sites-cohomology-lemma-olsson-laszlo-map-version-one

\begin{reference} This is a version of \cite[Lemma 2.1.10]{six-I} with slightly changed hypotheses. \end{reference} Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$ and $\mathcal{A}' \subset \textit{Mod}(\mathcal{O}')$ be weak Serre subcategories. Assume there is an integer $N$ such that \begin{enumerate} \item $\mathcal{C}, \mathcal{O}, \mathcal{A}$ satisfy the assumption of Situation \ref{situation-olsson-laszlo}, \item $\mathcal{C}', \mathcal{O}', \mathcal{A}'$ satisfy the assumption of Situation \ref{situation-olsson-laszlo}, \item $R^pf_*\mathcal{F} \in \Ob(\mathcal{A}')$ for $p \geq 0$ and $\mathcal{F} \in \Ob(\mathcal{A})$, \item $R^pf_*\mathcal{F} = 0$ for $p > N$ and $\mathcal{F} \in \Ob(\mathcal{A})$, \end{enumerate} Then for $K$ in $D_\mathcal{A}(\mathcal{O})$ we have \begin{enumerate} \item[(a)] $Rf_*K$ is in $D_{\mathcal{A}'}(\mathcal{O}')$, \item[(b)] the map $H^j(Rf_*K) \to H^j(Rf_*(\tau_{\geq -n}K))$ is an isomorphism for $j \geq N - n$. \end{enumerate}


sites-cohomology-lemma-olsson-laszlo-map-version-two

\begin{reference} This is a version of \cite[Lemma 2.1.10]{six-I} with slightly changed hypotheses. \end{reference} Let $f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')$ be a morphism of ringed sites. assume moreover there is an integer $N$ such that \begin{enumerate} \item $\mathcal{C}, \mathcal{O}, \mathcal{A}$ satisfy the assumption of Situation \ref{situation-olsson-laszlo}, \item $f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')$ and $\mathcal{A}$ satisfy the assumption of Situation \ref{situation-olsson-laszlo-prime}, \item $R^pf_*\mathcal{F} = 0$ for $p > N$ and $\mathcal{F} \in \Ob(\mathcal{A})$, \end{enumerate} Then for $K$ in $D_\mathcal{A}(\mathcal{O})$ the map $H^j(Rf_*K) \to H^j(Rf_*(\tau_{\geq -n}K))$ is an isomorphism for $j \geq N - n$.


sites-cohomology-lemma-c-square

In the situation above, choose a K-injective complex $\mathcal{I}^\bullet$ of $\mathcal{O}$-modules representing $K$. Using $-1$ times the canonical map for one of the four arrows we get maps of complexes $$ \mathcal{I}^\bullet(X) \xrightarrow{\alpha} \mathcal{I}^\bullet(Z) \oplus \mathcal{I}^\bullet(Y) \xrightarrow{\beta} \mathcal{I}^\bullet(E) $$ with $\beta \circ \alpha = 0$. Thus a canonical map $$ c^K_{X, Z, Y, E} : \mathcal{I}^\bullet(X) \longrightarrow C(\beta)^\bullet[-1] $$ This map is canonical in the sense that a different choice of K-injective complex representing $K$ determines an isomorphic arrow in the derived category of abelian groups. If $c^K_{X, Z, Y, E}$ is an isomorphism, then using its inverse we obtain a canonical distinguished triangle $$ R\Gamma(X, K) \to R\Gamma(Z, K) \oplus R\Gamma(Y, K) \to R\Gamma(E, K) \to R\Gamma(X, K)[1] $$ All of these constructions are functorial in $K$.


sites-cohomology-lemma-two-out-of-three-blow-up-square

In the situation above, let $K_1 \to K_2 \to K_3 \to K_1[1]$ be a distinguished triangle in $D(\mathcal{O})$. If $c^{K_i}_{X, Z, Y, E}$ is a quasi-isomorphism for two $i$ out of $\{1, 2, 3\}$, then it is a quasi-isomorphism for the third $i$.


sites-cohomology-lemma-square-triangle

In the situation above assume \begin{enumerate} \item $h_X^\# = h_Y^\# \amalg_{h_E^\#} h_Z^\#$, and \item $h_E^\# \to h_Y^\#$ is injective. \end{enumerate} Then the construction of Lemma \ref{lemma-c-square} produces a distinguished triangle $$ R\Gamma(X, K) \to R\Gamma(Z, K) \oplus R\Gamma(Y, K) \to R\Gamma(E, K) \to R\Gamma(X, K)[1] $$ functorial for $K$ in $D(\mathcal{C})$.


sites-cohomology-lemma-downstairs

Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. Consider the full subcategory $D' \subset D(\mathcal{O}_\mathcal{D})$ consisting of objects $K$ such that $$ K \longrightarrow Rf_*Lf^*K $$ is an isomorphism. Then $D'$ is a saturated triangulated strictly full subcategory of $D(\mathcal{O}_\mathcal{D})$ and the functor $Lf^* : D' \to D(\mathcal{O}_\mathcal{C})$ is fully faithful.


sites-cohomology-lemma-upstairs

Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. Consider the full subcategory $D' \subset D(\mathcal{O}_\mathcal{C})$ consisting of objects $K$ such that $$ Lf^*Rf_*K \longrightarrow K $$ is an isomorphism. Then $D'$ is a saturated triangulated strictly full subcategory of $D(\mathcal{O}_\mathcal{C})$ and the functor $Rf_* : D' \to D(\mathcal{O}_\mathcal{D})$ is fully faithful.


sites-cohomology-lemma-bounded-in-image-upstairs

Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. Let $K$ be an object of $D(\mathcal{O}_\mathcal{C})$. Assume \begin{enumerate} \item $f$ is flat, \item $K$ is bounded below, \item $f^*Rf_*H^q(K) \to H^q(K)$ is an isomorphism. \end{enumerate} Then $f^*Rf_*K \to K$ is an isomorphism.


sites-cohomology-lemma-bounded-in-image-downstairs

Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. Let $K$ be an object of $D(\mathcal{O}_\mathcal{D})$. Assume \begin{enumerate} \item $f$ is flat, \item $K$ is bounded below, \item $H^q(K) \to Rf_*f^*H^q(K)$ is an isomorphism. \end{enumerate} Then $K \to Rf_*f^*K$ is an isomorphism.


sites-cohomology-lemma-equivalence-bounded

Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$ and $\mathcal{A}' \subset \textit{Mod}(\mathcal{O}')$ be weak Serre subcategories. Assume \begin{enumerate} \item $f$ is flat, \item $f^*$ induces an equivalence of categories $\mathcal{A}' \to \mathcal{A}$, \item $\mathcal{F}' \to Rf_*f^*\mathcal{F}'$ is an isomorphism for $\mathcal{F}' \in \Ob(\mathcal{A}')$. \end{enumerate} Then $f^* : D_{\mathcal{A}'}^+(\mathcal{O}') \to D_\mathcal{A}^+(\mathcal{O})$ is an equivalence of categories with quasi-inverse given by $Rf_* : D_\mathcal{A}^+(\mathcal{O}) \to D_{\mathcal{A}'}^+(\mathcal{O}')$.


sites-cohomology-lemma-equivalence-unbounded-one

\begin{reference} This is analogous to \cite[Theorem 2.2.3]{six-I}. \end{reference} Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$ and $\mathcal{A}' \subset \textit{Mod}(\mathcal{O}')$ be weak Serre subcategories. Assume \begin{enumerate} \item $f$ is flat, \item $f^*$ induces an equivalence of categories $\mathcal{A}' \to \mathcal{A}$, \item $\mathcal{F}' \to Rf_*f^*\mathcal{F}'$ is an isomorphism for $\mathcal{F}' \in \Ob(\mathcal{A}')$, \item $\mathcal{C}, \mathcal{O}, \mathcal{A}$ satisfy the assumption of Situation \ref{situation-olsson-laszlo}, \item $\mathcal{C}', \mathcal{O}', \mathcal{A}'$ satisfy the assumption of Situation \ref{situation-olsson-laszlo}. \end{enumerate} Then $f^* : D_{\mathcal{A}'}(\mathcal{O}') \to D_\mathcal{A}(\mathcal{O})$ is an equivalence of categories with quasi-inverse given by $Rf_* : D_\mathcal{A}(\mathcal{O}) \to D_{\mathcal{A}'}(\mathcal{O}')$.


sites-cohomology-lemma-equivalence-unbounded-two

\begin{reference} This is analogous to \cite[Theorem 2.2.3]{six-I}. \end{reference} Let $f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')$ be a morphism of ringed sites. Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$ and $\mathcal{A}' \subset \textit{Mod}(\mathcal{O}')$ be weak Serre subcategories. Assume \begin{enumerate} \item $f$ is flat, \item $f^*$ induces an equivalence of categories $\mathcal{A}' \to \mathcal{A}$, \item $\mathcal{F}' \to Rf_*f^*\mathcal{F}'$ is an isomorphism for $\mathcal{F}' \in \Ob(\mathcal{A}')$, \item $\mathcal{C}, \mathcal{O}, \mathcal{A}$ satisfy the assumption of Situation \ref{situation-olsson-laszlo}, \item $f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')$ and $\mathcal{A}$ satisfy the assumption of Situation \ref{situation-olsson-laszlo-prime}. \end{enumerate} Then $f^* : D_{\mathcal{A}'}(\mathcal{O}') \to D_\mathcal{A}(\mathcal{O})$ is an equivalence of categories with quasi-inverse given by $Rf_* : D_\mathcal{A}(\mathcal{O}) \to D_{\mathcal{A}'}(\mathcal{O}')$.


sites-cohomology-lemma-compare-topologies-derived-adequate-modules

With $\epsilon : (\mathcal{C}_\tau, \mathcal{O}_\tau) \to (\mathcal{C}_{\tau'}, \mathcal{O}_{\tau'})$ as above. Let $\mathcal{B} \subset \Ob(\mathcal{C})$ be a subset. Let $\mathcal{A} \subset \textit{PMod}(\mathcal{O})$ be a full subcategory. Assume \begin{enumerate} \item every object of $\mathcal{A}$ is a sheaf for the $\tau$-topology, \item $\mathcal{A}$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_\tau)$, \item every object of $\mathcal{C}$ has a $\tau'$-covering whose members are elements of $\mathcal{B}$, and \item for every $U \in \mathcal{B}$ we have $H^p_\tau(U, \mathcal{F}) = 0$, $p > 0$ for all $\mathcal{F} \in \mathcal{A}$. \end{enumerate} Then $\mathcal{A}$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_{\tau'})$ and there is an equivalence of triangulated categories $D_\mathcal{A}(\mathcal{O}_\tau) = D_\mathcal{A}(\mathcal{O}_{\tau'})$ given by $\epsilon^*$ and $R\epsilon_*$.


sites-cohomology-lemma-descent-squares

With $\epsilon : (\mathcal{C}_\tau, \mathcal{O}_\tau) \to (\mathcal{C}_{\tau'}, \mathcal{O}_{\tau'})$ as above. Let $A$ be a set and for $\alpha \in A$ let $$ \xymatrix{ E_\alpha \ar[d] \ar[r] & Y_\alpha \ar[d] \\ Z_\alpha \ar[r] & X_\alpha } $$ be a commutative diagram in the category $\mathcal{C}$. Assume that \begin{enumerate} \item a $\tau'$-sheaf $\mathcal{F}'$ is a $\tau$-sheaf if $\mathcal{F}'(X_\alpha) = \mathcal{F}'(Z_\alpha) \times_{\mathcal{F}'(E_\alpha)}  \mathcal{F}'(Y_\alpha)$ for all $\alpha$, \item for $K'$ in $D(\mathcal{O}_{\tau'})$ in the essential image of $R\epsilon_*$ the maps $c^{K'}_{X_\alpha, Z_\alpha, Y_\alpha, E_\alpha}$ of Lemma \ref{lemma-c-square} are isomorphisms for all $\alpha$. \end{enumerate} Then $K' \in D^+(\mathcal{O}_{\tau'})$ is in the essential image of $R\epsilon_*$ if and only if the maps $c^{K'}_{X_\alpha, Z_\alpha, Y_\alpha, E_\alpha}$ are isomorphisms for all $\alpha$.


sites-cohomology-lemma-descent-squares-helper

With $\epsilon : (\mathcal{C}_\tau, \mathcal{O}_\tau) \to (\mathcal{C}_{\tau'}, \mathcal{O}_{\tau'})$ as above. Let $$ \xymatrix{ E \ar[d] \ar[r] & Y \ar[d] \\ Z \ar[r] & X } $$ be a commutative diagram in the category $\mathcal{C}$ such that \begin{enumerate} \item $h_X^\# = h_Y^\# \amalg_{h_E^\#} h_Z^\#$, and \item $h_E^\# \to h_Y^\#$ is injective \end{enumerate} where ${}^\#$ denotes $\tau$-sheafification. Then for $K' \in D(\mathcal{O}_{\tau'})$ in the essential image of $R\epsilon_*$ the map $c^{K'}_{X, Z, Y, E}$ of Lemma \ref{lemma-c-square} (using the $\tau'$-topology) is an isomorphism.


sites-cohomology-lemma-A

In Situation \ref{situation-compare} for $X$ in $\mathcal{C}$ denote $\mathcal{A}_X$ the objects of $\textit{Ab}(\mathcal{C}_\tau/X)$ of the form $\epsilon_X^{-1}\mathcal{F}'$ with $\mathcal{F}'$ in $\mathcal{A}'_X$. Then \begin{enumerate} \item for $\mathcal{F}$ in $\textit{Ab}(\mathcal{C}_\tau/X)$ we have $\mathcal{F} \in \mathcal{A}_X \Leftrightarrow \epsilon_{X, *}\mathcal{F} \in \mathcal{A}'_X$, and \item $f_\tau^{-1}$ sends $\mathcal{A}_Y$ into $\mathcal{A}_X$ for any morphism $f : X \to Y$ of $\mathcal{C}$. \end{enumerate}


sites-cohomology-lemma-V-implies-C-general

In Situation \ref{situation-compare} assume $(V_n)$ holds. For $f : X \to Y$ in $\mathcal{P}$ and $\mathcal{F}$ in $\mathcal{A}_X$ we have $R^if_{\tau', *}\epsilon_{X, *}\mathcal{F} = \epsilon_{Y, *}R^if_{\tau, *}\mathcal{F}$ for $i \leq n$.


sites-cohomology-lemma-V-implies-cohomology-general

In Situation \ref{situation-compare} if $(V_n)$ holds, then for $X$ in $\mathcal{C}$ and $L \in D(\mathcal{C}_{\tau'}/X)$ with $H^i(L) = 0$ for $i < 0$ and $H^i(L)$ in $\mathcal{A}'_X$ for $0 \leq i \leq n$ we have $H^n_{\tau'}(X, L) = H^n_\tau(X, \epsilon_X^{-1}L)$.


sites-cohomology-lemma-V-implies-cohomology-extra-general

In Situation \ref{situation-compare} if $(V_n)$ holds, then for $X$ in $\mathcal{C}$ and $\mathcal{F}$ in $\mathcal{A}_X$ the map $H^{n + 1}_{\tau'}(X, \epsilon_{X, *}\mathcal{F}) \to H^{n + 1}_\tau(X, \mathcal{F})$ is injective with image those classes which become trivial on a $\tau'$-covering of $X$.


sites-cohomology-lemma-make-class-zero-general

In Situation \ref{situation-compare} let $f : X \to Y$ be in $\mathcal{P}$ such that $\{X \to Y\}$ is a $\tau$-covering. Let $\mathcal{F}'$ be in $\mathcal{A}'_Y$. If $n \geq 0$ and $$ \theta \in \text{Equalizer}\left( \xymatrix{ H^{n + 1}_{\tau'}(X, \mathcal{F}') \ar@<1ex>[r] \ar@<-1ex>[r] & H^{n + 1}_{\tau'}(X \times_Y X, \mathcal{F}') } \right) $$ then there exists a $\tau'$-covering $\{Y_i \to Y\}$ such that $\theta$ restricts to zero in $H^{n + 1}_{\tau'}(Y_i \times_Y X, \mathcal{F}')$.


sites-cohomology-lemma-induction-step-V-C-general

In Situation \ref{situation-compare} we have $(V_n) \Rightarrow (V_{n + 1})$.


sites-cohomology-lemma-V-C-all-n-general

In Situation \ref{situation-compare} we have that $(V_n)$ is true for all $n$. Moreover: \begin{enumerate} \item For $X$ in $\mathcal{C}$ and $K' \in D^+_{\mathcal{A}'_X}(\mathcal{C}_{\tau'}/X)$ the map $K' \to R\epsilon_{X, *}(\epsilon_X^{-1}K')$ is an isomorphism. \item For $f : X \to Y$ in $\mathcal{P}$ and $K' \in D^+_{\mathcal{A}'_X}(\mathcal{C}_{\tau'}/X)$ we have $Rf_{\tau', *}K' \in D^+_{\mathcal{A}'_X}(\mathcal{C}_{\tau'}/Y)$ and $\epsilon_Y^{-1}(Rf_{\tau', *}K') = Rf_{\tau, *}(\epsilon_X^{-1}K')$. \end{enumerate}


sites-cohomology-lemma-LC-basic

The category $\textit{LC}$ has fibre products and a final object and hence has arbitrary finite limits. Given morphisms $X \to Z$ and $Y \to Z$ in $\textit{LC}$ with $X$ and $Y$ quasi-compact, then $X \times_Z Y$ is quasi-compact.


sites-cohomology-lemma-qc

Let $X$ be a Hausdorff and locally quasi-compact space, in other words, an object of $\textit{LC}$. \begin{enumerate} \item If $X' \to X$ is an isomorphism in $\textit{LC}$ then $\{X' \to X\}$ is a qc covering. \item If $\{f_i : X_i \to X\}_{i\in I}$ is a qc covering and for each $i$ we have a qc covering $\{g_{ij} : X_{ij} \to X_i\}_{j\in J_i}$, then $\{X_{ij} \to X\}_{i \in I, j\in J_i}$ is a qc covering. \item If $\{X_i \to X\}_{i\in I}$ is a qc covering and $X' \to X$ is a morphism of $\textit{LC}$ then $\{X' \times_X X_i \to X'\}_{i\in I}$ is a qc covering. \end{enumerate}


sites-cohomology-lemma-proper-surjective-is-qc-covering

Let $f : X \to Y$ be a morphism of $\textit{LC}$. If $f$ is proper and surjective, then $\{f : X \to Y\}$ is a qc covering.


sites-cohomology-lemma-describe-pullback-pi

Let $X$ be an object of $\textit{LC}_{qc}$. Let $\mathcal{F}$ be a sheaf on $X$. The rule $$ \textit{LC}_{qc}/X \longrightarrow \textit{Sets},\quad (f : Y \to X) \longmapsto \Gamma(Y, f^{-1}\mathcal{F}) $$ is a sheaf and a fortiori also a sheaf on $\textit{LC}_{Zar}/X$. This sheaf is equal to $\pi_X^{-1}\mathcal{F}$ on $\textit{LC}_{Zar}/X$ and $\epsilon_X^{-1}\pi_X^{-1}\mathcal{F}$ on $\textit{LC}_{qc}/X$.


sites-cohomology-lemma-collect-true-things-Zar

Let $X$ be an object of $\textit{LC}_{Zar}$. Then \begin{enumerate} \item for $\mathcal{F} \in \textit{Ab}(X)$ we have $H^n_{Zar}(X, \pi_X^{-1}\mathcal{F}) = H^n(X, \mathcal{F})$, \item $\pi_{X, *} : \textit{Ab}(\textit{LC}_{Zar}/X) \to \textit{Ab}(X)$ is exact, \item the unit $\text{id} \to \pi_{X, *} \circ \pi_X^{-1}$ of the adjunction is an isomorphism, and \item for $K \in D(X)$ the canonical map $K \to R\pi_{X, *} \pi_X^{-1}K$ is an isomorphism. \end{enumerate} Let $f : X \to Y$ be a morphism of $\textit{LC}_{Zar}$. Then \begin{enumerate} \item[(5)] there is a commutative diagram $$ \xymatrix{ \Sh(\textit{LC}_{Zar}/X) \ar[r]_{f_{Zar}} \ar[d]_{\pi_X} & \Sh(\textit{LC}_{Zar}/Y) \ar[d]^{\pi_Y} \\ \Sh(X_{Zar}) \ar[r]^f & \Sh(Y_{Zar}) } $$ of topoi, \item[(6)] for $L \in D^+(Y)$ we have $H^n_{Zar}(X, \pi_Y^{-1}L) = H^n(X, f^{-1}L)$, \item[(7)] if $f$ is proper, then we have \begin{enumerate} \item $\pi_Y^{-1} \circ f_* = f_{Zar, *} \circ \pi_X^{-1}$ as functors $\Sh(X) \to \Sh(\textit{LC}_{Zar}/Y)$, \item $\pi_Y^{-1} \circ Rf_* = Rf_{Zar, *} \circ \pi_X^{-1}$ as functors $D^+(X) \to D^+(\textit{LC}_{Zar}/Y)$. \end{enumerate} \end{enumerate}


sites-cohomology-lemma-push-pull-LC

Let $f : X \to Y$ be a morphism of $\textit{LC}_{qc}$. Then there are commutative diagrams of topoi $$ \vcenter{ \xymatrix{ \Sh(\textit{LC}_{qc}/X) \ar[r]_{f_{qc}} \ar[d]_{\epsilon_X} & \Sh(\textit{LC}_{qc}/Y) \ar[d]^{\epsilon_Y} \\ \Sh(\textit{LC}_{Zar}/X) \ar[r]^{f_{Zar}} & \Sh(\textit{LC}_{Zar}/Y) } } \quad\text{and}\quad \vcenter{ \xymatrix{ \Sh(\textit{LC}_{qc}/X) \ar[r]_{f_{qc}} \ar[d]_{a_X} & \Sh(\textit{LC}_{qc}/Y) \ar[d]^{a_Y} \\ \Sh(X) \ar[r]^f & \Sh(Y) } } $$ with $a_X = \pi_X \circ \epsilon_X$, $a_Y = \pi_X \circ \epsilon_X$. If $f$ is proper, then $a_Y^{-1} \circ f_* = f_{qc, *} \circ a_X^{-1}$.


sites-cohomology-lemma-compare-qc-zar

Consider the comparison morphism $\epsilon : \textit{LC}_{qc} \to \textit{LC}_{Zar}$. Let $\mathcal{P}$ denote the class of proper maps of topological spaces. For $X$ in $\textit{LC}_{Zar}$ denote $\mathcal{A}'_X \subset \textit{Ab}(\textit{LC}_{Zar}/X)$ the full subcategory consisting of sheaves of the form $\pi_X^{-1}\mathcal{F}$ with $\mathcal{F}$ in $\textit{Ab}(X)$. Then (\ref{item-base-change-P}), (\ref{item-restriction-A}), (\ref{item-A-sheaf}), (\ref{item-A-and-P}), and (\ref{item-refine-tau-by-P}) of Situation \ref{situation-compare} hold.


sites-cohomology-lemma-V-C-all-n

With notation as above. \begin{enumerate} \item For $X \in \Ob(\textit{LC}_{qc})$ and an abelian sheaf $\mathcal{F}$ on $X$ we have $\epsilon_{X, *}a_X^{-1}\mathcal{F} = \pi_X^{-1}\mathcal{F}$ and $R^i\epsilon_{X, *}(a_X^{-1}\mathcal{F}) = 0$ for $i > 0$. \item For a proper morphism $f : X \to Y$ in $\textit{LC}_{qc}$ and abelian sheaf $\mathcal{F}$ on $X$ we have $a_Y^{-1}(R^if_*\mathcal{F}) = R^if_{qc, *}(a_X^{-1}\mathcal{F})$ for all $i$. \item For $X \in \Ob(\textit{LC}_{qc})$ and $K$ in $D^+(X)$ the map $\pi_X^{-1}K \to R\epsilon_{X, *}(a_X^{-1}K)$ is an isomorphism. \item For a proper morphism $f : X \to Y$ in $\textit{LC}_{qc}$ and $K$ in $D^+(X)$ we have $a_Y^{-1}(Rf_*K) = Rf_{qc, *}(a_X^{-1}K)$. \end{enumerate}


sites-cohomology-lemma-cohomological-descent-LC

Let $X$ be an object of $\textit{LC}_{qc}$. For $K \in D^+(X)$ the map $$ K \longrightarrow Ra_{X, *}a_X^{-1}K $$ is an isomorphism with $a_X : \Sh(\textit{LC}_{qc}/X) \to \Sh(X)$ as above.


sites-cohomology-lemma-cup-product-associative

Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. The relative cup product of Remark \ref{remark-cup-product} is associative in the sense that the diagram $$ \xymatrix{ Rf_*K \otimes_{\mathcal{O}'}^\mathbf{L} Rf_*L \otimes_{\mathcal{O}'}^\mathbf{L} Rf_*M \ar[r] \ar[d] & Rf_*(K \otimes_\mathcal{O}^\mathbf{L} L) \otimes_{\mathcal{O}'}^\mathbf{L} Rf_*M \ar[d] \\ Rf_*K \otimes_{\mathcal{O}'}^\mathbf{L} Rf_*(L \otimes_\mathcal{O}^\mathbf{L} M) \ar[r] & Rf_*(K \otimes_\mathcal{O}^\mathbf{L}  L \otimes_\mathcal{O}^\mathbf{L} M) } $$ is commutative in $D(\mathcal{O}')$ for all $K, L, M$ in $D(\mathcal{O})$.


sites-cohomology-lemma-cup-product-commutative

Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. The relative cup product of Remark \ref{remark-cup-product} is commutative in the sense that the diagram $$ \xymatrix{ Rf_*K \otimes_{\mathcal{O}'}^\mathbf{L} Rf_*L \ar[r] \ar[d]_\psi & Rf_*(K \otimes_\mathcal{O}^\mathbf{L} L) \ar[d]^{Rf_*\psi} \\ Rf_*L \otimes_{\mathcal{O}'}^\mathbf{L} Rf_*K \ar[r] & Rf_*(L \otimes_\mathcal{O}^\mathbf{L} K) } $$ is commutative in $D(\mathcal{O}')$ for all $K, L$ in $D(\mathcal{O})$. Here $\psi$ is the commutativity constraint on the derived category (Lemma \ref{lemma-symmetric-monoidal-derived}).


sites-cohomology-lemma-compose

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given complexes $\mathcal{K}^\bullet, \mathcal{L}^\bullet, \mathcal{M}^\bullet$ of $\mathcal{O}$-modules there is an isomorphism $$ \SheafHom^\bullet(\mathcal{K}^\bullet, \SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{M}^\bullet)) = \SheafHom^\bullet(\text{Tot}(\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{L}^\bullet), \mathcal{M}^\bullet) $$ of complexes of $\mathcal{O}$-modules functorial in $\mathcal{K}^\bullet, \mathcal{L}^\bullet, \mathcal{M}^\bullet$.


sites-cohomology-lemma-composition

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given complexes $\mathcal{K}^\bullet, \mathcal{L}^\bullet, \mathcal{M}^\bullet$ of $\mathcal{O}$-modules there is a canonical morphism $$ \text{Tot}\left( \SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{M}^\bullet) \otimes_\mathcal{O} \SheafHom^\bullet(\mathcal{K}^\bullet, \mathcal{L}^\bullet) \right) \longrightarrow \SheafHom^\bullet(\mathcal{K}^\bullet, \mathcal{M}^\bullet) $$ of complexes of $\mathcal{O}$-modules.


sites-cohomology-lemma-diagonal-better

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given complexes $\mathcal{K}^\bullet, \mathcal{L}^\bullet, \mathcal{M}^\bullet$ of $\mathcal{O}$-modules there is a canonical morphism $$ \text{Tot}\left( \mathcal{K}^\bullet \otimes_\mathcal{O} \SheafHom^\bullet(\mathcal{M}^\bullet, \mathcal{L}^\bullet) \right) \longrightarrow \SheafHom^\bullet(\mathcal{M}^\bullet, \text{Tot}(\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{L}^\bullet)) $$ of complexes of $\mathcal{O}$-modules functorial in all three complexes.


sites-cohomology-lemma-diagonal

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given complexes $\mathcal{K}^\bullet, \mathcal{L}^\bullet, \mathcal{M}^\bullet$ of $\mathcal{O}$-modules there is a canonical morphism $$ \mathcal{K}^\bullet \longrightarrow \SheafHom^\bullet(\mathcal{L}^\bullet, \text{Tot}(\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{L}^\bullet)) $$ of complexes of $\mathcal{O}$-modules functorial in both complexes.


sites-cohomology-lemma-evaluate-and-more

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given complexes $\mathcal{K}^\bullet, \mathcal{L}^\bullet, \mathcal{M}^\bullet$ of $\mathcal{O}$-modules there is a canonical morphism $$ \text{Tot}(\SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{M}^\bullet) \otimes_\mathcal{O} \mathcal{K}^\bullet) \longrightarrow \SheafHom^\bullet(\SheafHom^\bullet(\mathcal{K}^\bullet, \mathcal{L}^\bullet), \mathcal{M}^\bullet) $$ of complexes of $\mathcal{O}$-modules functorial in all three complexes.


sites-cohomology-lemma-RHom-into-K-injective

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I}^\bullet$ be a K-injective complex of $\mathcal{O}$-modules. Let $\mathcal{L}^\bullet$ be a complex of $\mathcal{O}$-modules. Then $$ H^0(\Gamma(U, \SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{I}^\bullet))) = \Hom_{D(\mathcal{O}_U)}(L|_U, M|_U) $$ for all $U \in \Ob(\mathcal{C})$. Similarly, $H^0(\Gamma(\mathcal{C}, \SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{I}^\bullet))) = \Hom_{D(\mathcal{O}_U)}(L, M)$.


sites-cohomology-lemma-RHom-from-K-flat-into-K-injective

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I}^\bullet$ be a K-injective complex of $\mathcal{O}$-modules. Let $\mathcal{L}^\bullet$ be a K-flat complex of $\mathcal{O}$-modules. Then $\SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{I}^\bullet)$ is a K-injective complex of $\mathcal{O}$-modules.


sites-cohomology-lemma-internal-hom

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K, L, M$ be objects of $D(\mathcal{O})$. With the construction as described above there is a canonical isomorphism $$ R\SheafHom(K, R\SheafHom(L, M)) = R\SheafHom(K \otimes_\mathcal{O}^\mathbf{L} L, M) $$ in $D(\mathcal{O})$ functorial in $K, L, M$ which recovers (\ref{equation-internal-hom}) on taking $H^0(\mathcal{C}, -)$.


sites-cohomology-lemma-restriction-RHom-to-U

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K, L$ be objects of $D(\mathcal{O})$. The construction of $R\SheafHom(K, L)$ commutes with restrictions, i.e., for every object $U$ of $\mathcal{C}$ we have $R\SheafHom(K|_U, L|_U) = R\SheafHom(K, L)|_U$.


sites-cohomology-lemma-RHom-triangulated

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. The bifunctor $R\SheafHom(- , -)$ transforms distinguished triangles into distinguished triangles in both variables.


sites-cohomology-lemma-internal-hom-evaluate

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K, L, M$ be objects of $D(\mathcal{O})$. There is a canonical morphism $$ R\SheafHom(L, M) \otimes_\mathcal{O}^\mathbf{L} K \longrightarrow R\SheafHom(R\SheafHom(K, L), M) $$ in $D(\mathcal{O})$ functorial in $K, L, M$.


sites-cohomology-lemma-internal-hom-composition

\begin{slogan} Composition on RSheafHom. \end{slogan} Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given $K, L, M$ in $D(\mathcal{O})$ there is a canonical morphism $$ R\SheafHom(L, M) \otimes_\mathcal{O}^\mathbf{L} R\SheafHom(K, L) \longrightarrow R\SheafHom(K, M) $$ in $D(\mathcal{O})$.


sites-cohomology-lemma-internal-hom-diagonal-better

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given $K, L, M$ in $D(\mathcal{O})$ there is a canonical morphism $$ K \otimes_\mathcal{O}^\mathbf{L} R\SheafHom(M, L) \longrightarrow R\SheafHom(M, K \otimes_\mathcal{O}^\mathbf{L} L) $$ in $D(\mathcal{O})$ functorial in $K, L, M$.


sites-cohomology-lemma-dual

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $L$ be an object of $D(\mathcal{O})$. Set $L^\vee = R\SheafHom(L, \mathcal{O})$. For $M$ in $D(\mathcal{O})$ there is a canonical map \begin{equation} \label{equation-eval} M \otimes^\mathbf{L}_\mathcal{O} L^\vee \longrightarrow R\SheafHom(L, M) \end{equation} which induces a canonical map $$ H^0(\mathcal{C}, M \otimes_\mathcal{O}^\mathbf{L} L^\vee) \longrightarrow \Hom_{D(\mathcal{O})}(L, M) $$ functorial in $M$ in $D(\mathcal{O})$.


sites-cohomology-lemma-pullback-injective-pre-limp

Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous and cocontinuous functor of sites. Let $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be the corresponding morphism of topoi. Let $\mathcal{O}_\mathcal{D}$ be a sheaf of rings and let $\mathcal{I}$ be an injective $\mathcal{O}_\mathcal{D}$-module. Then $H^p(U, g^{-1}\mathcal{I}) = 0$ for all $p > 0$ and $U \in \Ob(\mathcal{C})$.


sites-cohomology-lemma-existence-derived-lower-shriek

Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous and cocontinuous functor of sites. Let $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be the corresponding morphism of topoi. Let $\mathcal{O}_\mathcal{D}$ be a sheaf of rings and set $\mathcal{O}_\mathcal{C} = g^{-1}\mathcal{O}_\mathcal{D}$. The functor $g_! : \textit{Mod}(\mathcal{O}_\mathcal{C}) \to \textit{Mod}(\mathcal{O}_\mathcal{D})$ (see Modules on Sites, Lemma \ref{sites-modules-lemma-lower-shriek-modules}) has a left derived functor $$ Lg_! : D(\mathcal{O}_\mathcal{C}) \longrightarrow D(\mathcal{O}_\mathcal{D}) $$ which is left adjoint to $g^*$. Moreover, for $U \in \Ob(\mathcal{C})$ we have $$ Lg_!(j_{U!}\mathcal{O}_U) = g_!j_{U!}\mathcal{O}_U = j_{u(U)!} \mathcal{O}_{u(U)}. $$ where $j_{U!}$ and $j_{u(U)!}$ are extension by zero associated to the localization morphism $j_U : \mathcal{C}/U \to \mathcal{C}$ and $j_{u(U)} : \mathcal{D}/u(U) \to \mathcal{D}$.


sites-cohomology-lemma-pullback-injective-limp

Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous and cocontinuous functor of sites. Let $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be the corresponding morphism of topoi. Let $\mathcal{O}_\mathcal{D}$ be a sheaf of rings and let $\mathcal{I}$ be an injective $\mathcal{O}_\mathcal{D}$-module. If $g_!^{Sh} : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ commutes with fibre products\footnote{Holds if $\mathcal{C}$ has finite connected limits and $u$ commutes with them, see Sites, Lemma \ref{sites-lemma-preserve-equalizers}.}, then $g^{-1}\mathcal{I}$ is totally acyclic.


sites-cohomology-lemma-pullback-same-cohomology

Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous and cocontinuous functor of sites. Let $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be the corresponding morphism of topoi. Let $U \in \Ob(\mathcal{C})$. \begin{enumerate} \item For $M$ in $D(\mathcal{D})$ we have $R\Gamma(U, g^{-1}M) = R\Gamma(u(U), M)$. \item If $\mathcal{O}_\mathcal{D}$ is a sheaf of rings and $\mathcal{O}_\mathcal{C} = g^{-1}\mathcal{O}_\mathcal{D}$, then for $M$ in $D(\mathcal{O}_\mathcal{D})$ we have $R\Gamma(U, g^*M) = R\Gamma(u(U), M)$. \end{enumerate}


sites-cohomology-lemma-fibred-category-with-object

Assumptions and notation as in Situation \ref{situation-fibred-category}. For $U \in \Ob(\mathcal{C})$ consider the induced morphism of topoi $$ \pi_U : \Sh(\mathcal{C}/U) \longrightarrow \Sh(\mathcal{D}/p(U)) $$ Then there exists a morphism of topoi $$ \sigma : \Sh(\mathcal{D}/p(U)) \to \Sh(\mathcal{C}/U) $$ such that $\pi_U \circ \sigma = \text{id}$ and $\sigma^{-1} = \pi_{U, *}$.


sites-cohomology-lemma-morphism-fibred-categories-with-object

Assumptions and notation as in Situation \ref{situation-morphism-fibred-categories}. For $U' \in \Ob(\mathcal{C}')$ set $U = u(U')$ and $V = p'(U')$ and consider the induced morphisms of ringed topoi $$ \xymatrix{ (\Sh(\mathcal{C}'/U'), \mathcal{O}_{U'}) \ar[rd]_{\pi'_{U'}} \ar[rr]_{g'} & & (\Sh(\mathcal{C}), \mathcal{O}_U) \ar[ld]^{\pi_U} \\ & (\Sh(\mathcal{D}/V), \mathcal{O}_V) } $$ Then there exists a morphism of topoi $$ \sigma' : \Sh(\mathcal{D}/V) \to \Sh(\mathcal{C}'/U'), $$ such that setting $\sigma = g' \circ \sigma'$ we have $\pi'_{U'} \circ \sigma' = \text{id}$, $\pi_U \circ \sigma = \text{id}$, $(\sigma')^{-1} = \pi'_{U', *}$, and $\sigma^{-1} = \pi_{U, *}$.


sites-cohomology-lemma-properties-lower-shriek-fibred-category

Assumption and notation as in Situation \ref{situation-morphism-fibred-categories}. \begin{enumerate} \item There are left adjoints $g_! : \textit{Mod}(\mathcal{O}_{\mathcal{C}'}) \to \textit{Mod}(\mathcal{O}_\mathcal{C})$ and $g_!^{\textit{Ab}} : \textit{Ab}(\mathcal{C}') \to \textit{Ab}(\mathcal{C})$ to $g^* = g^{-1}$ on modules and on abelian sheaves. \item The diagram $$ \xymatrix{ \textit{Mod}(\mathcal{O}_{\mathcal{C}'}) \ar[d] \ar[r]_{g_!} & \textit{Mod}(\mathcal{O}_\mathcal{C}) \ar[d] \\ \textit{Ab}(\mathcal{C}') \ar[r]^{g_!^{\textit{Ab}}} & \textit{Ab}(\mathcal{C}) } $$ commutes. \item There are left adjoints $Lg_! : D(\mathcal{O}_{\mathcal{C}'}) \to D(\mathcal{O}_\mathcal{C})$ and $Lg_!^{\textit{Ab}} : D(\mathcal{C}') \to D(\mathcal{C})$ to $g^* = g^{-1}$ on derived categories of modules and abelian sheaves. \item The diagram $$ \xymatrix{ D(\mathcal{O}_{\mathcal{C}'}) \ar[d] \ar[r]_{Lg_!} & D(\mathcal{O}_\mathcal{C}) \ar[d] \\ D(\mathcal{C}') \ar[r]^{Lg_!^{\textit{Ab}}} & D(\mathcal{C}) } $$ commutes. \end{enumerate}


sites-cohomology-lemma-compute-pi-shriek

Assumptions and notation as in Situation \ref{situation-fibred-category}. For $\mathcal{F}$ in $\textit{Ab}(\mathcal{C})$ the sheaf $\pi_!\mathcal{F}$ is the sheaf associated to the presheaf $$ V \longmapsto \colim_{\mathcal{C}_V^{opp}} \mathcal{F}|_{\mathcal{C}_V} $$ with restriction maps as indicated in the proof.


sites-cohomology-lemma-initial-final

Notation and assumptions as in Example \ref{example-category-to-point}. If $\mathcal{C}$ has either an initial or a final object, then $L\pi_! \circ \pi^{-1} = \text{id}$ on $D(\textit{Ab})$, resp.\ $D(B)$.


sites-cohomology-lemma-change-of-rings

Notation and assumptions as in Example \ref{example-category-to-point}. Let $B \to B'$ be a ring map. Consider the commutative diagram of ringed topoi $$ \xymatrix{ (\Sh(\mathcal{C}), \underline{B}) \ar[d]_\pi & (\Sh(\mathcal{C}), \underline{B'}) \ar[d]^{\pi'} \ar[l]^h \\ (*, B) & (*, B') \ar[l]_f } $$ Then $L\pi_! \circ Lh^* = Lf^* \circ L\pi'_!$.


sites-cohomology-lemma-compute-by-cosimplicial-resolution

Notation and assumptions as in Example \ref{example-category-to-point}. Let $U_\bullet$ be a cosimplicial object in $\mathcal{C}$ such that for every $U \in \Ob(\mathcal{C})$ the simplicial set $\Mor_\mathcal{C}(U_\bullet, U)$ is homotopy equivalent to the constant simplicial set on a singleton. Then $$ L\pi_!(\mathcal{F}) = \mathcal{F}(U_\bullet) $$ in $D(\textit{Ab})$, resp.\ $D(B)$ functorially in $\mathcal{F}$ in $\textit{Ab}(\mathcal{C})$, resp.\ $\textit{Mod}(\underline{B})$.


sites-cohomology-lemma-get-it-now

Notation and assumptions as in Example \ref{example-morphism-categories}. If there exists a cosimplicial object $U'_\bullet$ of $\mathcal{C}'$ such that Lemma \ref{lemma-compute-by-cosimplicial-resolution} applies to both $U'_\bullet$ in $\mathcal{C}'$ and $u(U'_\bullet)$ in $\mathcal{C}$, then we have $L\pi'_! \circ g^{-1} = L\pi_!$ as functors $D(\mathcal{C}) \to D(\textit{Ab})$, resp.\ $D(\mathcal{C}, \underline{B}) \to D(B)$.


sites-cohomology-lemma-product-categories

Let $\mathcal{C}_i$, $i = 1, 2$ be categories. Let $u_i : \mathcal{C}_1 \times \mathcal{C}_2 \to \mathcal{C}_i$ be the projection functors. Let $B$ be a ring. Let $g_i : (\Sh(\mathcal{C}_1 \times \mathcal{C}_2), \underline{B}) \to (\Sh(\mathcal{C}_i), \underline{B})$ be the corresponding morphisms of ringed topoi, see Example \ref{example-morphism-categories}. For $K_i \in D(\mathcal{C}_i, B)$ we have $$ L(\pi_1 \times \pi_2)_!( g_1^{-1}K_1 \otimes_{\underline{B}}^\mathbf{L} g_2^{-1}K_2) = L\pi_{1, !}(K_1) \otimes_B^\mathbf{L} L\pi_{2, !}(K_2) $$ in $D(B)$ with obvious notation.


sites-cohomology-lemma-eilenberg-zilber

Notation and assumptions as in Example \ref{example-category-to-point}. If there exists a cosimplicial object $U_\bullet$ of $\mathcal{C}$ such that Lemma \ref{lemma-compute-by-cosimplicial-resolution} applies, then $$ L\pi_!(K_1 \otimes^\mathbf{L}_{\underline{B}} K_2) = L\pi_!(K_1) \otimes^\mathbf{L}_B L\pi_!(K_2) $$ for all $K_i \in D(\underline{B})$.


sites-cohomology-lemma-O-homology-qis

Let $\mathcal{C}$ be a category (endowed with chaotic topology). Let $\mathcal{O} \to \mathcal{O}'$ be a map of sheaves of rings on $\mathcal{C}$. Assume \begin{enumerate} \item there exists a cosimplicial object $U_\bullet$ in $\mathcal{C}$ as in Lemma \ref{lemma-compute-by-cosimplicial-resolution}, and \item $L\pi_!\mathcal{O} \to L\pi_!\mathcal{O}'$ is an isomorphism. \end{enumerate} For $K$ in $D(\mathcal{O})$ we have $$ L\pi_!(K) = L\pi_!(K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}') $$ in $D(\textit{Ab})$.


sites-cohomology-lemma-compute-left-derived-pi-shriek-pre

Assumptions and notation as in Situation \ref{situation-fibred-category}. For $\mathcal{F}$ in $\textit{PAb}(\mathcal{C})$ and $n \geq 0$ consider the abelian sheaf $L_n(\mathcal{F})$ on $\mathcal{D}$ which is the sheaf associated to the presheaf $$ V \longmapsto H_n(\mathcal{C}_V, \mathcal{F}|_{\mathcal{C}_V}) $$ with restriction maps as indicated in the proof. Then $L_n(\mathcal{F}) = L_n(\mathcal{F}^\#)$.


sites-cohomology-lemma-compute-left-derived-pi-shriek

Assumptions and notation as in Situation \ref{situation-fibred-category}. For $\mathcal{F}$ in $\textit{Ab}(\mathcal{C})$ and $n \geq 0$ the sheaf $L_n\pi_!(\mathcal{F})$ is equal to the sheaf $L_n(\mathcal{F})$ constructed in Lemma \ref{lemma-compute-left-derived-pi-shriek-pre}.


sites-cohomology-lemma-compute-left-derived-g-shriek

Assumptions and notation as in Situation \ref{situation-morphism-fibred-categories}. For an abelian sheaf $\mathcal{F}'$ on $\mathcal{C}'$ the sheaf $L_ng_!(\mathcal{F}')$ is the sheaf associated to the presheaf $$ U \longmapsto H_n(\mathcal{I}_U, \mathcal{F}'_U) $$ For notation and restriction maps see proof.


sites-cohomology-lemma-base-change-by-qis

Let $\mathcal{C}$ be a site. Let $\mathcal{A}_\bullet \to \mathcal{B}_\bullet$ be a homomorphism of simplicial sheaves of rings on $\mathcal{C}$. If $L\pi_!\mathcal{A}_\bullet \to L\pi_!\mathcal{B}_\bullet$ is an isomorphism in $D(\mathcal{C})$, then we have $$ L\pi_!(K) = L\pi_!(K \otimes^\mathbf{L}_{\mathcal{A}_\bullet} \mathcal{B}_\bullet) $$ for all $K$ in $D(\mathcal{A}_\bullet)$.


sites-cohomology-lemma-cone

The cone on a morphism of strictly perfect complexes is strictly perfect.


sites-cohomology-lemma-tensor

The total complex associated to the tensor product of two strictly perfect complexes is strictly perfect.


sites-cohomology-lemma-strictly-perfect-pullback

Let $(f, f^\sharp) : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. If $\mathcal{F}^\bullet$ is a strictly perfect complex of $\mathcal{O}_\mathcal{D}$-modules, then $f^*\mathcal{F}^\bullet$ is a strictly perfect complex of $\mathcal{O}_\mathcal{C}$-modules.


sites-cohomology-lemma-local-lift-map

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. Given a solid diagram of $\mathcal{O}_U$-modules $$ \xymatrix{ \mathcal{E} \ar@{..>}[dr] \ar[r] & \mathcal{F} \\ & \mathcal{G} \ar[u]_p } $$ with $\mathcal{E}$ a direct summand of a finite free $\mathcal{O}_U$-module and $p$ surjective, then there exists a covering $\{U_i \to U\}$ such that a dotted arrow making the diagram commute exists over each $U_i$.


sites-cohomology-lemma-local-homotopy

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. \begin{enumerate} \item Let $\alpha : \mathcal{E}^\bullet \to \mathcal{F}^\bullet$ be a morphism of complexes of $\mathcal{O}_U$-modules with $\mathcal{E}^\bullet$ strictly perfect and $\mathcal{F}^\bullet$ acyclic. Then there exists a covering $\{U_i \to U\}$ such that each $\alpha|_{U_i}$ is homotopic to zero. \item Let $\alpha : \mathcal{E}^\bullet \to \mathcal{F}^\bullet$ be a morphism of complexes of $\mathcal{O}_U$-modules with $\mathcal{E}^\bullet$ strictly perfect, $\mathcal{E}^i = 0$ for $i < a$, and $H^i(\mathcal{F}^\bullet) = 0$ for $i \geq a$. Then there exists a covering $\{U_i \to U\}$ such that each $\alpha|_{U_i}$ is homotopic to zero. \end{enumerate}


sites-cohomology-lemma-lift-through-quasi-isomorphism

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. Given a solid diagram of complexes of $\mathcal{O}_U$-modules $$ \xymatrix{ \mathcal{E}^\bullet \ar@{..>}[dr] \ar[r]_\alpha & \mathcal{F}^\bullet \\ & \mathcal{G}^\bullet \ar[u]_f } $$ with $\mathcal{E}^\bullet$ strictly perfect, $\mathcal{E}^j = 0$ for $j < a$ and $H^j(f)$ an isomorphism for $j > a$ and surjective for $j = a$, then there exists a covering $\{U_i \to U\}$ and for each $i$ a dotted arrow over $U_i$ making the diagram commute up to homotopy.


sites-cohomology-lemma-local-actual

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. Let $\mathcal{E}^\bullet$, $\mathcal{F}^\bullet$ be complexes of $\mathcal{O}_U$-modules with $\mathcal{E}^\bullet$ strictly perfect. \begin{enumerate} \item For any element $\alpha \in \Hom_{D(\mathcal{O}_U)}(\mathcal{E}^\bullet, \mathcal{F}^\bullet)$ there exists a covering $\{U_i \to U\}$ such that $\alpha|_{U_i}$ is given by a morphism of complexes $\alpha_i : \mathcal{E}^\bullet|_{U_i} \to \mathcal{F}^\bullet|_{U_i}$. \item Given a morphism of complexes $\alpha : \mathcal{E}^\bullet \to \mathcal{F}^\bullet$ whose image in the group $\Hom_{D(\mathcal{O}_U)}(\mathcal{E}^\bullet, \mathcal{F}^\bullet)$ is zero, there exists a covering $\{U_i \to U\}$ such that $\alpha|_{U_i}$ is homotopic to zero. \end{enumerate}


sites-cohomology-lemma-Rhom-strictly-perfect

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{E}^\bullet$, $\mathcal{F}^\bullet$ be complexes of $\mathcal{O}$-modules with $\mathcal{E}^\bullet$ strictly perfect. Then the internal hom $R\SheafHom(\mathcal{E}^\bullet, \mathcal{F}^\bullet)$ is represented by the complex $\mathcal{H}^\bullet$ with terms $$ \mathcal{H}^n = \bigoplus\nolimits_{n = p + q} \SheafHom_\mathcal{O}(\mathcal{E}^{-q}, \mathcal{F}^p) $$ and differential as described in Section \ref{section-internal-hom}.


sites-cohomology-lemma-Rhom-complex-of-direct-summands-finite-free

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{E}^\bullet$, $\mathcal{F}^\bullet$ be complexes of $\mathcal{O}$-modules with \begin{enumerate} \item $\mathcal{F}^n = 0$ for $n \ll 0$, \item $\mathcal{E}^n = 0$ for $n \gg 0$, and \item $\mathcal{E}^n$ isomorphic to a direct summand of a finite free $\mathcal{O}$-module. \end{enumerate} Then the internal hom $R\SheafHom(\mathcal{E}^\bullet, \mathcal{F}^\bullet)$ is represented by the complex $\mathcal{H}^\bullet$ with terms $$ \mathcal{H}^n = \bigoplus\nolimits_{n = p + q} \SheafHom_\mathcal{O}(\mathcal{E}^{-q}, \mathcal{F}^p) $$ and differential as described in Section \ref{section-internal-hom}.


sites-cohomology-lemma-pseudo-coherent-independent-representative

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. \begin{enumerate} \item If $\mathcal{C}$ has a final object $X$ and if there exist a covering $\{U_i \to X\}$, strictly perfect complexes $\mathcal{E}_i^\bullet$ of $\mathcal{O}_{U_i}$-modules, and maps $\alpha_i : \mathcal{E}_i^\bullet \to E|_{U_i}$ in $D(\mathcal{O}_{U_i})$ with $H^j(\alpha_i)$ an isomorphism for $j > m$ and $H^m(\alpha_i)$ surjective, then $E$ is $m$-pseudo-coherent. \item If $E$ is $m$-pseudo-coherent, then any complex of $\mathcal{O}$-modules representing $E$ is $m$-pseudo-coherent. \item If for every object $U$ of $\mathcal{C}$ there exists a covering $\{U_i \to U\}$ such that $E|_{U_i}$ is $m$-pseudo-coherent, then $E$ is $m$-pseudo-coherent. \end{enumerate}


sites-cohomology-lemma-pseudo-coherent-pullback

Let $(f, f^\sharp) : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D})$ be a morphism of ringed sites. Let $E$ be an object of $D(\mathcal{O}_\mathcal{C})$. If $E$ is $m$-pseudo-coherent, then $Lf^*E$ is $m$-pseudo-coherent.


sites-cohomology-lemma-cone-pseudo-coherent

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site and $m \in \mathbf{Z}$. Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\mathcal{O})$. \begin{enumerate} \item If $K$ is $(m + 1)$-pseudo-coherent and $L$ is $m$-pseudo-coherent then $M$ is $m$-pseudo-coherent. \item If $K$ and $M$ are $m$-pseudo-coherent, then $L$ is $m$-pseudo-coherent. \item If $L$ is $(m + 1)$-pseudo-coherent and $M$ is $m$-pseudo-coherent, then $K$ is $(m + 1)$-pseudo-coherent. \end{enumerate}


sites-cohomology-lemma-tensor-pseudo-coherent

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K, L$ be objects of $D(\mathcal{O})$. \begin{enumerate} \item If $K$ is $n$-pseudo-coherent and $H^i(K) = 0$ for $i > a$ and $L$ is $m$-pseudo-coherent and $H^j(L) = 0$ for $j > b$, then $K \otimes_\mathcal{O}^\mathbf{L} L$ is $t$-pseudo-coherent with $t = \max(m + a, n + b)$. \item If $K$ and $L$ are pseudo-coherent, then $K \otimes_\mathcal{O}^\mathbf{L} L$ is pseudo-coherent. \end{enumerate}


sites-cohomology-lemma-summands-pseudo-coherent

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $m \in \mathbf{Z}$. If $K \oplus L$ is $m$-pseudo-coherent (resp.\ pseudo-coherent) in $D(\mathcal{O})$ so are $K$ and $L$.


sites-cohomology-lemma-finite-cohomology

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K$ be an object of $D(\mathcal{O})$. Let $m \in \mathbf{Z}$. \begin{enumerate} \item If $K$ is $m$-pseudo-coherent and $H^i(K) = 0$ for $i > m$, then $H^m(K)$ is a finite type $\mathcal{O}$-module. \item If $K$ is $m$-pseudo-coherent and $H^i(K) = 0$ for $i > m + 1$, then $H^{m + 1}(K)$ is a finitely presented $\mathcal{O}$-module. \end{enumerate}


sites-cohomology-lemma-last-one-flat

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{E}^\bullet$ be a bounded above complex of flat $\mathcal{O}$-modules with tor-amplitude in $[a, b]$. Then $\Coker(d_{\mathcal{E}^\bullet}^{a - 1})$ is a flat $\mathcal{O}$-module.


sites-cohomology-lemma-tor-amplitude

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. Let $a, b \in \mathbf{Z}$ with $a \leq b$. The following are equivalent \begin{enumerate} \item $E$ has tor-amplitude in $[a, b]$. \item $E$ is represented by a complex $\mathcal{E}^\bullet$ of flat $\mathcal{O}$-modules with $\mathcal{E}^i = 0$ for $i \not \in [a, b]$. \end{enumerate}


sites-cohomology-lemma-bounded-below-tor-amplitude

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. Let $a \in \mathbf{Z}$. The following are equivalent \begin{enumerate} \item $E$ has tor-amplitude in $[a, \infty]$. \item $E$ can be represented by a K-flat complex $\mathcal{E}^\bullet$ of flat $\mathcal{O}$-modules with $\mathcal{E}^i = 0$ for $i \not \in [a, \infty]$. \end{enumerate} Moreover, we can choose $\mathcal{E}^\bullet$ such that any pullback by a morphism of ringed sites is a K-flat complex with flat terms.


sites-cohomology-lemma-tor-amplitude-pullback

Let $(f, f^\sharp) : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D})$ be a morphism of ringed sites. Let $E$ be an object of $D(\mathcal{O}_\mathcal{D})$. If $E$ has tor amplitude in $[a, b]$, then $Lf^*E$ has tor amplitude in $[a, b]$.


sites-cohomology-lemma-cone-tor-amplitude

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\mathcal{O})$. Let $a, b \in \mathbf{Z}$. \begin{enumerate} \item If $K$ has tor-amplitude in $[a + 1, b + 1]$ and $L$ has tor-amplitude in $[a, b]$ then $M$ has tor-amplitude in $[a, b]$. \item If $K$ and $M$ have tor-amplitude in $[a, b]$, then $L$ has tor-amplitude in $[a, b]$. \item If $L$ has tor-amplitude in $[a + 1, b + 1]$ and $M$ has tor-amplitude in $[a, b]$, then $K$ has tor-amplitude in $[a + 1, b + 1]$. \end{enumerate}


sites-cohomology-lemma-tensor-tor-amplitude

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K, L$ be objects of $D(\mathcal{O})$. If $K$ has tor-amplitude in $[a, b]$ and $L$ has tor-amplitude in $[c, d]$ then $K \otimes_\mathcal{O}^\mathbf{L} L$ has tor amplitude in $[a + c, b + d]$.


sites-cohomology-lemma-summands-tor-amplitude

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $a, b \in \mathbf{Z}$. For $K$, $L$ objects of $D(\mathcal{O})$ if $K \oplus L$ has tor amplitude in $[a, b]$ so do $K$ and $L$.


sites-cohomology-lemma-bounded

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a sheaf of ideals. Let $K$ be an object of $D(\mathcal{O})$. \begin{enumerate} \item If $K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}/\mathcal{I}$ is bounded above, then $K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}/\mathcal{I}^n$ is uniformly bounded above for all $n$. \item If $K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}/\mathcal{I}$ as an object of $D(\mathcal{O}/\mathcal{I})$ has tor amplitude in $[a, b]$, then $K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}/\mathcal{I}^n$ as an object of $D(\mathcal{O}/\mathcal{I}^n)$ has tor amplitude in $[a, b]$ for all $n$. \end{enumerate}


sites-cohomology-lemma-tor-amplitude-stalk

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. Let $a, b \in \mathbf{Z}$. \begin{enumerate} \item If $E$ has tor amplitude in $[a, b]$, then for every point $p$ of the site $\mathcal{C}$ the object $E_p$ of $D(\mathcal{O}_p)$ has tor amplitude in $[a, b]$. \item If $\mathcal{C}$ has enough points, then the converse is true. \end{enumerate}


sites-cohomology-lemma-perfect-independent-representative

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. \begin{enumerate} \item If $\mathcal{C}$ has a final object $X$ and there exist a covering $\{U_i \to X\}$, strictly perfect complexes $\mathcal{E}_i^\bullet$ of $\mathcal{O}_{U_i}$-modules, and isomorphisms  $\alpha_i : \mathcal{E}_i^\bullet \to E|_{U_i}$ in $D(\mathcal{O}_{U_i})$, then $E$ is perfect. \item If $E$ is perfect, then any complex representing $E$ is perfect. \end{enumerate}


sites-cohomology-lemma-perfect-precise

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. Let $a \leq b$ be integers. If $E$ has tor amplitude in $[a, b]$ and is $(a - 1)$-pseudo-coherent, then $E$ is perfect.


sites-cohomology-lemma-perfect

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. The following are equivalent \begin{enumerate} \item $E$ is perfect, and \item $E$ is pseudo-coherent and locally has finite tor dimension. \end{enumerate}


sites-cohomology-lemma-perfect-pullback

Let $(f, f^\sharp) : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D})$ be a morphism of ringed sites. Let $E$ be an object of $D(\mathcal{O}_\mathcal{D})$. If $E$ is perfect in $D(\mathcal{O}_\mathcal{D})$, then $Lf^*E$ is perfect in $D(\mathcal{O}_\mathcal{C})$.


sites-cohomology-lemma-two-out-of-three-perfect

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\mathcal{O})$. If two out of three of $K, L, M$ are perfect then the third is also perfect.


sites-cohomology-lemma-symmetric-monoidal-cat-complexes

Let $(\mathcal{C}, \mathcal{O})$ be a ringed space. The category of complexes of $\mathcal{O}$-modules with tensor product defined by $\mathcal{F}^\bullet \otimes \mathcal{G}^\bullet = \text{Tot}(\mathcal{F}^\bullet \otimes_\mathcal{O} \mathcal{G}^\bullet)$ is a symmetric monoidal category.


sites-cohomology-lemma-dual-perfect-complex

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K$ be a perfect object of $D(\mathcal{O})$. Then $K^\vee = R\SheafHom(K, \mathcal{O})$ is a perfect object too and $(K^\vee)^\vee \cong K$. There are functorial isomorphisms $$ M \otimes^\mathbf{L}_\mathcal{O} K^\vee = R\SheafHom_\mathcal{O}(K, M) $$ and $$ H^0(\mathcal{C}, M \otimes^\mathbf{L}_\mathcal{O} K^\vee) = \Hom_{D(\mathcal{O})}(K, M) $$ for $M$ in $D(\mathcal{O})$.


sites-cohomology-lemma-symmetric-monoidal-derived

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. The derived category $D(\mathcal{O})$ is a symmetric monoidal category with tensor product given by derived tensor product with usual associativity and commutativity constraints (for sign rules, see More on Algebra, Section \ref{more-algebra-section-sign-rules}).


sites-cohomology-lemma-left-dual-derived

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $M$ be an object of $D(\mathcal{O})$. If $M$ has a left dual in the monoidal category $D(\mathcal{O})$ (Categories, Definition \ref{categories-definition-dual}) then $M$ is perfect and the left dual is as constructed in Example \ref{example-dual-derived}.


sites-cohomology-lemma-colim-and-lim-of-duals

\begin{slogan} Trivial duality for systems of perfect objects. \end{slogan} Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(K_n)_{n \in \mathbf{N}}$ be a system of perfect objects of $D(\mathcal{O})$. Let $K = \text{hocolim} K_n$ be the derived colimit (Derived Categories, Definition \ref{derived-definition-derived-colimit}). Then for any object $E$ of $D(\mathcal{O})$ we have $$ R\SheafHom(K, E) = R\lim E \otimes^\mathbf{L}_\mathcal{O} K_n^\vee $$ where $(K_n^\vee)$ is the inverse system of dual perfect complexes.


sites-cohomology-lemma-category-summands-finite-free

Let $(\mathcal{C}, \mathcal{O})$ be a ringed space. Set $R = \Gamma(\mathcal{C}, \mathcal{O})$. The category of $\mathcal{O}$-modules which are summands of finite free $\mathcal{O}$-modules is equivalent to the category of finite projective $R$-modules.


sites-cohomology-lemma-projection-formula

Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. Let $E \in D(\mathcal{O}_\mathcal{C})$ and $K \in D(\mathcal{O}_\mathcal{D})$. If $K$ is perfect, then $$ Rf_*E \otimes^\mathbf{L}_{\mathcal{O}_\mathcal{D}} K = Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_\mathcal{C}} Lf^*K) $$ in $D(\mathcal{O}_\mathcal{D})$.


sites-cohomology-lemma-w-contractible

Let $\mathcal{C}$ be a site. Let $U$ be a weakly contractible object of $\mathcal{C}$. Then \begin{enumerate} \item the functor $\mathcal{F} \mapsto \mathcal{F}(U)$ is an exact functor $\textit{Ab}(\mathcal{C}) \to \textit{Ab}$, \item $H^p(U, \mathcal{F}) = 0$ for every abelian sheaf $\mathcal{F}$ and all $p \geq 1$, and \item for any sheaf of groups $\mathcal{G}$ any $\mathcal{G}$-torsor has a section over $U$. \end{enumerate}


sites-cohomology-lemma-compact-in-terms-of-generators

Let $\mathcal{A}$ be a Grothendieck abelian category. Let $S \subset \Ob(\mathcal{A})$ be a set of objects such that \begin{enumerate} \item any object of $\mathcal{A}$ is a quotient of a direct sum of elements of $S$, and \item for any $E \in S$ the functor $\Hom_\mathcal{A}(E, -)$ commutes with direct sums. \end{enumerate} Then every compact object of $D(\mathcal{A})$ is a direct summand in $D(\mathcal{A})$ of a finite complex of finite direct sums of elements of $S$.


sites-cohomology-lemma-when-jshriek-lower-compact

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. Assume the functors $\mathcal{F} \mapsto H^p(U, \mathcal{F})$ commute with direct sums. Then $\mathcal{O}$-module $j_!\mathcal{O}_U$ is a compact object of $D^+(\mathcal{O})$ in the following sense: if $M = \bigoplus_{i \in I} M_i$ in $D(\mathcal{O})$ is bounded below, then $\Hom(j_{U!}\mathcal{O}_U, M) = \bigoplus_{i \in I} \Hom(j_{U!}\mathcal{O}_U, M_i)$.


sites-cohomology-lemma-when-jshriek-lower-compact-worked-out

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site with set of coverings $\text{Cov}_\mathcal{C}$. 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 for $U \in \mathcal{B}$ the object $j_{U!}\mathcal{O}_U$ is a compact object of $D^+(\mathcal{O})$ in the following sense: if $M = \bigoplus_{i \in I} M_i$ in $D(\mathcal{O})$ is bounded below, then $\Hom(j_{U!}\mathcal{O}_U, M) = \bigoplus_{i \in I} \Hom(j_{U!}\mathcal{O}_U, M_i)$.


sites-cohomology-lemma-when-jshriek-compact

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. The $\mathcal{O}$-module $j_!\mathcal{O}_U$ is a compact object of $D(\mathcal{O})$ if there exists an integer $d$ such that \begin{enumerate} \item $H^p(U, \mathcal{F}) = 0$ for all $p > d$, and \item the functors $\mathcal{F} \mapsto H^p(U, \mathcal{F})$ commute with direct sums. \end{enumerate}


sites-cohomology-lemma-quasi-compact-weakly-contractible-compact

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$ which is quasi-compact and weakly contractible. Then $j_!\mathcal{O}_U$ is a compact object of $D(\mathcal{O})$.


sites-cohomology-lemma-locally-constant

Let $\mathcal{C}$ be a site with final object $X$. Let $\Lambda$ be a Noetherian ring. Let $K \in D^b(\mathcal{C}, \Lambda)$ with $H^i(K)$ locally constant sheaves of $\Lambda$-modules of finite type. Then there exists a covering $\{U_i \to X\}$ such that each $K|_{U_i}$ is represented by a complex of locally constant sheaves of $\Lambda$-modules of finite type.


sites-cohomology-lemma-map-out-of-locally-constant

Let $\mathcal{C}$ be a site with final object $X$. Let $\Lambda$ be a ring. Let \begin{enumerate} \item $K$ a perfect object of $D(\Lambda)$, \item a finite complex $K^\bullet$ of finite projective $\Lambda$-modules representing $K$, \item $\mathcal{L}^\bullet$ a complex of sheaves of $\Lambda$-modules, and \item $\varphi : \underline{K} \to \mathcal{L}^\bullet$ a map in $D(\mathcal{C}, \Lambda)$. \end{enumerate} Then there exists a covering $\{U_i \to X\}$ and maps of complexes $\alpha_i : \underline{K}^\bullet|_{U_i} \to \mathcal{L}^\bullet|_{U_i}$ representing $\varphi|_{U_i}$.


sites-cohomology-lemma-locally-constant-map

Let $\mathcal{C}$ be a site with final object $X$. Let $\Lambda$ be a ring. Let $K, L$ be objects of $D(\Lambda)$ with $K$ perfect. Let $\varphi : \underline{K} \to \underline{L}$ be map in $D(\mathcal{C}, \Lambda)$. There exists a covering $\{U_i \to X\}$ such that $\varphi|_{U_i}$ is equal to $\underline{\alpha_i}$ for some map $\alpha_i : K \to L$ in $D(\Lambda)$.


sites-cohomology-lemma-locally-constant-tensor-product

Let $\mathcal{C}$ be a site. Let $\Lambda$ be a Noetherian ring. Let $K, L \in D^-(\mathcal{C}, \Lambda)$. If the cohomology sheaves of $K$ and $L$ are locally constant sheaves of $\Lambda$-modules of finite type, then the cohomology sheaves of $K \otimes_\Lambda^\mathbf{L} L$ are locally constant sheaves of $\Lambda$-modules of finite type.


sites-cohomology-lemma-locally-constant-bounded

Let $\mathcal{C}$ be a site. Let $\Lambda$ be a Noetherian ring. Let $I \subset \Lambda$ be an ideal. Let $K \in D^-(\mathcal{C}, \Lambda)$. If the cohomology sheaves of $K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I}$ are locally constant sheaves of $\Lambda/I$-modules of finite type, then the cohomology sheaves of $K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n}$ are locally constant sheaves of $\Lambda/I^n$-modules of finite type for all $n \geq 1$.


sites-cohomology-proposition-enough-weakly-contractibles

Let $\mathcal{C}$ be a site. Let $\mathcal{B} \subset \Ob(\mathcal{C})$ such that every $U \in \mathcal{B}$ is weakly contractible and every object of $\mathcal{C}$ has a covering by elements of $\mathcal{B}$. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. Then \begin{enumerate} \item A complex $\mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3$ of $\mathcal{O}$-modules is exact, if and only if $\mathcal{F}_1(U) \to \mathcal{F}_2(U) \to \mathcal{F}_3(U)$ is exact for all $U \in \mathcal{B}$. \item Every object $K$ of $D(\mathcal{O})$ is a derived limit of its canonical truncations: $K = R\lim \tau_{\geq -n} K$. \item Given an inverse system $\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1$ with surjective transition maps, the projection $\lim \mathcal{F}_n \to \mathcal{F}_1$ is surjective. \item Products are exact on $\textit{Mod}(\mathcal{O})$. \item Products on $D(\mathcal{O})$ can be computed by taking products of any representative complexes. \item If $(\mathcal{F}_n)$ is an inverse system of $\mathcal{O}$-modules, then $R^p\lim \mathcal{F}_n = 0$ for all $p > 1$ and $$ R^1\lim \mathcal{F}_n  = \Coker(\prod \mathcal{F}_n \to \prod \mathcal{F}_n) $$ where the map is $(x_n) \mapsto (x_n - f(x_{n + 1}))$. \item If $(K_n)$ is an inverse system of objects of $D(\mathcal{O})$, then there are short exact sequences $$ 0 \to R^1\lim H^{p - 1}(K_n) \to H^p(R\lim K_n) \to \lim H^p(K_n) \to 0 $$ \end{enumerate}


fields-theorem-existence-algebraic-closure

Every field has an algebraic closure.


fields-theorem-galois-theory

Let $L/K$ be a finite Galois extension with Galois group $G$. Then we have $K = L^G$ and the map $$ \{\text{subgroups of }G\} \longrightarrow \{\text{subextensions }K \subset M \subset L\},\quad H \longmapsto L^H $$ is a bijection whose inverse maps $M$ to $\text{Gal}(L/M)$. The normal subgroups $H$ of $G$ correspond exactly to those subextensions $M$ with $M/K$ Galois.


fields-theorem-inifinite-galois-theory

Let $L/K$ be a Galois extension. Let $G = \text{Gal}(L/K)$ be the Galois group viewed as a profinite topological group (Lemma \ref{lemma-galois-profinite}). Then we have $K = L^G$ and the map $$ \{\text{closed subgroups of }G\} \longrightarrow \{\text{subextensions }K \subset M \subset L\},\quad H \longmapsto L^H $$ is a bijection whose inverse maps $M$ to $\text{Gal}(L/M)$. The finite subextensions $M$ correspond exactly to the open subgroups $H \subset G$. The normal closed subgroups $H$ of $G$ correspond exactly to subextensions $M$ Galois over $K$.


fields-lemma-vector-space-is-free

If $k$ is a field, then every $k$-module is free.


fields-lemma-field-maps-injective

If $F$ is a field and $R$ is a nonzero ring, then any ring homomorphism $\varphi : F \to R$ is injective.


fields-lemma-field-extension-generated-by-one-element

If a field extension $F/k$ is generated by one element, then it is $k$-isomorphic either to the rational function field $k(t)/k$ or to one of the extensions $k[t]/(P)$ for $P \in k[t]$ irreducible.


fields-lemma-finite-goes-up

Let $K/E/F$ be a tower of algebraic field extensions. If $K$ is finite over $F$, then $K$ is finite over $E$.


fields-lemma-finite-finitely-generated

A finite extension of fields is a finitely generated field extension. The converse is not true.


fields-lemma-multiplicativity-degrees

Suppose given a tower of fields $F/E/k$. Then $$ [F:k] = [F:E][E:k] $$


fields-lemma-algebraic-goes-up

Let $K/E/F$ be a tower of field extensions. \begin{enumerate} \item If $\alpha \in K$ is algebraic over $F$, then $\alpha$ is algebraic over $E$. \item if $K$ is algebraic over $F$, then $K$ is algebraic over $E$. \end{enumerate}


fields-lemma-finite-is-algebraic

A finite extension is algebraic. In fact, an extension $E/k$ is algebraic if and only if every subextension $k(\alpha)/k$ generated by some $\alpha \in E$ is finite.


fields-lemma-algebraic-finitely-generated

\begin{slogan} A finitely generated algebraic extension is finite. \end{slogan} Let $k$ be a field, and let $\alpha_1, \alpha_2, \ldots, \alpha_n$ be elements of some extension field such that each $\alpha_i$ is algebraic over $k$. Then the extension $k(\alpha_1, \ldots, \alpha_n)/k$ is finite. That is, a finitely generated algebraic extension is finite.


fields-lemma-algebraic-permanence

Let $E/k$ and $F/E$ be algebraic extensions of fields. Then $F/k$ is an algebraic extension of fields.


fields-lemma-size-algebraic-extension

Let $E/F$ be an algebraic extension of fields. Then the cardinality $|E|$ of $E$ is at most $\max(\aleph_0, |F|)$.


fields-lemma-subalgebra-algebraic-extension-field

Let $E/F$ be a finite or more generally an algebraic extension of fields. Any subring $F \subset R \subset E$ is a field.


fields-lemma-algebraic-extension-self-map

Let $E/F$ an algebraic extension of fields. Any $F$-algebra map $f : E \to E$ is an automorphism.


fields-lemma-degree-minimal-polynomial

The degree of the minimal polynomial is $[k(\alpha) : k]$.


fields-lemma-algebraically-closed

Let $F$ be a field. The following are equivalent \begin{enumerate} \item $F$ is algebraically closed, \item every irreducible polynomial over $F$ is linear, \item every nonconstant polynomial over $F$ has a root, \item every nonconstant polynomial over $F$ is a product of linear factors. \end{enumerate}


fields-lemma-map-into-algebraic-closure

Let $F$ be a field. Let $\overline{F}$ be an algebraic closure of $F$. Let $M/F$ be an algebraic extension. Then there is a morphism of $F$-extensions $M \to \overline{F}$.


fields-lemma-relatively-prime-polynomials

Two polynomials in $k[x]$ are relatively prime precisely when they have no common roots in an algebraic closure $\overline{k}$ of $k$.


fields-lemma-irreducible-polynomials

Let $F$ be a field. Let $P \in F[x]$ be an irreducible polynomial over $F$. Let $P' = \text{d}P/\text{d}x$ be the derivative of $P$ with respect to $x$. Then one of the following two cases happens \begin{enumerate} \item $P$ and $P'$ are relatively prime, or \item $P'$ is the zero polynomial. \end{enumerate} The second case can only happen if $F$ has characteristic $p > 0$. In this case $P(x) = Q(x^q)$ where $q = p^f$ is a power of $p$ and $Q \in F[x]$ is an irreducible polynomial such that $Q$ and $Q'$ are relatively prime.


fields-lemma-separable-goes-up

Let $K/E/F$ be a tower of algebraic field extensions. \begin{enumerate} \item If $\alpha \in K$ is separable over $F$, then $\alpha$ is separable over $E$. \item if $K$ is separable over $F$, then $K$ is separable over $E$. \end{enumerate}


fields-lemma-recognize-separable

Let $F$ be a field. An irreducible polynomial $P$ over $F$ is separable if and only if $P$ has pairwise distinct roots in an algebraic closure of $F$.


fields-lemma-nr-roots-unchanged

Let $F$ be a field and let $\overline{F}$ be an algebraic closure of $F$. Let $p > 0$ be the characteristic of $F$. Let $P$ be a polynomial over $F$. Then the set of roots of $P$ and $P(x^p)$ in $\overline{F}$ have the same cardinality (not counting multiplicity).


fields-lemma-count-embeddings

In Situation \ref{situation-finitely-generated} the correspondence $$ \Mor_F(K, \overline{F}) \longrightarrow \{(\beta_1, \ldots, \beta_n)\text{ as below}\}, \quad \varphi \longmapsto (\varphi(\alpha_1), \ldots, \varphi(\alpha_n)) $$ is a bijection. Here the right hand side is the set of $n$-tuples $(\beta_1, \ldots, \beta_n)$ of elements of $\overline{F}$ such that $\beta_i$ is a root of $P_i^\varphi$.


fields-lemma-count-embeddings-explicitly

In Situation \ref{situation-finitely-generated} we have $|\Mor_F(K, \overline{F})| = \prod_{i = 1}^n \deg_s(P_i)$.


fields-lemma-separably-generated-separable

Assumptions and notation as in Situation \ref{situation-finitely-generated}. If each $P_i$ is separable, i.e., each $\alpha_i$ is separable over $K_{i - 1}$, then $$ |\Mor_F(K, \overline{F})| = [K : F] $$ and the field extension $K/F$ is separable. If one of the $\alpha_i$ is not separable over $K_{i - 1}$, then $|\Mor_F(K, \overline{F})| < [K : F]$.


fields-lemma-separable-equality

Let $K/F$ be a finite extension of fields. Let $\overline{F}$ be an algebraic closure of $F$. Then we have $$ |\Mor_F(K, \overline{F})| \leq [K : F] $$ with equality if and only if $K$ is separable over $F$.


fields-lemma-separable-permanence

Let $E/k$ and $F/E$ be separable algebraic extensions of fields. Then $F/k$ is a separable extension of fields.


fields-lemma-separable-elements

Let $E/k$ be a field extension. Then the elements of $E$ separable over $k$ form a subextension of $E/k$.


fields-lemma-independence-characters

Let $L$ be a field. Let $G$ be a monoid, for example a group. Let $\chi_1, \ldots, \chi_n : G \to L$ be pairwise distinct homomorphisms of monoids where $L$ is regarded as a monoid by multiplication. Then $\chi_1, \ldots, \chi_n$ are $L$-linearly independent: if $\lambda_1, \ldots, \lambda_n \in L$ not all zero, then $\sum \lambda_i\chi_i(g) \not = 0$ for some $g \in G$.


fields-lemma-sums-of-powers

Let $L$ be a field. Let $n \geq 1$ and $\alpha_1, \ldots, \alpha_n \in L$ pairwise distinct elements of $L$. Then there exists an $e \geq 0$ such that $\sum_{i = 1, \ldots, n} \alpha_i^e \not = 0$.


fields-lemma-independence-embeddings

Let $K/F$ and $L/F$ be field extensions. Let $\sigma_1, \ldots, \sigma_n : K \to L$ be pairwise distinct morphisms of $F$-extensions. Then $\sigma_1, \ldots, \sigma_n$ are $L$-linearly independent: if $\lambda_1, \ldots, \lambda_n \in L$ not all zero, then $\sum \lambda_i\sigma_i(\alpha) \not = 0$ for some $\alpha \in K$.


fields-lemma-finite-separable-tensor-alg-closed

Let $K/F$ and $L/F$ be field extensions with $K/F$ finite separable and $L$ algebraically closed. Then the map $$ K \otimes_F L \longrightarrow \prod\nolimits_{\sigma \in \Hom_F(K, L)} L,\quad \alpha \otimes \beta \mapsto (\sigma(\alpha)\beta)_\sigma $$ is an isomorphism of $L$-algebras.


fields-lemma-take-pth-root

Let $p$ be a prime number. Let $F$ be a field of characteristic $p$. Let $t \in F$ be an element which does not have a $p$th root in $F$. Then the polynomial $x^p - t$ is irreducible over $F$.


fields-lemma-purely-inseparable-permanence

Let $E/k$ and $F/E$ be purely inseparable extensions of fields. Then $F/k$ is a purely inseparable extension of fields.


fields-lemma-purely-inseparable-elements

Let $E/k$ be a field extension. Then the elements of $E$ purely-inseparable over $k$ form a subextension of $E/k$.


fields-lemma-finite-purely-inseparable

Let $E/F$ be a finite purely inseparable field extension of characteristic $p > 0$. Then there exists a sequence of elements $\alpha_1, \ldots, \alpha_n \in E$ such that we obtain a tower of fields $$ E = F(\alpha_1, \ldots, \alpha_n) \supset F(\alpha_1, \ldots, \alpha_{n - 1}) \supset \ldots \supset F(\alpha_1) \supset F $$ such that each intermediate extension is of degree $p$ and comes from adjoining a $p$th root. Namely, $\alpha_i^p \in F(\alpha_1, \ldots, \alpha_{i - 1})$ is an element which does not have a $p$th root in $F(\alpha_1, \ldots, \alpha_{i - 1})$ for $i = 1, \ldots, n$.


fields-lemma-separable-first

\begin{slogan} Any algebraic field extension is uniquely a separable field extension followed by a purely inseparable one. \end{slogan} Let $E/F$ be an algebraic field extension. There exists a unique subextension $E/E_{sep}/F$ such that $E_{sep}/F$ is separable and $E/E_{sep}$ is purely inseparable.


fields-lemma-separable-degree

Let $K/F$ be a finite extension. Let $\overline{F}$ be an algebraic closure of $F$. Then $[K : F]_s = |\Mor_F(K, \overline{F})|$.


fields-lemma-multiplicativity-all-degrees

Suppose given a tower of algebraic field extensions $K/E/F$. Then $$ [K : F]_s = [K : E]_s [E : F]_s \quad\text{and}\quad [K : F]_i = [K : E]_i [E : F]_i $$


fields-lemma-normal-goes-up

Let $K/E/F$ be a tower of algebraic field extensions. If $K$ is normal over $F$, then $K$ is normal over $E$.


fields-lemma-intersect-normal

Let $F$ be a field. Let $M/F$ be an algebraic extension. Let $F \subset E_i \subset M$, $i \in I$ be subextensions with $E_i/F$ normal. Then $\bigcap E_i$ is normal over $F$.


fields-lemma-separable-first-normal

Let $E/F$ be a normal algebraic field extension. Then the subextension $E/E_{sep}/F$ of Lemma \ref{lemma-separable-first} is normal.


fields-lemma-characterize-normal

Let $E/F$ be an algebraic extension of fields. Let $\overline{F}$ be an algebraic closure of $F$. The following are equivalent \begin{enumerate} \item $E$ is normal over $F$, and \item for every pair $\sigma, \sigma' \in \Mor_F(E, \overline{F})$ we have $\sigma(E) = \sigma'(E)$. \end{enumerate}


fields-lemma-normally-generated

Let $E/F$ be an algebraic extension of fields. If $E$ is generated by $\alpha_i \in E$, $i \in I$ over $F$ and if for each $i$ the minimal polynomial of $\alpha_i$ over $F$ splits completely in $E$, then $E/F$ is normal.


fields-lemma-lift-maps

Let $L/M/K$ be a tower of algebraic extensions. \begin{enumerate} \item If $M/K$ is normal, then any automorphism $\tau$ of $L/K$ induces an automorphism $\tau|_M : M \to M$. \item If $L/K$ is normal, then any $K$-algebra map $\sigma : M \to L$ extends to an automorphism of $L$. \end{enumerate}


fields-lemma-normal-and-automorphisms

Let $E/F$ be a finite extension. We have $$ |\text{Aut}(E/F)| \leq [E : F]_s $$ with equality if and only if $E$ is normal over $F$.


fields-lemma-normal-embeddings-differ-by-aut

Let $L/K$ be an algebraic normal extension of fields. Let $E/K$ be an extension of fields. Then either there is no $K$-embedding from $L$ to $E$ or there is one $\tau : L \to E$ and every other one is of the form $\tau \circ \sigma$ where $\sigma \in \text{Aut}(L/K)$.


fields-lemma-splitting-field

Let $F$ be a field. Let $P \in F[x]$ be a nonconstant polynomial. There exists a smallest field extension $E/F$ such that $P$ splits completely over $E$. Moreover, the field extension $E/F$ is normal and unique up to (nonunique) isomorphism.


fields-lemma-normal-closure

\begin{slogan} Existence of normal closure of finite extensions of fields. \end{slogan} Let $E/F$ be a finite extension of fields. There exists a unique smallest finite extension $K/E$ such that $K$ is normal over $F$.


fields-lemma-normal-closure-inside-normal

Let $L/K$ be an algebraic normal extension. \begin{enumerate} \item If $L/M/K$ is a subextension with $M/K$ finite, then there exists a tower $L/M'/M/K$ with $M'/K$ finite and normal. \item If $L/M'/M/K$ is a tower with $M/K$ normal and $M'/M$ finite, then there exists a tower $L/M''/M'/M/K$ with $M''/M$ finite and $M''/K$ normal. \end{enumerate}


fields-lemma-normal-closure-tensor-product

Let $L/K$ be a finite extension. Let $M/L$ be the normal closure of $L$ over $K$. Then there is a surjective map $$ L \otimes_K L \otimes_K \ldots \otimes_K L \longrightarrow M $$ of $K$-algebras where the number of tensors can be taken $[L : K]_s \leq [L : K]$.


fields-lemma-cyclic

Let $A$ be an abelian group of exponent dividing $n$ such that $\{x \in A \mid dx = 0\}$ has cardinality at most $d$ for all $d | n$. Then $A$ is cyclic of order dividing $n$.


fields-lemma-primitive-element

Let $E/F$ be a finite extension of fields. The following are equivalent \begin{enumerate} \item there exists a primitive element for $E$ over $F$, and \item there are finitely many subextensions $E/K/F$. \end{enumerate} Moreover, (1) and (2) hold if $E/F$ is separable.


fields-lemma-characteristic-vs-minimal-polynomial

Let $L/K$ be a finite extension of fields. Let $\alpha \in L$ and let $P$ be the minimal polynomial of $\alpha$ over $K$. Then the characteristic polynomial of the $K$-linear map $\alpha : L \to L$ is equal to $P^e$ with $e \deg(P) = [L : K]$.


fields-lemma-trace-and-norm-from-minimal-polynomial

Let $L/K$ be a finite extension of fields. Let $\alpha \in L$ and let $P = x^d + a_1 x^{d - 1} + \ldots + a_d$ be the minimal polynomial of $\alpha$ over $K$. Then $$ \text{Norm}_{L/K}(\alpha) = (-1)^{[L : K]} a_d^e \quad\text{and}\quad \text{Trace}_{L/K}(\alpha) = - e a_1 $$ where $e d = [L : K]$.


fields-lemma-trace-and-norm-linear

Let $L/K$ be a finite extension of fields. Let $V$ be a finite dimensional vector space over $L$. Let $\varphi : V \to V$ be an $L$-linear map. Then $$ \text{Trace}_K(\varphi : V \to V) = \text{Trace}_{L/K}(\text{Trace}_L(\varphi : V \to V)) $$ and $$ \det\nolimits_K(\varphi : V \to V) = \text{Norm}_{L/K}(\det\nolimits_L(\varphi : V \to V)) $$


fields-lemma-trace-and-norm-tower

Let $M/L/K$ be a tower of finite extensions of fields. Then $$ \text{Trace}_{M/K} = \text{Trace}_{L/K} \circ \text{Trace}_{M/L} \quad\text{and}\quad \text{Norm}_{M/K} = \text{Norm}_{L/K} \circ \text{Norm}_{M/L} $$


fields-lemma-separable-trace-pairing

Let $L/K$ be a finite extension of fields. The following are equivalent: \begin{enumerate} \item $L/K$ is separable, \item $\text{Trace}_{L/K}$ is not identically zero, and \item the trace pairing $Q_{L/K}$ is nondegenerate. \end{enumerate}


fields-lemma-finite-Galois

Let $E/F$ be a finite extension of fields. Then $E$ is Galois over $F$ if and only if $|\text{Aut}(E/F)| = [E : F]$.


fields-lemma-galois-goes-up

Let $K/E/F$ be a tower of algebraic field extensions. If $K$ is Galois over $F$, then $K$ is Galois over $E$.


fields-lemma-normal-closure-galois

Let $L/K$ be a finite separable extension of fields. Let $M$ be the normal closure of $L$ over $K$ (Definition \ref{definition-normal-closure}). Then $M/K$ is Galois.


fields-lemma-galois-over-fixed-field

Let $K$ be a field. Let $G$ be a finite group acting faithfully on $K$. Then the extension $K/K^G$ is Galois, we have $[K : K^G] = |G|$, and the Galois group of the extension is $G$.


fields-lemma-ses-galois

Let $L/M/K$ be a tower of fields. Assume $L/K$ and $M/K$ are finite Galois. Then we obtain a short exact sequence $$ 1 \to \text{Gal}(L/M) \to \text{Gal}(L/K) \to \text{Gal}(M/K) \to 1 $$ of finite groups.


fields-lemma-galois-profinite

Let $E/F$ be a Galois extension. Endow $\text{Gal}(E/F)$ with the coarsest topology such that $$ \text{Gal}(E/F) \times E \longrightarrow E $$ is continuous when $E$ is given the discrete topology. Then \begin{enumerate} \item for any topological space $X$ and map $X \to \text{Aut}(E/F)$ such that the action $X \times E \to E$ is continuous the induced map $X \to \text{Gal}(E/F)$ is continuous, \item this topology turns $\text{Gal}(E/F)$ into a profinite topological group. \end{enumerate}


fields-lemma-galois-infinite

Let $L/M/K$ be a tower of fields. Assume both $L/K$ and $M/K$ are Galois. Then there is a canonical surjective continuous homomorphism $c : \text{Gal}(L/K) \to \text{Gal}(M/K)$.


fields-lemma-infinite-galois-limit

Let $L/K$ be a Galois extension with Galois group $G$. Let $\Lambda$ be the set of finite Galois subextensions, i.e., $\lambda \in \Lambda$ corresponds to $L/L_\lambda/K$ with $L_\lambda/K$ finite Galois with Galois group $G_\lambda$. Define a partial ordering on $\Lambda$ by the rule $\lambda \geq \lambda'$ if and only if $L_\lambda \supset L_{\lambda'}$. Then \begin{enumerate} \item $\Lambda$ is a directed partially ordered set, \item $L_\lambda$ is a system of $K$-extensions over $\Lambda$ and $L = \colim L_\lambda$, \item $G_\lambda$ is an inverse system of finite groups over $\Lambda$, the transition maps are surjective, and $$ G = \lim_{\lambda \in \Lambda} G_\lambda $$ as a profinite group, and \item each of the projections $G \to G_\lambda$ is continuous and surjective. \end{enumerate}


fields-lemma-ses-infinite-galois

Let $L/M/K$ be a tower of fields. Assume $L/K$ and $M/K$ are Galois. Then we obtain a short exact sequence $$ 1 \to \text{Gal}(L/M) \to \text{Gal}(L/K) \to \text{Gal}(M/K) \to 1 $$ of profinite topological groups.


fields-lemma-C-algebraically-closed

The field $\mathbf{C}$ is algebraically closed.


fields-lemma-Kummer

Let $K \subset L$ be a Galois extension of fields whose Galois group is $\mathbf{Z}/n\mathbf{Z}$. Assume moreover that the characteristic of $K$ is prime to $n$ and that $K$ contains a primitive $n$th root of $1$. Then $L = K[z]$ with $z^n \in K$.


fields-lemma-adjoint-pth-root-unity

Let $K$ be a field with algebraic closure $\overline{K}$. Let $p$ be a prime different from the characteristic of $K$. Let $\zeta \in \overline{K}$ be a primitive $p$th root of $1$. Then $K(\zeta)/K$ is a Galois extension of degree dividing $p - 1$.


fields-lemma-subfields-kummer

Let $K$ be a field. Let $L/K$ be a finite extension of degree $e$ which is generated by an element $\alpha$ with $a = \alpha^e \in K$. Then any sub extension $L/L'/K$ is generated by $\alpha^d$ for some $d | e$.


fields-lemma-Artin-Schreier

Let $K \subset L$ be a Galois extension of fields of characteristic $p > 0$ with Galois group $\mathbf{Z}/p\mathbf{Z}$. Then $L = K[z]$ with $z^p - z \in K$.


fields-lemma-transcendence-degree

Let $E/F$ be a field extension. A transcendence basis of $E$ over $F$ exists. Any two transcendence bases have the same cardinality.


fields-lemma-transcendence-degree-tower

Let $k \subset K \subset L$ be field extensions. Then $$ \text{trdeg}_k(L) = \text{trdeg}_K(L) + \text{trdeg}_k(K). $$


fields-lemma-purely-transcendental-degree

Let $k'/k$ be a finite extension of fields. Let $k'(x_1, \ldots, x_r)/k(x_1, \ldots, x_r)$ be the induced extension of purely transcendental extensions. Then $[k'(x_1, \ldots, x_r) : k(x_1, \ldots, x_r)] = [k' : k] < \infty$.


fields-lemma-algebraic-closure-in-finitely-generated

Let $k \subset K$ be a finitely generated field extension. The algebraic closure of $k$ in $K$ is finite over $k$.


fields-lemma-normal-case

Let $E/F$ be a normal algebraic field extension. There exist subextensions $E / E_{sep} /F$ and $E / E_{insep} / F$ such that \begin{enumerate} \item $F \subset E_{sep}$ is Galois and $E_{sep} \subset E$ is purely inseparable, \item $F \subset E_{insep}$ is purely inseparable and $E_{insep} \subset E$ is Galois, \item $E = E_{sep} \otimes_F E_{insep}$. \end{enumerate}


fields-lemma-pth-root

Let $K$ be a field of characteristic $p > 0$. Let $K \subset L$ be a separable algebraic extension. Let $\alpha \in L$. \begin{enumerate} \item If the coefficients of the minimal polynomial of $\alpha$ over $K$ are $p$th powers in $K$ then $\alpha$ is a $p$th power in $L$. \item More generally, if $P \in K[T]$ is a polynomial such that (a) $\alpha$ is a root of $P$, (b) $P$ has pairwise distinct roots in an algebraic closure, and (c) all coefficients of $P$ are $p$th powers, then $\alpha$ is a $p$th power in $L$. \end{enumerate}


spaces-limits-lemma-characterize-relative-limit-preserving

Let $S$ be a scheme. Let $a : F \to G$ be a transformation of functors $(\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. The following are equivalent \begin{enumerate} \item $a : F \to G$ is limit preserving, and \item for every affine scheme $T$ over $S$ which is a limit $T = \lim T_i$ of a directed inverse system of affine schemes $T_i$ over $S$ the diagram of sets $$ \xymatrix{ \colim_i F(T_i) \ar[r] \ar[d]_a & F(T) \ar[d]^a \\ \colim_i G(T_i) \ar[r] & G(T) } $$ is a fibre product diagram. \end{enumerate}


spaces-limits-lemma-composition-locally-finite-presentation

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$, $b : G \to H$ be transformations of functors. If $a$ and $b$ are limit preserving, then $$ b \circ a : F \longrightarrow H $$ is limit preserving.


spaces-limits-lemma-locally-finite-presentation-permanence

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$, $b : G \to H$ be transformations of functors. If $b \circ a$ and $b$ are limit preserving, then $a$ is limit preserving.


spaces-limits-lemma-base-change-locally-finite-presentation

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$, $b : H \to G$ be transformations of functors. Consider the fibre product diagram $$ \xymatrix{ H \times_{b, G, a} F \ar[r]_-{b'} \ar[d]_{a'} & F \ar[d]^a \\ H \ar[r]^b & G } $$ If $a$ is limit preserving, then the base change $a'$ is limit preserving.


spaces-limits-lemma-fibre-product-locally-finite-presentation

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $E, F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$, $b : H \to G$, and $c : G \to E$ be transformations of functors. If $c$, $c \circ a$, and $c \circ b$ are limit preserving, then $F \times_G H \to E$ is too.


spaces-limits-lemma-sheafify-finite-presentation

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. If $F$ is limit preserving then its sheafification $F^\#$ is limit preserving.


spaces-limits-lemma-sheaf-finite-presentation

Let $S$ be a scheme. Let $F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Assume that \begin{enumerate} \item $F$ is a sheaf, and \item there exists an fppf covering $\{U_j \to S\}_{j \in J}$ such that $F|_{(\Sch/U_j)_{fppf}}$ is limit preserving. \end{enumerate} Then $F$ is limit preserving.


spaces-limits-lemma-surjection-is-enough

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If for every directed limit $T = \lim_{i \in I} T_i$ of affine schemes over $S$ the map $$ \colim X(T_i) \longrightarrow X(T) \times_{Y(T)} \colim Y(T_i) $$ is surjective, then $f$ is locally of finite presentation. In other words, in Proposition \ref{proposition-characterize-locally-finite-presentation} part (2) it suffices to check surjectivity in the criterion of Lemma \ref{lemma-characterize-relative-limit-preserving}.


spaces-limits-lemma-directed-inverse-system-has-limit

Let $S$ be a scheme. Let $I$ be a directed set. Let $(X_i, f_{ii'})$ be an inverse system over $I$ in the category of algebraic spaces over $S$. If the morphisms $f_{ii'} : X_i \to X_{i'}$ are affine, then the limit $X = \lim_i X_i$ (as an fppf sheaf) is an algebraic space. Moreover, \begin{enumerate} \item each of the morphisms $f_i : X \to X_i$ is affine, \item for any $i \in I$ and any morphism of algebraic spaces $T \to X_i$ we have $$ X \times_{X_i} T = \lim_{i' \geq i} X_{i'} \times_{X_i} T. $$ as algebraic spaces over $S$. \end{enumerate}


spaces-limits-lemma-space-over-limit

Let $S$ be a scheme. Let $I$ be a directed set. Let $(X_i, f_{ii'})$ be an inverse system over $I$ of algebraic spaces over $S$ with affine transition maps. Let $X = \lim_i X_i$. Let $0 \in I$. Suppose that $T \to X_0$ is a morphism of algebraic spaces. Then $$ T \times_{X_0} X = \lim_{i \geq 0} T \times_{X_0} X_i $$ as algebraic spaces over $S$.


spaces-limits-lemma-directed-inverse-system-closed-immersions

Let $S$ be a scheme. Let $I$ be a directed set. Let $(X_i, f_{i'i}) \to (Y_i, g_{i'i})$ be a morphism of inverse systems over $I$ of algebraic spaces over $S$. Assume \begin{enumerate} \item the morphisms $f_{i'i} : X_{i'} \to X_i$ are affine, \item the morphisms $g_{i'i} : Y_{i'} \to Y_i$ are affine, \item the morphisms $X_i \to Y_i$ are closed immersions. \end{enumerate} Then $\lim X_i \to \lim Y_i$ is a closed immersion.


spaces-limits-lemma-directed-inverse-system-reduced

Let $S$ be a scheme. Let $I$ be a directed set. Let $(X_i, f_{i'i})$ be an inverse systems over $I$ of algebraic spaces over $S$. If $X_i$ is reduced for all $i$, then $X$ is reduced.


spaces-limits-lemma-better-characterize-relative-limit-preserving

Let $S$ be a scheme. Let $X \to Y$ be a morphism of algebraic spaces over $S$. The equivalent conditions (1) and (2) of Proposition \ref{proposition-characterize-locally-finite-presentation} are also equivalent to \begin{enumerate} \item[(3)] for every directed limit $T = \lim T_i$ of quasi-compact and quasi-separated algebraic spaces $T_i$ over $S$ with affine transition morphisms the diagram of sets $$ \xymatrix{ \colim_i \Mor(T_i, X) \ar[r] \ar[d] & \Mor(T, X) \ar[d] \\ \colim_i \Mor(T_i, Y) \ar[r] & \Mor(T, Y) } $$ is a fibre product diagram. \end{enumerate}


spaces-limits-lemma-inverse-limit-sets

Let $S$ be a scheme. Let $X = \lim_{i \in I} X_i$ be the limit of a directed inverse system of algebraic spaces over $S$ with affine transition morphisms (Lemma \ref{lemma-directed-inverse-system-has-limit}). If each $X_i$ is decent (for example quasi-separated or locally separated) then $|X| = \lim_i |X_i|$ as sets.


spaces-limits-lemma-topology-limit

With same notation and assumptions as in Lemma \ref{lemma-inverse-limit-sets} we have $|X| = \lim_i |X_i|$ as topological spaces.


spaces-limits-lemma-limit-nonempty

Let $S$ be a scheme. Let $X = \lim_{i \in I} X_i$ be the limit of a directed inverse system of algebraic spaces over $S$ with affine transition morphisms (Lemma \ref{lemma-directed-inverse-system-has-limit}). If each $X_i$ is quasi-compact and nonempty, then $|X|$ is nonempty.


spaces-limits-lemma-inverse-limit-irreducibles

Let $S$ be a scheme. Let $X = \lim_{i \in I} X_i$ be the limit of a directed inverse system of algebraic spaces over $S$ with affine transition morphisms (Lemma \ref{lemma-directed-inverse-system-has-limit}). Let $x \in |X|$ with images $x_i \in |X_i|$. If each $X_i$ is decent, then $\overline{\{x\}} = \lim_i \overline{\{x_i\}}$ as sets and as algebraic spaces if endowed with reduced induced scheme structure.


spaces-limits-lemma-descend-opens

Notation and assumptions as in Situation \ref{situation-descent}. For any quasi-compact open subspace $U \subset X$ there exists an $i$ and a quasi-compact open $U_i \subset X_i$ whose inverse image in $X$ is $U$.


spaces-limits-lemma-descend-equality

Notation and assumptions as in Situation \ref{situation-descent}. Let $f_0 : Y_0 \to Z_0$ be a morphism of algebraic spaces over $X_0$. Assume (a) $Y_0 \to X_0$ and $Z_0 \to X_0$ are representable, (b) $Y_0$, $Z_0$ quasi-compact and quasi-separated, (c) $f_0$ locally of finite presentation, and (d) $Y_0 \times_{X_0} X \to Z_0 \times_{X_0} X$ an isomorphism. Then there exists an $i \geq 0$ such that $Y_0 \times_{X_0} X_i \to Z_0 \times_{X_0} X_i$ is an isomorphism.


spaces-limits-lemma-descend-separated

Notation and assumptions as in Situation \ref{situation-descent}. If $X$ is separated, then $X_i$ is separated for some $i \in I$.


spaces-limits-lemma-limit-is-affine

Notation and assumptions as in Situation \ref{situation-descent}. If $X$ is affine, then there exists an $i$ such that $X_i$ is affine.


spaces-limits-lemma-limit-is-scheme

Notation and assumptions as in Situation \ref{situation-descent}. If $X$ is a scheme, then there exists an $i$ such that $X_i$ is a scheme.


spaces-limits-lemma-finite-type-eventually-closed

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $X = \lim X_i$ be a directed limit of algebraic spaces over $B$ with affine transition morphisms. Let $Y \to X$ be a morphism of algebraic spaces over $B$. \begin{enumerate} \item If $Y \to X$ is a closed immersion, $X_i$ quasi-compact, and $Y \to B$ locally of finite type, then $Y \to X_i$ is a closed immersion for $i$ large enough. \item If $Y \to X$ is an immersion, $X_i$ quasi-separated, $Y \to B$ locally of finite type, and $Y$ quasi-compact, then $Y \to X_i$ is an immersion for $i$ large enough. \item If $Y \to X$ is an isomorphism, $X_i$ quasi-compact, $X_i \to B$ locally of finite type, the transition morphisms $X_{i'} \to X_i$ are closed immersions, and $Y \to B$ is locally of finite presentation, then $Y \to X_i$ is an isomorphism for $i$ large enough. \item If $Y \to X$ is a monomorphism, $X_i$ quasi-separated, $Y \to B$ locally of finite type, and $Y$ quasi-compact, then $Y \to X_i$ is a monomorphism for $i$ large enough. \end{enumerate}


spaces-limits-lemma-eventually-separated

Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$. Let $X = \lim X_i$ be a directed limit of algebraic spaces over $Y$ with affine transition morphisms. Assume \begin{enumerate} \item $Y$ is quasi-separated, \item $X_i$ is quasi-compact and quasi-separated, \item the morphism $X \to Y$ is separated. \end{enumerate} Then $X_i \to Y$ is separated for all $i$ large enough.


spaces-limits-lemma-eventually-closed-immersion

Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$. Let $X = \lim X_i$ be a directed limit of algebraic spaces over $Y$ with affine transition morphisms. Assume \begin{enumerate} \item $Y$ quasi-compact and quasi-separated, \item $X_i$ quasi-compact and quasi-separated, \item the transition morphisms $X_{i'} \to X_i$ are closed immersions, \item $X_i \to Y$ locally of finite type \item $X \to Y$ is a closed immersion. \end{enumerate} Then $X_i \to Y$ is a closed immersion for $i$ large enough.


spaces-limits-lemma-descend-etale

With notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is \'etale, \item $f_0$ is locally of finite presentation, \end{enumerate} then $f_i$ is \'etale for some $i \geq 0$.


spaces-limits-lemma-descend-smooth

With notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is smooth, \item $f_0$ is locally of finite presentation, \end{enumerate} then $f_i$ is smooth for some $i \geq 0$.


spaces-limits-lemma-descend-surjective

With notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is surjective, \item $f_0$ is locally of finite presentation, \end{enumerate} then $f_i$ is surjective for some $i \geq 0$.


spaces-limits-lemma-descend-universally-injective

Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is universally injective, \item $f_0$ is locally of finite type, \end{enumerate} then $f_i$ is universally injective for some $i \geq 0$.


spaces-limits-lemma-descend-finite

Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is finite, \item $f_0$ is locally of finite type, \end{enumerate} then $f_i$ is finite for some $i \geq 0$.


spaces-limits-lemma-descend-closed-immersion

Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is a closed immersion, \item $f_0$ is locally of finite type, \end{enumerate} then $f_i$ is a closed immersion for some $i \geq 0$.


spaces-limits-lemma-descend-separated-morphism

Notation and assumptions as in Situation \ref{situation-descent-property}. If $f$ is separated, then $f_i$ is separated for some $i \geq 0$.


spaces-limits-lemma-descend-isomorphism

Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is a isomorphism, \item $f_0$ is locally of finite presentation, \end{enumerate} then $f_i$ is a isomorphism for some $i \geq 0$.


spaces-limits-lemma-descend-flat

Notation and assumptions as in Situation \ref{situation-descent-property}. Let $\mathcal{F}_0$ be a quasi-coherent $\mathcal{O}_{X_0}$-module and denote $\mathcal{F}_i$ the pullback to $X_i$ and $\mathcal{F}$ the pullback to $X$. If \begin{enumerate} \item $\mathcal{F}$ is flat over $Y$, \item $\mathcal{F}_0$ is of finite presentation, and \item $f_0$ is locally of finite presentation, \end{enumerate} then $\mathcal{F}_i$ is flat over $Y_i$ for some $i \geq 0$. In particular, if $f_0$ is locally of finite presentation and $f$ is flat, then $f_i$ is flat for some $i \geq 0$.


spaces-limits-lemma-eventually-proper

Assumptions and notation as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is proper, and \item $f_0$ is locally of finite type, \end{enumerate} then there exists an $i$ such that $f_i$ is proper.


spaces-limits-lemma-eventually-relative-dimension

Assumptions and notation as in Situation \ref{situation-descent-property}. Let $d \geq 0$. If \begin{enumerate} \item $f$ has relative dimension $\leq d$ (Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-relative-dimension}), and \item $f_0$ is locally of finite type, \end{enumerate} then there exists an $i$ such that $f_i$ has relative dimension $\leq d$.


spaces-limits-lemma-descend-finite-presentation

Let $S$ be a scheme. Let $I$ be a directed set. Let $(X_i, f_{ii'})$ be an inverse system over $I$ of algebraic spaces over $S$. Assume \begin{enumerate} \item the morphisms $f_{ii'} : X_i \to X_{i'}$ are affine, \item the spaces $X_i$ are quasi-compact and quasi-separated. \end{enumerate} Let $X = \lim_i X_i$. Then the category of algebraic spaces of finite presentation over $X$ is the colimit over $I$ of the categories of algebraic spaces of finite presentation over $X_i$.


spaces-limits-lemma-descend-modules-finite-presentation

With notation and assumptions as in Lemma \ref{lemma-descend-finite-presentation}. The category of $\mathcal{O}_X$-modules of finite presentation is the colimit over $I$ of the categories $\mathcal{O}_{X_i}$-modules of finite presentation.


spaces-limits-lemma-descend-invertible-modules

With notation and assumptions as in Lemma \ref{lemma-descend-finite-presentation}. Then any invertible $\mathcal{O}_X$-module is the pullback of an invertible $\mathcal{O}_{X_i}$-module for some $i$.


spaces-limits-lemma-colimit-finitely-presented

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Every quasi-coherent $\mathcal{O}_X$-module is a filtered colimit of finitely presented $\mathcal{O}_X$-modules.


spaces-limits-lemma-directed-colimit-finite-type

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $\mathcal{F}$ is the directed colimit of its finite type quasi-coherent submodules.


spaces-limits-lemma-finite-directed-colimit-surjective-maps

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Then we can write $\mathcal{F} = \lim \mathcal{F}_i$ where each $\mathcal{F}_i$ is an $\mathcal{O}_X$-module of finite presentation and all transition maps $\mathcal{F}_i \to \mathcal{F}_{i'}$ surjective.


spaces-limits-lemma-algebra-directed-colimit-finite-presentation

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent $\mathcal{O}_X$-algebra. Then $\mathcal{A}$ is a directed colimit of finitely presented quasi-coherent $\mathcal{O}_X$-algebras.


spaces-limits-lemma-algebra-directed-colimit-finite-type

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent $\mathcal{O}_X$-algebra. Then $\mathcal{A}$ is the directed colimit of its finite type quasi-coherent $\mathcal{O}_X$-subalgebras.


spaces-limits-lemma-finite-algebra-directed-colimit-finite-finitely-presented

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{A}$ be a finite quasi-coherent $\mathcal{O}_X$-algebra. Then $\mathcal{A} = \colim \mathcal{A}_i$ is a directed colimit of finite and finitely presented quasi-coherent $\mathcal{O}_X$-algebras with surjective transition maps.


spaces-limits-lemma-integral-algebra-directed-colimit-finite

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{A}$ be an integral quasi-coherent $\mathcal{O}_X$-algebra. Then \begin{enumerate} \item $\mathcal{A}$ is the directed colimit of its finite quasi-coherent $\mathcal{O}_X$-subalgebras, and \item $\mathcal{A}$ is a directed colimit of finite and finitely presented $\mathcal{O}_X$-algebras. \end{enumerate}


spaces-limits-lemma-extend

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U \subset X$ be a quasi-compact open. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{G} \subset \mathcal{F}|_U$ be a quasi-coherent $\mathcal{O}_U$-submodule which is of finite type. Then there exists a quasi-coherent submodule $\mathcal{G}' \subset \mathcal{F}$ which is of finite type such that $\mathcal{G}'|_U = \mathcal{G}$.


spaces-limits-lemma-relative-approximation

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that \begin{enumerate} \item $X$ is quasi-compact and quasi-separated, and \item $Y$ is quasi-separated. \end{enumerate} Then $X = \lim X_i$ is a limit of a directed inverse system of algebraic spaces $X_i$ of finite presentation over $Y$ with affine transition morphisms over $Y$.


spaces-limits-lemma-affine-morphism-is-limit

Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$. If $Y$ quasi-compact and quasi-separated, then $X$ is a directed limit $X = \lim X_i$ with each $X_i$ affine and of finite presentation over $Y$.


spaces-limits-lemma-integral-limit-finite-and-finite-presentation

Let $S$ be a scheme. Let $f : X \to Y$ be an integral morphism of algebraic spaces over $S$. Assume $Y$ quasi-compact and quasi-separated. Then $X$ can be written as a directed limit $X = \lim X_i$ where $X_i$ are finite and of finite presentation over $Y$.


spaces-limits-lemma-finite-in-finite-and-finite-presentation

Let $S$ be a scheme. Let $f : X \to Y$ be a finite morphism of algebraic spaces over $S$. Assume $Y$ quasi-compact and quasi-separated. Then $X$ can be written as a directed limit $X = \lim X_i$ where the transition maps are closed immersions and the objects $X_i$ are finite and of finite presentation over $Y$.