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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{basics of etale cohomology} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology} [[!include cohomology - contents]] \hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory} [[!include topos theory - contents]] \hypertarget{tale_morphisms}{}\paragraph*{{\'E{}tale morphisms}}\label{tale_morphisms} [[!include etale morphisms - contents]] This page goes through some basics of [[étale cohomology]]. \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{tale_topos}{\'E{}tale topos}\dotfill \pageref*{tale_topos} \linebreak \noindent\hyperlink{tale_morphisms_2}{\'E{}tale morphisms}\dotfill \pageref*{tale_morphisms_2} \linebreak \noindent\hyperlink{SheafConditionAndExamples}{Descent theorem and examples of \'e{}tale sheaves}\dotfill \pageref*{SheafConditionAndExamples} \linebreak \noindent\hyperlink{BaseChange}{Base change and sheaf cohomology}\dotfill \pageref*{BaseChange} \linebreak \noindent\hyperlink{EtaleCohomology}{\'E{}tale cohomology}\dotfill \pageref*{EtaleCohomology} \linebreak \noindent\hyperlink{WithCoefficientsInCoherentModules}{With coefficients in coherent modules}\dotfill \pageref*{WithCoefficientsInCoherentModules} \linebreak \noindent\hyperlink{WithCoefficientsInACyclicGroup}{With coefficients in cyclic groups}\dotfill \pageref*{WithCoefficientsInACyclicGroup} \linebreak \noindent\hyperlink{WithCoefficientsInTheMultiplicativeGroup}{With coefficients in the multiplicative group}\dotfill \pageref*{WithCoefficientsInTheMultiplicativeGroup} \linebreak \noindent\hyperlink{MainTheorems}{Outlook: The main theorems}\dotfill \pageref*{MainTheorems} \linebreak \noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak \hypertarget{tale_topos}{}\subsection*{{\'E{}tale topos}}\label{tale_topos} To every [[scheme]] $X$ is assigned a [[site]] which is a geometric analog of the collection of [[étale spaces]] over a [[topological space]]. This is called the [[étale site]] $X_{et}$ of the scheme. The [[category of sheaves]] on that site is called the [[étale topos]] of the scheme. The intrinsic [[cohomology]] of that [[topos]], hence the [[abelian sheaf cohomology]] over the [[étale site]], is the \emph{[[étale cohomology]]} of $X$. This section starts with looking at some basic aspects of the [[étale topos]] as such, the basic definitions and the central [[descent theorem]] for characterizing its [[sheaves]]. The \hyperlink{EtaleCohomology}{next section} then genuinely considers the corresponding [[abelian sheaf cohomology]]. \'E{}tale cohomology is traditionally motivated by the route by which it was historically discovered, namely as a fix for technical problems encountered with the [[Zariski topology]]. You can find this historical motivation in all textbooks and lectures, see the \emph{\hyperlink{References}{References}} below. But [[étale cohomology]] has a more fundamental \emph{raison d'\^e{}tre} than this. As discussed at \emph{[[étale topos]]} it is induced in any context in which one has a ``[[reduction modality]]''. While fundamental, this is actually a simple point of view which leads to a simple characterization of [[étale morphisms]], and this is what we start with now. \hypertarget{tale_morphisms_2}{}\subsubsection*{{\'E{}tale morphisms}}\label{tale_morphisms_2} Of the many equivalent characterizations of [[étale morphisms]], here we will have use of the following incarnation: \begin{defn} \label{EtaleMoprhism}\hypertarget{EtaleMoprhism}{} A morphisms of [[schemes]] is an \emph{[[étale morphism of schemes]]} if it is \begin{enumerate}% \item [[formally étale morphism of schemes|formally étale]] -- recalled in a \hyperlink{ExplicitDefinition}{moment}; \item [[locally of finite presentation]]. \end{enumerate} \end{defn} \begin{remark} \label{}\hypertarget{}{} The first condition makes an [[étale morphism of schemes]] be like an [[étale space]] over its [[codomain]]. The second essentially just says demands this has [[finite set|finite]] [[fibers]]. \end{remark} \begin{defn} \label{EtaleSite}\hypertarget{EtaleSite}{} For $X$ a [[scheme]], its [[étale site]] has a [[objects]] the [[étale morphisms of schemes]] into $X$, as [[morphisms]] the morphisms of schemes [[over category|over]] $X$, and as [[coverings]] the jointly surjective [[étale morphisms of schemes|étale morphisms]] over $X$. The [[category of sheaves]] on $X_{et}$ is the \emph{[[étale topos]]} of $X$. The corresponding [[abelian sheaf cohomology]] is its \emph{[[étale cohomology]]}. \end{defn} The definition of [[formally étale morphisms of schemes|formally étale]] in components goes like this. \begin{defn} \label{ExplicitDefinition}\hypertarget{ExplicitDefinition}{} A [[morphism]] of [[commutative rings]] $R \longrightarrow A$ is called \emph{[[formally étale morphisms of schemes|formally étale]]} if for every ring $B$ and for every [[nilpotent ideal]] $I \subset B$ and for every [[commuting diagram]] of the form \begin{displaymath} \itexarray{ B/I &\leftarrow& A \\ \uparrow && \uparrow \\ B &\leftarrow& R } \end{displaymath} there is a unique diagonal morphism \begin{displaymath} \itexarray{ B/I &\leftarrow& A \\ \uparrow &\swarrow& \uparrow \\ B &\leftarrow& R } \end{displaymath} that makes both triangles commute. \end{defn} (e.g. \hyperlink{StackProject}{Stacks Project 57.9, 57.12}) \begin{remark} \label{}\hypertarget{}{} So [[Isbell duality|dually]] this means that $Spec(A) \to Spec(R)$ is formally \'e{}tale if it has the unique [[right lifting property]] against all [[infinitesimal object|infinitesimal extensions]] \begin{displaymath} \itexarray{ Spec(B_{red}) &\longrightarrow& Spec(A) \\ \downarrow &\nearrow& \downarrow \\ Spec(B) &\longrightarrow & Spec(R) } \,. \end{displaymath} \end{remark} and [[sheafification|locality]] this yields a notion of formally \'e{}tale morphisms of [[affine varieties]] and of [[schemes]]. It is useful to realize this equivalently but a bit more naturally as follows. \begin{defn} \label{ReductionOnRings}\hypertarget{ReductionOnRings}{} Write $CRing_{fin}$ for the [[category]] of [[finitely generated ring|finitely generated]] [[commutative rings]] and write $CRing_{fin}^{ext}$ for the category of [[infinitesimal ring extensions]]. Write \begin{displaymath} Red \;\colon\; CRing_{fin}^{ext} \longrightarrow CRing_{fin} \end{displaymath} for the [[functor]] which sends an [[infinitesimal ring extension]] to the underlying [[commutative ring]] (in the maximal case this sends a commutative ring to its [[reduced ring]], whence the name of the functor), and write \begin{displaymath} i \;\colon\; CRing_{fin} \hookrightarrow CRing_{fin}^{ext} \end{displaymath} for the [[full subcategory]] inclusion that regards a ring as the trivial infinitesimal extension over itself. \end{defn} \begin{prop} \label{DifferentialCohesionModality}\hypertarget{DifferentialCohesionModality}{} There is an [[adjoint triple]] of [[idempotent monad|idempotent]] ([[comonad|co]]-)[[monads]] \begin{displaymath} (Red \dashv \int_{inf} \dashv \flat_{inf}) \;\colon\; PSh((CRing_{fin}^{ext})^{op}) \longrightarrow PSh((CRing_{fin}^{ext})^{op}) \end{displaymath} where the [[left adjoint]] [[comonad]] $Red$ is given on [[representable functor|representables]] by the [[reduced scheme|reduction]] functor of def. \ref{ReductionOnRings} (followed by the inclusion). \end{prop} This statement and the following prop. \ref{FormalEtalenessBydeRhamSpace} is a slight paraphrase of an observation due to (\href{Q-category#KontsevichRosenbergSpaces}{Kontsevich-Rosenberg 04}). A closely related adjunction appeared in (\href{de+Rham+space#SimpsonTeleman}{Simpson-Teleman 13}) in the discussion of [[de Rham spaces]]. The general abstract situation of ``[[differential cohesion]]'' has been discussed in (\href{http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos}{Schreiber 13}). \begin{proof} The functors from def. \ref{ReductionOnRings} form an [[adjoint pair]] $(Red \dashv i)$ because an extension element can only map to an extension element; so for $\widehat R \to R$ an [[infinitesimal ring extension]] of $R = Red(\widehat R)$, and for $S$ a commutative ring with $i(S) = (S \to S)$ its trivial extension, there is a [[natural isomorphism]] \begin{displaymath} Hom_{CRing_{fin}^{ext}}(\widehat R, i(S)) \simeq Hom_{CRing_{fin}}(R,S) \,. \end{displaymath} This exhibits $CRing_{fin}$ as a [[reflective subcategory]] of $CRing_{fin}^{ext}$. \begin{displaymath} (Red \dashv i) \;\colon\; CRing_{fin} \stackrel{\overset{Red}{\leftarrow}}{\underset{i}{\hookrightarrow}} CRing_{fin}^{ext} \,. \end{displaymath} Via [[Kan extension]] this [[adjoint pair]] induces an [[adjoint quadruple]] of [[functors]] on [[categories of presheaves]] \begin{displaymath} PSh(CRing_{fin}^{op}) \stackrel{\overset{i_!}{\hookrightarrow}}{\stackrel{\overset{i^\ast = Red_!}{\leftarrow}}{\stackrel{\overset{Red^\ast}{\hookrightarrow}}{\underset{Red_\ast}{\leftarrow}}}} PSh((CRing_{fin}^{ext})^{op}) \,. \end{displaymath} The [[adjoint triple]] to be shown is obtained from composing these adjoints pairwise. That $Red$ coincides with the reduction functor on representables is a standard property of [[left Kan extension]] (see \href{Kan+extension#LeftKanOnRepresentables}{here} for details). \end{proof} \begin{remark} \label{}\hypertarget{}{} These considerations make sense in the general abstract context of ``[[differential cohesion]]'' where the [[adjoint triple]] of prop. \ref{DifferentialCohesionModality} would be called: ([[reduction modality]] $\dashv$ [[infinitesimal shape modality]] $\dashv$ [[infinitesimal flat modality]]). \end{remark} Due to the [[full subcategory]] inclusion $i_!$ in the proof of prop. \ref{DifferentialCohesionModality} we may equivalently regard presheaves on $(CRing_{fin})^{op}$ (e.g. [[schemes]]) as presheaves on $(CRing_{fin}^{ext})^{op}$ (e.g. [[formal schemes]]). This is what we do implicitly in the following. \begin{prop} \label{FormalEtalenessBydeRhamSpace}\hypertarget{FormalEtalenessBydeRhamSpace}{} A morphism $f \;\colon\; Spec A \to Spec R$ in $CRing_{fin}^{op} \hookrightarrow PSh(CRing_{fin}^{op})$ is formally \'e{}tale, def. \ref{ExplicitDefinition}, precisely if it is $\int_{inf}$-[[modal type|modal]] relative $Spec R$, hence if the [[natural transformation|naturality square]] of the [[infinitesimal shape modality]]-[[unit of a monad|unit]] \begin{displaymath} \itexarray{ Spec A &\longrightarrow& \int_{inf} Spec A \\ \downarrow && \downarrow \\ Spec R &\longrightarrow& \int_{inf} Spec R } \end{displaymath} is a [[pullback]] square. \end{prop} \begin{proof} Evaluated on $I \hookrightarrow R \to R/I \in CRing_{fin}^{ext}$ any object, by the [[Yoneda lemma]] and the $(Red \dashv \int_{inf})$-[[adjunction]] the naturality square becomes \begin{displaymath} \itexarray{ CRing(A,B) &\longrightarrow& CRing(A,B/I) \\ \downarrow && \downarrow \\ CRing(R,B) &\longrightarrow& CRing(R,B/I) } \,. \end{displaymath} in [[Set]]. Chasing elements through this shows that this is a [[pullback]] precisely if the condition in def. \ref{ExplicitDefinition} holds. \end{proof} The basic stability property of [[étale morphisms]], which we need in the following, immediately follows from this characterization: \begin{prop} \label{ClosureForFormallyEtale}\hypertarget{ClosureForFormallyEtale}{} For $\stackrel{f}{\to} \stackrel{g}{\to}$ two composable morphisms, then \begin{enumerate}% \item if $f$ and $g$ are both (formally) \'e{}tale, then so is their composite $g \circ f$; \item if $g$ and $g\circ f$ are (formally) \'e{}tale, then so is $f$; \item the [[pullback]] of a (formally) \'e{}tale morphism along any morphism is again (formally) \'e{}tale. \end{enumerate} \end{prop} \begin{proof} With prop. \ref{FormalEtalenessBydeRhamSpace} this is equivalently the statement of the [[pasting law]] for [[pullback]] diagrams. \end{proof} Apart from that, for the proofs in the following we need the following basic facts \begin{prop} \label{}\hypertarget{}{} \begin{itemize}% \item Every etale morphism is a [[flat morphism]]. \item Flat morphism between affines $Spec(B) \to Spec(A)$ is [[faithfully flat]] precisely if it is surjective \end{itemize} \end{prop} We repeatedly use the following example of \'e{}tale morphisms. \begin{prop} \label{OpenImmersionIsEtale}\hypertarget{OpenImmersionIsEtale}{} Every [[open immersion of schemes]] is an [[étale morphism of schemes]]. In particular a standard open inclusion (a [[cover]] in the [[Zariski topology]]) induced by the [[localization of a commutative ring]] \begin{displaymath} Spec(R[S^{-1}]) \longrightarrow Spec(R) \end{displaymath} is \'e{}tale. \end{prop} (e.g. [[The Stacks Project|Stacks Project, lemma 28.37.9]]) \begin{proof} By def. \ref{EtaleMoprhism} we need to check that the map $Spec(R[S^{-1}]) \longrightarrow Spec(R)$ is a [[formally étale morphism]] and [[locally of finite presentation]]. The latter is clear, since the very definition of [[localization of a commutative ring]] \begin{displaymath} R[S^{-1}] = R[s_1^{-1}, \cdots, s_n^{-1}](s_1 s_1^{-1} - 1, \cdots , s_n s_n^{-1} - 1) \end{displaymath} exhibits a [[finitely presented algebra]] over $R$. To see that it is formally \'e{}tale we need to check that for every [[commutative ring]] $T$ with [[nilpotent ideal]] $J$ we have a [[pullback]] diagram \begin{displaymath} \itexarray{ Hom(R[S^{-1}], T) &\longrightarrow& Hom(R[S^{-1}],T/J) \\ \downarrow && \downarrow \\ Hom(R, T) &\longrightarrow& Hom(R, T/J) } \,. \end{displaymath} Now by the [[universal property]] of the [[localization of a commutative ring|localization]], a homomorphism $R[S^{-1}] \longrightarrow T$ is a homomorphism $R \longrightarrow T$ which sends all elements in $S \hookrightarrow R$ to invertible elements in $T$. But no element in a [[nilpotent ideal]] can be invertible, Therefore the fiber product of the bottom and right map is the set of maps from $R$ to $T$ such that $S$ is taken to invertibles, which is indeed the top left set. \end{proof} \hypertarget{SheafConditionAndExamples}{}\subsubsection*{{Descent theorem and examples of \'e{}tale sheaves}}\label{SheafConditionAndExamples} Since there are ``many more'' [[étale morphisms of schemes]] than there are [[open immersions of schemes]], a priori the discussion of [[descent]] over the [[étale site]] is more intricate than that in, say, the [[Zariski topology]]. However, the following proposition drastically reduces the types of \'e{}tale [[covers]] over which [[descent]] has to be checked in addition to the [[open immersions of schemes|open immersions]]. Then the following [[descent theorem]] effectively solves the descent problem over these remaining covers. \begin{prop} \label{EtaleDescentDetectedOnOpenImmersionCovers}\hypertarget{EtaleDescentDetectedOnOpenImmersionCovers}{} For $X$ a [[scheme]], and $A \in PSh(X_{et})$ a [[presheaf]] on its [[étale site]], def. \ref{EtaleSite}, for checking the [[sheaf]] condition it is sufficient to check [[descent]] on the following two kinds of [[covers]] in the [[étale site]] \begin{enumerate}% \item jointly surjective collections of [[open immersions of schemes]]; \item single [[faithfully flat morphisms]] between [[affine schemes]] \end{enumerate} (all over $X$). \end{prop} (\hyperlink{Tamme}{Tamme, II Lemma (3.1.1)}, \hyperlink{Milne}{Milne, prop. 6.6}) \begin{proof} Suppose given an arbitrary \'e{}tale [[covering]] $\{X'_i \to X'\}$ over $X$. We show how to refine it to a more special cover which itslf is the composition of covers of the form as in the statement. To that end, first choose a cover $\{U'_j \to X'\}$ of $X_i$ by affine [[open immersions of schemes]]. Then pulling back the original cover along that one yields covers \begin{displaymath} \{X'_i \times_{X'} U'_j \to U'_j\} \end{displaymath} of each of the open affines. By pullback stability, prop. \ref{ClosureForFormallyEtale}, these are still \'e{}tale maps. Now these patches in turn we cover by open affines \begin{displaymath} \{ \{U'_{i j k} \to X'_i \times_{X'} U'_j \} \} \end{displaymath} leading to covers \begin{displaymath} \{ U'_{i j k} \to U'_j \} \end{displaymath} by affines. (Notice here crucially that while the $U'_{i j k}$ are affine open immersions in $X'_i \times_{X'} U'_j$, after this composition with an [[étale morphism]] they no longer need to be open immersions in $U'_j$, all we know is that the map is \'e{}tale. This is the source of the second condition in the proposition to be shown, as discussed now. ) Since each $U'_j$, being affine, is a [[quasi-compact scheme]], we may find a finite subcover \begin{displaymath} \{ U'_{j l} \to U'_j \} \,. \end{displaymath} Composed with the original $\{U'_j \to X'\}$ this yields a refinement of the original cover by open affines. Hence for checking descent it is sufficient to check it for these two kinds of overs. The latter is by open immersions. For the former, we may factor $\{U'_{j l} \to U'_j\}$ as a collection of open immersions \begin{displaymath} \{U'_{j i} \to \coprod U'_{j i}\} \end{displaymath} followed by the epimorphism of affines of the form \begin{displaymath} \{ \coprod U'_{j i} \to U'_j \} \,. \end{displaymath} Now this is morphism is etale, hence [[flat morphism|flat]], but also surjective. That makes it a [[faithfully flat morphism]]. \end{proof} Therefore we are led to consider [[descent]] along [[faithfully flat morphisms]] of affines. For these the \emph{[[descent theorem]]} says that they are [[effective epimorphisms]]: \begin{defn} \label{}\hypertarget{}{} Given a [[commutative ring]] $R$ and an $R$-[[associative algebra]] $A$, hence a [[ring]] [[homomorphism]] $f \colon R \longrightarrow A$, the \emph{[[Amitsur complex]]} is the [[Moore complex]] of the dual [[Cech nerve]] of $Spec(A) \to Spec(R)$, hence the [[chain complex]] \begin{displaymath} 0 \to R \stackrel{f}{\to} A \stackrel{1 \otimes id - id \otimes 1}{\longrightarrow} A \otimes_R A \to A \otimes_R A \otimes_R A \to \cdots \,. \end{displaymath} \end{defn} (See also at \emph{[[Sweedler coring]]} and at \emph{[[commutative Hopf algebroid]]} for the same or similar constructions.) \begin{prop} \label{DescentTheorem}\hypertarget{DescentTheorem}{} \textbf{(descent theorem)} If $A \to B$ is [[faithfully flat]] then its Amitsur complex is [[exact sequence|exact]]. \end{prop} This is due to ([[Grothendieck]], [[FGA]]1). The following reproduces the proof in low degree following (\hyperlink{Milne}{Milne, prop. 6.8}). \begin{proof} We show that \begin{displaymath} 0 \to A \stackrel{f}{\longrightarrow} B \stackrel{1 \otimes id - id \otimes 1}{\longrightarrow} B \otimes_A B \end{displaymath} is an [[exact sequence]] if $f \colon A \longrightarrow B$ is [[faithfully flat]]. First observe that the statement follows if $A \to B$ admits a [[section]] $s \colon B \to A$. Because then we can define a map \begin{displaymath} k \colon B \otimes_A B \longrightarrow B \end{displaymath} \begin{displaymath} k \;\colon\; b_1 \otimes b_2 \mapsto b_1 \cdot f(s(b_2)) \,. \end{displaymath} This is such that applied to a [[coboundary]] it yields \begin{displaymath} k(1 \otimes b - b \otimes 1) = f(s(b)) - b \end{displaymath} and hence it exhibits every [[cocycle]] $b$ as a coboundary $b = f(s(b))$. So the statement is true for the special morphism \begin{displaymath} B \to B \otimes_A B \end{displaymath} \begin{displaymath} b \mapsto b \otimes 1 \end{displaymath} because that has a section given by the multiplication map. But now observe that the morphism $B \to B \otimes_A B$ is the [[tensor product]] of the morphism $f$ with $B$ over $A$, hence the [[Amitsur complex]] of this morphism is [[exact sequence|exact]]. Finally, the fact that $A \to B$ is [[faithfully flat]] by assumption, hence that it exhibits $B$ as a [[faithfully flat module]] over $A$, means by definition that the [[Amitsur complex]] for $(A \to B)\otimes_A B$ is exact precisely if that for $A \to B$ is exact. \end{proof} \begin{prop} \label{XSchemesRepresentSheaves}\hypertarget{XSchemesRepresentSheaves}{} For $Z \to X$ any [[scheme]] over a [[scheme]] $X$, the induced [[presheaf]] on the [[étale site]] \begin{displaymath} (U_Y \to X) \mapsto Hom_X(U_Y, Z) \end{displaymath} is a [[sheaf]]. \end{prop} This is due to ([[Grothendieck]], [[SGA]]1 exp. XIII 5.3) A review is in (\hyperlink{Tamme}{Tamme, II theorem (3.1.2)}, \hyperlink{Milne}{Milne, 6.2}). \begin{proof} By prop. \ref{EtaleDescentDetectedOnOpenImmersionCovers} we are reduced to showing that the represented presheaf satisfies [[descent]] along collections of open immersions and along surjective maps of affines. For the first this is clear (it is [[Zariski topology]]-descent). For the second case of a [[faithfully flat]] cover of affines $Spec(B) \to Spec(A)$ it follows with the exactness of the corresponding [[Amitsur complex]], by the [[descent theorem]], prop. \ref{DescentTheorem}. \end{proof} \begin{remark} \label{}\hypertarget{}{} This map from $X$-schemes to sheaves on $X_{et}$ is not injective, different $X$-schemes may represent the same sheaf on $X_{et}$. Unique representatives are given by [[étale morphism of schemes|étale schemess]] over $X$. \end{remark} (e.g. \hyperlink{Tamme}{Tamme, II theorem 3.1}) We consider some examples of [[sheaves of abelian groups]] induced by prop. \ref{XSchemesRepresentSheaves} from [[group schemes]] over $X$. \begin{example} \label{}\hypertarget{}{} The [[additive group]] over $X$ is the [[group scheme]] \begin{displaymath} \mathbb{G}_a \coloneqq Spec(\mathbb{Z}[t]) \times_{Spec(\mathbb{Z})} X \,. \end{displaymath} By the [[universal property]] of the [[pullback]], the corresponding sheaf $(\mathbb{G}_a)_X$ is given by the assignment \begin{displaymath} \begin{aligned} (\mathbb{G}_a)_X(U_X \to X) & = Hom_X(U_X, Spec(\mathbb{Z}[t]) \times_{Spec(\mathbb{Z})} X) \\ & = Hom(U_X, Spec(\mathbb{Z}[t])) \\ & = Hom(\mathbb{Z}[t], \Gamma(U_X, \mathcal{O}_{U_X})) \\ & = \Gamma(U_X, \mathcal{O}_{U_X}) \end{aligned} \,. \end{displaymath} \end{example} \begin{remark} \label{}\hypertarget{}{} In other words, the sheaf represented by the [[additive group]] is the [[abelian sheaf]] underlying the [[structure sheaf]] of $X$, and in particular the structure sheaf is indeed an \'e{}tale sheaf. \end{remark} Similarly one finds: \begin{example} \label{}\hypertarget{}{} The [[multiplicative group]] over $X$ \begin{displaymath} \mathbb{G}_m \coloneqq Spec(\mathbb{Z}[t,t^{-1}]) \times_{Spec(\mathbb{Z})} X \end{displaymath} represents the sheaf $(\mathbb{G}_m)_X$ given by \begin{displaymath} (\mathbb{G}_m)_X(U_X) \mapsto \Gamma(U_X, \mathcal{O}_{U_X})^\times \,. \end{displaymath} \end{example} (e.g. \hyperlink{Tamme}{Tamme, II, 3}) \hypertarget{BaseChange}{}\subsubsection*{{Base change and sheaf cohomology}}\label{BaseChange} \begin{defn} \label{BaseChangeOnSites}\hypertarget{BaseChangeOnSites}{} For $f \colon X \longrightarrow Y$ a [[homomorphism]] of [[schemes]], there is induced a [[functor]] on the [[categories]] underlying the [[étale site]] \begin{displaymath} f^{-1} \;\colon\; Y_{et} \longrightarrow X_{et} \end{displaymath} given by sending an [[object]] $U_Y \to Y$ to the [[fiber product]]/[[pullback]] along $f$ \begin{displaymath} f^{-1} \colon (U_Y \to Y) \mapsto (X \times_Y U_Y \to X) \,. \end{displaymath} \end{defn} \begin{prop} \label{DirectAndInverseImageAlongMapOfBases}\hypertarget{DirectAndInverseImageAlongMapOfBases}{} The morphism in def. \ref{BaseChangeOnSites} is a [[morphism of sites]] and hence induces a [[geometric morphism]] between the \'e{}tale toposes \begin{displaymath} (f^\ast \dashv f_\ast) \;\colon\; Sh(X_{et}) \stackrel{\overset{f^\ast}{\leftarrow}}{\underset{f_\ast}{\longrightarrow}} Sh(Y_{et}) \,. \end{displaymath} Here the [[direct image]] is given on a [[sheaf]] $\mathcal{F} \in Sh(X_{et})$ by \begin{displaymath} f_\ast \mathcal{F} \;\colon\; (U_Y \to Y) \mapsto \mathcal{F}(f^{-1}(U_Y)) = \mathcal{F}(X \times_X U_Y) \end{displaymath} while the [[inverse image]] is given on a [[sheaf]] $\mathcal{F} \in Sh_(Y_{et})$ by \begin{displaymath} f^\ast \mathcal{F} \;\colon\; (U_X \to X) \mapsto \underset{\underset{U_X \to f^{-1}(U_Y)}{\longrightarrow}}{\lim} \mathcal{F}(U_Y) \,. \end{displaymath} \end{prop} By the discussion at \emph{\href{morphism+of+sites#RelationToGeometricMorphisms}{morphisms of sites -- Relation to geometric morphisms}}. See also for instance (\hyperlink{Tamme}{Tamme I 1.4}). \begin{prop} \label{DerivedDirectImageByCohomology}\hypertarget{DerivedDirectImageByCohomology}{} The $q$th [[derived functor]] $R^q f_\ast$ of the [[direct image]] functor of def. \ref{DirectAndInverseImageAlongMapOfBases} sends $\mathcal{F} \in Ab(Sh(X_{et}))$ to the [[sheafification]] of the [[presheaf]] \begin{displaymath} (U_Y \to Y) \mapsto H^q(X \times_Y U_Y, \mathcal{F}) \,, \end{displaymath} where on the right we have the degree $q$ [[abelian sheaf cohomology]] [[cohomology group|group]] with [[coefficients]] in the given $\mathcal{F}$ ([[étale cohomology]]). \end{prop} (e.g. \hyperlink{Tamme}{Tamme, I (3.7.1), II (1.3.4)}, \hyperlink{Milne}{Milne, 12.1}). \begin{proof} We have a [[commuting diagram]] \begin{displaymath} \itexarray{ Ab(PSh(X)) &\stackrel{(-)\circ f^{-1}}{\longrightarrow}& Ab(PSh(Y)) \\ \uparrow^{\mathrlap{inc}} && \downarrow^{L} \\ Ab(Sh(X)) &\stackrel{f_\ast}{\longrightarrow}& Ab(Sh(Y)) } \,, \end{displaymath} where the right vertical morphism is [[sheafification]]. Because $(-) \circ f^{-1}$ and $L$ are both [[exact functors]] it follows that for $\mathcal{F} \to I^\bullet$ an [[injective resolution]] that \begin{displaymath} \begin{aligned} R^p f_\ast(\mathcal{F}) & :\simeq H^p( f_\ast I) \\ & = H^p(L I^\bullet(f^{-1}(-))) \\ & = L (H^p(I^\bullet)(f^{-1}(-))) \end{aligned} \end{displaymath} \end{proof} \begin{remark} \label{GrothendieckAndLeraySpectralSequence}\hypertarget{GrothendieckAndLeraySpectralSequence}{} For $O_X \stackrel{f^{-1}}{\leftarrow} O_Y \stackrel{g^{-1}}{\leftarrow} O_Z$ two composable [[morphisms of sites]], the [[Grothendieck spectral sequence]] for the corresponding [[direct images]] is of the form \begin{displaymath} E^{p,q}_2 = R^p g_\ast(R^q f_\ast(\mathcal{F})) \Rightarrow E^{p+q} = R^{p+q}(g f)_\ast(\mathcal{F}) \,. \end{displaymath} For the special case that $S_Z = \ast$ and $g^{-1}$ includes an [[étale morphism of schemes|étale morphism]] $U_Y \to Y$ this yields the [[Leray spectral sequence]] \begin{displaymath} E^{p,q}_2 = H^p(U_Y, R^q f_\ast \mathcal{F}) \Rightarrow E^{p+q} = H^{p+q}(U_Y \times_Y X , \mathcal{F}) \,. \end{displaymath} \end{remark} \hypertarget{EtaleCohomology}{}\subsection*{{\'E{}tale cohomology}}\label{EtaleCohomology} With some basic facts about [[sheaves]] on the [[étale site]] in hand, we now consider basics of [[abelian sheaf cohomology]] with [[coefficients]] in some such sheaves. \begin{enumerate}% \item \hyperlink{WithCoefficientsInCoherentModules}{With coefficients in coherent modules} \item \hyperlink{WithCoefficientsInACyclicGroup}{With coefficients in cyclic groups} \item \hyperlink{WithCoefficientsInTheMultiplicativeGroup}{With coefficients in the multiplicative group} \end{enumerate} This may serve to give a first idea of the nature of [[étale cohomology]]. An outlook on the deep structurual theorems about [[étale cohomology]] is in the next section \hyperlink{MainTheorems}{below}. \hypertarget{WithCoefficientsInCoherentModules}{}\subsubsection*{{With coefficients in coherent modules}}\label{WithCoefficientsInCoherentModules} \begin{prop} \label{}\hypertarget{}{} For $X$ a [[scheme]] and $N$ a ([[flat module|flat]]) [[quasicoherent module]] over its [[structure sheaf]] $\mathcal{O}_X$, then this induces an [[abelian sheaf]] on the [[étale site]] by \begin{displaymath} N_{et} \;\colon\; (U_X \to X) \mapsto \Gamma(U_Y, N \otimes_{\mathcal{O}_X} \mathcal{O}_{U_Y}) \,. \end{displaymath} \end{prop} (e.g. \hyperlink{Tamme}{Tamme, II 3.2.1}) \begin{proof} By prop. \ref{EtaleDescentDetectedOnOpenImmersionCovers} it is sufficient to test the [[sheaf]] condition on open affine covers and on singleton covers by faithfully flat morphisms of affines. For the first case we have a sheaf since this is just the sheaf condition in the [[Zariski topology]]. For the second case the corresponding Cech complexes are the [[Amitsur complexes]] of a faithfully flat $A \to B$ [[tensor product|tensored]] with $N$. By the [[descent theorem]], prop. \ref{DescentTheorem} this is exact, hence verifies the sheaf condition. \end{proof} We consider now the \'e{}tale [[abelian sheaf cohomology]] with coefficients in such coherent modules. \begin{remark} \label{}\hypertarget{}{} A [[cover]] in the [[Zariski topology]] on [[schemes]] is an [[open immersion of schemes]] and hence is in particular an [[étale morphism of schemes]]. Hence the [[étale site]] is finer than the [[Zariski site]] and so every \'e{}tale [[sheaf]] is a Zarsiki sheaf, but not necessarily conversely. \end{remark} \begin{remark} \label{LerayForInclusionOfZariskiIntoEtale}\hypertarget{LerayForInclusionOfZariskiIntoEtale}{} For $X$ a [[scheme]], the inclusion \begin{displaymath} \epsilon \;\colon\; X_{Zar} \longrightarrow X_{et} \end{displaymath} of the [[Zariski site]] into the [[étale site]] is indeed a [[morphism of sites]]. Hence there is a [[Leray spectral sequence]], remark \ref{GrothendieckAndLeraySpectralSequence}, which computes \'e{}tale cohomology in terms of Zarsiki cohomology \begin{displaymath} E^{p,q}_2 = H^p(X_{Zar}, R^q \epsilon^\ast \mathcal{F}) \Rightarrow E^{p+q} = H^{p+q}(X_{et}, \mathcal{F}) \,. \end{displaymath} \end{remark} This is originally due to ([[Grothendieck]], [[SGA]] 4 (Chapter VII, p355)). Reviews include (\hyperlink{Tamme}{Tamme, II 1.3}). \begin{prop} \label{CohomologyWithCoeffsInCoherentModules}\hypertarget{CohomologyWithCoeffsInCoherentModules}{} For $N$ a [[quasi-coherent sheaf]] of $\mathcal{O}_X$-[[modules]] and $N_{et}$ the induced \'e{}tale sheaf (by the discussion at \href{etale+topos#QuasiCoherentModules}{\'e{}tale topos -- Quasicohetent sheaves}), then the [[edge morphism]] \begin{displaymath} H^p_{Zar}(X, N) \longrightarrow H^p_{et}(X,N_{et}) \end{displaymath} of the [[Leray spectral sequence]] of remark \ref{LerayForInclusionOfZariskiIntoEtale} is an [[isomorphism]] for all $p$, identifying the [[abelian sheaf cohomology]] on the [[Zariski site]] with [[coefficients]] in $N$ with the \'e{}tale cohomology with coefficients in $N_{et}$. Moreover, for $X$ affine we have \begin{displaymath} H^p_{et}(X, N_{et}) \simeq 0 \,. \end{displaymath} \end{prop} This is due to ([[Grothendieck]], [[FGA]] 1). See also for instance (\hyperlink{Tamme}{Tamme, II (4.1.2)}). \begin{proof} By the discussion at \emph{[[edge morphism]]} it suffices to show that \begin{displaymath} R^q \epsilon_\ast (N) = 0 \;\,,\;\;\; for \;\; p \gt 0 \,. \end{displaymath} By prop \ref{DerivedDirectImageByCohomology}, $R^q \epsilon_\ast N$ is the [[sheaf]] on the [[Zariski topology]] which is the [[sheafification]] of the [[presheaf]] given by \begin{displaymath} U \mapsto H^q(X_{et}|U, N) \,, \end{displaymath} hence it is sufficient that this vanishes, or rather, by locality ([[sheafification]]) it suffices to show this vanishes for $X = U = Spec(A)$ an affine [[algebraic variety]]. By the existence of \href{etale+site#CofinalAffineCovers}{cofinal affine \'e{}tale covers} the [[full subcategory]] $X_{et}^{a} \hookrightarrow X_{at}$ on the \'e{}tale maps with affien domains, equipped with the induced [[coverage]], is a [[dense subsite]]. Therefore it suffices to show the statement there. Moreover, by the finiteness condition on [[étale morphisms]] every cover of $X_{et}^{a}$ may be refined by a finite cover, hence by an affine covering map \begin{displaymath} Spec(B) \longrightarrow Spec(A) \,. \end{displaymath} It follows (by a discussion such as e.g. at [[Sweedler coring]]) that the corresponding [[Cech cohomology]] complex \begin{displaymath} N_{et}(Spec(A)) \to C^0(\{Spec(B) \to Spec(A)\}, N_{et}) \to C^1(\{Spec(B) \to Spec(A)\}, N_{et}) \to \cdots \end{displaymath} is of the form \begin{displaymath} 0 \to N \to N \otimes_A B \to N \otimes_{A} B \otimes_A B \to \cdots \,. \end{displaymath} known as the \emph{[[Amitsur complex]]} of $A \to B$, tensored with $N$. Since $A \to B$ is a [[faithfully flat morphism]], it follows again by the [[descent theorem]], prop. \ref{DescentTheorem} that this is [[exact sequence|exact]], hence that the cohomology indeed vanishes. \end{proof} \hypertarget{WithCoefficientsInACyclicGroup}{}\subsubsection*{{With coefficients in cyclic groups}}\label{WithCoefficientsInACyclicGroup} Let $X$ be a [[reduced scheme|reduced]] [[scheme]] of [[characteristic]] the [[prime number]] $p$, hence such that for all points $x \in X$ \begin{displaymath} p \cdot \mathcal{O}_{X,x} = 0 \,. \end{displaymath} Write \begin{displaymath} F - id \coloneqq (-)^p - (-) \;\colon\; (\mathbb{G}_a)_X \longrightarrow (\mathbb{G}_a)_X \end{displaymath} for the [[endomorphism]] of the [[additive group]] over the [[étale site]] $X_{et}$ of $X$ (the [[structure sheaf]] regarded as just a [[sheaf of abelian groups]]) which is the [[Frobenius endomorphism]] $F(-) \coloneqq (-)^p$ minus the identity. \begin{prop} \label{}\hypertarget{}{} There is a [[short exact sequence]] of [[abelian sheaves]] over the [[étale site]] \begin{displaymath} 0 \to (\mathbb{Z}/p\mathbb{Z})_X \to (\mathbb{G}_a)_X \stackrel{F-id}{\to} (\mathbb{G}_a)_X \to 0 \,. \end{displaymath} \end{prop} This is called the \emph{[[Artin-Schreier sequence]]} (e.g. \hyperlink{Tamme}{Tamme, section II 4.2}, \hyperlink{Milne}{Milne, example 7.9}). \begin{proof} By the discussion at \href{category+of+sheaves#EpiMonoIsomorphisms}{category of sheaves -- Epi-/Mono-morphisms} we need to show that the left morphism is an injection over any [[étale morphism]] $U_Y \to X$, and that for every element $s \in \mathcal{O}_X$ there exists an [[étale site]] [[covering]] $\{U_i \to X\}$ such that $(-)^p- (-)$ restricts on this to a morphism which hits the restriction of that element. The first statement is clear, since $s = s^p$ says that $s$ is a constant section, hence in the image of the [[constant sheaf]] $\mathbb{Z}/p\mathbb{Z}$ and hence for each connected $U_Y \to X$ the left morphism is the inclusion \begin{displaymath} \mathbb{Z}/p\mathbb{Z} \hookrightarrow \mathcal{O}_{X'} \end{displaymath} induced by including the unit [[section]] $e_{X'}$ and its multiples $r e_{X'}$ for $0 \leq r \lt p$. (This uses the ``[[freshman's dream]]''-fact that in [[characteristic]] $p$ we have $(a + b)^p = a^p + b^p$). This is injective by assumption that $X$ is of characteristic $p$. To show that $(-)^p - (-)$ is an epimorphism of sheaves, it is sufficient to find for each element $s \in \mathcal{O}_X = A$ an [[étale site|étale cover]] $Spec(B) \to Spec(A)$ such that its restriction along this cover is in the image of $(-)^p - (-) \colon B \to B$. The choice \begin{displaymath} B \coloneqq A[t]/(t- t^p - s) \end{displaymath} by construction has the desired property concerning $s$, the preimage of $s$ is the equivalence class of $t$. To see that with this choice $Spec(B) \to Spec(A)$ is indeed an [[étale morphism of schemes]] it is sufficient to observe that it is a [[morphism of finite presentation]] and a [[formally étale morphism]]. The first is true by construction. For the second observe that for a ring homomorphism $B \to T$ the generator $t$ cannot go to a nilpotent element since otherwise $s$ would have to be nilpotent. This implies [[formally étale morphism|formal étaleness]] analogous to the discussion at \href{etale+morphism+of+schemes#OpenImmersionIsEtale}{\'e{}tale morphism of schemes -- Open immersion is Etale}. \end{proof} \begin{prop} \label{}\hypertarget{}{} If $X = Spec(A)$ is an affine [[reduced scheme]] of [[characteristic]] a [[prime number]] $p$, then its [[étale cohomology]] with [[coefficients]] in $\mathbb{Z}/p\mathbb{Z}$ is \begin{displaymath} H^q(X, (\mathbb{Z}/p\mathbb{Z})_X) \simeq \left\{ \itexarray{ A/(F - id)A & if\; q = 1 \\ 0 & if \; q \gt 0 } \right. \,. \end{displaymath} \end{prop} \begin{proof} Under the given assumptions, the [[Artin-Schreier sequence]] (see there) induces a [[long exact sequence in cohomology]] of the form \begin{displaymath} \begin{aligned} 0 & \to H^0(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^0(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^0(X_{et}, \mathcal{O}_X) \\ & \to H^1(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^1(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^1(X_{et}, \mathcal{O}_X) \\ & \to H^2(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^2(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^2(X_{et}, \mathcal{O}_X) \to \cdots \end{aligned} \,, \end{displaymath} where $F(-) = (-)^p$ is the [[Frobenius endomorphism]]. By prop. \ref{CohomologyWithCoeffsInCoherentModules} the terms of the form $H^{p \geq 1}(X, \mathcal{O}_X)$ vanish, and so from [[exact sequence|exactness]] we find an [[isomorphism]] \begin{displaymath} H^0(X_{et}, \mathcal{O}_X)/(F-id)(H^0(X_{et}, \mathcal{O}_X)) \stackrel{\simeq}{\to} H^1(X_{et}, \mathbb{Z}/p\mathbb{Z}) \,, \end{displaymath} hence the claimed isomorphism \begin{displaymath} A/(F-id)(A) \stackrel{\simeq}{\to} H^1(X_{et}, \mathbb{Z}/p\mathbb{Z}) \,. \end{displaymath} By the same argument all the higher cohomology groups vanish, as claimed. \end{proof} \hypertarget{WithCoefficientsInTheMultiplicativeGroup}{}\subsubsection*{{With coefficients in the multiplicative group}}\label{WithCoefficientsInTheMultiplicativeGroup} the \'e{}tale cohomology groups with [[coefficients]] in the [[multiplicative group]] $\mathbb{G}_m$ in the first few degrees go by special names: \begin{itemize}% \item $H^0_{et}(-, \mathbb{G}_m)$: [[group of units]]; \item $H^1_{et}(-, \mathbb{G}_m)$: [[Picard group]] ([[Hilbert's theorem 90]], \hyperlink{Tamme}{Tamme, II 4.3.1}); \item $H^2_{et}(-, \mathbb{G}_m)$: [[Brauer group]]; \end{itemize} \hypertarget{MainTheorems}{}\subsection*{{Outlook: The main theorems}}\label{MainTheorems} What makes [[étale cohomology]] interesting in a broader context is that is verifies a collection of good structural theorems, which we just list now. In their totality these properties make [[étale cohomology]] (in its incarnation as [[ℓ-adic cohomology]]) qualify as a [[Weil cohomology theory]]. This in turn means that using [[étale cohomology]] one can give a [[proof]] of the [[Weil conjectures]] -- a number of [[conjectures]] about properties of the numbers of points in [[algebraic varieties]], hence of the numbers of solutions to certain [[polynomial]] [[equations]] over certain [[rings]] -- , and this was historically a central motivation for introducing [[étale cohomology]] in the first place. These theorems are \begin{enumerate}% \item [[proper base change theorem]] (\hyperlink{Milne}{Milne, section 17}) \item [[comparison theorem (étale cohomology)]] (\hyperlink{Milne}{Milne, section 21}) \item [[Künneth formula]] (\hyperlink{Milne}{Milne, section 22}) \item cycle map theorem (\hyperlink{Milne}{Milne, section 23}) \item [[Poincaré duality]] (\hyperlink{Milne}{Milne, section 24}) \end{enumerate} Together these imply the central ingredient for a proof of the [[Weil conjectures]], a Lefschetz fixed-point formula \begin{itemize}% \item \hyperlink{KünnethFormula}{K\"u{}nneth formula} + \hyperlink{CycleMap}{cycle map} + \hyperlink{PoincareDuality}{Poincar\'e{} duality} $\Rightarrow$ [[Lefschetz fixed-point formula]] (\hyperlink{Milne}{Milne, section 25}) \end{itemize} For more on this see\ldots{} elsewhere. \hypertarget{References}{}\subsection*{{References}}\label{References} \begin{itemize}% \item [[Günter Tamme]], \emph{[[Introduction to Étale Cohomology]]} \end{itemize} \begin{itemize}% \item [[James Milne]], \emph{[[Lectures on Étale Cohomology]]} \end{itemize} \begin{itemize}% \item [[The Stacks Project]], \emph{\'E{}tale cohomology} (\href{http://stacks.math.columbia.edu/download/etale-cohomology.pdf}{pdf}) \end{itemize} [[!redirects basics of étale cohomology]] \end{document}