pione-lemma-finite-etale-invariant-over-proper

Let $k \subset k'$ be an extension of algebraically closed fields. Let $X$ be a proper scheme over $k$. Then the functor $$ U \longmapsto U_{k'} $$ is an equivalence of categories between schemes finite \'etale over $X$ and schemes finite \'etale over $X_{k'}$.


pione-lemma-dense-faithful

Let $f : X \to Y$ be a morphism of schemes. If $f(X)$ is dense in $Y$ then the base change functor $\textit{F\'Et}_Y \to \textit{F\'Et}_X$ is faithful.


pione-lemma-same-etale-extensions

Let $(A, \mathfrak m)$ be a local ring. Set $X = \Spec(A)$ and let $U = X \setminus \{\mathfrak m\}$. If the punctured spectrum of the strict henselization of $A$ is connected, then $$ \textit{F\'Et}_X \longrightarrow \textit{F\'Et}_U,\quad Y \longmapsto Y \times_X U $$ is a fully faithful functor.


pione-lemma-quasi-compact-dense-open-connected-at-infinity-Noetherian

Let $X$ be a scheme. Let $U \subset X$ be a dense open. Assume \begin{enumerate} \item the underlying topological space of $X$ is Noetherian, and \item for every $x \in X \setminus U$ the punctured spectrum of the strict henselization of $\mathcal{O}_{X, x}$ is connected. \end{enumerate} Then $\textit{F\'Et}_X \to \textit{F\'et}_U$ is fully faithful.


pione-lemma-retrocompact-dense-open-connected-at-infinity-closed

Let $X$ be a scheme. Let $U \subset X$ be a dense open. Assume \begin{enumerate} \item $U \to X$ is quasi-compact, \item every point of $X \setminus U$ is closed, and \item for every $x \in X \setminus U$ the punctured spectrum of the strict henselization of $\mathcal{O}_{X, x}$ is connected. \end{enumerate} Then $\textit{F\'Et}_X \to \textit{F\'et}_U$ is fully faithful.


pione-lemma-quasi-compact-dense-open-connected-at-infinity

Let $X$ be a scheme. Let $U \subset X$ be a dense open. Assume \begin{enumerate} \item every quasi-compact open of $X$ has finitely many irreducible components, \item for every $x \in X \setminus U$ the punctured spectrum of the strict henselization of $\mathcal{O}_{X, x}$ is connected. \end{enumerate} Then $\textit{F\'Et}_X \to \textit{F\'et}_U$ is fully faithful.


pione-lemma-local-exact-sequence

Let $(A, \mathfrak m)$ be a local ring. Set $X = \Spec(A)$ and $U = X \setminus \{\mathfrak m\}$. Let $U^{sh}$ be the punctured spectrum of the strict henselization $A^{sh}$ of $A$. Assume $U$ is quasi-compact and $U^{sh}$ is connected. Then the sequence $$ \pi_1(U^{sh}, \overline{u}) \to \pi_1(U, \overline{u}) \to \pi_1(X, \overline{u}) \to 1 $$ is exact in the sense of Lemma \ref{lemma-functoriality-galois-ses} part (1).


pione-lemma-irreducible-geometrically-unibranch

Let $X$ be an irreducible, geometrically unibranch scheme. For any nonempty open $U \subset X$ the canonical map $$ \pi_1(U, \overline{u}) \longrightarrow \pi_1(X, \overline{u}) $$ is surjective. The map (\ref{equation-inclusion-generic-point}) $\pi_1(\eta, \overline{\eta}) \to \pi_1(X, \overline{\eta})$ is surjective as well.


pione-lemma-unramified-in-L

In the situation above the following are equivalent \begin{enumerate} \item $X$ is unramified in $L$, \item $Y \to X$ is \'etale, and \item $Y \to X$ is finite \'etale. \end{enumerate}


pione-lemma-finite-etale-covering-normal-unramified

Let $X$ be a normal integral scheme with function field $K$. Let $Y \to X$ be a finite \'etale morphism. If $Y$ is connected, then $Y$ is an integral normal scheme and $Y$ is the normalization of $X$ in the function field of $Y$.


pione-lemma-local-exact-sequence-normal

Let $(A, \mathfrak m)$ be a normal local ring. Set $X = \Spec(A)$. Let $A^{sh}$ be the strict henselization of $A$. Let $K$ and $K^{sh}$ be the fraction fields of $A$ and $A^{sh}$. Then the sequence $$ \pi_1(\Spec(K^{sh})) \to \pi_1(\Spec(K)) \to \pi_1(X) \to 1 $$ is exact in the sense of Lemma \ref{lemma-functoriality-galois-ses} part (1).


pione-lemma-get-algebraic-closure

Let $A$ be a normal domain whose fraction field $K$ is separably algebraically closed. Let $\mathfrak p \subset A$ be a nonzero prime ideal. Then the residue field $\kappa(\mathfrak p)$ is algebraically closed.


pione-lemma-normal-local-domain-separablly-closed-fraction-field

A normal local ring with separably closed fraction field is strictly henselian.


pione-lemma-inertia-base-change

Let $G$ be a finite group acting on a ring $R$. Let $R^G \to A$ be a ring map. Let $\mathfrak q' \subset A \otimes_{R^G} R$ be a prime lying over the prime $\mathfrak q \subset R$. Then $$ I_\mathfrak q = \{\sigma \in G \mid \sigma(\mathfrak q) = \mathfrak q\text{ and } \sigma \bmod \mathfrak q = \text{id}_{\kappa(\mathfrak q)}\} $$ is equal to $$ I_{\mathfrak q'} = \{\sigma \in G \mid \sigma(\mathfrak q') = \mathfrak q'\text{ and } \sigma \bmod \mathfrak q' = \text{id}_{\kappa(\mathfrak q')}\} $$


pione-lemma-inertia-invariants-etale

Let $G$ be a finite group acting on a ring $R$. Let $\mathfrak q \subset R$ be a prime. Set $$ I = \{\sigma \in G \mid \sigma(\mathfrak q) = \mathfrak q \text{ and } \sigma \bmod \mathfrak q = \text{id}_\mathfrak q\} $$ Then $R^G \to R^I$ is \'etale at $R^I \cap \mathfrak q$.


pione-lemma-inertial-invariants-unramified

Let $A$ be a normal domain with fraction field $K$. Let $L/K$ be a (possibly infinite) Galois extension. Let $G = \text{Gal}(L/K)$ and let $B$ be the integral closure of $A$ in $L$. Let $\mathfrak q \subset B$. Set $$ I = \{\sigma \in G \mid \sigma(\mathfrak q) = \mathfrak q \text{ and } \sigma \bmod \mathfrak q = \text{id}_{\kappa(\mathfrak q)}\} $$ Then $(B^I)_{B^I \cap \mathfrak q}$ is a filtered colimit of \'etale $A$-algebras.


pione-lemma-identify-inertia

In the situation described above, via the isomorphism $\pi_1(U) = \text{Gal}(K^{sep}/K)$ the diagram (\ref{equation-inertia-diagram-pione}) translates into the diagram $$ \xymatrix{ I \ar[r] \ar[rd]_1 & D \ar[d] \ar[r] & \text{Gal}(K^{sep}/K) \ar[d] \\ & \text{Gal}(\kappa(\mathfrak m^{sh})/\kappa) \ar[r] & \text{Gal}(M/K) } $$ where $K^{sep}/M/K$ is the maximal subextension unramified with respect to $A$. Moreover, the vertical arrows are surjective, the kernel of the left vertical arrow is $I$ and the kernel of the right vertical arrow is the smallest closed normal subgroup of $\text{Gal}(K^{sep}/K)$ containing $I$.


pione-lemma-structure-decomposition

Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$ be a (possibly infinite) Galois extension. Let $B$ be the integral closure of $A$ in $L$. Let $\mathfrak m$ be a maximal ideal of $B$. Let $G = \text{Gal}(L/K)$, $D = \{\sigma \in G \mid \sigma(\mathfrak m) = \mathfrak m\}$, and $I = \{\sigma \in D \mid \sigma \bmod \mathfrak m = \text{id}_{\kappa(\mathfrak m)}\}$. The decomposition group $D$ fits into a canonical exact sequence $$ 1 \to I \to D \to \text{Aut}(\kappa(\mathfrak m)/\kappa_A) \to 1 $$ The inertia group $I$ fits into a canonical exact sequence $$ 1 \to P \to I \to I_t \to 1 $$ such that \begin{enumerate} \item $P$ is a normal subgroup of $D$, \item $P$ is a pro-$p$-group if the characteristic of $\kappa_A$ is $p > 1$ and $P = \{1\}$ if the characteristic of $\kappa_A$ is zero, \item there is a multiplicatively directed $S \subset \mathbf{N}$ such that $\kappa(\mathfrak m)$ contains a primitive $n$th root of unity for each $n \in S$ (elements of $S$ are prime to $p$), \item there exists a canonical surjective map $$ \theta_{can} : I \to \lim_{n \in S} \mu_n(\kappa(\mathfrak m)) $$ whose kernel is $P$, which satisfies $\theta_{can}(\tau \sigma \tau^{-1}) = \tau(\theta_{can}(\sigma))$ for $\tau \in D$, $\sigma \in I$, and which induces an isomorphism $I_t \to \lim_{n \in S} \mu_n(\kappa(\mathfrak m))$. \end{enumerate}


pione-lemma-limit

Let $I$ be a directed set. Let $X_i$ be an inverse system of quasi-compact and quasi-separated schemes over $I$ with affine transition morphisms. Let $X = \lim X_i$ as in Limits, Section \ref{limits-section-limits}. Then there is an equivalence of categories $$ \colim \textit{F\'Et}_{X_i} = \textit{F\'Et}_X $$ If $X_i$ is connected for all sufficiently large $i$ and $\overline{x}$ is a geometric point of $X$, then $$ \pi_1(X, \overline{x}) = \lim \pi_1(X_i, \overline{x}) $$


pione-lemma-perfection

Let $k$ be a field with perfection $k^{perf}$. Let $X$ be a connected scheme over $k$. Then $X_{k^{perf}}$ is connected and $\pi_1(X_{k^{perf}}) \to \pi_1(X)$ is an isomorphism.


pione-lemma-stein-factorization-etale

\begin{reference} \cite[Expose X, Proposition 1.2, p. 262]{SGA1}. \end{reference} Let $f : X \to S$ be a proper morphism of schemes. Let $X \to S' \to S$ be the Stein factorization of $f$, see More on Morphisms, Theorem \ref{more-morphisms-theorem-stein-factorization-general}. If $f$ is of finite presentation, flat, with geometrically reduced fibres, then $S' \to S$ is finite \'etale.


pione-lemma-specialization-map-base-change

Consider a commutative diagram $$ \xymatrix{ Y \ar[d]_g \ar[r] & X \ar[d]^f \\ T \ar[r] & S } $$ of schemes where $f$ and $g$ are proper with geometrically connected fibres. Let $t' \leadsto t$ be a specialization of points in $T$ and consider a specialization map $sp : \pi_1(Y_{\overline{t}'}) \to \pi_1(Y_{\overline{t}})$ as above. Then there is a commutative diagram $$ \xymatrix{ \pi_1(Y_{\overline{t}'}) \ar[r]_{sp} \ar[d] & \pi_1(Y_{\overline{t}}) \ar[d] \\ \pi_1(X_{\overline{s}'}) \ar[r]^{sp} & \pi_1(X_{\overline{s}}) } $$ of specialization maps where $\overline{s}$ and $\overline{s}'$ are the images of $\overline{t}$ and $\overline{t}'$.


pione-lemma-specialization-map-discrete-valuation-ring

Let $f : X \to S$ be a proper morphism with geometrically connected fibres. Let $s' \leadsto s$ be a specialization of points of $S$ and let $sp : \pi_1(X_{\overline{s}'}) \to \pi_1(X_{\overline{s}})$ be a specialization map. If $S$ is Noetherian, then there exists a strictly henselian discrete valuation ring $R$ over $S$ such that $sp$ is isomorphic to $sp_R$ defined above.


pione-lemma-restriction-fully-faithful

Let $X$ be a Noetherian scheme and let $Y \subset X$ be a closed subscheme with ideal sheaf $\mathcal{I} \subset \mathcal{O}_X$. Assume the completion functor $$ \textit{Coh}(\mathcal{O}_X) \longrightarrow  \textit{Coh}(X, \mathcal{I}),\quad \mathcal{F} \longmapsto \mathcal{F}^\wedge $$ is fully faithful on the full subcategory of finite locally free objects (see above). Then the restriction functor $\textit{F\'Et}_X \to \textit{F\'Et}_Y$ is fully faithful.


pione-lemma-restriction-equivalence

Let $X$ be a Noetherian scheme and let $Y \subset X$ be a closed subscheme with ideal sheaf $\mathcal{I} \subset \mathcal{O}_X$. Assume the completion functor $$ \textit{Coh}(\mathcal{O}_X) \longrightarrow  \textit{Coh}(X, \mathcal{I}),\quad \mathcal{F} \longmapsto \mathcal{F}^\wedge $$ is an equivalence on full subcategories of finite locally free objects (see above). Then the restriction functor $\textit{F\'Et}_X \to \textit{F\'Et}_Y$ is an equivalence.


pione-lemma-restriction-fully-faithful-general

Let $X$ be a Noetherian scheme and let $Y \subset X$ be a closed subscheme with ideal sheaf $\mathcal{I} \subset \mathcal{O}_X$. Let $\mathcal{V}$ be the set of open subschemes $V \subset X$ containing $Y$ ordered by reverse inclusion. Assume the completion functor $$ \colim_\mathcal{V} \textit{Coh}(\mathcal{O}_V) \longrightarrow \textit{Coh}(X, \mathcal{I}), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge $$ defines is fully faithful on the full subcategory of finite locally free objects (see above). Then the restriction functor $\colim_\mathcal{V} \textit{F\'Et}_V \to \textit{F\'Et}_Y$ is fully faithful.


pione-lemma-restriction-equivalence-general

Let $X$ be a Noetherian scheme and let $Y \subset X$ be a closed subscheme with ideal sheaf $\mathcal{I} \subset \mathcal{O}_X$. Let $\mathcal{V}$ be the set of open subschemes $V \subset X$ containing $Y$ ordered by reverse inclusion. Assume the completion functor $$ \colim_\mathcal{V} \textit{Coh}(\mathcal{O}_V) \longrightarrow \textit{Coh}(X, \mathcal{I}), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge $$ defines an equivalence of the full subcategories of finite locally free objects (see explanation above). Then the restriction functor $$ \colim_\mathcal{V} \textit{F\'Et}_V \to \textit{F\'Et}_Y $$ is an equivalence.


pione-lemma-restriction-faithful

Let $X$ be a scheme and let $Y \subset X$ be a closed subscheme. If every connected component of $X$ meets $Y$, then the restriction functor $\textit{F\'Et}_X \to \textit{F\'Et}_Y$ is faithful.


pione-lemma-restriction-fully-faithful-special

Let $X$ be a Noetherian scheme and let $Y \subset X$ be a closed subscheme. Let $Y_n \subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$. Assume one of the following holds \begin{enumerate} \item $X$ is quasi-affine and $\Gamma(X, \mathcal{O}_X) \to \lim \Gamma(Y_n, \mathcal{O}_{Y_n})$ is an isomorphism, or \item $X$ has an ample invertible module $\mathcal{L}$ and $\Gamma(X, \mathcal{L}^{\otimes m}) \to \lim \Gamma(Y_n, \mathcal{L}^{\otimes m}|_{Y_n})$ is an isomorphism for all $m \gg 0$, or \item for every finite locally free $\mathcal{O}_X$-module $\mathcal{E}$ the map $\Gamma(X, \mathcal{E}) \to \lim \Gamma(Y_n, \mathcal{E}|_{Y_n})$ is an isomorphism. \end{enumerate} Then the restriction functor $\textit{F\'Et}_X \to \textit{F\'Et}_Y$ is fully faithful.


pione-lemma-fill-in-missing

In Situation \ref{situation-local-lefschetz}. Let $V \to U$ be a finite morphism.  Let $A^\wedge$ be the $\mathfrak m$-adic completion of $A$, let $X' = \Spec(A^\wedge)$ and let $U'$ and $V'$ be the base changes of $U$ and $V$ to $X'$. If $Y' \to X'$ is a finite morphism such that $V' = Y' \times_{X'} U'$, then there exists a finite morphism $Y \to X$ such that $V = Y \times_X U$ and $Y' = Y \times_X X'$.


pione-lemma-fully-faithful-henselian-completion

In Situation \ref{situation-local-lefschetz} assume $A$ is henselian or more generally that $(A, (f))$ is a henselian pair. Let $A^\wedge$ be the $\mathfrak m$-adic completion of $A$, let $X' = \Spec(A^\wedge)$ and let $U'$ and $U'_0$ be the base changes of $U$ and $U_0$ to $X'$. If $\textit{F\'Et}_{U'} \to \textit{F\'Et}_{U'_0}$ is fully faithful, then $\textit{F\'Et}_U \to \textit{F\'Et}_{U_0}$ is fully faithful.


pione-lemma-fully-faithful-simple

In Situation \ref{situation-local-lefschetz}. Assume \begin{enumerate} \item[(a)] $A$ has a dualizing complex, \item[(b)] the pair $(A, (f))$ is henselian, \item[(c)] one of the following is true \begin{enumerate} \item[(i)] $A_f$ is $(S_2)$ and every irreducible component of $X$ not contained in $X_0$ has dimension $\geq 3$, or \item[(ii)] for every prime $\mathfrak p \subset A$, $f \not \in \mathfrak p$ we have $\text{depth}(A_\mathfrak p) + \dim(A/\mathfrak p) > 2$. \end{enumerate} \end{enumerate} Then the restriction functor $\textit{F\'Et}_U \longrightarrow \textit{F\'Et}_{U_0}$ is fully faithful.


pione-lemma-fully-faithful-minimal

\begin{reference} \cite[Corollary 1.11]{Bhatt-local} \end{reference} In Situation \ref{situation-local-lefschetz}. Assume \begin{enumerate} \item $H^1_\mathfrak m(A)$ and $H^2_\mathfrak m(A)$ are annihilated by a power of $f$, and \item $A$ is henselian or more generally $(A, (f))$ is a henselian pair. \end{enumerate} Then the restriction functor $\textit{F\'Et}_U \longrightarrow \textit{F\'Et}_{U_0}$ is fully faithful.


pione-lemma-sections-over-punctured-spec

Let $(A, \mathfrak m)$ be a Noetherian local ring. Set $X = \Spec(A)$ and let $U = X \setminus \{\mathfrak m\}$. Let $\pi : Y \to X$ be a finite morphism such that $\text{depth}(\mathcal{O}_{Y, y}) \geq 2$ for all closed points $y \in Y$. Then $Y$ is the spectrum of $B = \mathcal{O}_Y(\pi^{-1}(U))$.


pione-lemma-reformulate-purity

Let $(A, \mathfrak m)$ be a Noetherian local ring. Set $X = \Spec(A)$ and let $U = X \setminus \{\mathfrak m\}$. Let $V$ be finite \'etale over $U$. Assume $A$ has depth $\geq 2$. The following are equivalent \begin{enumerate} \item $V = Y \times_X U$ for some $Y \to X$ finite \'etale, \item $B = \Gamma(V, \mathcal{O}_V)$ is finite \'etale over $A$. \end{enumerate}


pione-lemma-reformulate-purity-normal

Let $(A, \mathfrak m)$ be a Noetherian local ring. Set $X = \Spec(A)$ and let $U = X \setminus \{\mathfrak m\}$. Assume $A$ is normal of dimension $\geq 2$. The functor $$ \textit{F\'Et}_U \longrightarrow \left\{ \begin{matrix} \text{finite normal }A\text{-algebras }B\text{ such} \\ \text{that }\Spec(B) \to X\text{ is \'etale over }U \end{matrix} \right\}, \quad V \longmapsto \Gamma(V, \mathcal{O}_V) $$ is an equivalence. Moreover, $V = Y \times_X U$ for some $Y \to X$ finite \'etale if and only if $B = \Gamma(V, \mathcal{O}_V)$ is finite \'etale over $A$.


pione-lemma-purity-and-completion

Let $(A, \mathfrak m)$ be a Noetherian local ring. Set $X = \Spec(A)$ and let $U = X \setminus \{\mathfrak m\}$. Let $V$ be finite \'etale over $U$. Let $A^\wedge$ be the $\mathfrak m$-adic completion of $A$, let $X' = \Spec(A^\wedge)$ and let $U'$ and $V'$ be the base changes of $U$ and $V$ to $X'$. The following are equivalent \begin{enumerate} \item $V = Y \times_X U$ for some $Y \to X$ finite \'etale, and \item $V' = Y' \times_{X'} U'$ for some $Y' \to X'$ finite \'etale. \end{enumerate}


pione-lemma-lift-purity-general

In Situation \ref{situation-local-lefschetz}. Let $V$ be finite \'etale over $U$. Assume \begin{enumerate} \item $H^1_\mathfrak m(A)$ and $H^2_\mathfrak m(A)$ are annihilated by a power of $f$, \item $V_0 = V \times_U U_0$ is equal to $Y_0 \times_{X_0} U_0$ for some $Y_0 \to X_0$ finite \'etale. \end{enumerate} Then $V = Y \times_X U$ for some $Y \to X$ finite \'etale.


pione-lemma-lift-purity

In Situation \ref{situation-local-lefschetz}. Let $V$ be finite \'etale over $U$. Assume \begin{enumerate} \item $A$ has depth $\geq 3$, \item $V_0 = V \times_U U_0$ is equal to $Y_0 \times_{X_0} U_0$ for some $Y_0 \to X_0$ finite \'etale. \end{enumerate} Then $V = Y \times_X U$ for some $Y \to X$ finite \'etale.


pione-lemma-find-point-codim-1

Let $(A, \mathfrak m)$ be a Noetherian local ring with $\dim(A) \geq 1$. Let $f \in \mathfrak m$. Then there exist a $\mathfrak p \in V(f)$ with $\dim(A_\mathfrak p) = 1$.


pione-lemma-ramification-quasi-finite-flat

Let $f : X \to Y$ be a morphism of locally Noetherian schemes. Let $x \in X$. Assume \begin{enumerate} \item $f$ is flat, \item $f$ is quasi-finite at $x$, \item $x$ is not a generic point of an irreducible component of $X$, \item for specializations $x' \leadsto x$ with $\dim(\mathcal{O}_{X, x'}) = 1$ our $f$ is unramified at $x'$. \end{enumerate} Then $f$ is \'etale at $x$.


pione-lemma-local-purity

Let $(A, \mathfrak m)$ be a regular local ring of dimension $d \geq 2$. Set $X = \Spec(A)$ and $U = X \setminus \{\mathfrak m\}$. Then \begin{enumerate} \item the functor $\textit{F\'Et}_X \to \textit{F\'Et}_U$ is essentially surjective, i.e., purity holds for $A$, \item any finite $A \to B$ with $B$ normal which induces a finite \'etale morphism on punctured spectra is \'etale. \end{enumerate}


pione-lemma-purity

\begin{reference} \cite{Nagata-Purity} and \cite[Exp. X, Thm. 3.1]{SGA1} \end{reference} \begin{history} This result was first stated and proved by Zariski in geometric form in \cite{Zariski-Purity}. The generalization to nonperfect ground fields by Nagata was published as the next article in the same volume of the Proceedings of the National Academy of Sciences of the United States of America in \cite{Nagata-Remarks-Purity}. In the following year Nagata proved the result for Noetherian local rings in \cite{Nagata-Purity}. His proof uses a result of Chow which is a Bertini theorem for complete local domains, see \cite{Chow-Bertini}; the history of Bertini's theorems is discussed in Kleiman's historical article \cite{Kleiman-Bertini}. A few years later a completely different proof was found by Auslander, see \cite{Auslander-Purity}. \end{history} Let $f : X \to Y$ be a morphism of locally Noetherian schemes. Let $x \in X$ and set $y = f(x)$. Assume \begin{enumerate} \item $\mathcal{O}_{X, x}$ is normal, \item $\mathcal{O}_{Y, y}$ is regular, \item $f$ is quasi-finite at $x$, \item $\dim(\mathcal{O}_{X, x}) = \dim(\mathcal{O}_{Y, y}) \geq 1$ \item for specializations $x' \leadsto x$ with $\dim(\mathcal{O}_{X, x'}) = 1$ our $f$ is unramified at $x'$. \end{enumerate} Then $f$ is \'etale at $x$.


pione-lemma-extend-S2

Let $j : U \to X$ be an open immersion of locally Noetherian schemes such that $\text{depth}(\mathcal{O}_{X, x}) \geq 2$ for $x \not \in U$. Let $\pi : V \to U$ be finite \'etale. Then \begin{enumerate} \item $\mathcal{B} = j_*\pi_*\mathcal{O}_V$ is a reflexive coherent $\mathcal{O}_X$-algebra, set $Y = \underline{\Spec}_X(\mathcal{B})$, \item $Y \to X$ is the unique finite morphism such that $V = Y \times_X U$ and $\text{depth}(\mathcal{O}_{Y, y}) \geq 2$ for $y \in Y \setminus V$, \item $Y \to X$ is \'etale at $y$ if and only if $Y \to X$ is flat at $y$, and \item $Y \to X$ is \'etale if and only if $\mathcal{B}$ is finite locally free as an $\mathcal{O}_X$-module. \end{enumerate} Moreover, (a) the construction of $\mathcal{B}$ and $Y \to X$ commutes with base change by flat morphisms $X' \to X$ of locally Noetherian schemes, and (b) if $V' \to U'$ is a finite \'etale morphism with $U \subset U' \subset X$ open which restricts to $V \to U$ over $U$, then there is a unique isomorphism $Y' \times_X U' = V'$ over $U'$.


pione-lemma-extend-pure

Let $j : U \to X$ be an open immersion of Noetherian schemes such that purity holds for $\mathcal{O}_{X, x}$ for all $x \not \in U$. Then $$ \textit{F\'Et}_X \longrightarrow \textit{F\'Et}_U $$ is essentially surjective.


pione-lemma-essentially-surjective-general-better

In Situation \ref{situation-local-lefschetz} assume \begin{enumerate} \item $A$ has a dualizing complex and is $f$-adically complete, \item one of the following is true \begin{enumerate} \item $A_f$ is $(S_2)$ and every irreducible component of $X$ not contained in $X_0$ has dimension $\geq 4$, or \item if $\mathfrak p \not \in V(f)$ and $V(\mathfrak p) \cap V(f) \not = \{\mathfrak m\}$, then $\text{depth}(A_\mathfrak p) + \dim(A/\mathfrak p) > 3$. \end{enumerate} \end{enumerate} Then the restriction functor $$ \colim_{U_0 \subset U' \subset U\text{ open}} \textit{F\'Et}_{U'} \longrightarrow \textit{F\'Et}_{U_0} $$ is an equivalence.


pione-lemma-essentially-surjective-general

In Situation \ref{situation-local-lefschetz} assume \begin{enumerate} \item $A$ is $f$-adically complete, \item $f$ is a nonzerodivisor, \item $H^1_\mathfrak m(A/fA)$ and $H^2_\mathfrak m(A/fA)$ are finite $A$-modules. \end{enumerate} Then the restriction functor $$ \colim_{U_0 \subset U' \subset U\text{ open}} \textit{F\'Et}_{U'} \longrightarrow \textit{F\'Et}_{U_0} $$ is an equivalence.


pione-lemma-equivalence-better

In Situation \ref{situation-local-lefschetz} assume \begin{enumerate} \item $A$ has a dualizing complex and is $f$-adically complete, \item one of the following is true \begin{enumerate} \item $A_f$ is $(S_2)$ and every irreducible component of $X$ not contained in $X_0$ has dimension $\geq 4$, or \item if $\mathfrak p \not \in V(f)$ and $V(\mathfrak p) \cap V(f) \not = \{\mathfrak m\}$, then $\text{depth}(A_\mathfrak p) + \dim(A/\mathfrak p) > 3$. \end{enumerate} \item for every maximal ideal $\mathfrak p \subset A_f$ purity holds for $(A_f)_\mathfrak p$. \end{enumerate} Then the restriction functor $\textit{F\'Et}_U \to \textit{F\'Et}_{U_0}$ is essentially surjective.


pione-lemma-equivalence

Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $f \in \mathfrak m$. Assume \begin{enumerate} \item $A$ is $f$-adically complete, \item $f$ is a nonzerodivisor, \item $H^1_\mathfrak m(A/fA)$ and $H^2_\mathfrak m(A/fA)$ are finite $A$-modules, \item for every maximal ideal $\mathfrak p \subset A_f$ purity holds for $(A_f)_\mathfrak p$. \end{enumerate} Then the restriction functor $\textit{F\'Et}_U \to \textit{F\'Et}_{U_0}$ is essentially surjective.


pione-lemma-purity-inherited-by-hypersurface-better

Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $f \in \mathfrak m$. Assume \begin{enumerate} \item $A$ has a dualizing complex and is $f$-adically complete, \item one of the following is true \begin{enumerate} \item $A_f$ is $(S_2)$ and every irreducible component of $X$ not contained in $X_0$ has dimension $\geq 4$, or \item if $\mathfrak p \not \in V(f)$ and $V(\mathfrak p) \cap V(f) \not = \{\mathfrak m\}$, then $\text{depth}(A_\mathfrak p) + \dim(A/\mathfrak p) > 3$. \end{enumerate} \item for every maximal ideal $\mathfrak p \subset A_f$ purity holds for $(A_f)_\mathfrak p$, and \item purity holds for $A$. \end{enumerate} Then purity holds for $A/fA$.


pione-lemma-purity-inherited-by-hypersurface

Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $f \in \mathfrak m$. Assume \begin{enumerate} \item $A$ is $f$-adically complete, \item $f$ is a nonzerodivisor, \item $H^1_\mathfrak m(A/fA)$ and $H^2_\mathfrak m(A/fA)$ are finite $A$-modules, \item for every maximal ideal $\mathfrak p \subset A_f$ purity holds for $(A_f)_\mathfrak p$, \item purity holds for $A$. \end{enumerate} Then purity holds for $A/fA$.


pione-lemma-fully-faithful-power-series-over-depth2

Let $(A, \mathfrak m)$ be a Noetherian local ring of depth $\geq 2$. Let $B = A[[x_1, \ldots, x_d]]$ with $d \geq 1$. Set $Y = \Spec(B)$ and $Y_0 = V(x_1, \ldots, x_d)$. For any open subscheme $V \subset Y$ with $V_0 = V \cap Y_0$ equal to $Y_0 \setminus \{\mathfrak m_B\}$ the restriction functor $$ \textit{F\'Et}_V \longrightarrow \textit{F\'Et}_{V_0} $$ is fully faithful.


pione-lemma-purity-power-series-over-depth2

\begin{slogan} Ramanujam-Samuel for finite \'etale covers \end{slogan} Let $(A, \mathfrak m)$ be a Noetherian local ring of depth $\geq 2$. Let $B = A[[x_1, \ldots, x_d]]$ with $d \geq 1$. For any open $V \subset Y = \Spec(B)$ which contains \begin{enumerate} \item any prime $\mathfrak q \subset B$ such that $\mathfrak q \cap A \not = \mathfrak m$, \item the prime $\mathfrak m B$ \end{enumerate} the functor $ \textit{F\'Et}_Y \to \textit{F\'Et}_V $ is an equivalence. In particular purity holds for $B$.


pione-lemma-purity-smooth-over-depth2

Let $f : X \to S$ be a morphism of schemes. Let $U \subset X$ be an open subscheme. Assume \begin{enumerate} \item $f$ is smooth, \item $S$ is Noetherian, \item for $s \in S$ with $\text{depth}(\mathcal{O}_{S, s}) \leq 1$ we have $X_s = U_s$, \item $U_s \subset X_s$ is dense for all $s \in S$. \end{enumerate} Then $\textit{F\'Et}_X \to \textit{F\'Et}_U$ is an equivalence.


pione-lemma-characterize-rational-singularity

Let $A$ be a Noetherian normal local domain of dimension $2$. Assume $A$ is Nagata, has a dualizing module $\omega_A$, and has a resolution of singularities $f : X \to \Spec(A)$. Let $\omega_X$ be as in Resolution of Surfaces, Remark \ref{resolve-remark-dualizing-setup}. If $\omega_X \cong \mathcal{O}_X(E)$ for some effective Cartier divisor $E \subset X$ supported on the exceptional fibre, then $A$ defines a rational singularity. If $f$ is a minimal resolution, then $E = 0$.


pione-lemma-key-purity-ramification

Let $f : X \to \Spec(A)$ be a finite type morphism. Let $x \in X$ be a point. Assume \begin{enumerate} \item $A$ is an excellent regular local ring, \item $\mathcal{O}_{X, x}$ is normal of dimension $2$, \item $f$ is \'etale outside of $\overline{\{x\}}$. \end{enumerate} Then $f$ is \'etale at $x$.


pione-lemma-purity-ramification

\begin{reference} This result for complex spaces can be found on page 170 of \cite{Fischer}. In general this is \cite[Theorem 2.4]{Zong} attributed to Gabber. \end{reference} Let $f : X \to Y$ be a morphism of locally Noetherian schemes. Let $x \in X$ and set $y = f(x)$. Assume \begin{enumerate} \item $\mathcal{O}_{X, x}$ is normal of dimension $\geq 1$, \item $\mathcal{O}_{Y, y}$ is regular, \item $f$ is locally of finite type, and \item for specializations $x' \leadsto x$ with $\dim(\mathcal{O}_{X, x'}) = 1$ our $f$ is \'etale at $x'$. \end{enumerate} Then $f$ is \'etale at $x$.


pione-lemma-structure-cohomology

Let $(A, \mathfrak m)$ be a regular local ring which contains a field. Let $f : V \to \Spec(A)$ be \'etale and quasi-compact. Assume that $\mathfrak m \not \in f(V)$ and assume that $g : V \to \Spec(A) \setminus \{\mathfrak m\}$ is affine. Then $H^i(V, \mathcal{O}_V)$, $i > 0$ is isomorphic to a direct sum of copies of the injective hull of the residue field of $A$.


pione-lemma-conclude

In the situation of Lemma \ref{lemma-structure-cohomology} assume that $H^i(V, \mathcal{O}_V) = 0$ for $i \geq \dim(A) - 1$. Then $V$ is affine.


pione-lemma-specialization-map-surjective

Let $f : X \to S$ be a flat proper morphism with geometrically connected fibres. Let $s' \leadsto s$ be a specialization. If $X_s$ is geometrically reduced, then the specialization map $sp : \pi_1(X_{\overline{s}'}) \to \pi_1(X_{\overline{s}})$ is surjective.


pione-lemma-pullback-tame-codim1

Let $X' \to X$ be a morphism of locally Noetherian schemes. Let $U \subset X$ be a dense open. Assume \begin{enumerate} \item $U' = f^{-1}(U)$ is dense open in $X'$, \item for every prime divisor $Z \subset X$ with $Z \cap U = \emptyset$ the local ring $\mathcal{O}_{X, \xi}$ of $X$ at the generic point $\xi$ of $Z$ is a discrete valuation ring, \item for every prime divisor $Z' \subset X'$ with $Z' \cap U' = \emptyset$ the local ring $\mathcal{O}_{X', \xi'}$ of $X'$ at the generic point $\xi'$ of $Z'$ is a discrete valuation ring, \item if $\xi' \in X'$ is as in (3), then $\xi = f(\xi')$ is as in (2). \end{enumerate} Then if $f : Y \to U$ is finite \'etale and $Y$ is unramified, resp.\ tamely ramified over $X$ in codimension $1$, then $Y' = Y \times_X X' \to U'$ is finite \'etale and $Y'$ is unramified, resp.\ tamely ramified over $X'$ in codimension $1$.


pione-lemma-purity-one-divisor

Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be an effective Cartier divisor such that $D$ is a regular scheme. Let $Y \to X \setminus D$ be a finite \'etale morphism. If $Y$ is unramified over $X$ in codimension $1$, then there exists a finite \'etale morphism $Y' \to X$ whose restriction to $X \setminus D$ is $Y$.


pione-lemma-abhyankar-one-divisor

Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be an effective Cartier divisor such that $D$ is a regular scheme. Let $Y \to X \setminus D$ be a finite \'etale morphism. If $Y$ is tamely ramified over $X$ in codimension $1$, then \'etale locally on $X$ the morphism $Y \to X$ is as given as a finite disjoint union of standard tamely ramified morphisms as described in Example \ref{example-tamely-ramified}.


pione-lemma-extend-tame-covering-normal

In the situation of Lemma \ref{lemma-abhyankar-one-divisor} the normalization of $X$ in $Y$ is a finite locally free morphism $\pi : Y' \to X$ such that \begin{enumerate} \item the restriction of $Y'$ to $X \setminus D$ is isomorphic to $Y$, \item $D' = \pi^{-1}(D)_{red}$ is an effective Cartier divisor on $Y'$, and \item $D'$ is a regular scheme. \end{enumerate} Moreover, \'etale locally on $X$ the morphism $Y' \to X$ is a finite disjoint union of morphisms $$ \Spec(A[x]/(x^e - f)) \to \Spec(A) $$ where $A$ is a Noetherian ring, $f \in A$ is a nonzerodivisor with $A/fA$ regular, and $e \geq 1$ is invertible in $A$.


pione-lemma-tame-covering-split

In the situation of Lemma \ref{lemma-abhyankar-one-divisor} let $Y' \to X$ be as in Lemma \ref{lemma-extend-tame-covering-normal}. Let $R$ be a discrete valuation ring with fraction field $K$. Let $$ t : \Spec(R) \to X $$ be a morphism such that the scheme theoretic inverse image $t^{-1}D$ is the reduced closed point of $\Spec(R)$. \begin{enumerate} \item If $t|_{\Spec(K)}$ lifts to a point of $Y$, then we get a lift $t' : \Spec(R) \to Y'$ such that $Y' \to X$ is \'etale along $t'(\Spec(R))$. \item If $\Spec(K) \times_X Y$ is isomorphic to a disjoint union of copies of $\Spec(K)$, then $Y' \to X$ is finite \'etale over an open neighbourhood of $t(\Spec(R))$. \end{enumerate}


pione-lemma-extend-covering

Let $S$ be an integral normal Noetherian scheme with generic point $\eta$. Let $f : X \to S$ be a smooth morphism with geometrically connected fibres. Let $\sigma : S \to X$ be a section of $f$. Let $Z \to X_\eta$ be a finite \'etale Galois cover (Section \ref{section-finite-etale-under-galois}) with group $G$ of order invertible on $S$ such that $Z$ has a $\kappa(\eta)$-rational point mapping to $\sigma(\eta)$. Then there exists a finite \'etale Galois cover $Y \to X$ with group $G$ whose restriction to $X_\eta$ is $Z$.


pione-lemma-lower-invariant

In the situation above, if $NF(C, \varphi, \tau) > 0$, then there exist an \'etale $k$-algebra map $\varphi'$ and a surjective $k$-algebra map $\tau'$ fitting into the commutative diagram $$ \xymatrix{ & B \\ C \ar[r] & C/\varphi'(I)C \ar[u]_{\tau'} \\ k[x_1, \ldots, x_n] \ar[u]^{\varphi'} \ar[r] & A \ar[u] \ar@/_3em/[uu]_\pi } $$ with $NF(C, \varphi', \tau') < NF(C, \varphi, \tau)$.


pione-lemma-dominate-affine-space

Let $k$ be a field of characteristic $p > 0$. Let $Z \subset \mathbf{A}^n_k$ be a closed subscheme. Let $Y \to Z$ be finite \'etale. There exists a finite \'etale morphism $f : U \to \mathbf{A}^n_k$ such that there is an open and closed immersion $Y \to f^{-1}(Z)$ over $Z$.


pione-proposition-galois

\begin{reference} This is a weak version of \cite[Expos\'e V]{SGA1}. The proof is borrowed from \cite[Theorem 7.2.5]{BS}. \end{reference} Let $(\mathcal{C}, F)$ be a Galois category. Let $G = \text{Aut}(F)$ be as in Example \ref{example-from-C-F-to-G-sets}. The functor $F : \mathcal{C} \to \textit{Finite-}G\textit{-Sets}$ (\ref{equation-remember}) an equivalence.


pione-proposition-universal-homeomorphism

Let $f : X \to Y$ be a universal homeomorphism of schemes. Then $$ \textit{F\'Et}_Y \longrightarrow \textit{F\'Et}_X,\quad V \longmapsto V \times_Y X $$ is an equivalence. Thus if $X$ and $Y$ are connected, then $f$ induces an isomorphism $\pi_1(X, \overline{x}) \to \pi_1(Y, \overline{y})$ of fundamental groups.


pione-proposition-normal

Let $X$ be a normal integral scheme with function field $K$. Then the canonical map (\ref{equation-inclusion-generic-point}) $$ \text{Gal}(K^{sep}/K) = \pi_1(\eta, \overline{\eta}) \longrightarrow \pi_1(X, \overline{\eta}) $$ is identified with the quotient map $\text{Gal}(K^{sep}/K) \to \text{Gal}(M/K)$ where $M \subset K^{sep}$ is the union of the finite subextensions $L$ such that $X$ is unramified in $L$.


pione-proposition-first-homotopy-sequence

Let $f : X \to S$ be a flat proper morphism of finite presentation whose geometric fibres are connected and reduced. Assume $S$ is connected and let $\overline{s}$ be a geometric point of $S$. Then there is an exact sequence $$ \pi_1(X_{\overline{s}}) \to \pi_1(X) \to \pi_1(S) \to 1 $$ of fundamental groups.


pione-proposition-lefschetz-fully-faithful

Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{L})$. Let $Y = Z(s)$ be the zero scheme of $s$. Assume that for all $x \in X \setminus Y$ we have $$ \text{depth}(\mathcal{O}_{X, x}) + \dim(\overline{\{x\}}) > 1 $$ Then the restriction functor $\textit{F\'Et}_X \to \textit{F\'Et}_Y$ is fully faithful. In fact, for any open subscheme $V \subset X$ containing $Y$ the restriction functor $\textit{F\'Et}_V \to \textit{F\'Et}_Y$ is fully faithful.


pione-proposition-lefschetz-equivalence-general

Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{L})$. Let $Y = Z(s)$ be the zero scheme of $s$. Let $\mathcal{V}$ be the set of open subschemes of $X$ containing $Y$ ordered by reverse inclusion. Assume that for all $x \in X \setminus Y$ we have $$ \text{depth}(\mathcal{O}_{X, x}) + \dim(\overline{\{x\}}) > 2 $$ Then the restriction functor $$ \colim_\mathcal{V} \textit{F\'Et}_V \to \textit{F\'Et}_Y $$ is an equivalence.


pione-proposition-specialization-map-isomorphism

Let $f : X \to S$ be a smooth proper morphism with geometrically connected fibres. Let $s' \leadsto s$ be a specialization. If the characteristic to $\kappa(s)$ is zero, then the specialization map $$ sp : \pi_1(X_{\overline{s}'}) \to \pi_1(X_{\overline{s}}) $$ is an isomorphism.


stacks-cohomology-lemma-flat-pullback-quasi-coherent

If $f : \mathcal{X} \to \mathcal{Y}$ is a flat morphism of algebraic stacks then $f^* : \QCoh(\mathcal{O}_\mathcal{Y}) \to \QCoh(\mathcal{O}_\mathcal{X})$ is an exact functor.


stacks-cohomology-lemma-general-pushforward

Let $\mathcal{M}$ be a rule which associates to every algebraic stack $\mathcal{X}$ a subcategory $\mathcal{M}_\mathcal{X}$ of $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ such that \begin{enumerate} \item $\mathcal{M}_\mathcal{X}$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ (see Homology, Definition \ref{homology-definition-serre-subcategory}) for all algebraic stacks $\mathcal{X}$, \item for a smooth morphism of algebraic stacks $f : \mathcal{Y} \to \mathcal{X}$ the functor $f^*$ maps $\mathcal{M}_\mathcal{X}$ into $\mathcal{M}_\mathcal{Y}$, \item if $f_i : \mathcal{X}_i \to \mathcal{X}$ is a family of smooth morphisms of algebraic stacks with $|\mathcal{X}| = \bigcup |f_i|(|\mathcal{X}_i|)$, then an object $\mathcal{F}$ of $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ is in $\mathcal{M}_\mathcal{X}$ if and only if $f_i^*\mathcal{F}$ is in $\mathcal{M}_{\mathcal{X}_i}$ for all $i$, and \item if $f : \mathcal{Y} \to \mathcal{X}$ is a morphism of algebraic stacks such that $\mathcal{X}$ and $\mathcal{Y}$ are representable by affine schemes, then $R^if_*$ maps $\mathcal{M}_\mathcal{Y}$ into $\mathcal{M}_\mathcal{X}$. \end{enumerate} Then for any quasi-compact and quasi-separated morphism  $f : \mathcal{Y} \to \mathcal{X}$ of algebraic stacks $R^if_*$ maps $\mathcal{M}_\mathcal{Y}$ into $\mathcal{M}_\mathcal{X}$. (Higher direct images computed in \'etale topology.)


stacks-cohomology-lemma-check-lqc-on-etale-covering

Let $\mathcal{X}$ be an algebraic stack. Let $f_j : \mathcal{X}_j \to \mathcal{X}$ be a family of smooth morphisms of algebraic stacks with $|\mathcal{X}| =\bigcup |f_j|(|\mathcal{X}_j|)$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{X}$-modules on $\mathcal{X}_\etale$. If each $f_j^{-1}\mathcal{F}$ is locally quasi-coherent, then so is $\mathcal{F}$.


stacks-cohomology-lemma-pushforward-locally-quasi-coherent

Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. Let  $\mathcal{F}$ be a locally quasi-coherent $\mathcal{O}_\mathcal{X}$-module on $\mathcal{X}_\etale$. Then $R^if_*\mathcal{F}$ (computed in the \'etale topology) is locally quasi-coherent on $\mathcal{Y}_\etale$.


stacks-cohomology-lemma-check-lqc-on-flat-covering

Let $\mathcal{X}$ be an algebraic stack. Let $f_j : \mathcal{X}_j \to \mathcal{X}$ be a family of flat and locally finitely presented morphisms of algebraic stacks with $|\mathcal{X}| =\bigcup |f_j|(|\mathcal{X}_j|)$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{X}$-modules on $\mathcal{X}_{fppf}$. If each $f_j^{-1}\mathcal{F}$ is locally quasi-coherent, then so is $\mathcal{F}$.


stacks-cohomology-lemma-check-flat-comparison-on-etale-covering

Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal{X}$-module on $\mathcal{X}_\etale$. \begin{enumerate} \item If $\mathcal{F}$ has the flat base change property then for any morphism $g : \mathcal{Y} \to \mathcal{X}$ of algebraic stacks, the pullback $g^*\mathcal{F}$ does too. \item The full subcategory of $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ consisting of modules with the flat base change property is a weak Serre subcategory. \item  Let $f_i : \mathcal{X}_i \to \mathcal{X}$ be a family of smooth morphisms of algebraic stacks such that $|\mathcal{X}| = \bigcup_i |f_i|(|\mathcal{X}_i|)$. If each $f_i^*\mathcal{F}$ has the flat base change property then so does $\mathcal{F}$. \item The category of $\mathcal{O}_\mathcal{X}$-modules on $\mathcal{X}_\etale$ with the flat base change property has colimits and they agree with colimits in $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$. \end{enumerate}


stacks-cohomology-lemma-flat-comparison

Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. Let  $\mathcal{F}$ be an object of $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ which is locally quasi-coherent and has the flat base change property. Then each $R^if_*\mathcal{F}$ (computed in the \'etale topology) has the flat base change property.


stacks-cohomology-lemma-check-lqc-fbc-on-covering

Let $\mathcal{X}$ be an algebraic stack. With $\mathcal{M}_\mathcal{X}$ the category of locally quasi-coherent modules with the flat base change property. \begin{enumerate} \item Let $f_j : \mathcal{X}_j \to \mathcal{X}$ be a family of smooth morphisms of algebraic stacks with $|\mathcal{X}| =\bigcup |f_j|(|\mathcal{X}_j|)$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{X}$-modules on $\mathcal{X}_\etale$. If each $f_j^{-1}\mathcal{F}$ is in $\mathcal{M}_{\mathcal{X}_i}$, then $\mathcal{F}$ is in $\mathcal{M}_\mathcal{X}$. \item  Let $f_j : \mathcal{X}_j \to \mathcal{X}$ be a family of flat and locally finitely presented morphisms of algebraic stacks with $|\mathcal{X}| =\bigcup |f_j|(|\mathcal{X}_j|)$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{X}$-modules on $\mathcal{X}_{fppf}$. If each $f_j^{-1}\mathcal{F}$ is in $\mathcal{M}_{\mathcal{X}_i}$, then $\mathcal{F}$ is in $\mathcal{M}_\mathcal{X}$. \end{enumerate}


stacks-cohomology-lemma-parasitic

Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_\mathcal{X}$-modules. \begin{enumerate} \item If $\mathcal{F}$ is parasitic and $g : \mathcal{Y} \to \mathcal{X}$ is a flat morphism of algebraic stacks, then $g^*\mathcal{F}$ is parasitic. \item For $\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}$ we have \begin{enumerate} \item the $\tau$ sheafification of a parasitic presheaf of modules is parasitic, and \item the full subcategory of $\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$ consisting of parasitic modules is a Serre subcategory. \end{enumerate} \item Suppose $\mathcal{F}$ is a sheaf for the \'etale topology. Let $f_i : \mathcal{X}_i \to \mathcal{X}$ be a family of smooth morphisms of algebraic stacks such that $|\mathcal{X}| = \bigcup_i |f_i|(|\mathcal{X}_i|)$. If each $f_i^*\mathcal{F}$ is parasitic then so is $\mathcal{F}$. \item Suppose $\mathcal{F}$ is a sheaf for the fppf topology. Let $f_i : \mathcal{X}_i \to \mathcal{X}$ be a family of flat and locally finitely presented morphisms of algebraic stacks such that $|\mathcal{X}| = \bigcup_i |f_i|(|\mathcal{X}_i|)$. If each $f_i^*\mathcal{F}$ is parasitic then so is $\mathcal{F}$. \end{enumerate}


stacks-cohomology-lemma-pushforward-parasitic

Let $\tau \in \{\etale, fppf\}$. Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{F}$ be a parasitic object of $\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$. \begin{enumerate} \item $H^i_\tau(\mathcal{X}, \mathcal{F}) = 0$ for all $i$. \item Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Then $R^if_*\mathcal{F}$ (computed in $\tau$-topology) is a parasitic object of $\textit{Mod}(\mathcal{Y}_\tau, \mathcal{O}_\mathcal{Y})$. \end{enumerate}


stacks-cohomology-lemma-exact-sequence-quasi-coherent-parasitic-cohomology

Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{F}^\bullet$ be an exact complex in $\QCoh(\mathcal{O}_\mathcal{X})$. Then the cohomology sheaves of $\mathcal{F}^\bullet$ in either the \'etale or the fppf topology are parasitic $\mathcal{O}_\mathcal{X}$-modules.


stacks-cohomology-lemma-adjoint

Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{M}_\mathcal{X}$ be the category of locally quasi-coherent modules with the flat base change property, see Proposition \ref{proposition-lcq-flat-base-change}. The inclusion functor $i : \QCoh(\mathcal{O}_\mathcal{X}) \to \mathcal{M}_\mathcal{X}$ has a right adjoint $$ Q : \mathcal{M}_\mathcal{X} \to \QCoh(\mathcal{O}_\mathcal{X}) $$ such that $Q \circ i$ is the identity functor.


stacks-cohomology-lemma-adjoint-kernel-parasitic

Let $\mathcal{X}$ be an algebraic stack. Let $Q : \mathcal{M}_\mathcal{X} \to \QCoh(\mathcal{O}_\mathcal{X})$ be the functor constructed in Lemma \ref{lemma-adjoint}. \begin{enumerate} \item The kernel of $Q$ is exactly the collection of parasitic objects of $\mathcal{M}_\mathcal{X}$. \item For any object $\mathcal{F}$ of $\mathcal{M}_\mathcal{X}$ both the kernel and the cokernel of the adjunction map $Q(\mathcal{F}) \to \mathcal{F}$ are parasitic. \item The functor $Q$ is exact. \end{enumerate}


stacks-cohomology-lemma-lisse-etale

Let $\mathcal{X}$ be an algebraic stack. \begin{enumerate} \item The inclusion functor $\mathcal{X}_{lisse,\etale} \to \mathcal{X}_\etale$ is fully faithful, continuous and cocontinuous. It follows that \begin{enumerate} \item there is a morphism of topoi $$ g : \Sh(\mathcal{X}_{lisse,\etale}) \longrightarrow \Sh(\mathcal{X}_\etale) $$ with $g^{-1}$ given by restriction, \item the functor $g^{-1}$ has a left adjoint $g_!^{Sh}$ on sheaves of sets, \item the adjunction maps $g^{-1}g_* \to \text{id}$ and $\text{id} \to g^{-1}g_!^{Sh}$ are isomorphisms, \item the functor $g^{-1}$ has a left adjoint $g_!$ on abelian sheaves, \item the adjunction map $\text{id} \to g^{-1}g_!$ is an isomorphism, and \item we have $g^{-1}\mathcal{O}_\mathcal{X} = \mathcal{O}_{\mathcal{X}_{lisse,\etale}}$ hence $g$ induces a flat morphism of ringed topoi such that $g^{-1} = g^*$. \end{enumerate} \item The inclusion functor $\mathcal{X}_{flat,fppf} \to \mathcal{X}_{fppf}$ is fully faithful, continuous and cocontinuous. It follows that \begin{enumerate} \item there is a morphism of topoi $$ g : \Sh(\mathcal{X}_{flat,fppf}) \longrightarrow \Sh(\mathcal{X}_{fppf}) $$ with $g^{-1}$ given by restriction, \item the functor $g^{-1}$ has a left adjoint $g_!^{Sh}$ on sheaves of sets, \item the adjunction maps $g^{-1}g_* \to \text{id}$ and $\text{id} \to g^{-1}g_!^{Sh}$ are isomorphisms, \item the functor $g^{-1}$ has a left adjoint $g_!$ on abelian sheaves, \item the adjunction map $\text{id} \to g^{-1}g_!$ is an isomorphism, and \item we have $g^{-1}\mathcal{O}_\mathcal{X} = \mathcal{O}_{\mathcal{X}_{flat,fppf}}$ hence $g$ induces a flat morphism of ringed topoi such that $g^{-1} = g^*$. \end{enumerate} \end{enumerate}


stacks-cohomology-lemma-lisse-etale-modules

Let $\mathcal{X}$ be an algebraic stack. Notation as in Lemma \ref{lemma-lisse-etale}. \begin{enumerate} \item There exists a functor $$ g_! : \textit{Mod}(\mathcal{X}_{lisse,\etale}, \mathcal{O}_{\mathcal{X}_{lisse,\etale}}) \longrightarrow \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_{\mathcal{X}}) $$ which is left adjoint to $g^*$. Moreover it agrees with the functor $g_!$ on abelian sheaves and $g^*g_! = \text{id}$. \item There exists a functor $$ g_! : \textit{Mod}(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \longrightarrow \textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_{\mathcal{X}}) $$ which is left adjoint to $g^*$. Moreover it agrees with the functor $g_!$ on abelian sheaves and $g^*g_! = \text{id}$. \end{enumerate}


stacks-cohomology-lemma-lisse-etale-structure-sheaf

Let $\mathcal{X}$ be an algebraic stack. Notation as in Lemmas \ref{lemma-lisse-etale} and \ref{lemma-lisse-etale-modules}. \begin{enumerate} \item We have $g_!\mathcal{O}_{\mathcal{X}_{lisse,\etale}} = \mathcal{O}_\mathcal{X}$. \item We have $g_!\mathcal{O}_{\mathcal{X}_{flat, fppf}} = \mathcal{O}_\mathcal{X}$. \end{enumerate}


stacks-cohomology-lemma-parasitic-in-terms-flat-fppf

Let $\mathcal{X}$ be an algebraic stack. \begin{enumerate} \item Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal{X}$-module with the flat base change property on $\mathcal{X}_\etale$. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is parasitic, and \item $g^*\mathcal{F} = 0$ where $g : \Sh(\mathcal{X}_{lisse,\etale}) \to \Sh(\mathcal{X}_\etale)$ is as in Lemma \ref{lemma-lisse-etale}. \end{enumerate} \item Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal{X}$-module on $\mathcal{X}_{fppf}$. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is parasitic, and \item $g^*\mathcal{F} = 0$ where $g :  \Sh(\mathcal{X}_{flat,fppf}) \to \Sh(\mathcal{X}_{fppf})$ is as in Lemma \ref{lemma-lisse-etale}. \end{enumerate} \end{enumerate}


stacks-cohomology-lemma-lisse-etale-functorial

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. \begin{enumerate} \item If $f$ is smooth, then $f$ restricts to a continuous and cocontinuous functor $\mathcal{X}_{lisse,\etale} \to \mathcal{Y}_{lisse,\etale}$ which gives a morphism of ringed topoi fitting into the following commutative diagram $$ \xymatrix{ \Sh(\mathcal{X}_{lisse,\etale}) \ar[r]_{g'} \ar[d]_{f'} & \Sh(\mathcal{X}_\etale) \ar[d]^f \\ \Sh(\mathcal{Y}_{lisse,\etale}) \ar[r]^g & \Sh(\mathcal{Y}_\etale) } $$ We have $f'_*(g')^{-1} = g^{-1}f_*$ and $g'_!(f')^{-1} = f^{-1}g_!$. \item If $f$ is flat, then $f$ restricts to a continuous and cocontinuous functor $\mathcal{X}_{flat,fppf} \to \mathcal{Y}_{flat,fppf}$ which gives a morphism of ringed topoi fitting into the following commutative diagram $$ \xymatrix{ \Sh(\mathcal{X}_{flat,fppf}) \ar[r]_{g'} \ar[d]_{f'} & \Sh(\mathcal{X}_{fppf}) \ar[d]^f \\ \Sh(\mathcal{Y}_{flat,fppf}) \ar[r]^g & \Sh(\mathcal{Y}_{fppf}) } $$ We have $f'_*(g')^{-1} = g^{-1}f_*$ and $g'_!(f')^{-1} = f^{-1}g_!$. \end{enumerate}


stacks-cohomology-lemma-check-qc-on-etale-covering

Let $\mathcal{X}$ be an algebraic stack. \begin{enumerate} \item Let $f_j : \mathcal{X}_j \to \mathcal{X}$ be a family of smooth morphisms of algebraic stacks with $|\mathcal{X}| =\bigcup |f_j|(|\mathcal{X}_j|)$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{X}$-modules on $\mathcal{X}_\etale$. If each $f_j^{-1}\mathcal{F}$ is quasi-coherent, then so is $\mathcal{F}$. \item Let $f_j : \mathcal{X}_j \to \mathcal{X}$ be a family of flat and locally finitely presented morphisms of algebraic stacks with $|\mathcal{X}| =\bigcup |f_j|(|\mathcal{X}_j|)$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{X}$-modules on $\mathcal{X}_{fppf}$. If each $f_j^{-1}\mathcal{F}$ is quasi-coherent, then so is $\mathcal{F}$. \end{enumerate}


stacks-cohomology-lemma-shriek-quasi-coherent

Let $\mathcal{X}$ be an algebraic stack. Notation as in Lemma \ref{lemma-lisse-etale}. \begin{enumerate} \item Let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_{\mathcal{X}_{lisse,\etale}}$-module  on the lisse-\'etale site of $\mathcal{X}$. Then $g_!\mathcal{H}$ is a quasi-coherent module on $\mathcal{X}$. \item Let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_{\mathcal{X}_{flat,fppf}}$-module  on the flat-fppf site of $\mathcal{X}$. Then $g_!\mathcal{H}$ is a quasi-coherent module on $\mathcal{X}$. \end{enumerate}


stacks-cohomology-lemma-quasi-coherent

Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{M}_\mathcal{X}$ be the category of locally quasi-coherent $\mathcal{O}_\mathcal{X}$-modules with the flat base change property. \begin{enumerate} \item With $g$ as in Lemma \ref{lemma-lisse-etale} for the lisse-\'etale site we have \begin{enumerate} \item the functors $g^{-1}$ and $g_!$ define mutually inverse functors $$ \xymatrix{ \QCoh(\mathcal{O}_\mathcal{X}) \ar@<1ex>[r]^-{g^{-1}} & \QCoh(\mathcal{X}_{lisse,\etale}, \mathcal{O}_{\mathcal{X}_{lisse,\etale}}) \ar@<1ex>[l]^-{g_!} } $$ \item if $\mathcal{F}$ is in $\mathcal{M}_\mathcal{X}$ then $g^{-1}\mathcal{F}$ is in $\QCoh(\mathcal{X}_{lisse,\etale}, \mathcal{O}_{\mathcal{X}_{lisse,\etale}})$ and \item $Q(\mathcal{F}) = g_!g^{-1}\mathcal{F}$ where $Q$ is as in Lemma \ref{lemma-adjoint}. \end{enumerate} \item With $g$ as in Lemma \ref{lemma-lisse-etale} for the flat-fppf site we have \begin{enumerate} \item the functors $g^{-1}$ and $g_!$ define mutually inverse functors $$ \xymatrix{ \QCoh(\mathcal{O}_\mathcal{X}) \ar@<1ex>[r]^-{g^{-1}} & \QCoh(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \ar@<1ex>[l]^-{g_!} } $$ \item if $\mathcal{F}$ is in $\mathcal{M}_\mathcal{X}$ then $g^{-1}\mathcal{F}$ is in $\QCoh(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}})$ and \item $Q(\mathcal{F}) = g_!g^{-1}\mathcal{F}$ where $Q$ is as in Lemma \ref{lemma-adjoint}. \end{enumerate} \end{enumerate}


stacks-cohomology-proposition-lcq-flat-base-change

Summary of results on locally quasi-coherent modules having the flat base change property. \begin{enumerate} \item Let $\mathcal{X}$ be an algebraic stack. If $\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ which is locally quasi-coherent and has the flat base change property, then $\mathcal{F}$ is a sheaf for the fppf topology, i.e., it is an object of $\textit{Mod}(\mathcal{O}_\mathcal{X})$. \item The category of modules which are locally quasi-coherent and have the flat base change property is a weak Serre subcategory $\mathcal{M}_\mathcal{X}$ of both $\textit{Mod}(\mathcal{O}_\mathcal{X})$ and $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$. \item Pullback $f^*$ along any morphism of algebraic stacks $f : \mathcal{X} \to \mathcal{Y}$ induces a functor $f^* : \mathcal{M}_\mathcal{Y} \to \mathcal{M}_\mathcal{X}$. \item If $f : \mathcal{X} \to \mathcal{Y}$ is a quasi-compact and quasi-separated morphism of algebraic stacks and $\mathcal{F}$ is an object of $\mathcal{M}_\mathcal{X}$, then \begin{enumerate} \item the derived direct image $Rf_*\mathcal{F}$ and the higher direct images $R^if_*\mathcal{F}$ can be computed in either the \'etale or the fppf topology with the same result, and \item each $R^if_*\mathcal{F}$ is an object of $\mathcal{M}_\mathcal{Y}$. \end{enumerate} \item The category $\mathcal{M}_\mathcal{X}$ has colimits and they agree with colimits in $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ as well as in $\textit{Mod}(\mathcal{O}_\mathcal{X})$. \end{enumerate}


stacks-cohomology-proposition-direct-image-quasi-coherent

Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. The functor $f^* : \QCoh(\mathcal{O}_\mathcal{Y}) \to \QCoh(\mathcal{O}_\mathcal{X})$ has a right adjoint $$ f_{\QCoh, *} : \QCoh(\mathcal{O}_\mathcal{X}) \longrightarrow \QCoh(\mathcal{O}_\mathcal{Y}) $$ which can be defined as the composition $$ \QCoh(\mathcal{O}_\mathcal{X}) \to \mathcal{M}_\mathcal{X} \xrightarrow{f_*} \mathcal{M}_\mathcal{Y} \xrightarrow{Q} \QCoh(\mathcal{O}_\mathcal{Y}) $$ where the functors $f_*$ and $Q$ are as in Proposition \ref{proposition-lcq-flat-base-change} and Lemma \ref{lemma-adjoint}. Moreover, if we define $R^if_{\QCoh, *}$ as the composition $$ \QCoh(\mathcal{O}_\mathcal{X}) \to \mathcal{M}_\mathcal{X} \xrightarrow{R^if_*} \mathcal{M}_\mathcal{Y} \xrightarrow{Q} \QCoh(\mathcal{O}_\mathcal{Y}) $$ then the sequence of functors $\{R^if_{\QCoh, *}\}_{i \geq 0}$ forms a cohomological $\delta$-functor.


sites-cohomology-lemma-trivial-torsor

Let $\mathcal{C}$ be a site. Let $\mathcal{G}$ be a sheaf of (possibly non-commutative) groups on $\mathcal{C}$. A $\mathcal{G}$-torsor $\mathcal{F}$ is trivial if and only if $\Gamma(\mathcal{C}, \mathcal{F}) \not = \emptyset$.


sites-cohomology-lemma-torsors-h1

Let $\mathcal{C}$ be a site. Let $\mathcal{H}$ be an abelian sheaf on $\mathcal{C}$. There is a canonical bijection between the set of isomorphism classes of $\mathcal{H}$-torsors and $H^1(\mathcal{C}, \mathcal{H})$.


sites-cohomology-lemma-h1-extensions

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules on $\mathcal{C}$. There is a canonical bijection $$ \Ext^1_{\textit{Mod}(\mathcal{O})}(\mathcal{O}, \mathcal{F}) \longrightarrow H^1(\mathcal{C}, \mathcal{F}) $$ which associates to the extension $$ 0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{O} \to 0 $$ the image of $1 \in \Gamma(\mathcal{C}, \mathcal{O})$ in $H^1(\mathcal{C}, \mathcal{F})$.


sites-cohomology-lemma-h1-invertible

Let $(\mathcal{C}, \mathcal{O})$ be a locally ringed site. There is a canonical isomorphism $$ H^1(\mathcal{C}, \mathcal{O}^*) = \Pic(\mathcal{O}). $$ of abelian groups.


sites-cohomology-lemma-cohomology-of-open

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. \begin{enumerate} \item If $\mathcal{I}$ is an injective $\mathcal{O}$-module then $\mathcal{I}|_U$ is an injective $\mathcal{O}_U$-module. \item For any sheaf of $\mathcal{O}$-modules $\mathcal{F}$ we have $H^p(U, \mathcal{F}) = H^p(\mathcal{C}/U, \mathcal{F}|_U)$. \end{enumerate}


sites-cohomology-lemma-cohomology-bigger-site

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume $u$ satisfies the hypotheses of Sites, Lemma \ref{sites-lemma-bigger-site}. Let $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be the associated morphism of topoi. For any abelian sheaf $\mathcal{F}$ on $\mathcal{D}$ we have isomorphisms $$ R\Gamma(\mathcal{C}, g^{-1}\mathcal{F}) = R\Gamma(\mathcal{D}, \mathcal{F}), $$ in particular $H^p(\mathcal{C}, g^{-1}\mathcal{F}) = H^p(\mathcal{D}, \mathcal{F})$ and for any $U \in \Ob(\mathcal{C})$ we have isomorphisms $$ R\Gamma(U, g^{-1}\mathcal{F}) = R\Gamma(u(U), \mathcal{F}), $$ in particular $H^p(U, g^{-1}\mathcal{F}) = H^p(u(U), \mathcal{F})$. All of these isomorphisms are functorial in $\mathcal{F}$.


sites-cohomology-lemma-kill-cohomology-class-on-covering

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. Let $U$ be an object of $\mathcal{C}$. Let $n > 0$ and let $\xi \in H^n(U, \mathcal{F})$. Then there exists a covering $\{U_i \to U\}$ of $\mathcal{C}$ such that $\xi|_{U_i} = 0$ for all $i \in I$.


sites-cohomology-lemma-higher-direct-images

Let $f : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D})$ be a morphism of ringed sites corresponding to the continuous functor $u : \mathcal{D} \to \mathcal{C}$. For any $\mathcal{F} \in \Ob(\textit{Mod}(\mathcal{O}_\mathcal{C}))$ the sheaf $R^if_*\mathcal{F}$ is the sheaf associated to the presheaf $$ V \longmapsto H^i(u(V), \mathcal{F}) $$


sites-cohomology-lemma-cech-h0

Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be an abelian presheaf on $\mathcal{C}$. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is an abelian sheaf on $\mathcal{C}$ and \item for every covering $\mathcal{U} = \{U_i \to U\}_{i \in I}$ of the site $\mathcal{C}$ the natural map $$ \mathcal{F}(U) \to \check{H}^0(\mathcal{U}, \mathcal{F}) $$ (see Sites, Section \ref{sites-section-sheafification}) is bijective. \end{enumerate}


sites-cohomology-lemma-cech-exact-presheaves

The functor given by Equation (\ref{equation-cech-functor}) is an exact functor (see Homology, Lemma \ref{homology-lemma-exact-functor}).


sites-cohomology-lemma-cech-cohomology-delta-functor-presheaves

Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{U_i \to U\}_{i \in I}$ be a family of morphisms with fixed target such that all fibre products $U_{i_0} \times_U \ldots \times_U U_{i_p}$ exist in $\mathcal{C}$. The functors $\mathcal{F} \mapsto \check{H}^n(\mathcal{U}, \mathcal{F})$ form a $\delta$-functor from the abelian category $\textit{PAb}(\mathcal{C})$ to the category of $\mathbf{Z}$-modules (see Homology, Definition \ref{homology-definition-cohomological-delta-functor}).


sites-cohomology-lemma-cech-map-into

Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{U_i \to U\}_{i \in I}$ be a family of morphisms with fixed target such that all fibre products $U_{i_0} \times_U \ldots \times_U U_{i_p}$ exist in $\mathcal{C}$. Consider the chain complex $\mathbf{Z}_{\mathcal{U}, \bullet}$ of abelian presheaves $$ \ldots \to \bigoplus_{i_0i_1i_2} \mathbf{Z}_{U_{i_0} \times_U U_{i_1} \times_U U_{i_2}} \to \bigoplus_{i_0i_1} \mathbf{Z}_{U_{i_0} \times_U U_{i_1}} \to \bigoplus_{i_0} \mathbf{Z}_{U_{i_0}} \to 0 \to \ldots $$ where the last nonzero term is placed in degree $0$ and where the map $$ \mathbf{Z}_{U_{i_0} \times_U \ldots \times_u U_{i_{p + 1}}} \longrightarrow \mathbf{Z}_{U_{i_0} \times_U \ldots \widehat{U_{i_j}} \ldots \times_U U_{i_{p + 1}}} $$ is given by $(-1)^j$ times the canonical map. Then there is an isomorphism $$ \Hom_{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_{\mathcal{U}, \bullet}, \mathcal{F}) = \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) $$ functorial in $\mathcal{F} \in \Ob(\textit{PAb}(\mathcal{C}))$.


sites-cohomology-lemma-homology-complex

Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{f_i : U_i \to U\}_{i \in I}$ be a family of morphisms with fixed target such that all fibre products $U_{i_0} \times_U \ldots \times_U U_{i_p}$ exist in $\mathcal{C}$. The chain complex $\mathbf{Z}_{\mathcal{U}, \bullet}$ of presheaves of Lemma \ref{lemma-cech-map-into} above is exact in positive degrees, i.e., the homology presheaves $H_i(\mathbf{Z}_{\mathcal{U}, \bullet})$ are zero for $i > 0$.


sites-cohomology-lemma-complex-tensored-still-exact

\begin{slogan} The integral presheaf {\v C}ech complex is a flat resolution of the constant presheaf of integers. \end{slogan} Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{f_i : U_i \to U\}_{i \in I}$ be a family of morphisms with fixed target such that all fibre products $U_{i_0} \times_U \ldots \times_U U_{i_p}$ exist in $\mathcal{C}$. Let $\mathcal{O}$ be a presheaf of rings on $\mathcal{C}$. The chain complex $$ \mathbf{Z}_{\mathcal{U}, \bullet} \otimes_{p, \mathbf{Z}} \mathcal{O} $$ is exact in positive degrees. Here $\mathbf{Z}_{\mathcal{U}, \bullet}$ is the chain complex of Lemma \ref{lemma-cech-map-into}, and the tensor product is over the constant presheaf of rings with value $\mathbf{Z}$.


sites-cohomology-lemma-cech-cohomology-derived-presheaves

Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{f_i : U_i \to U\}_{i \in I}$ be a family of morphisms with fixed target such that all fibre products $U_{i_0} \times_U \ldots \times_U U_{i_p}$ exist in $\mathcal{C}$. The {\v C}ech cohomology functors $\check{H}^p(\mathcal{U}, -)$ are canonically isomorphic as a $\delta$-functor to the right derived functors of the functor $$ \check{H}^0(\mathcal{U}, -) : \textit{PAb}(\mathcal{C}) \longrightarrow \textit{Ab}. $$ Moreover, there is a functorial quasi-isomorphism $$ \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \longrightarrow R\check{H}^0(\mathcal{U}, \mathcal{F}) $$ where the right hand side indicates the derived functor $$ R\check{H}^0(\mathcal{U}, -) : D^{+}(\textit{PAb}(\mathcal{C})) \longrightarrow D^{+}(\mathbf{Z}) $$ of the left exact functor $\check{H}^0(\mathcal{U}, -)$.


sites-cohomology-lemma-injective-abelian-sheaf-injective-presheaf

Let $\mathcal{C}$ be a site. An injective abelian sheaf is also injective as an object in the category $\textit{PAb}(\mathcal{C})$.


sites-cohomology-lemma-injective-trivial-cech

Let $\mathcal{C}$ be a site. Let $\mathcal{U} = \{U_i \to U\}_{i \in I}$ be a covering of $\mathcal{C}$. Let $\mathcal{I}$ be an injective abelian sheaf, i.e., an injective object of $\textit{Ab}(\mathcal{C})$. Then $$ \check{H}^p(\mathcal{U}, \mathcal{I}) = \left\{ \begin{matrix} \mathcal{I}(U) & \text{if} & p = 0 \\ 0 & \text{if} & p > 0 \end{matrix} \right. $$


sites-cohomology-lemma-cech-h1

Let $\mathcal{C}$ be a site. Let $\mathcal{G}$ be an abelian sheaf on $\mathcal{C}$. Let $\mathcal{U} = \{U_i \to U\}_{i \in I}$ be a covering of $\mathcal{C}$. The map $$ \check{H}^1(\mathcal{U}, \mathcal{G}) \longrightarrow H^1(U, \mathcal{G}) $$ is injective and identifies $\check{H}^1(\mathcal{U}, \mathcal{G})$ via the bijection of Lemma \ref{lemma-torsors-h1} with the set of isomorphism classes of $\mathcal{G}|_U$-torsors which restrict to trivial torsors over each $U_i$.


sites-cohomology-lemma-include

Let $\mathcal{C}$ be a site. Consider the functor $i : \textit{Ab}(\mathcal{C}) \to \textit{PAb}(\mathcal{C})$. It is a left exact functor with right derived functors given by $$ R^pi(\mathcal{F}) = \underline{H}^p(\mathcal{F}) : U \longmapsto H^p(U, \mathcal{F}) $$ see discussion in Section \ref{section-locality}.


sites-cohomology-lemma-cech-spectral-sequence

Let $\mathcal{C}$ be a site. Let $\mathcal{U} = \{U_i \to U\}_{i \in I}$ be a covering of $\mathcal{C}$. For any abelian sheaf $\mathcal{F}$ there is a spectral sequence $(E_r, d_r)_{r \geq 0}$ with $$ E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F})) $$ converging to $H^{p + q}(U, \mathcal{F})$. This spectral sequence is functorial in $\mathcal{F}$.


sites-cohomology-lemma-cech-spectral-sequence-application

Let $\mathcal{C}$ be a site. Let $\mathcal{U} = \{U_i \to U\}_{i \in I}$ be a covering. Let $\mathcal{F} \in \Ob(\textit{Ab}(\mathcal{C}))$. Assume that $H^i(U_{i_0} \times_U \ldots \times_U U_{i_p}, \mathcal{F}) = 0$ for all $i > 0$, all $p \geq 0$ and all $i_0, \ldots, i_p \in I$. Then $\check{H}^p(\mathcal{U}, \mathcal{F}) = H^p(U, \mathcal{F})$.


sites-cohomology-lemma-ses-cech-h1

Let $\mathcal{C}$ be a site. Let $$ 0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0 $$ be a short exact sequence of abelian sheaves on $\mathcal{C}$. Let $U$ be an object of $\mathcal{C}$. If there exists a cofinal system of coverings $\mathcal{U}$ of $U$ such that $\check{H}^1(\mathcal{U}, \mathcal{F}) = 0$, then the map $\mathcal{G}(U) \to \mathcal{H}(U)$ is surjective.


sites-cohomology-lemma-cech-vanish-collection

(Variant of Cohomology, Lemma \ref{cohomology-lemma-cech-vanish}.) Let $\mathcal{C}$ be a site. Let $\text{Cov}_\mathcal{C}$ be the set of coverings of $\mathcal{C}$ (see Sites, Definition \ref{sites-definition-site}). Let $\mathcal{B} \subset \Ob(\mathcal{C})$, and $\text{Cov} \subset \text{Cov}_\mathcal{C}$ be subsets. Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}$. Assume that \begin{enumerate} \item For every $\mathcal{U} \in \text{Cov}$, $\mathcal{U} = \{U_i \to U\}_{i \in I}$ we have $U, U_i \in \mathcal{B}$ and every $U_{i_0} \times_U \ldots \times_U U_{i_p} \in \mathcal{B}$. \item For every $U \in \mathcal{B}$ the coverings of $U$ occurring in $\text{Cov}$ is a cofinal system of coverings of $U$. \item For every $\mathcal{U} \in \text{Cov}$ we have $\check{H}^p(\mathcal{U}, \mathcal{F}) = 0$ for all $p > 0$. \end{enumerate} Then $H^p(U, \mathcal{F}) = 0$ for all $p > 0$ and any $U \in \mathcal{B}$.


sites-cohomology-lemma-existence

Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a gerbe over a site whose automorphism sheaves are abelian. Let $\mathcal{G}$ be the sheaf of abelian groups constructed in Stacks, Lemma \ref{stacks-lemma-gerbe-abelian-auts}. Let $U$ be an object of $\mathcal{C}$ such that \begin{enumerate} \item there exists a cofinal system of coverings $\{U_i \to U\}$ of $U$ in $\mathcal{C}$ such that $H^1(U_i, \mathcal{G}) = 0$ and $H^1(U_i \times_U U_j, \mathcal{G}) = 0$ for all $i, j$, and \item $H^2(U, \mathcal{G}) = 0$. \end{enumerate} Then there exists an object of $\mathcal{S}$ lying over $U$.


sites-cohomology-lemma-injective-module-injective-presheaf

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. An injective sheaf of modules is also injective as an object in the category $\textit{PMod}(\mathcal{O})$.


sites-cohomology-lemma-include-modules

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Consider the functor $i : \textit{Mod}(\mathcal{C}) \to \textit{PMod}(\mathcal{C})$. It is a left exact functor with right derived functors given by $$ R^pi(\mathcal{F}) = \underline{H}^p(\mathcal{F}) : U \longmapsto H^p(U, \mathcal{F}) $$ see discussion in Section \ref{section-locality}.


sites-cohomology-lemma-injective-module-trivial-cech

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{U} = \{U_i \to U\}_{i \in I}$ be a covering of $\mathcal{C}$. Let $\mathcal{I}$ be an injective $\mathcal{O}$-module, i.e., an injective object of $\textit{Mod}(\mathcal{O})$. Then $$ \check{H}^p(\mathcal{U}, \mathcal{I}) = \left\{ \begin{matrix} \mathcal{I}(U) & \text{if} & p = 0 \\ 0 & \text{if} & p > 0 \end{matrix} \right. $$


sites-cohomology-lemma-cohomology-modules-abelian-agree

Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. Let $\mathcal{F}$ be an $\mathcal{O}$-module, and denote $\mathcal{F}_{ab}$ the underlying sheaf of abelian groups. Then we have $$ H^i(\mathcal{C}, \mathcal{F}_{ab}) = H^i(\mathcal{C}, \mathcal{F}) $$ and for any object $U$ of $\mathcal{C}$ we also have $$ H^i(U, \mathcal{F}_{ab}) = H^i(U, \mathcal{F}). $$ Here the left hand side is cohomology computed in $\textit{Ab}(\mathcal{C})$ and the right hand side is cohomology computed in $\textit{Mod}(\mathcal{O})$.


sites-cohomology-lemma-cohomology-products

Let $\mathcal{C}$ be a site. Let $I$ be a set. For $i \in I$ let  $\mathcal{F}_i$ be an abelian sheaf on $\mathcal{C}$. Let $U \in \Ob(\mathcal{C})$. The canonical map $$ H^p(U, \prod\nolimits_{i \in I} \mathcal{F}_i) \longrightarrow \prod\nolimits_{i \in I} H^p(U, \mathcal{F}_i) $$ is an isomorphism for $p = 0$ and injective for $p = 1$.


sites-cohomology-lemma-restriction-along-monomorphism-surjective

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $a : U' \to U$ be a monomorphism in $\mathcal{C}$. Then for any injective $\mathcal{O}$-module $\mathcal{I}$ the restriction mapping $\mathcal{I}(U) \to \mathcal{I}(U')$ is surjective.


sites-cohomology-lemma-cech-to-cohomology-sheaf-sets

Let $\mathcal{C}$ be a site. Let $K' \to K$ be a map of presheaves of sets on $\mathcal{C}$ whose sheafification is surjective. Set $K'_p = K' \times_K \ldots \times_K K'$ ($p + 1$-factors). For every abelian sheaf $\mathcal{F}$ there is a spectral sequence with $E_1^{p, q} = H^q(K'_p, \mathcal{F})$ converging to $H^{p + q}(K, \mathcal{F})$.


sites-cohomology-lemma-cohomology-on-sheaf-sets

Let $\mathcal{C}$ be a site. Let $K$ be a sheaf of sets on $\mathcal{C}$. Consider the morphism of topoi $j : \Sh(\mathcal{C}/K) \to \Sh(\mathcal{C})$, see Sites, Lemma \ref{sites-lemma-localize-topos-site}. Then $j^{-1}$ preserves injectives and $H^p(K, \mathcal{F}) = H^p(\mathcal{C}/K, j^{-1}\mathcal{F})$ for any abelian sheaf $\mathcal{F}$ on $\mathcal{C}$.


sites-cohomology-lemma-characterize-limp

Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be an abelian sheaf. If \begin{enumerate} \item $H^p(U, \mathcal{F}) = 0$ for $p > 0$ and $U \in \Ob(\mathcal{C})$, and \item for every surjection $K' \to K$ of sheaves of sets the extended {\v C}ech complex $$ 0 \to H^0(K, \mathcal{F}) \to H^0(K', \mathcal{F}) \to H^0(K' \times_K K', \mathcal{F}) \to \ldots $$ is exact, \end{enumerate} then $\mathcal{F}$ is totally acyclic (and the converse holds too).


sites-cohomology-lemma-direct-image-injective-sheaf

Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. Then for any injective object $\mathcal{I}$ in $\textit{Mod}(\mathcal{O}_\mathcal{C})$ the pushforward $f_*\mathcal{I}$ is totally acyclic.


sites-cohomology-lemma-pushforward-injective-flat

Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. If $f$ is flat, then $f_*\mathcal{I}$ is an injective $\mathcal{O}_\mathcal{D}$-module for any injective $\mathcal{O}_\mathcal{C}$-module $\mathcal{I}$.


sites-cohomology-lemma-limp-acyclic

Let $(\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})$ be a ringed topos. A totally acyclic sheaf is right acyclic for the following functors: \begin{enumerate} \item the functor $H^0(U, -)$ for any object $U$ of $\mathcal{C}$, \item the functor $\mathcal{F} \mapsto \mathcal{F}(K)$ for any presheaf of sets $K$, \item the functor $\Gamma(\mathcal{C}, -)$ of global sections, \item the functor $f_*$ for any morphism $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ of ringed topoi. \end{enumerate}


sites-cohomology-lemma-Leray

Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. Let $\mathcal{F}^\bullet$ be a bounded below complex of $\mathcal{O}_\mathcal{C}$-modules. There is a spectral sequence $$ E_2^{p, q} = H^p(\mathcal{D}, R^qf_*(\mathcal{F}^\bullet)) $$ converging to $H^{p + q}(\mathcal{C}, \mathcal{F}^\bullet)$.


sites-cohomology-lemma-apply-Leray

Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal{C}$-module. \begin{enumerate} \item If $R^qf_*\mathcal{F} = 0$ for $q > 0$, then $H^p(\mathcal{C}, \mathcal{F}) = H^p(\mathcal{D}, f_*\mathcal{F})$ for all $p$. \item If $H^p(\mathcal{D}, R^qf_*\mathcal{F}) = 0$ for all $q$ and $p > 0$, then $H^q(\mathcal{C}, \mathcal{F}) = H^0(\mathcal{D}, R^qf_*\mathcal{F})$ for all $q$. \end{enumerate}


sites-cohomology-lemma-relative-Leray

Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ and $g : (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) \to (\Sh(\mathcal{E}), \mathcal{O}_\mathcal{E})$ be morphisms of ringed topoi. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal{C}$-module. There is a spectral sequence with $$ E_2^{p, q} = R^pg_*(R^qf_*\mathcal{F}) $$ converging to $R^{p + q}(g \circ f)_*\mathcal{F}$. This spectral sequence is functorial in $\mathcal{F}$, and there is a version for bounded below complexes of $\mathcal{O}_\mathcal{C}$-modules.


sites-cohomology-lemma-base-change-map-flat-case

Let $$ \xymatrix{ (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_{g'} \ar[d]_{f'} & (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^f \\ (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^g & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) } $$ be a commutative diagram of ringed topoi. Let $\mathcal{F}^\bullet$ be a bounded below complex of $\mathcal{O}_\mathcal{C}$-modules. Assume both $g$ and $g'$ are flat. Then there exists a canonical base change map $$ g^*Rf_*\mathcal{F}^\bullet \longrightarrow R(f')_*(g')^*\mathcal{F}^\bullet $$ in $D^{+}(\mathcal{O}_{\mathcal{D}'})$.


sites-cohomology-lemma-colim-works-over-collection

Let $\mathcal{C}$ be a site. Let $\text{Cov}_\mathcal{C}$ be the set of coverings of $\mathcal{C}$ (see Sites, Definition \ref{sites-definition-site}). Let $\mathcal{B} \subset \Ob(\mathcal{C})$, and $\text{Cov} \subset \text{Cov}_\mathcal{C}$ be subsets. Assume that \begin{enumerate} \item For every $\mathcal{U} \in \text{Cov}$ we have $\mathcal{U} = \{U_i \to U\}_{i \in I}$ with $I$ finite, $U, U_i \in \mathcal{B}$ and every $U_{i_0} \times_U \ldots \times_U U_{i_p} \in \mathcal{B}$. \item For every $U \in \mathcal{B}$ the coverings of $U$ occurring in $\text{Cov}$ is a cofinal system of coverings of $U$. \end{enumerate} Then the map $$ \colim_i H^p(U, \mathcal{F}_i) \longrightarrow H^p(U, \colim_i \mathcal{F}_i) $$ is an isomorphism for every $p \geq 0$, every $U \in \mathcal{B}$, and every filtered diagram $\mathcal{I} \to \textit{Ab}(\mathcal{C})$.


sites-cohomology-lemma-colim-sites-injective

Let $\mathcal{I}$ be a cofiltered index category and let $(\mathcal{C}_i, f_a)$ be an inverse system of sites over $\mathcal{I}$ as in Sites, Situation \ref{sites-situation-inverse-limit-sites}. Set $\mathcal{C} = \colim \mathcal{C}_i$ as in Sites, Lemmas \ref{sites-lemma-colimit-sites} and \ref{sites-lemma-compute-pullback-to-limit}. Moreover, assume given \begin{enumerate} \item an abelian sheaf $\mathcal{F}_i$ on $\mathcal{C}_i$ for all $i \in \Ob(\mathcal{I})$, \item for $a : j \to i$ a map $\varphi_a : f_a^{-1}\mathcal{F}_i \to \mathcal{F}_j$ of abelian sheaves on $\mathcal{C}_j$ \end{enumerate} such that $\varphi_c = \varphi_b \circ f_b^{-1}\varphi_a$ whenever $c = a \circ b$. Then there exists a map of systems $(\mathcal{F}_i, \varphi_a) \to (\mathcal{G}_i, \psi_a)$ such that $\mathcal{F}_i \to \mathcal{G}_i$ is injective and $\mathcal{G}_i$ is an injective abelian sheaf.


sites-cohomology-lemma-colimit

In the situation of Lemma \ref{lemma-colim-sites-injective} set $\mathcal{F} = \colim f_i^{-1}\mathcal{F}_i$. Let $i \in \Ob(\mathcal{I})$, $X_i \in \text{Ob}(\mathcal{C}_i)$. Then $$ \colim_{a : j \to i} H^p(u_a(X_i), \mathcal{F}_j) = H^p(u_i(X_i), \mathcal{F}) $$ for all $p \geq 0$.


sites-cohomology-lemma-derived-tor-exact

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{G}^\bullet$ be a complex of $\mathcal{O}$-modules. The functors $$ K(\textit{Mod}(\mathcal{O})) \longrightarrow K(\textit{Mod}(\mathcal{O})), \quad \mathcal{F}^\bullet \longmapsto \text{Tot}(\mathcal{G}^\bullet \otimes_\mathcal{O} \mathcal{F}^\bullet) $$ and $$ K(\textit{Mod}(\mathcal{O})) \longrightarrow K(\textit{Mod}(\mathcal{O})), \quad \mathcal{F}^\bullet \longmapsto \text{Tot}(\mathcal{F}^\bullet \otimes_\mathcal{O} \mathcal{G}^\bullet) $$ are exact functors of triangulated categories.


sites-cohomology-lemma-K-flat-quasi-isomorphism

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{K}^\bullet$ be a K-flat complex. Then the functor $$ K(\textit{Mod}(\mathcal{O})) \longrightarrow K(\textit{Mod}(\mathcal{O})), \quad \mathcal{F}^\bullet \longmapsto \text{Tot}(\mathcal{F}^\bullet \otimes_\mathcal{O} \mathcal{K}^\bullet) $$ transforms quasi-isomorphisms into quasi-isomorphisms.


sites-cohomology-lemma-restriction-K-flat

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. If $\mathcal{K}^\bullet$ is a K-flat complex of $\mathcal{O}$-modules, then $\mathcal{K}^\bullet|_U$ is a K-flat complex of $\mathcal{O}_U$-modules.


sites-cohomology-lemma-tensor-product-K-flat

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. If $\mathcal{K}^\bullet$, $\mathcal{L}^\bullet$ are K-flat complexes of $\mathcal{O}$-modules, then $\text{Tot}(\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{L}^\bullet)$ is a K-flat complex of $\mathcal{O}$-modules.


sites-cohomology-lemma-K-flat-two-out-of-three

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{K}_1^\bullet, \mathcal{K}_2^\bullet, \mathcal{K}_3^\bullet)$ be a distinguished triangle in $K(\textit{Mod}(\mathcal{O}))$. If two out of three of $\mathcal{K}_i^\bullet$ are K-flat, so is the third.


sites-cohomology-lemma-K-flat-two-out-of-three-ses

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $0 \to \mathcal{K}_1^\bullet \to \mathcal{K}_2^\bullet \to \mathcal{K}_3^\bullet \to 0$ be a short exact sequence of complexes such that the terms of $\mathcal{K}_3^\bullet$ are flat $\mathcal{O}$-modules. If two out of three of $\mathcal{K}_i^\bullet$ are K-flat, so is the third.


sites-cohomology-lemma-bounded-flat-K-flat

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. A bounded above complex of flat $\mathcal{O}$-modules is K-flat.


sites-cohomology-lemma-colimit-K-flat

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{K}_1^\bullet \to \mathcal{K}_2^\bullet \to \ldots$ be a system of K-flat complexes. Then $\colim_i \mathcal{K}_i^\bullet$ is K-flat.


sites-cohomology-lemma-resolution-by-direct-sums-extensions-by-zero

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. For any complex $\mathcal{G}^\bullet$ of $\mathcal{O}$-modules there exists a commutative diagram of complexes of $\mathcal{O}$-modules $$ \xymatrix{ \mathcal{K}_1^\bullet \ar[d] \ar[r] & \mathcal{K}_2^\bullet \ar[d] \ar[r] & \ldots \\ \tau_{\leq 1}\mathcal{G}^\bullet \ar[r] & \tau_{\leq 2}\mathcal{G}^\bullet \ar[r] & \ldots } $$ with the following properties: (1) the vertical arrows are quasi-isomorphisms and termwise surjective, (2) each $\mathcal{K}_n^\bullet$ is a bounded above complex whose terms are direct sums of $\mathcal{O}$-modules of the form $j_{U!}\mathcal{O}_U$, and (3) the maps $\mathcal{K}_n^\bullet \to \mathcal{K}_{n + 1}^\bullet$ are termwise split injections whose cokernels are direct sums of $\mathcal{O}$-modules of the form $j_{U!}\mathcal{O}_U$. Moreover, the map $\colim \mathcal{K}_n^\bullet \to \mathcal{G}^\bullet$ is a quasi-isomorphism.


sites-cohomology-lemma-K-flat-resolution

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. For any complex $\mathcal{G}^\bullet$ there exists a $K$-flat complex $\mathcal{K}^\bullet$ whose terms are flat $\mathcal{O}$-modules and a quasi-isomorphism $\mathcal{K}^\bullet \to \mathcal{G}^\bullet$ which is termwise surjective.


sites-cohomology-lemma-derived-tor-quasi-isomorphism-other-side

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\alpha : \mathcal{P}^\bullet \to \mathcal{Q}^\bullet$ be a quasi-isomorphism of K-flat complexes of $\mathcal{O}$-modules. For every complex $\mathcal{F}^\bullet$ of $\mathcal{O}$-modules the induced map $$ \text{Tot}(\text{id}_{\mathcal{F}^\bullet} \otimes \alpha) : \text{Tot}(\mathcal{F}^\bullet \otimes_\mathcal{O} \mathcal{P}^\bullet) \longrightarrow \text{Tot}(\mathcal{F}^\bullet \otimes_\mathcal{O} \mathcal{Q}^\bullet) $$ is a quasi-isomorphism.


sites-cohomology-lemma-flat-tor-zero

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be an $\mathcal{O}$-module. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is a flat $\mathcal{O}$-module, and \item $\text{Tor}_1^\mathcal{O}(\mathcal{F}, \mathcal{G}) = 0$ for every $\mathcal{O}$-module $\mathcal{G}$. \end{enumerate}


sites-cohomology-lemma-K-flat-flat-acyclic

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{K}^\bullet$ be a K-flat, acyclic complex with flat terms. Then $\mathcal{F} = \Ker(\mathcal{K}^n \to \mathcal{K}^{n + 1})$ is a flat $\mathcal{O}$-module.


sites-cohomology-lemma-pullback-K-flat

Let $f : (\Sh(\mathcal{C}'), \mathcal{O}') \to (\Sh(\mathcal{C}), \mathcal{O})$ be a morphism of ringed topoi. Let $\mathcal{K}^\bullet$ be a K-flat complex of $\mathcal{O}$-modules whose terms are flat $\mathcal{O}$-modules. Then $f^*\mathcal{K}^\bullet$ is a K-flat complex of $\mathcal{O}'$-modules whose terms are flat $\mathcal{O}'$-modules.


sites-cohomology-lemma-derived-base-change

Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. There exists an exact functor $$ Lf^* : D(\mathcal{O}') \longrightarrow D(\mathcal{O}) $$ of triangulated categories so that $Lf^*\mathcal{K}^\bullet = f^*\mathcal{K}^\bullet$ for any K-flat complex $\mathcal{K}^\bullet$ with flat terms and in particular for any bounded above complex of flat $\mathcal{O}'$-modules.


sites-cohomology-lemma-derived-pullback-composition

Consider morphisms of ringed topoi $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ and $g : (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) \to (\Sh(\mathcal{E}), \mathcal{O}_\mathcal{E})$. Then $Lf^* \circ Lg^* = L(g \circ f)^*$ as functors $D(\mathcal{O}_\mathcal{E}) \to D(\mathcal{O}_\mathcal{C})$.


sites-cohomology-lemma-pullback-tensor-product

Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$ be a morphism of ringed topoi. There is a canonical bifunctorial isomorphism $$ Lf^*( \mathcal{F}^\bullet \otimes_{\mathcal{O}'}^{\mathbf{L}} \mathcal{G}^\bullet ) = Lf^*\mathcal{F}^\bullet  \otimes_{\mathcal{O}}^{\mathbf{L}} Lf^*\mathcal{G}^\bullet  $$ for $\mathcal{F}^\bullet, \mathcal{G}^\bullet \in \Ob(D(\mathcal{O}'))$.


sites-cohomology-lemma-variant-derived-pullback

Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. There is a canonical bifunctorial isomorphism $$ \mathcal{F}^\bullet \otimes_\mathcal{O}^{\mathbf{L}} Lf^*\mathcal{G}^\bullet = \mathcal{F}^\bullet  \otimes_{f^{-1}\mathcal{O}_Y}^{\mathbf{L}} f^{-1}\mathcal{G}^\bullet  $$ for $\mathcal{F}^\bullet$ in $D(\mathcal{O})$ and $\mathcal{G}^\bullet$ in $D(\mathcal{O}')$.


sites-cohomology-lemma-check-K-flat-stalks

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{K}^\bullet$ be a complex of $\mathcal{O}$-modules. \begin{enumerate} \item If $\mathcal{K}^\bullet$ is K-flat, then for every point $p$ of the site $\mathcal{C}$ the complex of $\mathcal{O}_p$-modules $\mathcal{K}_p^\bullet$ is K-flat in the sense of More on Algebra, Definition \ref{more-algebra-definition-K-flat} \item If $\mathcal{C}$ has enough points, then the converse is true. \end{enumerate}


sites-cohomology-lemma-pullback-K-flat-points

Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. If $\mathcal{C}$ has enough points, then the pullback of a K-flat complex of $\mathcal{O}'$-modules is a K-flat complex of $\mathcal{O}$-modules.


sites-cohomology-lemma-tensor-pull-compatibility

Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. Let $\mathcal{K}^\bullet$ and $\mathcal{M}^\bullet$ be complexes of $\mathcal{O}_\mathcal{D}$-modules. The diagram $$ \xymatrix{ Lf^*(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} \mathcal{M}^\bullet) \ar[r] \ar[d] & Lf^*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}} \mathcal{M}^\bullet) \ar[d] \\ Lf^*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} Lf^*\mathcal{M}^\bullet \ar[d] & f^*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}} \mathcal{M}^\bullet) \ar[d] \\ f^*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} f^*\mathcal{M}^\bullet \ar[r] & \text{Tot}(f^*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} f^*\mathcal{M}^\bullet) } $$ commutes.


sites-cohomology-lemma-adjoint

Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$ be a morphism of ringed topoi. The functor $Rf_*$ defined above and the functor $Lf^*$ defined in Lemma \ref{lemma-derived-base-change} are adjoint: $$ \Hom_{D(\mathcal{O})}(Lf^*\mathcal{G}^\bullet, \mathcal{F}^\bullet) = \Hom_{D(\mathcal{O}')}(\mathcal{G}^\bullet, Rf_*\mathcal{F}^\bullet) $$ bifunctorially in $\mathcal{F}^\bullet \in \Ob(D(\mathcal{O}))$ and $\mathcal{G}^\bullet \in \Ob(D(\mathcal{O}'))$.


sites-cohomology-lemma-derived-pushforward-composition

Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ and $g : (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) \to (\Sh(\mathcal{E}), \mathcal{O}_\mathcal{E})$ be morphisms of ringed topoi. Then $Rg_* \circ Rf_* = R(g \circ f)_*$ as functors $D(\mathcal{O}_\mathcal{C}) \to D(\mathcal{O}_\mathcal{E})$.


sites-cohomology-lemma-adjoints-push-pull-compatibility

Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. Let $\mathcal{K}^\bullet$ be a complex of $\mathcal{O}_\mathcal{C}$-modules. The diagram $$ \xymatrix{ Lf^*f_*\mathcal{K}^\bullet \ar[r] \ar[d] & f^*f_*\mathcal{K}^\bullet \ar[d] \\ Lf^*Rf_*\mathcal{K}^\bullet \ar[r] & \mathcal{K}^\bullet } $$ coming from $Lf^* \to f^*$ on complexes, $f_* \to Rf_*$ on complexes, and adjunction $Lf^* \circ Rf_* \to \text{id}$ commutes in $D(\mathcal{O}_\mathcal{C})$.


sites-cohomology-lemma-torsion

Let $\mathcal{C}$ be a site. Let $\mathcal{A} \subset \textit{Ab}(\mathcal{C})$ denote the Serre subcategory consisting of torsion abelian sheaves. Then the functor $D(\mathcal{A}) \to D_\mathcal{A}(\mathcal{C})$ is an equivalence.


sites-cohomology-lemma-restrict-K-injective-to-open

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. The restriction of a K-injective complex of $\mathcal{O}$-modules to $\mathcal{C}/U$ is a K-injective complex of $\mathcal{O}_U$-modules.


sites-cohomology-lemma-unbounded-cohomology-of-open

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \Ob(\mathcal{C})$. For $K$ in $D(\mathcal{O})$ we have $H^p(U, K) = H^p(\mathcal{C}/U, K|_{\mathcal{C}/U})$.


sites-cohomology-lemma-sheafification-cohomology

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K$ be an object of $D(\mathcal{O})$. The sheafification of $$ U \mapsto H^q(U, K) = H^q(\mathcal{C}/U, K|_{\mathcal{C}/U}) $$ is the $q$th cohomology sheaf $H^q(K)$ of $K$.


sites-cohomology-lemma-restrict-direct-image-open

Let $f : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D})$ be a morphism of ringed sites corresponding to the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Given $V \in \mathcal{D}$, set $U = u(V)$ and denote $g : (\mathcal{C}/U, \mathcal{O}_U) \to (\mathcal{D}/V, \mathcal{O}_V)$ the induced morphism of ringed sites (Modules on Sites, Lemma \ref{sites-modules-lemma-localize-morphism-ringed-sites}). Then $(Rf_*E)|_{\mathcal{D}/V} = Rg_*(E|_{\mathcal{C}/U})$ for $E$ in $D(\mathcal{O}_\mathcal{C})$.


sites-cohomology-lemma-Leray-unbounded

Let $f : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D})$ be a morphism of ringed sites corresponding to the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Then $R\Gamma(\mathcal{D}, -) \circ Rf_* = R\Gamma(\mathcal{C}, -)$ as functors $D(\mathcal{O}_\mathcal{C}) \to D(\Gamma(\mathcal{O}_\mathcal{D}))$. More generally, for $V \in \mathcal{D}$ with $U = u(V)$ we have $R\Gamma(U, -) = R\Gamma(V, -) \circ Rf_*$.


sites-cohomology-lemma-unbounded-describe-higher-direct-images

Let $f : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D})$ be a morphism of ringed sites corresponding to the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $K$ be in $D(\mathcal{O}_\mathcal{C})$. Then $H^i(Rf_*K)$ is the sheaf associated to the presheaf $$ V \mapsto H^i(u(V), K) = H^i(V, Rf_*K) $$


sites-cohomology-lemma-modules-abelian-unbounded

Let $(\mathcal{C}, \mathcal{O}_\mathcal{C})$ be a ringed site. Let $K$ be an object of $D(\mathcal{O}_\mathcal{C})$ and denote $K_{ab}$ its image in $D(\underline{\mathbf{Z}}_\mathcal{C})$. \begin{enumerate} \item There is a canonical map $R\Gamma(\mathcal{C}, K) \to R\Gamma(\mathcal{C}, K_{ab})$ which is an isomorphism in $D(\textit{Ab})$. \item For any $U \in \mathcal{C}$ there is a canonical map $R\Gamma(U, K) \to R\Gamma(U, K_{ab})$ which is an isomorphism in $D(\textit{Ab})$. \item Let $f : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D})$ be a morphism of ringed sites. There is a canonical map $Rf_*K \to Rf_*(K_{ab})$ which is an isomorphism in $D(\underline{\mathbf{Z}}_\mathcal{D})$. \end{enumerate}


sites-cohomology-lemma-adjoint-lower-shriek-restrict

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. Denote $j : (\Sh(\mathcal{C}/U), \mathcal{O}_U) \to (\Sh(\mathcal{C}), \mathcal{O})$ the corresponding localization morphism. The restriction functor $D(\mathcal{O}) \to D(\mathcal{O}_U)$ is a right adjoint to extension by zero $j_! : D(\mathcal{O}_U) \to D(\mathcal{O})$.


sites-cohomology-lemma-K-injective-flat

Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a flat morphism of ringed topoi. If $\mathcal{I}^\bullet$ is a K-injective complex of $\mathcal{O}_\mathcal{C}$-modules, then $f_*\mathcal{I}^\bullet$ is K-injective as a complex of $\mathcal{O}_\mathcal{D}$-modules.


sites-cohomology-lemma-hom-K-injective

Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}'$ be a map of sheaves of rings. If $\mathcal{I}^\bullet$ is a K-injective complex of $\mathcal{O}$-modules, then $\SheafHom_\mathcal{O}(\mathcal{O}', \mathcal{I}^\bullet)$ is a K-injective complex of $\mathcal{O}'$-modules.


sites-cohomology-lemma-localize-cartesian-square

Let $\mathcal{C}$ be a site. Let $$ \xymatrix{ X' \ar[d] \ar[r] & X \ar[d] \\ Y' \ar[r] & Y } $$ be a cartesian diagram of $\mathcal{C}$. Then we have $j_{Y'/Y}^{-1} \circ Rj_{X/Y, *} = Rj_{X'/Y', *} \circ j_{X'/X}^{-1}$ as functors $D(\mathcal{C}/X) \to D(\mathcal{C}/Y')$.


sites-cohomology-lemma-localize-cartesian-square-modules

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $$ \xymatrix{ X' \ar[d] \ar[r] & X \ar[d] \\ Y' \ar[r] & Y } $$ be a cartesian diagram of $\mathcal{C}$. Then we have $j_{Y'/Y}^* \circ Rj_{X/Y, *} = Rj_{X'/Y', *} \circ j_{X'/X}^*$ as functors $D(\mathcal{O}_X) \to D(\mathcal{O}_{Y'})$.


sites-cohomology-lemma-derived-limit-is-ok

Let $\mathcal{C}$ be a site. Let $K$ be an object of $D(\mathcal{C} \times \mathbf{N})$. Set $K_n = i_n^{-1}K$ as above. Then $$ R\lim K \cong R\lim K_n $$ in $D(\mathcal{C})$.


sites-cohomology-lemma-RGamma-commutes-with-Rlim

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. The functors $R\Gamma(\mathcal{C}, -)$ and $R\Gamma(U, -)$ for $U \in \Ob(\mathcal{C})$ commute with $R\lim$. Moreover, there are short exact sequences $$ 0 \to R^1\lim H^{m - 1}(U, K_n) \to H^m(U, R\lim K_n) \to \lim H^m(U, K_n) \to 0 $$ for any inverse system $(K_n)$ in $D(\mathcal{O})$ and $m \in \mathbf{Z}$. Similar for $H^m(\mathcal{C}, R\lim K_n)$.


sites-cohomology-lemma-Rf-commutes-with-Rlim

Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. Then $Rf_*$ commutes with $R\lim$, i.e., $Rf_*$ commutes with derived limits.


sites-cohomology-lemma-inverse-limit-is-derived-limit

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{F}_n)$ be an inverse system of $\mathcal{O}$-modules. Let $\mathcal{B} \subset \Ob(\mathcal{C})$ be a subset. Assume \begin{enumerate} \item every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$, \item $H^p(U, \mathcal{F}_n) = 0$ for $p > 0$ and $U \in \mathcal{B}$, \item the inverse system $\mathcal{F}_n(U)$ has vanishing $R^1\lim$ for $U \in \mathcal{B}$. \end{enumerate} Then $R\lim \mathcal{F}_n = \lim \mathcal{F}_n$ and we have $H^p(U, \lim \mathcal{F}_n) = 0$ for $p > 0$ and $U \in \mathcal{B}$.


sites-cohomology-lemma-cohomology-derived-limit-injective

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(K_n)$ be an inverse system in $D(\mathcal{O})$. Let $V \in \Ob(\mathcal{C})$ and $m \in \mathbf{Z}$. Assume there exist an integer $n(V)$ and a cofinal system $\text{Cov}_V$ of coverings of $V$ such that for $\{V_i \to V\} \in \text{Cov}_V$ \begin{enumerate} \item $R^1\lim H^{m - 1}(V_i, K_n) = 0$, and \item $H^m(V_i, K_n) \to H^m(V_i, K_{n(V)})$ is injective for $n \geq n(V)$. \end{enumerate} Then the map on sections $H^m(R\lim K_n)(V) \to H^m(K_{n(V)})(V)$ is injective.


sites-cohomology-lemma-is-limit-per-object

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E \in D(\mathcal{O})$. Let $\mathcal{B} \subset \Ob(\mathcal{C})$ be a subset. Assume \begin{enumerate} \item every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$, and \item for every $V \in \mathcal{B}$ there exist a function $p(V, -) : \mathbf{Z} \to \mathbf{Z}$ and a cofinal system $\text{Cov}_V$ of coverings of $V$ such that $$ H^p(V_i, H^{m - p}(E)) = 0 $$ for all $\{V_i \to V\} \in \text{Cov}_V$ and all integers $p, m$ satisfying $p > p(V, m)$. \end{enumerate} Then the canonical map $E \to R\lim \tau_{\geq -n} E$ is an isomorphism in $D(\mathcal{O})$.


sites-cohomology-lemma-is-limit-spaltenstein

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E \in D(\mathcal{O})$. Let $\mathcal{B} \subset \Ob(\mathcal{C})$ be a subset. Assume \begin{enumerate} \item every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$, and \item for every $V \in \mathcal{B}$ there exist an integer $d_V \geq 0$ and a cofinal system $\text{Cov}_V$ of coverings of $V$ such that $$ H^p(V_i, H^q(E)) = 0 \text{ for } \{V_i \to V\} \in \text{Cov}_V,\ p > d_V, \text{ and }q < 0 $$ \end{enumerate} Then the canonical map $E \to R\lim \tau_{\geq -n} E$ is an isomorphism in $D(\mathcal{O})$.


sites-cohomology-lemma-is-limit

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E \in D(\mathcal{O})$. Assume there exists a function $p(-) : \mathbf{Z} \to \mathbf{Z}$ and a subset $\mathcal{B} \subset \Ob(\mathcal{C})$ such that \begin{enumerate} \item every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$, \item $H^p(V, H^{m - p}(E)) = 0$ for $p > p(m)$ and $V \in \mathcal{B}$. \end{enumerate} Then the canonical map $E \to R\lim \tau_{\geq -n} E$ is an isomorphism in $D(\mathcal{O})$.


sites-cohomology-lemma-is-limit-dimension

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E \in D(\mathcal{O})$. Assume there exists an integer $d \geq 0$ and a subset $\mathcal{B} \subset \Ob(\mathcal{C})$ such that \begin{enumerate} \item every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$, \item $H^p(V, H^q(E)) = 0$ for $p > d$, $q < 0$, and $V \in \mathcal{B}$. \end{enumerate} Then the canonical map $E \to R\lim \tau_{\geq -n} E$ is an isomorphism in $D(\mathcal{O})$.


sites-cohomology-lemma-cohomology-over-U-trivial

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K$ be an object of $D(\mathcal{O})$. Let $\mathcal{B} \subset \Ob(\mathcal{C})$ be a subset. Assume \begin{enumerate} \item every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$, \item $H^p(U, H^q(K)) = 0$ for all $p > 0$, $q \in \mathbf{Z}$, and $U \in \mathcal{B}$. \end{enumerate} Then $H^q(U, K) = H^0(U, H^q(K))$ for $q \in \mathbf{Z}$ and $U \in \mathcal{B}$.


sites-cohomology-lemma-derived-limit-suitable-system

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(K_n)$ be an inverse system of objects of $D(\mathcal{O})$. Let $\mathcal{B} \subset \Ob(\mathcal{C})$ be a subset. Assume \begin{enumerate} \item every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$, \item for all $U \in \mathcal{B}$ and all $q \in \mathbf{Z}$ we have \begin{enumerate} \item $H^p(U, H^q(K_n)) = 0$ for $p > 0$, \item the inverse system $H^0(U, H^q(K_n))$ has vanishing $R^1\lim$. \end{enumerate} \end{enumerate} Then $H^q(R\lim K_n) = \lim H^q(K_n)$ for $q \in \mathbf{Z}$.


sites-cohomology-lemma-K-injective

In the situation described above. Denote $\mathcal{H}^m = H^m(\mathcal{F}^\bullet)$ the $m$th cohomology sheaf. Let $\mathcal{B} \subset \Ob(\mathcal{C})$ be a subset. Let $d \in \mathbf{N}$. Assume \begin{enumerate} \item every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$, \item for every $U \in \mathcal{B}$ we have $H^p(U, \mathcal{H}^q) = 0$ for $p > d$ and $q < 0$\footnote{It suffices if $\forall m$, $\exists p(m)$, $H^p(U. \mathcal{H}^{m - p}) = 0$ for $p > p(m)$, see Lemma \ref{lemma-is-limit}.}. \end{enumerate} Then (\ref{equation-into-candidate-K-injective}) is a quasi-isomorphism.


sites-cohomology-lemma-olsson-laszlo

\begin{reference} This is \cite[Proposition 2.1.4]{six-I} with slightly changed hypotheses; it is the analogue of \cite[Proposition 3.13]{Spaltenstein} for sites. \end{reference} In Situation \ref{situation-olsson-laszlo} for any $E \in D_\mathcal{A}(\mathcal{O})$ the canonical map $E \to R\lim \tau_{\geq -n} E$ is an isomorphism in $D(\mathcal{O})$.


sites-cohomology-lemma-olsson-laszlo-modified

In Situation \ref{situation-olsson-laszlo} let $(K_n)$ be an inverse system in $D_\mathcal{A}^+(\mathcal{O})$. Assume that for every $j$ the inverse system $(H^j(K_n))$ in $\mathcal{A}$ is eventually constant with value $\mathcal{H}^j$. Then $H^j(R\lim K_n) = \mathcal{H}^j$ for all $j$.