etale-cohomology-lemma-morphism-locally-ringed

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


etale-cohomology-lemma-2-morphism

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


etale-cohomology-lemma-faithful

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


etale-cohomology-lemma-morphism-ringed-etale-topoi-affines

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


etale-cohomology-lemma-property-A-implies

Let $f : X \to Y$ be a morphism of schemes. Assume (A). \begin{enumerate} \item $f_{small, *} : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)$ reflects injections and surjections, \item $f_{small}^{-1}f_{small, *}\mathcal{F} \to \mathcal{F}$ is surjective for any abelian sheaf $\mathcal{F}$ on $X_\etale$, \item $f_{small, *} : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)$ is faithful. \end{enumerate}


etale-cohomology-lemma-locally-quasi-finite-A

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


etale-cohomology-lemma-integral-A

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


etale-cohomology-lemma-when-push-pull-surjective

Let $f : X \to Y$ be a morphism of schemes. Denote $f_{small} : \Sh(X_\etale) \to \Sh(Y_\etale)$ the associated morphism of small \'etale topoi. Assume at least one of the following \begin{enumerate} \item $f$ is integral, or \item $f$ is separated and locally quasi-finite. \end{enumerate} Then the functor $f_{small, *} : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)$ has the following properties \begin{enumerate} \item the map $f_{small}^{-1}f_{small, *}\mathcal{F} \to \mathcal{F}$ is always surjective, \item $f_{small, *}$ is faithful, and \item $f_{small, *}$ reflects injections and surjections. \end{enumerate}


etale-cohomology-lemma-property-B-implies

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


etale-cohomology-lemma-simplify-B

Let $f : X \to Y$ be a morphism of schemes. Suppose \begin{enumerate} \item $V \to Y$ is an \'etale morphism of schemes, \item $\{U_i \to X \times_Y V\}$ is an \'etale covering, and \item $v \in V$ is a point. \end{enumerate} Assume that for any such data there exists an \'etale neighbourhood $(V', v') \to (V, v)$, a disjoint union decomposition $X \times_Y V' = \coprod W'_i$, and morphisms $W'_i \to U_i$ over $X \times_Y V$. Then property (B) holds.


etale-cohomology-lemma-finite-B

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


etale-cohomology-lemma-integral-B

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


etale-cohomology-lemma-what-integral

Let $f : X \to Y$ be a morphism of schemes. Assume $f$ is integral (for example finite). Then \begin{enumerate} \item $f_{small, *}$ transforms surjections into surjections (on sheaves of sets and on abelian sheaves), \item $f_{small}^{-1}f_{small, *}\mathcal{F} \to \mathcal{F}$ is surjective for any abelian sheaf $\mathcal{F}$ on $X_\etale$, \item $f_{small, *} : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)$ is faithful and reflects injections and surjections, and \item $f_{small, *} : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)$ is exact. \end{enumerate}


etale-cohomology-lemma-property-C-implies

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


etale-cohomology-lemma-property-C-closed-implies

Let $f : X \to Y$ be a morphism of schemes. Assume that for any $V \to Y$ \'etale we have that \begin{enumerate} \item $X \times_Y V \to V$ has property (C), and \item $X \times_Y V \to V$ is closed. \end{enumerate} Then the functor $Y_\etale \to X_\etale$, $V \mapsto X \times_Y V$ is almost cocontinuous, see Sites, Definition \ref{sites-definition-almost-cocontinuous}.


etale-cohomology-lemma-integral-homeo-onto-image-C

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


etale-cohomology-lemma-closed-immersion-almost-full

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


etale-cohomology-lemma-closed-immersion-almost-essentially-surjective

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


etale-cohomology-lemma-stalk-pushforward-closed-immersion

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


etale-cohomology-lemma-monomorphism-big-push-pull

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


etale-cohomology-lemma-closed-immersion-cover-from-below

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


etale-cohomology-lemma-prepare-closed-immersion-almost-cocontinuous

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


etale-cohomology-lemma-closed-immersion-almost-cocontinuous

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


etale-cohomology-lemma-closed-immersion-pushforward-exact

Let $f : X \to Y$ be a closed immersion of schemes. Let $\tau \in \{syntomic, smooth, \etale\}$. \begin{enumerate} \item The pushforward $f_{big, *} : \Sh((\Sch/X)_\tau) \to \Sh((\Sch/Y)_\tau)$ commutes with coequalizers and pushouts. \item The pushforward $f_{big, *} : \textit{Ab}((\Sch/X)_\tau) \to \textit{Ab}((\Sch/Y)_\tau)$ is exact. \end{enumerate}


etale-cohomology-lemma-compare-structure-sheaves

Let $X$ be a scheme. Let $\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}$. Let $\mathcal{C}_1 \subset \mathcal{C}_2 \subset (\Sch/X)_\tau$ be full subcategories with the following properties: \begin{enumerate} \item For an object $U/X$ of $\mathcal{C}_t$, \begin{enumerate} \item if $\{U_i \to U\}$ is a covering of $(\Sch/X)_\tau$, then $U_i/X$ is an object of $\mathcal{C}_t$, \item $U \times \mathbf{A}^1/X$ is an object of $\mathcal{C}_t$. \end{enumerate} \item $X/X$ is an object of $\mathcal{C}_t$. \end{enumerate} We endow $\mathcal{C}_t$ with the structure of a site whose coverings are exactly those coverings $\{U_i \to U\}$ of $(\Sch/X)_\tau$ with $U \in \Ob(\mathcal{C}_t)$. Then \begin{enumerate} \item[(a)] The functor $\mathcal{C}_1 \to \mathcal{C}_2$ is fully faithful, continuous, and cocontinuous. \end{enumerate} Denote $g : \Sh(\mathcal{C}_1) \to \Sh(\mathcal{C}_2)$ the corresponding morphism of topoi. Denote $\mathcal{O}_t$ the restriction of $\mathcal{O}$ to $\mathcal{C}_t$. Denote $g_!$ the functor of Modules on Sites, Definition \ref{sites-modules-definition-g-shriek}. \begin{enumerate} \item[(b)] The canonical map $g_!\mathcal{O}_1 \to \mathcal{O}_2$ is an isomorphism. \end{enumerate}


etale-cohomology-lemma-mayer-vietoris

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


etale-cohomology-lemma-relative-mayer-vietoris

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


etale-cohomology-lemma-colimit-affine-sites

Let $I$ be a directed set. Let $(X_i, f_{i'i})$ be an inverse system of schemes over $I$ with affine transition morphisms. Let $X = \lim_{i \in I} X_i$. With notation as in Lemma \ref{lemma-alternative} we have $$ X_{affine, \etale} = \colim (X_i)_{affine, \etale} $$ as sites in the sense of Sites, Lemma \ref{sites-lemma-colimit-sites}.


etale-cohomology-lemma-colimit

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


etale-cohomology-lemma-directed-colimit-cohomology

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


etale-cohomology-lemma-higher-direct-images

Let $f: X\to Y$ be a morphism of schemes and $\mathcal{F}\in \textit{Ab}(X_\etale)$. Then $R^pf_*\mathcal{F}$ is the sheaf associated to the presheaf $$ (V \to Y) \longmapsto H_\etale^p(X \times_Y V, \mathcal{F}|_{X \times_Y V}). $$


etale-cohomology-lemma-relative-colimit

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


etale-cohomology-lemma-relative-colimit-general

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


etale-cohomology-lemma-linus-hamann

Let $X = \lim_{i \in I} X_i$ be a directed limit of schemes with affine transition morphisms $f_{i'i}$ and projection morphisms $f_i : X \to X_i$. Let $\mathcal{F}$ be a sheaf on $X_\etale$. Then \begin{enumerate} \item there are canonical maps $\varphi_{i'i} : f_{i'i}^{-1}f_{i, *}\mathcal{F} \to f_{i', *}\mathcal{F}$ such that $(f_{i, *}\mathcal{F}, \varphi_{i'i})$ is a system of sheaves on $(X_i, f_{i'i})$ as in Definition \ref{definition-inverse-system-sheaves}, and \item $\mathcal{F} = \colim f_i^{-1}f_{i, *}\mathcal{F}$. \end{enumerate}


etale-cohomology-lemma-prepare-leray

Let $f: X \to Y$ be a morphism and $\mathcal{I}$ an injective object of $\textit{Ab}(X_\etale)$. Let $V \in \Ob(Y_\etale)$. Then \begin{enumerate} \item for any covering $\mathcal{V} = \{V_j\to V\}_{j \in J}$ we have $\check H^p(\mathcal{V}, f_*\mathcal{I}) = 0$ for all $p > 0$, \item $f_*\mathcal{I}$ is acyclic for the functor $\Gamma(V, -)$, and \item if $g : Y \to Z$, then $f_*\mathcal{I}$ is acyclic for $g_*$. \end{enumerate}


etale-cohomology-lemma-vanishing-etale-cohomology-strictly-henselian

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


etale-cohomology-lemma-finite-pushforward-commutes-with-base-change

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


etale-cohomology-lemma-integral-pushforward-commutes-with-base-change

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


etale-cohomology-lemma-cohomological-descent-finite

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


etale-cohomology-lemma-global-sections-point

Assumptions and notations as in Theorem \ref{theorem-equivalence-sheaves-point}. There is a functorial bijection $$ \Gamma(S, \mathcal{F}) = (\mathcal{F}_{\overline{s}})^G $$


etale-cohomology-lemma-ext-modules-hom

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


etale-cohomology-lemma-finite-dim-group-cohomology

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


etale-cohomology-lemma-profinite-group-cohomology-torsion

Let $G$ be a profinite topological group. Then \begin{enumerate} \item $H^i(G, M)$ is torsion for $i > 0$ and any $G$-module $M$, and \item $H^i(G, M) = 0$ if $M$ is a $\mathbf{Q}$-vector space. \end{enumerate}


etale-cohomology-lemma-equivalence-abelian-sheaves-point

Let $S = \Spec(K)$ with $K$ a field. Let $\overline{s}$ be a geometric point of $S$. Let $G = \text{Gal}_{\kappa(s)}$ denote the absolute Galois group. The stalk functor induces an equivalence of categories $$ \textit{Ab}(S_\etale) \longrightarrow \text{Mod}_G, \quad \mathcal{F} \longmapsto \mathcal{F}_{\overline{s}}. $$


etale-cohomology-lemma-compare-cohomology-point

Notation and assumptions as in Lemma \ref{lemma-equivalence-abelian-sheaves-point}. Let $\mathcal{F}$ be an abelian sheaf on $\Spec(K)_\etale$ which corresponds to the $G$-module $M$. Then \begin{enumerate} \item in $D(\textit{Ab})$ we have a canonical isomorphism $R\Gamma(S, \mathcal{F}) = R\Gamma_G(M)$, \item $H_\etale^0(S, \mathcal{F}) = M^G$, and \item $H_\etale^q(S, \mathcal{F}) = H^q(G, M)$. \end{enumerate}


etale-cohomology-lemma-end-unique-up-to-invertible

Let $S$ be a scheme. Let $\mathcal{F}$ and $\mathcal{G}$ be finite locally free sheaves of $\mathcal{O}_S$-modules of positive rank. If there exists an isomorphism $\SheafHom_{\mathcal{O}_S}(\mathcal{F}, \mathcal{F}) \cong \SheafHom_{\mathcal{O}_S}(\mathcal{G}, \mathcal{G})$ of $\mathcal{O}_S$-algebras, then there exists an invertible sheaf $\mathcal{L}$ on $S$ such that $\mathcal{F} \otimes_{\mathcal{O}_S} \mathcal{L} \cong \mathcal{G}$ and such that this isomorphism induces the given isomorphism of endomorphism algebras.


etale-cohomology-lemma-vanishing-affine-char-p-p

Let $p$ be a prime. Let $S$ be a scheme of characteristic $p$. \begin{enumerate} \item If $S$ is affine, then $H_\etale^q(S, \underline{\mathbf{Z}/p\mathbf{Z}}) = 0$ for all $q \geq 2$. \item If $S$ is a quasi-compact and quasi-separated scheme of dimension $d$, then $H_\etale^q(S, \underline{\mathbf{Z}/p\mathbf{Z}}) = 0$ for all $q \geq 2 + d$. \end{enumerate}


etale-cohomology-lemma-F-1

Let $k$ be an algebraically closed field of characteristic $p > 0$. Let $V$ be a finite dimensional $k$-vector space. Let $F : V \to V$ be a frobenius linear map, i.e., an additive map such that $F(\lambda v) = \lambda^p F(v)$ for all $\lambda \in k$ and $v \in V$. Then $F - 1 : V \to V$ is surjective with kernel a finite dimensional $\mathbf{F}_p$-vector space of dimension $\leq \dim_k(V)$.


etale-cohomology-lemma-top-cohomology-coherent

Let $X$ be a separated scheme of finite type over a field $k$. Let $\mathcal{F}$ be a coherent sheaf of $\mathcal{O}_X$-modules. Then $\dim_k H^d(X, \mathcal{F}) < \infty$ where $d = \dim(X)$.


etale-cohomology-lemma-vanishing-variety-char-p-p

Let $X$ be separated of finite type over an algebraically closed field $k$ of characteristic $p > 0$. Then $H_\etale^q(X, \underline{\mathbf{Z}/p\mathbf{Z}}) = 0$ for $q \geq dim(X) + 1$.


etale-cohomology-lemma-finiteness-proper-variety-char-p-p

Let $X$ be a proper scheme over an algebraically closed field $k$ of characteristic $p > 0$. Then \begin{enumerate} \item $H_\etale^q(X, \underline{\mathbf{Z}/p\mathbf{Z}})$ is a finite $\mathbf{Z}/p\mathbf{Z}$-module for all $q$, and \item $H^q_\etale(X, \underline{\mathbf{Z}/p\mathbf{Z}}) \to H^q_\etale(X_{k'}, \underline{\mathbf{Z}/p\mathbf{Z}}))$ is an isomorphism if $k \subset k'$ is an extension of algebraically closed fields. \end{enumerate}


etale-cohomology-lemma-pullback-locally-constant

Let $f : X \to Y$ be a morphism of schemes. If $\mathcal{G}$ is a locally constant sheaf of sets, abelian groups, or $\Lambda$-modules on $Y_\etale$, the same is true for $f^{-1}\mathcal{G}$ on $X_\etale$.


etale-cohomology-lemma-pushforward-locally-constant

Let $f : X \to Y$ be a finite \'etale morphism of schemes. If $\mathcal{F}$ is a (finite) locally constant sheaf of sets, (finite) locally constant sheaf of abelian groups, or (finite type) locally constant sheaf of $\Lambda$-modules on $X_\etale$, the same is true for $f_*\mathcal{F}$ on $Y_\etale$.


etale-cohomology-lemma-characterize-finite-locally-constant

Let $X$ be a scheme and $\mathcal{F}$ a sheaf of sets on $X_\etale$. Then the following are equivalent \begin{enumerate} \item $\mathcal{F}$ is finite locally constant, and \item $\mathcal{F} = h_U$ for some finite \'etale morphism $U \to X$. \end{enumerate}


etale-cohomology-lemma-morphism-locally-constant

Let $X$ be a scheme. \begin{enumerate} \item Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of locally constant sheaves of sets on $X_\etale$. If $\mathcal{F}$ is finite locally constant, there exists an \'etale covering $\{U_i \to X\}$ such that $\varphi|_{U_i}$ is the map of constant sheaves associated to a map of sets. \item Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of locally constant sheaves of abelian groups on $X_\etale$. If $\mathcal{F}$ is finite locally constant, there exists an \'etale covering $\{U_i \to X\}$ such that $\varphi|_{U_i}$ is the map of constant abelian sheaves associated to a map of abelian groups. \item Let $\Lambda$ be a ring. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of locally constant sheaves of $\Lambda$-modules on $X_\etale$. If $\mathcal{F}$ is of finite type, then there exists an \'etale covering $\{U_i \to X\}$ such that $\varphi|_{U_i}$ is the map of constant sheaves of $\Lambda$-modules associated to a map of $\Lambda$-modules. \end{enumerate}


etale-cohomology-lemma-kernel-finite-locally-constant

Let $X$ be a scheme. \begin{enumerate} \item The category of finite locally constant sheaves of sets is closed under finite limits and colimits inside $\Sh(X_\etale)$. \item The category of finite locally constant abelian sheaves is a weak Serre subcategory of $\textit{Ab}(X_\etale)$. \item Let $\Lambda$ be a Noetherian ring. The category of finite type, locally constant sheaves of $\Lambda$-modules on $X_\etale$ is a weak Serre subcategory of $\textit{Mod}(X_\etale, \Lambda)$. \end{enumerate}


etale-cohomology-lemma-tensor-product-locally-constant

Let $X$ be a scheme. Let $\Lambda$ be a ring. The tensor product of two locally constant sheaves of $\Lambda$-modules on $X_\etale$ is a locally constant sheaf of $\Lambda$-modules.


etale-cohomology-lemma-connected-locally-constant

Let $X$ be a connected scheme. Let $\Lambda$ be a ring and let $\mathcal{F}$ be a locally constant sheaf of $\Lambda$-modules. Then there exists a $\Lambda$-module $M$ and an \'etale covering $\{U_i \to X\}$ such that $\mathcal{F}|_{U_i} \cong \underline{M}|_{U_i}$.


etale-cohomology-lemma-locally-constant-on-connected

Let $X$ be a connected scheme. Let $\overline{x}$ be a geometric point of $X$. \begin{enumerate} \item There is an equivalence of categories $$ \left\{ \begin{matrix} \text{finite locally constant}\\ \text{sheaves of sets on }X_\etale \end{matrix} \right\} \longleftrightarrow \left\{ \begin{matrix} \text{finite }\pi_1(X, \overline{x})\text{-sets} \end{matrix} \right\} $$ \item There is an equivalence of categories $$ \left\{ \begin{matrix} \text{finite locally constant}\\ \text{sheaves of abelian groups on }X_\etale \end{matrix} \right\} \longleftrightarrow \left\{ \begin{matrix} \text{finite }\pi_1(X, \overline{x})\text{-modules} \end{matrix} \right\} $$ \item Let $\Lambda$ be a finite ring. There is an equivalence of categories $$ \left\{ \begin{matrix} \text{finite type, locally constant}\\ \text{sheaves of }\Lambda\text{-modules on }X_\etale \end{matrix} \right\} \longleftrightarrow \left\{ \begin{matrix} \text{finite }\pi_1(X, \overline{x})\text{-modules endowed}\\ \text{with commuting }\Lambda\text{-module structure} \end{matrix} \right\} $$ \end{enumerate}


etale-cohomology-lemma-pullback-filtered

Let $S$ be a connected scheme. Let $\ell$ be a prime number. Let $\mathcal{F}$ be a finite type, locally constant sheaf of $\mathbf{F}_\ell$-vector spaces on $S_\etale$. Then there exists a finite \'etale morphism $f : T \to S$ of degree prime to $\ell$ such that $f^{-1}\mathcal{F}$ has a finite filtration whose successive quotients are $\underline{\mathbf{Z}/\ell\mathbf{Z}}_T$.


etale-cohomology-lemma-nonvanishing-inherited

Let $\ell$ be a prime number and $n$ an integer $> 0$. Let $S$ be a quasi-compact and quasi-separated scheme. Let $X = \lim_{i \in I} X_i$ be the limit of a directed system of $S$-schemes each $X_i \to S$ being finite \'etale of constant degree relatively prime to $\ell$. The following are equivalent: \begin{enumerate} \item there exists an $\ell$-power torsion sheaf $\mathcal{G}$ on $S$ such that $H_\etale^n(S, \mathcal{G}) \neq 0$ and \item there exists an $\ell$-power torsion sheaf $\mathcal{F}$ on $X$ such that $H_\etale^n(X, \mathcal{F}) \neq 0$. \end{enumerate} In fact, given $\mathcal{G}$ we can take $\mathcal{F} = g^{-1}\mathcal{F}$ and given $\mathcal{F}$ we can take $\mathcal{G} = g_*\mathcal{F}$.


etale-cohomology-lemma-reduce-to-l-group

Let $\ell$ be a prime number and $n$ an integer $> 0$. Let $K$ be a field with $G = Gal(K^{sep}/K)$ and let $H \subset G$ be a maximal pro-$\ell$ subgroup with $L/K$ being the corresponding field extension. Then $H^n_\etale(\Spec(K), \mathcal{F}) = 0$ for all $\ell$-power torsion $\mathcal{F}$ if and only if $H^n_\etale(\Spec(L), \underline{\mathbf{Z}/\ell\mathbf{Z}}) = 0$.


etale-cohomology-lemma-reduce-to-l-group-higher

Let $\ell$ be a prime number and $n$ an integer $> 0$. Let $K$ be a field with $G = Gal(K^{sep}/K)$ and let $H \subset G$ be a maximal pro-$\ell$ subgroup  with $L/K$ being the corresponding field extension. Then $H^q_\etale(\Spec(K),\mathcal{F}) = 0$ for $q \geq n$ and all $\ell$-torsion sheaves $\mathcal{F}$ if  and only if $H^n_\etale(\Spec(L), \underline{\mathbf{Z}/\ell\mathbf{Z}}) = 0$.


etale-cohomology-lemma-algebraically-closed-find-solutions

Let $k$ be an algebraically closed field. Let $f_1, \ldots, f_s \in k[T_1, \ldots, T_n]$ be homogeneous polynomials of degree $d_1, \ldots, d_s$ with $d_i > 0$. If $s < n$, then $f_1 = \ldots = f_s = 0$ have a common nontrivial solution.


etale-cohomology-lemma-curve-brauer-zero

Let $C$ be a curve over an algebraically closed field $k$. Then the Brauer group of the function field of $C$ is zero: $\text{Br}(k(C)) = 0$.


etale-cohomology-lemma-cohomology-Gm-function-field-curve

Let $k$ be an algebraically closed field and $k \subset K$ a field extension of transcendence degree 1. Then for all $q \geq 1$, $H_\etale^q(\Spec(K), \mathbf{G}_m) = 0$.


etale-cohomology-lemma-cohomology-smooth-projective-curve

Let $X$ be a smooth projective curve of genus $g$ over an algebraically closed field $k$ and let $n \geq 1$ be invertible in $k$. Then there are canonical identifications $$ H_\etale^q(X, \mu_n) = \left\{ \begin{matrix} \mu_n(k) & \text{ if }q = 0, \\ \Pic^0(X)[n] & \text{ if }q = 1, \\ \mathbf{Z}/n\mathbf{Z} & \text{ if }q = 2, \\ 0 & \text{ if }q \geq 3. \end{matrix} \right. $$ Since $\mu_n \cong \underline{\mathbf{Z}/n\mathbf{Z}}$, this gives (noncanonical) identifications $$ H_\etale^q(X, \underline{\mathbf{Z}/n\mathbf{Z}}) \cong \left\{ \begin{matrix} \mathbf{Z}/n\mathbf{Z} & \text{ if }q = 0, \\ (\mathbf{Z}/n\mathbf{Z})^{2g} & \text{ if }q = 1, \\ \mathbf{Z}/n\mathbf{Z} & \text{ if }q = 2, \\ 0 & \text{ if }q \geq 3. \end{matrix} \right. $$


etale-cohomology-lemma-vanishing-cohomology-mu-smooth-curve

Let $X$ be an affine smooth curve over an algebraically closed field $k$ and $n\in k^*$. Then \begin{enumerate} \item $H_\etale^0(X, \mu_n) = \mu_n(k)$; \item $H_\etale^1(X, \mu_n) \cong \left(\mathbf{Z}/n\mathbf{Z}\right)^{2g+r-1}$, where $r$ is the number of points in $\bar X - X$ for some smooth projective compactification $\bar X$ of $X$, and \item for all $q\geq 2$, $H_\etale^q(X, \mu_n) = 0$. \end{enumerate}


etale-cohomology-lemma-jshriek-open

Let $j : U \to X$ be an open immersion of schemes. For any abelian sheaf $\mathcal{F}$ on $U_\etale$, the adjunction mappings $j^{-1}j_*\mathcal{F} \to \mathcal{F}$ and $\mathcal{F} \to j^{-1}j_!\mathcal{F}$ are isomorphisms. In fact, $j_!\mathcal{F}$ is the unique abelian sheaf on $X_\etale$ whose restriction to $U$ is $\mathcal{F}$ and whose stalks at geometric points of $X \setminus U$ are zero.


etale-cohomology-lemma-shriek-base-change

Let $f: Y \to X$ be a morphism of schemes. Let $j: V \to X$ be an \'etale morphism. Consider the fibre product $$ \xymatrix{ V' = Y \times_X V \ar[d]_{f'} \ar[r]_-{j'} & Y \ar[d]^f \\ V \ar[r]^j & X } $$ Then we have $j'_! f'^{-1} = f^{-1} j_!$ on abelian sheaves and on sheaves of modules.


etale-cohomology-lemma-shriek-into-star-separated-etale

Let $j : U \to X$ be separated and \'etale. Then there is a functorial injective map $j_!\mathcal{F} \to j_*\mathcal{F}$ on abelian sheaves and sheaves of $\Lambda$-modules.


etale-cohomology-lemma-shriek-equals-star-finite-etale

Let $j : U \to X$ be finite and \'etale. Then the map $j_! \to j_*$ of Lemma \ref{lemma-shriek-into-star-separated-etale} is an isomorphism on abelian sheaves and sheaves of $\Lambda$-modules.


etale-cohomology-lemma-ses-associated-to-open

Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme and let $U \subset X$ be the complement. Denote $i : Z \to X$ and $j : U \to X$ the inclusion morphisms. For every abelian sheaf $\mathcal{F}$ on $X_\etale$ there is a canonical short exact sequence $$ 0 \to j_!j^{-1}\mathcal{F} \to \mathcal{F} \to i_*i^{-1}\mathcal{F} \to 0 $$ on $X_\etale$.


etale-cohomology-lemma-constructible-quasi-compact-quasi-separated

Let $X$ be a quasi-compact and quasi-separated scheme. Let $\mathcal{F}$ be a sheaf of sets on $X_\etale$. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is constructible, \item there exists an open covering $X = \bigcup U_i$ such that $\mathcal{F}|_{U_i}$ is constructible, and \item there exists a partition $X = \bigcup X_i$ by constructible locally closed subschemes such that $\mathcal{F}|_{X_i}$ is finite locally constant. \end{enumerate} A similar statement holds for abelian sheaves and sheaves of $\Lambda$-modules if $\Lambda$ is Noetherian.


etale-cohomology-lemma-constructible-local

Let $X$ be a scheme. Checking constructibility of a sheaf of sets, abelian groups, $\Lambda$-modules (with $\Lambda$ Noetherian) can be done Zariski locally on $X$.


etale-cohomology-lemma-pullback-constructible

Let $f : X \to Y$ be a morphism of schemes. If $\mathcal{F}$ is a constructible sheaf of sets, abelian groups, or $\Lambda$-modules (with $\Lambda$ Noetherian) on $Y_\etale$, the same is true for $f^{-1}\mathcal{F}$ on $X_\etale$.


etale-cohomology-lemma-constructible-abelian

Let $X$ be a scheme. \begin{enumerate} \item The category of constructible sheaves of sets is closed under finite limits and colimits inside $\Sh(X_\etale)$. \item The category of constructible abelian sheaves is a weak Serre subcategory of $\textit{Ab}(X_\etale)$. \item Let $\Lambda$ be a Noetherian ring. The category of constructible sheaves of $\Lambda$-modules on $X_\etale$ is a weak Serre subcategory of $\textit{Mod}(X_\etale, \Lambda)$. \end{enumerate}


etale-cohomology-lemma-tensor-product-constructible

Let $X$ be a scheme. Let $\Lambda$ be a Noetherian ring. The tensor product of two constructible sheaves of $\Lambda$-modules on $X_\etale$ is a constructible sheaf of $\Lambda$-modules.


etale-cohomology-lemma-support-constructible

Let $X$ be a quasi-compact and quasi-separated scheme. \begin{enumerate} \item Let $\mathcal{F} \to \mathcal{G}$ be a map of constructible sheaves of sets on $X_\etale$. Then the set of points $x \in X$ where $\mathcal{F}_{\overline{x}} \to \mathcal{G}_{\overline{x}}$ is surjective, resp.\ injective, resp.\ is isomorphic to a given map of sets, is constructible in $X$. \item Let $\mathcal{F}$ be a constructible abelian sheaf on $X_\etale$. The support of $\mathcal{F}$ is constructible. \item Let $\Lambda$ be a Noetherian ring. Let $\mathcal{F}$ be a constructible sheaf of $\Lambda$-modules on $X_\etale$. The support of $\mathcal{F}$ is constructible. \end{enumerate}


etale-cohomology-lemma-colimit-constructible

Let $X$ be a quasi-compact and quasi-separated scheme. Let $\mathcal{F} = \colim_{i \in I} \mathcal{F}_i$ be a filtered colimit of sheaves of sets, abelian sheaves, or sheaves of modules. \begin{enumerate} \item If $\mathcal{F}$ and $\mathcal{F}_i$ are constructible sheaves of sets, then the ind-object $\mathcal{F}_i$ is essentially constant with value $\mathcal{F}$. \item If $\mathcal{F}$ and $\mathcal{F}_i$ are constructible sheaves of abelian groups, then the ind-object $\mathcal{F}_i$ is essentially constant with value $\mathcal{F}$. \item Let $\Lambda$ be a Noetherian ring. If $\mathcal{F}$ and $\mathcal{F}_i$ are constructible sheaves of $\Lambda$-modules, then the ind-object $\mathcal{F}_i$ is essentially constant with value $\mathcal{F}$. \end{enumerate}


etale-cohomology-lemma-etale-stratified-finite

Let $U \to X$ be an \'etale morphism of quasi-compact and quasi-separated schemes (for example an \'etale morphism of Noetherian schemes). Then there exists a partition $X = \coprod_i X_i$ by constructible locally closed subschemes such that $X_i \times_X U \to X_i$ is finite \'etale for all $i$.


etale-cohomology-lemma-generically-finite

Let $f: X \to Y$ be a morphism of schemes which is quasi-compact, quasi-separated, and locally of finite type. If $\eta$ is a generic point of an irreducible component of $Y$ such that $f^{-1}(\eta)$ is finite, then there exists an open $V \subset Y$ containing $\eta$ such that $f^{-1}(V) \to V$ is finite.


etale-cohomology-lemma-decompose-quasi-finite-morphism

Let $f : Y \to X$ be a quasi-finite and finitely presented morphism of affine schemes. \begin{enumerate} \item There exists a surjective morphism of affine schemes $X' \to X$ and a closed subscheme $Z' \subset Y' = X' \times_X Y$ such that \begin{enumerate} \item $Z' \subset Y'$ is a thickening, and \item $Z' \to X'$ is a finite \'etale morphism. \end{enumerate} \item There exists a finite partition $X = \coprod X_i$ by locally closed, constructible, affine strata, and surjective finite locally free morphisms $X'_i \to X_i$ such that the reduction of $Y'_i = X'_i \times_X Y \to X'_i$ is isomorphic to $\coprod_{j = 1}^{n_i} (X'_i)_{red} \to (X'_i)_{red}$ for some $n_i$. \end{enumerate}


etale-cohomology-lemma-jshriek-constructible

Let $j : U \to X$ be an \'etale morphism of quasi-compact and quasi-separated schemes. \begin{enumerate} \item The sheaf $h_U$ is a constructible sheaf of sets. \item The sheaf $j_!\underline{M}$ is a constructible abelian sheaf for a finite abelian group $M$. \item If $\Lambda$ is a Noetherian ring and $M$ is a finite $\Lambda$-module, then $j_!\underline{M}$ is a constructible sheaf of $\Lambda$-modules on $X_\etale$. \end{enumerate}


etale-cohomology-lemma-torsion-colimit-constructible

Let $X$ be a quasi-compact and quasi-separated scheme. \begin{enumerate} \item Let $\mathcal{F}$ be a sheaf of sets on $X_\etale$. Then $\mathcal{F}$ is a filtered colimit of constructible sheaves of sets. \item Let $\mathcal{F}$ be a torsion abelian sheaf on $X_\etale$. Then $\mathcal{F}$ is a filtered colimit of constructible abelian sheaves. \item Let $\Lambda$ be a Noetherian ring and $\mathcal{F}$ a sheaf of $\Lambda$-modules on $X_\etale$. Then $\mathcal{F}$ is a filtered colimit of constructible sheaves of $\Lambda$-modules. \end{enumerate}


etale-cohomology-lemma-check-constructible

Let $f : X \to Y$ be a surjective morphism of quasi-compact and quasi-separated schemes. \begin{enumerate} \item Let $\mathcal{F}$ be a sheaf of sets on $Y_\etale$. Then $\mathcal{F}$ is constructible if and only if $f^{-1}\mathcal{F}$ is constructible. \item Let $\mathcal{F}$ be an abelian sheaf on $Y_\etale$. Then $\mathcal{F}$ is constructible if and only if $f^{-1}\mathcal{F}$ is constructible. \item Let $\Lambda$ be a Noetherian ring. Let $\mathcal{F}$ be sheaf of $\Lambda$-modules on $Y_\etale$. Then $\mathcal{F}$ is constructible if and only if $f^{-1}\mathcal{F}$ is constructible. \end{enumerate}


etale-cohomology-lemma-pushforward-constructible

Let $f : X \to Y$ be a finite \'etale morphism of schemes. Let $\Lambda$ be a Noetherian ring. If $\mathcal{F}$ is a constructible sheaf of sets, constructible sheaf of abelian groups, or constructible sheaf of $\Lambda$-modules on $X_\etale$, the same is true for $f_*\mathcal{F}$ on $Y_\etale$.


etale-cohomology-lemma-category-constructible-sets

Let $X$ be a quasi-compact and quasi-separated scheme. The category of constructible sheaves of sets is the full subcategory of $\Sh(X_\etale)$ consisting of sheaves $\mathcal{F}$ which are coequalizers $$ \xymatrix{ \mathcal{F}_1 \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}_0 \ar[r] & \mathcal{F}} $$ such that $\mathcal{F}_i$, $i = 0, 1$ is a finite coproduct of sheaves of the form $h_U$ with $U$ a quasi-compact and quasi-separated object of $X_\etale$.


etale-cohomology-lemma-category-constructible-modules

Let $X$ be a quasi-compact and quasi-separated scheme. Let $\Lambda$ be a Noetherian ring. The category of constructible sheaves of $\Lambda$-modules is exactly the category of modules of the form $$ \Coker\left( \bigoplus\nolimits_{j = 1, \ldots, m} j_{V_j!}\underline{\Lambda}_{V_j} \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} j_{U_i!}\underline{\Lambda}_{U_i} \right) $$ with $V_j$ and $U_i$ quasi-compact and quasi-separated objects of $X_\etale$. In fact, we can even assume $U_i$ and $V_j$ affine.


etale-cohomology-lemma-category-constructible-abelian

Let $X$ be a quasi-compact and quasi-separated scheme. The category of constructible abelian sheaves is exactly the category of abelian sheaves of the form $$ \Coker\left( \bigoplus\nolimits_{j = 1, \ldots, m} j_{V_j!}\underline{\mathbf{Z}/m_j\mathbf{Z}}_{V_j} \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} j_{U_i!}\underline{\mathbf{Z}/n_i\mathbf{Z}}_{U_i} \right) $$ with $V_j$ and $U_i$ quasi-compact and quasi-separated objects of $X_\etale$ and $m_j$, $n_i$ positive integers. In fact, we can even assume $U_i$ and $V_j$ affine.


etale-cohomology-lemma-constructible-is-compact

Let $X$ be a quasi-compact and quasi-separated scheme. Let $\Lambda$ be a Noetherian ring. Let $\mathcal{F}$ be a constructible sheaf of sets, abelian groups, or $\Lambda$-modules on $X_\etale$. Let $\mathcal{G} = \colim \mathcal{G}_i$ be a filtered colimit of sheaves of sets, abelian groups, or $\Lambda$-modules. Then $$ \Mor(\mathcal{F}, \mathcal{G}) = \colim \Mor(\mathcal{F}, \mathcal{G}_i) $$ in the category of sheaves of sets, abelian groups, or $\Lambda$-modules on $X_\etale$.


etale-cohomology-lemma-finite-pushforward-constructible

Let $f : X \to Y$ be a finite and finitely presented morphism of schemes. Let $\Lambda$ be a Noetherian ring. If $\mathcal{F}$ is a constructible sheaf of sets, abelian groups, or $\Lambda$-modules on $X_\etale$, then $f_*\mathcal{F}$ is too.


etale-cohomology-lemma-category-is-colimit

Let $X = \lim_{i \in I} X_i$ be a limit of a directed system of schemes with affine transition morphisms. We assume that $X_i$ is quasi-compact and quasi-separated for all $i \in I$. \begin{enumerate} \item The category of constructible sheaves of sets on $X_\etale$ is the colimit of the categories of constructible sheaves of sets on $(X_i)_\etale$. \item The category of constructible abelian sheaves on $X_\etale$ is the colimit of the categories of constructible abelian sheaves on $(X_i)_\etale$. \item Let $\Lambda$ be a Noetherian ring. The category of constructible sheaves of $\Lambda$-modules on $X_\etale$ is the colimit of the categories of constructible sheaves of $\Lambda$-modules on $(X_i)_\etale$. \end{enumerate}


etale-cohomology-lemma-irreducible-subsheaf-constant-zero

Let $X$ be an irreducible scheme with generic point $\eta$. \begin{enumerate} \item Let $S' \subset S$ be an inclusion of sets. If we have $\underline{S'} \subset \mathcal{G} \subset \underline{S}$ in $\Sh(X_\etale)$ and $S' = \mathcal{G}_{\overline{\eta}}$, then $\mathcal{G} = \underline{S'}$. \item Let $A' \subset A$ be an inclusion of abelian groups. If we have $\underline{A'} \subset \mathcal{G} \subset \underline{A}$ in $\textit{Ab}(X_\etale)$ and $A' = \mathcal{G}_{\overline{\eta}}$, then $\mathcal{G} = \underline{A'}$. \item Let $M' \subset M$ be an inclusion of modules over a ring $\Lambda$. If we have $\underline{M'} \subset \mathcal{G} \subset \underline{M}$ in $\textit{Mod}(X_\etale, \underline{\Lambda})$ and $M' = \mathcal{G}_{\overline{\eta}}$, then $\mathcal{G} = \underline{M'}$. \end{enumerate}


etale-cohomology-lemma-push-constant-sheaf-from-open

Let $X$ be an integral normal scheme with function field $K$. Let $E$ be a set. \begin{enumerate} \item Let $g : \Spec(K) \to X$ be the inclusion of the generic point. Then $g_*\underline{E} = \underline{E}$. \item Let $j : U \to X$ be the inclusion of a nonempty open. Then $j_*\underline{E} = \underline{E}$. \end{enumerate}


etale-cohomology-lemma-constructible-over-noetherian-noetherian

Let $X$ be a Noetherian scheme. Let $\Lambda$ be a Noetherian ring. Consider inclusions $$ \mathcal{F}_1 \subset \mathcal{F}_2 \subset \mathcal{F}_3 \subset \ldots \subset \mathcal{F} $$ in the category of sheaves of sets, abelian groups, or $\Lambda$-modules. If $\mathcal{F}$ is constructible, then for some $n$ we have $\mathcal{F}_n = \mathcal{F}_{n + 1} = \mathcal{F}_{n + 2} = \ldots$.


etale-cohomology-lemma-constructible-maps-into-constant

Let $X$ be a Noetherian scheme. \begin{enumerate} \item Let $\mathcal{F}$ be a constructible sheaf of sets on $X_\etale$. There exist an injective map of sheaves $$ \mathcal{F} \longrightarrow \prod\nolimits_{i = 1, \ldots, n} f_{i, *}\underline{E_i} $$ where $f_i : Y_i \to X$ is a finite morphism and $E_i$ is a finite set. \item Let $\mathcal{F}$ be a constructible abelian sheaf on $X_\etale$. There exist an injective map of abelian sheaves $$ \mathcal{F} \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} f_{i, *}\underline{M_i} $$ where $f_i : Y_i \to X$ is a finite morphism and $M_i$ is a finite abelian group. \item Let $\Lambda$ be a Noetherian ring. Let $\mathcal{F}$ be a constructible sheaf of $\Lambda$-modules on $X_\etale$. There exist an injective map of sheaves of modules $$ \mathcal{F} \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} f_{i, *}\underline{M_i} $$ where $f_i : Y_i \to X$ is a finite morphism and $M_i$ is a finite $\Lambda$-module. \end{enumerate} Moreover, we may assume each $Y_i$ is irreducible, reduced, maps onto an irreducible and reduced closed subscheme $Z_i \subset X$ such that $Y_i \to Z_i$ is finite \'etale over a nonempty open of $Z_i$.


etale-cohomology-lemma-constructible-maps-into-constant-general

\begin{reference} \cite[Exposee IX, Proposition 2.14]{SGA4} \end{reference} Let $X$ be a quasi-compact and quasi-separated scheme. \begin{enumerate} \item Let $\mathcal{F}$ be a constructible sheaf of sets on $X_\etale$. There exist an injective map of sheaves $$ \mathcal{F} \longrightarrow \prod\nolimits_{i = 1, \ldots, n} f_{i, *}\underline{E_i} $$ where $f_i : Y_i \to X$ is a finite and finitely presented morphism and $E_i$ is a finite set. \item Let $\mathcal{F}$ be a constructible abelian sheaf on $X_\etale$. There exist an injective map of abelian sheaves $$ \mathcal{F} \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} f_{i, *}\underline{M_i} $$ where $f_i : Y_i \to X$ is a finite and finitely presented morphism and $M_i$ is a finite abelian group. \item Let $\Lambda$ be a Noetherian ring. Let $\mathcal{F}$ be a constructible sheaf of $\Lambda$-modules on $X_\etale$. There exist an injective map of sheaves of modules $$ \mathcal{F} \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} f_{i, *}\underline{M_i} $$ where $f_i : Y_i \to X$ is a finite and finitely presented morphism and $M_i$ is a finite $\Lambda$-module. \end{enumerate}


etale-cohomology-lemma-support-in-subset

Let $X$ be a Noetherian scheme. Let $E \subset X$ be a subset closed under specialization. \begin{enumerate} \item Let $\mathcal{F}$ be a torsion abelian sheaf on $X_\etale$ whose support is contained in $E$. Then $\mathcal{F} = \colim \mathcal{F}_i$ is a filtered colimit of constructible abelian sheaves $\mathcal{F}_i$ such that for each $i$ the support of $\mathcal{F}_i$ is contained in a closed subset contained in $E$. \item Let $\Lambda$ be a Noetherian ring and $\mathcal{F}$ a sheaf of $\Lambda$-modules on $X_\etale$ whose support is contained in $E$. Then $\mathcal{F} = \colim \mathcal{F}_i$ is a filtered colimit of constructible sheaves of $\Lambda$-modules $\mathcal{F}_i$ such that for each $i$ the support of $\mathcal{F}_i$ is contained in a closed subset contained in $E$. \end{enumerate}


etale-cohomology-lemma-one-constructible

Let $X$ be a quasi-compact and quasi-separated scheme. Let $\Lambda$ be a Noetherian ring. Let $K \in D(X_\etale, \Lambda)$ and $b \in \mathbf{Z}$ such that $H^b(K)$ is constructible. Then there exist a sheaf $\mathcal{F}$ which is a finite direct sum of $j_{U!}\underline{\Lambda}$ with $U \in \Ob(X_\etale)$ affine and a map $\mathcal{F}[-b] \to K$ in $D(X_\etale, \Lambda)$ inducing a surjection $\mathcal{F} \to H^b(K)$.


etale-cohomology-lemma-bounded-above-c

Let $X$ be a quasi-compact and quasi-separated scheme. Let $\Lambda$ be a Noetherian ring. Let $K \in D^-(X_\etale, \Lambda)$. Then the following are equivalent \begin{enumerate} \item $K$ is in $D_c(X_\etale, \Lambda)$, \item $K$ can be represented by a bounded above complex whose terms are finite direct sums of $j_{U!}\underline{\Lambda}$ with $U \in \Ob(X_\etale)$ affine, \item $K$ can be represented by a bounded above complex of flat constructible sheaves of $\Lambda$-modules. \end{enumerate}


etale-cohomology-lemma-tensor-c

Let $X$ be a scheme. Let $\Lambda$ be a Noetherian ring. Let $K, L \in D_c^-(X_\etale, \Lambda)$. Then $K \otimes_\Lambda^\mathbf{L} L$ is in $D_c^-(X_\etale, \Lambda)$.


etale-cohomology-lemma-when-ctf

Let $\Lambda$ be a Noetherian ring. Let $X$ be a quasi-compact and quasi-separated scheme. Let $K \in D(X_\etale, \Lambda)$. The following are equivalent \begin{enumerate} \item $K \in D_{ctf}(X_\etale, \Lambda)$, and \item $K$ can be represented by a finite complex of constructible flat sheaves of $\Lambda$-modules. \end{enumerate} In fact, if $K$ has tor amplitude in $[a, b]$ then we can represent $K$ by a complex $\mathcal{F}^a \to \ldots \to \mathcal{F}^b$ with $\mathcal{F}^p$ a constructible flat sheaf of $\Lambda$-modules.


etale-cohomology-lemma-connected-ctf-locally-constant

Let $X$ be a connected scheme. Let $\Lambda$ be a Noetherian ring. Let $K \in D_{ctf}(X_\etale, \Lambda)$ have locally constant cohomology sheaves. Then there exists a finite complex of finite projective $\Lambda$-modules $M^\bullet$ and an \'etale covering $\{U_i \to X\}$ such that $K|_{U_i} \cong \underline{M^\bullet}|_{U_i}$ in $D(U_{i, \etale}, \Lambda)$.


etale-cohomology-lemma-torsion-cohomology

Let $X$ be a quasi-compact and quasi-separated scheme. \begin{enumerate} \item If $\mathcal{F}$ is a torsion abelian sheaf on $X_\etale$, then $H^n_\etale(X, \mathcal{F})$ is a torsion abelian group for all $n$. \item If $K$ in $D^+(X_\etale)$ has torsion cohomology sheaves, then $H^n_\etale(X, K)$ is a torsion abelian group for all $n$. \end{enumerate}


etale-cohomology-lemma-torsion-direct-image

Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of schemes. \begin{enumerate} \item If $\mathcal{F}$ is a torsion abelian sheaf on $X_\etale$, then $R^nf_*\mathcal{F}$ is a torsion abelian sheaf on $Y_\etale$ for all $n$. \item If $K$ in $D^+(X_\etale)$ has torsion cohomology sheaves, then $Rf_*K$ is an object of $D^+(Y_\etale)$ whose cohomology sheaves are torsion abelian sheaves. \end{enumerate}


etale-cohomology-lemma-sections-with-support-acyclic

Let $i : Z \to X$ be a closed immersion of schemes. Let $\mathcal{I}$ be an injective abelian sheaf on $X_\etale$. Then $\mathcal{H}_Z(\mathcal{I})$ is an injective abelian sheaf on $Z_\etale$.


etale-cohomology-lemma-cohomology-with-support-triangle

Let $i : Z \to X$ be a closed immersion of schemes. Let $j : U \to X$ be the inclusion of the complement of $Z$. Let $\mathcal{F}$ be an abelian sheaf on $X_\etale$. There is a distinguished triangle $$ i_*R\mathcal{H}_Z(\mathcal{F}) \to \mathcal{F} \to Rj_*(\mathcal{F}|_U) \to i_*R\mathcal{H}_Z(\mathcal{F})[1] $$ in $D(X_\etale)$. This produces an exact sequence $$ 0 \to i_*\mathcal{H}_Z(\mathcal{F}) \to \mathcal{F} \to j_*(\mathcal{F}|_U) \to i_*\mathcal{H}^1_Z(\mathcal{F}) \to 0 $$ and isomorphisms $R^pj_*(\mathcal{F}|_U) \cong i_*\mathcal{H}^{p + 1}_Z(\mathcal{F})$ for $p \geq 1$.


etale-cohomology-lemma-cohomology-with-support-quasi-coherent

Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module and denote $\mathcal{F}^a$ the associated quasi-coherent sheaf on the small \'etale site of $X$ (Proposition \ref{proposition-quasi-coherent-sheaf-fpqc}). Then \begin{enumerate} \item $H^q_Z(X, \mathcal{F})$ agrees with $H^q_Z(X_\etale, \mathcal{F}^a)$, \item if the complement of $Z$ is retrocompact in $X$, then $i_*\mathcal{H}^q_Z(\mathcal{F}^a)$ is a quasi-coherent sheaf of $\mathcal{O}_X$-modules equal to $(i_*\mathcal{H}^q_Z(\mathcal{F}))^a$. \end{enumerate}


etale-cohomology-lemma-local-rings-strictly-henselian

Let $S$ be a scheme all of whose local rings are strictly henselian. Then for any abelian sheaf $\mathcal{F}$ on $S_\etale$ we have $H^i(S_\etale, \mathcal{F}) = H^i(S_{Zar}, \mathcal{F})$.


etale-cohomology-lemma-normal-scheme-with-alg-closed-function-field

Let $X$ be an integral normal scheme with separably closed function field. \begin{enumerate} \item A separated \'etale morphism $U \to X$ is a disjoint union of open immersions. \item All local rings of $X$ are strictly henselian. \end{enumerate}


etale-cohomology-lemma-Rf-star-zero-normal-with-alg-closed-function-field

Let $f : X \to Y$ be a morphism of schemes where $X$ is an integral normal scheme with separably closed function field. Then $R^qf_*\underline{M} = 0$ for $q > 0$ and any abelian group $M$.


etale-cohomology-lemma-closed-of-affine-normal-scheme-with-alg-closed-function-field

Let $X$ be an affine integral normal scheme with separably closed function field. Let $Z \subset X$ be a closed subscheme. Let $V \to Z$ be an \'etale morphism with $V$ affine. Then $V$ is a finite disjoint union of open subschemes of $Z$. If $V \to Z$ is surjective and finite \'etale, then $V \to Z$ has a section.


etale-cohomology-lemma-gabber-for-h1-absolutely-algebraically-closed

Let $X$ be a normal integral affine scheme with separably closed function field. Let $Z \subset X$ be a closed subscheme. For any finite abelian group $M$ we have $H^1_\etale(Z, \underline{M}) = 0$.


etale-cohomology-lemma-gabber-for-absolutely-algebraically-closed

Let $X$ be a normal integral affine scheme with separably closed function field. Let $Z \subset X$ be a closed subscheme. For any finite abelian group $M$ we have $H^q_\etale(Z, \underline{M}) = 0$ for $q \geq 1$.


etale-cohomology-lemma-integral-cover-trivial-cohomology

Let $X$ be an affine scheme. \begin{enumerate} \item There exists an integral surjective morphism $X' \to X$ such that for every closed subscheme $Z' \subset X'$, every finite abelian group $M$, and every $q \geq 1$ we have $H^q_\etale(Z', \underline{M}) = 0$. \item For any closed subscheme $Z \subset X$, finite abelian group $M$, $q \geq 1$, and $\xi \in H^q_\etale(Z, \underline{M})$ there exists a finite surjective morphism $X' \to X$ of finite presentation such that $\xi$ pulls back to zero in $H^q_\etale(X' \times_X Z, \underline{M})$. \end{enumerate}


etale-cohomology-lemma-efface-cohomology-on-closed-by-finite-cover

Let $X$ be an affine scheme. Let $\mathcal{F}$ be a torsion abelian sheaf on  $X_\etale$. Let $Z \subset X$ be a closed subscheme. Let $\xi \in H^q_\etale(Z, \mathcal{F}|_Z)$ for some $q > 0$. Then there exists an injective map $\mathcal{F} \to \mathcal{F}'$ of torsion abelian sheaves on $X_\etale$ such that the image of $\xi$ in $H^q_\etale(Z, \mathcal{F}'|_Z)$ is zero.


etale-cohomology-lemma-gabber-h0

Let $X$ be a quasi-compact and quasi-separated scheme. Let $i : Z \to X$ be a closed immersion. Assume that \begin{enumerate} \item for any sheaf $\mathcal{F}$ on $X_{Zar}$ the map $\Gamma(X, \mathcal{F}) \to \Gamma(Z, i^{-1}\mathcal{F})$ is bijective, and \item for any finite morphism $X' \to X$ assumption (1) holds for $Z \times_X X' \to X'$. \end{enumerate} Then for any sheaf $\mathcal{F}$ on $X_\etale$ we have $\Gamma(X, \mathcal{F}) = \Gamma(Z, i^{-1}_{small}\mathcal{F})$.


etale-cohomology-lemma-connected-topological

Let $Z \subset X$ be a closed subset of a topological space $X$. Assume \begin{enumerate} \item $X$ is a spectral space (Topology, Definition \ref{topology-definition-spectral-space}), and \item for $x \in X$ the intersection $Z \cap \overline{\{x\}}$ is connected (in particular nonempty). \end{enumerate} If $Z = Z_1 \amalg Z_2$ with $Z_i$ closed in $Z$, then there exists a decomposition $X = X_1 \amalg X_2$ with $X_i$ closed in $X$ and $Z_i = Z \cap X_i$.


etale-cohomology-lemma-h0-topological

Let $Z \subset X$ be a closed subset of a topological space $X$. Assume \begin{enumerate} \item $X$ is a spectral space (Topology, Definition \ref{topology-definition-spectral-space}), and \item for $x \in X$ the intersection $Z \cap \overline{\{x\}}$ is connected (in particular nonempty). \end{enumerate} Then for any sheaf $\mathcal{F}$ on $X$ we have $\Gamma(X, \mathcal{F}) = \Gamma(Z, \mathcal{F}|_Z)$.


etale-cohomology-lemma-h0-henselian-pair

Let $(A, I)$ be a henselian pair. Set $X = \Spec(A)$ and $Z = \Spec(A/I)$. For any sheaf $\mathcal{F}$ on $X_\etale$ we have $\Gamma(X, \mathcal{F}) = \Gamma(Z, \mathcal{F}|_Z)$.


etale-cohomology-lemma-vanishing-restriction-injective

Let $X$ be a scheme with affine diagonal which can be covered by $n + 1$ affine opens. Let $Z \subset X$ be a closed subscheme. Let $\mathcal{A}$ be a torsion sheaf of rings on $X_\etale$ and let $\mathcal{I}$ be an injective sheaf of $\mathcal{A}$-modules on $X_\etale$. Then $H^q_\etale(Z, \mathcal{I}|_Z) = 0$ for $q > n$.


etale-cohomology-lemma-constant-smooth-statements

In Situation \ref{situation-what-to-prove} assume $X$ is smooth and $\mathcal{F} = \underline{\mathbf{Z}/\ell\mathbf{Z}}$ for some prime number $\ell$. Then statements (\ref{item-vanishing}) -- (\ref{item-surjective}) hold for $\mathcal{F}$.


etale-cohomology-lemma-ses-statements

Let $k$ be an algebraically closed field. Let $X$ be a separated finite type scheme over $k$ of dimension $\leq 1$. Let $0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0$ be a short exact sequence of torsion abelian sheaves on $X$. If statements (\ref{item-vanishing}) -- (\ref{item-surjective}) hold for $\mathcal{F}_1$ and $\mathcal{F}_2$, then they hold for $\mathcal{F}$.


etale-cohomology-lemma-finite-pushforward-statements

Let $k$ be an algebraically closed field. Let $f : X \to Y$ be a finite morphism of separated finite type schemes over $k$ of dimension $\leq 1$. Let $\mathcal{F}$ be a torsion abelian sheaf on $X$. If statements (\ref{item-vanishing}) -- (\ref{item-surjective}) hold for $\mathcal{F}$, then they hold for $f_*\mathcal{F}$.


etale-cohomology-lemma-even-easier

In Situation \ref{situation-what-to-prove} assume $X$ is smooth. Let $j : U \to X$ an open immersion. Let $\ell$ be a prime number. Let $\mathcal{F} = j_!\underline{\mathbf{Z}/\ell\mathbf{Z}}$. Then statements (\ref{item-vanishing}) -- (\ref{item-surjective}) hold for $\mathcal{F}$.


etale-cohomology-lemma-somewhat-easier

In Situation \ref{situation-what-to-prove} assume $X$ reduced. Let $j : U \to X$ an open immersion. Let $\ell$ be a prime number and $\mathcal{F} = j_! \underline{\mathbf{Z}/\ell\mathbf{Z}}$. Then statements (\ref{item-vanishing}) -- (\ref{item-surjective}) hold for $\mathcal{F}$.


etale-cohomology-lemma-vanishing-easier

In Situation \ref{situation-what-to-prove} assume $X$ reduced. Let $j : U \to X$ an open immersion with $U$ connected. Let $\ell$ be a prime number. Let $\mathcal{G}$ a finite locally constant sheaf of $\mathbf{F}_\ell$-vector spaces on $U$. Let $\mathcal{F} = j_!\mathcal{G}$. Then statements (\ref{item-vanishing}) -- (\ref{item-surjective}) hold for $\mathcal{F}$.


etale-cohomology-lemma-base-change-dim-1-separably-closed

Let $k \subset k'$ be an extension of separably closed fields. Let $X$ be a proper scheme over $k$ of dimension $\leq 1$. Let $\mathcal{F}$ be a torsion abelian sheaf on $X$. Then the map $H^q_\etale(X, \mathcal{F}) \to H^q_\etale(X_{k'}, \mathcal{F}|_{X_{k'}})$ is an isomorphism for $q \geq 0$.


etale-cohomology-lemma-proper-over-henselian-and-h1

Let $A$ be a henselian local ring. Let $X$ be a proper scheme over $A$ with closed fibre $X_0$. Let $M$ be a finite abelian group. Then $H^1_\etale(X, \underline{M}) = H^1_\etale(X_0, \underline{M})$.


etale-cohomology-lemma-efface-cohomology-on-fibre-by-finite-cover

Let $A$ be a henselian local ring. Let $X = \mathbf{P}^1_A$. Let $X_0 \subset X$ be the closed fibre. Let $\ell$ be a prime number. Let $\mathcal{I}$ be an injective sheaf of $\mathbf{Z}/\ell\mathbf{Z}$-modules on $X_\etale$. Then $H^q_\etale(X_0, \mathcal{I}|_{X_0}) = 0$ for $q > 0$.


etale-cohomology-lemma-base-change-Rf-star-colim

Let $I$ be a directed set. Consider an inverse system of cartesian diagrams of schemes $$ \xymatrix{ X_i \ar[d]_{f_i} & Y_i \ar[l]^{h_i} \ar[d]^{e_i} \\ S_i & T_i \ar[l]_{g_i} } $$ with affine transition morphisms and with $g_i$ quasi-compact and quasi-separated. Set $X = \lim X_i$, $S = \lim S_i$, $T = \lim T_i$ and $Y = \lim Y_i$ to obtain the cartesian diagram $$ \xymatrix{ X \ar[d]_f & Y \ar[l]^h \ar[d]^e \\ S & T \ar[l]_g } $$ Let $(\mathcal{F}_i, \varphi_{i'i})$ be a system of sheaves on $(T_i)$ as in Definition \ref{definition-inverse-system-sheaves}. Set $\mathcal{F} = \colim p_i^{-1}\mathcal{F}_i$ on $T$ where $p_i : T \to T_i$ is the projection. Then we have the following \begin{enumerate} \item If $f_i^{-1}g_{i, *}\mathcal{F}_i = h_{i, *}e_i^{-1}\mathcal{F}_i$ for all $i$, then $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$. \item If $\mathcal{F}_i$ is an abelian sheaf for all $i$ and $f_i^{-1}R^qg_{i, *}\mathcal{F}_i = R^qh_{i, *}e_i^{-1}\mathcal{F}_i$ for all $i$, then $f^{-1}R^qg_*\mathcal{F} = R^qh_*e^{-1}\mathcal{F}$. \end{enumerate}


etale-cohomology-lemma-base-change-f-star-general

Consider the cartesian diagram of schemes $$ \xymatrix{ X \ar[d]_f & Y \ar[l]^h \ar[d]^e \\ S & T \ar[l]_g } $$ Assume that $f$ is flat and every object $U$ of $X_\etale$ has a covering $\{U_i \to U\}$ such that $U_i \to S$ factors as $U_i \to V_i \to S$ with $V_i \to S$ \'etale and $U_i \to V_i$ quasi-compact with geometrically connected fibres. Then for any sheaf $\mathcal{F}$ of sets on $T_\etale$ we have $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$.


etale-cohomology-lemma-fppf-reduced-fibres-base-change-f-star

Consider a cartesian diagram of schemes $$ \xymatrix{ X \ar[d]_f & Y \ar[l]^h \ar[d]^e \\ S & T \ar[l]_g } $$ where $f$ is flat and locally of finite presentation with geometrically reduced fibres. Then $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$ for any sheaf $\mathcal{F}$ on $T_\etale$.


etale-cohomology-lemma-base-change-q-injective

With $f : X \to S$ and $n$ as in Remark \ref{remark-base-change-holds} assume for some $q \geq 1$ we have $BC(f, n, q - 1)$. Then for every commutative diagram $$ \xymatrix{ X \ar[d]_f & X' \ar[l] \ar[d]_{f'} & Y \ar[l]^h \ar[d]^e \\ S & S' \ar[l] & T \ar[l]_g } $$ with $X' = X \times_S S'$ and $Y = X' \times_{S'} T$ and $g$ quasi-compact and quasi-separated, and every abelian sheaf $\mathcal{F}$ on $T_\etale$ annihilated by $n$ \begin{enumerate} \item the base change map $(f')^{-1}R^qg_*\mathcal{F}\to R^qh_*e^{-1}\mathcal{F}$ is injective, \item if $\mathcal{F} \subset \mathcal{G}$ where $\mathcal{G}$ on $T_\etale$ is annihilated by $n$, then $$ \Coker\left( (f')^{-1}R^qg_*\mathcal{F}\to R^qh_*e^{-1}\mathcal{F} \right) \subset \Coker\left( (f')^{-1}R^qg_*\mathcal{G}\to R^qh_*e^{-1}\mathcal{G} \right) $$ \item if in (2) the sheaf $\mathcal{G}$ is an injective sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules, then $$ \Coker\left((f')^{-1}R^qg_*\mathcal{F}\to R^qh_*e^{-1}\mathcal{F} \right) \subset R^qh_*e^{-1}\mathcal{G} $$ \end{enumerate}


etale-cohomology-lemma-base-change-q-integral-top

With $f : X \to S$ and $n$ as in Remark \ref{remark-base-change-holds} assume for some $q \geq 1$ we have $BC(f, n, q - 1)$. Consider commutative diagrams $$ \vcenter{ \xymatrix{ X \ar[d]_f & X' \ar[d]_{f'} \ar[l] & Y \ar[l]^h \ar[d]^e & Y' \ar[l]^{\pi'} \ar[d]^{e'} \\ S & S' \ar[l] & T \ar[l]_g & T' \ar[l]_\pi } } \quad\text{and}\quad \vcenter{ \xymatrix{ X' \ar[d]_{f'} & & Y' \ar[ll]^{h' = h \circ \pi'} \ar[d]^{e'} \\ S' & & T' \ar[ll]_{g' = g \circ \pi} } } $$ where all squares are cartesian, $g$ quasi-compact and quasi-separated, and $\pi$ is integral surjective. Let $\mathcal{F}$ be an abelian sheaf on $T_\etale$ annihilated by $n$ and set $\mathcal{F}' = \pi^{-1}\mathcal{F}$. If the base change map $$ (f')^{-1}R^qg'_*\mathcal{F}' \longrightarrow R^qh'_*(e')^{-1}\mathcal{F}' $$ is an isomorphism, then the base change map $(f')^{-1}R^qg_*\mathcal{F} \to R^qh_*e^{-1}\mathcal{F}$ is an isomorphism.


etale-cohomology-lemma-base-change-q-integral-bottom

With $f : X \to S$ and $n$ as in Remark \ref{remark-base-change-holds} assume for some $q \geq 1$ we have $BC(f, n, q - 1)$. Consider commutative diagrams $$ \vcenter{ \xymatrix{ X \ar[d]_f & X' \ar[d]_{f'} \ar[l] & X'' \ar[l]^{\pi'} \ar[d]_{f''} & Y \ar[l]^{h'} \ar[d]^e \\ S & S' \ar[l] & S'' \ar[l]_\pi & T \ar[l]_{g'} } } \quad\text{and}\quad \vcenter{ \xymatrix{ X' \ar[d]_{f'} & & Y \ar[ll]^{h = h' \circ \pi'} \ar[d]^e \\ S' & & T \ar[ll]_{g = g' \circ \pi} } } $$ where all squares are cartesian, $g'$ quasi-compact and quasi-separated, and $\pi$ is integral. Let $\mathcal{F}$ be an abelian sheaf on $T_\etale$ annihilated by $n$. If the base change map $$ (f')^{-1}R^qg_*\mathcal{F} \longrightarrow R^qh_*e^{-1}\mathcal{F} $$ is an isomorphism, then the base change map $(f'')^{-1}R^qg'_*\mathcal{F} \to R^qh'_*e^{-1}\mathcal{F}$ is an isomorphism.


etale-cohomology-lemma-formal-argument

Let $T$ be a quasi-compact and quasi-separated scheme. Let $P$ be a property for quasi-compact and quasi-separated schemes over $T$. Assume \begin{enumerate} \item If $T'' \to T'$ is a thickening of quasi-compact and quasi-separated schemes over $T$, then $P(T'')$ if and only if $P(T')$. \item If $T' = \lim T_i$ is a limit of an inverse system of quasi-compact and quasi-separated schemes over $T$ with affine transition morphisms and $P(T_i)$ holds for all $i$, then $P(T')$ holds. \item If $Z \subset T'$ is a closed subscheme with quasi-compact complement $V \subset T'$ and $P(T')$ holds, then either $P(V)$ or $P(Z)$ holds. \end{enumerate} Then $P(T)$ implies $P(\Spec(K))$ for some morphism $\Spec(K) \to T$ where $K$ is a field.


etale-cohomology-lemma-base-change-does-not-hold-pre

With $f : X \to S$ and $n$ as in Remark \ref{remark-base-change-holds} assume for some $q \geq 1$ we have that $BC(f, n, q - 1)$ is true, but $BC(f, n, q)$ is not. Then there exist a commutative diagram $$ \xymatrix{ X \ar[d]_f & X' \ar[d]_{f'} \ar[l] & Y \ar[l]^h \ar[d]^e \\ S & S' \ar[l] & \Spec(K) \ar[l]_g } $$ where $X' = X \times_S S'$, $Y = X' \times_{S'} \Spec(K)$, $K$ is a field, and $\mathcal{F}$ is an abelian sheaf on $\Spec(K)$ annihilated by $n$ such that $(f')^{-1}R^qg_*\mathcal{F} \to R^qh_*e^{-1}\mathcal{F}$ is not an isomorphism.


etale-cohomology-lemma-base-change-does-not-hold

With $f : X \to S$ and $n$ as in Remark \ref{remark-base-change-holds} assume for some $q \geq 1$ we have that $BC(f, n, q - 1)$ is true, but $BC(f, n, q)$ is not. Then there exist a commutative diagram $$ \xymatrix{ X \ar[d]_f & X' \ar[d] \ar[l] & Y \ar[l]^h \ar[d] \\ S & S' \ar[l] & \Spec(K) \ar[l] } $$ with both squares cartesian, where \begin{enumerate} \item $S'$ is affine, integral, and normal with algebraically closed function field, \item $K$ is algebraically closed and $\Spec(K) \to S'$ is dominant (in other words $K$ is an extension of the function field of $S'$) \end{enumerate} and there exists an integer $d | n$ such that $R^qh_*(\mathbf{Z}/d\mathbf{Z})$ is nonzero.


etale-cohomology-lemma-smooth-base-change-fields

Let $K/k$ be an extension of fields. Let $X$ be a smooth affine curve over $k$ with a rational point $x \in X(k)$. Let $\mathcal{F}$ be an abelian sheaf on $\Spec(K)$ annihilated by an integer $n$ invertible in $k$. Let $q > 0$ and $$ \xi \in H^q(X_K, (X_K \to \Spec(K))^{-1}\mathcal{F}) $$ There exist \begin{enumerate} \item finite extensions $K'/K$ and $k'/k$ with $k' \subset K'$, \item a finite \'etale Galois cover $Z \to X_{k'}$ with group $G$ \end{enumerate} such that the order of $G$ divides a power of $n$, such that $Z \to X_{k'}$ is split over $x_{k'}$, and such that $\xi$ dies in $H^q(Z_{K'}, (Z_{K'} \to \Spec(K))^{-1}\mathcal{F})$.


etale-cohomology-lemma-smooth-base-change-general

Let $S$ be a scheme. Let $S' = \lim S_i$ be a directed inverse limit of schemes $S_i$ smooth over $S$ with affine transition morphisms. Let $f : X \to S$ be quas-compact and quasi-separated and form the fibre square $$ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ S' \ar[r]^g & S } $$ Then $$ g^{-1}Rf_*E = R(f')_*(g')^{-1}E $$ for any $E \in D^+(X_\etale)$ whose cohomology sheaves $H^q(E)$ have stalks which are torsion of orders invertible on $S$.


etale-cohomology-lemma-base-change-field-extension

Let $L/K$ be an extension of fields. Let $g : T \to S$ be a quasi-compact and quasi-separated morphism of schemes over $K$. Denote $g_L : T_L \to S_L$ the base change of $g$ to $\Spec(L)$. Let $E \in D^+(T_\etale)$ have cohomology sheaves whose stalks are torsion of orders invertible in $K$. Let $E_L$ be the pullback of $E$ to $(T_L)_\etale$. Then $Rg_{L, *}E_L$ is the pullback of $Rg_*E$ to $S_L$.


etale-cohomology-lemma-smooth-base-change-separably-closed

Let $K/k$ be an extension of separably closed fields. Let $X$ be a quasi-compact and quasi-separated scheme over $k$. Let $E \in D^+(X_\etale)$ have cohomology sheaves whose stalks are torsion of orders invertible in $k$. Then \begin{enumerate} \item the maps $H^q_\etale(X, E) \to H^q_\etale(X_K, E|_{X_K})$ are isomorphisms, and \item $E \to R(X_K \to X)_*E|_{X_K}$ is an isomorphism. \end{enumerate}


etale-cohomology-lemma-base-change-does-not-hold-post

With $f : X \to S$ and $n$ as in Remark \ref{remark-base-change-holds} assume $n$ is invertible on $S$ and that for some $q \geq 1$ we have that $BC(f, n, q - 1)$ is true, but $BC(f, n, q)$ is not. Then there exist a commutative diagram $$ \xymatrix{ X \ar[d]_f & X' \ar[d] \ar[l] & Y \ar[l]^h \ar[d] \\ S & S' \ar[l] & \Spec(K) \ar[l] } $$ with both squares cartesian, where $S'$ is affine, integral, and normal with algebraically closed function field $K$ and there exists an integer $d | n$ such that $R^qh_*(\mathbf{Z}/d\mathbf{Z})$ is nonzero.


etale-cohomology-lemma-zariski-h0-proper-over-henselian-pair

Let $(A, I)$ be a henselian pair. Let $f : X \to \Spec(A)$ be a proper morphism of schemes. Let $Z = X \times_{\Spec(A)} \Spec(A/I)$. For any sheaf $\mathcal{F}$ on the topological space associated to $X$ we have $\Gamma(X, \mathcal{F}) = \Gamma(Z, \mathcal{F}|_Z)$.


etale-cohomology-lemma-h0-proper-over-henselian-pair

Let $(A, I)$ be a henselian pair. Let $f : X \to \Spec(A)$ be a proper morphism of schemes. Let $i : Z \to X$ be the closed immersion of $X \times_{\Spec(A)} \Spec(A/I)$ into $X$. For any sheaf $\mathcal{F}$ on $X_\etale$ we have $\Gamma(X, \mathcal{F}) = \Gamma(Z, i_{small}^{-1}\mathcal{F})$.


etale-cohomology-lemma-h0-proper-over-henselian-local

Let $A$ be a henselian local ring. Let $f : X \to \Spec(A)$ be a proper morphism of schemes. Let $X_0 \subset X$ be the fibre of $f$ over the closed point. For any sheaf $\mathcal{F}$ on $X_\etale$ we have $\Gamma(X, \mathcal{F}) = \Gamma(X_0, \mathcal{F}|_{X_0})$.


etale-cohomology-lemma-proper-pushforward-stalk

Let $f : X \to S$ be a proper morphism of schemes. Let $\overline{s} \to S$ be a geometric point. For any sheaf $\mathcal{F}$ on $X_\etale$ the canonical map $$ (f_*\mathcal{F})_{\overline{s}} \longrightarrow \Gamma(X_{\overline{s}}, \mathcal{F}_{\overline{s}}) $$ is bijective.


etale-cohomology-lemma-proper-base-change-f-star

Let $f : X \to Y$ be a proper morphism of schemes. Let $g : Y' \to Y$ be a morphism of schemes. Set $X' = Y' \times_Y X$ with projections $f' : X' \to Y'$ and $g' : X' \to X$. Let $\mathcal{F}$ be any sheaf on $X_\etale$. Then $g^{-1}f_*\mathcal{F} = f'_*(g')^{-1}\mathcal{F}$.


etale-cohomology-lemma-proper-base-change-in-terms-of-injectives

Let $f : X \to Y$ be a proper morphism of schemes. The following are equivalent \begin{enumerate} \item cohomology commutes with base change for $f$ (see above), \item for every prime number $\ell$ and every injective sheaf of $\mathbf{Z}/\ell\mathbf{Z}$-modules $\mathcal{I}$ on $X_\etale$ and every diagram (\ref{equation-base-change-diagram}) where $X' = Y' \times_Y X$ the sheaves $R^qf'_*(g')^{-1}\mathcal{I}$ are zero for $q > 0$. \end{enumerate}


etale-cohomology-lemma-sandwich

Let $f : X \to Y$ and $g : Y \to Z$ be proper morphisms of schemes. Assume \begin{enumerate} \item cohomology commutes with base change for $f$, \item cohomology commutes with base change for $g \circ f$, and \item $f$ is surjective. \end{enumerate} Then cohomology commutes with base change for $g$.


etale-cohomology-lemma-composition

Let $f : X \to Y$ and $g : Y \to Z$ be proper morphisms of schemes. Assume \begin{enumerate} \item cohomology commutes with base change for $f$, and \item cohomology commutes with base change for $g$. \end{enumerate} Then cohomology commutes with base change for $g \circ f$.


etale-cohomology-lemma-finite

\begin{slogan} Proper base change for \'etale cohomology holds for finite morphisms. \end{slogan} Let $f : X \to Y$ be a finite morphism of schemes. Then cohomology commutes with base change for $f$.


etale-cohomology-lemma-reduce-to-P1

To prove that cohomology commutes with base change for every proper morphism of schemes it suffices to prove it holds for the morphism $\mathbf{P}^1_S \to S$ for every scheme $S$.


etale-cohomology-lemma-proper-base-change

Let $f : X \to Y$ be a proper morphism of schemes. Let $g : Y' \to Y$ be a morphism of schemes. Set $X' = Y' \times_Y X$ and denote $f' : X' \to Y'$ and $g' : X' \to X$ the projections. Let $E \in D^+(X_\etale)$ have torsion cohomology sheaves. Then the base change map (\ref{equation-base-change}) $g^{-1}Rf_*E \to Rf'_*(g')^{-1}E$ is an isomorphism.


etale-cohomology-lemma-proper-base-change-stalk

Let $f : X \to Y$ be a proper morphism of schemes. Let $\overline{y} \to Y$ be a geometric point. \begin{enumerate} \item For a torsion abelian sheaf $\mathcal{F}$ on $X_\etale$ we have $(R^nf_*\mathcal{F})_{\overline{y}} = H^n_\etale(X_{\overline{y}}, \mathcal{F}_{\overline{y}})$. \item For $E \in D^+(X_\etale)$ with torsion cohomology sheaves we have $(R^nf_*E)_{\overline{y}} = H^n_\etale(X_{\overline{y}}, E_{\overline{y}})$. \end{enumerate}


etale-cohomology-lemma-cohomological-dimension-proper

Let $f : X \to Y$ be a proper morphism of schemes all of whose fibres have dimension $\leq n$. Then for any abelian torsion sheaf $\mathcal{F}$ on $X_\etale$ we have $R^qf_*\mathcal{F} = 0$ for $q > 2n$.


etale-cohomology-lemma-proper-base-change-mod-n

Let $f : X \to Y$ be a proper morphism of schemes. Let $g : Y' \to Y$ be a morphism of schemes. Set $X' = Y' \times_Y X$ and denote $f' : X' \to Y'$ and $g' : X' \to X$ the projections. Let $n \geq 1$ be an integer. Let $E \in D(X_\etale, \mathbf{Z}/n\mathbf{Z})$. Then the base change map (\ref{equation-base-change}) $g^{-1}Rf_*E \to Rf'_*(g')^{-1}E$ is an isomorphism.


etale-cohomology-lemma-pull-out-constant

Let $X$ be a quasi-compact and quasi-separated scheme. Let $E \in D^+(X_\etale)$ and $K \in D^+(\mathbf{Z})$. Then $$ R\Gamma(X, E \otimes_\mathbf{Z}^\mathbf{L} \underline{K}) = R\Gamma(X, E) \otimes_\mathbf{Z}^\mathbf{L} K $$


etale-cohomology-lemma-projection-formula-proper

Let $f : X \to Y$ be a proper morphism of schemes. Let $E \in D^+(X_\etale)$ have torsion cohomology sheaves. Let $K \in D^+(Y_\etale)$. Then $$ Rf_*E \otimes_\mathbf{Z}^\mathbf{L} K = Rf_*(E \otimes_\mathbf{Z}^\mathbf{L} f^{-1}K) $$ in $D^+(Y_\etale)$.


etale-cohomology-lemma-cd-limit

Let $X = \lim X_i$ be a directed limit of a system of quasi-compact and quasi-separated schemes with affine transition morphisms. Then $\text{cd}(X) \leq \max \text{cd}(X_i)$.


etale-cohomology-lemma-cd-curve-over-field

Let $K$ be a field. Let $X$ be a $1$-dimensional affine scheme of finite type over $K$. Then $\text{cd}(X) \leq 1 + \text{cd}(K)$.


etale-cohomology-lemma-cd-field-extension

Let $L/K$ be a field extension. Then we have $\text{cd}(L) \leq \text{cd}(K) + \text{trdeg}_K(L)$.


etale-cohomology-lemma-strictly-henselian

Let $K$ be a field. Let $X$ be a scheme of finite type over $K$. Let $x \in X$. Set $a = \text{trdeg}_K(\kappa(x))$ and $d = \dim_x(X)$. Then there is a map $$ K(t_1, \ldots, t_a)^{sep} \longrightarrow \mathcal{O}_{X, x}^{sh} $$ such that \begin{enumerate} \item the residue field of $\mathcal{O}_{X, x}^{sh}$ is a purely inseparable extension of $K(t_1, \ldots, t_a)^{sep}$, \item $\mathcal{O}_{X, x}^{sh}$ is a filtered colimit of finite type $K(t_1, \ldots, t_a)^{sep}$-algebras of dimension $\leq d - a$. \end{enumerate}


etale-cohomology-lemma-interlude-II

Let $K$ be a field. Let $X$ be an affine scheme of finite type over $K$. Let $E_a \subset X$ be the set of points $x \in X$ with $\text{trdeg}_K(\kappa(x)) \leq a$. Let $\mathcal{F}$ be an abelian torsion sheaf on $X_\etale$ whose support is contained in $E_a$. Then $H^b_\etale(X, \mathcal{F}) = 0$ for $b > a + \text{cd}(K)$.


etale-cohomology-lemma-interlude-I

Let $f : X \to Y$ be an affine morphism of schemes of finite type over a field $K$. Let $E_a(X)$ be the set of points $x \in X$ with $\text{trdeg}_K(\kappa(x)) \leq a$. Let $\mathcal{F}$ be an abelian torsion sheaf on $X_\etale$ whose support is contained in $E_a$. Then $R^qf_*\mathcal{F}$ has support contained in $E_{a - q}(Y)$.


etale-cohomology-lemma-finite-cd

Let $K$ be a field. \begin{enumerate} \item If $f : X \to Y$ is a morphism of finite type schemes over $K$, then $\text{cd}(f) < \infty$. \item If $\text{cd}(K) < \infty$, then $\text{cd}(X) < \infty$ for any finite type scheme $X$ over $K$. \end{enumerate}


etale-cohomology-lemma-finite-cd-mod-n-direct-sums

Cohomology and direct sums. Let $n \geq 1$ be an integer. \begin{enumerate} \item Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of schemes with $\text{cd}(f) < \infty$. Then the functor $$ Rf_* : D(X_\etale, \mathbf{Z}/n\mathbf{Z}) \longrightarrow D(Y_\etale, \mathbf{Z}/n\mathbf{Z}) $$ commutes with direct sums. \item Let $X$ be a quasi-compact and quasi-separated scheme with $\text{cd}(X) < \infty$. Then the functor $$ R\Gamma(X, -) : D(X_\etale, \mathbf{Z}/n\mathbf{Z}) \longrightarrow D(\mathbf{Z}/n\mathbf{Z}) $$ commutes with direct sums. \end{enumerate}


etale-cohomology-lemma-proper-mod-n-direct-sums

Let $f : X \to Y$ be a proper morphism of schemes. Let $n \geq 1$ be an integer. Then the functor $$ Rf_* : D(X_\etale, \mathbf{Z}/n\mathbf{Z}) \longrightarrow D(Y_\etale, \mathbf{Z}/n\mathbf{Z}) $$ commutes with direct sums.


etale-cohomology-lemma-pull-out-constant-mod-n

Let $X$ be a quasi-compact and quasi-separated scheme such that $\text{cd}(X) < \infty$. Let $n \geq 1$ be an integer. Let $E \in D(X_\etale, \mathbf{Z}/n\mathbf{Z})$ and $K \in D(\mathbf{Z}/n\mathbf{Z})$. Then $$ R\Gamma(X, E \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} \underline{K}) = R\Gamma(X, E) \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} K $$


etale-cohomology-lemma-projection-formula-proper-mod-n

Let $f : X \to Y$ be a proper morphism of schemes. Let $n \geq 1$ be an integer. Let $E \in D(X_\etale, \mathbf{Z}/n\mathbf{Z})$ and $K \in D(Y_\etale, \mathbf{Z}/n\mathbf{Z})$. Then $$ Rf_*E \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} K = Rf_*(E \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} f^{-1}K) $$ in $D(Y_\etale, \mathbf{Z}/n\mathbf{Z})$.


etale-cohomology-lemma-kunneth-one-proper

Let $k$ be a separably closed field. Let $X$ be a proper scheme over $k$. Let $Y$ be a quasi-compact and quasi-separated scheme over $k$. \begin{enumerate} \item If $E \in D^+(X_\etale)$ has torsion cohomology sheaves and $K \in D^+(Y_\etale)$, then $$ R\Gamma(X \times_{\Spec(k)} Y, \text{pr}_1^{-1}E \otimes_\mathbf{Z}^\mathbf{L} \text{pr}_2^{-1}K ) = R\Gamma(X, E) \otimes_\mathbf{Z}^\mathbf{L} R\Gamma(Y, K) $$ \item If $n \geq 1$ is an integer, $Y$ is of finite type over $k$, $E \in D(X_\etale, \mathbf{Z}/n\mathbf{Z})$, and $K \in D(Y_\etale, \mathbf{Z}/n\mathbf{Z})$, then $$ R\Gamma(X \times_{\Spec(k)} Y, \text{pr}_1^{-1}E \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} \text{pr}_2^{-1}K ) = R\Gamma(X, E) \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} R\Gamma(Y, K) $$ \end{enumerate}


etale-cohomology-lemma-supported-in-closed-points

Let $K$ be a separably closed field. Let $X$ be a scheme of finite type over $K$. Let $\mathcal{F}$ be an abelian sheaf on $X_\etale$ whose support is contained in the set of closed points of $X$. Then $H^q(X, \mathcal{F}) = 0$ for $q > 0$ and $\mathcal{F}$ is globally generated.


etale-cohomology-lemma-vanishing-closed-points

Let $K$ be a separably closed field. Let $X$ be a scheme of finite type over $K$. Let $Q \in D(X_\etale)$. Assume that $Q_{\overline{x}}$ is nonzero only if $x$ is a closed point of $X$. Then $$ Q = 0 \Leftrightarrow H^i(X, Q) = 0 \text{ for all }i $$


etale-cohomology-lemma-kunneth-localize-on-X

Let $K$ be a field. Let $j : U \to X$ be an open immersion of schemes of finite type over $K$. Let $Y$ be a scheme of finite type over $K$. Consider the diagram $$ \xymatrix{ Y \times_{\Spec(K)} X \ar[d]_q  & Y \times_{\Spec(K)} U \ar[l]^h \ar[d]^p \\ X & U \ar[l]_j } $$ Then the base change map $q^{-1}Rj_*\mathcal{F} \to Rh_*p^{-1}\mathcal{F}$ is an isomorphism for $\mathcal{F}$ an abelian sheaf on $U_\etale$ whose stalks are torsion of orders invertible in $K$.


etale-cohomology-lemma-punctual-base-change

Let $K$ be a field. For any commutative diagram $$ \xymatrix{ X \ar[d] & X' \ar[l] \ar[d]_{f'} & Y \ar[l]^h \ar[d]^e \\ \Spec(K) & S' \ar[l] & T \ar[l]_g } $$ of schemes over $K$ with $X' = X \times_{\Spec(K)} S'$ and $Y = X' \times_{S'} T$ and $g$ quasi-compact and quasi-separated, and every abelian sheaf $\mathcal{F}$ on $T_\etale$ whose stalks are torsion of orders invertible in $K$ the base change map $$ (f')^{-1}Rg_*\mathcal{F} \longrightarrow Rh_*e^{-1}\mathcal{F} $$ is an isomorphism.


etale-cohomology-lemma-punctual-base-change-upgrade

Let $K$ be a field. Let $n \geq 1$ be invertible in $K$. Consider a commutative diagram $$ \xymatrix{ X \ar[d] & X' \ar[l]^p \ar[d]_{f'} & Y \ar[l]^h \ar[d]^e \\ \Spec(K) & S' \ar[l] & T \ar[l]_g } $$ of schemes with $X' = X \times_{\Spec(K)} S'$ and $Y = X' \times_{S'} T$ and $g$ quasi-compact and quasi-separated. The canonical map $$ p^{-1}E \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} (f')^{-1}Rg_*F \longrightarrow Rh_*(h^{-1}p^{-1}E \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} e^{-1}F) $$ is an isomorphism if $E$ in $D^+(X_\etale, \mathbf{Z}/n\mathbf{Z})$ has tor amplitude in $[a, \infty]$ for some $a \in \mathbf{Z}$ and $F$ in $D^+(T_\etale, \mathbf{Z}/n\mathbf{Z})$.


etale-cohomology-lemma-punctual-base-change-upgrade-unbounded

Let $K$ be a field. Let $n \geq 1$ be invertible in $K$. Consider a commutative diagram $$ \xymatrix{ X \ar[d] & X' \ar[l]^p \ar[d]_{f'} & Y \ar[l]^h \ar[d]^e \\ \Spec(K) & S' \ar[l] & T \ar[l]_g } $$ of schemes of finite type over $K$ with $X' = X \times_{\Spec(K)} S'$ and $Y = X' \times_{S'} T$. The canonical map $$ p^{-1}E \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} (f')^{-1}Rg_*F \longrightarrow Rh_*(h^{-1}p^{-1}E \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} e^{-1}F) $$ is an isomorphism for $E$ in $D(X_\etale, \mathbf{Z}/n\mathbf{Z})$ and $F$ in $D(T_\etale, \mathbf{Z}/n\mathbf{Z})$.


etale-cohomology-lemma-compare-injectives

Let $S$ be a scheme. Let $T$ be an object of $(\Sch/S)_\etale$. \begin{enumerate} \item If $\mathcal{I}$ is injective in $\textit{Ab}((\Sch/S)_\etale)$, then \begin{enumerate} \item $i_f^{-1}\mathcal{I}$ is injective in $\textit{Ab}(T_\etale)$, \item $\mathcal{I}|_{S_\etale}$ is injective in $\textit{Ab}(S_\etale)$, \end{enumerate} \item If $\mathcal{I}^\bullet$ is a K-injective complex in $\textit{Ab}((\Sch/S)_\etale)$, then \begin{enumerate} \item $i_f^{-1}\mathcal{I}^\bullet$ is a K-injective complex in $\textit{Ab}(T_\etale)$, \item $\mathcal{I}^\bullet|_{S_\etale}$ is a K-injective complex in $\textit{Ab}(S_\etale)$, \end{enumerate} \end{enumerate} The corresponding statements for modules do not hold.


etale-cohomology-lemma-compare-higher-direct-image

Let $f : T \to S$ be a morphism of schemes. \begin{enumerate} \item For $K$ in $D((\Sch/T)_\etale)$ we have $ (Rf_{big, *}K)|_{S_\etale} = Rf_{small, *}(K|_{T_\etale}) $ in $D(S_\etale)$. \item For $K$ in $D((\Sch/T)_\etale, \mathcal{O})$ we have $ (Rf_{big, *}K)|_{S_\etale} = Rf_{small, *}(K|_{T_\etale}) $ in $D(\textit{Mod}(S_\etale, \mathcal{O}_S))$. \end{enumerate} More generally, let $g : S' \to S$ be an object of $(\Sch/S)_\etale$. Consider the fibre product $$ \xymatrix{ T' \ar[r]_{g'} \ar[d]_{f'} & T \ar[d]^f \\ S' \ar[r]^g & S } $$ Then \begin{enumerate} \item[(3)] For $K$ in $D((\Sch/T)_\etale)$ we have $i_g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K)$ in $D(S'_\etale)$. \item[(4)] For $K$ in $D((\Sch/T)_\etale, \mathcal{O})$ we have $i_g^*(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^*K)$ in $D(\textit{Mod}(S'_\etale, \mathcal{O}_{S'}))$. \item[(5)] For $K$ in $D((\Sch/T)_\etale)$ we have $g_{big}^{-1}(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^{-1}K)$ in $D((\Sch/S')_\etale)$. \item[(6)] For $K$ in $D((\Sch/T)_\etale, \mathcal{O})$ we have $g_{big}^*(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^*K)$ in $D(\textit{Mod}(S'_\etale, \mathcal{O}_{S'}))$. \end{enumerate}


etale-cohomology-lemma-cohomological-descent-etale

Let $S$ be a scheme. For $K \in D(S_\etale)$ the map $$ K \longrightarrow R\pi_{S, *}\pi_S^{-1}K $$ is an isomorphism.


etale-cohomology-lemma-compare-higher-direct-image-proper

Let $f : T \to S$ be a proper morphism of schemes. Then we have \begin{enumerate} \item $\pi_S^{-1} \circ f_{small, *} = f_{big, *} \circ \pi_T^{-1}$ as functors $\Sh(T_\etale) \to \Sh((\Sch/S)_\etale)$, \item $\pi_S^{-1}Rf_{small, *}K = Rf_{big, *}\pi_T^{-1}K$ for $K$ in $D^+(T_\etale)$ whose cohomology sheaves are torsion, \item $\pi_S^{-1}Rf_{small, *}K = Rf_{big, *}\pi_T^{-1}K$ for $K$ in $D(T_\etale, \mathbf{Z}/n\mathbf{Z})$, and \item $\pi_S^{-1}Rf_{small, *}K = Rf_{big, *}\pi_T^{-1}K$ for all $K$ in $D(T_\etale)$ if $f$ is finite. \end{enumerate}


etale-cohomology-lemma-describe-pullback-pi-fppf

With notation as above. Let $\mathcal{F}$ be a sheaf on $S_\etale$. The rule $$ (\Sch/S)_{fppf} \longrightarrow \textit{Sets},\quad (f : X \to S) \longmapsto \Gamma(X, f_{small}^{-1}\mathcal{F}) $$ is a sheaf and a fortiori a sheaf on $(\Sch/S)_\etale$. In fact this sheaf is equal to $\pi_S^{-1}\mathcal{F}$ on $(\Sch/S)_\etale$ and $\epsilon_S^{-1}\pi_S^{-1}\mathcal{F}$ on $(\Sch/S)_{fppf}$.


etale-cohomology-lemma-push-pull-fppf-etale

With notation as above. Let $f : X \to Y$ be a morphism of $(\Sch/S)_{fppf}$. Then there are commutative diagrams of topoi $$ \xymatrix{ \Sh((\Sch/X)_{fppf}) \ar[rr]_{f_{big, fppf}} \ar[d]_{\epsilon_X} & & \Sh((\Sch/Y)_{fppf}) \ar[d]^{\epsilon_Y} \\ \Sh((\Sch/X)_\etale) \ar[rr]^{f_{big, \etale}} & & \Sh((\Sch/Y)_\etale) } $$ and $$ \xymatrix{ \Sh((\Sch/X)_{fppf}) \ar[rr]_{f_{big, fppf}} \ar[d]_{a_X} & & \Sh((\Sch/Y)_{fppf}) \ar[d]^{a_Y} \\ \Sh(X_\etale) \ar[rr]^{f_{small}} & & \Sh(Y_\etale) } $$ with $a_X = \pi_X \circ \epsilon_X$ and $a_Y = \pi_X \circ \epsilon_X$.


etale-cohomology-lemma-proper-push-pull-fppf-etale

In Lemma \ref{lemma-push-pull-fppf-etale} if $f$ is proper, then we have $a_Y^{-1} \circ f_{small, *} = f_{big, fppf, *} \circ a_X^{-1}$.


etale-cohomology-lemma-descent-sheaf-fppf-etale

In Lemma \ref{lemma-push-pull-fppf-etale} assume $f$ is flat, locally of finite presentation, and surjective. Then the functor $$ \Sh(Y_\etale) \longrightarrow \left\{ (\mathcal{G}, \mathcal{H}, \alpha) \middle| \begin{matrix} \mathcal{G} \in \Sh(X_\etale),\ \mathcal{H} \in \Sh((\Sch/Y)_{fppf}), \\ \alpha : a_X^{-1}\mathcal{G} \to f_{big, fppf}^{-1}\mathcal{H} \text{ an isomorphism} \end{matrix} \right\} $$ sending $\mathcal{F}$ to $(f_{small}^{-1}\mathcal{F}, a_Y^{-1}\mathcal{F}, can)$ is an equivalence.


etale-cohomology-lemma-compare-fppf-etale

Consider the comparison morphism $\epsilon : (\Sch/S)_{fppf} \to (\Sch/S)_\etale$. Let $\mathcal{P}$ denote the class of finite morphisms of schemes. For $X$ in $(\Sch/S)_\etale$ denote $\mathcal{A}'_X \subset \textit{Ab}((\Sch/X)_\etale)$ the full subcategory consisting of sheaves of the form $\pi_X^{-1}\mathcal{F}$ with $\mathcal{F}$ in $\textit{Ab}(X_\etale)$. Then Cohomology on Sites, Properties (\ref{sites-cohomology-item-base-change-P}), (\ref{sites-cohomology-item-restriction-A}), (\ref{sites-cohomology-item-A-sheaf}), (\ref{sites-cohomology-item-A-and-P}), and (\ref{sites-cohomology-item-refine-tau-by-P}) of Cohomology on Sites, Situation \ref{sites-cohomology-situation-compare} hold.


etale-cohomology-lemma-V-C-all-n-etale-fppf

With notation as above. \begin{enumerate} \item For $X \in \Ob((\Sch/S)_{fppf})$ and an abelian sheaf $\mathcal{F}$ on $X_\etale$ we have $\epsilon_{X, *}a_X^{-1}\mathcal{F} = \pi_X^{-1}\mathcal{F}$ and $R^i\epsilon_{X, *}(a_X^{-1}\mathcal{F}) = 0$ for $i > 0$. \item For a finite morphism $f : X \to Y$ in $(\Sch/S)_{fppf}$ and abelian sheaf $\mathcal{F}$ on $X$ we have $a_Y^{-1}(R^if_{small, *}\mathcal{F}) = R^if_{big, fppf, *}(a_X^{-1}\mathcal{F})$ for all $i$. \item For a scheme $X$ and $K$ in $D^+(X_\etale)$ the map $\pi_X^{-1}K \to R\epsilon_{X, *}(a_X^{-1}K)$ is an isomorphism. \item For a finite morphism $f : X \to Y$ of schemes and $K$ in $D^+(X_\etale)$ we have $a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)$. \item For a proper morphism $f : X \to Y$ of schemes and $K$ in $D^+(X_\etale)$ with torsion cohomology sheaves we have $a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)$. \end{enumerate}


etale-cohomology-lemma-cohomological-descent-etale-fppf

Let $X$ be a scheme. For $K \in D^+(X_\etale)$ the map $$ K \longrightarrow Ra_{X, *}a_X^{-1}K $$ is an isomorphism with $a_X : \Sh((\Sch/X)_{fppf}) \to \Sh(X_\etale)$ as above.


etale-cohomology-lemma-review-quasi-coherent

Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_S$-module on $S_\etale$. \begin{enumerate} \item The rule $$ \mathcal{F}^a : (\Sch/S)_\etale \longrightarrow \textit{Ab},\quad (f : T \to S) \longmapsto \Gamma(T, f_{small}^*\mathcal{F}) $$ satisfies the sheaf condition for fppf and a fortiori \'etale coverings, \item $\mathcal{F}^a = \pi_S^*\mathcal{F}$ on $(\Sch/S)_\etale$, \item $\mathcal{F}^a = a_S^*\mathcal{F}$ on $(\Sch/S)_{fppf}$, \item the rule $\mathcal{F} \mapsto \mathcal{F}^a$ defines an equivalence between quasi-coherent $\mathcal{O}_S$-modules and quasi-coherent modules on $((\Sch/S)_\etale, \mathcal{O})$, \item the rule $\mathcal{F} \mapsto \mathcal{F}^a$ defines an equivalence between quasi-coherent $\mathcal{O}_S$-modules and quasi-coherent modules on $((\Sch/S)_{fppf}, \mathcal{O})$, \item we have $\epsilon_{S, *}a_S^*\mathcal{F} = \pi_S^*\mathcal{F}$ and $a_{S, *}a_S^*\mathcal{F} = \mathcal{F}$, \item we have $R^i\epsilon_{S, *}(a_S^*\mathcal{F}) = 0$ and $R^ia_{S, *}(a_S^*\mathcal{F}) = 0$ for $i > 0$. \end{enumerate}


etale-cohomology-lemma-describe-pullback-pi-ph

With notation as above. Let $\mathcal{F}$ be a sheaf on $S_\etale$. The rule $$ (\Sch/S)_{ph} \longrightarrow \textit{Sets},\quad (f : X \to S) \longmapsto \Gamma(X, f_{small}^{-1}\mathcal{F}) $$ is a sheaf and a fortiori a sheaf on $(\Sch/S)_\etale$. In fact this sheaf is equal to $\pi_S^{-1}\mathcal{F}$ on $(\Sch/S)_\etale$ and $\epsilon_S^{-1}\pi_S^{-1}\mathcal{F}$ on $(\Sch/S)_{ph}$.


etale-cohomology-lemma-push-pull-ph-etale

With notation as above. Let $f : X \to Y$ be a morphism of $(\Sch/S)_{ph}$. Then there are commutative diagrams of topoi $$ \xymatrix{ \Sh((\Sch/X)_{ph}) \ar[rr]_{f_{big, ph}} \ar[d]_{\epsilon_X} & & \Sh((\Sch/Y)_{ph}) \ar[d]^{\epsilon_Y} \\ \Sh((\Sch/X)_\etale) \ar[rr]^{f_{big, \etale}} & & \Sh((\Sch/Y)_\etale) } $$ and $$ \xymatrix{ \Sh((\Sch/X)_{ph}) \ar[rr]_{f_{big, ph}} \ar[d]_{a_X} & & \Sh((\Sch/Y)_{ph}) \ar[d]^{a_Y} \\ \Sh(X_\etale) \ar[rr]^{f_{small}} & & \Sh(Y_\etale) } $$ with $a_X = \pi_X \circ \epsilon_X$ and $a_Y = \pi_X \circ \epsilon_X$.


etale-cohomology-lemma-compare-ph-etale

Consider the comparison morphism $\epsilon : (\Sch/S)_{ph} \to (\Sch/S)_\etale$. Let $\mathcal{P}$ denote the class of proper morphisms of schemes. For $X$ in $(\Sch/S)_\etale$ denote $\mathcal{A}'_X \subset \textit{Ab}((\Sch/X)_\etale)$ the full subcategory consisting of sheaves of the form $\pi_X^{-1}\mathcal{F}$ where $\mathcal{F}$ is a torsion abelian sheaf on $X_\etale$ Then Cohomology on Sites, Properties (\ref{sites-cohomology-item-base-change-P}), (\ref{sites-cohomology-item-restriction-A}), (\ref{sites-cohomology-item-A-sheaf}), (\ref{sites-cohomology-item-A-and-P}), and (\ref{sites-cohomology-item-refine-tau-by-P}) of Cohomology on Sites, Situation \ref{sites-cohomology-situation-compare} hold.


etale-cohomology-lemma-V-C-all-n-etale-ph

With notation as above. \begin{enumerate} \item For $X \in \Ob((\Sch/S)_{ph})$ and an abelian torsion sheaf $\mathcal{F}$ on $X_\etale$ we have $\epsilon_{X, *}a_X^{-1}\mathcal{F} = \pi_X^{-1}\mathcal{F}$ and $R^i\epsilon_{X, *}(a_X^{-1}\mathcal{F}) = 0$ for $i > 0$. \item For a proper morphism $f : X \to Y$ in $(\Sch/S)_{ph}$ and abelian torsion sheaf $\mathcal{F}$ on $X$ we have $a_Y^{-1}(R^if_{small, *}\mathcal{F}) = R^if_{big, ph, *}(a_X^{-1}\mathcal{F})$ for all $i$. \item For a scheme $X$ and $K$ in $D^+(X_\etale)$ with torsion cohomology sheaves the map $\pi_X^{-1}K \to R\epsilon_{X, *}(a_X^{-1}K)$ is an isomorphism. \item For a proper morphism $f : X \to Y$ of schemes and $K$ in $D^+(X_\etale)$ with torsion cohomology sheaves we have $a_Y^{-1}(Rf_{small, *}K) = Rf_{big, ph, *}(a_X^{-1}K)$. \end{enumerate}


etale-cohomology-lemma-cohomological-descent-etale-ph

Let $X$ be a scheme. For $K \in D^+(X_\etale)$ with torsion cohomology sheaves the map $$ K \longrightarrow Ra_{X, *}a_X^{-1}K $$ is an isomorphism with $a_X : \Sh((\Sch/X)_{ph}) \to \Sh(X_\etale)$ as above.


etale-cohomology-lemma-describe-pullback-pi-h

With notation as above. Let $\mathcal{F}$ be a sheaf on $S_\etale$. The rule $$ (\Sch/S)_h \longrightarrow \textit{Sets},\quad (f : X \to S) \longmapsto \Gamma(X, f_{small}^{-1}\mathcal{F}) $$ is a sheaf and a fortiori a sheaf on $(\Sch/S)_\etale$. In fact this sheaf is equal to $\pi_S^{-1}\mathcal{F}$ on $(\Sch/S)_\etale$ and $\epsilon_S^{-1}\pi_S^{-1}\mathcal{F}$ on $(\Sch/S)_h$.


etale-cohomology-lemma-push-pull-h-etale

With notation as above. Let $f : X \to Y$ be a morphism of $(\Sch/S)_h$. Then there are commutative diagrams of topoi $$ \xymatrix{ \Sh((\Sch/X)_h) \ar[rr]_{f_{big, h}} \ar[d]_{\epsilon_X} & & \Sh((\Sch/Y)_h) \ar[d]^{\epsilon_Y} \\ \Sh((\Sch/X)_\etale) \ar[rr]^{f_{big, \etale}} & & \Sh((\Sch/Y)_\etale) } $$ and $$ \xymatrix{ \Sh((\Sch/X)_h) \ar[rr]_{f_{big, h}} \ar[d]_{a_X} & & \Sh((\Sch/Y)_h) \ar[d]^{a_Y} \\ \Sh(X_\etale) \ar[rr]^{f_{small}} & & \Sh(Y_\etale) } $$ with $a_X = \pi_X \circ \epsilon_X$ and $a_Y = \pi_X \circ \epsilon_X$.


etale-cohomology-lemma-compare-h-etale

Consider the comparison morphism $\epsilon : (\Sch/S)_h \to (\Sch/S)_\etale$. Let $\mathcal{P}$ denote the class of proper morphisms. For $X$ in $(\Sch/S)_\etale$ denote $\mathcal{A}'_X \subset \textit{Ab}((\Sch/X)_\etale)$ the full subcategory consisting of sheaves of the form $\pi_X^{-1}\mathcal{F}$ where $\mathcal{F}$ is a torsion abelian sheaf on $X_\etale$ Then Cohomology on Sites, Properties (\ref{sites-cohomology-item-base-change-P}), (\ref{sites-cohomology-item-restriction-A}), (\ref{sites-cohomology-item-A-sheaf}), (\ref{sites-cohomology-item-A-and-P}), and (\ref{sites-cohomology-item-refine-tau-by-P}) of Cohomology on Sites, Situation \ref{sites-cohomology-situation-compare} hold.


etale-cohomology-lemma-V-C-all-n-etale-h

With notation as above. \begin{enumerate} \item For $X \in \Ob((\Sch/S)_{h})$ and an abelian torsion sheaf $\mathcal{F}$ on $X_\etale$ we have $\epsilon_{X, *}a_X^{-1}\mathcal{F} = \pi_X^{-1}\mathcal{F}$ and $R^i\epsilon_{X, *}(a_X^{-1}\mathcal{F}) = 0$ for $i > 0$. \item For a proper morphism $f : X \to Y$ in $(\Sch/S)_h$ and abelian torsion sheaf $\mathcal{F}$ on $X$ we have $a_Y^{-1}(R^if_{small, *}\mathcal{F}) = R^if_{big, h, *}(a_X^{-1}\mathcal{F})$ for all $i$. \item For a scheme $X$ and $K$ in $D^+(X_\etale)$ with torsion cohomology sheaves the map $\pi_X^{-1}K \to R\epsilon_{X, *}(a_X^{-1}K)$ is an isomorphism. \item For a proper morphism $f : X \to Y$ of schemes and $K$ in $D^+(X_\etale)$ with torsion cohomology sheaves we have $a_Y^{-1}(Rf_{small, *}K) = Rf_{big, h, *}(a_X^{-1}K)$. \end{enumerate}


etale-cohomology-lemma-cohomological-descent-etale-h

Let $X$ be a scheme. For $K \in D^+(X_\etale)$ with torsion cohomology sheaves the map $$ K \longrightarrow Ra_{X, *}a_X^{-1}K $$ is an isomorphism with $a_X : \Sh((\Sch/X)_h) \to \Sh(X_\etale)$ as above.


etale-cohomology-lemma-glue-etale-sheaf-section

Let $f : X \to Y$ be a morphism of schemes which has a section. Then the functor $$ \Sh(Y_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{X \to Y\} $$ sending $\mathcal{G}$ in $\Sh(Y_\etale)$ to the canonical descent datum is an equivalence of categories.


etale-cohomology-lemma-glue-etale-sheaf-integral-surjective

Let $f : X \to Y$ be a surjective integral morphism of schemes. The functor $$ \Sh(Y_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{X \to Y\} $$ is an equivalence of categories.


etale-cohomology-lemma-glue-etale-sheaf-proper-surjective

Let $f : X \to Y$ be a surjective proper morphism of schemes. The functor $$ \Sh(Y_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{X \to Y\} $$ is an equivalence of categories.


etale-cohomology-lemma-glue-etale-sheaf-check-after-base-change

Let $f : X \to Y$ be a morphism of schemes. Let $Z \to Y$ be a surjective integral morphism of schemes or a surjective proper morphism of schemes. If the functors $$ \Sh(Z_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{X \times_Y Z \to Z\} $$ and $$ \Sh((Z \times_Y Z)_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt } \{X \times_Y (Z \times_Y Z) \to Z \times_Y Z\} $$ are equivalences of categories, then $$ \Sh(Y_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{X \to Y\} $$ is an equivalence.


etale-cohomology-lemma-glue-etale-sheaf-fppf-cover

Let $f : X \to Y$ be a morphism of schemes which is surjective, flat, locally of finite presentation. The functor $$ \Sh(Y_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{X \to Y\} $$ is an equivalence of categories.


etale-cohomology-lemma-glue-etale-sheaf-modification

Let $f : X' \to X$ be a proper morphism of schemes. Let $i : Z \to X$ be a closed immersion. Set $E = Z \times_X X'$. Picture $$ \xymatrix{ E \ar[d]_g \ar[r]_j & X' \ar[d]^f \\ Z \ar[r]^i & X } $$ If $f$ is an isomorphism over $X \setminus Z$, then the functor $$ \Sh(X_\etale) \longrightarrow \Sh(X'_\etale) \times_{\Sh(E_\etale)} \Sh(Z_\etale) $$ is an equivalence of categories.


etale-cohomology-lemma-blow-up-square-cohomological-descent

Let $X$ be a scheme and let $Z \subset X$ be a closed subscheme cut out by a quasi-coherent ideal of finite type. Consider the corresponding blow up square $$ \xymatrix{ E \ar[d]_\pi \ar[r]_j & X' \ar[d]^b \\ Z \ar[r]^i & X } $$ For $K \in D^+(X_\etale)$ with torsion cohomology sheaves we have a distinguished triangle $$ K \to Ri_*(K|_Z) \oplus Rb_*(K|_{X'}) \to Rc_*(K|_E) \to K[1] $$ in $D(X_\etale)$ where $c = i \circ \pi = b \circ j$.


etale-cohomology-lemma-blow-up-square-h

With notation as above, if $K$ is in the essential image of $R\epsilon_*$, then the maps $c^K_{X, Z, X', E}$ of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-c-square} are quasi-isomorphisms.


etale-cohomology-lemma-refine-check-h

Let $K$ be an object of $D^+((\Sch/S)_{fppf})$. Then $K$ is in the essential image of $R\epsilon_* : D((\Sch/S)_h) \to D((\Sch/S)_{fppf})$ if and only if $c^K_{X, X', Z, E}$ is a quasi-isomorphism for every almost blow up square as in More on Flatness, Examples \ref{flat-example-one-generator} and \ref{flat-example-two-generators}.


etale-cohomology-lemma-h-sheaf-colim-F

Let $p$ be a prime number. Let $S$ be a scheme over $\mathbf{F}_p$. Consider the sheaf $\mathcal{O}^{perf} = \colim_F \mathcal{O}$ on $(\Sch/S)_{fppf}$. Then $\mathcal{O}^{perf}$ is in the essential image of $R\epsilon_* : D((\Sch/S)_h) \to D((\Sch/S)_{fppf})$.


etale-cohomology-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.