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 \input{preamble} % OK, start here. % \begin{document} \title{\'Etale Cohomology} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent These are the notes of a course on \'etale 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. Over time we will add references to background material in the rest of the Stacks project and provide rigorous proofs of all the statements. \section{Which sections to skip on a first reading?} \label{section-skip} \noindent 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 \'etale 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. \medskip\noindent In particular, we suggest that the reader skip the following sections: \begin{enumerate} \item Comparing big and small topoi, Section \ref{section-compare}. \item Recovering morphisms, Section \ref{section-morphisms}. \item Push and pull, Section \ref{section-monomorphisms}. \item Property (A), Section \ref{section-A}. \item Property (B), Section \ref{section-B}. \item Property (C), Section \ref{section-C}. \item Topological invariance of the small \'etale site, Section \ref{section-topological-invariance}. \item Integral universally injective morphisms, Section \ref{section-integral-universally-injective}. \item Big sites and pushforward, Section \ref{section-big}. \item Exactness of big lower shriek, Section \ref{section-exactness-lower-shriek}. \end{enumerate} Besides these sections there are some sporadic results that may be skipped that the reader can recognize by the keywords given above. %9.08.09 \section{Prologue} \label{section-prologue} \noindent 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 {\it flasque}. In this case, all higher {\v C}ech 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. \medskip\noindent 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. \section{The \'etale topology} \label{section-etale-topology} \noindent 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 \'etale topology, one considers all morphisms $\varphi : U \to X$ which are \'etale. If $X$ is a smooth projective variety over $\mathbf{C}$, then this means \begin{enumerate} \item $U$ is a disjoint union of smooth varieties, and \item $\varphi$ is (analytically) locally an isomorphism. \end{enumerate} 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 $X$). \medskip\noindent 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 \'etale 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 \'etale 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 \'etale. But we can remove other points from the source of the morphism also, and the morphism will still be \'etale. To consider the \'etale 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. \begin{definition} \label{definition-etale-covering-initial} A family of morphisms $\{ \varphi_i : U_i \to X\}_{i \in I}$ is called an {\it \'etale covering} if each $\varphi_i$ is an \'etale morphism and their images cover $X$, i.e., $X = \bigcup_{i \in I} \varphi_i(U_i)$. \end{definition} \noindent This defines'' the \'etale topology. In other words, we can now say what the sheaves are. An {\it \'etale sheaf} $\mathcal{F}$ of sets (resp.\ abelian groups, vector spaces, etc) on $X$ is the data: \begin{enumerate} \item for each \'etale morphism $\varphi : U \to X$ a set (resp.\ abelian group, vector space, etc) $\mathcal{F}(U)$, \item for each pair $U, \ U'$ of \'etale schemes over $X$, and each morphism $U \to U'$ over $X$ (which is automatically \'etale) a restriction map $\rho^{U'}_U : \mathcal{F}(U') \to \mathcal{F}(U)$ \end{enumerate} 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 \'etale over $X$ as well as the following {\it sheaf axiom}: \begin{itemize} \item[(*)] for every \'etale covering $\{ \varphi_i : U_i \to U\}_{i \in I}$, the diagram $$\xymatrix{ \emptyset \ar[r] & \mathcal{F} (U) \ar[r] & \Pi_{i \in I} \mathcal{F} (U_i) \ar@<1ex>[r] \ar@<-1ex>[r] & \Pi_{i, j \in I} \mathcal{F} (U_i \times_U U_j) }$$ is exact in the category of sets (resp.\ abelian groups, vector spaces, etc). \end{itemize} \begin{remark} \label{remark-i-is-j} 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. \end{remark} \noindent Since the identity is an \'etale morphism, we can compute the global sections of an \'etale 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. \section{Feats of the \'etale topology} \label{section-feats} \noindent 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 \'etale 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. \section{A computation} \label{section-computation} \noindent How do we compute the cohomology of $\mathbf{P}^1_\mathbf{C}$ with coefficients $\Lambda = \mathbf{Z}/n\mathbf{Z}$? We use {\v C}ech 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 \'etale cohomology. This gives an exact sequence $$H_\etale^{i-1}(U_0\cap U_1, \Lambda) \to H_\etale^i(\mathbf{P}^1_C, \Lambda) \to H_\etale^i(U_0, \Lambda) \oplus H_\etale^i(U_1, \Lambda) \to H_\etale^i(U_0\cap U_1, \Lambda).$$ 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 \'etale cohomology is at our disposal. \medskip\noindent {\bf 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. \medskip\noindent {\bf 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 \'etale 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. \medskip\noindent The curve $\mathbf{A}^1_\mathbf{C}$ has no nontrivial finite \'etale 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. \medskip\noindent Let us describe the finite \'etale 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 \'etale 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 $\mathbf{A}^1_{\mathbf{C}} \setminus \{0\}$. \medskip\noindent 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 \'etale 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 \'etale 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 \ref{remark-normalize-H1-Gm}. 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). \section{Nontorsion coefficients} \label{section-nontorsion} \noindent 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 {\it limit} of the system of cohomology groups $H_\etale^i(X, \mathbf{Z}/\ell^n\mathbf{Z})$ indexed by $n$, see Categories, Section \ref{categories-section-posets-limits}. Thus we will need to study systems of sheaves satisfying some compatibility conditions. \section{Sheaf theory} \label{section-sheaf-theory} %9.10.09 \noindent 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. \section{Presheaves} \label{section-presheaves} \noindent A reference for this section is Sites, Section \ref{sites-section-presheaves}. \begin{definition} \label{definition-presheaf} Let $\mathcal{C}$ be a category. A {\it presheaf of sets} (respectively, an {\it abelian presheaf}) on $\mathcal{C}$ is a functor $\mathcal{C}^{opp} \to \textit{Sets}$ (resp.\ $\textit{Ab}$). \end{definition} \noindent {\bf Terminology.} If $U \in \Ob(\mathcal{C})$, then elements of $\mathcal{F}(U)$ are called {\it 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 {\it 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 $\Gamma(U, \mathcal{F}) = \mathcal{F}(U)$. \medskip\noindent 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. \medskip\noindent 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 \ref{categories-remark-big-categories}.) \begin{example} \label{example-representable-presheaf} Given an object $X \in \Ob(\mathcal{C})$, we consider the functor $$\begin{matrix} h_X : & \mathcal{C}^{opp} & \longrightarrow & \textit{Sets} \\ & U & \longmapsto & h_X(U) = \Mor_\mathcal{C}(U, X) \\ & V \xrightarrow{\varphi} U & \longmapsto & \varphi \circ - : h_X(U) \to h_X(V). \end{matrix}$$ It is a presheaf, called the {\it representable presheaf associated to $X$.} It is not true that representable presheaves are sheaves in every topology on every site. \end{example} \begin{lemma}[Yoneda] \label{lemma-yoneda} \begin{slogan} Morphisms between objects are in bijection with natural transformations between the functors they represent. \end{slogan} Let $\mathcal{C}$ be a category, and $X, Y \in \Ob(\mathcal{C})$. There is a natural bijection $$\begin{matrix} \Mor_\mathcal{C}(X, Y) & \longrightarrow & \Mor_{\textit{PSh}(\mathcal{C})} (h_X, h_Y) \\ \psi & \longmapsto & h_\psi = \psi \circ - : h_X \to h_Y. \end{matrix}$$ \end{lemma} \begin{proof} See Categories, Lemma \ref{categories-lemma-yoneda}. \end{proof} \section{Sites} \label{section-sites} \begin{definition} \label{definition-family-morphisms-fixed-target} Let $\mathcal{C}$ be a category. A {\it family of morphisms with fixed target} $\mathcal{U} = \{\varphi_i : U_i \to U\}_{i\in I}$ is the data of \begin{enumerate} \item an object $U \in \mathcal{C}$, \item a set $I$ (possibly empty), and \item for all $i\in I$, a morphism $\varphi_i : U_i \to U$ of $\mathcal{C}$ with target $U$. \end{enumerate} \end{definition} \noindent There is a notion of a {\it morphism of families of morphisms with fixed target}. A special case of that is the notion of a {\it refinement}. A reference for this material is Sites, Section \ref{sites-section-refinements}. \begin{definition} \label{definition-site} A {\it site}\footnote{What we call a site is a called a category endowed with a pretopology in \cite[Expos\'e II, D\'efinition 1.3]{SGA4}. In \cite{ArtinTopologies} it is called a category with a Grothendieck topology.} consists of a category $\mathcal{C}$ and a set $\text{Cov}(\mathcal{C})$ consisting of families of morphisms with fixed target called {\it coverings}, such that \begin{enumerate} \item (isomorphism) if $\varphi : V \to U$ is an isomorphism in $\mathcal{C}$, then $\{\varphi : V \to U\}$ is a covering, \item (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 \item (base change) if $\{U_i \to U\}_{i\in I}$ is a covering and $V \to U$ is a morphism in $\mathcal{C}$, then \begin{enumerate} \item for all $i \in I$ the fibre product $U_i \times_U V$ exists in $\mathcal{C}$, and \item $\{U_i \times_U V \to V\}_{i\in I}$ is a covering. \end{enumerate} \end{enumerate} \end{definition} \noindent 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 \ref{sites-remark-no-big-sites}. \begin{example} \label{example-site-topological-space} 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. \end{example} \section{Sheaves} \label{section-sheaves} \begin{definition} \label{definition-sheaf} A presheaf $\mathcal{F}$ of sets (resp. abelian presheaf) on a site $\mathcal{C}$ is said to be a {\it 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 {\it sheaf} if for all coverings $\{\varphi_i : U_i \to U\}_{i\in I} \in \text{Cov} (\mathcal{C})$, the diagram \begin{equation} \label{equation-sheaf-axiom} \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). \end{definition} \begin{remark} \label{remark-empty-covering} 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 $\textit{Ab}$). \end{remark} \begin{example} \label{example-sheaf-site-space} Working this out for the site $X_{Zar}$ associated to a topological space, see Example \ref{example-site-topological-space}, gives the usual notion of sheaves. \end{example} \begin{definition} \label{definition-category-sheaves} 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 {\it category of sheaves of sets} (resp.\ {\it abelian sheaves}) on $\mathcal{C}$. \end{definition} \section{The example of G-sets} \label{section-G-sets} \noindent 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 $U = \bigcup_{i\in I} \varphi_i(U_i)$. \medskip\noindent 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 \ref{sites-section-example-sheaf-G-sets}) let us try to explain why this is true. \begin{enumerate} \item 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 $\mathcal{F}(S) = \prod_{i\in I} \mathcal{F}(O_i)$. \item 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 g_2^{-1}) \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. \end{enumerate} 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 \ref{sites-section-example-sheaf-G-sets}. \section{Sheafification} \label{section-sheafification} \begin{definition} \label{definition-0-cech} 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 {\it zeroth {\v C}ech cohomology group} of $\mathcal{F}$ with respect to $\mathcal{U}$ by $$\check H^0 (\mathcal{U}, \mathcal{F}) = \left\{ (s_i)_{i\in I} \in \prod\nolimits_{i\in I }\mathcal{F}(U_i) \text{ such that } s_i|_{U_i \times_U U_j} = s_j |_{U_i \times_U U_j} \right\}.$$ \end{definition} \noindent 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 {\it 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 \begin{enumerate} \item the map is well-defined, and \item depends only on $\chi$ and is independent of the choice of $\alpha, \{\chi_j\}_{j \in J}$. \end{enumerate} 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 $s \in \check H(\mathcal{U}, \mathcal{F})$ 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. \begin{theorem} \label{theorem-sheafification} Let $\mathcal{C}$ be a site and $\mathcal{F}$ a presheaf on $\mathcal{C}$. \begin{enumerate} \item 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. \item There is a canonical map of presheaves $\mathcal{F} \to \mathcal{F}^+$. \item If $\mathcal{F}$ is a separated presheaf then $\mathcal{F}^+$ is a sheaf and the map in (2) is injective. \item $\mathcal{F}^+$ is a separated presheaf. \item $\mathcal{F}^\# = (\mathcal{F}^+)^+$ 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 $\mathcal{G} \in \Sh(\mathcal{C})$. \end{enumerate} \end{theorem} \begin{proof} See Sites, Theorem \ref{sites-theorem-plus}. \end{proof} \noindent In other words, this means that the natural map $\mathcal{F} \to \mathcal{F}^\#$ is a left adjoint to the forgetful functor $\Sh(\mathcal{C}) \to \textit{PSh}(\mathcal{C})$. \section{Cohomology} \label{section-cohomology} \noindent The following is the basic result that makes it possible to define cohomology for abelian sheaves on sites. \begin{theorem} \label{theorem-enough-injectives} The category of abelian sheaves on a site is an abelian category which has enough injectives. \end{theorem} \begin{proof} See Modules on Sites, Lemma \ref{sites-modules-lemma-abelian-abelian} and Injectives, Theorem \ref{injectives-theorem-sheaves-injectives}. \end{proof} \noindent 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 \ref{sites-modules-lemma-abelian-abelian}. For more general discussion of cohomology on sites (including the global sections functor and its right derived functors), see Cohomology on Sites, Section \ref{sites-cohomology-section-cohomology-sheaves}. \section{The fpqc topology} \label{section-fpqc} %9.15.09 \noindent Before doing \'etale cohomology we study a bit the fpqc topology, since it works well for quasi-coherent sheaves. \begin{definition} \label{definition-fpqc-covering} Let $T$ be a scheme. An {\it fpqc covering} of $T$ is a family $\{ \varphi_i : T_i \to T\}_{i \in I}$ such that \begin{enumerate} \item each $\varphi_i$ is a flat morphism and $\bigcup_{i\in I} \varphi_i(T_i) = T$, and \item 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 $U = \bigcup_{k \in K} \varphi_{\mathbf{i}(k)}(U_{\mathbf{i}(k)})$. \end{enumerate} \end{definition} \begin{remark} \label{remark-fpqc} The first condition corresponds to fp, which stands for {\it fid\element plat}, faithfully flat in french, and the second to qc, {\it quasi-compact}. The second part of the first condition is unnecessary when the second condition holds. \end{remark} \begin{example} \label{example-fpqc-coverings} Examples of fpqc coverings. \begin{enumerate} \item Any Zariski open covering of $T$ is an fpqc covering. \item A family $\{\Spec(B) \to \Spec(A)\}$ is an fpqc covering if and only if $A \to B$ is a faithfully flat ring map. \item If $f: X \to Y$ is flat, surjective and quasi-compact, then $\{ f: X\to Y\}$ is an fpqc covering. \item 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. \item Write $\mathbf{A}^2_k = \Spec(k[x, y])$. Denote $i_x : D(x) \to \mathbf{A}^2_k$ and $i_y : D(y) \hookrightarrow \mathbf{A}^2_k$ 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. \end{enumerate} \end{example} \begin{lemma} \label{lemma-site-fpqc} The collection of fpqc coverings on the category of schemes satisfies the axioms of site. \end{lemma} \begin{proof} See Topologies, Lemma \ref{topologies-lemma-fpqc}. \end{proof} \noindent 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 \ref{topologies-section-fpqc}. 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. \begin{definition} \label{definition-sheaf-property-fpqc} 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 $\mathcal{F}$ {\it 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 (\ref{equation-sheaf-axiom}) is an equalizer diagram. \end{definition} \noindent We similarly say that $\mathcal{F}$ {\it satisfies the sheaf property for the Zariski topology} if for every open covering $U = \bigcup_{i \in I} U_i$ the diagram (\ref{equation-sheaf-axiom}) is an equalizer diagram. See Schemes, Definition \ref{schemes-definition-representable-by-open-immersions}. 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. \begin{lemma} \label{lemma-fpqc-sheaves} Let $\mathcal{F}$ be a presheaf on $\Sch/S$. Then $\mathcal{F}$ satisfies the sheaf property for the fpqc topology if and only if \begin{enumerate} \item $\mathcal{F}$ satisfies the sheaf property with respect to the Zariski topology, and \item 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. \end{enumerate} \end{lemma} \begin{proof} See Topologies, Lemma \ref{topologies-lemma-sheaf-property-fpqc}. \end{proof} \noindent 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: \begin{enumerate} \item for each $T/S$, a usual (i.e., Zariski) sheaf $\mathcal{F}_T$ on $T_{Zar}$, \item for every map $f : T' \to T$ over $S$, a restriction mapping $f^{-1}\mathcal{F}_T \to \mathcal{F}_{T'}$ \end{enumerate} such that \begin{enumerate} \item[(a)] the restriction mappings are functorial, \item[(b)] 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 \item[(c)] 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. \end{enumerate} 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 \ref{lemma-fpqc-sheaves} condition (c) then suffices to get the sheaf condition for the fpqc topology. \begin{example} \label{example-quasi-coherent} 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 \'etale and smooth sites. \end{example} \begin{lemma} \label{lemma-representable-sheaf-fpqc} Any representable presheaf on $\Sch/S$ satisfies the sheaf condition for the fpqc topology. \end{lemma} \begin{proof} See Descent, Lemma \ref{descent-lemma-fpqc-universal-effective-epimorphisms}. \end{proof} \noindent 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 {\it the fpqc topology is weaker than the canonical topology}, or that the fpqc topology is {\it subcanonical}. In the setting of sites this is discussed in Sites, Section \ref{sites-section-representable-sheaves}. \begin{remark} \label{remark-fpqc-finest} The fpqc is the finest topology that we will see. Hence any presheaf satisfying the sheaf condition for the fpqc topology will be a sheaf in the subsequent sites (\'etale, smooth, etc). In particular representable presheaves will be sheaves on the \'etale site of a scheme for example. \end{remark} \begin{example} \label{example-additive-group-sheaf} Let $S$ be a scheme. Consider the additive group scheme $\mathbf{G}_{a, S} = \mathbf{A}^1_S$ over $S$, see Groupoids, Example \ref{groupoids-example-additive-group}. 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. \end{example} \section{Faithfully flat descent} \label{section-fpqc-descent} \begin{definition} \label{definition-descent-datum} Let $\mathcal{U} = \{ t_i : T_i \to T\}_{i \in I}$ be a family of morphisms of schemes with fixed target. A {\it descent datum} for quasi-coherent sheaves with respect to $\mathcal{U}$ is a family $(\mathcal{F}_i, \varphi_{ij})_{i, j\in I}$ where \begin{enumerate} \item for all $i$, $\mathcal{F}_i$ is a quasi-coherent sheaf on $T_i$, and \item for all $i, j \in I$ the map $\varphi_{ij} : \text{pr}_0^* \mathcal{F}_i \cong \text{pr}_1^* \mathcal{F}_j$ is an isomorphism on $T_i \times_T T_j$ such that 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 $T_i \times_T T_j \times_T T_k$. \end{enumerate} This descent datum is called {\it effective} if there exist a quasi-coherent sheaf $\mathcal{F}$ over $T$ and $\mathcal{O}_{T_i}$-module isomorphisms $\varphi_i : t_i^* \mathcal{F} \cong \mathcal{F}_i$ satisfying the cocycle condition, namely $$\varphi_{ij} = \text{pr}_1^* (\varphi_j) \circ \text{pr}_0^* (\varphi_i)^{-1}.$$ \end{definition} \noindent In this and the next section we discuss some ingredients of the proof of the following theorem, as well as some related material. \begin{theorem} \label{theorem-descent-quasi-coherent} 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. \end{theorem} \begin{proof} See Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent}. \end{proof} \noindent 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 \ref{sheaves-section-glueing-sheaves}) 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. \medskip\noindent {\bf 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*} \begin{lemma} \label{lemma-algebra-descent} If $A \to B$ is faithfully flat, then the complex $(B/A)_\bullet$ is exact in positive degrees, and $H^0((B/A)_\bullet) = A$. \end{lemma} \begin{proof} See Descent, Lemma \ref{descent-lemma-ff-exact}. \end{proof} \noindent 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. \medskip\noindent The same strategy proves the following lemma. \begin{lemma} \label{lemma-descent-modules} 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 $H^0((B/A)_\bullet \otimes_A M) = M$. \end{lemma} \begin{proof} See Descent, Lemma \ref{descent-lemma-ff-exact}. \end{proof} \begin{definition} \label{definition-descent-datum-modules} Let $A \to B$ be a ring map and $N$ a $B$-module. A {\it 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 $\varphi_{02}$. \end{definition} \noindent If $N' = B \otimes_A M$ for some $A$-module M, then it has a canonical descent datum given by the map $$\begin{matrix} \varphi_\text{can}: & N' \otimes_A B & \to & B \otimes_A N' \\ & b_0 \otimes m \otimes b_1 & \mapsto & b_0 \otimes b_1 \otimes m. \end{matrix}$$ \begin{definition} \label{definition-effective-modules} A descent datum $(N, \varphi)$ is called {\it 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. \end{definition} \noindent Theorem \ref{theorem-descent-quasi-coherent} is a consequence the following result. \begin{theorem} \label{theorem-descent-modules} If $A \to B$ is faithfully flat then descent data with respect to $A\to B$ are effective. \end{theorem} \begin{proof} See Descent, Proposition \ref{descent-proposition-descent-module}. See also Descent, Remark \ref{descent-remark-homotopy-equivalent-cosimplicial-algebras} for an alternative view of the proof. \end{proof} \begin{remarks} \label{remarks-theorem-modules-exactness} The results on descent of modules have several applications: \begin{enumerate} \item The exactness of the {\v C}ech 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. \item 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 \ref{descent-proposition-descent-module}. \end{enumerate} \end{remarks} %9.17.09 \section{Quasi-coherent sheaves} \label{section-quasi-coherent} \noindent We can apply the descent of modules to study quasi-coherent sheaves. \begin{proposition} \label{proposition-quasi-coherent-sheaf-fpqc} 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. \end{proposition} \begin{proof} This is proved in Descent, Lemma \ref{descent-lemma-sheaf-condition-holds}. We indicate the proof here. As established in Lemma \ref{lemma-fpqc-sheaves}, 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. \medskip\noindent 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 \ref{lemma-descent-modules}. \end{proof} \noindent 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 \ref{sites-modules-section-local}. 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 \ref{categories-example-category-over-X}. 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 {\it restriction} $\mathcal{F}|_{\mathcal{C}/U}$ (defined in the obvious manner) is a sheaf as well. See Sites, Section \ref{sites-section-localize} for details. \begin{definition} \label{definition-ringed-site} Let $\mathcal{C}$ be a {\it 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 {\it 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. \end{definition} \noindent Although it is useful to be able to give a general definition as above this notion is not well behaved in general. \begin{remark} \label{remark-final-object} 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 \ref{sites-modules-lemma-local-final-object}. \end{remark} \begin{theorem}[Meta theorem on quasi-coherent sheaves] \label{theorem-quasi-coherent} Let $S$ be a scheme. Let $\mathcal{C}$ be a site. Assume that \begin{enumerate} \item the underlying category $\mathcal{C}$ is a full subcategory of $\Sch/S$, \item any Zariski covering of $T \in \Ob(\mathcal{C})$ can be refined by a covering of $\mathcal{C}$, \item $S/S$ is an object of $\mathcal{C}$, \item every covering of $\mathcal{C}$ is an fpqc covering of schemes. \end{enumerate} 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 $S$. \end{theorem} \begin{proof} After some formal arguments this is exactly Theorem \ref{theorem-descent-quasi-coherent}. Details omitted. In Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent} we prove a more precise version of the theorem for the big Zariski, fppf, \'etale, smooth, and syntomic sites of $S$, as well as the small Zariski and \'etale sites of $S$. \end{proof} \noindent 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, Section \ref{descent-section-quasi-coherent-sheaves}. In this chapter we will sometimes refer to a site as in Theorem \ref{theorem-quasi-coherent}'' in order to conveniently state results which hold in any of those situations. \section{{\v C}ech cohomology} \label{section-cech-cohomology} \noindent 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 \ref{theorem-quasi-coherent}. To this end, we introduce {\v C}ech cohomology on sites. See \cite{ArtinTopologies} and Cohomology on Sites, Sections \ref{sites-cohomology-section-cech}, \ref{sites-cohomology-section-cech-functor} and \ref{sites-cohomology-section-cech-cohomology-cohomology} for more details. \begin{definition} \label{definition-cech-complex} Let $\mathcal{C}$ be a category, $\mathcal{U} = \{U_i \to U\}_{i \in I}$ a family of morphisms of $\mathcal{C}$ with fixed target, and $\mathcal{F} \in \textit{PAb}(\mathcal{C})$ an abelian presheaf. We define the {\it {\v C}ech 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. Again, it is essential to allow the case $i_0 = i_1$ etc. The {\it {\v C}ech cohomology groups} are defined by $$\check{H}^p(\mathcal{U}, \mathcal{F}) = H^p(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})).$$ \end{definition} \begin{lemma} \label{lemma-cech-presheaves} The functor $\check{\mathcal{C}}^\bullet(\mathcal{U}, -)$ is exact on the category $\textit{PAb}(\mathcal{C})$. \end{lemma} \noindent 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. \begin{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 \ref{sites-cohomology-lemma-cech-exact-presheaves}. \end{proof} \noindent 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 {\it effaceable} functor. We start by recalling the Yoneda lemma. \begin{lemma}[Yoneda Lemma] \label{lemma-yoneda-presheaf} For any presheaf $\mathcal{F}$ on a category $\mathcal{C}$ there is a functorial isomorphism $$\Hom_{\textit{PSh}(\mathcal{C})}(h_U, \mathcal{F}) = \mathcal{F}(U).$$ \end{lemma} \begin{proof} See Categories, Lemma \ref{categories-lemma-yoneda}. \end{proof} \noindent 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. \begin{definition} \label{definition-free-abelian-presheaf} Let $\mathcal{C}$ be a category. Given a presheaf of sets $\mathcal{G}$, we define the {\it 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 $\mathbf{Z}_U = \mathbf{Z}_{h_U}$. \end{definition} \noindent 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. \begin{lemma} \label{lemma-cech-complex-describe} The {\v C}ech 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*} \end{lemma} \begin{proof} This follows from the formula above. See Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-map-into}. \end{proof} \noindent This reduces us to studying only the complex in the first argument of the last $\Hom$. \begin{lemma} \label{lemma-exact} 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 $\textit{PAb}(\mathcal{C})$. \end{lemma} \begin{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 \ref{sites-cohomology-lemma-homology-complex} for more details. \end{proof} \begin{lemma} \label{lemma-hom-injective} Let $\mathcal{C}$ be a category. If $\mathcal{I}$ is an injective object of $\textit{PAb}(\mathcal{C})$ and $\mathcal{U}$ is a family of morphisms with fixed target in $\mathcal{C}$, then $\check H^p(\mathcal{U}, \mathcal{I}) = 0$ for all $p>0$. \end{lemma} \begin{proof} The {\v C}ech 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. \end{proof} \begin{theorem} \label{theorem-cech-derived} On $\textit{PAb}(\mathcal{C})$ the functors $\check{H}^p(\mathcal{U}, -)$ are the right derived functors of $\check{H}^0(\mathcal{U}, -)$. \end{theorem} \begin{proof} By the Lemma \ref{lemma-hom-injective}, 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 \ref{sites-cohomology-lemma-cech-cohomology-derived-presheaves}. \end{proof} \begin{remark} \label{remark-presheaves-no-topology} Observe that all of the preceding statements are about presheaves so we haven't made use of the topology yet. \end{remark} \section{The {\v C}ech-to-cohomology spectral sequence} \label{section-cech-ss} \noindent This spectral sequence is fundamental in proving foundational results on cohomology of sheaves. \begin{lemma} \label{lemma-forget-injectives} The forgetful functor $\textit{Ab}(\mathcal{C})\to \textit{PAb}(\mathcal{C})$ transforms injectives into injectives. \end{lemma} \begin{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 \ref{sites-cohomology-lemma-injective-abelian-sheaf-injective-presheaf}. \end{proof} \begin{theorem} \label{theorem-cech-ss} 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 $V \mapsto H^q(V, \mathcal{F})$. \end{theorem} \begin{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 \ref{lemma-forget-injectives}, it is a complex of injectives in $\textit{PAb}(\mathcal{C})$. Thus, the rows of the double complex are exact in positive degrees (Lemma \ref{lemma-hom-injective}), 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 $H^0(U, \mathcal{F})$. \medskip\noindent 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 \ref{sites-cohomology-lemma-cech-spectral-sequence}. \end{proof} \begin{remark} \label{remark-grothendieck-ss} This is a Grothendieck spectral sequence for the composition of functors $$\textit{Ab}(\mathcal{C}) \longrightarrow \textit{PAb}(\mathcal{C}) \xrightarrow{\check H^0} \textit{Ab}.$$ \end{remark} \section{Big and small sites of schemes} \label{section-big-small} \noindent 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 \'etale topology, then you can simply assume $\tau = \etale$ throughout. Moreover, we will discuss \'etale morphisms, \'etale coverings, and \'etale sites in more detail starting in Section \ref{section-etale-site}. 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. \begin{definition} \label{definition-tau-covering} (See Topologies, Definitions \ref{topologies-definition-fppf-covering}, \ref{topologies-definition-syntomic-covering}, \ref{topologies-definition-smooth-covering}, \ref{topologies-definition-etale-covering}, and \ref{topologies-definition-zariski-covering}.) 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 {\it $\tau$-covering} if and only if each $f_i$ is flat of finite presentation, syntomic, smooth, \'etale, resp.\ an open immersion, and we have $\bigcup f_i(T_i) = T$. \end{definition} \noindent It turns out that the class of all $\tau$-coverings satisfies the axioms (1), (2) and (3) of Definition \ref{definition-site} (our definition of a site), see Topologies, Lemmas \ref{topologies-lemma-fppf}, \ref{topologies-lemma-syntomic}, \ref{topologies-lemma-smooth}, \ref{topologies-lemma-etale}, and \ref{topologies-lemma-zariski}. In order to be able to compare any of these new topologies to the fpqc topology and to use the preceding results on descent on modules we single out a special class of $\tau$-coverings of affine schemes called standard coverings. \begin{definition} \label{definition-standard-tau} (See Topologies, Definitions \ref{topologies-definition-standard-fppf}, \ref{topologies-definition-standard-syntomic}, \ref{topologies-definition-standard-smooth}, \ref{topologies-definition-standard-etale}, and \ref{topologies-definition-standard-Zariski}.) Let $\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}$. Let $T$ be an affine scheme. A {\it 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, \'etale, resp.\ the immersion of a standard principal open in $T$ and $T = \bigcup f_j(U_j)$. \end{definition} \begin{lemma} \label{lemma-tau-affine} Let $\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}$. Any $\tau$-covering of an affine scheme can be refined by a standard $\tau$-covering. \end{lemma} \begin{proof} See Topologies, Lemmas \ref{topologies-lemma-fppf-affine}, \ref{topologies-lemma-syntomic-affine}, \ref{topologies-lemma-smooth-affine}, \ref{topologies-lemma-etale-affine}, and \ref{topologies-lemma-zariski-affine}. \end{proof} \noindent Finally, we come to our definition of the sites we will be working with. This is actually somewhat involved since, contrary to what happens in \cite{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 \ref{sets-section-sets-hierarchy}), 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 \'etale (or other) cohomology of $S$ (just as in \cite{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. \begin{definition} \label{definition-tau-site} Let $S$ be a scheme. Let $\tau \in \{fppf, syntomic, smooth, \etale, \linebreak[0] Zariski\}$. \begin{enumerate} \item A {\it big $\tau$-site of $S$} is any of the sites $(\Sch/S)_\tau$ constructed as explained above and in more detail in Topologies, Definitions \ref{topologies-definition-big-small-fppf}, \ref{topologies-definition-big-small-syntomic}, \ref{topologies-definition-big-small-smooth}, \ref{topologies-definition-big-small-etale}, and \ref{topologies-definition-big-small-Zariski}. \item If $\tau \in \{\etale, Zariski\}$, then the {\it 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 \'etale, 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 $S_\tau$. \end{enumerate} \end{definition} \noindent 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 \ref{definition-tau-covering}) 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 \ref{topologies-lemma-fppf-induced}, \ref{topologies-lemma-syntomic-induced}, \ref{topologies-lemma-smooth-induced}, \ref{topologies-lemma-etale-induced}, and \ref{topologies-lemma-zariski-induced}. We will mostly ignore these issues in this chapter. \medskip\noindent 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 \ref{section-cohomology}. 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 \'etale or Zariski site. However, this ambiguity is harmless by the following lemma. \begin{lemma} \label{lemma-compare-cohomology-big-small} 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 $S$. \end{lemma} \begin{proof} By Topologies, Lemmas \ref{topologies-lemma-at-the-bottom} and \ref{topologies-lemma-at-the-bottom-etale} the functors $S_\tau \to (\Sch/S)_\tau$ satisfy the hypotheses of Sites, Lemma \ref{sites-lemma-bigger-site}. Hence our lemma follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-bigger-site}. \end{proof} \noindent 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 \'etale topos in Lemma \ref{lemma-etale-topos-independent-partial-universe} below. For notation and terminology used in this lemma we refer to Topologies, Section \ref{topologies-section-change-alpha}. \begin{lemma} \label{lemma-cohomology-enlarge-partial-universe} 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 \begin{enumerate} \item 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 $U$. \item 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 $\mathcal{F}$. \end{enumerate} \end{lemma} \begin{proof} By Topologies, Lemma \ref{topologies-lemma-change-alpha} the inclusion functor $(\Sch/S)_\tau \to (\Sch'/S)_\tau$ satisfies the assumptions of Sites, Lemma \ref{sites-lemma-bigger-site}. This implies (2) and (1) follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-bigger-site}. \end{proof} \section{The \'etale topos} \label{section-etale-topos} \noindent A {\it topos} is the category of sheaves of sets on a site, see Sites, Definition \ref{sites-definition-topos}. Hence it is customary to refer to the use the phrase \'etale topos of a scheme'' to refer to the category of sheaves on the small \'etale site of a scheme. Here is the formal definition. \begin{definition} \label{definition-etale-topos} Let $S$ be a scheme. \begin{enumerate} \item The {\it \'etale topos}, or the {\it small \'etale topos} of $S$ is the category $\Sh(S_\etale)$ of sheaves of sets on the small \'etale site of $S$. \item The {\it Zariski topos}, or the {\it small Zariski topos} of $S$ is the category $\Sh(S_{Zar})$ of sheaves of sets on the small Zariski site of $S$. \item For $\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}$ a {\it big $\tau$-topos} is the category of sheaves of set on a big $\tau$-topos of $S$. \end{enumerate} \end{definition} \noindent 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 \ref{topologies-lemma-Zariski-usual}. Whereas the small \'etale topos does not depend on the choices made in the construction of the small \'etale site, in general the big topoi do depend on those choices. \medskip\noindent Here is a lemma, which is one of many possible lemmas expressing the fact that it doesn't matter too much which site we choose to define the small \'etale topos of a scheme. \begin{lemma} \label{lemma-alternative} Let $S$ be a scheme. Let $S_{affine, \etale}$ denote the full subcategory of $S_\etale$ whose objects are those $U/S \in \Ob(S_\etale)$ with $U$ affine. A covering of $S_{affine, \etale}$ will be a standard \'etale covering, see Topologies, Definition \ref{topologies-definition-standard-etale}. Then restriction $$\mathcal{F} \longmapsto \mathcal{F}|_{S_{affine, \etale}}$$ defines an equivalence of topoi $\Sh(S_\etale) \cong \Sh(S_{affine, \etale})$. \end{lemma} \begin{proof} This you can show directly from the definitions, and is a good exercise. But it also follows immediately from Sites, Lemma \ref{sites-lemma-equivalence} by checking that the inclusion functor $S_{affine, \etale} \to S_\etale$ is a special cocontinuous functor (see Sites, Definition \ref{sites-definition-special-cocontinuous-functor}). \end{proof} \begin{lemma} \label{lemma-etale-topos-independent-partial-universe} Let $S$ be a scheme. The \'etale topos of $S$ is independent (up to canonical equivalence) of the construction of the small \'etale site in Definition \ref{definition-tau-site}. \end{lemma} \begin{proof} We have to show, given two big \'etale sites $\Sch_\etale$ and $\Sch_\etale'$ containing $S$, then $\Sh(S_\etale) \cong \Sh(S_\etale')$ with obvious notation. By Topologies, Lemma \ref{topologies-lemma-contained-in} we may assume $\Sch_\etale \subset \Sch_\etale'$. By Sets, Lemma \ref{sets-lemma-what-is-in-it} any affine scheme \'etale 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 \'etale coverings into standard \'etale coverings. Hence the result follows from Lemma \ref{lemma-alternative}. \end{proof} \section{Cohomology of quasi-coherent sheaves} \label{section-cohomology-quasi-coherent} %9.22.09 \noindent We start with a simple lemma (which holds in greater generality than stated). It says that the {\v C}ech complex of a standard covering is equal to the {\v C}ech complex of an fpqc covering of the form $\{\Spec(B) \to \Spec(A)\}$ with $A \to B$ faithfully flat. \begin{lemma} \label{lemma-cech-complex} 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 \begin{enumerate} \item $V$ is an affine scheme, \item $\mathcal{V} = \{V \to U\}$ is a $\tau$-covering and an fpqc covering, \item the {\v C}ech complexes $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ and $\check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F})$ agree. \end{enumerate} \end{lemma} \begin{proof} As the covering is a standard $\tau$-covering 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 \ref{example-fpqc-coverings}. Note that $\mathcal{U}$ is a refinement of $\mathcal{V}$ and hence there is a map of {\v C}ech complexes $\check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$, see Cohomology on Sites, Equation (\ref{sites-cohomology-equation-map-cech-complexes}). 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 \begin{equation} \label{equation-sheaf-coprod} \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 {\v C}ech 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. \end{proof} \noindent Note that Equality (\ref{equation-sheaf-coprod}) 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). \begin{remark} \label{remark-refinement} In the statement of Lemma \ref{lemma-cech-complex} 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 {\v C}ech 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 {\v C}ech 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. \end{remark} \begin{lemma}[Locality of cohomology] \label{lemma-locality-cohomology} 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 $i \in I$. \end{lemma} \begin{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 \ref{sites-cohomology-lemma-kill-cohomology-class-on-covering}. \end{proof} \begin{theorem} \label{theorem-zariski-fpqc-quasi-coherent} 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 \begin{enumerate} \item the left hand side indicates the usual cohomology of the sheaf $\mathcal{F}$ on the underlying topological space of the scheme $S$, and \item the right hand side indicates cohomology of the abelian sheaf $\mathcal{F}^a$ (see Proposition \ref{proposition-quasi-coherent-sheaf-fpqc}) on the site $\mathcal{C}$. \end{enumerate} \end{theorem} \begin{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. \medskip\noindent 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 $p$. \begin{enumerate} \item[p = 1] Pick $\xi \in H^1_\tau(U, \mathcal{F}^a)$. By Lemma \ref{lemma-locality-cohomology}, 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 \ref{theorem-cech-ss}, 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 \ref{lemma-cech-complex}, $\check H^\bullet(\mathcal{U}, \mathcal{F}^a) = \check H^\bullet(\mathcal{V}, \mathcal{F}^a)$. 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 {\v C}ech complex $\check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F})$ is none other than the complex $(B/A)_\bullet \otimes_A M$. Now by Lemma \ref{lemma-descent-modules}, $H^p((B/A)_\bullet \otimes_A M) = 0$ for $p>0$, hence $\check \xi = 0$ and so $\xi = 0$. \item[p > 1] Pick $\xi \in H^p_\tau(U, \mathcal{F}^a)$. By Lemma \ref{lemma-locality-cohomology}, 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 \ref{theorem-cech-ss}. 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 \ref{lemma-descent-modules} again, we see that the {\v C}ech cohomology class $\check \xi$ must be zero, hence $\xi = 0$. \end{enumerate} 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 \ref{theorem-cech-ss} 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 \ref{coherent-lemma-cech-cohomology-quasi-coherent}. \medskip\noindent 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 \ref{descent-proposition-same-cohomology-quasi-coherent} (whose proof takes a slightly different approach) combined with Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-of-open}. \end{proof} \begin{remark} \label{remark-right-derived-global-sections} Comment on Theorem \ref{theorem-zariski-fpqc-quasi-coherent}. 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 $\mathcal{C}$. \end{remark} \section{Examples of sheaves} \label{section-examples-sheaves} \noindent Let $S$ and $\tau$ be as in Section \ref{section-big-small}. We have already seen that any representable presheaf is a sheaf on $(\Sch/S)_\tau$ or $S_\tau$, see Lemma \ref{lemma-representable-sheaf-fpqc} and Remark \ref{remark-fpqc-finest}. Here are some special cases. \begin{definition} \label{definition-additive-sheaf} On any of the sites $(\Sch/S)_\tau$ or $S_\tau$ of Section \ref{section-big-small}. \begin{enumerate} \item 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. \item 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. \item The {\it constant sheaf} $\underline{\mathbf{Z}/n\mathbf{Z}}$ on any site is the sheafification of the constant presheaf $U \mapsto \mathbf{Z}/n\mathbf{Z}$. \end{enumerate} \end{definition} \noindent The first is a sheaf by Theorem \ref{theorem-quasi-coherent} 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 \ref{groupoids-section-group-schemes}. The third by the finite \'etale 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 \ref{groupoids-example-constant-group} and the following remark. \begin{remark} \label{remark-constant-locally-constant-maps} Let $G$ be an abstract group. On any of the sites $(\Sch/S)_\tau$ or $S_\tau$ of Section \ref{section-big-small} the sheafification $\underline{G}$ of the constant presheaf associated to $G$ in the {\it 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 \ref{groupoids-example-constant-group}. By Lemma \ref{lemma-representable-sheaf-fpqc} 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. \end{remark} \begin{definition} \label{definition-structure-sheaf} Let $S$ be a scheme. The {\it 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. \end{definition} \noindent 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 \'etale site of $S$. \begin{remark} \label{remark-special-case-fpqc-cohomology-quasi-coherent} In the terminology introduced above a special case of Theorem \ref{theorem-zariski-fpqc-quasi-coherent} 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 $X$. \end{remark} \section{Picard groups} \label{section-picard-groups} \noindent The following theorem is sometimes called Hilbert 90''. \begin{theorem} \label{theorem-picard-group} 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*} \end{theorem} \begin{proof} Let $\tau$ be one of the topologies considered in Section \ref{section-big-small}. By Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-h1-invertible} 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 \ref{theorem-quasi-coherent} or the more precise Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent} we see that $\Pic(\mathcal{O}_\tau) = \Pic(X)$. The last equality is proved in the same way. \end{proof} \section{The \'etale site} \label{section-etale-site} \noindent At this point we start exploring the \'etale site of a scheme in more detail. As a first step we discuss a little the notion of an \'etale morphism. \section{\'Etale morphisms} \label{section-etale-morphism} \noindent For more details, see Morphisms, Section \ref{morphisms-section-etale} for the formal definition and \'Etale Morphisms, Sections \ref{etale-section-etale-morphisms}, \ref{etale-section-structure-etale-map}, \ref{etale-section-etale-smooth}, \ref{etale-section-topological-etale}, \ref{etale-section-functorial-etale}, and \ref{etale-section-properties-permanence} for a survey of interesting properties of \'etale morphisms. \medskip\noindent Recall that an algebra $A$ over an algebraically closed field $k$ is {\it 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 {\it 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 an $A$-algebra is said to be 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 {\it smooth} if it is flat, locally of finite presentation, and the geometric fibers are smooth. \medskip\noindent For these facts please see Morphisms, Section \ref{morphisms-section-smooth}. Using this we may define an \'etale morphism as follows. \begin{definition} \label{definition-etale-morphism} A morphism of schemes is {\it \'etale} if it is smooth of relative dimension 0. \end{definition} \noindent In particular, a morphism of schemes $X \to S$ is \'etale if it is smooth and $\Omega_{X/S} = 0$. \begin{proposition} \label{proposition-etale-morphisms} Facts on \'etale morphisms. \begin{enumerate} \item Let $k$ be a field. A morphism of schemes $U \to \Spec(k)$ is \'etale 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 $k$. \item Let $\varphi : U \to S$ be a morphism of schemes. The following conditions are equivalent: \begin{enumerate} \item $\varphi$ is \'etale, \item $\varphi$ is locally finitely presented, flat, and all its fibres are \'etale, \item $\varphi$ is flat, unramified and locally of finite presentation. \end{enumerate} \item A ring map $A \to B$ is \'etale 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 $B$. \item The base change of an \'etale morphism is \'etale. \item Compositions of \'etale morphisms are \'etale. \item Fibre products and products of \'etale morphisms are \'etale. \item An \'etale morphism has relative dimension 0. \item Let $Y \to X$ be an \'etale morphism. If $X$ is reduced (respectively regular) then so is $Y$. \item \'Etale morphisms are open. \item If $X \to S$ and $Y \to S$ are \'etale, then any $S$-morphism $X \to Y$ is also \'etale. \end{enumerate} \end{proposition} \begin{proof} We have proved these facts (and more) in the preceding chapters. Here is a list of references: (1) Morphisms, Lemma \ref{morphisms-lemma-etale-over-field}. (2) Morphisms, Lemmas \ref{morphisms-lemma-etale-flat-etale-fibres} and \ref{morphisms-lemma-flat-unramified-etale}. (3) Algebra, Lemma \ref{algebra-lemma-etale-standard-smooth}. (4) Morphisms, Lemma \ref{morphisms-lemma-base-change-etale}. (5) Morphisms, Lemma \ref{morphisms-lemma-composition-etale}. (6) Follows formally from (4) and (5). (7) Morphisms, Lemmas \ref{morphisms-lemma-etale-locally-quasi-finite} and \ref{morphisms-lemma-locally-quasi-finite-rel-dimension-0}. (8) See Algebra, Lemmas \ref{algebra-lemma-reduced-goes-up} and \ref{algebra-lemma-Rk-goes-up}, see also more results of this kind in \'Etale Morphisms, Section \ref{etale-section-properties-permanence}. (9) See Morphisms, Lemma \ref{morphisms-lemma-fppf-open} and \ref{morphisms-lemma-etale-flat}. (10) See Morphisms, Lemma \ref{morphisms-lemma-etale-permanence}. \end{proof} \begin{definition} \label{definition-standard-etale} A ring map $A \to B$ is called {\it standard \'etale} 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 $B$. \end{definition} \noindent It is true that a standard \'etale ring map is \'etale. 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 \'etale 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 $e_1, \ldots, e_b > 0$. 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 $\kappa(\mathfrak p) \subset \kappa_i$ is finite separable as desired. \medskip\noindent It turns out that any \'etale ring map is locally standard \'etale. To formulate this we introduce the following notation. A ring map $A \to B$ is {\it \'etale 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 \'etale. Here is the result. \begin{theorem} \label{theorem-standard-etale} A ring map $A \to B$ is \'etale 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 \'etale over $A$. \end{theorem} \begin{proof} See Algebra, Proposition \ref{algebra-proposition-etale-locally-standard}. \end{proof} \section{\'Etale coverings} \label{section-etale-covering} \noindent We recall the definition. \begin{definition} \label{definition-etale-covering} An {\it \'etale covering} of a scheme $U$ is a family of morphisms of schemes $\{\varphi_i : U_i \to U\}_{i \in I}$ such that \begin{enumerate} \item each $\varphi_i$ is an \'etale morphism, \item the $U_i$ cover $U$, i.e., $U = \bigcup_{i\in I}\varphi_i(U_i)$. \end{enumerate} \end{definition} \begin{lemma} \label{lemma-etale-fpqc} Any \'etale covering is an fpqc covering. \end{lemma} \begin{proof} (See also Topologies, Lemma \ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc}.) Let $\{\varphi_i : U_i \to U\}_{i \in I}$ be an \'etale covering. Since an \'etale 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 \'etale 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. \end{proof} \noindent So any statement which is true for fpqc coverings remains true {\it a fortiori} for \'etale coverings. For instance, the \'etale site is subcanonical. \begin{definition} \label{definition-big-etale-site} (For more details see Section \ref{section-big-small}, or Topologies, Section \ref{topologies-section-etale}.) Let $S$ be a scheme. The {\it big \'etale site over $S$} is the site $(\Sch/S)_\etale$, see Definition \ref{definition-tau-site}. The {\it small \'etale site over $S$} is the site $S_\etale$, see Definition \ref{definition-tau-site}. We define similarly the {\it big} and {\it small Zariski sites} on $S$, denoted $(\Sch/S)_{Zar}$ and $S_{Zar}$. \end{definition} \noindent Loosely speaking the big \'etale site of $S$ is made up out of schemes over $S$ and coverings the \'etale coverings. The small \'etale site of $S$ is made up out of schemes \'etale over $S$ with coverings the \'etale coverings. Actually any morphism between objects of $S_\etale$ is \'etale, in virtue of Proposition \ref{proposition-etale-morphisms}, 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. \medskip\noindent The small \'etale site has fewer objects than the big \'etale site, it contains only the opens'' of the \'etale topology on $S$. It is a full subcategory of the big \'etale site, and its topology is induced from the topology on the big site. Hence it is true that the restriction functor from the big \'etale site to the small one is exact and maps injectives to injectives. This has the following consequence. \begin{proposition} \label{proposition-cohomology-restrict-small-site} 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 $p \geq 0$. \end{proposition} \begin{proof} This is a special case of Lemma \ref{lemma-compare-cohomology-big-small}. \end{proof} \noindent In accordance with the general notation introduced in Section \ref{section-big-small} we write $H_\etale^p(S, \mathcal{F})$ for the above cohomology group. %9.24.09 \section{Kummer theory} \label{section-kummer} \noindent 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 \ref{groupoids-example-roots-of-unity} this is a representable functor, and the scheme representing it is denoted $\mu_n$ also. By Lemma \ref{lemma-representable-sheaf-fpqc} this functor satisfies the sheaf condition for the fpqc topology (in particular, it also satisfies the sheaf condition for the \'etale, Zariski, etc topology). \begin{lemma} \label{lemma-kummer-sequence} 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 \'etale site of $S$. \end{lemma} \begin{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 \'etale 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 \ref{constructions-section-spec-via-glueing} or \ref{constructions-section-spec} 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 \'etale, in particular \'etale. Since this holds for every affine open of $U$ we conclude that $\pi$ is \'etale. Hence $\mathcal{U} = \{\pi : U' \to U\}$ is an \'etale 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. \end{proof} \begin{remark} \label{remark-no-kummer-sequence-zariski} Lemma \ref{lemma-kummer-sequence} is false when \'etale'' is replaced with `Zariski''. Since the \'etale topology is coarser than the smooth topology, see Topologies, Lemma \ref{topologies-lemma-zariski-etale-smooth} it follows that the sequence is also exact in the smooth topology. \end{remark} \noindent By Theorem \ref{theorem-picard-group} and Lemma \ref{lemma-kummer-sequence} 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. \begin{lemma} \label{lemma-kummer-sequence-syntomic} 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 $(\Sch/S)_{syntomic}$. \end{lemma} \begin{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 \ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf}, 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 \ref{constructions-section-spec-via-glueing} or \ref{constructions-section-spec} 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. \end{proof} \begin{remark} \label{remark-no-kummer-sequence-smooth-etale-zariski} Lemma \ref{lemma-kummer-sequence-syntomic} is false for the smooth, \'etale, or Zariski topology. \end{remark} \noindent By Theorem \ref{theorem-picard-group} and Lemma \ref{lemma-kummer-sequence-syntomic} 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. \medskip\noindent 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 \begin{equation} \label{equation-isomorphisms-pairs} \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 {\bf not} the $n$-torsion in $\Pic(S)$. \begin{lemma} \label{lemma-describe-h1-mun} 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. \end{lemma} \begin{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 $\varphi_i : (\mathcal{L}, \alpha)|_{V_i} \to (\mathcal{O}_{V_i}, 1)$. \medskip\noindent 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. \medskip\noindent First construction: using {\v C}ech 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 (\ref{equation-isomorphisms-pairs}) 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 \ref{theorem-cech-ss} it maps to a cohomology class in $H^1_{syntomic}(S, \mu_{n, S})$. \medskip\noindent 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 \ref{sites-modules-section-internal-hom}). Since the conditions defining this subpresheaf are local, we see that it is a sheaf. By (\ref{equation-isomorphisms-pairs}) 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 \ref{sites-cohomology-lemma-torsors-h1} we obtain a corresponding element of $H^1_{syntomic}(S, \mu_{n, S})$. \medskip\noindent Ok, now we have to still show the following \begin{enumerate} \item The two constructions give the same cohomology class. \item Isomorphic pairs give rise to the same cohomology class. \item The cohomology class of $(\mathcal{L}, \alpha) \otimes (\mathcal{L}', \alpha')$ is the sum of the cohomology classes of $(\mathcal{L}, \alpha)$ and $(\mathcal{L}', \alpha')$. \item If the cohomology class is trivial, then the pair is trivial. \item Any element of $H^1_{syntomic}(S, \mu_{n, S})$ is the cohomology class of a pair. \end{enumerate} 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 {\v C}ech 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 \ref{sites-cohomology-lemma-trivial-torsor}. \medskip\noindent 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 \ref{theorem-picard-group} 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). \medskip\noindent 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 \'etale covering (compare with the proof of Lemma \ref{lemma-kummer-sequence}). 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 \ref{lemma-kummer-sequence}. \end{proof} \section{Neighborhoods, stalks and points} \label{section-stalks} \noindent 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 \'etale site of a scheme $S$ has enough points. It also turns out that any point of the small \'etale topos of $S$ (an abstract notion) is given by a geometric point. Thus in some sense the small \'etale topos of $S$ can be understood in terms of geometric points and neighbourhoods. \begin{definition} \label{definition-geometric-point} Let $S$ be a scheme. \begin{enumerate} \item A {\it 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 $k$. \item We say $\overline{s}$ {\it lies over} $s$ to indicate that $s \in S$ is the image of $\overline{s}$. \item An {\it \'etale 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 \'etale morphism of schemes. We write $(U, \overline{u}) \to (S, \overline{s})$. \item A {\it morphism of \'etale neighborhoods} $(U, \overline{u}) \to (U', \overline{u}')$ is an $S$-morphism $h: U \to U'$ such that $\overline{u}' = h \circ \overline{u}$. \end{enumerate} \end{definition} \begin{remark} \label{remark-etale-between-etale} Since $U$ and $U'$ are \'etale over $S$, any $S$-morphism between them is also \'etale, see Proposition \ref{proposition-etale-morphisms}. In particular all morphisms of \'etale neighborhoods are \'etale. \end{remark} \begin{remark} \label{remark-etale-neighbourhoods} Let $S$ be a scheme and $s \in S$ a point. In More on Morphisms, Definition \ref{more-morphisms-definition-etale-neighbourhood} we defined the notion of an \'etale neighbourhood $(U, u) \to (S, s)$ of $(S, s)$. If $\overline{s}$ is a geometric point of $S$ lying over $s$, then any \'etale neighbourhood $(U, \overline{u}) \to (S, \overline{s})$ gives rise to an \'etale 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 \'etale neighbourhood $(U, u)$ of $(S, s)$ the residue field extension $\kappa(s) \subset \kappa(u)$ is finite separable (see Proposition \ref{proposition-etale-morphisms}) 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 \'etale neighbourhood of $(S, \overline{s})$. We will use these observations to go between the two types of \'etale neighbourhoods. \end{remark} \begin{lemma} \label{lemma-cofinal-etale} Let $S$ be a scheme, and let $\overline{s}$ be a geometric point of $S$. The category of \'etale neighborhoods is cofiltered. More precisely: \begin{enumerate} \item Let $(U_i, \overline{u}_i)_{i = 1, 2}$ be two \'etale neighborhoods of $\overline{s}$ in $S$. Then there exists a third \'etale neighborhood $(U, \overline{u})$ and morphisms $(U, \overline{u}) \to (U_i, \overline{u}_i)$, $i = 1, 2$. \item Let $h_1, h_2: (U, \overline{u}) \to (U', \overline{u}')$ be two morphisms between \'etale neighborhoods of $\overline{s}$. Then there exist an \'etale 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 $h_1 \circ h = h_2 \circ h$. \end{enumerate} \end{lemma} \begin{proof} For part (1), consider the fibre product $U = U_1 \times_S U_2$. It is \'etale over both $U_1$ and $U_2$ because \'etale morphisms are preserved under base change, see Proposition \ref{proposition-etale-morphisms}. The map $\overline{s} \to U$ defined by $(\overline{u}_1, \overline{u}_2)$ gives it the structure of an \'etale 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 \'etale over $S$, so is the fibre product $U'\times_S U'$ (see Proposition \ref{proposition-etale-morphisms}). Hence the vertical arrow $(h_1, h_2)$ is \'etale by Remark \ref{remark-etale-between-etale} above. Therefore $U''$ is \'etale over $U'$ by base change, and hence also \'etale over $S$ (because compositions of \'etale morphisms are \'etale). Thus $(U'', \overline{u}'')$ is a solution to the problem. \end{proof} \begin{lemma} \label{lemma-geometric-lift-to-cover} Let $S$ be a scheme. Let $\overline{s}$ be a geometric point of $S$. Let $(U, \overline{u})$ be an \'etale neighborhood of $\overline{s}$. Let $\mathcal{U} = \{\varphi_i : U_i \to U \}_{i\in I}$ be an \'etale 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 \'etale neighborhoods. \end{lemma} \begin{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 \'etale morphisms so it is \'etale, see Proposition \ref{proposition-etale-morphisms}. Therefore, $\overline{s} \times_{\overline{u}, U, \varphi_i} U_i$ is a disjoint union of finite separable extensions of $k$, by Proposition \ref{proposition-etale-morphisms}. 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 $\overline{u}$. \end{proof} \begin{definition} \label{definition-stalk} Let $S$ be a scheme. Let $\mathcal{F}$ be a presheaf on $S_\etale$. Let $\overline{s}$ be a geometric point of $S$. The {\it 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 \'etale neighborhoods of $\overline{s}$ in $S$. \end{definition} \noindent By Lemma \ref{lemma-cofinal-etale}, this colimit is over a filtered index category, namely the opposite of the category of \'etale 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 \'etale neighbourhood $(U'', \overline{u}'')$ and morphisms of \'etale 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 \ref{categories-section-directed-colimits}. \begin{lemma} \label{lemma-stalk-gives-point} 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 \ref{sites-definition-point}) and its associated stalk functor $\mathcal{F} \mapsto \mathcal{F}_p$ (Sites, Equation \ref{sites-equation-stalk}) is the functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$ defined above. \end{lemma} \begin{proof} In the proof of Lemma \ref{lemma-geometric-lift-to-cover} 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 \ref{definition-geometric-point}. 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 \ref{lemma-geometric-lift-to-cover}. Hence Sites, Proposition \ref{sites-proposition-point-limits} 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 \ref{sites-equation-stalk} which proves the final assertion. \end{proof} \begin{remark} \label{remark-map-stalks} 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 {\it morphism $a : \overline{s} \to \overline{s}'$ of geometric points} is simply a morphism $a : \Spec(k) \to \Spec(k')$ such that $a \circ \overline{s}' = \overline{s}$. Given such a morphism we obtain a functor from the category of \'etale neighbourhoods of $\overline{s}'$ to the category of \'etale 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 \ref{categories-lemma-functorial-colimit}. 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. \medskip\noindent 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 \ref{sites-definition-morphism-points}, 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 \ref{lemma-stalk-gives-point}. 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 (insert future reference here). \end{remark} \begin{lemma} \label{lemma-stalk-exact} Let $S$ be a scheme. Let $\overline{s}$ be a geometric point of $S$. \begin{enumerate} \item The stalk functor $\textit{PAb}(S_\etale) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$ is exact. \item We have $(\mathcal{F}^\#)_{\overline{s}} = \mathcal{F}_{\overline{s}}$ for any presheaf of sets $\mathcal{F}$ on $S_\etale$. \item The functor $\textit{Ab}(S_\etale) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$ is exact. \item 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}_{\o