\), we may assume that \(\mathcal{U}\) is a standard \(\tau\)-covering. Applying the spectral sequence of Theorem 03OW, we see that \(\xi\) comes from a cohomology class \(\check \xi \in \check H^1(\mathcal{U}, \mathcal{F}^a)\). Consider the covering \(\mathcal{V} = \{\coprod_{i\in I} U_i \to U\}\). By Lemma 03OZ, \(\check H^\bullet(\mathcal{U}, \mathcal{F}^a) = \check H^\bullet(\mathcal{V}, \mathcal{F}^a)\). On the other hand, since \(\mathcal{V}\) is a covering of the form \(\{\Spec(B) \to \Spec(A)\}\) and \(f^*\mathcal{F} = \widetilde{M}\) for some \(A\)-module \(M\), we see the Čech complex \(\check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F})\) is none other than the complex \((B/A)_\bullet \otimes_A M\). Now by Lemma 03OA, \(H^p((B/A)_\bullet \otimes_A M) = 0\) for \(p > 0\), hence \(\check \xi = 0\) and so \(\xi = 0\).

2. Pick \(\xi \in H^p_\tau(U, \mathcal{F}^a)\). By Lemma 03P1, there exists an fpqc covering \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) such that \(\xi|_{U_i} = 0\) for all \(i \in I\). Up to refining \(\mathcal{U}\), we may assume that \(\mathcal{U}\) is a standard \(\tau\)-covering. We apply the spectral sequence of Theorem 03OW. Observe that the intersections \(U_{i_0} \times_U \ldots \times_U U_{i_p}\) are affine, so that by induction hypothesis the cohomology groups \[E_2^{p, q} = \check H^p(\mathcal{U}, \underline{H}^q(\mathcal{F}^a))\] vanish for all \(0 < q < p\). We see that \(\xi\) must come from a \(\check \xi \in \check H^p(\mathcal{U}, \mathcal{F}^a)\). Replacing \(\mathcal{U}\) with the covering \(\mathcal{V}\) containing only one morphism and using Lemma 03OA again, we see that the Čech cohomology class \(\check \xi\) must be zero, hence \(\xi = 0\).

Next, assume that \(U\) is separated. Choose an affine open covering \(U = \bigcup_{i \in I} U_i\) of \(U\). The family \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) is then an fpqc covering, and all the intersections \(U_{i_0} \times_U \ldots \times_U U_{i_p}\) are affine since \(U\) is separated. So all rows of the spectral sequence of Theorem 03OW are zero, except the zeroth row. Therefore \[H^p_\tau(U, \mathcal{F}^a) = \check H^p(\mathcal{U}, \mathcal{F}^a) = \check H^p(\mathcal{U}, \mathcal{F}) = H^p(U, \mathcal{F})\] where the last equality results from standard scheme theory, see Cohomology of Schemes, Lemma 01XD.

The general case is technical and (to extend the proof as given here) requires a discussion about maps of spectral sequences, so we won’t treat it. It follows from Descent, Proposition 03DW (whose proof takes a slightly different approach) combined with Cohomology on Sites, Lemma 03F3.

Remark

Comment on Theorem 03P2. Since \(S\) is a final object in the category \(\mathcal{C}\), the cohomology groups on the right-hand side are merely the right derived functors of the global sections functor. In fact the proof shows that \(H^p(U, f^*\mathcal{F}) = H^p_\tau(U, \mathcal{F}^a)\) for any object \(f : U \to S\) of the site \(\mathcal{C}\).

Examples of sheaves

Let \(S\) and \(\tau\) be as in Section 03X7. We have already seen that any representable presheaf is a sheaf on \((\Sch/S)_\tau\) or \(S_\tau\), see Lemma 03O3 and Remark 03O4. Here are some special cases.

Definition

On any of the sites \((\Sch/S)_\tau\) or \(S_\tau\) of Section 03X7.

1. The sheaf \(T \mapsto \Gamma(T, \mathcal{O}_T)\) is denoted \(\mathcal{O}_S\), or \(\mathbf{G}_a\), or \(\mathbf{G}_{a, S}\) if we want to indicate the base scheme.

2. Similarly, the sheaf \(T \mapsto \Gamma(T, \mathcal{O}^*_T)\) is denoted \(\mathcal{O}_S^*\), or \(\mathbf{G}_m\), or \(\mathbf{G}_{m, S}\) if we want to indicate the base scheme.

3. The constant sheaf \(\underline{\mathbf{Z}/n\mathbf{Z}}\) on any site is the sheafification of the constant presheaf \(U \mapsto \mathbf{Z}/n\mathbf{Z}\).

The first is a sheaf by Theorem 03OJ for example. The second is a sub presheaf of the first, which is easily seen to be a sheaf itself. The third is a sheaf by definition. Note that each of these sheaves is representable. The first and second by the schemes \(\mathbf{G}_{a, S}\) and \(\mathbf{G}_{m, S}\), see Groupoids, Section 022R. The third by the finite étale group scheme \(\mathbf{Z}/n\mathbf{Z}_S\) sometimes denoted \((\mathbf{Z}/n\mathbf{Z})_S\) which is just \(n\) copies of \(S\) endowed with the obvious group scheme structure over \(S\), see Groupoids, Example 03YW and the following remark.

Remark

Let \(G\) be an abstract group. On any of the sites \((\Sch/S)_\tau\) or \(S_\tau\) of Section 03X7 the sheafification \(\underline{G}\) of the constant presheaf associated to \(G\) in the Zariski topology of the site already gives \[\Gamma(U, \underline{G}) = \{\text{Zariski locally constant maps }U \to G\}\] This Zariski sheaf is representable by the group scheme \(G_S\) according to Groupoids, Example 03YW. By Lemma 03O3 any representable presheaf satisfies the sheaf condition for the \(\tau\)-topology as well, and hence we conclude that the Zariski sheafification \(\underline{G}\) above is also the \(\tau\)-sheafification.

Definition

Let \(S\) be a scheme. The structure sheaf of \(S\) is the sheaf of rings \(\mathcal{O}_S\) on any of the sites \(S_{Zar}\), \(S_\etale\), or \((\Sch/S)_\tau\) discussed above.

If there is some possible confusion as to which site we are working on then we will indicate this by using indices. For example we may use \(\mathcal{O}_{S_\etale}\) to stress the fact that we are working on the small étale site of \(S\).

Remark

In the terminology introduced above a special case of Theorem 03P2 is \[H_{fppf}^p(X, \mathbf{G}_a) = H_\etale^p(X, \mathbf{G}_a) = H_{Zar}^p(X, \mathbf{G}_a) = H^p(X, \mathcal{O}_X)\] for all \(p \geq 0\). Moreover, we could use the notation \(H^p_{fppf}(X, \mathcal{O}_X)\) to indicate the cohomology of the structure sheaf on the big fppf site of \(X\).

Picard groups

The following theorem is sometimes called “Hilbert 90”.

Theorem

For any scheme \(X\) we have canonical identifications \[\begin{align*} H_{fppf}^1(X, \mathbf{G}_m) & = H^1_{syntomic}(X, \mathbf{G}_m) \\ & = H^1_{smooth}(X, \mathbf{G}_m) \\ & = H_\etale^1(X, \mathbf{G}_m) \\ & = H^1_{Zar}(X, \mathbf{G}_m) \\ & = \Pic(X) \\ & = H^1(X, \mathcal{O}_X^*) \end{align*}\]

Proof

Let \(\tau\) be one of the topologies considered in Section 03X7. By Cohomology on Sites, Lemma 040E we see that \(H^1_\tau(X, \mathbf{G}_m) = H^1_\tau(X, \mathcal{O}_\tau^*) = \Pic(\mathcal{O}_\tau)\) where \(\mathcal{O}_\tau\) is the structure sheaf of the site \((\Sch/X)_\tau\). Now an invertible \(\mathcal{O}_\tau\)-module is a quasi-coherent \(\mathcal{O}_\tau\)-module. By Theorem 03OJ or the more precise Descent, Proposition 03DX we see that \(\Pic(\mathcal{O}_\tau) = \Pic(X)\). The last equality is proved in the same way.

The étale site

At this point we start exploring the étale site of a scheme in more detail. As a first step we discuss a little the notion of an étale morphism.

Étale morphisms

For more details, see Morphisms, Section 02GH for the formal definition and Étale Morphisms, Sections 0257, 025A, 039P, 025F, 025J, and 025L for a survey of interesting properties of étale morphisms.

Recall that an algebra \(A\) over an algebraically closed field \(k\) is smooth if it is of finite type and the module of differentials \(\Omega_{A/k}\) is finite locally free of rank equal to the dimension. A scheme \(X\) over \(k\) is smooth over \(k\) if it is locally of finite type and each affine open is the spectrum of a smooth \(k\)-algebra. If \(k\) is not algebraically closed then a \(k\)-algebra \(A\) is a smooth \(k\)-algebra if \(A \otimes_k \overline{k}\) is a smooth \(\overline{k}\)-algebra. A ring map \(A \to B\) is smooth if it is flat, finitely presented, and for all primes \(\mathfrak p \subset A\) the fibre ring \(\kappa(\mathfrak p) \otimes_A B\) is smooth over the residue field \(\kappa(\mathfrak p)\). More generally, a morphism of schemes is smooth if it is flat, locally of finite presentation, and the geometric fibers are smooth.

For these facts please see Morphisms, Section 01V4. Using this we may define an étale morphism as follows.

Definition

A morphism of schemes is étale if it is smooth of relative dimension 0.

In particular, a morphism of schemes \(X \to S\) is étale if it is smooth and \(\Omega_{X/S} = 0\).

Proposition

Facts on étale morphisms.

1. Let \(k\) be a field. A morphism of schemes \(U \to \Spec(k)\) is étale if and only if \(U \cong \coprod_{i \in I} \Spec(k_i)\) such that for each \(i \in I\) the ring \(k_i\) is a field which is a finite separable extension of \(k\).

2. Let \(\varphi : U \to S\) be a morphism of schemes. The following conditions are equivalent:

   1. \(\varphi\) is étale,

   2. \(\varphi\) is locally finitely presented, flat, and all its fibres are étale,

   3. \(\varphi\) is flat, unramified and locally of finite presentation.

3. A ring map \(A \to B\) is étale if and only if \(B \cong A[x_1, \ldots, x_n]/(f_1, \ldots, f_n)\) such that \(\Delta = \det \left( \frac{\partial f_i}{\partial x_j} \right)\) is invertible in \(B\).

4. The base change of an étale morphism is étale.

5. Compositions of étale morphisms are étale.

6. Fibre products and products of étale morphisms are étale.

7. An étale morphism has relative dimension 0.

8. Let \(Y \to X\) be an étale morphism. If \(X\) is reduced (respectively regular) then so is \(Y\).

9. Étale morphisms are open.

10. If \(X \to S\) and \(Y \to S\) are étale, then any \(S\)-morphism \(X \to Y\) is also étale.

Proof

We have proved these facts (and more) in the preceding chapters. Here is a list of references: (1) Morphisms, Lemma 02GL. (2) Morphisms, Lemmas 02GM and 02GV. (3) Algebra, Lemma 00U9. (4) Morphisms, Lemma 02GO. (5) Morphisms, Lemma 02GN. (6) Follows formally from (4) and (5). (7) Morphisms, Lemmas 03WS and 0397. (8) See Algebra, Lemmas 033B and 033A, see also more results of this kind in Étale Morphisms, Section 025L. (9) See Morphisms, Lemma 01UA and 02GS. (10) See Morphisms, Lemma 02GW.

Definition

A ring map \(A \to B\) is called standard étale if \(B \cong \left(A[t]/(f)\right)_g\) with \(f, g \in A[t]\), with \(f\) monic, and \(\text{d}f/\text{d}t\) invertible in \(B\).

It is true that a standard étale ring map is étale. Namely, suppose that \(B = \left(A[t]/(f)\right)_g\) with \(f, g \in A[t]\), with \(f\) monic, and \(\text{d}f/\text{d}t\) invertible in \(B\). Then \(A[t]/(f)\) is a finite free \(A\)-module of rank equal to the degree of the monic polynomial \(f\). Hence \(B\), as a localization of this free algebra is finitely presented and flat over \(A\). To finish the proof that \(B\) is étale it suffices to show that the fibre rings \[\kappa(\mathfrak p) \otimes_A B \cong \kappa(\mathfrak p) \otimes_A (A[t]/(f))_g \cong \kappa(\mathfrak p)[t, 1/\overline{g}]/(\overline{f})\] are finite products of finite separable field extensions. Here \(\overline{f}, \overline{g} \in \kappa(\mathfrak p)[t]\) are the images of \(f\) and \(g\). Let \[\overline{f} = \overline{f}_1 \ldots \overline{f}_a \overline{f}_{a + 1}^{e_1} \ldots \overline{f}_{a + b}^{e_b}\] be the factorization of \(\overline{f}\) into powers of pairwise distinct irreducible monic factors \(\overline{f}_i\) with \(e_1, \ldots, e_b > 1\). By assumption \(\text{d}\overline{f}/\text{d}t\) is invertible in \(\kappa(\mathfrak p)[t, 1/\overline{g}]\). Hence we see that at least all the \(\overline{f}_i\), \(i > a\) are invertible. We conclude that \[\kappa(\mathfrak p)[t, 1/\overline{g}]/(\overline{f}) \cong \prod\nolimits_{i \in I} \kappa(\mathfrak p)[t]/(\overline{f}_i)\] where \(I \subset \{1, \ldots, a\}\) is the subset of indices \(i\) such that \(\overline{f}_i\) does not divide \(\overline{g}\). Moreover, the image of \(\text{d}\overline{f}/\text{d}t\) in the factor \(\kappa(\mathfrak p)[t]/(\overline{f}_i)\) is clearly equal to a unit times \(\text{d}\overline{f}_i/\text{d}t\). Hence we conclude that \(\kappa_i = \kappa(\mathfrak p)[t]/(\overline{f}_i)\) is a finite field extension of \(\kappa(\mathfrak p)\) generated by one element whose minimal polynomial is separable, i.e., the field extension \(\kappa_i/\kappa(\mathfrak p)\) is finite separable as desired.

It turns out that any étale ring map is locally standard étale. To formulate this we introduce the following notation. A ring map \(A \to B\) is étale at a prime \(\mathfrak q\) of \(B\) if there exists \(h \in B\), \(h \not \in \mathfrak q\) such that \(A \to B_h\) is étale. Here is the result.

Theorem

A ring map \(A \to B\) is étale at a prime \(\mathfrak q\) if and only if there exists \(g \in B\), \(g \not \in \mathfrak q\) such that \(B_g\) is standard étale over \(A\).

Proof

See Algebra, Proposition 00UE.

Étale coverings

We recall the definition.

Definition

An étale covering of a scheme \(U\) is a family of morphisms of schemes \(\{\varphi_i : U_i \to U\}_{i \in I}\) such that

1. each \(\varphi_i\) is an étale morphism,

2. the \(U_i\) cover \(U\), i.e., \(U = \bigcup_{i\in I}\varphi_i(U_i)\).

Lemma

Any étale covering is an fpqc covering.

Proof

(See also Topologies, Lemma 022C.) Let \(\{\varphi_i : U_i \to U\}_{i \in I}\) be an étale covering. Since an étale morphism is flat, and the elements of the covering should cover its target, the property fp (faithfully flat) is satisfied. To check the property qc (quasi-compact), let \(V \subset U\) be an affine open, and write \(\varphi_i^{-1}(V) = \bigcup_{j \in J_i} V_{ij}\) for some affine opens \(V_{ij} \subset U_i\). Since \(\varphi_i\) is open (as étale morphisms are open), we see that \(V = \bigcup_{i\in I} \bigcup_{j \in J_i} \varphi_i(V_{ij})\) is an open covering of \(V\). Further, since \(V\) is quasi-compact, this covering has a finite refinement.

So any statement which is true for fpqc coverings remains true a fortiori for étale coverings. For instance, the étale site is subcanonical.

Definition

(For more details see Section 03X7, or Topologies, Section 0214.) Let \(S\) be a scheme. The big étale site over \(S\) is the site \((\Sch/S)_\etale\), see Definition 03XB. The small étale site over \(S\) is the site \(S_\etale\), see Definition 03XB. We define similarly the big and small Zariski sites on \(S\), denoted \((\Sch/S)_{Zar}\) and \(S_{Zar}\).

Loosely speaking the big étale site of \(S\) is made up out of schemes over \(S\) and coverings the étale coverings. The small étale site of \(S\) is made up out of schemes étale over \(S\) with coverings the étale coverings. Actually any morphism between objects of \(S_\etale\) is étale, in virtue of Proposition 03PC, hence to check that \(\{U_i \to U\}_{i \in I}\) in \(S_\etale\) is a covering it suffices to check that \(\coprod U_i \to U\) is surjective.

The small étale site has fewer objects than the big étale site, it contains only the “opens” of the étale topology on \(S\). It is a full subcategory of the big étale site, and its topology is induced from the topology on the big site. Hence it is true that the restriction functor from the big étale site to the small one is exact and maps injectives to injectives. This has the following consequence.

Proposition

Let \(S\) be a scheme and \(\mathcal{F}\) an abelian sheaf on \((\Sch/S)_\etale\). Then \(\mathcal{F}|_{S_\etale}\) is a sheaf on \(S_\etale\) and \[H^p_\etale(S, \mathcal{F}|_{S_\etale}) = H^p_\etale(S, \mathcal{F})\] for all \(p \geq 0\).

Proof

This is a special case of Lemma 03YX.

In accordance with the general notation introduced in Section 03X7 we write \(H_\etale^p(S, \mathcal{F})\) for the above cohomology group.

Kummer theory

Let \(n \in \mathbf{N}\) and consider the functor \(\mu_n\) defined by \[\begin{matrix} \Sch^{opp} & \longrightarrow & \textit{Ab} \\ S & \longmapsto & \mu_n(S) = \{t \in \Gamma(S, \mathcal{O}_S^*) \mid t^n = 1 \}. \end{matrix}\] By Groupoids, Example 040M this is a representable functor, and the scheme representing it is denoted \(\mu_n\) also. By Lemma 03O3 this functor satisfies the sheaf condition for the fpqc topology (in particular, it also satisfies the sheaf condition for the étale, Zariski, etc topology).

Lemma

If \(n\in \mathcal{O}_S^*\) then \[0 \to \mu_{n, S} \to \mathbf{G}_{m, S} \xrightarrow{(\cdot)^n} \mathbf{G}_{m, S} \to 0\] is a short exact sequence of sheaves on both the small and big étale site of \(S\).

Proof

By definition the sheaf \(\mu_{n, S}\) is the kernel of the map \((\cdot)^n\). Hence it suffices to show that the last map is surjective. Let \(U\) be a scheme over \(S\). Let \(f \in \mathbf{G}_m(U) = \Gamma(U, \mathcal{O}_U^*)\). We need to show that we can find an étale cover of \(U\) over the members of which the restriction of \(f\) is an \(n\)th power. Set \[U' = \underline{\Spec}_U(\mathcal{O}_U[T]/(T^n-f)) \xrightarrow{\pi} U.\] (See Constructions, Section 01LL or 01LQ for a discussion of the relative spectrum.) Let \(\Spec(A) \subset U\) be an affine open, and say \(f|_{\Spec(A)}\) corresponds to the unit \(a \in A^*\). Then \(\pi^{-1}(\Spec(A)) = \Spec(B)\) with \(B = A[T]/(T^n - a)\). The ring map \(A \to B\) is finite free of rank \(n\), hence it is faithfully flat, and hence we conclude that \(\Spec(B) \to \Spec(A)\) is surjective. Since this holds for every affine open in \(U\) we conclude that \(\pi\) is surjective. In addition, \(n\) and \(T^{n - 1}\) are invertible in \(B\), so \(nT^{n-1} \in B^*\) and the ring map \(A \to B\) is standard étale, in particular étale. Since this holds for every affine open of \(U\) we conclude that \(\pi\) is étale. Hence \(\mathcal{U} = \{\pi : U' \to U\}\) is an étale covering. Moreover, \(f|_{U'} = (f')^n\) where \(f'\) is the class of \(T\) in \(\Gamma(U', \mathcal{O}_{U'}^*)\), so \(\mathcal{U}\) has the desired property.

Remark

Lemma 03PL is false when “étale” is replaced with “Zariski”. Since the étale topology is coarser than the smooth topology, see Topologies, Lemma 0220 it follows that the sequence is also exact in the smooth topology.

By Theorem 03P8 and Lemma 03PL and general properties of cohomology we obtain the long exact cohomology sequence \[\xymatrix{ 0 \ar[r] & H_\etale^0(S, \mu_{n, S}) \ar[r] & \Gamma(S, \mathcal{O}_S^*) \ar[r]^{(\cdot)^n} & \Gamma(S, \mathcal{O}_S^*) \ar@(rd, ul)[rdllllr] \\ & H_\etale^1(S, \mu_{n, S}) \ar[r] & \Pic(S) \ar[r]^{(\cdot)^n} & \Pic(S) \ar@(rd, ul)[rdllllr] \\ & H_\etale^2(S, \mu_{n, S}) \ar[r] & \ldots }\] at least if \(n\) is invertible on \(S\). When \(n\) is not invertible on \(S\) we can apply the following lemma.

Lemma

For any \(n \in \mathbf{N}\) the sequence \[0 \to \mu_{n, S} \to \mathbf{G}_{m, S} \xrightarrow{(\cdot)^n} \mathbf{G}_{m, S} \to 0\] is a short exact sequence of sheaves on the site \((\Sch/S)_{fppf}\) and \((\Sch/S)_{syntomic}\).

Proof

By definition the sheaf \(\mu_{n, S}\) is the kernel of the map \((\cdot)^n\). Hence it suffices to show that the last map is surjective. Since the syntomic topology is weaker than the fppf topology, see Topologies, Lemma 021N, it suffices to prove this for the syntomic topology. Let \(U\) be a scheme over \(S\). Let \(f \in \mathbf{G}_m(U) = \Gamma(U, \mathcal{O}_U^*)\). We need to show that we can find a syntomic cover of \(U\) over the members of which the restriction of \(f\) is an \(n\)th power. Set \[U' = \underline{\Spec}_U(\mathcal{O}_U[T]/(T^n-f)) \xrightarrow{\pi} U.\] (See Constructions, Section 01LL or 01LQ for a discussion of the relative spectrum.) Let \(\Spec(A) \subset U\) be an affine open, and say \(f|_{\Spec(A)}\) corresponds to the unit \(a \in A^*\). Then \(\pi^{-1}(\Spec(A)) = \Spec(B)\) with \(B = A[T]/(T^n - a)\). The ring map \(A \to B\) is finite free of rank \(n\), hence it is faithfully flat, and hence we conclude that \(\Spec(B) \to \Spec(A)\) is surjective. Since this holds for every affine open in \(U\) we conclude that \(\pi\) is surjective. In addition, \(B\) is a global relative complete intersection over \(A\), so the ring map \(A \to B\) is standard syntomic, in particular syntomic. Since this holds for every affine open of \(U\) we conclude that \(\pi\) is syntomic. Hence \(\mathcal{U} = \{\pi : U' \to U\}\) is a syntomic covering. Moreover, \(f|_{U'} = (f')^n\) where \(f'\) is the class of \(T\) in \(\Gamma(U', \mathcal{O}_{U'}^*)\), so \(\mathcal{U}\) has the desired property.

Remark

Lemma 040N is false for the smooth, étale, or Zariski topology.

By Theorem 03P8 and Lemma 040N and general properties of cohomology we obtain the long exact cohomology sequence \[\xymatrix{ 0 \ar[r] & H_{fppf}^0(S, \mu_{n, S}) \ar[r] & \Gamma(S, \mathcal{O}_S^*) \ar[r]^{(\cdot)^n} & \Gamma(S, \mathcal{O}_S^*) \ar@(rd, ul)[rdllllr] \\ & H_{fppf}^1(S, \mu_{n, S}) \ar[r] & \Pic(S) \ar[r]^{(\cdot)^n} & \Pic(S) \ar@(rd, ul)[rdllllr] \\ & H_{fppf}^2(S, \mu_{n, S}) \ar[r] & \ldots }\] for any scheme \(S\) and any integer \(n\). Of course there is a similar sequence with syntomic cohomology.

Let \(n \in \mathbf{N}\) and let \(S\) be any scheme. There is another more direct way to describe the first cohomology group with values in \(\mu_n\). Consider pairs \((\mathcal{L}, \alpha)\) where \(\mathcal{L}\) is an invertible sheaf on \(S\) and \(\alpha : \mathcal{L}^{\otimes n} \to \mathcal{O}_S\) is a trivialization of the \(n\)th tensor power of \(\mathcal{L}\). Let \((\mathcal{L}', \alpha')\) be a second such pair. An isomorphism \(\varphi : (\mathcal{L}, \alpha) \to (\mathcal{L}', \alpha')\) is an isomorphism \(\varphi : \mathcal{L} \to \mathcal{L}'\) of invertible sheaves such that the diagram \[\xymatrix{ \mathcal{L}^{\otimes n} \ar[d]_{\varphi^{\otimes n}} \ar[r]_\alpha & \mathcal{O}_S \ar[d]^1 \\ (\mathcal{L}')^{\otimes n} \ar[r]^{\alpha'} & \mathcal{O}_S \\ }\] commutes. Thus we have [040P]\[\begin{equation} \mathit{Isom}_S((\mathcal{L}, \alpha), (\mathcal{L}', \alpha')) = \left\{ \begin{matrix} \emptyset & \text{if} & \text{they are not isomorphic} \\ H^0(S, \mu_{n, S})\cdot \varphi & \text{if} & \varphi \text{ isomorphism of pairs} \end{matrix} \right. \end{equation}\] Moreover, given two pairs \((\mathcal{L}, \alpha)\), \((\mathcal{L}', \alpha')\) the tensor product \[(\mathcal{L}, \alpha) \otimes (\mathcal{L}', \alpha') = (\mathcal{L} \otimes \mathcal{L}', \alpha \otimes \alpha')\] is another pair. The pair \((\mathcal{O}_S, 1)\) is an identity for this tensor product operation, and an inverse is given by \[(\mathcal{L}, \alpha)^{-1} = (\mathcal{L}^{\otimes -1}, \alpha^{\otimes -1}).\] Hence the collection of isomorphism classes of pairs forms an abelian group. Note that \[(\mathcal{L}, \alpha)^{\otimes n} = (\mathcal{L}^{\otimes n}, \alpha^{\otimes n}) \xrightarrow{\alpha} (\mathcal{O}_S, 1)\] is an isomorphism hence every element of this group has order dividing \(n\). We warn the reader that this group is in general not the \(n\)-torsion in \(\Pic(S)\).

Lemma

Let \(S\) be a scheme. There is a canonical identification \[H_\etale^1(S, \mu_n) = \text{group of pairs }(\mathcal{L}, \alpha)\text{ up to isomorphism as above}\] if \(n\) is invertible on \(S\). In general we have \[H_{fppf}^1(S, \mu_n) = \text{group of pairs }(\mathcal{L}, \alpha)\text{ up to isomorphism as above}.\] The same result holds with fppf replaced by syntomic.

Proof

We first prove the second isomorphism. Let \((\mathcal{L}, \alpha)\) be a pair as above. Choose an affine open covering \(S = \bigcup U_i\) such that \(\mathcal{L}|_{U_i} \cong \mathcal{O}_{U_i}\). Say \(s_i \in \mathcal{L}(U_i)\) is a generator. Then \(\alpha(s_i^{\otimes n}) = f_i \in \mathcal{O}_S^*(U_i)\). Writing \(U_i = \Spec(A_i)\) we see there exists a global relative complete intersection \(A_i \to B_i = A_i[T]/(T^n - f_i)\) such that \(f_i\) maps to an \(n\)th power in \(B_i\). In other words, setting \(V_i = \Spec(B_i)\) we obtain a syntomic covering \(\mathcal{V} = \{V_i \to S\}_{i \in I}\) and trivializations \(\varphi_i : (\mathcal{L}, \alpha)|_{V_i} \to (\mathcal{O}_{V_i}, 1)\).

We will use this result (the existence of the covering \(\mathcal{V}\)) to associate to this pair a cohomology class in \(H^1_{syntomic}(S, \mu_{n, S})\). We give two (equivalent) constructions.

First construction: using Čech cohomology. Over the double overlaps \(V_i \times_S V_j\) we have the isomorphism \[(\mathcal{O}_{V_i \times_S V_j}, 1) \xrightarrow{\text{pr}_0^*\varphi_i^{-1}} (\mathcal{L}|_{V_i \times_S V_j}, \alpha|_{V_i \times_S V_j}) \xrightarrow{\text{pr}_1^*\varphi_j} (\mathcal{O}_{V_i \times_S V_j}, 1)\] of pairs. By (040P) this is given by an element \(\zeta_{ij} \in \mu_n(V_i \times_S V_j)\). We omit the verification that these \(\zeta_{ij}\)’s give a \(1\)-cocycle, i.e., give an element \((\zeta_{i_0i_1}) \in \check C(\mathcal{V}, \mu_n)\) with \(d(\zeta_{i_0i_1}) = 0\). Thus its class is an element in \(\check H^1(\mathcal{V}, \mu_n)\) and by Theorem 03OW it maps to a cohomology class in \(H^1_{syntomic}(S, \mu_{n, S})\).

Second construction: Using torsors. Consider the presheaf \[\mu_n(\mathcal{L}, \alpha) : U \longmapsto \mathit{Isom}_U((\mathcal{O}_U, 1), (\mathcal{L}, \alpha)|_U)\] on \((\Sch/S)_{syntomic}\). We may view this as a subpresheaf of \(\SheafHom_\mathcal{O}(\mathcal{O}, \mathcal{L})\) (internal hom sheaf, see Modules on Sites, Section 04TT). Since the conditions defining this subpresheaf are local, we see that it is a sheaf. By (040P) this sheaf has a free action of the sheaf \(\mu_{n, S}\). Hence the only thing we have to check is that it locally has sections. This is true because of the existence of the trivializing cover \(\mathcal{V}\). Hence \(\mu_n(\mathcal{L}, \alpha)\) is a \(\mu_{n, S}\)-torsor and by Cohomology on Sites, Lemma 03AJ we obtain a corresponding element of \(H^1_{syntomic}(S, \mu_{n, S})\).

Ok, now we have to still show the following

1. The two constructions give the same cohomology class.

2. Isomorphic pairs give rise to the same cohomology class.

3. The cohomology class of \((\mathcal{L}, \alpha) \otimes (\mathcal{L}', \alpha')\) is the sum of the cohomology classes of \((\mathcal{L}, \alpha)\) and \((\mathcal{L}', \alpha')\).

4. If the cohomology class is trivial, then the pair is trivial.

5. Any element of \(H^1_{syntomic}(S, \mu_{n, S})\) is the cohomology class of a pair.

We omit the proof of (1). Part (2) is clear from the second construction, since isomorphic torsors give the same cohomology classes. Part (3) is clear from the first construction, since the resulting Čech classes add up. Part (4) is clear from the second construction since a torsor is trivial if and only if it has a global section, see Cohomology on Sites, Lemma 03AI.

Part (5) can be seen as follows (although a direct proof would be preferable). Suppose \(\xi \in H^1_{syntomic}(S, \mu_{n, S})\). Then \(\xi\) maps to an element \(\overline{\xi} \in H^1_{syntomic}(S, \mathbf{G}_{m, S})\) with \(n \overline{\xi} = 0\). By Theorem 03P8 we see that \(\overline{\xi}\) corresponds to an invertible sheaf \(\mathcal{L}\) whose \(n\)th tensor power is isomorphic to \(\mathcal{O}_S\). Hence there exists a pair \((\mathcal{L}, \alpha')\) whose cohomology class \(\xi'\) has the same image \(\overline{\xi'}\) in \(H^1_{syntomic}(S, \mathbf{G}_{m, S})\). Thus it suffices to show that \(\xi - \xi'\) is the class of a pair. By construction, and the long exact cohomology sequence above, we see that \(\xi - \xi' = \partial(f)\) for some \(f \in H^0(S, \mathcal{O}_S^*)\). Consider the pair \((\mathcal{O}_S, f)\). We omit the verification that the cohomology class of this pair is \(\partial(f)\), which finishes the proof of the first identification (with fppf replaced with syntomic).

To see the first, note that if \(n\) is invertible on \(S\), then the covering \(\mathcal{V}\) constructed in the first part of the proof is actually an étale covering (compare with the proof of Lemma 03PL). The rest of the proof is independent of the topology, apart from the very last argument which uses that the Kummer sequence is exact, i.e., uses Lemma 03PL.

Neighborhoods, stalks and points

We can associate to any geometric point of \(S\) a stalk functor which is exact. A map of sheaves on \(S_\etale\) is an isomorphism if and only if it is an isomorphism on all these stalks. A complex of abelian sheaves is exact if and only if the complex of stalks is exact at all geometric points. Altogether this means that the small étale site of a scheme \(S\) has enough points. It also turns out that any point of the small étale topos of \(S\) (an abstract notion) is given by a geometric point. Thus in some sense the small étale topos of \(S\) can be understood in terms of geometric points and neighbourhoods.

Definition

Let \(S\) be a scheme.

1. A geometric point of \(S\) is a morphism \(\Spec(k) \to S\) where \(k\) is algebraically closed. Such a point is usually denoted \(\overline{s}\), i.e., by an overlined small case letter. We often use \(\overline{s}\) to denote the scheme \(\Spec(k)\) as well as the morphism, and we use \(\kappa(\overline{s})\) to denote \(k\).

2. We say \(\overline{s}\) lies over \(s\) to indicate that \(s \in S\) is the image of \(\overline{s}\).

3. An étale neighborhood of a geometric point \(\overline{s}\) of \(S\) is a commutative diagram \[\xymatrix{ & U \ar[d]^\varphi \\ {\overline{s}} \ar[r]^{\overline{s}} \ar[ur]^{\bar u} & S }\] where \(\varphi\) is an étale morphism of schemes. We write \((U, \overline{u}) \to (S, \overline{s})\).

4. A morphism of étale neighborhoods \((U, \overline{u}) \to (U', \overline{u}')\) is an \(S\)-morphism \(h: U \to U'\) such that \(\overline{u}' = h \circ \overline{u}\).

Remark

Since \(U\) and \(U'\) are étale over \(S\), any \(S\)-morphism between them is also étale, see Proposition 03PC. In particular all morphisms of étale neighborhoods are étale.

Remark

Let \(S\) be a scheme and \(s \in S\) a point. In More on Morphisms, Definition 02LE we defined the notion of an étale neighbourhood \((U, u) \to (S, s)\) of \((S, s)\). If \(\overline{s}\) is a geometric point of \(S\) lying over \(s\), then any étale neighbourhood \((U, \overline{u}) \to (S, \overline{s})\) gives rise to an étale neighbourhood \((U, u)\) of \((S, s)\) by taking \(u \in U\) to be the unique point of \(U\) such that \(\overline{u}\) lies over \(u\). Conversely, given an étale neighbourhood \((U, u)\) of \((S, s)\) the residue field extension \(\kappa(u)/\kappa(s)\) is finite separable (see Proposition 03PC) and hence we can find an embedding \(\kappa(u) \subset \kappa(\overline{s})\) over \(\kappa(s)\). In other words, we can find a geometric point \(\overline{u}\) of \(U\) lying over \(u\) such that \((U, \overline{u})\) is an étale neighbourhood of \((S, \overline{s})\). We will use these observations to go between the two types of étale neighbourhoods.

Lemma

Let \(S\) be a scheme, and let \(\overline{s}\) be a geometric point of \(S\). The category of étale neighborhoods is cofiltered. More precisely:

1. Let \((U_i, \overline{u}_i)_{i = 1, 2}\) be two étale neighborhoods of \(\overline{s}\) in \(S\). Then there exists a third étale neighborhood \((U, \overline{u})\) and morphisms \((U, \overline{u}) \to (U_i, \overline{u}_i)\), \(i = 1, 2\).

2. Let \(h_1, h_2: (U, \overline{u}) \to (U', \overline{u}')\) be two morphisms between étale neighborhoods of \(\overline{s}\). Then there exist an étale neighborhood \((U'', \overline{u}'')\) and a morphism \(h : (U'', \overline{u}'') \to (U, \overline{u})\) which equalizes \(h_1\) and \(h_2\), i.e., such that \(h_1 \circ h = h_2 \circ h\).

Proof

For part (1), consider the fibre product \(U = U_1 \times_S U_2\). It is étale over both \(U_1\) and \(U_2\) because étale morphisms are preserved under base change, see Proposition 03PC. The map \(\overline{s} \to U\) defined by \((\overline{u}_1, \overline{u}_2)\) gives it the structure of an étale neighborhood mapping to both \(U_1\) and \(U_2\). For part (2), define \(U''\) as the fibre product \[\xymatrix{ U'' \ar[r] \ar[d] & U \ar[d]^{(h_1, h_2)} \\ U' \ar[r]^-\Delta & U' \times_S U'. }\] Since \(\overline{u}\) and \(\overline{u}'\) agree over \(S\) with \(\overline{s}\), we see that \(\overline{u}'' = (\overline{u}, \overline{u}')\) is a geometric point of \(U''\). In particular \(U'' \not = \emptyset\). Moreover, since \(U'\) is étale over \(S\), so is the fibre product \(U'\times_S U'\) (see Proposition 03PC). Hence the vertical arrow \((h_1, h_2)\) is étale by Remark 03PP above. Therefore \(U''\) is étale over \(U'\) by base change, and hence also étale over \(S\) (because compositions of étale morphisms are étale). Thus \((U'', \overline{u}'')\) is a solution to the problem.

Lemma

Let \(S\) be a scheme. Let \(\overline{s}\) be a geometric point of \(S\). Let \((U, \overline{u})\) be an étale neighborhood of \(\overline{s}\). Let \(\mathcal{U} = \{\varphi_i : U_i \to U \}_{i\in I}\) be an étale covering. Then there exist \(i \in I\) and \(\overline{u}_i : \overline{s} \to U_i\) such that \(\varphi_i : (U_i, \overline{u}_i) \to (U, \overline{u})\) is a morphism of étale neighborhoods.

Proof

As \(U = \bigcup_{i\in I} \varphi_i(U_i)\), the fibre product \(\overline{s} \times_{\overline{u}, U, \varphi_i} U_i\) is not empty for some \(i\). Then look at the cartesian diagram \[\xymatrix{ \overline{s} \times_{\overline{u}, U, \varphi_i} U_i \ar[d]^{\text{pr}_1} \ar[r]_-{\text{pr}_2} & U_i \ar[d]^{\varphi_i} \\ \Spec(k) = \overline{s} \ar@/^1pc/[u]^\sigma \ar[r]^-{\overline{u}} & U }\] The projection \(\text{pr}_1\) is the base change of an étale morphisms so it is étale, see Proposition 03PC. Therefore, \(\overline{s} \times_{\overline{u}, U, \varphi_i} U_i\) is a disjoint union of finite separable extensions of \(k\), by Proposition 03PC. Here \(\overline{s} = \Spec(k)\). But \(k\) is algebraically closed, so all these extensions are trivial, and there exists a section \(\sigma\) of \(\text{pr}_1\). The composition \(\text{pr}_2 \circ \sigma\) gives a map compatible with \(\overline{u}\).

Definition

Let \(S\) be a scheme. Let \(\mathcal{F}\) be a presheaf on \(S_\etale\). Let \(\overline{s}\) be a geometric point of \(S\). The stalk of \(\mathcal{F}\) at \(\overline{s}\) is \[\mathcal{F}_{\overline{s}} = \colim_{(U, \overline{u})} \mathcal{F}(U)\] where \((U, \overline{u})\) runs over all étale neighborhoods of \(\overline{s}\) in \(S\).

By Lemma 03PQ, this colimit is over a filtered index category, namely the opposite of the category of étale neighbourhoods. In other words, an element of \(\mathcal{F}_{\overline{s}}\) can be thought of as a triple \((U, \overline{u}, \sigma)\) where \(\sigma \in \mathcal{F}(U)\). Two triples \((U, \overline{u}, \sigma)\), \((U', \overline{u}', \sigma')\) define the same element of the stalk if there exists a third étale neighbourhood \((U'', \overline{u}'')\) and morphisms of étale neighbourhoods \(h : (U'', \overline{u}'') \to (U, \overline{u})\), \(h' : (U'', \overline{u}'') \to (U', \overline{u}')\) such that \(h^*\sigma = (h')^*\sigma'\) in \(\mathcal{F}(U'')\). See Categories, Section 04AX.

Lemma

Let \(S\) be a scheme. Let \(\overline{s}\) be a geometric point of \(S\). Consider the functor \[\begin{align*} u : S_\etale & \longrightarrow \textit{Sets}, \\ U & \longmapsto |U_{\overline{s}}| = \{\overline{u} \text{ such that }(U, \overline{u}) \text{ is an \'etale neighbourhood of }\overline{s}\}. \end{align*}\] Here \(|U_{\overline{s}}|\) denotes the underlying set of the geometric fibre. Then \(u\) defines a point \(p\) of the site \(S_\etale\) (Sites, Definition 00Y5) and its associated stalk functor \(\mathcal{F} \mapsto \mathcal{F}_p\) (Sites, Equation 04EH) is the functor \(\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}\) defined above.

Proof

In the proof of Lemma 03PR we have seen that the scheme \(U_{\overline{s}}\) is a disjoint union of schemes isomorphic to \(\overline{s}\). Thus we can also think of \(|U_{\overline{s}}|\) as the set of geometric points of \(U\) lying over \(\overline{s}\), i.e., as the collection of morphisms \(\overline{u} : \overline{s} \to U\) fitting into the diagram of Definition 03PO. From this it follows that \(u(S)\) is a singleton, and that \(u(U \times_V W) = u(U) \times_{u(V)} u(W)\) whenever \(U \to V\) and \(W \to V\) are morphisms in \(S_\etale\). And, given a covering \(\{U_i \to U\}_{i \in I}\) in \(S_\etale\) we see that \(\coprod u(U_i) \to u(U)\) is surjective by Lemma 03PR. Hence Sites, Proposition 00YC applies, so \(p\) is a point of the site \(S_\etale\). Finally, our functor \(\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}\) is given by exactly the same colimit as the functor \(\mathcal{F} \mapsto \mathcal{F}_p\) associated to \(p\) in Sites, Equation 04EH which proves the final assertion.

Remark

Let \(S\) be a scheme and let \(\overline{s} : \Spec(k) \to S\) and \(\overline{s}' : \Spec(k') \to S\) be two geometric points of \(S\). A morphism \(a : \overline{s} \to \overline{s}'\) of geometric points is simply a morphism \(a : \Spec(k) \to \Spec(k')\) such that \(\overline{s}' \circ a = \overline{s}\). Given such a morphism we obtain a functor from the category of étale neighbourhoods of \(\overline{s}'\) to the category of étale neighbourhoods of \(\overline{s}\) by the rule \((U, \overline{u}') \mapsto (U, \overline{u}' \circ a)\). Hence we obtain a canonical map \[\mathcal{F}_{\overline{s}'} = \colim_{(U, \overline{u}')} \mathcal{F}(U) \longrightarrow \colim_{(U, \overline{u})} \mathcal{F}(U) = \mathcal{F}_{\overline{s}}\] from Categories, Lemma 002K. Using the description of elements of stalks as triples this maps the element of \(\mathcal{F}_{\overline{s}'}\) represented by the triple \((U, \overline{u}', \sigma)\) to the element of \(\mathcal{F}_{\overline{s}}\) represented by the triple \((U, \overline{u}' \circ a, \sigma)\). Since the functor above is clearly an equivalence we conclude that this canonical map is an isomorphism of stalk functors.

Let us make sure we have the map of stalks corresponding to \(a\) pointing in the correct direction. Note that the above means, according to Sites, Definition 00YI, that \(a\) defines a morphism \(a : p \to p'\) between the points \(p, p'\) of the site \(S_\etale\) associated to \(\overline{s}, \overline{s}'\) by Lemma 04FM. There are more general morphisms of points (corresponding to specializations of points of \(S\)) which we will describe later, and which will not be isomorphisms, see Section 0GJ2.

Lemma

Let \(S\) be a scheme. Let \(\overline{s}\) be a geometric point of \(S\).

1. The stalk functor \(\textit{PAb}(S_\etale) \to \textit{Ab}\), \(\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}\) is exact.

2. We have \((\mathcal{F}^\#)_{\overline{s}} = \mathcal{F}_{\overline{s}}\) for any presheaf of sets \(\mathcal{F}\) on \(S_\etale\).

3. The functor \(\textit{Ab}(S_\etale) \to \textit{Ab}\), \(\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}\) is exact.

4. Similarly the functors \(\textit{PSh}(S_\etale) \to \textit{Sets}\) and \(\Sh(S_\etale) \to \textit{Sets}\) given by the stalk functor \(\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}\) are exact (see Categories, Definition 0034) and commute with arbitrary colimits.

Proof

Before we indicate how to prove this by direct arguments we note that the result follows from the general material in Modules on Sites, Section 04EM. This is true because \(\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}\) comes from a point of the small étale site of \(S\), see Lemma 04FM. We will only give a direct proof of (1), (2) and (3), and omit a direct proof of (4).

Exactness as a functor on \(\textit{PAb}(S_\etale)\) is formal from the fact that directed colimits commute with all colimits and with finite limits. The identification of the stalks in (2) is via the map \[\kappa : \mathcal{F}_{\overline{s}} \longrightarrow (\mathcal{F}^\#)_{\overline{s}}\] induced by the natural morphism \(\mathcal{F}\to \mathcal{F}^\#\), see Theorem 03NS. We claim that this map is an isomorphism of abelian groups. We will show injectivity and omit the proof of surjectivity.

Let \(\sigma\in \mathcal{F}_{\overline{s}}\). There exists an étale neighborhood \((U, \overline{u})\to (S, \overline{s})\) such that \(\sigma\) is the image of some section \(s \in \mathcal{F}(U)\). If \(\kappa(\sigma) = 0\) in \((\mathcal{F}^\#)_{\overline{s}}\) then there exists a morphism of étale neighborhoods \((U', \overline{u}')\to (U, \overline{u})\) such that \(s|_{U'}\) is zero in \(\mathcal{F}^\#(U')\). It follows there exists an étale covering \(\{U_i'\to U'\}_{i\in I}\) such that \(s|_{U_i'}=0\) in \(\mathcal{F}(U_i')\) for all \(i\). By Lemma 03PR there exist \(i \in I\) and a morphism \(\overline{u}_i': \overline{s} \to U_i'\) such that \((U_i', \overline{u}_i') \to (U', \overline{u}')\to (U, \overline{u})\) are morphisms of étale neighborhoods. Hence \(\sigma = 0\) since \((U_i', \overline{u}_i') \to (U, \overline{u})\) is a morphism of étale neighbourhoods such that we have \(s|_{U'_i}=0\). This proves \(\kappa\) is injective.

To show that the functor \(\textit{Ab}(S_\etale) \to \textit{Ab}\) is exact, consider any short exact sequence in \(\textit{Ab}(S_\etale)\): \(0\to \mathcal{F}\to \mathcal{G}\to \mathcal H \to 0.\) This gives us the exact sequence of presheaves \[0 \to \mathcal{F} \to \mathcal{G} \to \mathcal H \to \mathcal H/^p\mathcal{G} \to 0,\] where \(/^p\) denotes the quotient in \(\textit{PAb}(S_\etale)\). Taking stalks at \(\overline{s}\), we see that \((\mathcal H /^p\mathcal{G})_{\bar{s}} = (\mathcal H /\mathcal{G})_{\bar{s}} = 0\), since the sheafification of \(\mathcal H/^p\mathcal{G}\) is \(0\). Therefore, \[0\to \mathcal{F}_{\overline{s}} \to \mathcal{G}_{\overline{s}} \to \mathcal{H}_{\overline{s}} \to 0 = (\mathcal H/^p\mathcal{G})_{\overline{s}}\] is exact, since taking stalks is exact as a functor from presheaves.

Theorem

Let \(S\) be a scheme. A map \(a : \mathcal{F} \to \mathcal{G}\) of sheaves of sets is injective (resp. surjective) if and only if the map on stalks \(a_{\overline{s}} : \mathcal{F}_{\overline{s}} \to \mathcal{G}_{\overline{s}}\) is injective (resp. surjective) for all geometric points of \(S\). A sequence of abelian sheaves on \(S_\etale\) is exact if and only if it is exact on all stalks at geometric points of \(S\).

Proof

The necessity of exactness on stalks follows from Lemma 03PT. For the converse, it suffices to show that a map of sheaves is surjective (respectively injective) if and only if it is surjective (respectively injective) on all stalks. We prove this in the case of surjectivity, and omit the proof in the case of injectivity.

Let \(\alpha : \mathcal{F} \to \mathcal{G}\) be a map of sheaves such that \(\mathcal{F}_{\overline{s}} \to \mathcal{G}_{\overline{s}}\) is surjective for all geometric points. Fix \(U \in \Ob(S_\etale)\) and \(s \in \mathcal{G}(U)\). For every \(u \in U\) choose some \(\overline{u} \to U\) lying over \(u\) and an étale neighborhood \((V_u , \overline{v}_u) \to (U, \overline{u})\) such that \(s|_{V_u} = \alpha(s_{V_u})\) for some \(s_{V_u} \in \mathcal{F}(V_u)\). This is possible since \(\alpha\) is surjective on stalks. Then \(\{V_u \to U\}_{u \in U}\) is an étale covering on which the restrictions of \(s\) are in the image of the map \(\alpha\). Thus, \(\alpha\) is surjective, see Sites, Section 00WL.

Remarks

On points of the geometric sites.

1. Theorem 03PU says that the family of points of \(S_\etale\) given by the geometric points of \(S\) (Lemma 04FM) is conservative, see Sites, Definition 00YK. In particular \(S_\etale\) has enough points.

2. Suppose \(\mathcal{F}\) is a sheaf on the big étale site [04FP] of \(S\). Let \(T \to S\) be an object of the big étale site of \(S\), and let \(\overline{t}\) be a geometric point of \(T\). Then we define \(\mathcal{F}_{\overline{t}}\) as the stalk of the restriction \(\mathcal{F}|_{T_\etale}\) of \(\mathcal{F}\) to the small étale site of \(T\). In other words, we can define the stalk of \(\mathcal{F}\) at any geometric point of any scheme \(T/S \in \Ob((\Sch/S)_\etale)\).

3. The big étale site of \(S\) also has enough points, by considering all geometric points of all objects of this site, see (04FP).

The following lemma should be skipped on a first reading.

Lemma

Let \(S\) be a scheme.

1. Let \(p\) be a point of the small étale site \(S_\etale\) of \(S\) given by a functor \(u : S_\etale \to \textit{Sets}\). Then there exists a geometric point \(\overline{s}\) of \(S\) such that \(p\) is isomorphic to the point of \(S_\etale\) associated to \(\overline{s}\) in Lemma 04FM.

2. Let \(p : \Sh(pt) \to \Sh(S_\etale)\) be a point of the small étale topos of \(S\). Then \(p\) comes from a geometric point of \(S\), i.e., the stalk functor \(\mathcal{F} \mapsto \mathcal{F}_p\) is isomorphic to a stalk functor as defined in Definition 040R.

Proof

By Sites, Lemma 00YA there is a one to one correspondence between points of the site and points of the associated topos, hence it suffices to prove (1). By Sites, Proposition 00YC the functor \(u\) has the following properties: (a) \(u(S) = \{*\}\), (b) \(u(U \times_V W) = u(U) \times_{u(V)} u(W)\), and (c) if \(\{U_i \to U\}\) is an étale covering, then \(\coprod u(U_i) \to u(U)\) is surjective. In particular, if \(U' \subset U\) is an open subscheme, then \(u(U') \subset u(U)\). Moreover, by Sites, Lemma 00YA we can write \(u(U) = p^{-1}(h_U^\#)\), in other words \(u(U)\) is the stalk of the representable sheaf \(h_U\). If \(U = V \amalg W\), then we see that \(h_U = (h_V \amalg h_W)^\#\) and we get \(u(U) = u(V) \amalg u(W)\) since \(p^{-1}\) is exact.

Consider the restriction of \(u\) to \(S_{Zar}\). By Sites, Examples 00YE and 04EJ there exists a unique point \(s \in S\) such that for \(S' \subset S\) open we have \(u(S') = \{*\}\) if \(s \in S'\) and \(u(S') = \emptyset\) if \(s \not \in S'\). Note that if \(\varphi : U \to S\) is an object of \(S_\etale\) then \(\varphi(U) \subset S\) is open (see Proposition 03PC) and \(\{U \to \varphi(U)\}\) is an étale covering. Hence we conclude that \(u(U) = \emptyset \Leftrightarrow s \not \in \varphi(U)\).

Pick a geometric point \(\overline{s} : \overline{s} \to S\) lying over \(s\), see Definition 03PO for customary abuse of notation. Suppose that \(\varphi : U \to S\) is an object of \(S_\etale\) with \(U\) affine. Note that \(\varphi\) is separated, and that the fibre \(U_s\) of \(\varphi\) over \(s\) is an affine scheme over \(\Spec(\kappa(s))\) which is the spectrum of a finite product of finite separable extensions \(k_i\) of \(\kappa(s)\). Hence we may apply Étale Morphisms, Lemma 04HM to get an étale neighbourhood \((V, \overline{v})\) of \((S, \overline{s})\) such that \[U \times_S V = U_1 \amalg \ldots \amalg U_n \amalg W\] with \(U_i \to V\) an isomorphism and \(W\) having no point lying over \(\overline{v}\). Thus we conclude that \[u(U) \times u(V) = u(U \times_S V) = u(U_1) \amalg \ldots \amalg u(U_n) \amalg u(W)\] and of course also \(u(U_i) = u(V)\). After shrinking \(V\) a bit we can assume that \(V\) has exactly one point lying over \(s\), and hence \(W\) has no point lying over \(s\). By the above this then gives \(u(W) = \emptyset\). Hence we obtain \[u(U) \times u(V) = u(U_1) \amalg \ldots \amalg u(U_n) = \coprod\nolimits_{i = 1, \ldots, n} u(V)\] where the equality signs are compatible with the maps to the set \(u(V)\). Note that \(u(V) \not = \emptyset\) as \(s\) is in the image of \(V \to S\). In particular, we see that in this situation \(u(U)\) is a finite set with \(n\) element, namely the left hand side has fibre \(u(U)\) over any element of \(u(V)\) and the right hand side has fibre of cardinality \(n\).

Consider the limit \[\lim_{(V, \overline{v})} u(V)\] over the category of étale neighbourhoods \((V, \overline{v})\) of \(\overline{s}\). It is clear that we get the same value when taking the limit over the subcategory of \((V, \overline{v})\) with \(V\) affine. By the previous paragraph (applied with the roles of \(V\) and \(U\) switched) we see that in this case \(u(V)\) is always a finite nonempty set. Moreover, the limit is cofiltered, see Lemma 03PQ. Hence by Categories, Section 04AY the limit is nonempty. Pick an element \(x\) from this limit. This means we obtain a \(x_{V, \overline{v}} \in u(V)\) for every étale neighbourhood \((V, \overline{v})\) of \((S, \overline{s})\) such that for every morphism of étale neighbourhoods \(\varphi : (V', \overline{v}') \to (V, \overline{v})\) we have \(u(\varphi)(x_{V', \overline{v}'}) = x_{V, \overline{v}}\).

We will use the choice of \(x\) to construct a functorial bijective map \[c : |U_{\overline{s}}| \longrightarrow u(U)\] for \(U \in \Ob(S_\etale)\) which will conclude the proof. See Lemma 04FM and its proof for a description of \(|U_{\overline{s}}|\). First we claim that it suffices to construct the map for \(U\) affine. We omit the proof of this claim. Assume \(U \to S\) in \(S_\etale\) with \(U\) affine, and let \(\overline{u} : \overline{s} \to U\) be an element of \(|U_{\overline{s}}|\). Choose a \((V, \overline{v})\) such that \(U \times_S V\) decomposes as in the third paragraph of the proof. Then the pair \((\overline{u}, \overline{v})\) gives a geometric point of \(U \times_S V\) lying over \(\overline{v}\) and determines one of the components \(U_i\) of \(U \times_S V\). More precisely, there exists a section \(\sigma : V \to U \times_S V\) of the projection \(\text{pr}_U\) such that \((\overline{u}, \overline{v}) = \sigma \circ \overline{v}\). Set \(c(\overline{u}) = u(\text{pr}_U)(u(\sigma)(x_{V, \overline{v}})) \in u(U)\). We have to check this is independent of the choice of \((V, \overline{v})\). By Lemma 03PQ the category of étale neighbourhoods is cofiltered. Hence it suffice to show that given a morphism of étale neighbourhood \(\varphi : (V', \overline{v}') \to (V, \overline{v})\) and a choice of a section \(\sigma' : V' \to U \times_S V'\) of the projection such that \((\overline{u}, \overline{v'}) = \sigma' \circ \overline{v}'\) we have \(u(\sigma')(x_{V', \overline{v}'}) = u(\sigma)(x_{V, \overline{v}})\). Consider the diagram \[\xymatrix{ V' \ar[d]^{\sigma'} \ar[r]_\varphi & V \ar[d]^\sigma \\ U \times_S V' \ar[r]^{1 \times \varphi} & U \times_S V }\] Now, it may not be the case that this diagram commutes. The reason is that the schemes \(V'\) and \(V\) may not be connected, and hence the decompositions used to construct \(\sigma'\) and \(\sigma\) above may not be unique. But we do know that \(\sigma \circ \varphi \circ \overline{v}' = (1 \times \varphi) \circ \sigma' \circ \overline{v}'\) by construction. Hence, since \(U \times_S V\) is étale over \(S\), there exists an open neighbourhood \(V'' \subset V'\) of \(\overline{v'}\) such that the diagram does commute when restricted to \(V''\), see Morphisms, Lemma 04HB. This means we may extend the diagram above to \[\xymatrix{ V'' \ar[r] \ar[d]^{\sigma'|_{V''}} & V' \ar[d]^{\sigma'} \ar[r]_\varphi & V \ar[d]^\sigma \\ U \times_S V'' \ar[r] & U \times_S V' \ar[r]^{1 \times \varphi} & U \times_S V }\] such that the left square and the outer rectangle commute. Since \(u\) is a functor this implies that \(x_{V'', \overline{v}'}\) maps to the same element in \(u(U \times_S V)\) no matter which route we take through the diagram. On the other hand, it maps to the elements \(x_{V', \overline{v}'}\) and \(x_{V, \overline{v}}\) in \(u(V')\) and \(u(V)\). This implies the desired equality \(u(\sigma')(x_{V', \overline{v}'}) = u(\sigma)(x_{V, \overline{v}})\).

In a similar manner one proves that the construction \(c : |U_{\overline{s}}| \to u(U)\) is functorial in \(U\); details omitted. And finally, by the results of the third paragraph it is clear that the map \(c\) is bijective which ends the proof of the lemma.

Points in other topologies

In this section we briefly discuss the existence of points for some sites other than the étale site of a scheme. We refer to Sites, Section 00YJ and Topologies, Section 020M ff for the terminology used in this section. All of the geometric sites have enough points.

Lemma

Let \(S\) be a scheme. All of the following sites have enough points \(S_{affine, Zar}\), \(S_{Zar}\), \(S_{affine, \etale}\), \(S_\etale\), \((\Sch/S)_{Zar}\), \((\textit{Aff}/S)_{Zar}\), \((\Sch/S)_\etale\), \((\textit{Aff}/S)_\etale\), \((\Sch/S)_{smooth}\), \((\textit{Aff}/S)_{smooth}\), \((\Sch/S)_{syntomic}\), \((\textit{Aff}/S)_{syntomic}\), \((\Sch/S)_{fppf}\), and \((\textit{Aff}/S)_{fppf}\).

Proof

For each of the big sites the associated topos is equivalent to the topos defined by the site \((\textit{Aff}/S)_\tau\), see Topologies, Lemmas 020W, 021E, 06VC, 06VD, and 021V. The result for the sites \((\textit{Aff}/S)_\tau\) follows immediately from Deligne’s result Sites, Lemma 0DW0.

The result for \(S_{Zar}\) is clear. The result for \(S_{affine, Zar}\) follows from Deligne’s result. The result for \(S_\etale\) either follows from (the proof of) Theorem 03PU or from Topologies, Lemma 04HR and Deligne’s result applied to \(S_{affine, \etale}\).

The lemma above guarantees the existence of points, but it doesn’t tell us what these points look like. We can explicitly construct some points as follows. Suppose \(\overline{s} : \Spec(k) \to S\) is a geometric point with \(k\) algebraically closed. Consider the functor \[u : (\Sch/S)_{fppf} \longrightarrow \textit{Sets}, \quad u(U) = U(k) = \Mor_S(\Spec(k), U).\] Note that \(U \mapsto U(k)\) commutes with finite limits as \(S(k) = \{\overline{s}\}\) and \((U_1 \times_U U_2)(k) = U_1(k) \times_{U(k)} U_2(k)\). Moreover, if \(\{U_i \to U\}\) is an fppf covering, then \(\coprod U_i(k) \to U(k)\) is surjective. By Sites, Proposition 00YC we see that \(u\) defines a point \(p\) of \((\Sch/S)_{fppf}\) with stalks \[\mathcal{F}_p = \colim_{(U, x)} \mathcal{F}(U)\] where the colimit is over pairs \(U \to S\), \(x \in U(k)\) as usual. But... this category has an initial object, namely \((\Spec(k), \text{id})\), hence we see that \[\mathcal{F}_p = \mathcal{F}(\Spec(k))\] which isn’t terribly interesting! In fact, in general these points won’t form a conservative family of points. A more interesting type of point is described in the following remark.

Remark

Let \(S = \Spec(A)\) be an affine scheme. Let \((p, u)\) be a point of the site \((\textit{Aff}/S)_{fppf}\), see Sites, Sections 00Y3 and 05UZ. Let \(B = \mathcal{O}_p\) be the stalk of the structure sheaf at the point \(p\). Recall that \[B = \colim_{(U, x)} \mathcal{O}(U) = \colim_{(\Spec(C), x_C)} C\] where \(x_C \in u(\Spec(C))\). It can happen that \(\Spec(B)\) is an object of \((\textit{Aff}/S)_{fppf}\) and that there is an element \(x_B \in u(\Spec(B))\) mapping to the compatible system \(x_C\). In this case the system of neighbourhoods has an initial object and it follows that \(\mathcal{F}_p = \mathcal{F}(\Spec(B))\) for any sheaf \(\mathcal{F}\) on \((\textit{Aff}/S)_{fppf}\). It is straightforward to see that if \(\mathcal{F} \mapsto \mathcal{F}(\Spec(B))\) defines a point of \(\Sh((\textit{Aff}/S)_{fppf})\), then \(B\) has to be a local \(A\)-algebra such that for every faithfully flat, finitely presented ring map \(B \to B'\) there is a section \(B' \to B\). Conversely, for any such \(A\)-algebra \(B\) the functor \(\mathcal{F} \mapsto \mathcal{F}(\Spec(B))\) is the stalk functor of a point. Details omitted. It is not clear what a general point of the site \((\textit{Aff}/S)_{fppf}\) looks like.

Supports of abelian sheaves

First we talk about supports of local sections.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{F}\) be a subsheaf of the final object of the étale topos of \(S\) (see Sites, Example 00W3). Then there exists a unique open \(W \subset S\) such that \(\mathcal{F} = h_W\).

Proof

The condition means that \(\mathcal{F}(U)\) is a singleton or empty for all \(\varphi : U \to S\) in \(\Ob(S_\etale)\). In particular local sections always glue. If \(\mathcal{F}(U) \not = \emptyset\), then \(\mathcal{F}(\varphi(U)) \not = \emptyset\) because \(\{\varphi : U \to \varphi(U)\}\) is a covering. Hence we can take \(W = \bigcup_{\varphi : U \to S, \mathcal{F}(U) \not = \emptyset} \varphi(U)\).

Lemma

Let \(S\) be a scheme. Let \(\mathcal{F}\) be an abelian sheaf on \(S_\etale\). Let \(\sigma \in \mathcal{F}(U)\) be a local section. There exists an open subset \(W \subset U\) such that

1. \(W \subset U\) is the largest Zariski open subset of \(U\) such that \(\sigma|_W = 0\),

2. for every \(\varphi : V \to U\) in \(S_\etale\) we have \[\sigma|_V = 0 \Leftrightarrow \varphi(V) \subset W,\]

3. for every geometric point \(\overline{u}\) of \(U\) we have \[(U, \overline{u}, \sigma) = 0\text{ in }\mathcal{F}_{\overline{s}} \Leftrightarrow \overline{u} \in W\] where \(\overline{s} = (U \to S) \circ \overline{u}\).

Proof

Since \(\mathcal{F}\) is a sheaf in the étale topology the restriction of \(\mathcal{F}\) to \(U_{Zar}\) is a sheaf on \(U\) in the Zariski topology. Hence there exists a Zariski open \(W\) having property (1), see Modules, Lemma 01AU. Let \(\varphi : V \to U\) be an arrow of \(S_\etale\). Note that \(\varphi(V) \subset U\) is an open subset and that \(\{V \to \varphi(V)\}\) is an étale covering. Hence if \(\sigma|_V = 0\), then by the sheaf condition for \(\mathcal{F}\) we see that \(\sigma|_{\varphi(V)} = 0\). This proves (2). To prove (3) we have to show that if \((U, \overline{u}, \sigma)\) defines the zero element of \(\mathcal{F}_{\overline{s}}\), then \(\overline{u} \in W\). This is true because the assumption means there exists a morphism of étale neighbourhoods \((V, \overline{v}) \to (U, \overline{u})\) such that \(\sigma|_V = 0\). Hence by (2) we see that \(V \to U\) maps into \(W\), and hence \(\overline{u} \in W\).

Let \(S\) be a scheme. Let \(s \in S\). Let \(\mathcal{F}\) be a sheaf on \(S_\etale\). By Remark 04FN the isomorphism class of the stalk of the sheaf \(\mathcal{F}\) at a geometric points lying over \(s\) is well defined.

Definition

Let \(S\) be a scheme. Let \(\mathcal{F}\) be an abelian sheaf on \(S_\etale\).

1. The support of \(\mathcal{F}\) is the set of points \(s \in S\) such that \(\mathcal{F}_{\overline{s}} \not = 0\) for any (some) geometric point \(\overline{s}\) lying over \(s\).

2. Let \(\sigma \in \mathcal{F}(U)\) be a section. The support of \(\sigma\) is the closed subset \(U \setminus W\), where \(W \subset U\) is the largest open subset of \(U\) on which \(\sigma\) restricts to zero (see Lemma 04FR).

In general the support of an abelian sheaf is not closed. For example, suppose that \(S = \mathbf{A}^1_{\mathbf{C}}\). Let \(i_t : \Spec(\mathbf{C}) \to S\) be the inclusion of the point \(t \in \mathbf{C}\). We will see later that \(\mathcal{F}_t = i_{t, *}(\mathbf{Z}/2\mathbf{Z})\) is an abelian sheaf whose support is exactly \(\{t\}\), see Section 04E1. Then \[\bigoplus\nolimits_{n \in \mathbf{N}} \mathcal{F}_n\] is an abelian sheaf with support \(\{1, 2, 3, \ldots\} \subset S\). This is true because taking stalks commutes with colimits, see Lemma 03PT. Thus an example of an abelian sheaf whose support is not closed. Here are some basic facts on supports of sheaves and sections.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{F}\) be an abelian sheaf on \(S_\etale\). Let \(U \in \Ob(S_\etale)\) and \(\sigma \in \mathcal{F}(U)\).

1. The support of \(\sigma\) is closed in \(U\).

2. The support of \(\sigma + \sigma'\) is contained in the union of the supports of \(\sigma, \sigma' \in \mathcal{F}(U)\).

3. If \(\varphi : \mathcal{F} \to \mathcal{G}\) is a map of abelian sheaves on \(S_\etale\), then the support of \(\varphi(\sigma)\) is contained in the support of \(\sigma \in \mathcal{F}(U)\).

4. The support of \(\mathcal{F}\) is the union of the images of the supports of all local sections of \(\mathcal{F}\).

5. If \(\mathcal{F} \to \mathcal{G}\) is surjective then the support of \(\mathcal{G}\) is a subset of the support of \(\mathcal{F}\).

6. If \(\mathcal{F} \to \mathcal{G}\) is injective then the support of \(\mathcal{F}\) is a subset of the support of \(\mathcal{G}\).

Proof

Part (1) holds by definition. Parts (2) and (3) hold because they holds for the restriction of \(\mathcal{F}\) and \(\mathcal{G}\) to \(U_{Zar}\), see Modules, Lemma 01AU. Part (4) is a direct consequence of Lemma 04FR part (3). Parts (5) and (6) follow from the other parts.

Lemma

The support of a sheaf of rings on \(S_\etale\) is closed.

Proof

This is true because (according to our conventions) a ring is \(0\) if and only if \(1 = 0\), and hence the support of a sheaf of rings is the support of the unit section.

Henselian rings

We begin by stating a theorem which has already been used many times in the Stacks project. There are many versions of this result; here we just state the algebraic version.

Theorem

Let \(A\to B\) be finite type ring map and \(\mathfrak p \subset A\) a prime ideal. Then there exist an étale ring map \(A \to A'\) and a prime \(\mathfrak p' \subset A'\) lying over \(\mathfrak p\) such that

1. \(\kappa(\mathfrak p) = \kappa(\mathfrak p')\),

2. \(B \otimes_A A' = B_1\times \ldots \times B_r \times C\),

3. \(A'\to B_i\) is finite and there exists a unique prime \(q_i\subset B_i\) lying over \(\mathfrak p'\), and

4. all irreducible components of the fibre \(\Spec(C \otimes_{A'} \kappa(\mathfrak p'))\) of \(C\) over \(\mathfrak p'\) have dimension at least 1.

Proof

See Algebra, Lemma 00UK, or see [EGA4, Théorème 18.12.1]. For a slew of versions in terms of morphisms of schemes, see More on Morphisms, Section 04HF.

Recall Hensel’s lemma. There are many versions of this lemma. Here are two:

1. if \(f\in \mathbf{Z}_p[T]\) monic and \(f \bmod p = g_0 h_0\) with \(gcd(g_0, h_0) = 1\) then \(f\) factors as \(f = gh\) with \(\bar g = g_0\) and \(\bar h = h_0\),

2. if \(f \in \mathbf{Z}_p[T]\), monic \(a_0 \in \mathbf{F}_p\), \(\bar f(a_0) =0\) but \(\bar f'(a_0) \neq 0\) then there exists \(a \in \mathbf{Z}_p\) with \(f(a) = 0\) and \(\bar a = a_0\).

Both versions are true (we will see this later). The first version asks for lifts of factorizations into coprime parts, and the second version asks for lifts of simple roots modulo the maximal ideal. It turns out that requiring these conditions for a general local ring are equivalent, and are equivalent to many other conditions. We use the root lifting property as the definition of a henselian local ring as it is often the easiest one to check.

Definition

(See Algebra, Definition 04GF.) A local ring \((R, \mathfrak m, \kappa)\) is called henselian if for all \(f \in R[T]\) monic, for all \(a_0 \in \kappa\) such that \(\bar f(a_0) = 0\) and \(\bar f'(a_0) \neq 0\), there exists an \(a \in R\) such that \(f(a) = 0\) and \(a \bmod \mathfrak m = a_0\).

A good example of henselian local rings to keep in mind is complete local rings. Recall (Algebra, Definition 0324) that a complete local ring is a local ring \((R, \mathfrak m)\) such that \(R \cong \lim_n R/\mathfrak m^n\), i.e., it is complete and separated for the \(\mathfrak m\)-adic topology.

Theorem

Complete local rings are henselian.

Proof

Newton’s method. See Algebra, Lemma 04GM.

Theorem

Let \((R, \mathfrak m, \kappa)\) be a local ring. The following are equivalent:

1. \(R\) is henselian,

2. for any \(f\in R[T]\) and any factorization \(\bar f = g_0 h_0\) in \(\kappa[T]\) with \(\gcd(g_0, h_0)=1\), there exists a factorization \(f = gh\) in \(R[T]\) with \(\bar g = g_0\) and \(\bar h = h_0\),

3. any finite \(R\)-algebra \(S\) is isomorphic to a finite product of local rings finite over \(R\),

4. any finite type \(R\)-algebra \(A\) is isomorphic to a product \(A \cong A' \times C\) where \(A' \cong A_1 \times \ldots \times A_r\) is a product of finite local \(R\)-algebras and all the irreducible components of \(C \otimes_R \kappa\) have dimension at least 1,

5. if \(A\) is an étale \(R\)-algebra and \(\mathfrak n\) is a maximal ideal of \(A\) lying over \(\mathfrak m\) such that \(\kappa \cong A/\mathfrak n\), then there exists an isomorphism \(\varphi : A \cong R \times A'\) such that \(\varphi(\mathfrak n) = \mathfrak m \times A' \subset R \times A'\).

Proof

This is just a subset of the results from Algebra, Lemma 04GG. Note that part (5) above corresponds to part (8) of Algebra, Lemma 04GG but is formulated slightly differently.

Lemma

If \(R\) is henselian and \(A\) is a finite \(R\)-algebra, then \(A\) is a finite product of henselian local rings.

Proof

See Algebra, Lemma 04GH.

Definition

A local ring \(R\) is called strictly henselian if it is henselian and its residue field is separably closed.

Example

In the case \(R = \mathbf{C}[[t]]\), the étale \(R\)-algebras are finite products of the trivial extension \(R \to R\) and the extensions \(R \to R[X, X^{-1}]/(X^n-t)\). The latter ones factor through the open \(D(t) \subset \Spec(R)\), so any étale covering can be refined by the covering \(\{\text{id} : \Spec(R) \to \Spec(R)\}\). We will see below that this is a somewhat general fact on étale coverings of spectra of henselian rings. This will show that higher étale cohomology of the spectrum of a strictly henselian ring is zero.

Theorem

Let \((R, \mathfrak m, \kappa)\) be a local ring and \(\kappa\subset\kappa^{sep}\) a separable algebraic closure. There exist canonical flat local ring maps \(R \to R^h \to R^{sh}\) where

1. \(R^h\), \(R^{sh}\) are filtered colimits of étale \(R\)-algebras,

2. \(R^h\) is henselian, \(R^{sh}\) is strictly henselian,

3. \(\mathfrak m R^h\) (resp. \(\mathfrak m R^{sh}\)) is the maximal ideal of \(R^h\) (resp. \(R^{sh}\)), and

4. \(\kappa = R^h/\mathfrak m R^h\), and \(\kappa^{sep} = R^{sh}/\mathfrak m R^{sh}\) as extensions of \(\kappa\).

Proof

The structure of \(R^h\) and \(R^{sh}\) is described in Algebra, Lemmas 04GN and 04GP.

The rings constructed in Theorem 03QL are called respectively the henselization and the strict henselization of the local ring \(R\), see Algebra, Definition 04GQ. Many of the properties of \(R\) are reflected in its (strict) henselization, see More on Algebra, Section 07QL.

Stalks of the structure sheaf

In this section we identify the stalk of the structure sheaf at a geometric point with the strict henselization of the local ring at the corresponding “usual” point.

Lemma

Let \(S\) be a scheme. Let \(\overline{s}\) be a geometric point of \(S\) lying over \(s \in S\). Let \(\kappa = \kappa(s)\) and let \(\kappa \subset \kappa^{sep} \subset \kappa(\overline{s})\) denote the separable algebraic closure of \(\kappa\) in \(\kappa(\overline{s})\). Then there is a canonical identification \[(\mathcal{O}_{S, s})^{sh} \cong (\mathcal{O}_S)_{\overline{s}}\] where the left hand side is the strict henselization of the local ring \(\mathcal{O}_{S, s}\) as described in Theorem 03QL and right hand side is the stalk of the structure sheaf \(\mathcal{O}_S\) on \(S_\etale\) at the geometric point \(\overline{s}\).

Proof

Let \(\Spec(A) \subset S\) be an affine neighbourhood of \(s\). Let \(\mathfrak p \subset A\) be the prime ideal corresponding to \(s\). With these choices we have canonical isomorphisms \(\mathcal{O}_{S, s} = A_{\mathfrak p}\) and \(\kappa(s) = \kappa(\mathfrak p)\). Thus we have \(\kappa(\mathfrak p) \subset \kappa^{sep} \subset \kappa(\overline{s})\). Recall that \[(\mathcal{O}_S)_{\overline{s}} = \colim_{(U, \overline{u})} \mathcal{O}(U)\] where the limit is over the étale neighbourhoods of \((S, \overline{s})\). A cofinal system is given by those étale neighbourhoods \((U, \overline{u})\) such that \(U\) is affine and \(U \to S\) factors through \(\Spec(A)\). In other words, we see that \[(\mathcal{O}_S)_{\overline{s}} = \colim_{(B, \mathfrak q, \phi)} B\] where the colimit is over étale \(A\)-algebras \(B\) endowed with a prime \(\mathfrak q\) lying over \(\mathfrak p\) and a \(\kappa(\mathfrak p)\)-algebra map \(\phi : \kappa(\mathfrak q) \to \kappa(\overline{s})\). Note that since \(\kappa(\mathfrak q)\) is finite separable over \(\kappa(\mathfrak p)\) the image of \(\phi\) is contained in \(\kappa^{sep}\). Via these translations the result of the lemma is equivalent to the result of Algebra, Lemma 04GW.

Definition

Let \(S\) be a scheme. Let \(\overline{s}\) be a geometric point of \(S\) lying over the point \(s \in S\).

1. The étale local ring of \(S\) at \(\overline{s}\) is the stalk of the structure sheaf \(\mathcal{O}_S\) on \(S_\etale\) at \(\overline{s}\). We sometimes call this the strict henselization of \(\mathcal{O}_{S, s}\) relative to the geometric point \(\overline{s}\). Notation used: \(\mathcal{O}_{S, \overline{s}}^{sh}\).

2. The henselization of \(\mathcal{O}_{S, s}\) is the henselization of the local ring of \(S\) at \(s\). See Algebra, Definition 04GQ, and Theorem 03QL. Notation: \(\mathcal{O}_{S, s}^h\).

3. The strict henselization of \(S\) at \(\overline{s}\) is the scheme \(\Spec(\mathcal{O}_{S, \overline{s}}^{sh})\).

4. The henselization of \(S\) at \(s\) is the scheme \(\Spec(\mathcal{O}_{S, s}^h)\).

Let \(f : T \to S\) be a morphism of schemes. Let \(\overline{t}\) be a geometric point of \(T\) with image \(\overline{s}\) in \(S\). Let \(t \in T\) and \(s \in S\) be their images. Then we obtain a canonical commutative diagram \[\xymatrix{ \Spec(\mathcal{O}^{sh}_{T, \overline{t}}) \ar[r] \ar[d] & \Spec(\mathcal{O}^h_{T, t}) \ar[r] \ar[d] & T \ar[d]^f \\ \Spec(\mathcal{O}^{sh}_{S, \overline{s}}) \ar[r] & \Spec(\mathcal{O}^h_{S, s}) \ar[r] & S }\] of henselizations and strict henselizations of \(T\) and \(S\). You can prove this by choosing affine neighbourhoods of \(t\) and \(s\) and using the functoriality of (strict) henselizations given by Algebra, Lemmas 08HU and 08HV.

Lemma

Let \(S\) be a scheme. Let \(s \in S\). Then we have \[\mathcal{O}_{S, s}^h = \colim_{(U, u)} \mathcal{O}(U)\] where the colimit is over the filtered category of étale neighbourhoods \((U, u)\) of \((S, s)\) such that \(\kappa(s) = \kappa(u)\).

Proof

This lemma is a copy of More on Morphisms, Lemma 05KS.

Remark

Let \(S\) be a scheme. Let \(s \in S\). If \(S\) is locally Noetherian then \(\mathcal{O}_{S, s}^h\) is also Noetherian and it has the same completion: \[\widehat{\mathcal{O}_{S, s}} \cong \widehat{\mathcal{O}_{S, s}^h}.\] In particular, \(\mathcal{O}_{S, s} \subset \mathcal{O}_{S, s}^h \subset \widehat{\mathcal{O}_{S, s}}\). The henselization of \(\mathcal{O}_{S, s}\) is in general much smaller than its completion and inherits many of its properties. For example, if \(\mathcal{O}_{S, s}\) is reduced, then so is \(\mathcal{O}_{S, s}^h\), but this is not true for the completion in general. Insert future references here.

Lemma

Let \(S\) be a scheme. The small étale site \(S_\etale\) endowed with its structure sheaf \(\mathcal{O}_S\) is a locally ringed site, see Modules on Sites, Definition 04EU.

Proof

This follows because the stalks \((\mathcal{O}_S)_{\overline{s}} = \mathcal{O}^{sh}_{S, \overline{s}}\) are local, and because \(S_\etale\) has enough points, see Lemma 04HX, Theorem 03PU, and Remarks 040S. See Modules on Sites, Lemmas 04ET and 05D8 for the fact that this implies the small étale site is locally ringed.

Functoriality of small étale topos

So far we haven’t yet discussed the functoriality of the étale site, in other words what happens when given a morphism of schemes. A precise formal discussion can be found in Topologies, Section 0214. In this and the next sections we discuss this material briefly specifically in the setting of small étale sites.

Let \(f : X \to Y\) be a morphism of schemes. We obtain a functor [04I1]\[\begin{equation} u : Y_\etale \longrightarrow X_\etale, \quad V/Y \longmapsto X \times_Y V/X. \end{equation}\] This functor has the following important properties

1. \(u(\text{final object}) = \text{final object}\),

2. \(u\) preserves fibre products,

3. if \(\{V_j \to V\}\) is a covering in \(Y_\etale\), then \(\{u(V_j) \to u(V)\}\) is a covering in \(X_\etale\).

Each of these is easy to check (omitted). As a consequence we obtain what is called a morphism of sites \[f_{small} : X_\etale \longrightarrow Y_\etale,\] see Sites, Definition 00X1 and Sites, Proposition 00X6. It is not necessary to know about the abstract notion in detail in order to work with étale sheaves and étale cohomology. It usually suffices to know that there are functors \(f_{small, *}\) (pushforward) and \(f_{small}^{-1}\) (pullback) on étale sheaves, and to know some of their simple properties. We will discuss these properties in the next sections, but we will sometimes refer to the more abstract material for proofs since that is often the natural setting to prove them.

Direct images

Let us define the pushforward of a presheaf.

Definition

Let \(f: X\to Y\) be a morphism of schemes. Let \(\mathcal{F}\) a presheaf of sets on \(X_\etale\). The direct image, or pushforward of \(\mathcal{F}\) (under \(f\)) is \[f_*\mathcal{F} : Y_\etale^{opp} \longrightarrow \textit{Sets}, \quad (V/Y) \longmapsto \mathcal{F}(X \times_Y V/X).\] We sometimes write \(f_* = f_{small, *}\) to distinguish from other direct image functors (such as usual Zariski pushforward or \(f_{big, *}\)).

This is a well-defined étale presheaf since the base change of an étale morphism is again étale. A more categorical way of saying this is that \(f_*\mathcal{F}\) is the composition of functors \(\mathcal{F} \circ u\) where \(u\) is as in Equation (04I1). This makes it clear that the construction is functorial in the presheaf \(\mathcal{F}\) and hence we obtain a functor \[f_* = f_{small, *} : \textit{PSh}(X_\etale) \longrightarrow \textit{PSh}(Y_\etale)\] Note that if \(\mathcal{F}\) is a presheaf of abelian groups, then \(f_*\mathcal{F}\) is also a presheaf of abelian groups and we obtain \[f_* = f_{small, *} : \textit{PAb}(X_\etale) \longrightarrow \textit{PAb}(Y_\etale)\] as before (i.e., defined by exactly the same rule).

Remark

We claim that the direct image of a sheaf is a sheaf. Namely, if \(\{V_j \to V\}\) is an étale covering in \(Y_\etale\) then \(\{X \times_Y V_j \to X \times_Y V\}\) is an étale covering in \(X_\etale\). Hence the sheaf condition for \(\mathcal{F}\) with respect to \(\{X \times_Y V_i \to X \times_Y V\}\) is equivalent to the sheaf condition for \(f_*\mathcal{F}\) with respect to \(\{V_i \to V\}\). Thus if \(\mathcal{F}\) is a sheaf, so is \(f_*\mathcal{F}\).

Definition

Let \(f: X\to Y\) be a morphism of schemes. Let \(\mathcal{F}\) a sheaf of sets on \(X_\etale\). The direct image, or pushforward of \(\mathcal{F}\) (under \(f\)) is \[f_*\mathcal{F} : Y_\etale^{opp} \longrightarrow \textit{Sets}, \quad (V/Y) \longmapsto \mathcal{F}(X \times_Y V/X)\] which is a sheaf by Remark 03PX. We sometimes write \(f_* = f_{small, *}\) to distinguish from other direct image functors (such as usual Zariski pushforward or \(f_{big, *}\)).

The exact same discussion as above applies and we obtain functors \[f_* = f_{small, *} : \Sh(X_\etale) \longrightarrow \Sh(Y_\etale)\] and \[f_* = f_{small, *} : \textit{Ab}(X_\etale) \longrightarrow \textit{Ab}(Y_\etale)\] called direct image again.

The functor \(f_*\) on abelian sheaves is left exact. (See Homology, Section 010M for what it means for a functor between abelian categories to be left exact.) Namely, if \(0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3\) is exact on \(X_\etale\), then for every \(U/X \in \Ob(X_\etale)\) the sequence of abelian groups \(0 \to \mathcal{F}_1(U) \to \mathcal{F}_2(U) \to \mathcal{F}_3(U)\) is exact. Hence for every \(V/Y \in \Ob(Y_\etale)\) the sequence of abelian groups \(0 \to f_*\mathcal{F}_1(V) \to f_*\mathcal{F}_2(V) \to f_*\mathcal{F}_3(V)\) is exact, because this is the previous sequence with \(U = X \times_Y V\).

Definition

Let \(f: X \to Y\) be a morphism of schemes. The right derived functors \(\{R^pf_*\}_{p \geq 1}\) of \(f_* : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)\) are called higher direct images.

The higher direct images and their derived category variants are discussed in more detail in (insert future reference here).

Inverse image

In this section we briefly discuss pullback of sheaves on the small étale sites. The precise construction of this is in Topologies, Section 0214.

Definition

Let \(f: X\to Y\) be a morphism of schemes. The inverse image, or pullback² functors are the functors \[f^{-1} = f_{small}^{-1} : \Sh(Y_\etale) \longrightarrow \Sh(X_\etale)\] and \[f^{-1} = f_{small}^{-1} : \textit{Ab}(Y_\etale) \longrightarrow \textit{Ab}(X_\etale)\] which are left adjoint to \(f_* = f_{small, *}\). Thus \(f^{-1}\) is characterized by the fact that \[\Hom_{{\Sh(X_\etale)}} (f^{-1}\mathcal{G}, \mathcal{F}) = \Hom_{\Sh(Y_\etale)} (\mathcal{G}, f_*\mathcal{F})\] functorially, for any \(\mathcal{F} \in \Sh(X_\etale)\) and \(\mathcal{G} \in \Sh(Y_\etale)\). We similarly have \[\Hom_{{\textit{Ab}(X_\etale)}} (f^{-1}\mathcal{G}, \mathcal{F}) = \Hom_{\textit{Ab}(Y_\etale)} (\mathcal{G}, f_*\mathcal{F})\] for \(\mathcal{F} \in \textit{Ab}(X_\etale)\) and \(\mathcal{G} \in \textit{Ab}(Y_\etale)\).

It is not trivial that such an adjoint exists. On the other hand, it exists in a fairly general setting, see Remark 03Q2 below. The general machinery shows that \(f^{-1}\mathcal{G}\) is the sheaf associated to the presheaf [04I3]\[\begin{equation} U/X \longmapsto \colim_{U \to X \times_Y V} \mathcal{G}(V/Y) \end{equation}\] where the colimit is over the category of pairs \((V/Y, \varphi : U/X \to X \times_Y V/X)\). To see this apply Sites, Proposition 00X6 to the functor \(u\) of Equation (04I1) and use the description of \(u_s = (u_p\ )^\#\) in Sites, Sections 00WU and 00VC. We will occasionally use this formula for the pullback in order to prove some of its basic properties.

Lemma

Let \(f : X \to Y\) be a morphism of schemes.

1. The functor \(f^{-1} : \textit{Ab}(Y_\etale) \to \textit{Ab}(X_\etale)\) is exact.

2. The functor \(f^{-1} : \Sh(Y_\etale) \to \Sh(X_\etale)\) is exact, i.e., it commutes with finite limits and colimits, see Categories, Definition 0034.

3. Let \(\overline{x} \to X\) be a geometric point. Let \(\mathcal{G}\) be a sheaf on \(Y_\etale\). Then there is a canonical identification \[(f^{-1}\mathcal{G})_{\overline{x}} = \mathcal{G}_{\overline{y}}.\] where \(\overline{y} = f \circ \overline{x}\).

4. For any \(V \to Y\) étale we have \(f^{-1}h_V = h_{X \times_Y V}\).

Proof

The exactness of \(f^{-1}\) on sheaves of sets is a consequence of Sites, Proposition 00X6 applied to our functor \(u\) of Equation (04I1). In fact the exactness of pullback is part of the definition of a morphism of topoi (or sites if you like). Thus we see (2) holds. It implies part (1) since given an abelian sheaf \(\mathcal{G}\) on \(Y_\etale\) the underlying sheaf of sets of \(f^{-1}\mathcal{F}\) is the same as \(f^{-1}\) of the underlying sheaf of sets of \(\mathcal{F}\), see Sites, Section 00YR. See also Modules on Sites, Lemma 04JC. In the literature (1) and (2) are sometimes deduced from (3) via Theorem 03PU.

Part (3) is a general fact about stalks of pullbacks, see Sites, Lemma 05V1. We will also prove (3) directly as follows. Note that by Lemma 03PT taking stalks commutes with sheafification. Now recall that \(f^{-1}\mathcal{G}\) is the sheaf associated to the presheaf \[U \longrightarrow \colim_{U \to X \times_Y V} \mathcal{G}(V),\] see Equation (04I3). Thus we have \[\begin{align*} (f^{-1}\mathcal{G})_{\overline{x}} & = \colim_{(U, \overline{u})} f^{-1}\mathcal{G}(U) \\ & = \colim_{(U, \overline{u})} \colim_{a : U \to X \times_Y V} \mathcal{G}(V) \\ & = \colim_{(V, \overline{v})} \mathcal{G}(V) \\ & = \mathcal{G}_{\overline{y}} \end{align*}\] in the third equality the pair \((U, \overline{u})\) and the map \(a : U \to X \times_Y V\) corresponds to the pair \((V, a \circ \overline{u})\).

Part (4) can be proved in a similar manner by identifying the colimits which define \(f^{-1}h_V\). Or you can use Yoneda’s lemma (Categories, Lemma 001P) and the functorial equalities \[\Mor_{\Sh(X_\etale)}(f^{-1}h_V, \mathcal{F}) = \Mor_{\Sh(Y_\etale)}(h_V, f_*\mathcal{F}) = f_*\mathcal{F}(V) = \mathcal{F}(X \times_Y V)\] combined with the fact that representable presheaves are sheaves. See also Sites, Lemma 04D3 for a completely general result.

The pair of functors \((f_*, f^{-1})\) define a morphism of small étale topoi \[f_{small} : \Sh(X_\etale) \longrightarrow \Sh(Y_\etale)\] Many generalities on cohomology of sheaves hold for topoi and morphisms of topoi. We will try to point out when results are general and when they are specific to the étale topos.

Remark

More generally, let \(\mathcal{C}_1, \mathcal{C}_2\) be sites, and assume they have final objects and fibre products. Let \(u: \mathcal{C}_2 \to \mathcal{C}_1\) be a functor satisfying:

1. if \(\{V_i \to V\}\) is a covering of \(\mathcal{C}_2\), then \(\{u(V_i) \to u(V)\}\) is a covering of \(\mathcal{C}_1\) (we say that \(u\) is continuous), and

2. \(u\) commutes with finite limits (i.e., \(u\) is left exact, i.e., \(u\) preserves fibre products and final objects).

Then one can define \(f_*: \Sh(\mathcal{C}_1) \to \Sh(\mathcal{C}_2)\) by \(f_* \mathcal{F}(V) = \mathcal{F}(u(V))\). Moreover, there exists an exact functor \(f^{-1}\) which is left adjoint to \(f_*\), see Sites, Definition 00X1 and Proposition 00X6. Warning: It is not enough to require simply that \(u\) is continuous and commutes with fibre products in order to get a morphism of topoi.

Functoriality of big topoi

Given a morphism of schemes \(f : X \to Y\) there are a whole host of morphisms of topoi associated to \(f\), see Topologies, Section 03FE for a list. Perhaps the most used ones are the morphisms of topoi \[f_{big} = f_{big, \tau} : \Sh((\Sch/X)_\tau) \longrightarrow \Sh((\Sch/Y)_\tau)\] where \(\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}\). These each correspond to a continuous functor \[(\Sch/Y)_\tau \longrightarrow (\Sch/X)_\tau, \quad V/Y \longmapsto X \times_Y V/X\] which preserves final objects, fibre products and covering, and hence defines a morphism of sites \[f_{big} : (\Sch/X)_\tau \longrightarrow (\Sch/Y)_\tau.\] See Topologies, Sections 020N, 0214, 021Y, 0224, and 021L. In particular, pushforward along \(f_{big}\) is given by the rule \[(f_{big, *}\mathcal{F})(V/Y) = \mathcal{F}(X \times_Y V/X)\] It turns out that these morphisms of topoi have an inverse image functor \(f_{big}^{-1}\) which is very easy to describe. Namely, we have \[(f_{big}^{-1}\mathcal{G})(U/X) = \mathcal{G}(U/Y)\] where the structure morphism of \(U/Y\) is the composition of the structure morphism \(U \to X\) with \(f\), see Topologies, Lemmas 0210, 021H, 04HC, 04HD, and 021W.

Functoriality and sheaves of modules

In this section we are going to reformulate some of the material explained in Descent, Sections 03DR, 0GN8, and 0GN9 in the setting of étale topologies. Let \(f : X \to Y\) be a morphism of schemes. We have seen above, see Sections 04I0, 03PV, and 03PZ that this induces a morphism \(f_{small}\) of small étale sites. In Descent, Remark 070R we have seen that \(f\) also induces a natural map \[f_{small}^\sharp : \mathcal{O}_{Y_\etale} \longrightarrow f_{small, *}\mathcal{O}_{X_\etale}\] of sheaves of rings on \(Y_\etale\) such that \((f_{small}, f_{small}^\sharp)\) is a morphism of ringed sites. See Modules on Sites, Definition 03AD for the definition of a morphism of ringed sites. Let us just recall here that \(f_{small}^\sharp\) is defined by the compatible system of maps \[\text{pr}_V^\sharp : \mathcal{O}(V) \longrightarrow \mathcal{O}(X \times_Y V)\] for \(V\) varying over the objects of \(Y_\etale\).

It is clear that this construction is compatible with compositions of morphisms of schemes. More precisely, if \(f : X \to Y\) and \(g : Y \to Z\) are morphisms of schemes, then we have \[(g_{small}, g_{small}^\sharp) \circ (f_{small}, f_{small}^\sharp) = ((g \circ f)_{small}, (g \circ f)_{small}^\sharp)\] as morphisms of ringed topoi. Moreover, by Modules on Sites, Definition 03D6 we see that given a morphism \(f : X \to Y\) of schemes we get well defined pullback and direct image functors \[\begin{align*} f_{small}^* : \textit{Mod}(\mathcal{O}_{Y_\etale}) \longrightarrow \textit{Mod}(\mathcal{O}_{X_\etale}), \\ f_{small, *} : \textit{Mod}(\mathcal{O}_{X_\etale}) \longrightarrow \textit{Mod}(\mathcal{O}_{Y_\etale}) \end{align*}\] which are adjoint in the usual way. If \(g : Y \to Z\) is another morphism of schemes, then we have \((g \circ f)_{small}^* = f_{small}^* \circ g_{small}^*\) and \((g \circ f)_{small, *} = g_{small, *} \circ f_{small, *}\) because of what we said about compositions.

There is quite a bit of difference between the category of all \(\mathcal{O}_X\) modules on \(X\) and the category between all \(\mathcal{O}_{X_\etale}\)-modules on \(X_\etale\). But the results of Descent, Sections 03DR, 0GN8, and 0GN9 tell us that there is not much difference between considering quasi-coherent modules on \(S\) and quasi-coherent modules on \(S_\etale\). (We have already seen this in Theorem 03OJ for example.) In particular, if \(f : X \to Y\) is any morphism of schemes, then the pullback functors \(f_{small}^*\) and \(f^*\) match for quasi-coherent sheaves, see Descent, Proposition 03LC. Moreover, the same is true for pushforward provided \(f\) is quasi-compact and quasi-separated, see Descent, Lemma 071N.

A few words about functoriality of the structure sheaf on big sites. Let \(f : X \to Y\) be a morphism of schemes. Choose any of the topologies \(\tau \in \{Zariski,\linebreak[0] \etale,\linebreak[0] smooth,\linebreak[0] syntomic,\linebreak[0] fppf\}\). Then the morphism \(f_{big} : (\Sch/X)_\tau \to (\Sch/Y)_\tau\) becomes a morphism of ringed sites by a map \[f_{big}^\sharp : \mathcal{O}_Y \longrightarrow f_{big, *}\mathcal{O}_X\] see Descent, Remark 070R. In fact it is given by the same construction as in the case of small sites explained above.

Comparing topologies

In this section we start studying what happens when you compare sheaves with respect to different topologies.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{F}\) be a sheaf of sets on \(S_\etale\). Let \(s, t \in \mathcal{F}(S)\). Then there exists an open \(W \subset S\) characterized by the following property: A morphism \(f : T \to S\) factors through \(W\) if and only if \(s|_T = t|_T\) (restriction is pullback by \(f_{small}\)).

Proof

Consider the presheaf which assigns to \(U \in \Ob(S_\etale)\) the empty set if \(s|_U \not = t|_U\) and a singleton else. It is clear that this is a subsheaf of the final object of \(\Sh(S_\etale)\). By Lemma 04HV we find an open \(W \subset S\) representing this presheaf. For a geometric point \(\overline{x}\) of \(S\) we see that \(\overline{x} \in W\) if and only if the stalks of \(s\) and \(t\) at \(\overline{x}\) agree. By the description of stalks of pullbacks in Lemma 03Q1 we see that \(W\) has the desired property.

Lemma

Let \(S\) be a scheme. Let \(\tau \in \{Zariski, \etale\}\). Consider the morphism \[\pi_S : (\Sch/S)_\tau \longrightarrow S_\tau\] of Topologies, Lemma 020Z or 021G. Let \(\mathcal{F}\) be a sheaf on \(S_\tau\). Then \(\pi_S^{-1}\mathcal{F}\) is given by the rule \[(\pi_S^{-1}\mathcal{F})(T) = \Gamma(T_\tau, f_{small}^{-1}\mathcal{F})\] where \(f : T \to S\). Moreover, \(\pi_S^{-1}\mathcal{F}\) satisfies the sheaf condition with respect to fpqc coverings.

Proof

Observe that we have a morphism \(i_f : \Sh(T_\tau) \to \Sh(\Sch/S)_\tau)\) such that \(\pi_S \circ i_f = f_{small}\) as morphisms \(T_\tau \to S_\tau\), see Topologies, Lemmas 020Y, 0211, 021F, and 021I. Since pullback is transitive we see that \(i_f^{-1} \pi_S^{-1}\mathcal{F} = f_{small}^{-1}\mathcal{F}\) as desired.

Let \(\{g_i : T_i \to T\}_{i \in I}\) be an fpqc covering. The final statement means the following: Given a sheaf \(\mathcal{G}\) on \(T_\tau\) and given sections \(s_i \in \Gamma(T_i, g_{i, small}^{-1}\mathcal{G})\) whose pullbacks to \(T_i \times_T T_j\) agree, there is a unique section \(s\) of \(\mathcal{G}\) over \(T\) whose pullback to \(T_i\) agrees with \(s_i\).

Let \(V \to T\) be an object of \(T_\tau\) and let \(t \in \mathcal{G}(V)\). For every \(i\) there is a largest open \(W_i \subset T_i \times_T V\) such that the pullbacks of \(s_i\) and \(t\) agree as sections of the pullback of \(\mathcal{G}\) to \(W_i \subset T_i \times_T V\), see Lemma 09XM. Because \(s_i\) and \(s_j\) agree over \(T_i \times_T T_j\) we find that \(W_i\) and \(W_j\) pullback to the same open over \(T_i \times_T T_j \times_T V\). By Descent, Lemma 03N0 we find an open \(W \subset V\) whose inverse image to \(T_i \times_T V\) recovers \(W_i\).

By construction of \(g_{i, small}^{-1}\mathcal{G}\) there exists a \(\tau\)-covering \(\{T_{ij} \to T_i\}_{j \in J_i}\), for each \(j\) an open immersion or étale morphism \(V_{ij} \to T\), a section \(t_{ij} \in \mathcal{G}(V_{ij})\), and commutative diagrams \[\xymatrix{ T_{ij} \ar[r] \ar[d] & V_{ij} \ar[d] \\ T_i \ar[r] & T }\] such that \(s_i|_{T_{ij}}\) is the pullback of \(t_{ij}\). In other words, after replacing the covering \(\{T_i \to T\}\) by \(\{T_{ij} \to T\}\) we may assume there are factorizations \(T_i \to V_i \to T\) with \(V_i \in \Ob(T_\tau)\) and sections \(t_i \in \mathcal{G}(V_i)\) pulling back to \(s_i\) over \(T_i\). By the result of the previous paragraph we find opens \(W_i \subset V_i\) such that \(t_i|_{W_i}\) “agrees with” every \(s_j\) over \(T_j \times_T W_i\). Note that \(T_i \to V_i\) factors through \(W_i\). Hence \(\{W_i \to T\}\) is a \(\tau\)-covering and the lemma is proven.

Lemma

Let \(S\) be a scheme. Let \(f : T \to S\) be a morphism such that

1. \(f\) is flat and quasi-compact, and

2. the geometric fibres of \(f\) are connected.

Let \(\mathcal{F}\) be a sheaf on \(S_\etale\). Then \(\Gamma(S, \mathcal{F}) = \Gamma(T, f^{-1}_{small}\mathcal{F})\).

Proof

There is a canonical map \(\Gamma(S, \mathcal{F}) \to \Gamma(T, f_{small}^{-1}\mathcal{F})\). Since \(f\) is surjective (because its fibres are connected) we see that this map is injective.

To show that the map is surjective, let \(\alpha \in \Gamma(T, f_{small}^{-1}\mathcal{F})\). Since \(\{T \to S\}\) is an fpqc covering we can use Lemma 09XN to see that suffices to prove that \(\alpha\) pulls back to the same section over \(T \times_S T\) by the two projections. Let \(\overline{s} \to S\) be a geometric point. It suffices to show the agreement holds over \((T \times_S T)_{\overline{s}}\) as every geometric point of \(T \times_S T\) is contained in one of these geometric fibres. In other words, we are trying to show that \(\alpha|_{T_{\overline{s}}}\) pulls back to the same section over \[(T \times_S T)_{\overline{s}} = T_{\overline{s}} \times_{\overline{s}} T_{\overline{s}}\] by the two projections to \(T_{\overline{s}}\). However, since \(\mathcal{F}|_{T_{\overline{s}}}\) is the pullback of \(\mathcal{F}|_{\overline{s}}\) it is a constant sheaf with value \(\mathcal{F}_{\overline{s}}\). Since \(T_{\overline{s}}\) is connected by assumption, any section of a constant sheaf is constant. Hence \(\alpha|_{T_{\overline{s}}}\) corresponds to an element of \(\mathcal{F}_{\overline{s}}\). Thus the two pullbacks to \((T \times_S T)_{\overline{s}}\) both correspond to this same element and we conclude.

Here is a version of Lemma 0A3H where we do not assume that the morphism is flat.

Lemma

Let \(S\) be a scheme. Let \(f : X \to S\) be a morphism such that

1. \(f\) is submersive, and

2. the geometric fibres of \(f\) are connected.

Let \(\mathcal{F}\) be a sheaf on \(S_\etale\). Then \(\Gamma(S, \mathcal{F}) = \Gamma(X, f^{-1}_{small}\mathcal{F})\).

Proof

There is a canonical map \(\Gamma(S, \mathcal{F}) \to \Gamma(X, f_{small}^{-1}\mathcal{F})\). Since \(f\) is surjective (because its fibres are connected) we see that this map is injective.

To show that the map is surjective, let \(\tau \in \Gamma(X, f_{small}^{-1}\mathcal{F})\). It suffices to find an étale covering \(\{U_i \to S\}\) and sections \(\sigma_i \in \mathcal{F}(U_i)\) such that \(\sigma_i\) pulls back to \(\tau|_{X \times_S U_i}\). Namely, the injectivity shown above guarantees that \(\sigma_i\) and \(\sigma_j\) restrict to the same section of \(\mathcal{F}\) over \(U_i \times_S U_j\). Thus we obtain a unique section \(\sigma \in \mathcal{F}(S)\) which restricts to \(\sigma_i\) over \(U_i\). Then the pullback of \(\sigma\) to \(X\) is \(\tau\) because this is true locally.

Let \(\overline{x}\) be a geometric point of \(X\) with image \(\overline{s}\) in \(S\). Consider the image of \(\tau\) in the stalk \[(f_{small}^{-1}\mathcal{F})_{\overline{x}} = \mathcal{F}_{\overline{s}}\] See Lemma 03Q1. We can find an étale neighbourhood \(U \to S\) of \(\overline{s}\) and a section \(\sigma \in \mathcal{F}(U)\) mapping to this image in the stalk. Thus after replacing \(S\) by \(U\) and \(X\) by \(X \times_S U\) we may assume there exits a section \(\sigma\) of \(\mathcal{F}\) over \(S\) whose image in \((f_{small}^{-1}\mathcal{F})_{\overline{x}}\) is the same as \(\tau\).

By Lemma 09XM there exists a maximal open \(W \subset X\) such that \(f_{small}^{-1}\sigma\) and \(\tau\) agree over \(W\) and the formation of \(W\) commutes with further pullback. Observe that the pullback of \(\mathcal{F}\) to the geometric fibre \(X_{\overline{s}}\) is the pullback of \(\mathcal{F}_{\overline{s}}\) viewed as a sheaf on \(\overline{s}\) by \(X_{\overline{s}} \to \overline{s}\). Hence we see that \(\tau\) and \(\sigma\) give sections of the constant sheaf with value \(\mathcal{F}_{\overline{s}}\) on \(X_{\overline{s}}\) which agree in one point. Since \(X_{\overline{s}}\) is connected by assumption, we conclude that \(W\) contains \(X_s\). The same argument for different geometric fibres shows that \(W\) contains every fibre it meets. Since \(f\) is submersive, we conclude that \(W\) is the inverse image of an open neighbourhood of \(s\) in \(S\). This finishes the proof.

Lemma

Let \(K/k\) be an extension of fields with \(k\) separably algebraically closed. Let \(S\) be a scheme over \(k\). Denote \(p : S_K = S \times_{\Spec(k)} \Spec(K) \to S\) the projection. Let \(\mathcal{F}\) be a sheaf on \(S_\etale\). Then \(\Gamma(S, \mathcal{F}) = \Gamma(S_K, p^{-1}_{small}\mathcal{F})\).

Proof

Follows from Lemma 0A3H. Namely, it is clear that \(p\) is flat and quasi-compact as the base change of \(\Spec(K) \to \Spec(k)\). On the other hand, if \(\overline{s} : \Spec(L) \to S\) is a geometric point, then the fibre of \(p\) over \(\overline{s}\) is the spectrum of \(K \otimes_k L\) which is irreducible hence connected by Algebra, Lemma 00I7.

Recovering morphisms

In this section we prove that the rule which associates to a scheme its locally ringed small étale topos is fully faithful in a suitable sense, see Theorem 04I7.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. The morphism of ringed sites \((f_{small}, f_{small}^\sharp)\) associated to \(f\) is a morphism of locally ringed sites, see Modules on Sites, Definition 04HA.

Proof

Note that the assertion makes sense since we have seen that \((X_\etale, \mathcal{O}_{X_\etale})\) and \((Y_\etale, \mathcal{O}_{Y_\etale})\) are locally ringed sites, see Lemma 04HZ. Moreover, we know that \(X_\etale\) has enough points, see Theorem 03PU and Remarks 040S. Hence it suffices to prove that \((f_{small}, f_{small}^\sharp)\) satisfies condition (3) of Modules on Sites, Lemma 04H9. To see this take a point \(p\) of \(X_\etale\). By Lemma 04HU \(p\) corresponds to a geometric point \(\overline{x}\) of \(X\). By Lemma 03Q1 the point \(q = f_{small} \circ p\) corresponds to the geometric point \(\overline{y} = f \circ \overline{x}\) of \(Y\). Hence the assertion we have to prove is that the induced map of stalks \[(\mathcal{O}_Y)_{\overline{y}} \longrightarrow (\mathcal{O}_X)_{\overline{x}}\] is a local ring map. Suppose that \(a \in (\mathcal{O}_Y)_{\overline{y}}\) is an element of the left hand side which maps to an element of the maximal ideal of the right hand side. Suppose that \(a\) is the equivalence class of a triple \((V, \overline{v}, a)\) with \(V \to Y\) étale, \(\overline{v} : \overline{x} \to V\) over \(Y\), and \(a \in \mathcal{O}(V)\). It maps to the equivalence class of \((X \times_Y V, \overline{x} \times \overline{v}, \text{pr}_V^\sharp(a))\) in the local ring \((\mathcal{O}_X)_{\overline{x}}\). But it is clear that being in the maximal ideal means that pulling back \(\text{pr}_V^\sharp(a)\) to an element of \(\kappa(\overline{x})\) gives zero. Hence also pulling back \(a\) to \(\kappa(\overline{x})\) is zero. Which means that \(a\) lies in the maximal ideal of \((\mathcal{O}_Y)_{\overline{y}}\).

Lemma

Let \(X\), \(Y\) be schemes. Let \(f : X \to Y\) be a morphism of schemes. Let \(t\) be a \(2\)-morphism from \((f_{small}, f_{small}^\sharp)\) to itself, see Modules on Sites, Definition 04IC. Then \(t = \text{id}\).

Proof

This means that \(t : f^{-1}_{small} \to f^{-1}_{small}\) is a transformation of functors such that the diagram \[\xymatrix{ f_{small}^{-1}\mathcal{O}_Y \ar[rd]_{f_{small}^\sharp} & & f_{small}^{-1}\mathcal{O}_Y \ar[ll]^t \ar[ld]^{f_{small}^\sharp} \\ & \mathcal{O}_X }\] is commutative. Suppose \(V \to Y\) is étale with \(V\) affine. By Morphisms, Lemma 04II we may choose an immersion \(i : V \to \mathbf{A}^n_Y\) over \(Y\). In terms of sheaves this means that \(i\) induces an injection \(h_i : h_V \to \prod_{j = 1, \ldots, n} \mathcal{O}_Y\) of sheaves. The base change \(i'\) of \(i\) to \(X\) is an immersion (Schemes, Lemma 01JY). Hence \(i' : X \times_Y V \to \mathbf{A}^n_X\) is an immersion, which in turn means that \(h_{i'} : h_{X \times_Y V} \to \prod_{j = 1, \ldots, n} \mathcal{O}_X\) is an injection of sheaves. Via the identification \(f_{small}^{-1}h_V = h_{X \times_Y V}\) of Lemma 03Q1 the map \(h_{i'}\) is equal to \[\xymatrix{ f_{small}^{-1}h_V \ar[r]^-{f^{-1}h_i} & \prod_{j = 1, \ldots, n} f_{small}^{-1}\mathcal{O}_Y \ar[r]^{\prod f^\sharp} & \prod_{j = 1, \ldots, n} \mathcal{O}_X }\] (verification omitted). This means that the map \(t : f_{small}^{-1}h_V \to f_{small}^{-1}h_V\) fits into the commutative diagram \[\xymatrix{ f_{small}^{-1}h_V \ar[r]^-{f^{-1}h_i} \ar[d]^t & \prod_{j = 1, \ldots, n} f_{small}^{-1}\mathcal{O}_Y \ar[r]^-{\prod f^\sharp} \ar[d]^{\prod t} & \prod_{j = 1, \ldots, n} \mathcal{O}_X \ar[d]^{\text{id}}\\ f_{small}^{-1}h_V \ar[r]^-{f^{-1}h_i} & \prod_{j = 1, \ldots, n} f_{small}^{-1}\mathcal{O}_Y \ar[r]^-{\prod f^\sharp} & \prod_{j = 1, \ldots, n} \mathcal{O}_X }\] The commutativity of the right square holds by our assumption on \(t\) explained above. Since the composition of the horizontal arrows is injective by the discussion above we conclude that the left vertical arrow is the identity map as well. Any sheaf of sets on \(Y_\etale\) admits a surjection from a (huge) coproduct of sheaves of the form \(h_V\) with \(V\) affine (combine Topologies, Lemma 04HR with Sites, Lemma 00WS). Thus we conclude that \(t : f_{small}^{-1} \to f_{small}^{-1}\) is the identity transformation as desired.

Lemma

Let \(X\), \(Y\) be schemes. Any two morphisms \(a, b : X \to Y\) of schemes for which there exists a \(2\)-isomorphism \((a_{small}, a_{small}^\sharp) \cong (b_{small}, b_{small}^\sharp)\) in the \(2\)-category of ringed topoi are equal.

Proof

Let us argue this carefully since it is a bit confusing. Let \(t : a_{small}^{-1} \to b_{small}^{-1}\) be the \(2\)-isomorphism. Consider any open \(V \subset Y\). Note that \(h_V\) is a subsheaf of the final sheaf \(*\). Thus both \(a_{small}^{-1}h_V = h_{a^{-1}(V)}\) and \(b_{small}^{-1}h_V = h_{b^{-1}(V)}\) are subsheaves of the final sheaf. Thus the isomorphism \[t : a_{small}^{-1}h_V = h_{a^{-1}(V)} \to b_{small}^{-1}h_V = h_{b^{-1}(V)}\] has to be the identity, and \(a^{-1}(V) = b^{-1}(V)\). It follows that \(a\) and \(b\) are equal on underlying topological spaces. Next, take a section \(f \in \mathcal{O}_Y(V)\). This determines and is determined by a map of sheaves of sets \(f : h_V \to \mathcal{O}_Y\). Pull this back and apply \(t\) to get a commutative diagram \[\xymatrix{ h_{b^{-1}(V)} \ar@{=}[r] & b_{small}^{-1}h_V \ar[d]^{b_{small}^{-1}(f)} & & a_{small}^{-1}h_V \ar[d]^{a_{small}^{-1}(f)} \ar[ll]^t & h_{a^{-1}(V)} \ar@{=}[l] \\ & b_{small}^{-1}\mathcal{O}_Y \ar[rd]_{b^\sharp} & & a_{small}^{-1}\mathcal{O}_Y \ar[ll]^t \ar[ld]^{a^\sharp} \\ & & \mathcal{O}_X }\] where the triangle is commutative by definition of a \(2\)-isomorphism in Modules on Sites, Section 04IB. Above we have seen that the composition of the top horizontal arrows comes from the identity \(a^{-1}(V) = b^{-1}(V)\). Thus the commutativity of the diagram tells us that \(a_{small}^\sharp(f) = b_{small}^\sharp(f)\) in \(\mathcal{O}_X(a^{-1}(V)) = \mathcal{O}_X(b^{-1}(V))\). Since this holds for every open \(V\) and every \(f \in \mathcal{O}_Y(V)\) we conclude that \(a = b\) as morphisms of schemes.

Lemma

Let \(X\), \(Y\) be affine schemes. Let \[(g, g^\#) : (\Sh(X_\etale), \mathcal{O}_X) \longrightarrow (\Sh(Y_\etale), \mathcal{O}_Y)\] be a morphism of locally ringed topoi. Then there exists a unique morphism of schemes \(f : X \to Y\) such that \((g, g^\#)\) is \(2\)-isomorphic to \((f_{small}, f_{small}^\sharp)\), see Modules on Sites, Definition 04IC.

Proof

In this proof we write \(\mathcal{O}_X\) for the structure sheaf of the small étale site \(X_\etale\), and similarly for \(\mathcal{O}_Y\). Say \(Y = \Spec(B)\) and \(X = \Spec(A)\). Since \(B = \Gamma(Y_\etale, \mathcal{O}_Y)\), \(A = \Gamma(X_\etale, \mathcal{O}_X)\) we see that \(g^\sharp\) induces a ring map \(\varphi : B \to A\). Let \(f = \Spec(\varphi) : X \to Y\) be the corresponding morphism of affine schemes. We will show this \(f\) does the job.

Let \(V \to Y\) be an affine scheme étale over \(Y\). Thus we may write \(V = \Spec(C)\) with \(C\) an étale \(B\)-algebra. We can write \[C = B[x_1, \ldots, x_n]/(P_1, \ldots, P_n)\] with \(P_i\) polynomials such that \(\Delta = \det(\partial P_i/ \partial x_j)\) is invertible in \(C\), see for example Algebra, Lemma 00U9. If \(T\) is a scheme over \(Y\), then a \(T\)-valued point of \(V\) is given by \(n\) sections of \(\Gamma(T, \mathcal{O}_T)\) which satisfy the polynomial equations \(P_1 = 0, \ldots, P_n = 0\). In other words, the sheaf \(h_V\) on \(Y_\etale\) is the equalizer of the two maps \[\xymatrix{ \prod\nolimits_{i = 1, \ldots, n} \mathcal{O}_Y \ar@<1ex>[r]^a \ar@<-1ex>[r]_b & \prod\nolimits_{j = 1, \ldots, n} \mathcal{O}_Y }\] where \(b(h_1, \ldots, h_n) = 0\) and \(a(h_1, \ldots, h_n) = (P_1(h_1, \ldots, h_n), \ldots, P_n(h_1, \ldots, h_n))\). Since \(g^{-1}\) is exact we conclude that the top row of the following solid commutative diagram is an equalizer diagram as well: \[\xymatrix{ g^{-1}h_V \ar[r] \ar@{..>}[d] & \prod\nolimits_{i = 1, \ldots, n} g^{-1}\mathcal{O}_Y \ar@<1ex>[r]^{g^{-1}a} \ar@<-1ex>[r]_{g^{-1}b} \ar[d]^{\prod g^\sharp} & \prod\nolimits_{j = 1, \ldots, n} g^{-1}\mathcal{O}_Y \ar[d]^{\prod g^\sharp}\\ h_{X \times_Y V} \ar[r] & \prod\nolimits_{i = 1, \ldots, n} \mathcal{O}_X \ar@<1ex>[r]^{a'} \ar@<-1ex>[r]_{b'} & \prod\nolimits_{j = 1, \ldots, n} \mathcal{O}_X \\ }\] Here \(b'\) is the zero map and \(a'\) is the map defined by the images \(P'_i = \varphi(P_i) \in A[x_1, \ldots, x_n]\) via the same rule \(a'(h_1, \ldots, h_n) = (P'_1(h_1, \ldots, h_n), \ldots, P'_n(h_1, \ldots, h_n))\). that \(a\) was defined by. The commutativity of the diagram follows from the fact that \(\varphi = g^\sharp\) on global sections. The lower row is an equalizer diagram also, by exactly the same arguments as before since \(X \times_Y V\) is the affine scheme \(\Spec(A \otimes_B C)\) and \(A \otimes_B C = A[x_1, \ldots, x_n]/(P'_1, \ldots, P'_n)\). Thus we obtain a unique dotted arrow \(g^{-1}h_V \to h_{X \times_Y V}\) fitting into the diagram

We claim that the map of sheaves \(g^{-1}h_V \to h_{X \times_Y V}\) is an isomorphism. Since the small étale site of \(X\) has enough points (Theorem 03PU) it suffices to prove this on stalks. Hence let \(\overline{x}\) be a geometric point of \(X\), and denote \(p\) the associate point of the small étale topos of \(X\). Set \(q = g \circ p\). This is a point of the small étale topos of \(Y\). By Lemma 04HU we see that \(q\) corresponds to a geometric point \(\overline{y}\) of \(Y\). Consider the map of stalks \[(g^\sharp)_p : (\mathcal{O}_Y)_{\overline{y}} = \mathcal{O}_{Y, q} = (g^{-1}\mathcal{O}_Y)_p \longrightarrow \mathcal{O}_{X, p} = (\mathcal{O}_X)_{\overline{x}}\] Since \((g, g^\sharp)\) is a morphism of locally ringed topoi \((g^\sharp)_p\) is a local ring homomorphism of strictly henselian local rings. Applying localization to the big commutative diagram above and Algebra, Lemma 04GX we conclude that \((g^{-1}h_V)_p \to (h_{X \times_Y V})_p\) is an isomorphism as desired.

We claim that the isomorphisms \(g^{-1}h_V \to h_{X \times_Y V}\) are functorial. Namely, suppose that \(V_1 \to V_2\) is a morphism of affine schemes étale over \(Y\). Write \(V_i = \Spec(C_i)\) with \[C_i = B[x_{i, 1}, \ldots, x_{i, n_i}]/(P_{i, 1}, \ldots, P_{i, n_i})\] The morphism \(V_1 \to V_2\) is given by a \(B\)-algebra map \(C_2 \to C_1\) which in turn is given by some polynomials \(Q_j \in B[x_{1, 1}, \ldots, x_{1, n_1}]\) for \(j = 1, \ldots, n_2\). Then it is an easy matter to show that the diagram of sheaves \[\xymatrix{ h_{V_1} \ar[d] \ar[r] & \prod_{i = 1, \ldots, n_1} \mathcal{O}_Y \ar[d]^{Q_1, \ldots, Q_{n_2}}\\ h_{V_2} \ar[r] & \prod_{i = 1, \ldots, n_2} \mathcal{O}_Y }\] is commutative, and pulling back to \(X_\etale\) we obtain the solid commutative diagram \[\xymatrix{ g^{-1}h_{V_1} \ar@{..>}[dd] \ar[rrd] \ar[r] & \prod_{i = 1, \ldots, n_1} g^{-1}\mathcal{O}_Y \ar[dd]^{g^\sharp} \ar[rrd]^{Q_1, \ldots, Q_{n_2}} \\ & & g^{-1}h_{V_2} \ar@{..>}[dd] \ar[r] & \prod_{i = 1, \ldots, n_2} g^{-1}\mathcal{O}_Y \ar[dd]^{g^\sharp} \\ h_{X \times_Y V_1} \ar[r] \ar[rrd] & \prod\nolimits_{i = 1, \ldots, n_1} \mathcal{O}_X \ar[rrd]^{Q'_1, \ldots, Q'_{n_2}} \\ & & h_{X \times_Y V_2} \ar[r] & \prod\nolimits_{i = 1, \ldots, n_2} \mathcal{O}_X }\] where \(Q'_j \in A[x_{1, 1}, \ldots, x_{1, n_1}]\) is the image of \(Q_j\) via \(\varphi\). Since the dotted arrows exist, make the two squares commute, and the horizontal arrows are injective we see that the whole diagram commutes. This proves functoriality (and also that the construction of \(g^{-1}h_V \to h_{X \times_Y V}\) is independent of the choice of the presentation, although we strictly speaking do not need to show this).

At this point we are able to show that \(f_{small, *} \cong g_*\). Namely, let \(\mathcal{F}\) be a sheaf on \(X_\etale\). For every \(V \in \Ob(X_\etale)\) affine we have \[\begin{align*} (g_*\mathcal{F})(V) & = \Mor_{\Sh(Y_\etale)}(h_V, g_*\mathcal{F}) \\ & = \Mor_{\Sh(X_\etale)}(g^{-1}h_V, \mathcal{F}) \\ & = \Mor_{\Sh(X_\etale)}(h_{X \times_Y V}, \mathcal{F}) \\ & = \mathcal{F}(X \times_Y V) \\ & = f_{small, *}\mathcal{F}(V) \end{align*}\] where in the third equality we use the isomorphism \(g^{-1}h_V \cong h_{X \times_Y V}\) constructed above. These isomorphisms are clearly functorial in \(\mathcal{F}\) and functorial in \(V\) as the isomorphisms \(g^{-1}h_V \cong h_{X \times_Y V}\) are functorial. Now any sheaf on \(Y_\etale\) is determined by the restriction to the subcategory of affine schemes (Topologies, Lemma 04HR), and hence we obtain an isomorphism of functors \(f_{small, *} \cong g_*\) as desired.

Finally, we have to check that, via the isomorphism \(f_{small, *} \cong g_*\) above, the maps \(f_{small}^\sharp\) and \(g^\sharp\) agree. By construction this is already the case for the global sections of \(\mathcal{O}_Y\), i.e., for the elements of \(B\). We only need to check the result on sections over an affine \(V\) étale over \(Y\) (by Topologies, Lemma 04HR again). Writing \(V = \Spec(C)\), \(C = B[x_i]/(P_j)\) as before it suffices to check that the coordinate functions \(x_i\) are mapped to the same sections of \(\mathcal{O}_X\) over \(X \times_Y V\). And this is exactly what it means that the diagram \[\xymatrix{ g^{-1}h_V \ar[r] \ar@{..>}[d] & \prod\nolimits_{i = 1, \ldots, n} g^{-1}\mathcal{O}_Y \ar[d]^{\prod g^\sharp} \\ h_{X \times_Y V} \ar[r] & \prod\nolimits_{i = 1, \ldots, n} \mathcal{O}_X }\] commutes. Thus the lemma is proved.

Here is a version for general schemes.

Theorem

Let \(X\), \(Y\) be schemes. Let \[(g, g^\#) : (\Sh(X_\etale), \mathcal{O}_X) \longrightarrow (\Sh(Y_\etale), \mathcal{O}_Y)\] be a morphism of locally ringed topoi. Then there exists a unique morphism of schemes \(f : X \to Y\) such that \((g, g^\#)\) is isomorphic to \((f_{small}, f_{small}^\sharp)\). In other words, the construction \[\Sch \longrightarrow \textit{Locally ringed topoi}, \quad X \longrightarrow (X_\etale, \mathcal{O}_X)\] is fully faithful (morphisms up to \(2\)-isomorphisms on the right hand side).

Proof

You can prove this theorem by carefully adjusting the arguments of the proof of Lemma 04I6 to the global setting. However, we want to indicate how we can glue the result of that lemma to get a global morphism due to the rigidity provided by the result of Lemma 04IJ. Unfortunately, this is a bit messy.

Let us prove existence when \(Y\) is affine. In this case choose an affine open covering \(X = \bigcup U_i\). For each \(i\) the inclusion morphism \(j_i : U_i \to X\) induces a morphism of locally ringed topoi \((j_{i, small}, j_{i, small}^\sharp) : (\Sh(U_{i, \etale}), \mathcal{O}_{U_i}) \to (\Sh(X_\etale), \mathcal{O}_X)\) by Lemma 04I5. We can compose this with \((g, g^\sharp)\) to obtain a morphism of locally ringed topoi \[(g, g^\sharp) \circ (j_{i, small}, j_{i, small}^\sharp) : (\Sh(U_{i, \etale}), \mathcal{O}_{U_i}) \to (\Sh(Y_\etale), \mathcal{O}_Y)\] see Modules on Sites, Lemma 04IG. By Lemma 04I6 there exists a unique morphism of schemes \(f_i : U_i \to Y\) and a \(2\)-isomorphism \[t_i : (f_{i, small}, f_{i, small}^\sharp) \longrightarrow (g, g^\sharp) \circ (j_{i, small}, j_{i, small}^\sharp).\] Set \(U_{i, i'} = U_i \cap U_{i'}\), and denote \(j_{i, i'} : U_{i, i'} \to U_i\) the inclusion morphism. Since we have \(j_i \circ j_{i, i'} = j_{i'} \circ j_{i', i}\) we see that \[\begin{align*} (g, g^\sharp) \circ (j_{i, small}, j_{i, small}^\sharp) \circ (j_{i, i', small}, j_{i, i', small}^\sharp) = \\ (g, g^\sharp) \circ (j_{i', small}, j_{i', small}^\sharp) \circ (j_{i', i, small}, j_{i', i, small}^\sharp) \end{align*}\] Hence by uniqueness (see Lemma 04LW) we conclude that \(f_i \circ j_{i, i'} = f_{i'} \circ j_{i', i}\), in other words the morphisms of schemes \(f_i = f \circ j_i\) are the restrictions of a global morphism of schemes \(f : X \to Y\). Consider the diagram of \(2\)-isomorphisms (where we drop the components \({}^\sharp\) to ease the notation) \[\xymatrix{ g \circ j_{i, small} \circ j_{i, i', small} \ar[rr]^{t_i \star \text{id}_{j_{i, i', small}}} \ar@{=}[d] & & f_{small} \circ j_{i, small} \circ j_{i, i', small} \ar@{=}[d] \\ g \circ j_{i', small} \circ j_{i', i, small} \ar[rr]^{t_{i'} \star \text{id}_{j_{i', i, small}}} & & f_{small} \circ j_{i', small} \circ j_{i', i, small} }\] The notation \(\star\) indicates horizontal composition, see Categories, Definition 003H in general and Sites, Section 04I9 for our particular case. By the result of Lemma 04IJ this diagram commutes. Hence for any sheaf \(\mathcal{G}\) on \(Y_\etale\) the isomorphisms \(t_i : f_{small}^{-1}\mathcal{G}|_{U_i} \to g^{-1}\mathcal{G}|_{U_i}\) agree over \(U_{i, i'}\) and we obtain a global isomorphism \(t : f_{small}^{-1}\mathcal{G} \to g^{-1}\mathcal{G}\). It is clear that this isomorphism is functorial in \(\mathcal{G}\) and is compatible with the maps \(f_{small}^\sharp\) and \(g^\sharp\) (because it is compatible with these maps locally). This proves the theorem in case \(Y\) is affine.

In the general case, let \(V \subset Y\) be an affine open. Then \(h_V\) is a subsheaf of the final sheaf \(*\) on \(Y_\etale\). As \(g\) is exact we see that \(g^{-1}h_V\) is a subsheaf of the final sheaf on \(X_\etale\). Hence by Lemma 04HV there exists an open subscheme \(W \subset X\) such that \(g^{-1}h_V = h_W\). By Modules on Sites, Lemma 04IH there exists a commutative diagram of morphisms of locally ringed topoi \[\xymatrix{ (\Sh(W_\etale), \mathcal{O}_W) \ar[r] \ar[d]_{g'} & (\Sh(X_\etale), \mathcal{O}_X) \ar[d]^g \\ (\Sh(V_\etale), \mathcal{O}_V) \ar[r] & (\Sh(Y_\etale), \mathcal{O}_Y) }\] where the horizontal arrows are the localization morphisms (induced by the inclusion morphisms \(V \to Y\) and \(W \to X\)) and where \(g'\) is induced from \(g\). By the result of the preceding paragraph we obtain a morphism of schemes \(f' : W \to V\) and a \(2\)-isomorphism \(t : (f'_{small}, (f'_{small})^\sharp) \to (g', (g')^\sharp)\). Exactly as before these morphisms \(f'\) (for varying affine opens \(V \subset Y\)) agree on overlaps by uniqueness, so we get a morphism \(f : X \to Y\). Moreover, the \(2\)-isomorphisms \(t\) are compatible on overlaps by Lemma 04IJ again and we obtain a global \(2\)-isomorphism \((f_{small}, (f_{small})^\sharp) \to (g, (g)^\sharp)\). as desired. Some details omitted.

Push and pull

Let \(f : X \to Y\) be a morphism of schemes. Here is a list of conditions we will consider in the following:

1. For every étale morphism \(U \to X\) and \(u \in U\) there exist an étale morphism \(V \to Y\) and a disjoint union decomposition \(X \times_Y V = W \amalg W'\) and a morphism \(h : W \to U\) over \(X\) with \(u\) in the image of \(h\).

2. For every \(V \to Y\) étale, and every étale covering \(\{U_i \to X \times_Y V\}\) there exists an étale covering \(\{V_j \to V\}\) such that for each \(j\) we have \(X \times_Y V_j = \coprod W_{ij}\) where \(W_{ij} \to X \times_Y V\) factors through \(U_i \to X \times_Y V\) for some \(i\).

3. For every \(U \to X\) étale, there exists a \(V \to Y\) étale and a surjective morphism \(X \times_Y V \to U\) over \(X\).

It turns out that each of these properties has meaning in terms of the behaviour of the functor \(f_{small, *}\). We will work this out in the next few sections.

Property (A)

Please see Section 04C6 for the definition of property (A).

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume (A).

1. \(f_{small, *} : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)\) reflects injections and surjections,

2. \(f_{small}^{-1}f_{small, *}\mathcal{F} \to \mathcal{F}\) is surjective for any abelian sheaf \(\mathcal{F}\) on \(X_\etale\),

3. \(f_{small, *} : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)\) is faithful.

Proof

Let \(\mathcal{F}\) be an abelian sheaf on \(X_\etale\). Let \(U\) be an object of \(X_\etale\). By assumption we can find a covering \(\{W_i \to U\}\) in \(X_\etale\) such that each \(W_i\) is an open and closed subscheme of \(X \times_Y V_i\) for some object \(V_i\) of \(Y_\etale\). The sheaf condition shows that \[\mathcal{F}(U) \subset \prod \mathcal{F}(W_i)\] and that \(\mathcal{F}(W_i)\) is a direct summand of \(\mathcal{F}(X \times_Y V_i) = f_{small, *}\mathcal{F}(V_i)\). Hence it is clear that \(f_{small, *}\) reflects injections.

Next, suppose that \(a : \mathcal{G} \to \mathcal{F}\) is a map of abelian sheaves such that \(f_{small, *}a\) is surjective. Let \(s \in \mathcal{F}(U)\) with \(U\) as above. With \(W_i\), \(V_i\) as above we see that it suffices to show that \(s|_{W_i}\) is étale locally the image of a section of \(\mathcal{G}\) under \(a\). Since \(\mathcal{F}(W_i)\) is a direct summand of \(\mathcal{F}(X \times_Y V_i)\) it suffices to show that for any \(V \in \Ob(Y_\etale)\) any element \(s \in \mathcal{F}(X \times_Y V)\) is étale locally on \(X \times_Y V\) the image of a section of \(\mathcal{G}\) under \(a\). Since \(\mathcal{F}(X \times_Y V) = f_{small, *}\mathcal{F}(V)\) we see by assumption that there exists a covering \(\{V_j \to V\}\) such that \(s\) is the image of \(s_j \in f_{small, *}\mathcal{G}(V_j) = \mathcal{G}(X \times_Y V_j)\). This proves \(f_{small, *}\) reflects surjections.

Parts (2), (3) follow formally from part (1), see Modules on Sites, Lemma 04DA.

Lemma

Let \(f : X \to Y\) be a separated locally quasi-finite morphism of schemes. Then property (A) above holds.

Proof

Let \(U \to X\) be an étale morphism and \(u \in U\). The geometric statement (A) reduces directly to the case where \(U\) and \(Y\) are affine schemes. Denote \(x \in X\) and \(y \in Y\) the images of \(u\). Since \(X \to Y\) is locally quasi-finite, and \(U \to X\) is locally quasi-finite (see Morphisms, Lemma 03WS) we see that \(U \to Y\) is locally quasi-finite (see Morphisms, Lemma 01TL). Moreover both \(X \to Y\) and \(U \to Y\) are separated. Thus More on Morphisms, Lemma 02LO applies to both morphisms. This means we may pick an étale neighbourhood \((V, v) \to (Y, y)\) such that \[X \times_Y V = W \amalg R, \quad U \times_Y V = W' \amalg R'\] and points \(w \in W\), \(w' \in W'\) such that

1. \(W\), \(R\) are open and closed in \(X \times_Y V\),

2. \(W'\), \(R'\) are open and closed in \(U \times_Y V\),

3. \(W \to V\) and \(W' \to V\) are finite,

4. \(w\), \(w'\) map to \(v\),

5. \(\kappa(v) \subset \kappa(w)\) and \(\kappa(v) \subset \kappa(w')\) are purely inseparable, and

6. no other point of \(W\) or \(W'\) maps to \(v\).

Here is a commutative diagram \[\xymatrix{ U \ar[d] & U \times_Y V \ar[l] \ar[d] & W' \amalg R' \ar[d] \ar[l] \\ X \ar[d] & X \times_Y V \ar[l] \ar[d] & W \amalg R \ar[l] \\ Y & V \ar[l] }\] After shrinking \(V\) we may assume that \(W'\) maps into \(W\): just remove the image the inverse image of \(R\) in \(W'\); this is a closed set (as \(W' \to V\) is finite) not containing \(v\). Then \(W' \to W\) is finite because both \(W \to V\) and \(W' \to V\) are finite. Hence \(W' \to W\) is finite étale, and there is exactly one point in the fibre over \(w\) with \(\kappa(w) = \kappa(w')\). Hence \(W' \to W\) is an isomorphism in an open neighbourhood \(W^\circ\) of \(w\), see Étale Morphisms, Lemma 04DH. Since \(W \to V\) is finite the image of \(W \setminus W^\circ\) is a closed subset \(T\) of \(V\) not containing \(v\). Thus after replacing \(V\) by \(V \setminus T\) we may assume that \(W' \to W\) is an isomorphism. Now the decomposition \(X \times_Y V = W \amalg R\) and the morphism \(W \to U\) are as desired and we win.

Lemma

Let \(f : X \to Y\) be an integral morphism of schemes. Then property (A) holds.

Proof

Let \(U \to X\) be étale, and let \(u \in U\) be a point. We have to find \(V \to Y\) étale, a disjoint union decomposition \(X \times_Y V = W \amalg W'\) and an \(X\)-morphism \(W \to U\) with \(u\) in the image. We may shrink \(U\) and \(Y\) and assume \(U\) and \(Y\) are affine. In this case also \(X\) is affine, since an integral morphism is affine by definition. Write \(Y = \Spec(A)\), \(X = \Spec(B)\) and \(U = \Spec(C)\). Then \(A \to B\) is an integral ring map, and \(B \to C\) is an étale ring map. By Algebra, Lemma 00U2 we can find a finite \(A\)-subalgebra \(B' \subset B\) and an étale ring map \(B' \to C'\) such that \(C = B \otimes_{B'} C'\). Thus the question reduces to the étale morphism \(U' = \Spec(C') \to X' = \Spec(B')\) over the finite morphism \(X' \to Y\). In this case the result follows from Lemma 04DL.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Denote \(f_{small} : \Sh(X_\etale) \to \Sh(Y_\etale)\) the associated morphism of small étale topoi. Assume at least one of the following

1. \(f\) is integral, or

2. \(f\) is separated and locally quasi-finite.

Then the functor \(f_{small, *} : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)\) has the following properties

1. the map \(f_{small}^{-1}f_{small, *}\mathcal{F} \to \mathcal{F}\) is always surjective,

2. \(f_{small, *}\) is faithful, and

3. \(f_{small, *}\) reflects injections and surjections.

Proof

Combine Lemmas 04DL, 04DM, and 04DK.

Property (B)

Please see Section 04C6 for the definition of property (B).

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume (B) holds. Then the functor \(f_{small, *} : \Sh(X_\etale) \to \Sh(Y_\etale)\) transforms surjections into surjections.

Proof

This follows from Sites, Lemma 04D7.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Suppose

1. \(V \to Y\) is an étale morphism of schemes,

2. \(\{U_i \to X \times_Y V\}\) is an étale covering, and

3. \(v \in V\) is a point.

Assume that for any such data there exists an étale neighbourhood \((V', v') \to (V, v)\), a disjoint union decomposition \(X \times_Y V' = \coprod W'_i\), and morphisms \(W'_i \to U_i\) over \(X \times_Y V\). Then property (B) holds.

Proof

Omitted.

Lemma

Let \(f : X \to Y\) be a finite morphism of schemes. Then property (B) holds.

Proof

Consider \(V \to Y\) étale, \(\{U_i \to X \times_Y V\}\) an étale covering, and \(v \in V\). We have to find a \(V' \to V\) and decomposition and maps as in Lemma 04DP. We may shrink \(V\) and \(Y\), hence we may assume that \(V\) and \(Y\) are affine. Since \(X\) is finite over \(Y\), this also implies that \(X\) is affine. During the proof we may (finitely often) replace \((V, v)\) by an étale neighbourhood \((V', v')\) and correspondingly the covering \(\{U_i \to X \times_Y V\}\) by \(\{V' \times_V U_i \to X \times_Y V'\}\).

Since \(X \times_Y V \to V\) is finite there exist finitely many (pairwise distinct) points \(x_1, \ldots, x_n \in X \times_Y V\) mapping to \(v\). We may apply More on Morphisms, Lemma 02LO to \(X \times_Y V \to V\) and the points \(x_1, \ldots, x_n\) lying over \(v\) and find an étale neighbourhood \((V', v') \to (V, v)\) such that \[X \times_Y V' = R \amalg \coprod T_a\] with \(T_a \to V'\) finite with exactly one point \(p_a\) lying over \(v'\) and moreover \(\kappa(v') \subset \kappa(p_a)\) purely inseparable, and such that \(R \to V'\) has empty fibre over \(v'\). Because \(X \to Y\) is finite, also \(R \to V'\) is finite. Hence after shrinking \(V'\) we may assume that \(R = \emptyset\). Thus we may assume that \(X \times_Y V = X_1 \amalg \ldots \amalg X_n\) with exactly one point \(x_l \in X_l\) lying over \(v\) with moreover \(\kappa(v) \subset \kappa(x_l)\) purely inseparable. Note that this property is preserved under refinement of the étale neighbourhood \((V, v)\).

For each \(l\) choose an \(i_l\) and a point \(u_l \in U_{i_l}\) mapping to \(x_l\). Now we apply property (A) for the finite morphism \(X \times_Y V \to V\) and the étale morphisms \(U_{i_l} \to X \times_Y V\) and the points \(u_l\). This is permissible by Lemma 04DM This gives produces an étale neighbourhood \((V', v') \to (V, v)\) and decompositions \[X \times_Y V' = W_l \amalg R_l\] and \(X\)-morphisms \(a_l : W_l \to U_{i_l}\) whose image contains \(u_{i_l}\). Here is a picture: \[\xymatrix{ & & & U_{i_l} \ar[d] & \\ W_l \ar[rrru] \ar[r] & W_l \amalg R_l \ar@{=}[r] & X \times_Y V' \ar[r] \ar[d] & X \times_Y V \ar[r] \ar[d] & X \ar[d] \\ & & V' \ar[r] & V \ar[r] & Y }\] After replacing \((V, v)\) by \((V', v')\) we conclude that each \(x_l\) is contained in an open and closed neighbourhood \(W_l\) such that the inclusion morphism \(W_l \to X \times_Y V\) factors through \(U_i \to X \times_Y V\) for some \(i\). Replacing \(W_l\) by \(W_l \cap X_l\) we see that these open and closed sets are disjoint and moreover that \(\{x_1, \ldots, x_n\} \subset W_1 \cup \ldots \cup W_n\). Since \(X \times_Y V \to V\) is finite we may shrink \(V\) and assume that \(X \times_Y V = W_1 \amalg \ldots \amalg W_n\) as desired.

Lemma

Let \(f : X \to Y\) be an integral morphism of schemes. Then property (B) holds.

Proof

Consider \(V \to Y\) étale, \(\{U_i \to X \times_Y V\}\) an étale covering, and \(v \in V\). We have to find a \(V' \to V\) and decomposition and maps as in Lemma 04DP. We may shrink \(V\) and \(Y\), hence we may assume that \(V\) and \(Y\) are affine. Since \(X\) is integral over \(Y\), this also implies that \(X\) and \(X \times_Y V\) are affine. We may refine the covering \(\{U_i \to X \times_Y V\}\), and hence we may assume that \(\{U_i \to X \times_Y V\}_{i = 1, \ldots, n}\) is a standard étale covering. Write \(Y = \Spec(A)\), \(X = \Spec(B)\), \(V = \Spec(C)\), and \(U_i = \Spec(B_i)\). Then \(A \to B\) is an integral ring map, and \(B \otimes_A C \to B_i\) are étale ring maps. By Algebra, Lemma 00U2 we can find a finite \(A\)-subalgebra \(B' \subset B\) and an étale ring map \(B' \otimes_A C \to B'_i\) for \(i = 1, \ldots, n\) such that \(B_i = B \otimes_{B'} B'_i\). Thus the question reduces to the étale covering \(\{\Spec(B'_i) \to X' \times_Y V\}_{i = 1, \ldots, n}\) with \(X' = \Spec(B')\) finite over \(Y\). In this case the result follows from Lemma 04DQ.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume \(f\) is integral (for example finite). Then

1. \(f_{small, *}\) transforms surjections into surjections (on sheaves of sets and on abelian sheaves),

2. \(f_{small}^{-1}f_{small, *}\mathcal{F} \to \mathcal{F}\) is surjective for any abelian sheaf \(\mathcal{F}\) on \(X_\etale\),

3. \(f_{small, *} : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)\) is faithful and reflects injections and surjections, and

4. \(f_{small, *} : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)\) is exact.

Proof

Parts (2), (3) we have seen in Lemma 04C9. Part (1) follows from Lemmas 04DR and 04DO. Part (4) is a consequence of part (1), see Modules on Sites, Lemma 04DB.

Property (C)

Please see Section 04C6 for the definition of property (C).

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume (C) holds. Then the functor \(f_{small, *} : \Sh(X_\etale) \to \Sh(Y_\etale)\) reflects injections and surjections.

Proof

Follows from Sites, Lemma 04D9. We omit the verification that property (C) implies that the functor \(Y_\etale \to X_\etale\), \(V \mapsto X \times_Y V\) satisfies the assumption of Sites, Lemma 04D9.

Remark

Property (C) holds if \(f : X \to Y\) is an open immersion. Namely, if \(U \in \Ob(X_\etale)\), then we can view \(U\) also as an object of \(Y_\etale\) and \(U \times_Y X = U\). Hence property (C) does not imply that \(f_{small, *}\) is exact as this is not the case for open immersions (in general).

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume that for any \(V \to Y\) étale we have that

1. \(X \times_Y V \to V\) has property (C), and

2. \(X \times_Y V \to V\) is closed.

Then the functor \(Y_\etale \to X_\etale\), \(V \mapsto X \times_Y V\) is almost cocontinuous, see Sites, Definition 04B7.

Proof

Let \(V \to Y\) be an object of \(Y_\etale\) and let \(\{U_i \to X \times_Y V\}_{i \in I}\) be a covering of \(X_\etale\). By assumption (1) for each \(i\) we can find an étale morphism \(h_i : V_i \to V\) and a surjective morphism \(X \times_Y V_i \to U_i\) over \(X \times_Y V\). Note that \(\bigcup h_i(V_i) \subset V\) is an open set containing the closed set \(Z = \Im(X \times_Y V \to V)\). Let \(h_0 : V_0 = V \setminus Z \to V\) be the open immersion. It is clear that \(\{V_i \to V\}_{i \in I \cup \{0\}}\) is an étale covering such that for each \(i \in I \cup \{0\}\) we have either \(V_i \times_Y X = \emptyset\) (namely if \(i = 0\)), or \(V_i \times_Y X \to V \times_Y X\) factors through \(U_i \to X \times_Y V\) (if \(i \not = 0\)). Hence the functor \(Y_\etale \to X_\etale\) is almost cocontinuous.

Lemma

Let \(f : X \to Y\) be an integral morphism of schemes which defines a homeomorphism of \(X\) with a closed subset of \(Y\). Then property (C) holds.

Proof

Let \(g : U \to X\) be an étale morphism. We need to find an object \(V \to Y\) of \(Y_\etale\) and a surjective morphism \(X \times_Y V \to U\) over \(X\). Suppose that for every \(u \in U\) we can find an object \(V_u \to Y\) of \(Y_\etale\) and a morphism \(h_u : X \times_Y V_u \to U\) over \(X\) with \(u \in \Im(h_u)\). Then we can take \(V = \coprod V_u\) and \(h = \coprod h_u\) and we win. Hence given a point \(u \in U\) we find a pair \((V_u, h_u)\) as above. To do this we may shrink \(U\) and assume that \(U\) is affine. In this case \(g : U \to X\) is locally quasi-finite. Let \(g^{-1}(g(\{u\})) = \{u, u_2, \ldots, u_n\}\). Since there are no specializations \(u_i \leadsto u\) we may replace \(U\) by an affine neighbourhood so that \(g^{-1}(g(\{u\})) = \{u\}\).

The image \(g(U) \subset X\) is open, hence \(f(g(U))\) is locally closed in \(Y\). Choose an open \(V \subset Y\) such that \(f(g(U)) = f(X) \cap V\). It follows that \(g\) factors through \(X \times_Y V\) and that the resulting \(\{U \to X \times_Y V\}\) is an étale covering. Since \(f\) has property (B) , see Lemma 04DR, we see that there exists an étale covering \(\{V_j \to V\}\) such that \(X \times_Y V_j \to X \times_Y V\) factor through \(U\). This implies that \(V' = \coprod V_j\) is étale over \(Y\) and that there is a morphism \(h : X \times_Y V' \to U\) whose image surjects onto \(g(U)\). Since \(u\) is the only point in its fibre it must be in the image of \(h\) and we win.

We urge the reader to think of the following lemma as a way station³ on the journey towards the ultimate truth regarding \(f_{small, *}\) for integral universally injective morphisms.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume that \(f\) is universally injective and integral (for example a closed immersion). Then

1. \(f_{small, *} : \Sh(X_\etale) \to \Sh(Y_\etale)\) reflects injections and surjections,

2. \(f_{small, *} : \Sh(X_\etale) \to \Sh(Y_\etale)\) commutes with pushouts and coequalizers (and more generally finite connected colimits),

3. \(f_{small, *}\) transforms surjections into surjections (on sheaves of sets and on abelian sheaves),

4. the map \(f_{small}^{-1}f_{small, *}\mathcal{F} \to \mathcal{F}\) is surjective for any sheaf (of sets or of abelian groups) \(\mathcal{F}\) on \(X_\etale\),

5. the functor \(f_{small, *}\) is faithful (on sheaves of sets and on abelian sheaves),

6. \(f_{small, *} : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)\) is exact, and

7. the functor \(Y_\etale \to X_\etale\), \(V \mapsto X \times_Y V\) is almost cocontinuous.

Proof

By Lemmas 04DM, 04DR and 04DW we know that the morphism \(f\) has properties (A), (B), and (C). Moreover, by Lemma 04DV we know that the functor \(Y_\etale \to X_\etale\) is almost cocontinuous. Now we have

1. property (C) implies (1) by Lemma 04DT,

2. almost continuous implies (2) by Sites, Lemma 04BA,

3. property (B) implies (3) by Lemma 04DO.

Properties (4), (5), and (6) follow formally from the first three, see Sites, Lemma 04D6 and Modules on Sites, Lemma 04DB. Property (7) we saw above.

Topological invariance of the small étale site

In the following theorem we show that the small étale site is a topological invariant in the following sense: If \(f : X \to Y\) is a morphism of schemes which is a universal homeomorphism, then \(X_\etale \cong Y_\etale\) as sites. This improves the result of Étale Morphisms, Theorem 039R. We first prove the result for morphisms and then we state the result for categories.

Theorem

Let \(X\) and \(Y\) be two schemes over a base scheme \(S\). Let \(S' \to S\) be a universal homeomorphism. Denote \(X'\) (resp. \(Y'\)) the base change to \(S'\). If \(X\) is étale over \(S\), then the map \[\Mor_S(Y, X) \longrightarrow \Mor_{S'}(Y', X')\] is bijective.

Proof

After base changing via \(Y \to S\), we may assume that \(Y = S\). Thus we may and do assume both \(X\) and \(Y\) are étale over \(S\). In other words, the theorem states that the base change functor is a fully faithful functor from the category of schemes étale over \(S\) to the category of schemes étale over \(S'\).

Consider the forgetful functor [0BTZ]\[\begin{equation} \begin{matrix} \text{descent data }(X', \varphi')\text{ relative to }S'/S \\ \text{ with }X'\text{ \'etale over }S' \end{matrix} \longrightarrow \text{schemes }X'\text{ \'etale over }S' \end{equation}\] We claim this functor is an equivalence. On the other hand, the functor [0BU0]\[\begin{equation} \text{schemes }X\text{ \'etale over }S \longrightarrow \begin{matrix} \text{descent data }(X', \varphi')\text{ relative to }S'/S \\ \text{ with }X'\text{ \'etale over }S' \end{matrix} \end{equation}\] is fully faithful by Étale Morphisms, Lemma 0BTL. Thus the claim implies the theorem.

Proof of the claim. Recall that a universal homeomorphism is the same thing as an integral, universally injective, surjective morphism, see Morphisms, Lemma 04DF. In particular, the diagonal \(\Delta : S' \to S' \times_S S'\) is a thickening by Morphisms, Lemma 01S4. Thus by Étale Morphisms, Theorem 025H we see that given \(X' \to S'\) étale there is a unique isomorphism \[\varphi' : X' \times_S S' \to S' \times_S X'\] of schemes étale over \(S' \times_S S'\) which pulls back under \(\Delta\) to \(\text{id} : X' \to X'\) over \(S'\). Since \(S' \to S' \times_S S' \times_S S'\) is a thickening as well (it is bijective and a closed immersion) we conclude that \((X', \varphi')\) is a descent datum relative to \(S'/S\). The canonical nature of the construction of \(\varphi'\) shows that it is compatible with morphisms between schemes étale over \(S'\). In other words, we obtain a quasi-inverse \(X' \mapsto (X', \varphi')\) of the functor (0BTZ). This proves the claim and finishes the proof of the theorem.

Theorem

Let \(f : X \to Y\) be a morphism of schemes. Assume \(f\) is integral, universally injective and surjective (i.e., \(f\) is a universal homeomorphism, see Morphisms, Lemma 04DF). The functor \[V \longmapsto V_X = X \times_Y V\] defines an equivalence of categories \[\{ \text{schemes }V\text{ \'etale over }Y \} \leftrightarrow \{ \text{schemes }U\text{ \'etale over }X \}\]

We give two proofs. The first uses effectivity of descent for quasi-compact, separated, étale morphisms relative to surjective integral morphisms. The second uses the material on properties (A), (B), and (C) discussed earlier in the chapter.

Proof

By Theorem 0BTY we see that the functor is fully faithful. It remains to show that the functor is essentially surjective. Let \(U \to X\) be an étale morphism of schemes.

Suppose that the result holds if \(U\) and \(Y\) are affine. In that case, we choose an affine open covering \(U = \bigcup U_i\) such that each \(U_i\) maps into an affine open of \(Y\). By assumption (affine case) we can find étale morphisms \(V_i \to Y\) such that \(X \times_Y V_i \cong U_i\) as schemes over \(X\). Let \(V_{i, i'} \subset V_i\) be the open subscheme whose underlying topological space corresponds to \(U_i \cap U_{i'}\). Because we have isomorphisms \[X \times_Y V_{i, i'} \cong U_i \cap U_{i'} \cong X \times_Y V_{i', i}\] as schemes over \(X\) we see by fully faithfulness that we obtain isomorphisms \(\theta_{i, i'} : V_{i, i'} \to V_{i', i}\) of schemes over \(Y\). We omit the verification that these isomorphisms satisfy the cocycle condition of Schemes, Section 01JA. Applying Schemes, Lemma 01JC we obtain a scheme \(V \to Y\) by glueing the schemes \(V_i\) along the identifications \(\theta_{i, i'}\). It is clear that \(V \to Y\) is étale and \(X \times_Y V \cong U\) by construction.

Thus it suffices to show the lemma in case \(U\) and \(Y\) are affine. Recall that in the proof of Theorem 0BTY we showed that \(U\) comes with a unique descent datum \((U, \varphi)\) relative to \(X/Y\). By Étale Morphisms, Proposition 0BTP (which applies because \(U \to X\) is quasi-compact and separated as well as étale by our reduction to the affine case) there exists an étale morphism \(V \to Y\) such that \(X \times_Y V \cong U\) and the proof is complete.

Proof

By Theorem 0BTY we see that the functor is fully faithful. It remains to show that the functor is essentially surjective. Let \(U \to X\) be an étale morphism of schemes.

Suppose that the result holds if \(U\) and \(Y\) are affine. In that case, we choose an affine open covering \(U = \bigcup U_i\) such that each \(U_i\) maps into an affine open of \(Y\). By assumption (affine case) we can find étale morphisms \(V_i \to Y\) such that \(X \times_Y V_i \cong U_i\) as schemes over \(X\). Let \(V_{i, i'} \subset V_i\) be the open subscheme whose underlying topological space corresponds to \(U_i \cap U_{i'}\). Because we have isomorphisms \[X \times_Y V_{i, i'} \cong U_i \cap U_{i'} \cong X \times_Y V_{i', i}\] as schemes over \(X\) we see by fully faithfulness that we obtain isomorphisms \(\theta_{i, i'} : V_{i, i'} \to V_{i', i}\) of schemes over \(Y\). We omit the verification that these isomorphisms satisfy the cocycle condition of Schemes, Section 01JA. Applying Schemes, Lemma 01JC we obtain a scheme \(V \to Y\) by glueing the schemes \(V_i\) along the identifications \(\theta_{i, i'}\). It is clear that \(V \to Y\) is étale and \(X \times_Y V \cong U\) by construction.

Thus it suffices to prove that the functor [04E0]\[\begin{equation} \{ \text{affine schemes }V\text{ \'etale over }Y \} \leftrightarrow \{ \text{affine schemes }U\text{ \'etale over }X \} \end{equation}\] is essentially surjective when \(X\) and \(Y\) are affine.

Let \(U \to X\) be an affine scheme étale over \(X\). We have to find \(V \to Y\) étale (and affine) such that \(X \times_Y V\) is isomorphic to \(U\) over \(X\). Note that an étale morphism of affines has universally bounded fibres, see Morphisms, Lemmas 03WS and 03JA. Hence we can do induction on the integer \(n\) bounding the degree of the fibres of \(U \to X\). See Morphisms, Lemma 03WU for a description of this integer in the case of an étale morphism. If \(n = 1\), then \(U \to X\) is an open immersion (see Étale Morphisms, Theorem 025G), and the result is clear. Assume \(n > 1\).

By Lemma 04DW there exists an étale morphism of schemes \(W \to Y\) and a surjective morphism \(W_X \to U\) over \(X\). As \(U\) is quasi-compact we may replace \(W\) by a disjoint union of finitely many affine opens of \(W\), hence we may assume that \(W\) is affine as well. Here is a diagram \[\xymatrix{ U \ar[d] & U \times_Y W \ar[l] \ar[d] & W_X \amalg R \ar@{=}[l]\\ X \ar[d] & W_X \ar[l] \ar[d] \\ Y & W \ar[l] }\] The disjoint union decomposition arises because by construction the étale morphism of affine schemes \(U \times_Y W \to W_X\) has a section. OK, and now we see that the morphism \(R \to X \times_Y W\) is an étale morphism of affine schemes whose fibres have degree universally bounded by \(n - 1\). Hence by induction assumption there exists a scheme \(V' \to W\) étale such that \(R \cong W_X \times_W V'\). Taking \(V'' = W \amalg V'\) we find a scheme \(V''\) étale over \(W\) whose base change to \(W_X\) is isomorphic to \(U \times_Y W\) over \(X \times_Y W\).

At this point we can use descent to find \(V\) over \(Y\) whose base change to \(X\) is isomorphic to \(U\) over \(X\). Namely, by the fully faithfulness of the functor (04E0) corresponding to the universal homeomorphism \(X \times_Y (W \times_Y W) \to (W \times_Y W)\) there exists a unique isomorphism \(\varphi : V'' \times_Y W \to W \times_Y V''\) whose base change to \(X \times_Y (W \times_Y W)\) is the canonical descent datum for \(U \times_Y W\) over \(X \times_Y W\). In particular \(\varphi\) satisfies the cocycle condition. Hence by Descent, Lemma 0245 we see that \(\varphi\) is effective (recall that all schemes above are affine). Thus we obtain \(V \to Y\) and an isomorphism \(V'' \cong W \times_Y V\) such that the canonical descent datum on \(W \times_Y V/W/Y\) agrees with \(\varphi\). Note that \(V \to Y\) is étale, by Descent, Lemma 02VN. Moreover, there is an isomorphism \(V_X \cong U\) which comes from descending the isomorphism \[V_X \times_X W_X = X \times_Y V \times_Y W = (X \times_Y W) \times_W (W \times_Y V) \cong W_X \times_W V'' \cong U \times_Y W\] which we have by construction. Some details omitted.

Remark

In the situation of Theorem 04DZ it is also true that \(V \mapsto V_X\) induces an equivalence between those étale morphisms \(V \to Y\) with \(V\) affine and those étale morphisms \(U \to X\) with \(U\) affine. This follows for example from Limits, Proposition 05YU.

Proposition

Let \(X_0 \to X\) be a universal homeomorphism of schemes (for example the closed immersion defined by a nilpotent sheaf of ideals). Then

1. the étale sites \(X_\etale\) and \((X_0)_\etale\) are isomorphic,

2. the étale topoi \(\Sh(X_\etale)\) and \(\Sh((X_0)_\etale)\) are equivalent, and

3. \(H^q_\etale(X, \mathcal{F}) = H^q_\etale(X_0, \mathcal{F}|_{X_0})\) for all \(q\) and for any abelian sheaf \(\mathcal{F}\) on \(X_\etale\).

Proof

The equivalence of categories \(X_\etale \to (X_0)_\etale\) is given by Theorem 04DZ. We omit the proof that under this equivalence the étale coverings correspond. Hence (1) holds. Parts (2) and (3) follow formally from (1).

Closed immersions and pushforward

Before stating and proving Proposition 04CA in its correct generality we briefly state and prove it for closed immersions. Namely, some of the preceding arguments are quite a bit easier to follow in the case of a closed immersion and so we repeat them here in their simplified form.

In the rest of this section \(i : Z \to X\) is a closed immersion. The functor \[\Sch/X \longrightarrow \Sch/Z, \quad U \longmapsto U_Z = Z \times_X U\] will be denoted \(U \mapsto U_Z\) as indicated. Since being a closed immersion is preserved under arbitrary base change the scheme \(U_Z\) is a closed subscheme of \(U\).

Lemma

Let \(i : Z \to X\) be a closed immersion of schemes. Let \(U, U'\) be schemes étale over \(X\). Let \(h : U_Z \to U'_Z\) be a morphism over \(Z\). Then there exists a diagram \[\xymatrix{ U & W \ar[l]_a \ar[r]^b & U' }\] in \(X_\etale\) such that \(a_Z : W_Z \to U_Z\) is an isomorphism and \(h = b_Z \circ (a_Z)^{-1}\).

Proof

Consider the scheme \(M = U \times_X U'\). The graph \(\Gamma_h \subset M_Z\) of \(h\) is open. This is true for example as \(\Gamma_h\) is the image of a section of the étale morphism \(\text{pr}_{1, Z} : M_Z \to U_Z\), see Étale Morphisms, Proposition 024T. Hence there exists an open subscheme \(W \subset M\) whose intersection with the closed subset \(M_Z\) is \(\Gamma_h\). Set \(a = \text{pr}_1|_W\) and \(b = \text{pr}_2|_W\).

Lemma

Let \(i : Z \to X\) be a closed immersion of schemes. Let \(V \to Z\) be an étale morphism of schemes. There exist étale morphisms \(U_i \to X\) and morphisms \(U_{i, Z} \to V\) such that \(\{U_{i, Z} \to V\}\) is a Zariski covering of \(V\).

Proof

Since we only have to find a Zariski covering of \(V\) consisting of schemes of the form \(U_Z\) with \(U\) étale over \(X\), we may Zariski localize on \(X\) and \(V\). Hence we may assume \(X\) and \(V\) affine. In the affine case this is Algebra, Lemma 04D1.

If \(\overline{x} : \Spec(k) \to X\) is a geometric point of \(X\), then either \(\overline{x}\) factors (uniquely) through the closed subscheme \(Z\), or \(Z_{\overline{x}} = \emptyset\). If \(\overline{x}\) factors through \(Z\) we say that \(\overline{x}\) is a geometric point of \(Z\) (because it is) and we use the notation “\(\overline{x} \in Z\)” to indicate this.

Lemma

Let \(i : Z \to X\) be a closed immersion of schemes. Let \(\mathcal{G}\) be a sheaf of sets on \(Z_\etale\). Let \(\overline{x}\) be a geometric point of \(X\). Then \[(i_{small, *}\mathcal{G})_{\overline{x}} = \left\{ \begin{matrix} * & \text{if} & \overline{x} \not \in Z \\ \mathcal{G}_{\overline{x}} & \text{if} & \overline{x} \in Z \end{matrix} \right.\] where \(*\) denotes a singleton set.

Proof

Let \(U = X \setminus Z\). Note that \(i_{small, *}\mathcal{G}|_{U_\etale} = *\) is the final object in the category of étale sheaves on \(U\), i.e., the sheaf which associates a singleton set to each scheme étale over \(U\). This explains the value of \((i_{small, *}\mathcal{G})_{\overline{x}}\) if \(\overline{x} \not \in Z\).

Next, suppose that \(\overline{x} \in Z\). Note that \[(i_{small, *}\mathcal{G})_{\overline{x}} = \colim_{(U, \overline{u})} \mathcal{G}(U_Z)\] and on the other hand \[\mathcal{G}_{\overline{x}} = \colim_{(V, \overline{v})} \mathcal{G}(V).\] Let \(\mathcal{C}_1 = \{(U, \overline{u})\}^{opp}\) be the opposite of the category of étale neighbourhoods of \(\overline{x}\) in \(X\), and let \(\mathcal{C}_2 = \{(V, \overline{v})\}^{opp}\) be the opposite of the category of étale neighbourhoods of \(\overline{x}\) in \(Z\). The canonical map \[\mathcal{G}_{\overline{x}} \longrightarrow (i_{small, *}\mathcal{G})_{\overline{x}}\] corresponds to the functor \(F : \mathcal{C}_1 \to \mathcal{C}_2\), \(F(U, \overline{u}) = (U_Z, \overline{x})\). Now Lemmas 04FW and 04FV imply that \(\mathcal{C}_1\) is cofinal in \(\mathcal{C}_2\), see Categories, Definition 04E6. Hence it follows that the displayed arrow is an isomorphism, see Categories, Lemma 04E7.

Proposition

Let \(i : Z \to X\) be a closed immersion of schemes.

1. The functor \[i_{small, *} : \Sh(Z_\etale) \longrightarrow \Sh(X_\etale)\] is fully faithful and its essential image is those sheaves of sets \(\mathcal{F}\) on \(X_\etale\) whose restriction to \(X \setminus Z\) is isomorphic to \(*\), and

2. the functor \[i_{small, *} : \textit{Ab}(Z_\etale) \longrightarrow \textit{Ab}(X_\etale)\] is fully faithful and its essential image is those abelian sheaves on \(X_\etale\) whose support is contained in \(Z\).

In both cases \(i_{small}^{-1}\) is a left inverse to the functor \(i_{small, *}\).

Proof

Let’s discuss the case of sheaves of sets. For any sheaf \(\mathcal{G}\) on \(Z\) the morphism \(i_{small}^{-1}i_{small, *}\mathcal{G} \to \mathcal{G}\) is an isomorphism by Lemma 04FX (and Theorem 03PU). This implies formally that \(i_{small, *}\) is fully faithful, see Sites, Lemma 04D6. It is clear that \(i_{small, *}\mathcal{G}|_{U_\etale} \cong *\) where \(U = X \setminus Z\). Conversely, suppose that \(\mathcal{F}\) is a sheaf of sets on \(X\) such that \(\mathcal{F}|_{U_\etale} \cong *\). Consider the adjunction mapping \[\mathcal{F} \longrightarrow i_{small, *}i_{small}^{-1}\mathcal{F}\] Combining Lemmas 04FX and 03Q1 we see that it is an isomorphism. This finishes the proof of (1). The proof of (2) is identical.

Integral universally injective morphisms

Here is the general version of Proposition 04CA.

Proposition

Let \(f : X \to Y\) be a morphism of schemes which is integral and universally injective.

1. The functor \[f_{small, *} : \Sh(X_\etale) \longrightarrow \Sh(Y_\etale)\] is fully faithful and its essential image is those sheaves of sets \(\mathcal{F}\) on \(Y_\etale\) whose restriction to \(Y \setminus f(X)\) is isomorphic to \(*\), and

2. the functor \[f_{small, *} : \textit{Ab}(X_\etale) \longrightarrow \textit{Ab}(Y_\etale)\] is fully faithful and its essential image is those abelian sheaves on \(Y_\etale\) whose support is contained in \(f(X)\).

In both cases \(f_{small}^{-1}\) is a left inverse to the functor \(f_{small, *}\).

Proof

We may factor \(f\) as \[\xymatrix{ X \ar[r]^h & Z \ar[r]^i & Y }\] where \(h\) is integral, universally injective and surjective and \(i : Z \to Y\) is a closed immersion. Apply Proposition 04CA to \(i\) and apply Theorem 04DZ to \(h\).

Big sites and pushforward

In this section we prove some technical results on \(f_{big, *}\) for certain types of morphisms of schemes.

Lemma

Let \(\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}\). Let \(f : X \to Y\) be a monomorphism of schemes. Then the canonical map \(f_{big}^{-1}f_{big, *}\mathcal{F} \to \mathcal{F}\) is an isomorphism for any sheaf \(\mathcal{F}\) on \((\Sch/X)_\tau\).

Proof

In this case the functor \((\Sch/X)_\tau \to (\Sch/Y)_\tau\) is continuous, cocontinuous and fully faithful. Hence the result follows from Sites, Lemma 00XT.

Remark

In the situation of Lemma 04C7 it is true that the canonical map \(\mathcal{F} \to f_{big}^{-1}f_{big!}\mathcal{F}\) is an isomorphism for any sheaf of sets \(\mathcal{F}\) on \((\Sch/X)_\tau\). The proof is the same. This also holds for sheaves of abelian groups. However, note that the functor \(f_{big!}\) for sheaves of abelian groups is defined in Modules on Sites, Section 04BE and is in general different from \(f_{big!}\) on sheaves of sets. The result for sheaves of abelian groups follows from Modules on Sites, Lemma 077I.

Lemma

Let \(f : X \to Y\) be a closed immersion of schemes. Let \(U \to X\) be a syntomic (resp. smooth, resp. étale) morphism. Then there exist syntomic (resp. smooth, resp. étale) morphisms \(V_i \to Y\) and morphisms \(V_i \times_Y X \to U\) such that \(\{V_i \times_Y X \to U\}\) is a Zariski covering of \(U\).

Proof

Let us prove the lemma when \(\tau = syntomic\). The question is local on \(U\). Thus we may assume that \(U\) is an affine scheme mapping into an affine of \(Y\). Hence we reduce to proving the following case: \(Y = \Spec(A)\), \(X = \Spec(A/I)\), and \(U = \Spec(\overline{B})\), where \(A/I \to \overline{B}\) be a syntomic ring map. By Algebra, Lemma 00T0 we can find elements \(\overline{g}_i \in \overline{B}\) such that \(\overline{B}_{\overline{g}_i} = A_i/IA_i\) for certain syntomic ring maps \(A \to A_i\). This proves the lemma in the syntomic case. The proof of the smooth case is the same except it uses Algebra, Lemma 04B1. In the étale case use Algebra, Lemma 04D1.

Lemma

Let \(f : X \to Y\) be a closed immersion of schemes. Let \(\{U_i \to X\}\) be a syntomic (resp. smooth, resp. étale) covering. There exists a syntomic (resp. smooth, resp. étale) covering \(\{V_j \to Y\}\) such that for each \(j\), either \(V_j \times_Y X = \emptyset\), or the morphism \(V_j \times_Y X \to X\) factors through \(U_i\) for some \(i\).

Proof

For each \(i\) we can choose syntomic (resp. smooth, resp. étale) morphisms \(g_{ij} : V_{ij} \to Y\) and morphisms \(V_{ij} \times_Y X \to U_i\) over \(X\), such that \(\{V_{ij} \times_Y X \to U_i\}\) are Zariski coverings, see Lemma 04E3. This in particular implies that \(\bigcup_{ij} g_{ij}(V_{ij})\) contains the closed subset \(f(X)\). Hence the family of syntomic (resp. smooth, resp. étale) maps \(g_{ij}\) together with the open immersion \(Y \setminus f(X) \to Y\) forms the desired syntomic (resp. smooth, resp. étale) covering of \(Y\).

Lemma

Let \(f : X \to Y\) be a closed immersion of schemes. Let \(\tau \in \{syntomic, smooth, \etale\}\). The functor \(V \mapsto X \times_Y V\) defines an almost cocontinuous functor (see Sites, Definition 04B7) \((\Sch/Y)_\tau \to (\Sch/X)_\tau\) between big \(\tau\) sites.

Proof

We have to show the following: given a morphism \(V \to Y\) and any syntomic (resp. smooth, resp. étale) covering \(\{U_i \to X \times_Y V\}\), there exists a smooth (resp. smooth, resp. étale) covering \(\{V_j \to V\}\) such that for each \(j\), either \(X \times_Y V_j\) is empty, or \(X \times_Y V_j \to Z \times_Y V\) factors through one of the \(U_i\). This follows on applying Lemma 04E4 above to the closed immersion \(X \times_Y V \to V\).

Lemma

Let \(f : X \to Y\) be a closed immersion of schemes. Let \(\tau \in \{syntomic, smooth, \etale\}\).

1. The pushforward \(f_{big, *} : \Sh((\Sch/X)_\tau) \to \Sh((\Sch/Y)_\tau)\) commutes with coequalizers and pushouts.

2. The pushforward \(f_{big, *} : \textit{Ab}((\Sch/X)_\tau) \to \textit{Ab}((\Sch/Y)_\tau)\) is exact.

Proof

This follows from Sites, Lemma 04BA, Modules on Sites, Lemma 04BD, and Lemma 04C3 above.

Remark

In Lemma 04C4 the case \(\tau = fppf\) is missing. The reason is that given a ring \(A\), an ideal \(I\) and a faithfully flat, finitely presented ring map \(A/I \to \overline{B}\), there is no reason to think that one can find any flat finitely presented ring map \(A \to B\) with \(B/IB \not = 0\) such that \(A/I \to B/IB\) factors through \(\overline{B}\). Hence the proof of Lemma 04C3 does not work for the fppf topology. In fact it is likely false that \(f_{big, *} : \textit{Ab}((\Sch/X)_{fppf}) \to \textit{Ab}((\Sch/Y)_{fppf})\) is exact when \(f\) is a closed immersion. If you know an example, please email stacks.project@gmail.com.

Exactness of big lower shriek

This is just the following technical result. Note that the functor \(f_{big!}\) has nothing whatsoever to do with cohomology with compact support in general.

Lemma

Let \(\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}\). Let \(f : X \to Y\) be a morphism of schemes. Let \[f_{big} : \Sh((\Sch/X)_\tau) \longrightarrow \Sh((\Sch/Y)_\tau)\] be the corresponding morphism of topoi as in Topologies, Lemma 0210, 021H, 04HC, 04HD, or 021W.

1. The functor \(f_{big}^{-1} : \textit{Ab}((\Sch/Y)_\tau) \to \textit{Ab}((\Sch/X)_\tau)\) has a left adjoint \[f_{big!} : \textit{Ab}((\Sch/X)_\tau) \to \textit{Ab}((\Sch/Y)_\tau)\] which is exact.

2. The functor \(f_{big}^* : \textit{Mod}((\Sch/Y)_\tau, \mathcal{O}) \to \textit{Mod}((\Sch/X)_\tau, \mathcal{O})\) has a left adjoint \[f_{big!} : \textit{Mod}((\Sch/X)_\tau, \mathcal{O}) \to \textit{Mod}((\Sch/Y)_\tau, \mathcal{O})\] which is exact.

Moreover, the two functors \(f_{big!}\) agree on underlying sheaves of abelian groups.

Proof

Recall that \(f_{big}\) is the morphism of topoi associated to the continuous and cocontinuous functor \(u : (\Sch/X)_\tau \to (\Sch/Y)_\tau\), \(U/X \mapsto U/Y\). Moreover, we have \(f_{big}^{-1}\mathcal{O} = \mathcal{O}\). Hence the existence of \(f_{big!}\) follows from Modules on Sites, Lemma 04BG, respectively Modules on Sites, Lemma 0797. Note that if \(U\) is an object of \((\Sch/X)_\tau\) then the functor \(u\) induces an equivalence of categories \[u' : (\Sch/X)_\tau/U \longrightarrow (\Sch/Y)_\tau/U\] because both sides of the arrow are equal to \((\Sch/U)_\tau\). Hence the agreement of \(f_{big!}\) on underlying abelian sheaves follows from the discussion in Modules on Sites, Remark 0798. The exactness of \(f_{big!}\) follows from Modules on Sites, Lemma 04BH as the functor \(u\) above which commutes with fibre products and equalizers.

Next, we prove a technical lemma that will be useful later when comparing sheaves of modules on different sites associated to algebraic stacks.

Lemma

Let \(X\) be a scheme. Let \(\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}\). Let \(\mathcal{C}_1 \subset \mathcal{C}_2 \subset (\Sch/X)_\tau\) be full subcategories with the following properties:

1. For an object \(U/X\) of \(\mathcal{C}_t\),

   1. if \(\{U_i \to U\}\) is a covering of \((\Sch/X)_\tau\), then \(U_i/X\) is an object of \(\mathcal{C}_t\),

   2. \(U \times \mathbf{A}^1/X\) is an object of \(\mathcal{C}_t\).

2. \(X/X\) is an object of \(\mathcal{C}_t\).

We endow \(\mathcal{C}_t\) with the structure of a site whose coverings are exactly those coverings \(\{U_i \to U\}\) of \((\Sch/X)_\tau\) with \(U \in \Ob(\mathcal{C}_t)\). Then

1. The functor \(\mathcal{C}_1 \to \mathcal{C}_2\) is fully faithful, continuous, and cocontinuous.

Denote \(g : \Sh(\mathcal{C}_1) \to \Sh(\mathcal{C}_2)\) the corresponding morphism of topoi. Denote \(\mathcal{O}_t\) the restriction of \(\mathcal{O}\) to \(\mathcal{C}_t\). Denote \(g_!\) the functor of Modules on Sites, Definition 04BF.

1. The canonical map \(g_!\mathcal{O}_1 \to \mathcal{O}_2\) is an isomorphism.

Proof

Assertion (a) is immediate from the definitions. In this proof all schemes are schemes over \(X\) and all morphisms of schemes are morphisms of schemes over \(X\). Note that \(g^{-1}\) is given by restriction, so that for an object \(U\) of \(\mathcal{C}_1\) we have \(\mathcal{O}_1(U) = \mathcal{O}_2(U) = \mathcal{O}(U)\). Recall that \(g_!\mathcal{O}_1\) is the sheaf associated to the presheaf \(g_{p!}\mathcal{O}_1\) which associates to \(V\) in \(\mathcal{C}_2\) the group \[\colim_{V \to U} \mathcal{O}(U)\] where \(U\) runs over the objects of \(\mathcal{C}_1\) and the colimit is taken in the category of abelian groups. Below we will use frequently that if \[V \to U \to U'\] are morphisms with \(U, U' \in \Ob(\mathcal{C}_1)\) and if \(f' \in \mathcal{O}(U')\) restricts to \(f \in \mathcal{O}(U)\), then \((V \to U, f)\) and \((V \to U', f')\) define the same element of the colimit. Also, \(g_!\mathcal{O}_1 \to \mathcal{O}_2\) maps the element \((V \to U, f)\) simply to the pullback of \(f\) to \(V\).

Surjectivity. Let \(V\) be a scheme and let \(h \in \mathcal{O}(V)\). Then we obtain a morphism \(V \to X \times \mathbf{A}^1\) induced by \(h\) and the structure morphism \(V \to X\). Writing \(\mathbf{A}^1 = \Spec(\mathbf{Z}[x])\) we see the element \(x \in \mathcal{O}(X \times \mathbf{A}^1)\) pulls back to \(h\). Since \(X \times \mathbf{A}^1\) is an object of \(\mathcal{C}_1\) by assumptions (1)(b) and (2) we obtain the desired surjectivity.

Injectivity. Let \(V\) be a scheme. Let \(s = \sum_{i = 1, \ldots, n} (V \to U_i, f_i)\) be an element of the colimit displayed above. For any \(i\) we can use the morphism \(f_i : U_i \to X \times \mathbf{A}^1\) to see that \((V \to U_i, f_i)\) defines the same element of the colimit as \((f_i : V \to X \times \mathbf{A}^1, x)\). Then we can consider \[f_1 \times \ldots \times f_n : V \to X \times \mathbf{A}^n\] and we see that \(s\) is equivalent in the colimit to \[\sum\nolimits_{i = 1, \ldots, n} (f_1 \times \ldots \times f_n : V \to X \times \mathbf{A}^n, x_i) = (f_1 \times \ldots \times f_n : V \to X \times \mathbf{A}^n, x_1 + \ldots + x_n)\] Now, if \(x_1 + \ldots + x_n\) restricts to zero on \(V\), then we see that \(f_1 \times \ldots \times f_n\) factors through \(X \times \mathbf{A}^{n - 1} = V(x_1 + \ldots + x_n)\). Hence we see that \(s\) is equivalent to zero in the colimit.

Étale cohomology

In the following sections we prove some basic results on étale cohomology. Here is an example of something we know for cohomology of topological spaces which also holds for étale cohomology.

Lemma

Let \(X\) be a scheme. Suppose that \(X = U \cup V\) is a union of two opens. For any abelian sheaf \(\mathcal{F}\) on \(X_\etale\) there exists a long exact cohomology sequence \[\begin{matrix} 0 \to H^0_\etale(X, \mathcal{F}) \to H^0_\etale(U, \mathcal{F}) \oplus H^0_\etale(V, \mathcal{F}) \to H^0_\etale(U \cap V, \mathcal{F}) \phantom{\to \ldots} \\ \phantom{0} \to H^1_\etale(X, \mathcal{F}) \to H^1_\etale(U, \mathcal{F}) \oplus H^1_\etale(V, \mathcal{F}) \to H^1_\etale(U \cap V, \mathcal{F}) \to \ldots \end{matrix}\] This long exact sequence is functorial in \(\mathcal{F}\).

Proof

Observe that if \(\mathcal{I}\) is an injective abelian sheaf, then \[0 \to \mathcal{I}(X) \to \mathcal{I}(U) \oplus \mathcal{I}(V) \to \mathcal{I}(U \cap V) \to 0\] is exact. This is true in the first and middle spots as \(\mathcal{I}\) is a sheaf. It is true on the right, because \(\mathcal{I}(U) \to \mathcal{I}(U \cap V)\) is surjective by Cohomology on Sites, Lemma 093X. Another way to prove it would be to show that the cokernel of the map \(\mathcal{I}(U) \oplus \mathcal{I}(V) \to \mathcal{I}(U \cap V)\) is the first Čech cohomology group of \(\mathcal{I}\) with respect to the covering \(X = U \cup V\) which vanishes by Lemmas 03OR and 03OV. Thus, if \(\mathcal{F} \to \mathcal{I}^\bullet\) is an injective resolution, then \[0 \to \mathcal{I}^\bullet(X) \to \mathcal{I}^\bullet(U) \oplus \mathcal{I}^\bullet(V) \to \mathcal{I}^\bullet(U \cap V) \to 0\] is a short exact sequence of complexes and the associated long exact cohomology sequence is the sequence of the statement of the lemma.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Suppose that \(X = U \cup V\) is a union of two open subschemes. Denote \(a = f|_U : U \to Y\), \(b = f|_V : V \to Y\), and \(c = f|_{U \cap V} : U \cap V \to Y\). For every abelian sheaf \(\mathcal{F}\) on \(X_\etale\) there exists a long exact sequence \[0 \to f_*\mathcal{F} \to a_*(\mathcal{F}|_U) \oplus b_*(\mathcal{F}|_V) \to c_*(\mathcal{F}|_{U \cap V}) \to R^1f_*\mathcal{F} \to \ldots\] on \(Y_\etale\). This long exact sequence is functorial in \(\mathcal{F}\).

Proof

Let \(\mathcal{F} \to \mathcal{I}^\bullet\) be an injective resolution of \(\mathcal{F}\) on \(X_\etale\). We claim that we get a short exact sequence of complexes \[0 \to f_*\mathcal{I}^\bullet \to a_*\mathcal{I}^\bullet|_U \oplus b_*\mathcal{I}^\bullet|_V \to c_*\mathcal{I}^\bullet|_{U \cap V} \to 0.\] Namely, for any \(W\) in \(Y_\etale\), and for any \(n \geq 0\) the corresponding sequence of groups of sections over \(W\) \[0 \to \mathcal{I}^n(W \times_Y X) \to \mathcal{I}^n(W \times_Y U) \oplus \mathcal{I}^n(W \times_Y V) \to \mathcal{I}^n(W \times_Y (U \cap V)) \to 0\] was shown to be short exact in the proof of Lemma 0A50. The lemma follows by taking cohomology sheaves and using the fact that \(\mathcal{I}^\bullet|_U\) is an injective resolution of \(\mathcal{F}|_U\) and similarly for \(\mathcal{I}^\bullet|_V\), \(\mathcal{I}^\bullet|_{U \cap V}\).

Colimits

We recall that if \((\mathcal{F}_i, \varphi_{ii'})\) is a diagram of sheaves on a site \(\mathcal{C}\) its colimit (in the category of sheaves) is the sheafification of the presheaf \(U \mapsto \colim_i \mathcal{F}_i(U)\). See Sites, Lemma 00WI. If the system is directed, \(U\) is a quasi-compact object of \(\mathcal{C}\) which has a cofinal system of coverings by quasi-compact objects, then \(\mathcal{F}(U) = \colim \mathcal{F}_i(U)\), see Sites, Lemma 0738. See Cohomology on Sites, Lemma 0739 for a result dealing with higher cohomology groups of colimits of abelian sheaves.

In Cohomology on Sites, Lemma 09YP we generalize this result to a system of sheaves on an inverse system of sites. Here is the corresponding notion in the case of a system of étale sheaves living on an inverse system of schemes.

Definition

Let \(I\) be a preordered set. Let \((X_i, f_{i'i})\) be an inverse system of schemes over \(I\). A system \((\mathcal{F}_i, \varphi_{i'i})\) of sheaves on \((X_i, f_{i'i})\) is given by

1. a sheaf \(\mathcal{F}_i\) on \((X_i)_\etale\) for all \(i \in I\),

2. for \(i' \geq i\) a map \(\varphi_{i'i} : f_{i'i}^{-1}\mathcal{F}_i \to \mathcal{F}_{i'}\) of sheaves on \((X_{i'})_\etale\)

such that \(\varphi_{i''i} = \varphi_{i''i'} \circ f_{i'' i'}^{-1}\varphi_{i'i}\) whenever \(i'' \geq i' \geq i\).

In the situation of Definition 0EZL, assume \(I\) is a directed set and the transition morphisms \(f_{i'i}\) affine. Let \(X = \lim X_i\) be the limit in the category of schemes, see Limits, Section 01YV. Denote \(f_i : X \to X_i\) the projection morphisms and consider the maps \[f_i^{-1}\mathcal{F}_i = f_{i'}^{-1}f_{i'i}^{-1}\mathcal{F}_i \xrightarrow{f_{i'}^{-1}\varphi_{i'i}} f_{i'}^{-1}\mathcal{F}_{i'}\] This turns \(f_i^{-1}\mathcal{F}_i\) into a system of sheaves on \(X_\etale\) over \(I\) (it is a good exercise to check this). We often want to know whether there is an isomorphism \[H^q_\etale(X, \colim f_i^{-1}\mathcal{F}_i) = \colim H^q_\etale(X_i, \mathcal{F}_i)\] It will turn out this is true if \(X_i\) is quasi-compact and quasi-separated for all \(i\), see Theorem 09YQ.

Lemma

Let \(I\) be a directed set. Let \((X_i, f_{i'i})\) be an inverse system of schemes over \(I\) with affine transition morphisms. Let \(X = \lim_{i \in I} X_i\). With notation as in Topologies, Lemma 04HR we have \[X_{affine, \etale} = \colim (X_i)_{affine, \etale}\] as sites in the sense of Sites, Lemma 09YL.

Proof

Let us first prove this when \(X\) and \(X_i\) are quasi-compact and quasi-separated for all \(i\) (as this is true in all cases of interest). In this case any object of \(X_{affine, \etale}\), resp. \((X_i)_{affine, \etale}\) is of finite presentation over \(X\). Moreover, the category of schemes of finite presentation over \(X\) is the colimit of the categories of schemes of finite presentation over \(X_i\), see Limits, Lemma 01ZM. The same holds for the subcategories of affine objects étale over \(X\) by Limits, Lemmas 01Z6 and 07RP. Finally, if \(\{U^j \to U\}\) is a covering of \(X_{affine, \etale}\) and if \(U_i^j \to U_i\) is morphism of affine schemes étale over \(X_i\) whose base change to \(X\) is \(U^j \to U\), then we see that the base change of \(\{U^j_i \to U_i\}\) to some \(X_{i'}\) is a covering for \(i'\) large enough, see Limits, Lemma 07RR.

In the general case, let \(U\) be an object of \(X_{affine, \etale}\). Then \(U \to X\) is étale and separated (as \(U\) is separated) but in general not quasi-compact. Still, \(U \to X\) is locally of finite presentation and hence by Limits, Lemma 0EY1 there exists an \(i\), a quasi-compact and quasi-separated scheme \(U_i\), and a morphism \(U_i \to X_i\) which is locally of finite presentation whose base change to \(X\) is \(U \to X\). Then \(U = \lim_{i' \geq i} U_{i'}\) where \(U_{i'} = U_i \times_{X_i} X_{i'}\). After increasing \(i\) we may assume \(U_i\) is affine, see Limits, Lemma 01Z6. To check that \(U_i \to X_i\) is étale for \(i\) sufficiently large, choose a finite affine open covering \(U_i = U_{i, 1} \cup \ldots \cup U_{i, m}\) such that \(U_{i, j} \to U_i \to X_i\) maps into an affine open \(W_{i, j} \subset X_i\). Then we can apply Limits, Lemma 07RP to see that \(U_{i, j} \to W_{i, j}\) is étale after possibly increasing \(i\). In this way we see that the functor \(\colim (X_i)_{affine, \etale} \to X_{affine, \etale}\) is essentially surjective. Fully faithfulness follows directly from the already used Limits, Lemma 0EY1. The statement on coverings is proved in exactly the same manner as done in the first paragraph of the proof.

Using the above we get the following general result on colimits and cohomology.

Theorem

Let \(X = \lim_{i \in I} X_i\) be a limit of a directed system of schemes with affine transition morphisms \(f_{i'i} : X_{i'} \to X_i\). We assume that \(X_i\) is quasi-compact and quasi-separated for all \(i \in I\). Let \((\mathcal{F}_i, \varphi_{i'i})\) be a system of abelian sheaves on \((X_i, f_{i'i})\). Denote \(f_i : X \to X_i\) the projection and set \(\mathcal{F} = \colim f_i^{-1}\mathcal{F}_i\). Then \[\colim_{i\in I} H_\etale^p(X_i, \mathcal{F}_i) = H_\etale^p(X, \mathcal{F}).\] for all \(p \geq 0\).

Proof

By Topologies, Lemma 04HR we can compute the cohomology of \(\mathcal{F}\) on \(X_{affine, \etale}\). Thus the result by a combination of Lemma 0EYL and Cohomology on Sites, Lemma 09YP.

The following two results are special cases of the theorem above.

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(I\) be a directed set. Let \((\mathcal{F}_i, \varphi_{ij})\) be a system of abelian sheaves on \(X_\etale\) over \(I\). Then \[\colim_{i\in I} H_\etale^p(X, \mathcal{F}_i) = H_\etale^p(X, \colim_{i\in I} \mathcal{F}_i).\]

Proof

This is a special case of Theorem 09YQ. We also sketch a direct proof. We prove it for all \(X\) at the same time, by induction on \(p\).

1. For any quasi-compact and quasi-separated scheme \(X\) and any étale covering \(\mathcal{U}\) of \(X\), show that there exists a refinement \(\mathcal{V} = \{V_j \to X\}_{j\in J}\) with \(J\) finite and each \(V_j\) quasi-compact and quasi-separated such that all \(V_{j_0} \times_X \ldots \times_X V_{j_p}\) are also quasi-compact and quasi-separated.

2. Using the previous step and the definition of colimits in the category of sheaves, show that the theorem holds for \(p = 0\) and all \(X\).

3. Using the locality of cohomology (Lemma 03P1), the Čech-to-cohomology spectral sequence (Theorem 03OW) and the fact that the induction hypothesis applies to all \(V_{j_0} \times_X \ldots \times_X V_{j_p}\) in the above situation, prove the induction step \(p \to p + 1\).

Lemma

Let \(A\) be a ring, \((I, \leq)\) a directed set and \((B_i, \varphi_{ij})\) a system of \(A\)-algebras. Set \(B = \colim_{i\in I} B_i\). Let \(X \to \Spec(A)\) be a quasi-compact and quasi-separated morphism of schemes. Let \(\mathcal{F}\) an abelian sheaf on \(X_\etale\). Denote \(Y_i = X \times_{\Spec(A)} \Spec(B_i)\), \(Y = X \times_{\Spec(A)} \Spec(B)\), \(\mathcal{G}_i = (Y_i \to X)^{-1}\mathcal{F}\) and \(\mathcal{G} = (Y \to X)^{-1}\mathcal{F}\). Then \[H_\etale^p(Y, \mathcal{G}) = \colim_{i\in I} H_\etale^p (Y_i, \mathcal{G}_i).\]

Proof

This is a special case of Theorem 09YQ. We also outline a direct proof as follows.

1. Given \(V \to Y\) étale with \(V\) quasi-compact and quasi-separated, there exist \(i\in I\) and \(V_i \to Y_i\) such that \(V = V_i \times_{Y_i} Y\). If all the schemes considered were affine, this would correspond to the following algebra statement: if \(B = \colim B_i\) and \(B \to C\) is étale, then there exist \(i \in I\) and \(B_i \to C_i\) étale such that \(C \cong B \otimes_{B_i} C_i\). This is proved in Algebra, Lemma 00U2.

2. In the situation of (1) show that \(\mathcal{G}(V) = \colim_{i' \geq i} \mathcal{G}_{i'}(V_{i'})\) where \(V_{i'}\) is the base change of \(V_i\) to \(Y_{i'}\).

3. By (1), we see that for every étale covering \(\mathcal{V} = \{V_j \to Y\}_{j\in J}\) with \(J\) finite and the \(V_j\)s quasi-compact and quasi-separated, there exists \(i \in I\) and an étale covering \(\mathcal{V}_i = \{V_{ij} \to Y_i\}_{j \in J}\) such that \(\mathcal{V} \cong \mathcal{V}_i \times_{Y_i} Y\).

4. Show that (2) and (3) imply \[\check H^*(\mathcal{V}, \mathcal{G})= \colim_{i\in I} \check H^*(\mathcal{V}_i, \mathcal{G}_i).\]

5. Cleverly use the Čech-to-cohomology spectral sequence (Theorem 03OW).

Lemma

Let \(f: X\to Y\) be a morphism of schemes and \(\mathcal{F}\in \textit{Ab}(X_\etale)\). Then \(R^pf_*\mathcal{F}\) is the sheaf associated to the presheaf \[(V \to Y) \longmapsto H_\etale^p(X \times_Y V, \mathcal{F}|_{X \times_Y V}).\] More generally, for \(K \in D(X_\etale)\) we have that \(R^pf_*K\) is the sheaf associated to the presheaf \[(V \to Y) \longmapsto H_\etale^p(X \times_Y V, K|_{X \times_Y V}).\]

Proof

This lemma is valid for topological spaces, and the proof in this case is the same. See Cohomology on Sites, Lemma 072W for the case of a sheaf and see Cohomology on Sites, Lemma 0BKV for the case of a complex of abelian sheaves.

Lemma

Let \(S\) be a scheme. Let \(X = \lim_{i \in I} X_i\) be a limit of a directed system of schemes over \(S\) with affine transition morphisms \(f_{i'i} : X_{i'} \to X_i\). We assume the structure morphisms \(g_i : X_i \to S\) and \(g : X \to S\) are quasi-compact and quasi-separated. Let \((\mathcal{F}_i, \varphi_{i'i})\) be a system of abelian sheaves on \((X_i, f_{i'i})\). Denote \(f_i : X \to X_i\) the projection and set \(\mathcal{F} = \colim f_i^{-1}\mathcal{F}_i\). Then \[\colim_{i\in I} R^p g_{i, *} \mathcal{F}_i = R^p g_* \mathcal{F}\] for all \(p \geq 0\).

Proof

Recall (Lemma 03Q8) that \(R^p g_{i, *} \mathcal{F}_i\) is the sheaf associated to the presheaf \(U \mapsto H^p_\etale(U \times_S X_i, \mathcal{F}_i)\) and similarly for \(R^pg_*\mathcal{F}\). Moreover, the colimit of a system of sheaves is the sheafification of the colimit on the level of presheaves. Note that every object of \(S_\etale\) has a covering by quasi-compact and quasi-separated objects (e.g., affine schemes). Moreover, if \(U\) is a quasi-compact and quasi-separated object, then we have \[\colim H^p_\etale(U \times_S X_i, \mathcal{F}_i) = H^p_\etale(U \times_S X, \mathcal{F})\] by Theorem 09YQ. Thus the lemma follows.

Lemma

Let \(I\) be a directed set. Let \(g_i : X_i \to S_i\) be an inverse system of morphisms of schemes over \(I\). Assume \(g_i\) is quasi-compact and quasi-separated and for \(i' \geq i\) the transition morphisms \(f_{i'i} : X_{i'} \to X_i\) and \(h_{i'i} : S_{i'} \to S_i\) are affine. Let \(g : X \to S\) be the limit of the morphisms \(g_i\), see Limits, Section 01YV. Denote \(f_i : X \to X_i\) and \(h_i : S \to S_i\) the projections. Let \((\mathcal{F}_i, \varphi_{i'i})\) be a system of sheaves on \((X_i, f_{i'i})\). Set \(\mathcal{F} = \colim f_i^{-1}\mathcal{F}_i\). Then \[R^p g_* \mathcal{F} = \colim_{i \in I} h_i^{-1}R^p g_{i, *} \mathcal{F}_i\] for all \(p \geq 0\).

Proof

How is the map of the lemma constructed? For \(i' \geq i\) we have a commutative diagram \[\xymatrix{ X \ar[r]_{f_{i'}} \ar[d]_g & X_{i'} \ar[r]_{f_{i'i}} \ar[d]_{g_{i'}} & X_i \ar[d]^{g_i} \\ S \ar[r]^{h_{i'}} & S_{i'} \ar[r]^{h_{i'i}} & S_i }\] If we combine the base change map \(h_{i'i}^{-1}Rg_{i, *}\mathcal{F}_i \to Rg_{i', *}f_{i'i}^{-1}\mathcal{F}_i\) (Cohomology on Sites, Lemma 0736 or Remark 07A7) with the map \(Rg_{i', *}\varphi_{i'i}\), then we obtain \(\psi_{i'i} : h_{i' i}^{-1} R^p g_{i, *} \mathcal{F}_i \to R^pg_{i', *} \mathcal{F}_{i'}\). Similarly, using the left square in the diagram we obtain maps \(\psi_i : h_i^{-1}R^pg_{i, *}\mathcal{F}_i \to R^pg_*\mathcal{F}\). The maps \(h_{i'}^{-1}\psi_{i'i}\) and \(\psi_i\) are the maps used in the statement of the lemma. For this to make sense, we have to check that \(\psi_{i''i} = \psi_{i''i'} \circ h_{i''i'}^{-1}\psi_{i'i}\) and \(\psi_{i'} \circ h_{i'}^{-1}\psi_{i'i} = \psi_i\); this follows from Cohomology on Sites, Remark 0E47.

Proof of the equality. First proof using dimension shifting⁴. For any \(U\) affine and étale over \(X\) by Theorem 09YQ we have \[g_*\mathcal{F}(U) = H^0(U \times_S X, \mathcal{F}) = \colim H^0(U_i \times_{S_i} X_i, \mathcal{F}_i) = \colim g_{i, *}\mathcal{F}_i(U_i)\] where the colimit is over \(i\) large enough such that there exists an \(i\) and \(U_i\) affine étale over \(S_i\) whose base change is \(U\) over \(S\) (see Lemma 0EYL). The right hand side is equal to \((\colim h_i^{-1}g_{i, *}\mathcal{F}_i)(U)\) by Sites, Lemma 09YN. This proves the lemma for \(p = 0\). If \((\mathcal{G}_i, \varphi_{i'i})\) is a system with \(\mathcal{G} = \colim f_i^{-1}\mathcal{G}_i\) such that \(\mathcal{G}_i\) is an injective abelian sheaf on \(X_i\) for all \(i\), then for any \(U\) affine and étale over \(X\) by Theorem 09YQ we have \[H^p(U \times_S X, \mathcal{G}) = \colim H^p(U_i \times_{S_i} X_i, \mathcal{G}_i) = 0\] for \(p > 0\) (same colimit as before). Hence \(R^pg_*\mathcal{G} = 0\) and we get the result for \(p > 0\) for such a system. In general we may choose a short exact sequence of systems \[0 \to (\mathcal{F}_i, \varphi_{i'i}) \to (\mathcal{G}_i, \varphi_{i'i}) \to (\mathcal{Q}_i, \varphi_{i'i}) \to 0\] where \((\mathcal{G}_i, \varphi_{i'i})\) is as above, see Cohomology on Sites, Lemma 0EXZ. By induction the lemma holds for \(p - 1\) and by the above we have vanishing for \(p\) and \((\mathcal{G}_i, \varphi_{i'i})\). Hence the result for \(p\) and \((\mathcal{F}_i, \varphi_{i'i})\) by the long exact sequence of cohomology.

Second proof. Recall that \(S_{affine, \etale} = \colim (S_i)_{affine, \etale}\), see Lemma 0EYL. Thus if \(U\) is an object of \(S_{affine, \etale}\), then we can write \(U = U_i \times_{S_i} S\) for some \(i\) and some \(U_i\) in \((S_i)_{affine, \etale}\) and \[(\colim_{i \in I} h_i^{-1}R^p g_{i, *} \mathcal{F}_i)(U) = \colim_{i' \geq i} (R^p g_{i', *}\mathcal{F}_{i'})(U_i \times_{S_i} S_{i'})\] by Sites, Lemma 09YN and the construction of the transition maps in the system described above. Since \(R^pg_{i', *}\mathcal{F}_{i'}\) is the sheaf associated to the presheaf \(U_{i'} \mapsto H^p(U_{i'} \times_{S_{i'}} X_{i'}, \mathcal{F}_{i'})\) and since \(R^pg_*\mathcal{F}\) is the sheaf associated to the presheaf \(U \mapsto H^p(U \times_S X, \mathcal{F})\) (Lemma 03Q8) we obtain a canonical commutative diagram \[\xymatrix{ \colim_{i' \geq i} H^p(U_i \times_{S_i} X_{i'}, \mathcal{F}_{i'}) \ar[r] \ar[d] & \colim_{i' \geq i} (R^p g_{i', *}\mathcal{F}_{i'})(U_i \times_{S_i} S_{i'}) \ar[d] \\ H^p(U \times_S X, \mathcal{F}) \ar[r] & R^pg_*\mathcal{F}(U) }\] Observe that the left hand vertical arrow is an isomorphism by Theorem 09YQ. We’re trying to show that the right hand vertical arrow is an isomorphism. However, we already know that the source and target of this arrow are sheaves on \(S_{affine, \etale}\). Hence it suffices to show: (1) an element in the target, locally comes from an element in the source and (2) an element in the source which maps to zero in the target locally vanishes. Part (1) follows immediately from the above and the fact that the lower horizontal arrow comes from a map of presheaves which becomes an isomorphism after sheafification. For part (2), say \(\xi \in \colim_{i' \geq i} (R^p g_{i', *}\mathcal{F}_{i'})(U_i \times_{S_i} S_{i'})\) is in the kernel. Choose an \(i' \geq i\) and \(\xi_{i'} \in (R^p g_{i', *}\mathcal{F}_{i'})(U_i \times_{S_i} S_{i'})\) representing \(\xi\). Choose a standard étale covering \(\{U_{i', k} \to U_i \times_{S_i} S_{i'}\}_{k = 1, \ldots, m}\) such that \(\xi_{i'}|_{U_{i', k}}\) comes from \(\xi_{i', k} \in H^p(U_{i', k} \times_{S_{i'}} X_{i'}, \mathcal{F}_{i'})\). Since it is enough to prove that \(\xi\) dies locally, we may replace \(U\) by the members of the étale covering \(\{U_{i', k} \times_{S_{i'}} S \to U = U_i \times_{S_i} S\}\). After this replacement we see that \(\xi\) is the image of an element \(\xi'\) of the group \(\colim_{i' \geq i} H^p(U_i \times_{S_i} X_{i'}, \mathcal{F}_{i'})\) in the diagram above. Since \(\xi'\) maps to zero in \(R^pg_*\mathcal{F}(U)\) we can do another replacement and assume that \(\xi'\) maps to zero in \(H^p(U \times_S X, \mathcal{F})\). However, since the left vertical arrow is an isomorphism we then conclude \(\xi' = 0\) hence \(\xi = 0\) as desired.

Lemma

Let \(X = \lim_{i \in I} X_i\) be a directed limit of schemes with affine transition morphisms \(f_{i'i}\) and projection morphisms \(f_i : X \to X_i\). Let \(\mathcal{F}\) be a sheaf on \(X_\etale\). Then

1. there are canonical maps \(\varphi_{i'i} : f_{i'i}^{-1}f_{i, *}\mathcal{F} \to f_{i', *}\mathcal{F}\) such that \((f_{i, *}\mathcal{F}, \varphi_{i'i})\) is a system of sheaves on \((X_i, f_{i'i})\) as in Definition 0EZL, and

2. \(\mathcal{F} = \colim f_i^{-1}f_{i, *}\mathcal{F}\).

Proof

Via Topologies, Lemma 04HR and Lemma 0EYL this is a special case of Sites, Lemma 0EXJ.

Lemma

Let \(I\) be a directed set. Let \(g_i : X_i \to S_i\) be an inverse system of morphisms of schemes over \(I\). Assume \(g_i\) is quasi-compact and quasi-separated and for \(i' \geq i\) the transition morphisms \(X_{i'} \to X_i\) and \(S_{i'} \to S_i\) are affine. Let \(g : X \to S\) be the limit of the morphisms \(g_i\), see Limits, Section 01YV. Denote \(f_i : X \to X_i\) and \(h_i : S \to S_i\) the projections. Let \(\mathcal{F}\) be an abelian sheaf on \(X\). Then we have \[R^pg_*\mathcal{F} = \colim_{i \in I} h_i^{-1}R^pg_{i, *}(f_{i, *}\mathcal{F})\]

Proof

Formal combination of Lemmas 0EYM and 0EYN.

Colimits and complexes

In this section we discuss taking cohomology of systems of complexes in various settings, continuing the discussion for sheaves started in Section 03Q4. We strongly urge the reader not to read this section unless absolutely necessary.

Lemma

Let \(X = \lim_{i \in I} X_i\) be a limit of a directed system of schemes with affine transition morphisms \(f_{i'i} : X_{i'} \to X_i\). We assume that \(X_i\) is quasi-compact and quasi-separated for all \(i \in I\). Let \(\mathcal{F}_i^\bullet\) be a complex of abelian sheaves on \(X_{i, \etale}\). Let \(\varphi_{i'i} : f_{i'i}^{-1}\mathcal{F}_i^\bullet \to \mathcal{F}_{i'}^\bullet\) be a map of complexes on \(X_{i, \etale}\) such that \(\varphi_{i''i} = \varphi_{i''i'} \circ f_{i'' i'}^{-1}\varphi_{i'i}\) whenever \(i'' \geq i' \geq i\). Assume there is an integer \(a\) such that \(\mathcal{F}_i^n = 0\) for \(n < a\) and all \(i \in I\). Then we have \[H^p_\etale(X, \colim f_i^{-1}\mathcal{F}_i^\bullet) = \colim H^p_\etale(X_i, \mathcal{F}^\bullet_i)\] where \(f_i : X \to X_i\) is the projection.

Proof

This is a consequence of Theorem 09YQ. Set \(\mathcal{F}^\bullet = \colim f_i^{-1}\mathcal{F}_i^\bullet\). The theorem tells us that \[\colim_{i \in I} H_\etale^p(X_i, \mathcal{F}_i^n) = H_\etale^p(X, \mathcal{F}^n)\] for all \(n, p \in \mathbf{Z}\). Let us use the spectral sequences \[E_{1, i}^{s, t} = H_\etale^t(X_i, \mathcal{F}_i^s) \Rightarrow H_\etale^{s + t}(X_i, \mathcal{F}_i^\bullet)\] and \[E_1^{s, t} = H_\etale^t(X, \mathcal{F}^s) \Rightarrow H_\etale^{s + t}(X, \mathcal{F}^\bullet)\] of Derived Categories, Lemma 015J. Since \(\mathcal{F}_i^n = 0\) for \(n < a\) (with \(a\) independent of \(i\)) we see that only a fixed finite number of terms \(E_{1, i}^{s, t}\) (independent of \(i\)) and \(E_1^{s, t}\) contribute to \(H^q_\etale(X_i, \mathcal{F}_i^\bullet)\) and \(H^q_\etale(X, \mathcal{F}^\bullet)\) and \(E_1^{s, t} = \colim E_{i, i}^{s, t}\). This implies what we want. Some details omitted. (There is an alternative argument using “stupid” truncations of complexes which avoids using spectral sequences.)

Lemma

Let \(X\) be a quasi-compact and quasi-sepated scheme. Let \(K_i \in D(X_\etale)\), \(i \in I\) be a family of objects. Assume given \(a \in \mathbf{Z}\) such that \(H^n(K_i) = 0\) for \(n < a\) and \(i \in I\). Then \(R\Gamma(X, \bigoplus_i K_i) = \bigoplus_i R\Gamma(X, K_i)\).

Proof

We have to show that \(H^p(X, \bigoplus_i K_i) = \bigoplus_i H^p(X, K_i)\) for all \(p \in \mathbf{Z}\). Choose complexes \(\mathcal{F}_i^\bullet\) representing \(K_i\) such that \(\mathcal{F}_i^n = 0\) for \(n < a\). The direct sum of the complexes \(\mathcal{F}_i^\bullet\) represents the object \(\bigoplus K_i\) by Injectives, Lemma 07D9. Since \(\bigoplus \mathcal{F}^\bullet\) is the filtered colimit of the finite direct sums, the result follows from Lemma 0EZM.

Lemma

Let \(S\) be a scheme. Let \(X = \lim_{i \in I} X_i\) be a limit of a directed system of schemes over \(S\) with affine transition morphisms \(f_{i'i} : X_{i'} \to X_i\). We assume that \(X_i\) is quasi-compact and quasi-separated for all \(i \in I\). Let \(K \in D^+(S_\etale)\). Then \[\colim_{i \in I} H_\etale^p(X_i, K|_{X_i}) = H_\etale^p(X, K|_X).\] for all \(p \in \mathbf{Z}\) where \(K|_{X_i}\) and \(K|_X\) are the pullbacks of \(K\) to \(X_i\) and \(X\).

Proof

We may represent \(K\) by a bounded below complex \(\mathcal{G}^\bullet\) of abelian sheaves on \(S_\etale\). Say \(\mathcal{G}^n = 0\) for \(n < a\). Denote \(\mathcal{F}^\bullet_i\) and \(\mathcal{F}^\bullet\) the pullbacks of this complex of \(X_i\) and \(X\). These complexes represent the objects \(K|_{X_i}\) and \(K|_X\) and we have \(\mathcal{F}^\bullet = \colim f_i^{-1}\mathcal{F}_i^\bullet\) termwise. Hence the lemma follows from Lemma 0EZM.

Lemma

Let \(I\), \(g_i : X_i \to S_i\), \(g : X \to S\), \(f_i\), \(g_i\), \(h_i\) be as in Lemma 0EYM. Let \(0 \in I\) and \(K_0 \in D^+(X_{0, \etale})\). For \(i \geq 0\) denote \(K_i\) the pullback of \(K_0\) to \(X_i\). Denote \(K\) the pullback of \(K_0\) to \(X\). Then \[R^pg_*K = \colim_{i \geq 0} h_i^{-1}R^pg_{i, *}K_i\] for all \(p \in \mathbf{Z}\).

Proof

Fix an integer \(p_0 \in \mathbf{Z}\). Let \(a\) be an integer such that \(H^j(K_0) = 0\) for \(j < a\). We will prove the formula holds for all \(p \leq p_0\) by descending induction on \(a\). If \(a > p_0\), then we see that the left and right hand side of the formula are zero for \(p \leq p_0\) by trivial vanishing, see Derived Categories, Lemma 05TC. Assume \(a \leq p_0\). Consider the distinguished triangle \[H^a(K_0)[-a] \to K_0 \to \tau_{\geq a + 1}K_0\] Pulling back this distinguished triangle to \(X_i\) and \(X\) gives compatible distinguished triangles for \(K_i\) and \(K\). For \(p \leq p_0\) we consider the commutative diagram \[\xymatrix{ \colim_{i \geq 0} h_i^{-1}R^{p - 1}g_{i, *}(\tau_{\geq a + 1}K_i) \ar[r]_-\alpha \ar[d] & R^{p - 1}g_*(\tau_{\geq a + 1}K) \ar[d] \\ \colim_{i \geq 0} h_i^{-1}R^pg_{i, *}(H^a(K_i)[-a]) \ar[r]_-\beta \ar[d] & R^pg_*(H^a(K)[-a]) \ar[d] \\ \colim_{i \geq 0} h_i^{-1}R^pg_{i, *}K_i \ar[r]_-\gamma \ar[d] & R^pg_*K \ar[d] \\ \colim_{i \geq 0} R^pg_{i, *}\tau_{\geq a + 1}K_i \ar[r]_-\delta \ar[d] & R^pg_*\tau_{\geq a + 1}K \ar[d] \\ \colim_{i \geq 0} R^{p + 1}g_{i, *}(H^a(K_i)[-a]) \ar[r]^-\epsilon & R^{p + 1}g_*(H^a(K)[-a]) }\] with exact columns. The arrows \(\beta\) and \(\epsilon\) are isomorphisms by Lemma 0EYM. The arrows \(\alpha\) and \(\delta\) are isomorphisms by induction hypothesis. Hence \(\gamma\) is an isomorphism as desired.

Lemma

Let \(I\), \(g_i : X_i \to S_i\), \(g : X \to S\), \(f_{ii'}\), \(f_i\), \(g_i\), \(h_i\) be as in Lemma 0EYM. Let \(\mathcal{F}_i^\bullet\) be a complex of abelian sheaves on \(X_{i, \etale}\). Let \(\varphi_{i'i} : f_{i'i}^{-1}\mathcal{F}_i^\bullet \to \mathcal{F}_{i'}^\bullet\) be a map of complexes on \(X_{i, \etale}\) such that \(\varphi_{i''i} = \varphi_{i''i'} \circ f_{i'' i'}^{-1}\varphi_{i'i}\) whenever \(i'' \geq i' \geq i\). Assume there is an integer \(a\) such that \(\mathcal{F}_i^n = 0\) for \(n < a\) and all \(i \in I\). Then \[R^pg_*(\colim f_i^{-1}\mathcal{F}_i^\bullet) = \colim_{i \geq 0} h_i^{-1}R^pg_{i, *}\mathcal{F}_i^\bullet\] for all \(p \in \mathbf{Z}\).

Proof

This is a consequence of Lemma 0EYM. Set \(\mathcal{F}^\bullet = \colim f_i^{-1}\mathcal{F}_i^\bullet\). The lemma tells us that \[\colim_{i \in I} h_i^{-1}R^pg_{i, *}\mathcal{F}_i^n = R^pg_*\mathcal{F}^n\] for all \(n, p \in \mathbf{Z}\). Let us use the spectral sequences \[E_{1, i}^{s, t} = R^tg_{i, *}\mathcal{F}_i^s \Rightarrow R^{s + t}g_{i, *}\mathcal{F}_i^\bullet\] and \[E_1^{s, t} = R^tg_*\mathcal{F}^s \Rightarrow R^{s + t}g_*\mathcal{F}^\bullet\] of Derived Categories, Lemma 015J. Since \(\mathcal{F}_i^n = 0\) for \(n < a\) (with \(a\) independent of \(i\)) we see that only a fixed finite number of terms \(E_{1, i}^{s, t}\) (independent of \(i\)) and \(E_1^{s, t}\) contribute and \(E_1^{s, t} = \colim E_{i, i}^{s, t}\). This implies what we want. Some details omitted. (There is an alternative argument using “stupid” truncations of complexes which avoids using spectral sequences.)

Lemma

Let \(f : X \to Y\) be a quasi-compact and quasi-sepated morphism of schemes. Let \(K_i \in D(X_\etale)\), \(i \in I\) be a family of objects. Assume given \(a \in \mathbf{Z}\) such that \(H^n(K_i) = 0\) for \(n < a\) and \(i \in I\). Then \(Rf_*(\bigoplus_i K_i) = \bigoplus_i Rf_*K_i\).

Proof

We have to show that \(R^pf_*(\bigoplus_i K_i) = \bigoplus_i R^pf_*K_i\) for all \(p \in \mathbf{Z}\). Choose complexes \(\mathcal{F}_i^\bullet\) representing \(K_i\) such that \(\mathcal{F}_i^n = 0\) for \(n < a\). The direct sum of the complexes \(\mathcal{F}_i^\bullet\) represents the object \(\bigoplus K_i\) by Injectives, Lemma 07D9. Since \(\bigoplus \mathcal{F}^\bullet\) is the filtered colimit of the finite direct sums, the result follows from Lemma 0GIV.

Stalks of higher direct images

The stalks of higher direct images can often be computed as follows.

Theorem

Let \(f: X \to S\) be a quasi-compact and quasi-separated morphism of schemes, \(\mathcal{F}\) an abelian sheaf on \(X_\etale\), and \(\overline{s}\) a geometric point of \(S\) lying over \(s \in S\). Then \[\left(R^nf_* \mathcal{F}\right)_{\overline{s}} = H_\etale^n( X \times_S \Spec(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}\mathcal{F})\] where \(p : X \times_S \Spec(\mathcal{O}_{S, \overline{s}}^{sh}) \to X\) is the projection. For \(K \in D^+(X_\etale)\) and \(n \in \mathbf{Z}\) we have \[\left(R^nf_*K\right)_{\overline{s}} = H_\etale^n(X \times_S \Spec(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}K)\] In fact, we have \[\left(Rf_*K\right)_{\overline{s}} = R\Gamma_\etale(X \times_S \Spec(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}K)\] in \(D^+(\textit{Ab})\).

Proof

Let \(\mathcal{I}\) be the category of étale neighborhoods of \(\overline{s}\) on \(S\). By Lemma 03Q8 we have \[(R^nf_*\mathcal{F})_{\overline{s}} = \colim_{(V, \overline{v}) \in \mathcal{I}^{opp}} H_\etale^n(X \times_S V, \mathcal{F}|_{X \times_S V}).\] We may replace \(\mathcal{I}\) by the initial subcategory consisting of affine étale neighbourhoods of \(\overline{s}\). Observe that \[\Spec(\mathcal{O}_{S, \overline{s}}^{sh}) = \lim_{(V, \overline{v}) \in \mathcal{I}} V\] by Lemma 04HX and Limits, Lemma 01YW. Since fibre products commute with limits we also obtain \[X \times_S \Spec(\mathcal{O}_{S, \overline{s}}^{sh}) = \lim_{(V, \overline{v}) \in \mathcal{I}} X \times_S V\] We conclude by Lemma 03Q6. For the second variant, use the same argument using Lemma 0GIT instead of Lemma 03Q6.

To see that the last statement is true, it suffices to produce a map \(\left(Rf_*K\right)_{\overline{s}} \to R\Gamma_\etale(X \times_S \Spec(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}K)\) in \(D^+(\textit{Ab})\) which realizes the ismorphisms on cohomology groups in degree \(n\) above for all \(n\). To do this, choose a bounded below complex \(\mathcal{J}^\bullet\) of injective abelian sheaves on \(X_\etale\) representing \(K\). The complex \(f_*\mathcal{J}^\bullet\) represents \(Rf_*K\). Thus the complex \[(f_*\mathcal{J}^\bullet)_{\overline{s}} = \colim_{(V, \overline{v}) \in \mathcal{I}^{opp}} (f_*\mathcal{J}^\bullet)(V)\] represents \((Rf_*K)_{\overline{s}}\). For each \(V\) we have maps \[(f_*\mathcal{J}^\bullet)(V) = \Gamma(X \times_S V, \mathcal{J}^\bullet) \longrightarrow \Gamma(X \times_S \Spec(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}\mathcal{J}^\bullet)\] and the target complex represents \(R\Gamma_\etale(X \times_S \Spec(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}K)\) in \(D^+(\textit{Ab})\). Taking the colimit of these maps we obtain the result.

Remark

Let \(f : X \to S\) be a morphism of schemes. Let \(K \in D(X_\etale)\). Let \(\overline{s}\) be a geometric point of \(S\). There are always canonical maps \[(Rf_*K)_{\overline{s}} \longrightarrow R\Gamma(X \times_S \Spec(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}K) \longrightarrow R\Gamma(X_{\overline{s}}, K|_{X_{\overline{s}}})\] where \(p : X \times_S \Spec(\mathcal{O}_{S, \overline{s}}^{sh}) \to X\) is the projection. Namely, consider the commutative diagram \[\xymatrix{ X_{\overline{s}} \ar[r] \ar[d]^{f_{\overline{s}}} & X \times_S \Spec(\mathcal{O}_{S, \overline{s}}^{sh}) \ar[r]_-p \ar[d]^{f'} & X \ar[d]^f \\ \overline{s} \ar[r]^-i & \Spec(\mathcal{O}_{S, \overline{s}}^{sh}) \ar[r]^-j & S }\] We have the base change maps \[i^{-1}Rf'_*(p^{-1}K) \to Rf_{\overline{s}, *}(K|_{X_{\overline{s}}}) \quad\text{and}\quad j^{-1}Rf_*K \to Rf'_*(p^{-1}K)\] (Cohomology on Sites, Remark 07A7) for the two squares in this diagram. Taking global sections we obtain the desired maps. By Cohomology on Sites, Remark 0E47 the composition of these two maps is the usual (base change) map \((Rf_*K)_{\overline{s}} \to R\Gamma(X_{\overline{s}}, K|_{X_{\overline{s}}})\).

The Leray spectral sequence

Lemma

Let \(f: X \to Y\) be a morphism and \(\mathcal{I}\) an injective object of \(\textit{Ab}(X_\etale)\). Let \(V \in \Ob(Y_\etale)\). Then

1. for any covering \(\mathcal{V} = \{V_j\to V\}_{j \in J}\) we have \(\check H^p(\mathcal{V}, f_*\mathcal{I}) = 0\) for all \(p > 0\),

2. \(f_*\mathcal{I}\) is acyclic for the functor \(\Gamma(V, -)\), and

3. if \(g : Y \to Z\), then \(f_*\mathcal{I}\) is acyclic for \(g_*\).

Proof

Observe that \(\check{\mathcal{C}}^\bullet(\mathcal{V}, f_*\mathcal{I}) = \check{\mathcal{C}}^\bullet(\mathcal{V} \times_Y X, \mathcal{I})\) which has vanishing higher cohomology groups by Lemma 03OR. This proves (1). The second statement follows as a sheaf which has vanishing higher Čech cohomology groups for any covering has vanishing higher cohomology groups. This a wonderful exercise in using the Čech-to-cohomology spectral sequence, but see Cohomology on Sites, Lemma 03F9 for details and a more precise and general statement. Part (3) is a consequence of (2) and the description of \(R^pg_*\) in Lemma 03Q8.

Using the formalism of Grothendieck spectral sequences, this gives the following.

Proposition

Let \(f: X \to Y\) be a morphism of schemes and \(\mathcal{F}\) an étale sheaf on \(X\). Then there is a spectral sequence \[E_2^{p, q} = H_\etale^p(Y, R^qf_*\mathcal{F}) \Rightarrow H_\etale^{p+q}(X, \mathcal{F}).\]

Proof

See Lemma 03QB and see Derived Categories, Section 015L.

Vanishing of finite higher direct images

The next goal is to prove that the higher direct images of a finite morphism of schemes vanish.

Lemma

Let \(R\) be a strictly henselian local ring. Set \(S = \Spec(R)\) and let \(\overline{s}\) be its closed point. Then the global sections functor \(\Gamma(S, -) : \textit{Ab}(S_\etale) \to \textit{Ab}\) is exact. In fact we have \(\Gamma(S, \mathcal{F}) = \mathcal{F}_{\overline{s}}\) for any sheaf of sets \(\mathcal{F}\). In particular \[\forall p\geq 1, \quad H_\etale^p(S, \mathcal{F})=0\] for all \(\mathcal{F}\in \textit{Ab}(S_\etale)\).

Proof

If we show that \(\Gamma(S, \mathcal{F}) = \mathcal{F}_{\overline{s}}\) then \(\Gamma(S, -)\) is exact as the stalk functor is exact. Let \((U, \overline{u})\) be an étale neighbourhood of \(\overline{s}\). Pick an affine open neighborhood \(\Spec(A)\) of \(\overline{u}\) in \(U\). Then \(R \to A\) is étale and \(\kappa(\overline{s}) = \kappa(\overline{u})\). By Theorem 03QH we see that \(A \cong R \times A'\) as an \(R\)-algebra compatible with maps to \(\kappa(\overline{s}) = \kappa(\overline{u})\). Hence we get a section \[\xymatrix{ \Spec(A) \ar[r] & U \ar[d]\\ & S \ar[ul] }\] It follows that in the system of étale neighbourhoods of \(\overline{s}\) the identity map \((S, \overline{s}) \to (S, \overline{s})\) is cofinal. Hence \(\Gamma(S, \mathcal{F}) = \mathcal{F}_{\overline{s}}\). The final statement of the lemma follows as the higher derived functors of an exact functor are zero, see Derived Categories, Lemma 015F.

Proposition

Let \(f : X \to Y\) be a finite morphism of schemes.

1. For any geometric point \(\overline{y} : \Spec(k) \to Y\) we have \[(f_*\mathcal{F})_{\overline{y}} = \prod\nolimits_{\overline{x} : \Spec(k) \to X,\ f(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}}.\] for \(\mathcal{F}\) in \(\Sh(X_\etale)\) and \[(f_*\mathcal{F})_{\overline{y}} = \bigoplus\nolimits_{\overline{x} : \Spec(k) \to X,\ f(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}}.\] for \(\mathcal{F}\) in \(\textit{Ab}(X_\etale)\).

2. For any \(q \geq 1\) we have \(R^q f_*\mathcal{F} = 0\) for \(\mathcal{F}\) in \(\textit{Ab}(X_\etale)\).

Proof

Let \(X_{\overline{y}}^{sh}\) denote the fiber product \(X \times_Y \Spec(\mathcal{O}_{Y, \overline{y}}^{sh})\). By Theorem 03Q9 the stalk of \(R^qf_*\mathcal{F}\) at \(\overline{y}\) is computed by \(H_\etale^q(X_{\overline{y}}^{sh}, \mathcal{F})\). Since \(f\) is finite, \(X_{\bar y}^{sh}\) is finite over \(\Spec(\mathcal{O}_{Y, \overline{y}}^{sh})\), thus \(X_{\bar y}^{sh} = \Spec(A)\) for some ring \(A\) finite over \(\mathcal{O}_{Y, \bar y}^{sh}\). Since the latter is strictly henselian, Lemma 03QJ implies that \(A\) is a finite product of henselian local rings \(A = A_1 \times \ldots \times A_r\). Since the residue field of \(\mathcal{O}_{Y, \overline{y}}^{sh}\) is separably closed the same is true for each \(A_i\). Hence \(A_i\) is strictly henselian. This implies that \(X_{\overline{y}}^{sh} = \coprod_{i = 1}^r \Spec(A_i)\). The vanishing of Lemma 03QO implies that \((R^qf_*\mathcal{F})_{\overline{y}} = 0\) for \(q > 0\) which implies (2) by Theorem 03PU. Part (1) follows from the corresponding statement of Lemma 03QO.

Lemma

Consider a cartesian square \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] of schemes with \(f\) a finite morphism. For any sheaf of sets \(\mathcal{F}\) on \(X_\etale\) we have \(f'_*(g')^{-1}\mathcal{F} = g^{-1}f_*\mathcal{F}\).

Proof

In great generality there is a pullback map \(g^{-1}f_*\mathcal{F} \to f'_*(g')^{-1}\mathcal{F}\), see Sites, Section 06UM. It suffices to check on stalks (Theorem 03PU). Let \(\overline{y}' : \Spec(k) \to Y'\) be a geometric point. We have \[\begin{align*} (f'_*(g')^{-1}\mathcal{F})_{\overline{y}'} & = \prod\nolimits_{\overline{x}' : \Spec(k) \to X',\ f' \circ \overline{x}' = \overline{y}'} ((g')^{-1}\mathcal{F})_{\overline{x}'} \\ & = \prod\nolimits_{\overline{x}' : \Spec(k) \to X',\ f' \circ \overline{x}' = \overline{y}'} \mathcal{F}_{g' \circ \overline{x}'} \\ & = \prod\nolimits_{\overline{x} : \Spec(k) \to X,\ f \circ \overline{x} = g \circ \overline{y}'} \mathcal{F}_{\overline{x}} \\ & = (f_*\mathcal{F})_{g \circ \overline{y}'} \\ & = (g^{-1}f_*\mathcal{F})_{\overline{y}'} \end{align*}\] The first equality by Proposition 03QP. The second equality by Lemma 03Q1. The third equality holds because the diagram is a cartesian square and hence the map \[\{\overline{x}' : \Spec(k) \to X',\ f' \circ \overline{x}' = \overline{y}'\} \longrightarrow \{\overline{x} : \Spec(k) \to X,\ f \circ \overline{x} = g \circ \overline{y}'\}\] sending \(\overline{x}'\) to \(g' \circ \overline{x}'\) is a bijection. The fourth equality by Proposition 03QP. The fifth equality by Lemma 03Q1.

Lemma

Consider a cartesian square \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] of schemes with \(f\) an integral morphism. For any sheaf of sets \(\mathcal{F}\) on \(X_\etale\) we have \(f'_*(g')^{-1}\mathcal{F} = g^{-1}f_*\mathcal{F}\).

Proof

The question is local on \(Y\) and hence we may assume \(Y\) is affine. Then we can write \(X = \lim X_i\) with \(f_i : X_i \to Y\) finite (this is easy in the affine case, but see Limits, Lemma 09YZ for a reference). Denote \(p_{i'i} : X_{i'} \to X_i\) the transition morphisms and \(p_i : X \to X_i\) the projections. Setting \(\mathcal{F}_i = p_{i, *}\mathcal{F}\) we obtain from Lemma 0EYN a system \((\mathcal{F}_i, \varphi_{i'i})\) with \(\mathcal{F} = \colim p_i^{-1}\mathcal{F}_i\). We get \(f_*\mathcal{F} = \colim f_{i, *}\mathcal{F}_i\) from Lemma 09Z1. Set \(X'_i = Y' \times_Y X_i\) with projections \(f'_i\) and \(g'_i\). Then \(X' = \lim X'_i\) as limits commute with limits. Denote \(p'_i : X' \to X'_i\) the projections. We have \[\begin{align*} g^{-1}f_*\mathcal{F} & = g^{-1} \colim f_{i, *}\mathcal{F}_i \\ & = \colim g^{-1}f_{i, *}\mathcal{F}_i \\ & = \colim f'_{i, *}(g'_i)^{-1}\mathcal{F}_i \\ & = f'_*(\colim (p'_i)^{-1}(g'_i)^{-1}\mathcal{F}_i) \\ & = f'_*(\colim (g')^{-1}p_i^{-1}\mathcal{F}_i) \\ & = f'_*(g')^{-1} \colim p_i^{-1}\mathcal{F}_i \\ & = f'_*(g')^{-1}\mathcal{F} \end{align*}\] as desired. For the first equality see above. For the second use that pullback commutes with colimits. For the third use the finite case, see Lemma 0959. For the fourth use Lemma 09Z1. For the fifth use that \(g'_i \circ p'_i = p_i \circ g'\). For the sixth use that pullback commutes with colimits. For the seventh use \(\mathcal{F} = \colim p_i^{-1}\mathcal{F}_i\).

The following lemma is a case of cohomological descent dealing with étale sheaves and finite surjective morphisms. We will significantly generalize this result once we prove the proper base change theorem.

Lemma

Let \(f : X \to Y\) be a surjective finite morphism of schemes. Set \(f_n : X_n \to Y\) equal to the \((n + 1)\)-fold fibre product of \(X\) over \(Y\). For \(\mathcal{F} \in \textit{Ab}(Y_\etale)\) set \(\mathcal{F}_n = f_{n, *}f_n^{-1}\mathcal{F}\). There is an exact sequence \[0 \to \mathcal{F} \to \mathcal{F}_0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \ldots\] on \(Y_\etale\). Moreover, there is a spectral sequence \[E_1^{p, q} = H^q_\etale(X_p, f_p^{-1}\mathcal{F})\] converging to \(H^{p + q}(Y_\etale, \mathcal{F})\). This spectral sequence is functorial in \(\mathcal{F}\).

Proof

If we prove the first statement of the lemma, then we obtain a spectral sequence with \(E_1^{p, q} = H^q_\etale(Y, \mathcal{F}_p)\) converging to \(H^{p + q}(Y_\etale, \mathcal{F})\), see Derived Categories, Lemma 015J. On the other hand, since \(R^if_{p, *}f_p^{-1}\mathcal{F} = 0\) for \(i > 0\) (Proposition 03QP) we get \[H^q_\etale(X_p, f_p^{-1}\mathcal{F}) = H^q_\etale(Y, f_{p, *}f_p^{-1} \mathcal{F}) = H^q_\etale(Y, \mathcal{F}_p)\] by Proposition 03QC and we get the spectral sequence of the lemma.

To prove the first statement of the lemma, observe that \(X_n\) forms a simplicial scheme over \(Y\), see Simplicial, Example 016E. Observe moreover, that for each of the projections \(d_j : X_{n + 1} \to X_n\) there is a map \(d_j^{-1} f_n^{-1}\mathcal{F} \to f_{n + 1}^{-1}\mathcal{F}\). These maps induce maps \[\delta_j : \mathcal{F}_n \to \mathcal{F}_{n + 1}\] for \(j = 0, \ldots, n + 1\). We use the alternating sum of these maps to define the differentials \(\mathcal{F}_n \to \mathcal{F}_{n + 1}\). Similarly, there is a canonical augmentation \(\mathcal{F} \to \mathcal{F}_0\), namely this is just the canonical map \(\mathcal{F} \to f_*f^{-1}\mathcal{F}\). To check that this sequence of sheaves is an exact complex it suffices to check on stalks at geometric points (Theorem 03PU). Thus we let \(\overline{y} : \Spec(k) \to Y\) be a geometric point. Let \(E = \{\overline{x} : \Spec(k) \to X \mid f(\overline{x}) = \overline{y}\}\). Then \(E\) is a finite nonempty set and we see that \[(\mathcal{F}_n)_{\overline{y}} = \bigoplus\nolimits_{e \in E^{n + 1}} \mathcal{F}_{\overline{y}}\] by Proposition 03QP and Lemma 03Q1. Thus we have to see that given an abelian group \(M\) the sequence \[0 \to M \to \bigoplus\nolimits_{e \in E} M \to \bigoplus\nolimits_{e \in E^2} M \to \ldots\] is exact. Here the first map is the diagonal map and the map \(\bigoplus_{e \in E^{n + 1}} M \to \bigoplus_{e \in E^{n + 2}} M\) is the alternating sum of the maps induced by the \((n + 2)\) projections \(E^{n + 2} \to E^{n + 1}\). This can be shown directly or deduced by applying Simplicial, Lemma 019P to the map \(E \to \{*\}\).

Remark

In the situation of Lemma 09Z2 if \(\mathcal{G}\) is a sheaf of sets on \(Y_\etale\), then we have \[\Gamma(Y, \mathcal{G}) = \text{Equalizer}( \xymatrix{ \Gamma(X_0, f_0^{-1}\mathcal{G}) \ar@<1ex>[r] \ar@<-1ex>[r] & \Gamma(X_1, f_1^{-1}\mathcal{G}) } )\] This is proved in exactly the same way, by showing that the sheaf \(\mathcal{G}\) is the equalizer of the two maps \(f_{0, *}f_0^{-1}\mathcal{G} \to f_{1, *}f_1^{-1}\mathcal{G}\).

Galois action on stalks

In this section we define an action of the absolute Galois group of a residue field of a point \(s\) of \(S\) on the stalk functor at any geometric point lying over \(s\).

Galois action on stalks. Let \(S\) be a scheme. Let \(\overline{s}\) be a geometric point of \(S\). Let \(\sigma \in \text{Aut}(\kappa(\overline{s})/\kappa(s))\). Define an action of \(\sigma\) on the stalk \(\mathcal{F}_{\overline{s}}\) of a sheaf \(\mathcal{F}\) as follows [04JK]\[\begin{equation} \begin{matrix} \mathcal{F}_{\overline{s}} & \longrightarrow & \mathcal{F}_{\overline{s}} \\ (U, \overline{u}, t) & \longmapsto & (U, \overline{u} \circ \Spec(\sigma), t). \end{matrix} \end{equation}\] where we use the description of elements of the stalk in terms of triples as in the discussion following Definition 040R. This is a left action, since if \(\sigma_i \in \text{Aut}(\kappa(\overline{s})/\kappa(s))\) then \[\begin{align*} \sigma_1 \cdot (\sigma_2 \cdot (U, \overline{u}, t)) & = \sigma_1 \cdot (U, \overline{u} \circ \Spec(\sigma_2), t) \\ & = (U, \overline{u} \circ \Spec(\sigma_2) \circ \Spec(\sigma_1), t) \\ & = (U, \overline{u} \circ \Spec(\sigma_1 \circ \sigma_2), t) \\ & = (\sigma_1 \circ \sigma_2) \cdot (U, \overline{u}, t) \end{align*}\] It is clear that this action is functorial in the sheaf \(\mathcal{F}\). We note that we could have defined this action by referring directly to Remark 04FN.

Definition

Let \(S\) be a scheme. Let \(\overline{s}\) be a geometric point lying over the point \(s\) of \(S\). Let \(\kappa(s) \subset \kappa(s)^{sep} \subset \kappa(\overline{s})\) denote the separable algebraic closure of \(\kappa(s)\) in the algebraically closed field \(\kappa(\overline{s})\).

1. In this situation the absolute Galois group of \(\kappa(s)\) is \(\text{Gal}(\kappa(s)^{sep}/\kappa(s))\). It is sometimes denoted \(\text{Gal}_{\kappa(s)}\).

2. The geometric point \(\overline{s}\) is called algebraic if \(\kappa(s) \subset \kappa(\overline{s})\) is an algebraic closure of \(\kappa(s)\).

Example

The geometric point \(\Spec(\mathbf{C}) \to \Spec(\mathbf{Q})\) is not algebraic.

Let \(\kappa(s) \subset \kappa(s)^{sep} \subset \kappa(\overline{s})\) be as in the definition. Note that as \(\kappa(\overline{s})\) is algebraically closed the map \[\text{Aut}(\kappa(\overline{s})/\kappa(s)) \longrightarrow \text{Gal}(\kappa(s)^{sep}/\kappa(s)) = \text{Gal}_{\kappa(s)}\] is surjective. Suppose \((U, \overline{u})\) is an étale neighbourhood of \(\overline{s}\), and say \(\overline{u}\) lies over the point \(u\) of \(U\). Since \(U \to S\) is étale, the residue field extension \(\kappa(u)/\kappa(s)\) is finite separable. This implies the following

1. If \(\sigma \in \text{Aut}(\kappa(\overline{s})/\kappa(s)^{sep})\) then \(\sigma\) acts trivially on \(\mathcal{F}_{\overline{s}}\).

2. More precisely, the action of \(\text{Aut}(\kappa(\overline{s})/\kappa(s))\) determines and is determined by an action of the absolute Galois group \(\text{Gal}_{\kappa(s)}\) on \(\mathcal{F}_{\overline{s}}\).

3. Given \((U, \overline{u}, t)\) representing an element \(\xi\) of \(\mathcal{F}_{\overline{s}}\) any element of \(\text{Gal}(\kappa(s)^{sep}/K)\) acts trivially, where \(\kappa(s) \subset K \subset \kappa(s)^{sep}\) is the image of \(\overline{u}^\sharp : \kappa(u) \to \kappa(\overline{s})\).

Altogether we see that \(\mathcal{F}_{\overline{s}}\) becomes a \(\text{Gal}_{\kappa(s)}\)-set (see Fundamental Groups, Definition 04JJ). Hence we may think of the stalk functor as a functor \[\Sh(S_\etale) \longrightarrow \text{Gal}_{\kappa(s)}\textit{-Sets}, \quad \mathcal{F} \longmapsto \mathcal{F}_{\overline{s}}\] and from now on we usually do think about the stalk functor in this way.

Theorem

Let \(S = \Spec(K)\) with \(K\) a field. Let \(\overline{s}\) be a geometric point of \(S\). Let \(G = \text{Gal}_{\kappa(s)}\) denote the absolute Galois group. Taking stalks induces an equivalence of categories \[\Sh(S_\etale) \longrightarrow G\textit{-Sets}, \quad \mathcal{F} \longmapsto \mathcal{F}_{\overline{s}}.\]

Proof

Let us construct the inverse to this functor. In Fundamental Groups, Lemma 03QR we have seen that given a \(G\)-set \(M\) there exists an étale morphism \(X \to \Spec(K)\) such that \(\Mor_K(\Spec(K^{sep}), X)\) is isomorphic to \(M\) as a \(G\)-set. Consider the sheaf \(\mathcal{F}\) on \(\Spec(K)_\etale\) defined by the rule \(U \mapsto \Mor_K(U, X)\). This is a sheaf as the étale topology is subcanonical. Then we see that \(\mathcal{F}_{\overline{s}} = \Mor_K(\Spec(K^{sep}), X) = M\) as \(G\)-sets (details omitted). This gives the inverse of the functor and we win.

Remark

Another way to state the conclusion of Theorem 03QT and Fundamental Groups, Lemma 03QR is to say that every sheaf on \(\Spec(K)_\etale\) is representable by a scheme \(X\) étale over \(\Spec(K)\). This does not mean that every sheaf is representable in the sense of Sites, Definition 00WR. The reason is that in our construction of \(\Spec(K)_\etale\) we chose a sufficiently large set of schemes étale over \(\Spec(K)\), whereas sheaves on \(\Spec(K)_\etale\) form a proper class.

Lemma

Assumptions and notations as in Theorem 03QT. There is a functorial bijection \[\Gamma(S, \mathcal{F}) = (\mathcal{F}_{\overline{s}})^G\]

Proof

We can prove this using formal arguments and the result of Theorem 03QT as follows. Given a sheaf \(\mathcal{F}\) corresponding to the \(G\)-set \(M = \mathcal{F}_{\overline{s}}\) we have \[\begin{eqnarray*} \Gamma(S, \mathcal{F}) & = & \Mor_{\Sh(S_\etale)}(h_{\Spec(K)}, \mathcal{F}) \\ & = & \Mor_{G\textit{-Sets}}(\{*\}, M) \\ & = & M^G \end{eqnarray*}\] Here the first identification is explained in Sites, Sections 00V1 and 00WO, the second results from Theorem 03QT and the third is clear. We will also give a direct proof⁵.

Suppose that \(t \in \Gamma(S, \mathcal{F})\) is a global section. Then the triple \((S, \overline{s}, t)\) defines an element of \(\mathcal{F}_{\overline{s}}\) which is clearly invariant under the action of \(G\). Conversely, suppose that \((U, \overline{u}, t)\) defines an element of \(\mathcal{F}_{\overline{s}}\) which is invariant. Then we may shrink \(U\) and assume \(U = \Spec(L)\) for some finite separable field extension of \(K\), see Proposition 03PC. In this case the map \(\mathcal{F}(U) \to \mathcal{F}_{\overline{s}}\) is injective, because for any morphism of étale neighbourhoods \((U', \overline{u}') \to (U, \overline{u})\) the restriction map \(\mathcal{F}(U) \to \mathcal{F}(U')\) is injective since \(U' \to U\) is a covering of \(S_\etale\). After enlarging \(L\) a bit we may assume \(K \subset L\) is a finite Galois extension. At this point we use that \[\Spec(L) \times_{\Spec(K)} \Spec(L) = \coprod\nolimits_{\sigma \in \text{Gal}(L/K)} \Spec(L)\] where the maps \(\Spec(L) \to \Spec(L \otimes_K L)\) come from the ring maps \(a \otimes b \mapsto a\sigma(b)\). Hence we see that the condition that \((U, \overline{u}, t)\) is invariant under all of \(G\) implies that \(t \in \mathcal{F}(\Spec(L))\) maps to the same element of \(\mathcal{F}(\Spec(L) \times_{\Spec(K)} \Spec(L))\) via restriction by either projection (this uses the injectivity mentioned above; details omitted). Hence the sheaf condition of \(\mathcal{F}\) for the étale covering \(\{\Spec(L) \to \Spec(K)\}\) kicks in and we conclude that \(t\) comes from a unique section of \(\mathcal{F}\) over \(\Spec(K)\).

Remark

Let \(S\) be a scheme and let \(\overline{s} : \Spec(k) \to S\) be a geometric point of \(S\). By definition this means that \(k\) is algebraically closed. In particular the absolute Galois group of \(k\) is trivial. Hence by Theorem 03QT the category of sheaves on \(\Spec(k)_\etale\) is equivalent to the category of sets. The equivalence is given by taking sections over \(\Spec(k)\). This finally provides us with an alternative definition of the stalk functor. Namely, the functor \[\Sh(S_\etale) \longrightarrow \textit{Sets}, \quad \mathcal{F} \longmapsto \mathcal{F}_{\overline{s}}\] is isomorphic to the functor \[\Sh(S_\etale) \longrightarrow \Sh(\Spec(k)_\etale) = \textit{Sets}, \quad \mathcal{F} \longmapsto \overline{s}^*\mathcal{F}\] To prove this rigorously one can use Lemma 03Q1 part (3) with \(f = \overline{s}\). Moreover, having said this the general case of Lemma 03Q1 part (3) follows from functoriality of pullbacks.

Group cohomology

In the following, if we write \(H^i(G, M)\) we will mean that \(G\) is a topological group and \(M\) a discrete \(G\)-module with continuous \(G\)-action and \(H^i(G, -)\) is the \(i\)th right derived functor on the category \(\text{Mod}_G\) of such \(G\)-modules, see Definitions 04JP and 04JR. This includes the case of an abstract group \(G\), which simply means that \(G\) is viewed as a topological group with the discrete topology.

When the module has a nondiscrete topology, we will use the notation \(H^i_{cont}(G, M)\) to indicate the continuous cohomology groups introduced in [Tate], see Section 0DVG.

Definition

Let \(G\) be a topological group.

1. A \(G\)-module, sometimes called a discrete \(G\)-module, is an abelian group \(M\) endowed with a left action \(a : G \times M \to M\) by group homomorphisms such that \(a\) is continuous when \(M\) is given the discrete topology.

2. A morphism of \(G\)-modules \(f : M \to N\) is a \(G\)-equivariant homomorphism from \(M\) to \(N\).

3. The category of \(G\)-modules is denoted \(\text{Mod}_G\).

Let \(R\) be a ring.

1. An \(R\text{-}G\)-module is an \(R\)-module \(M\) endowed with a left action \(a : G \times M \to M\) by \(R\)-linear maps such that \(a\) is continuous when \(M\) is given the discrete topology.

2. A morphism of \(R\text{-}G\)-modules \(f : M \to N\) is a \(G\)-equivariant \(R\)-module map from \(M\) to \(N\).

3. The category of \(R\text{-}G\)-modules is denoted \(\text{Mod}_{R, G}\).

The condition that \(a : G \times M \to M\) is continuous is equivalent with the condition that the stabilizer of any \(x \in M\) is open in \(G\). If \(G\) is an abstract group then this corresponds to the notion of an abelian group endowed with a \(G\)-action provided we endow \(G\) with the discrete topology. Observe that \(\text{Mod}_{\mathbf{Z}, G} = \text{Mod}_G\).

The category \(\text{Mod}_G\) has enough injectives, see Injectives, Lemma 04JF. Consider the left exact functor \[\text{Mod}_G \longrightarrow \textit{Ab}, \quad M \longmapsto M^G = \{x \in M \mid g \cdot x = x\ \forall g \in G\}\] We sometimes denote \(M^G = H^0(G, M)\) and sometimes we write \(M^G = \Gamma_G(M)\). This functor has a total right derived functor \(R\Gamma_G(M)\) and \(i\)th right derived functor \(R^i\Gamma_G(M) = H^i(G, M)\) for any \(i \geq 0\).

The same construction works for \(H^0(G, -) : \text{Mod}_{R, G} \to \text{Mod}_R\). We will see in Lemma 0DVD that this agrees with the cohomology of the underlying \(G\)-module.

Definition

Let \(G\) be a topological group. Let \(M\) be a discrete \(G\)-module with continuous \(G\)-action. In other words, \(M\) is an object of the category \(\text{Mod}_G\) introduced in Definition 04JP.

1. The right derived functors \(H^i(G, M)\) of \(H^0(G, M)\) on the category \(\text{Mod}_G\) are called the continuous group cohomology groups of \(M\).

2. If \(G\) is an abstract group endowed with the discrete topology then the \(H^i(G, M)\) are called the group cohomology groups of \(M\).

3. If \(G\) is a Galois group, then the groups \(H^i(G, M)\) are called the Galois cohomology groups of \(M\).

4. If \(G\) is the absolute Galois group of a field \(K\), then the groups \(H^i(G, M)\) are sometimes called the Galois cohomology groups of \(K\) with coefficients in \(M\). In this case we sometimes write \(H^i(K, M)\) instead of \(H^i(G, M)\).

Lemma

Let \(G\) be a topological group. Let \(R\) be a ring. For every \(i \geq 0\) the diagram \[\xymatrix{ \text{Mod}_{R, G} \ar[rr]_{H^i(G, -)} \ar[d] & & \text{Mod}_R \ar[d] \\ \text{Mod}_G \ar[rr]^{H^i(G, -)} & & \textit{Ab} }\] whose vertical arrows are the forgetful functors is commutative.

Proof

Let us denote the forgetful functor \(F : \text{Mod}_{R, G} \to \text{Mod}_G\). Then \(F\) has a left adjoint \(H : \text{Mod}_G \to \text{Mod}_{R, G}\) given by \(H(M) = M \otimes_\mathbf{Z} R\). Observe that every object of \(\text{Mod}_G\) is a quotient of a direct sum of modules of the form \(\mathbf{Z}[G/U]\) where \(U \subset G\) is an open subgroup. Here \(\mathbf{Z}[G/U]\) denotes the \(G\)-modules of finite \(\mathbf{Z}\)-linear combinations of right \(U\) congruence classes in \(G\) endowed with left \(G\)-action. Thus every bounded above complex in \(\text{Mod}_G\) is quasi-isomorphic to a bounded above complex in \(\text{Mod}_G\) whose underlying terms are flat \(\mathbf{Z}\)-modules (Derived Categories, Lemma 05T7). Thus it is clear that \(LH\) exists on \(D^-(\text{Mod}_G)\) and is computed by evaluating \(H\) on any complex whose terms are flat \(\mathbf{Z}\)-modules; this follows from Derived Categories, Lemma 05T9 and Proposition 05TA. We conclude from Derived Categories, Lemma 0DVC that \[\text{Ext}^i(\mathbf{Z}, F(M)) = \text{Ext}^i(R, M)\] for \(M\) in \(\text{Mod}_{R, G}\). Observe that \(H^0(G, -) = \Hom(\mathbf{Z}, -)\) on \(\text{Mod}_G\) where \(\mathbf{Z}\) denotes the \(G\)-module with trivial action. Hence \(H^i(G, -) = \text{Ext}^i(\mathbf{Z}, -)\) on \(\text{Mod}_G\). Similarly we have \(H^i(G, -) = \text{Ext}^i(R, -)\) on \(\text{Mod}_{R, G}\). Combining everything we see that the lemma is true.

Lemma

Let \(G\) be a topological group. Let \(R\) be a ring. Let \(M\), \(N\) be \(R\text{-}G\)-modules. If \(M\) is finite projective as an \(R\)-module, then \(\text{Ext}^i(M, N) = H^i(G, M^\vee \otimes_R N)\) (for notation see proof).

Proof

The module \(M^\vee = \Hom_R(M, R)\) endowed with the contragredient action of \(G\). Namely \((g \cdot \lambda)(m) = \lambda(g^{-1} \cdot m)\) for \(g \in G\), \(\lambda \in M^\vee\), \(m \in M\). The action of \(G\) on \(M^\vee \otimes_R N\) is the diagonal one, i.e., given by \(g \cdot (\lambda \otimes n) = g \cdot \lambda \otimes g \cdot n\). Note that for a third \(R\text{-}G\)-module \(E\) we have \(\Hom(E, M^\vee \otimes_R N) = \Hom(M \otimes_R E, N)\). Namely, this is true on the level of \(R\)-modules by Algebra, Lemmas 00DE and 0DVB and the definitions of \(G\)-actions are chosen such that it remains true for \(R\text{-}G\)-modules. It follows that \(M^\vee \otimes_R N\) is an injective \(R\text{-}G\)-module if \(N\) is an injective \(R\text{-}G\)-module. Hence if \(N \to N^\bullet\) is an injective resolution, then \(M^\vee \otimes_R N \to M^\vee \otimes_R N^\bullet\) is an injective resolution. Then \[\Hom(M, N^\bullet) = \Hom(R, M^\vee \otimes_R N^\bullet) = (M^\vee \otimes_R N^\bullet)^G\] Since the left hand side computes \(\text{Ext}^i(M, N)\) and the right hand side computes \(H^i(G, M^\vee \otimes_R N)\) the proof is complete.

Lemma

Let \(G\) be a topological group. Let \(k\) be a field. Let \(V\) be a \(k\text{-}G\)-module. If \(G\) is topologically finitely generated and \(\dim_k(V) < \infty\), then \(\dim_k H^1(G, V) < \infty\).

Proof

Let \(g_1, \ldots, g_r \in G\) be elements which topologically generate \(G\), i.e., this means that the subgroup generated by \(g_1, \ldots, g_r\) is dense. By Lemma 0DVE we see that \(H^1(G, V)\) is the \(k\)-vector space of extensions \[0 \to V \to E \to k \to 0\] of \(k\text{-}G\)-modules. Choose \(e \in E\) mapping to \(1 \in k\). Write \[g_i \cdot e = v_i + e\] for some \(v_i \in V\). This is possible because \(g_i \cdot 1 = 1\). We claim that the list of elements \(v_1, \ldots, v_r \in V\) determine the isomorphism class of the extension \(E\). Once we prove this the lemma follows as this means that our Ext vector space is isomorphic to a subquotient of the \(k\)-vector space \(V^{\oplus r}\); some details omitted. Since \(E\) is an object of the category defined in Definition 04JP we know there is an open subgroup \(U\) such that \(u \cdot e = e\) for all \(u \in U\). Now pick any \(g \in G\). Then \(gU\) contains a word \(w\) in the elements \(g_1, \ldots, g_r\). Say \(gu = w\). Since the element \(w \cdot e\) is determined by \(v_1, \ldots, v_r\), we see that \(g \cdot e = (gu) \cdot e = w \cdot e\) is too.

Lemma

Let \(G\) be a profinite topological group. Then

1. \(H^i(G, M)\) is torsion for \(i > 0\) and any \(G\)-module \(M\), and

2. \(H^i(G, M) = 0\) if \(M\) is a \(\mathbf{Q}\)-vector space.

Proof

Proof of (1). By dimension shifting we see that it suffices to show that \(H^1(G, M)\) is torsion for every \(G\)-module \(M\). Choose an exact sequence \(0 \to M \to I \to N \to 0\) with \(I\) an injective object of the category of \(G\)-modules. Then any element of \(H^1(G, M)\) is the image of an element \(y \in N^G\). Choose \(x \in I\) mapping to \(y\). The stabilizer \(U \subset G\) of \(x\) is open, hence has finite index \(r\). Let \(g_1, \ldots, g_r \in G\) be a system of representatives for \(G/U\). Then \(\sum g_i(x)\) is an invariant element of \(I\) which maps to \(ry\). Thus \(r\) kills the element of \(H^1(G, M)\) we started with. Part (2) follows as then \(H^i(G, M)\) is both a \(\mathbf{Q}\)-vector space and torsion.