sites-modules-lemma-limits-colimits

Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos. All limits and colimits exist in $\textit{Mod}(\mathcal{O})$ and the forgetful functor $\textit{Mod}(\mathcal{O}) \to \textit{Ab}(\mathcal{C})$ commutes with them. Moreover, filtered colimits are exact.


sites-modules-lemma-exactness-pushforward-pullback

Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. \begin{enumerate} \item The functor $f_*$ is left exact. In fact it commutes with all limits. \item The functor $f^*$ is right exact. In fact it commutes with all colimits. \end{enumerate}


sites-modules-lemma-check-exactness-stalks

Let $\mathcal{C}$ be a site. If $\{p_i\}_{i \in I}$ is a conservative family of points, then we may check exactness of a sequence of abelian sheaves on the stalks at the points $p_i$, $i \in I$. If $\mathcal{C}$ has enough points, then exactness of a sequence of abelian sheaves may be checked on stalks.


sites-modules-lemma-reflect-surjections

Let $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be a morphism of topoi. The following are equivalent: \begin{enumerate} \item $f^{-1}f_*\mathcal{F} \to \mathcal{F}$ is surjective for all $\mathcal{F}$ in $\textit{Ab}(\mathcal{C})$, and \item $f_* : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})$ reflects surjections. \end{enumerate} In this case the functor $f_* : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})$ is faithful.


sites-modules-lemma-exactness

Let $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be a morphism of topoi. Assume at least one of the following properties holds \begin{enumerate} \item $f_*$ transforms surjections of sheaves of sets into surjections, \item $f_*$ transforms surjections of abelian sheaves into surjections, \item $f_*$ commutes with coequalizers on sheaves of sets, \item $f_*$ commutes with pushouts on sheaves of sets, \end{enumerate} Then $f_* : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})$ is exact.


sites-modules-lemma-morphism-ringed-sites-almost-cocontinuous

Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites associated to the continuous functor $u : \mathcal{C} \to \mathcal{D}$. Assume $u$ is almost cocontinuous. Then \begin{enumerate} \item $f_* : \textit{Ab}(\mathcal{D}) \to \textit{Ab}(\mathcal{C})$ is exact. \item if $f^\sharp : f^{-1}\mathcal{O}_\mathcal{C} \to \mathcal{O}_\mathcal{D}$ is given so that $f$ becomes a morphism of ringed sites, then $f_* : \textit{Mod}(\mathcal{O}_\mathcal{D}) \to \textit{Mod}(\mathcal{O}_\mathcal{C})$ is exact. \end{enumerate}


sites-modules-lemma-g-shriek-adjoint

The functor $g_{p!}$ is a left adjoint to the functor $u^p$. The functor $g_!$ is a left adjoint to the functor $g^{-1}$. In other words the formulas \begin{align*} \Mor_{\textit{PAb}(\mathcal{C})}(\mathcal{F}, u^p\mathcal{G}) & = \Mor_{\textit{PAb}(\mathcal{D})}(g_{p!}\mathcal{F}, \mathcal{G}), \\ \Mor_{\textit{Ab}(\mathcal{C})}(\mathcal{F}, g^{-1}\mathcal{G}) & = \Mor_{\textit{Ab}(\mathcal{D})}(g_!\mathcal{F}, \mathcal{G}) \end{align*} hold bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.


sites-modules-lemma-exactness-lower-shriek

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that \begin{enumerate} \item[(a)] $u$ is cocontinuous, \item[(b)] $u$ is continuous, and \item[(c)] fibre products and equalizers exist in $\mathcal{C}$ and $u$ commutes with them. \end{enumerate} In this case the functor $g_! : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})$ is exact.


sites-modules-lemma-back-and-forth

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that \begin{enumerate} \item[(a)] $u$ is cocontinuous, \item[(b)] $u$ is continuous, and \item[(c)] $u$ is fully faithful. \end{enumerate} For $g_!, g^{-1}, g_*$ as above the canonical maps $\mathcal{F} \to g^{-1}g_!\mathcal{F}$ and $g^{-1}g_*\mathcal{F} \to \mathcal{F}$ are isomorphisms for all abelian sheaves $\mathcal{F}$ on $\mathcal{C}$.


sites-modules-lemma-have-left-adjoint

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be the morphism of topoi associated to a continuous and cocontinuous functor $u : \mathcal{C} \to \mathcal{D}$. \begin{enumerate} \item If $u$ has a left adjoint $w$, then $g_!$ agrees with $g_!^{\Sh}$ on underlying sheaves of sets and $g_!$ is exact. \item If in addition $w$ is cocontinuous, then $g_! = h^{-1}$ and $g^{-1} = h_*$ where $h : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$ is the morphism of topoi associated to $w$. \end{enumerate}


sites-modules-lemma-global-pullback

Let $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal{D}$-module. \begin{enumerate} \item If $\mathcal{F}$ is free then $f^*\mathcal{F}$ is free. \item If $\mathcal{F}$ is finite free then $f^*\mathcal{F}$ is finite free. \item If $\mathcal{F}$ is generated by global sections then $f^*\mathcal{F}$ is generated by global sections. \item Given $r \geq 0$ if $\mathcal{F}$ is generated by $r$ global sections, then $f^*\mathcal{F}$ is generated by $r$ global sections. \item If $\mathcal{F}$ is generated by finitely many global sections then $f^*\mathcal{F}$ is generated by finitely many global sections. \item If $\mathcal{F}$ has a global presentation then $f^*\mathcal{F}$ has a global presentation. \item If $\mathcal{F}$ has a finite global presentation then $f^*\mathcal{F}$ has a finite global presentation. \end{enumerate}


sites-modules-lemma-extension-by-zero

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \Ob(\mathcal{C})$. The restriction functor $j_U^* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_U)$ has a left adjoint $j_{U!} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})$. So $$ \Mor_{\textit{Mod}(\mathcal{O}_U)}(\mathcal{G}, j_U^*\mathcal{F}) = \Mor_{\textit{Mod}(\mathcal{O})}(j_{U!}\mathcal{G}, \mathcal{F}) $$ for $\mathcal{F} \in \Ob(\textit{Mod}(\mathcal{O}))$ and $\mathcal{G} \in \Ob(\textit{Mod}(\mathcal{O}_U))$. Moreover, the extension by zero $j_{U!}\mathcal{G}$ of $\mathcal{G}$ is the sheaf associated to the presheaf $$ V \longmapsto \bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U) $$ with obvious restriction mappings and an obvious $\mathcal{O}$-module structure.


sites-modules-lemma-extension-by-zero-exact

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \Ob(\mathcal{C})$. The functor $j_{U!} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})$ is exact.


sites-modules-lemma-j-shriek-reflects-exactness

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \Ob(\mathcal{C})$. A complex of $\mathcal{O}_U$-modules $\mathcal{G}_1 \to \mathcal{G}_2 \to \mathcal{G}_3$ is exact if and only if $j_{U!}\mathcal{G}_1 \to j_{U!}\mathcal{G}_2 \to j_{U!}\mathcal{G}_3$ is exact as a sequence of $\mathcal{O}$-modules.


sites-modules-lemma-relocalize

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $f : V \to U$ be a morphism of $\mathcal{C}$. Then there exists a commutative diagram $$ \xymatrix{ (\Sh(\mathcal{C}/V), \mathcal{O}_V) \ar[rd]_{(j_V, j_V^\sharp)} \ar[rr]_{(j, j^\sharp)} & & (\Sh(\mathcal{C}/U), \mathcal{O}_U) \ar[ld]^{(j_U, j_U^\sharp)} \\ & (\Sh(\mathcal{C}), \mathcal{O}) & } $$ of ringed topoi. Here $(j, j^\sharp)$ is the localization morphism associated to the object $V/U$ of the ringed site $(\mathcal{C}/V, \mathcal{O}_V)$.


sites-modules-lemma-restrict-back

Let $\mathcal{C}$ be a site. Let $U \in \Ob(\mathcal{C})$. Assume that every $X$ in $\mathcal{C}$ has at most one morphism to $U$. Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}/U$. The canonical maps $\mathcal{F} \to j_U^{-1}j_{U!}\mathcal{F}$ and $j_U^{-1}j_{U*}\mathcal{F} \to \mathcal{F}$ are isomorphisms.


sites-modules-lemma-localize-morphism-ringed-sites

Let $(f, f^\sharp) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{D}, \mathcal{O}')$ be a morphism of ringed sites where $f$ is given by the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V$ be an object of $\mathcal{D}$ and set $U = u(V)$. Then there is a canonical map of sheaves of rings $(f')^\sharp$ such that the diagram of Sites, Lemma \ref{sites-lemma-localize-morphism} is turned into a commutative diagram of ringed topoi $$ \xymatrix{ (\Sh(\mathcal{C}/U), \mathcal{O}_U) \ar[rr]_{(j_U, j_U^\sharp)} \ar[d]_{(f', (f')^\sharp)} & & (\Sh(\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp)} \\ (\Sh(\mathcal{D}/V), \mathcal{O}'_V) \ar[rr]^{(j_V, j_V^\sharp)} & & (\Sh(\mathcal{D}), \mathcal{O}'). } $$ Moreover, in this situation we have $f'_*j_U^{-1} = j_V^{-1}f_*$ and $f'_*j_U^* = j_V^*f_*$.


sites-modules-lemma-relocalize-morphism-ringed-sites

Let $(f, f^\sharp) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{D}, \mathcal{O}')$ be a morphism of ringed sites where $f$ is given by the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V \in \Ob(\mathcal{D})$, $U \in \Ob(\mathcal{C})$ and $c : U \to u(V)$ a morphism of $\mathcal{C}$. There exists a commutative diagram of ringed topoi $$ \xymatrix{ (\Sh(\mathcal{C}/U), \mathcal{O}_U) \ar[rr]_{(j_U, j_U^\sharp)} \ar[d]_{(f_c, f_c^\sharp)} & & (\Sh(\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp)} \\ (\Sh(\mathcal{D}/V), \mathcal{O}'_V) \ar[rr]^{(j_V, j_V^\sharp)} & & (\Sh(\mathcal{D}), \mathcal{O}'). } $$ The morphism $(f_c, f_c^\sharp)$ is equal to the composition of the morphism $$ (f', (f')^\sharp) : (\Sh(\mathcal{C}/u(V)), \mathcal{O}_{u(V)}) \longrightarrow (\Sh(\mathcal{D}/V), \mathcal{O}'_V) $$ of Lemma \ref{lemma-localize-morphism-ringed-sites} and the morphism $$ (j, j^\sharp) : (\Sh(\mathcal{C}/U), \mathcal{O}_U) \to (\Sh(\mathcal{C}/u(V)), \mathcal{O}_{u(V)}) $$ of Lemma \ref{lemma-relocalize}. Given any morphisms $b : V' \to V$, $a : U' \to U$ and $c' : U' \to u(V')$ such that $$ \xymatrix{ U' \ar[r]_-{c'} \ar[d]_a & u(V') \ar[d]^{u(b)} \\ U \ar[r]^-c & u(V) } $$ commutes, then the following diagram of ringed topoi $$ \xymatrix{ (\Sh(\mathcal{C}/U'), \mathcal{O}_{U'}) \ar[rr]_{(j_{U'/U}, j_{U'/U}^\sharp)} \ar[d]_{(f_{c'}, f_{c'}^\sharp)} & & (\Sh(\mathcal{C}/U), \mathcal{O}_U) \ar[d]^{(f_c, f_c^\sharp)} \\ (\Sh(\mathcal{D}/V'), \mathcal{O}'_{V'}) \ar[rr]^{(j_{V'/V}, j_{V'/V}^\sharp)} & & (\Sh(\mathcal{D}/V), \mathcal{O}'_{V'}) } $$ commutes.


sites-modules-lemma-localize-ringed-topos

Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $\mathcal{F} \in \Sh(\mathcal{C})$ be a sheaf. For a sheaf $\mathcal{H}$ on $\mathcal{C}$ denote $\mathcal{H}_\mathcal{F}$ the sheaf $\mathcal{H} \times \mathcal{F}$ seen as an object of the category $\Sh(\mathcal{C})/\mathcal{F}$. The pair $(\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F})$ is a ringed topos and there is a canonical morphism of ringed topoi $$ (j_\mathcal{F}, j_\mathcal{F}^\sharp) : (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) \longrightarrow (\Sh(\mathcal{C}), \mathcal{O}) $$ which is a localization as in Section \ref{section-localize} such that \begin{enumerate} \item the functor $j_\mathcal{F}^{-1}$ is the functor $\mathcal{H} \mapsto \mathcal{H}_\mathcal{F}$, \item the functor $j_\mathcal{F}^*$ is the functor $\mathcal{H} \mapsto \mathcal{H}_\mathcal{F}$, \item the functor $j_{\mathcal{F}!}$ on sheaves of sets is the forgetful functor $\mathcal{G}/\mathcal{F} \mapsto \mathcal{G}$, \item the functor $j_{\mathcal{F}!}$ on sheaves of modules associates to the $\mathcal{O}_\mathcal{F}$-module $\varphi : \mathcal{G} \to \mathcal{F}$ the $\mathcal{O}$-module which is the sheafification of the presheaf $$ V \longmapsto \bigoplus\nolimits_{s \in \mathcal{F}(V)} \{\sigma \in \mathcal{G}(V) \mid \varphi(\sigma) = s \} $$ for $V \in \Ob(\mathcal{C})$. \end{enumerate}


sites-modules-lemma-localize-compare

With $(\Sh(\mathcal{C}), \mathcal{O})$ and $\mathcal{F} \in \Sh(\mathcal{C})$ as in Lemma \ref{lemma-localize-ringed-topos}. If $\mathcal{F} = h_U^\#$ for some object $U$ of $\mathcal{C}$ then via the identification $\Sh(\mathcal{C}/U) = \Sh(\mathcal{C})/h_U^\#$ of Sites, Lemma \ref{sites-lemma-essential-image-j-shriek} we have \begin{enumerate} \item canonically $\mathcal{O}_U = \mathcal{O}_\mathcal{F}$, and \item with these identifications we have $(j_\mathcal{F}, j_\mathcal{F}^\sharp) = (j_U, j_U^\sharp)$. \end{enumerate}


sites-modules-lemma-relocalize-ringed-topos

Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos. If $s : \mathcal{G} \to \mathcal{F}$ is a morphism of sheaves on $\mathcal{C}$ then there exists a natural commutative diagram of morphisms of ringed topoi $$ \xymatrix{ (\Sh(\mathcal{C})/\mathcal{G}, \mathcal{O}_\mathcal{G}) \ar[rd]_{(j_\mathcal{G}, j_\mathcal{G}^\sharp)} \ar[rr]_{(j, j^\sharp)} & & (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) \ar[ld]^{(j_\mathcal{F}, j_\mathcal{F}^\sharp)} \\ & (\Sh(\mathcal{C}), \mathcal{O}) & } $$ where $(j, j^\sharp)$ is the localization morphism of the ringed topos $(\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F})$ at the object $\mathcal{G}/\mathcal{F}$.


sites-modules-lemma-relocalize-compare

With $(\Sh(\mathcal{C}), \mathcal{O})$, $s : \mathcal{G} \to \mathcal{F}$ as in Lemma \ref{lemma-relocalize-ringed-topos}. If there exist a morphism $f : V \to U$ of $\mathcal{C}$ such that $\mathcal{G} = h_V^\#$ and $\mathcal{F} = h_U^\#$ and $s$ is induced by $f$, then the diagrams of Lemma \ref{lemma-relocalize} and Lemma \ref{lemma-relocalize-ringed-topos} agree via the identifications $(j_\mathcal{F}, j_\mathcal{F}^\sharp) = (j_U, j_U^\sharp)$ and $(j_\mathcal{G}, j_\mathcal{G}^\sharp) = (j_V, j_V^\sharp)$ of Lemma \ref{lemma-localize-compare}.


sites-modules-lemma-localize-morphism-ringed-topoi

Let $$ f : (\Sh(\mathcal{C}), \mathcal{O}) \longrightarrow (\Sh(\mathcal{D}), \mathcal{O}') $$ be a morphism of ringed topoi. Let $\mathcal{G}$ be a sheaf on $\mathcal{D}$. Set $\mathcal{F} = f^{-1}\mathcal{G}$. Then there exists a commutative diagram of ringed topoi $$ \xymatrix{ (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) \ar[rr]_{(j_\mathcal{F}, j_\mathcal{F}^\sharp)} \ar[d]_{(f', (f')^\sharp)} & & (\Sh(\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp)} \\ (\Sh(\mathcal{D})/\mathcal{G}, \mathcal{O}'_\mathcal{G}) \ar[rr]^{(j_\mathcal{G}, j_\mathcal{G}^\sharp)} & & (\Sh(\mathcal{D}), \mathcal{O}') } $$ We have $f'_*j_\mathcal{F}^{-1} = j_\mathcal{G}^{-1}f_*$ and $f'_*j_\mathcal{F}^* = j_\mathcal{G}^*f_*$. Moreover, the morphism $f'$ is characterized by the rule $$ (f')^{-1}(\mathcal{H} \xrightarrow{\varphi} \mathcal{G}) = (f^{-1}\mathcal{H} \xrightarrow{f^{-1}\varphi} \mathcal{F}). $$


sites-modules-lemma-localize-morphism-compare

Let $$ f : (\Sh(\mathcal{C}), \mathcal{O}) \longrightarrow (\Sh(\mathcal{D}), \mathcal{O}') $$ be a morphism of ringed topoi. Let $\mathcal{G}$ be a sheaf on $\mathcal{D}$. Set $\mathcal{F} = f^{-1}\mathcal{G}$. If $f$ is given by a continuous functor $u : \mathcal{D} \to \mathcal{C}$ and $\mathcal{G} = h_V^\#$, then the commutative diagrams of Lemma \ref{lemma-localize-morphism-ringed-sites} and Lemma \ref{lemma-localize-morphism-ringed-topoi} agree via the identifications of Lemma \ref{lemma-localize-compare}.


sites-modules-lemma-relocalize-morphism-ringed-topoi

Let $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$ be a morphism of ringed topoi. Let $\mathcal{G}$ be a sheaf on $\mathcal{D}$, let $\mathcal{F}$ be a sheaf on $\mathcal{C}$, and let $s : \mathcal{F} \to f^{-1}\mathcal{G}$ a morphism of sheaves. There exists a commutative diagram of ringed topoi $$ \xymatrix{ (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) \ar[rr]_{(j_\mathcal{F}, j_\mathcal{F}^\sharp)} \ar[d]_{(f_c, f_c^\sharp)} & & (\Sh(\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp)} \\ (\Sh(\mathcal{D})/\mathcal{G}, \mathcal{O}'_\mathcal{G}) \ar[rr]^{(j_\mathcal{G}, j_\mathcal{G}^\sharp)} & & (\Sh(\mathcal{D}), \mathcal{O}'). } $$ The morphism $(f_s, f_s^\sharp)$ is equal to the composition of the morphism $$ (f', (f')^\sharp) : (\Sh(\mathcal{C})/f^{-1}\mathcal{G}, \mathcal{O}_{f^{-1}\mathcal{G}}) \longrightarrow (\Sh(\mathcal{D})/{\mathcal{G}}, \mathcal{O}'_\mathcal{G}) $$ of Lemma \ref{lemma-localize-morphism-ringed-topoi} and the morphism $$ (j, j^\sharp) : (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) \to (\Sh(\mathcal{C})/f^{-1}\mathcal{G}, \mathcal{O}_{f^{-1}\mathcal{G}}) $$ of Lemma \ref{lemma-relocalize-ringed-topos}. Given any morphisms $b : \mathcal{G}' \to \mathcal{G}$, $a : \mathcal{F}' \to \mathcal{F}$, and $s' : \mathcal{F}' \to f^{-1}\mathcal{G}'$ such that $$ \xymatrix{ \mathcal{F}' \ar[r]_-{s'} \ar[d]_a & f^{-1}\mathcal{G}' \ar[d]^{f^{-1}b} \\ \mathcal{F} \ar[r]^-s & f^{-1}\mathcal{G} } $$ commutes, then the following diagram of ringed topoi $$ \xymatrix{ (\Sh(\mathcal{C})/\mathcal{F}', \mathcal{O}_{\mathcal{F}'}) \ar[rr]_{(j_{\mathcal{F}'/\mathcal{F}}, j_{\mathcal{F}'/\mathcal{F}}^\sharp)} \ar[d]_{(f_{s'}, f_{s'}^\sharp)} & & (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) \ar[d]^{(f_s, f_s^\sharp)} \\ (\Sh(\mathcal{D})/\mathcal{G}', \mathcal{O}'_{\mathcal{G}'}) \ar[rr]^{(j_{\mathcal{G}'/\mathcal{G}}, j_{\mathcal{G}'/\mathcal{G}}^\sharp)} & & (\Sh(\mathcal{D})/\mathcal{G}, \mathcal{O}'_{\mathcal{G}'}) } $$ commutes.


sites-modules-lemma-relocalize-morphism-compare

Let $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$, $s : \mathcal{F} \to f^{-1}\mathcal{G}$ be as in Lemma \ref{lemma-relocalize-morphism-ringed-topoi}. If $f$ is given by a continuous functor $u : \mathcal{D} \to \mathcal{C}$ and $\mathcal{G} = h_V^\#$, $\mathcal{F} = h_U^\#$ and $s$ comes from a morphism $c : U \to u(V)$, then the commutative diagrams of Lemma \ref{lemma-relocalize-morphism-ringed-sites} and Lemma \ref{lemma-relocalize-morphism-ringed-topoi} agree via the identifications of Lemma \ref{lemma-localize-compare}.


sites-modules-lemma-local-final-object

Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $\mathcal{F}$ be an $\mathcal{O}$-module. Assume that the site $\mathcal{C}$ has a final object $X$. Then \begin{enumerate} \item The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is locally free, \item there exists a covering $\{X_i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_i}$ is a locally free $\mathcal{O}_{X_i}$-module, and \item there exists a covering $\{X_i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_i}$ is a free $\mathcal{O}_{X_i}$-module. \end{enumerate} \item The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is finite locally free, \item there exists a covering $\{X_i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_i}$ is a finite locally free $\mathcal{O}_{X_i}$-module, and \item there exists a covering $\{X_i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_i}$ is a finite free $\mathcal{O}_{X_i}$-module. \end{enumerate} \item The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is locally generated by sections, \item there exists a covering $\{X_i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_i}$ is an $\mathcal{O}_{X_i}$-module locally generated by sections, and \item there exists a covering $\{X_i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_i}$ is an $\mathcal{O}_{X_i}$-module globally generated by sections. \end{enumerate} \item Given $r \geq 0$, the following are equivalent \begin{enumerate} \item $\mathcal{F}$ is locally generated by $r$ sections, \item there exists a covering $\{X_i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_i}$ is an $\mathcal{O}_{X_i}$-module locally generated by $r$ sections, and \item there exists a covering $\{X_i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_i}$ is an $\mathcal{O}_{X_i}$-module globally generated by $r$ sections. \end{enumerate} \item The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is of finite type, \item there exists a covering $\{X_i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_i}$ is an $\mathcal{O}_{X_i}$-module of finite type, and \item there exists a covering $\{X_i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_i}$ is an $\mathcal{O}_{X_i}$-module globally generated by finitely many sections. \end{enumerate} \item The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is quasi-coherent, \item there exists a covering $\{X_i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_i}$ is a quasi-coherent $\mathcal{O}_{X_i}$-module, and \item there exists a covering $\{X_i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_i}$ is an $\mathcal{O}_{X_i}$-module which has a global presentation. \end{enumerate} \item The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is of finite presentation, \item there exists a covering $\{X_i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_i}$ is an $\mathcal{O}_{X_i}$-module of finite presentation, and \item there exists a covering $\{X_i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_i}$ is an $\mathcal{O}_{X_i}$-module has a finite global presentation. \end{enumerate} \item The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is coherent, and \item there exists a covering $\{X_i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_i}$ is a coherent $\mathcal{O}_{X_i}$-module. \end{enumerate} \end{enumerate}


sites-modules-lemma-local-pullback

Let $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal{D}$-module. \begin{enumerate} \item If $\mathcal{F}$ is locally free then $f^*\mathcal{F}$ is locally free. \item If $\mathcal{F}$ is finite locally free then $f^*\mathcal{F}$ is finite locally free. \item If $\mathcal{F}$ is locally generated by sections then $f^*\mathcal{F}$ is locally generated by sections. \item If $\mathcal{F}$ is locally generated by $r$ sections then $f^*\mathcal{F}$ is locally generated by $r$ sections. \item If $\mathcal{F}$ is of finite type then $f^*\mathcal{F}$ is of finite type. \item If $\mathcal{F}$ is quasi-coherent then $f^*\mathcal{F}$ is quasi-coherent. \item If $\mathcal{F}$ is of finite presentation then $f^*\mathcal{F}$ is of finite presentation. \end{enumerate}


sites-modules-lemma-kernel-surjection-finite-onto-finite-presentation

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\theta : \mathcal{G} \to \mathcal{F}$ be a surjective $\mathcal{O}$-module map with $\mathcal{F}$ of finite presentation and $\mathcal{G}$ of finite type. Then $\Ker(\theta)$ is of finite type.


sites-modules-lemma-i-star-equivalence

Let $i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$ be a morphism of ringed topoi. Assume $i$ is a closed immersion of topoi and $i^\sharp : \mathcal{O}' \to i_*\mathcal{O}$ is surjective. Denote $\mathcal{I} \subset \mathcal{O}'$ the kernel of $i^\sharp$. The functor $$ i_* : \textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}') $$ is exact, fully faithful, with essential image those $\mathcal{O}'$-modules $\mathcal{G}$ such that $\mathcal{I}\mathcal{G} = 0$.


sites-modules-lemma-tensor-product-pullback

Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}_\mathcal{D}$-modules. Then $f^*(\mathcal{F} \otimes_{\mathcal{O}_\mathcal{D}} \mathcal{G}) = f^*\mathcal{F} \otimes_{\mathcal{O}_\mathcal{C}} f^*\mathcal{G}$ functorially in $\mathcal{F}$, $\mathcal{G}$.


sites-modules-lemma-internal-hom

If $\mathcal{C}$ is a site, $\mathcal{O}$ is a sheaf of rings, $\mathcal{F}$ is a presheaf of $\mathcal{O}$-modules, and $\mathcal{G}$ is a sheaf of $\mathcal{O}$-modules, then $\SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G})$ is a sheaf of $\mathcal{O}$-modules.


sites-modules-lemma-internal-hom-restriction

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}, \mathcal{G}$ be sheaves of $\mathcal{O}$-modules. Then formation of $\SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G})$ commutes with restriction to $U$ for $U \in \Ob(\mathcal{C})$.


sites-modules-lemma-internal-hom-commute-limits

Internal hom and (co)limits. Let $\mathcal{C}$ be a category and let $\mathcal{O}$ be a presheaf of rings. \begin{enumerate} \item For any presheaf of $\mathcal{O}$-modules $\mathcal{F}$ the functor $$ \textit{PMod}(\mathcal{O}) \longrightarrow \textit{PMod}(\mathcal{O}) , \quad \mathcal{G} \longmapsto \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G}) $$ commutes with arbitrary limits. \item For any presheaf of $\mathcal{O}$-modules $\mathcal{G}$ the functor $$ \textit{PMod}(\mathcal{O}) \longrightarrow \textit{PMod}(\mathcal{O})^{opp} , \quad \mathcal{F} \longmapsto \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G}) $$ commutes with arbitrary colimits, in a formula $$ \SheafHom_\mathcal{O}(\colim_i \mathcal{F}_i, \mathcal{G}) = \lim_i \SheafHom_\mathcal{O}(\mathcal{F}_i, \mathcal{G}). $$ \end{enumerate} Suppose that $\mathcal{C}$ is a site, and $\mathcal{O}$ is a sheaf of rings. \begin{enumerate} \item[(3)] For any sheaf of $\mathcal{O}$-modules $\mathcal{F}$ the functor $$ \textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}) , \quad \mathcal{G} \longmapsto \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G}) $$ commutes with arbitrary limits. \item[(4)] For any sheaf of $\mathcal{O}$-modules $\mathcal{G}$ the functor $$ \textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O})^{opp} , \quad \mathcal{F} \longmapsto \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G}) $$ commutes with arbitrary colimits, in a formula $$ \SheafHom_\mathcal{O}(\colim_i \mathcal{F}_i, \mathcal{G}) = \lim_i \SheafHom_\mathcal{O}(\mathcal{F}_i, \mathcal{G}). $$ \end{enumerate}


sites-modules-lemma-internal-hom-adjoint-tensor

Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings. \begin{enumerate} \item Let $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$ be presheaves of $\mathcal{O}$-modules. There is a canonical isomorphism $$ \SheafHom_\mathcal{O} (\mathcal{F} \otimes_{p, \mathcal{O}} \mathcal{G}, \mathcal{H}) \longrightarrow \SheafHom_\mathcal{O} (\mathcal{F}, \SheafHom_\mathcal{O}(\mathcal{G}, \mathcal{H})) $$ which is functorial in all three entries (sheaf Hom in all three spots). In particular, $$ \Mor_{\textit{PMod}(\mathcal{O})}( \mathcal{F} \otimes_{p, \mathcal{O}} \mathcal{G}, \mathcal{H}) = \Mor_{\textit{PMod}(\mathcal{O})}( \mathcal{F}, \SheafHom_\mathcal{O}(\mathcal{G}, \mathcal{H})) $$ \item Suppose that $\mathcal{C}$ is a site, $\mathcal{O}$ is a sheaf of rings, and $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$ are sheaves of $\mathcal{O}$-modules. There is a canonical isomorphism $$ \SheafHom_\mathcal{O} (\mathcal{F} \otimes_\mathcal{O} \mathcal{G}, \mathcal{H}) \longrightarrow \SheafHom_\mathcal{O} (\mathcal{F}, \SheafHom_\mathcal{O}(\mathcal{G}, \mathcal{H})) $$ which is functorial in all three entries (sheaf Hom in all three spots). In particular, $$ \Mor_{\textit{Mod}(\mathcal{O})}( \mathcal{F} \otimes_\mathcal{O} \mathcal{G}, \mathcal{H}) = \Mor_{\textit{Mod}(\mathcal{O})}( \mathcal{F}, \SheafHom_\mathcal{O}(\mathcal{G}, \mathcal{H})) $$ \end{enumerate}


sites-modules-lemma-tensor-commute-colimits

Tensor product and colimits. Let $\mathcal{C}$ be a category and let $\mathcal{O}$ be a presheaf of rings. \begin{enumerate} \item For any presheaf of $\mathcal{O}$-modules $\mathcal{F}$ the functor $$ \textit{PMod}(\mathcal{O}) \longrightarrow \textit{PMod}(\mathcal{O}) , \quad \mathcal{G} \longmapsto \mathcal{F} \otimes_{p, \mathcal{O}} \mathcal{G} $$ commutes with arbitrary colimits. \item Suppose that $\mathcal{C}$ is a site, and $\mathcal{O}$ is a sheaf of rings. For any sheaf of $\mathcal{O}$-modules $\mathcal{F}$ the functor $$ \textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}) , \quad \mathcal{G} \longmapsto \mathcal{F} \otimes_\mathcal{O} \mathcal{G} $$ commutes with arbitrary colimits. \end{enumerate}


sites-modules-lemma-adjoint-hom-restrict

Let $\mathcal{C}$ be a category, resp.\ a site Let $\mathcal{O} \to \mathcal{O}'$ be a map of presheaves, resp.\ sheaves of rings. Then $$ \Hom_\mathcal{O}(\mathcal{G}, \mathcal{F}) = \Hom_{\mathcal{O}'}(\mathcal{G}, \SheafHom_\mathcal{O}(\mathcal{O}', \mathcal{F})) $$ for any $\mathcal{O}'$-module $\mathcal{G}$ and $\mathcal{O}$-module $\mathcal{F}$.


sites-modules-lemma-j-shriek-and-tensor

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \Ob(\mathcal{C})$. For $\mathcal{G}$ in $\textit{Mod}(\mathcal{O}_U)$ and $\mathcal{F}$ in $\textit{Mod}(\mathcal{O})$ we have $j_{U!}\mathcal{G} \otimes_\mathcal{O} \mathcal{F} = j_{U!}(\mathcal{G} \otimes_{\mathcal{O}_U} \mathcal{F}|_U)$.


sites-modules-lemma-flatness-presheaves

Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}$-modules. If each $\mathcal{F}(U)$ is a flat $\mathcal{O}(U)$-module, then $\mathcal{F}$ is flat.


sites-modules-lemma-flatness-sheafification

Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}$-modules. If $\mathcal{F}$ is a flat $\mathcal{O}$-module, then $\mathcal{F}^\#$ is a flat $\mathcal{O}^\#$-module.


sites-modules-lemma-colimits-flat

Colimits and tensor product. \begin{enumerate} \item A filtered colimit of flat presheaves of modules is flat. A direct sum of flat presheaves of modules is flat. \item A filtered colimit of flat sheaves of modules is flat. A direct sum of flat sheaves of modules is flat. \end{enumerate}


sites-modules-lemma-restriction-flat

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. If $\mathcal{F}$ is a flat $\mathcal{O}$-module, then $\mathcal{F}|_U$ is a flat $\mathcal{O}_U$-module.


sites-modules-lemma-j-shriek-flat

Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings. Let $U$ be an object of $\mathcal{C}$. Consider the functor $j_U : \mathcal{C}/U \to \mathcal{C}$. \begin{enumerate} \item The presheaf of $\mathcal{O}$-modules $j_{U!}\mathcal{O}_U$ (see Remark \ref{remark-localize-presheaves}) is flat. \item If $\mathcal{C}$ is a site, $\mathcal{O}$ is a sheaf of rings, $j_{U!}\mathcal{O}_U$ is a flat sheaf of $\mathcal{O}$-modules. \end{enumerate}


sites-modules-lemma-module-quotient-flat

Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings. \begin{enumerate} \item Any presheaf of $\mathcal{O}$-modules is a quotient of a direct sum $\bigoplus j_{U_i!}\mathcal{O}_{U_i}$. \item Any presheaf of $\mathcal{O}$-modules is a quotient of a flat presheaf of $\mathcal{O}$-modules. \item If $\mathcal{C}$ is a site, $\mathcal{O}$ is a sheaf of rings, then any sheaf of $\mathcal{O}$-modules is a quotient of a direct sum $\bigoplus j_{U_i!}\mathcal{O}_{U_i}$. \item If $\mathcal{C}$ is a site, $\mathcal{O}$ is a sheaf of rings, then any sheaf of $\mathcal{O}$-modules is a quotient of a flat sheaf of $\mathcal{O}$-modules. \end{enumerate}


sites-modules-lemma-flat-tor-zero

Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings. Let $$ 0 \to \mathcal{F}'' \to \mathcal{F}' \to \mathcal{F} \to 0 $$ be a short exact sequence of presheaves of $\mathcal{O}$-modules. Let $\mathcal{G}$ be a presheaf of $\mathcal{O}$-modules. \begin{enumerate} \item If $\mathcal{F}$ is a flat presheaf of modules, then the sequence $$ 0 \to \mathcal{F}'' \otimes_{p, \mathcal{O}} \mathcal{G} \to \mathcal{F}' \otimes_{p, \mathcal{O}} \mathcal{G} \to \mathcal{F} \otimes_{p, \mathcal{O}} \mathcal{G} \to 0 $$ is exact. \item If $\mathcal{C}$ is a site, $\mathcal{O}$, $\mathcal{F}$, $\mathcal{F}'$, $\mathcal{F}''$, and $\mathcal{G}$ are sheaves, and $\mathcal{F}$ is flat as a sheaf of modules, then the sequence $$ 0 \to \mathcal{F}'' \otimes_\mathcal{O} \mathcal{G} \to \mathcal{F}' \otimes_\mathcal{O} \mathcal{G} \to \mathcal{F} \otimes_\mathcal{O} \mathcal{G} \to 0 $$ is exact. \end{enumerate}


sites-modules-lemma-flat-ses

Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings. Let $$ 0 \to \mathcal{F}_2 \to \mathcal{F}_1 \to \mathcal{F}_0 \to 0 $$ be a short exact sequence of presheaves of $\mathcal{O}$-modules. \begin{enumerate} \item If $\mathcal{F}_2$ and $\mathcal{F}_0$ are flat so is $\mathcal{F}_1$. \item If $\mathcal{F}_1$ and $\mathcal{F}_0$ are flat so is $\mathcal{F}_2$. \end{enumerate} If $\mathcal{C}$ is a site and $\mathcal{O}$ is a sheaf of rings then the same result holds in $\textit{Mod}(\mathcal{O})$.


sites-modules-lemma-flat-resolution-of-flat

Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings. Let $$ \ldots \to \mathcal{F}_2 \to \mathcal{F}_1 \to \mathcal{F}_0 \to \mathcal{Q} \to 0 $$ be an exact complex of presheaves of $\mathcal{O}$-modules. If $\mathcal{Q}$ and all $\mathcal{F}_i$ are flat $\mathcal{O}$-modules, then for any presheaf $\mathcal{G}$ of $\mathcal{O}$-modules the complex $$ \ldots \to \mathcal{F}_2 \otimes_{p, \mathcal{O}} \mathcal{G} \to \mathcal{F}_1 \otimes_{p, \mathcal{O}} \mathcal{G} \to \mathcal{F}_0 \otimes_{p, \mathcal{O}} \mathcal{G} \to \mathcal{Q} \otimes_{p, \mathcal{O}} \mathcal{G} \to 0 $$ is exact also. If $\mathcal{C}$ is a site and $\mathcal{O}$ is a sheaf of rings then the same result holds $\textit{Mod}(\mathcal{O})$.


sites-modules-lemma-tensor-flats

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. If $\mathcal{G}$ and $\mathcal{F}$ are flat $\mathcal{O}$-modules, then $\mathcal{G} \otimes_\mathcal{O} \mathcal{F}$ is a flat $\mathcal{O}$-module.


sites-modules-lemma-flat-change-of-rings

Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a map of sheaves of rings on a site $\mathcal{C}$. If $\mathcal{G}$ is a flat $\mathcal{O}_1$-module, then $\mathcal{G} \otimes_{\mathcal{O}_1} \mathcal{O}_2$ is a flat $\mathcal{O}_2$-module.


sites-modules-lemma-flat-eq

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be an $\mathcal{O}$-module. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is a flat $\mathcal{O}$-module. \item Let $U$ be an object of $\mathcal{C}$ and let $$ \mathcal{O}_U \xrightarrow{(f_1, \ldots, f_n)} \mathcal{O}_U^{\oplus n} \xrightarrow{(s_1, \ldots, s_n)} \mathcal{F}|_U $$ be a complex of $\mathcal{O}_U$-modules. Then there exists a covering $\{U_i \to U\}$ and for each $i$ a factorization $$ \mathcal{O}_{U_i}^{\oplus n} \xrightarrow{B_i} \mathcal{O}_{U_i}^{\oplus l_i} \xrightarrow{(t_{i1}, \ldots, t_{il_i})} \mathcal{F}|_{U_i} $$ of $(s_1, \ldots, s_n)|_{U_i}$ such that $B_i \circ (f_1, \ldots, f_n)|_{U_i} = 0$. \item Let $U$ be an object of $\mathcal{C}$ and let $$ \mathcal{O}_U^{\oplus m} \xrightarrow{A} \mathcal{O}_U^{\oplus n} \xrightarrow{(s_1, \ldots, s_n)} \mathcal{F}|_U $$ be a complex of $\mathcal{O}_U$-modules. Then there exists a covering $\{U_i \to U\}$ and for each $i$ a factorization $$ \mathcal{O}_{U_i}^{\oplus n} \xrightarrow{B_i} \mathcal{O}_{U_i}^{\oplus l_i} \xrightarrow{(t_{i1}, \ldots, t_{il_i})} \mathcal{F}|_{U_i} $$ of $(s_1, \ldots, s_n)|_{U_i}$ such that $B_i \circ A|_{U_i} = 0$. \end{enumerate}


sites-modules-lemma-flat-over-thickening

Let $\mathcal{C}$ be a site. Let $\mathcal{O}' \to \mathcal{O}$ be a surjection of sheaves of rings whose kernel $\mathcal{I}$ is an ideal of square zero. Let $\mathcal{F}'$ be an $\mathcal{O}'$-module and set $\mathcal{F} = \mathcal{F}'/\mathcal{I}\mathcal{F}'$. The following are equivalent \begin{enumerate} \item $\mathcal{F}'$ is a flat $\mathcal{O}'$-module, and \item $\mathcal{F}$ is a flat $\mathcal{O}$-module and $\mathcal{I} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{F}'$ is injective. \end{enumerate}


sites-modules-lemma-left-dual-module

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be a $\mathcal{O}$-module. Let $\mathcal{G}, \eta, \epsilon$ be a left dual of $\mathcal{F}$ in the monoidal category of $\mathcal{O}$-modules, see Categories, Definition \ref{categories-definition-dual}. Then \begin{enumerate} \item for every object $U$ of $\mathcal{C}$ there exists a covering $\{U_i \to U\}$ such that $\mathcal{F}|_{U_i}$ is a direct summand of a finite free $\mathcal{O}|_{U_i}$-module, \item the map $e : \SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{O}) \to \mathcal{G}$ sending a local section $\lambda$ to $(\lambda \otimes 1)(\eta)$ is an isomorphism, \item we have $\epsilon(f, g) = e^{-1}(g)(f)$ for local sections $f$ and $g$ of $\mathcal{F}$ and $\mathcal{G}$. \end{enumerate}


sites-modules-lemma-flat-locally-finite-presentation

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be locally of finite presentation and flat. Then given an object $U$ of $\mathcal{C}$ there exists a covering $\{U_i \to U\}$ such that $\mathcal{F}|_{U_i}$ is a direct summand of a finite free $\mathcal{O}_{U_i}$-module.


sites-modules-lemma-covering-gives-surjection

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\{U_i \to U\}$ be a covering of $\mathcal{C}$. Then the sequence $$ \bigoplus j_{U_i \times_U U_j!}\mathcal{O}_{U_i \times_U U_j} \to \bigoplus j_{U_i!}\mathcal{O}_{U_i} \to j_!\mathcal{O}_U \to 0 $$ is exact.


sites-modules-lemma-silly-quasi-compact

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{U} = \{U_i \to U\}_{i \in I}$ be covering of $\mathcal{C}$. If $U$ is quasi-compact, then there exist a finite subset $I' \subset I$ such that the sequence $$ \bigoplus\nolimits_{i, i' \in I'} j_{U_i \times_U U_{i'}!}\mathcal{O}_{U_i \times_U U_{i'}} \to \bigoplus\nolimits_{i \in I'} j_{U_i!}\mathcal{O}_{U_i} \to j_!\mathcal{O}_U \to 0 $$ is exact.


sites-modules-lemma-sections-over-quasi-compact

Let $\mathcal{C}$ be a site. Let $W$ be a quasi-compact object of $\mathcal{C}$. \begin{enumerate} \item The functor $\Sh(\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}(W)$ commutes with coproducts. \item Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. The functor $\textit{Mod}(\mathcal{O}) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}(W)$ commutes with direct sums. \end{enumerate}


sites-modules-lemma-quasi-compact-hom-from

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be a quasi-compact object of $\mathcal{C}$. Then the functor $\Hom_\mathcal{O}(j_!\mathcal{O}_U, -)$ commutes with direct sums.


sites-modules-lemma-module-quotient-direct-sum

In Situation \ref{situation-quasi-compact-objects} assume (\ref{item-enough}) holds. \begin{enumerate} \item Every sheaf of sets is the target of a surjective map whose source is a coproduct $\coprod h_{U_i}^\#$ with $U_i$ in $\mathcal{B}$. \item If $\mathcal{O}$ is a sheaf of rings, then every $\mathcal{O}$-module is a quotient of a direct sum $\bigoplus\nolimits j_{U_i!}\mathcal{O}_{U_i}$ with $U_i$ in $\mathcal{B}$. \end{enumerate}


sites-modules-lemma-module-filtered-colimit-constructibles

In Situation \ref{situation-quasi-compact-objects} assume (\ref{item-enough}) and (\ref{item-enough-qc}) hold. \begin{enumerate} \item Every sheaf of sets is a filtered colimit of sheaves of the form \begin{equation} \label{equation-towards-constructible-sets} \text{Coequalizer}\left( \xymatrix{ \coprod\nolimits_{j = 1, \ldots, m} h_{V_j}^\# \ar@<1ex>[r] \ar@<-1ex>[r] & \coprod\nolimits_{i = 1, \ldots, n} h_{U_i}^\# } \right) \end{equation} with $U_i$ and $V_j$ in $\mathcal{B}$. \item If $\mathcal{O}$ is a sheaf of rings, then every $\mathcal{O}$-module is a filtered colimit of sheaves of the form \begin{equation} \label{equation-towards-constructible} \Coker\left( \bigoplus\nolimits_{j = 1, \ldots, m} j_{V_j!}\mathcal{O}_{V_j} \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} j_{U_i!}\mathcal{O}_{U_i} \right) \end{equation} with $U_i$ and $V_j$ in $\mathcal{B}$. \end{enumerate}


sites-modules-lemma-cokernel-map-towards-constructibles

In Situation \ref{situation-quasi-compact-objects} assume (\ref{item-enough}) and (\ref{item-enough-qc}) hold. Let $\mathcal{O}$ be a sheaf of rings. Then a cokernel of a map between modules as in (\ref{equation-towards-constructible}) is another module as in (\ref{equation-towards-constructible}).


sites-modules-lemma-change-presentation-towards-constructibles

In Situation \ref{situation-quasi-compact-objects} assume (\ref{item-enough}), (\ref{item-enough-qc}), and (\ref{item-enough-qc-qs}) hold. Let $\mathcal{O}$ be a sheaf of rings. Assume given a map $$ \bigoplus\nolimits_{j = 1, \ldots, m} j_{V_j!}\mathcal{O}_{V_j} \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} j_{U_i!}\mathcal{O}_{U_i} $$ with $U_i$ and $V_j$ in $\mathcal{B}$, and coverings $\{U_{ik} \to U_i\}_{k \in K_i}$ with $U_{ik} \in \mathcal{B}$. Then there exist finite subsets $K'_i \subset K_i$ and a finite set $L$ of $W_l \in \mathcal{B}$ and a commutative diagram $$ \xymatrix{ \bigoplus_{l \in L} j_{W_l!}\mathcal{O}_{W_l} \ar[d] \ar[r] & \bigoplus_{i = 1, \ldots, n} \bigoplus_{k \in K'_i} j_{U_{ik}!}\mathcal{O}_{U_{ik}} \ar[d] \\ \bigoplus_{j = 1, \ldots, m} j_{V_j!}\mathcal{O}_{V_j} \ar[r] & \bigoplus_{i = 1, \ldots, n} j_{U_i!}\mathcal{O}_{U_i} } $$ inducing an isomorphism on cokernels of the horizontal maps.


sites-modules-lemma-extension-towards-constructibles

In Situation \ref{situation-quasi-compact-objects} assume (\ref{item-enough}), (\ref{item-enough-qc}), and (\ref{item-enough-qc-qs}) hold. Let $\mathcal{O}$ be a sheaf of rings. Then an extension of modules as in (\ref{equation-towards-constructible}) is another module as in (\ref{equation-towards-constructible}).


sites-modules-lemma-flat-pullback-exact

Let $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{C}')$ be a morphism of ringed topoi. Then $$ f^{-1} : \textit{Ab}(\mathcal{C}') \longrightarrow \textit{Ab}(\mathcal{C}), \quad \mathcal{F} \longmapsto f^{-1}\mathcal{F} $$ is exact. If $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')$ is a flat morphism of ringed topoi then $$ f^* : \textit{Mod}(\mathcal{O}') \longrightarrow \textit{Mod}(\mathcal{O}), \quad \mathcal{F} \longmapsto f^*\mathcal{F} $$ is exact.


sites-modules-lemma-invertible

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{L}$ be an $\mathcal{O}$-module. The following are equivalent: \begin{enumerate} \item $\mathcal{L}$ is invertible, and \item there exists an $\mathcal{O}$-module $\mathcal{N}$ such that $\mathcal{L} \otimes_\mathcal{O} \mathcal{N} \cong \mathcal{O}$. \end{enumerate} In this case we have \begin{enumerate} \item[(a)] $\mathcal{L}$ is a flat $\mathcal{O}$-module of finite presentation, \item[(b)] for every object $U$ of $\mathcal{C}$ there exists a covering $U\{U_i \to U\}$ such that $\mathcal{L}|_{U_i}$ is a direct summand of a finite free module, and \item[(c)] the module $\mathcal{N}$ in (2) is isomorphic to $\SheafHom_\mathcal{O}(\mathcal{L}, \mathcal{O})$. \end{enumerate}


sites-modules-lemma-universal-module

Let $\mathcal{C}$ be a site. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. The functor $$ \textit{Mod}(\mathcal{O}_2) \longrightarrow \textit{Ab}, \quad \mathcal{F} \longmapsto \text{Der}_{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{F}) $$ is representable.


sites-modules-lemma-differentials-sheafify

Let $\mathcal{C}$ be a site. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of presheaves of rings. Then $\Omega_{\mathcal{O}_2^\#/\mathcal{O}_1^\#}$ is the sheaf associated to the presheaf $U \mapsto \Omega_{\mathcal{O}_2(U)/\mathcal{O}_1(U)}$.


sites-modules-lemma-pullback-differentials

Let $f : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$ be a morphism of topoi. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings on $\mathcal{C}$. Then there is a canonical identification $f^{-1}\Omega_{\mathcal{O}_2/\mathcal{O}_1} = \Omega_{f^{-1}\mathcal{O}_2/f^{-1}\mathcal{O}_1}$ compatible with universal derivations.


sites-modules-lemma-localize-differentials

Let $\mathcal{C}$ be a site. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. For any object $U$ of $\mathcal{C}$ there is a canonical isomorphism $$ \Omega_{\mathcal{O}_2/\mathcal{O}_1}|_U = \Omega_{(\mathcal{O}_2|_U)/(\mathcal{O}_1|_U)} $$ compatible with universal derivations.


sites-modules-lemma-functoriality-differentials

Let $\mathcal{C}$ be a site. Let $$ \xymatrix{ \mathcal{O}_2 \ar[r]_\varphi & \mathcal{O}_2' \\ \mathcal{O}_1 \ar[r] \ar[u] & \mathcal{O}'_1 \ar[u] } $$ be a commutative diagram of sheaves of rings on $\mathcal{C}$. The map $\mathcal{O}_2 \to \mathcal{O}'_2$ composed with the map $\text{d} : \mathcal{O}'_2 \to \Omega_{\mathcal{O}'_2/\mathcal{O}'_1}$ is a $\mathcal{O}_1$-derivation. Hence we obtain a canonical map of $\mathcal{O}_2$-modules $\Omega_{\mathcal{O}_2/\mathcal{O}_1} \to \Omega_{\mathcal{O}'_2/\mathcal{O}'_1}$. It is uniquely characterized by the property that $\text{d}(f)$ mapsto $\text{d}(\varphi(f))$ for any local section $f$ of $\mathcal{O}_2$. In this way $\Omega_{-/-}$ becomes a functor on the category of arrows of sheaves of rings.


sites-modules-lemma-differential-seq

In Lemma \ref{lemma-functoriality-differentials} suppose that $\mathcal{O}_2 \to \mathcal{O}'_2$ is surjective with kernel $\mathcal{I} \subset \mathcal{O}_2$ and assume that $\mathcal{O}_1 = \mathcal{O}'_1$. Then there is a canonical exact sequence of $\mathcal{O}'_2$-modules $$ \mathcal{I}/\mathcal{I}^2 \longrightarrow \Omega_{\mathcal{O}_2/\mathcal{O}_1} \otimes_{\mathcal{O}_2} \mathcal{O}'_2 \longrightarrow \Omega_{\mathcal{O}'_2/\mathcal{O}_1} \longrightarrow 0 $$ The leftmost map is characterized by the rule that a local section $f$ of $\mathcal{I}$ maps to $\text{d}f \otimes 1$.


sites-modules-lemma-double-structure-gives-derivation

Let $\mathcal{C}$ be a site. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. Consider a short exact sequence $$ 0 \to \mathcal{F} \to \mathcal{A} \to \mathcal{O}_2 \to 0 $$ Here $\mathcal{A}$ is a sheaf of $\mathcal{O}_1$-algebras, $\pi : \mathcal{A} \to \mathcal{O}_2$ is a surjection of sheaves of $\mathcal{O}_1$-algebras, and $\mathcal{F} = \Ker(\pi)$ is its kernel. Assume $\mathcal{F}$ an ideal sheaf with square zero in $\mathcal{A}$. So $\mathcal{F}$ has a natural structure of an $\mathcal{O}_2$-module. A section $s : \mathcal{O}_2 \to \mathcal{A}$ of $\pi$ is a $\mathcal{O}_1$-algebra map such that $\pi \circ s = \text{id}$. Given any section $s : \mathcal{O}_2 \to \mathcal{F}$ of $\pi$ and any $\varphi$-derivation $D : \mathcal{O}_1 \to \mathcal{F}$ the map $$ s + D : \mathcal{O}_1 \to \mathcal{A} $$ is a section of $\pi$ and every section $s'$ is of the form $s + D$ for a unique $\varphi$-derivation $D$.


sites-modules-lemma-functoriality-differentials-ringed-topoi

Let $X = (\Sh(\mathcal{C}_X), \mathcal{O}_X)$, $Y = (\Sh(\mathcal{C}_Y), \mathcal{O}_Y)$, $X' = (\Sh(\mathcal{C}_{X'}), \mathcal{O}_{X'})$, and $Y' = (\Sh(\mathcal{C}_{Y'}), \mathcal{O}_{Y'})$ be ringed topoi. Let $$ \xymatrix{ X' \ar[d] \ar[r]_f & X \ar[d] \\ Y' \ar[r] & Y } $$ be a commutative diagram of morphisms of ringed topoi. The map $f^\sharp : \mathcal{O}_X \to f_*\mathcal{O}_{X'}$ composed with the map $f_*\text{d}_{X'/Y'} : f_*\mathcal{O}_{X'} \to f_*\Omega_{X'/Y'}$ is a $Y$-derivation. Hence we obtain a canonical map of $\mathcal{O}_X$-modules $\Omega_{X/Y} \to f_*\Omega_{X'/Y'}$, and by adjointness of $f_*$ and $f^*$ a canonical $\mathcal{O}_{X'}$-module homomorphism $$ c_f : f^*\Omega_{X/Y} \longrightarrow \Omega_{X'/Y'}. $$ It is uniquely characterized by the property that $f^*\text{d}_{X/Y}(t)$ mapsto $\text{d}_{X'/Y'}(f^* t)$ for any local section $t$ of $\mathcal{O}_X$.


sites-modules-lemma-composition-differential-operators

Let $\mathcal{C}$ be a site. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a map of sheaves of rings. Let $\mathcal{E}, \mathcal{F}, \mathcal{G}$ be sheaves of $\mathcal{O}_2$-modules. If $D : \mathcal{E} \to \mathcal{F}$ and $D' : \mathcal{F} \to \mathcal{G}$ are differential operators of order $k$ and $k'$, then $D' \circ D$ is a differential operator of order $k + k'$.


sites-modules-lemma-module-principal-parts

Let $\mathcal{C}$ be a site. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a map of sheaves of rings. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_2$-modules. Let $k \geq 0$. There exists a sheaf of $\mathcal{O}_2$-modules $\mathcal{P}^k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F})$ and a canonical isomorphism $$ \text{Diff}^k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}, \mathcal{G}) = \Hom_{\mathcal{O}_2}( \mathcal{P}^k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}), \mathcal{G}) $$ functorial in the $\mathcal{O}_2$-module $\mathcal{G}$.


sites-modules-lemma-differential-operators-sheafify

Let $\mathcal{C}$ be a site. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of presheaves of rings. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_2$-modules. Then $\mathcal{P}^k_{\mathcal{O}_2^\#/\mathcal{O}_1^\#}(\mathcal{F}^\#)$ is the sheaf associated to the presheaf $U \mapsto P^k_{\mathcal{O}_2(U)/\mathcal{O}_1(U)}(\mathcal{F}(U))$.


sites-modules-lemma-sequence-of-principal-parts

Let $\mathcal{C}$ be a site. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_2$-modules. There is a canonical short exact sequence $$ 0 \to \Omega_{\mathcal{O}_2/\mathcal{O}_1} \otimes_{\mathcal{O}_2} \mathcal{F} \to \mathcal{P}^1_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}) \to \mathcal{F} \to 0 $$ functorial in $\mathcal{F}$ called the {\it sequence of principal parts}.


sites-modules-lemma-NL-up-to-qis

In the situation above there is a canonical isomorphism $\NL(\alpha) = \NL_{\mathcal{B}/\mathcal{A}}$ in $D(\mathcal{B})$.


sites-modules-lemma-pullback-NL

Let $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be morphism of topoi. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $\mathcal{D}$. Then $f^{-1}\NL_{\mathcal{B}/\mathcal{A}} = \NL_{f^{-1}\mathcal{B}/f^{-1}\mathcal{A}}$.


sites-modules-lemma-stalk-exact

Let $\mathcal{C}$ be a site. Let $p$ be a point of $\mathcal{C}$. \begin{enumerate} \item We have $(\mathcal{F}^\#)_p = \mathcal{F}_p$ for any presheaf of sets on $\mathcal{C}$. \item The stalk functor $\Sh(\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}_p$ is exact (see Categories, Definition \ref{categories-definition-exact}) and commutes with arbitrary colimits. \item The stalk functor $\textit{PSh}(\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}_p$ is exact (see Categories, Definition \ref{categories-definition-exact}) and commutes with arbitrary colimits. \end{enumerate}


sites-modules-lemma-stalk-exact-abelian

Let $\mathcal{C}$ be a site. Let $p$ be a point of $\mathcal{C}$. \begin{enumerate} \item The functor $\textit{Ab}(\mathcal{C}) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}_p$ is exact. \item The stalk functor $\textit{PAb}(\mathcal{C}) \to \textit{Ab}$, $\mathcal{F}  \mapsto  \mathcal{F}_p$ is exact. \item For $\mathcal{F} \in \Ob(\textit{PAb}(\mathcal{C}))$ we have $\mathcal{F}_p = \mathcal{F}^\#_p$. \end{enumerate}


sites-modules-lemma-stalk-exact-modules

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $p$ be a point of $\mathcal{C}$. \begin{enumerate} \item The functor $\textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_p)$, $\mathcal{F} \mapsto \mathcal{F}_p$ is exact. \item The stalk functor $\textit{PMod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_p)$, $\mathcal{F} \mapsto \mathcal{F}_p$ is exact. \item For $\mathcal{F} \in \Ob(\textit{PMod}(\mathcal{O}))$ we have $\mathcal{F}_p = \mathcal{F}^\#_p$. \end{enumerate}


sites-modules-lemma-pullback-stalk

Let $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi or ringed sites. Let $p$ be a point of $\mathcal{C}$ or $\Sh(\mathcal{C})$ and set $q = f \circ p$. Then $$ (f^*\mathcal{F})_p = \mathcal{F}_q \otimes_{\mathcal{O}_{\mathcal{D}, q}} \mathcal{O}_{\mathcal{C}, p} $$ for any $\mathcal{O}_\mathcal{D}$-module $\mathcal{F}$.


sites-modules-lemma-skyscraper-exact

Let $\mathcal{C}$ be a site. Let $p$ be a point of $\mathcal{C}$ or of its associated topos. \begin{enumerate} \item The functor $p_* : \textit{Ab} \to \textit{Ab}(\mathcal{C})$, $A \mapsto p_*A$ is exact. \item There is a functorial direct sum decomposition $$ p^{-1}p_*A = A \oplus I(A) $$ for $A \in \Ob(\textit{Ab})$. \end{enumerate}


sites-modules-lemma-stalk-j-shriek

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $p$ be a point of $\mathcal{C}$. Let $U$ be an object of $\mathcal{C}$. For $\mathcal{G}$ in $\textit{Mod}(\mathcal{O}_U)$ we have $$ (j_{U!}\mathcal{G})_p = \bigoplus\nolimits_q \mathcal{G}_q $$ where the coproduct is over the points $q$ of $\mathcal{C}/U$ lying over $p$, see Sites, Lemma \ref{sites-lemma-points-above-point}.


sites-modules-lemma-pullback-flat

\begin{reference} \cite[Expos\'e V, Corollary 1.7.1]{SGA4} \end{reference} Let $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi or ringed sites. Then $f^*\mathcal{F}$ is a flat $\mathcal{O}_\mathcal{C}$-module whenever $\mathcal{F}$ is a flat $\mathcal{O}_\mathcal{D}$-module.


sites-modules-lemma-stalk-flat

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $p$ be a point of $\mathcal{C}$. If $\mathcal{F}$ is a flat $\mathcal{O}$-module, then $\mathcal{F}_p$ is a flat $\mathcal{O}_p$-module.


sites-modules-lemma-check-flat-stalks

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. Let $\{p_i\}_{i \in I}$ be a conservative family of points of $\mathcal{C}$. Then $\mathcal{F}$ is flat if and only if $\mathcal{F}_{p_i}$ is a flat $\mathcal{O}_{p_i}$-module for all $i \in I$.


sites-modules-lemma-pullback-ses

Let $f : (\Sh(\mathcal{C}'), \mathcal{O}') \to (\Sh(\mathcal{C}'), \mathcal{O})$ be a morphism of ringed topoi. Let $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ be a short exact sequence of $\mathcal{O}$-modules with $\mathcal{H}$ a flat $\mathcal{O}$-module. Then the sequence $0 \to f^*\mathcal{F} \to f^*\mathcal{G} \to f^*\mathcal{H} \to 0$ is exact as well.


sites-modules-lemma-locally-ringed

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. The following are equivalent \begin{enumerate} \item For every object $U$ of $\mathcal{C}$ and $f \in \mathcal{O}(U)$ there exists a covering $\{U_j \to U\}$ such that for each $j$ either $f|_{U_j}$ is invertible or $(1 - f)|_{U_j}$ is invertible. \item For $U \in \Ob(\mathcal{C})$, $n \geq 1$, and $f_1, \ldots, f_n \in \mathcal{O}(U)$ which generate the unit ideal in $\mathcal{O}(U)$ there exists a covering $\{U_j \to U\}$ such that for each $j$ there exists an $i$ such that $f_i|_{U_j}$ is invertible. \item The map of sheaves of sets $$ (\mathcal{O} \times \mathcal{O}) \amalg (\mathcal{O} \times \mathcal{O}) \longrightarrow \mathcal{O} \times \mathcal{O} $$ which maps $(f, a)$ in the first component to $(f, af)$ and $(f, b)$ in the second component to $(f, b(1 - f))$ is surjective. \end{enumerate}


sites-modules-lemma-locally-ringed-stalk

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Consider the following conditions \begin{enumerate} \item For every object $U$ of $\mathcal{C}$ and $f \in \mathcal{O}(U)$ there exists a covering $\{U_j \to U\}$ such that for each $j$ either $f|_{U_j}$ is invertible or $(1 - f)|_{U_j}$ is invertible. \item For every point $p$ of $\mathcal{C}$ the stalk $\mathcal{O}_p$ is either the zero ring or a local ring. \end{enumerate} We always have (1) $\Rightarrow$ (2). If $\mathcal{C}$ has enough points then (1) and (2) are equivalent.


sites-modules-lemma-ringed-stalk-not-zero

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Consider the statements \begin{enumerate} \item (\ref{equation-one-is-never-zero}) is an isomorphism, and \item for every point $p$ of $\mathcal{C}$ the stalk $\mathcal{O}_p$ is not the zero ring. \end{enumerate} We always have (1) $\Rightarrow$ (2) and if $\mathcal{C}$ has enough points then (1) $\Leftrightarrow$ (2).


sites-modules-lemma-locally-ringed-intrinsic

Being a locally ringed site is an intrinsic property. More precisely, \begin{enumerate} \item if $f : \Sh(\mathcal{C}') \to \Sh(\mathcal{C})$ is a morphism of topoi and $(\mathcal{C}, \mathcal{O})$ is a locally ringed site, then $(\mathcal{C}', f^{-1}\mathcal{O})$ is a locally ringed site, and \item if $(f, f^\sharp) : (\Sh(\mathcal{C}'), \mathcal{O}') \to (\Sh(\mathcal{C}), \mathcal{O})$ is an equivalence of ringed topoi, then $(\mathcal{C}, \mathcal{O})$ is locally ringed if and only if $(\mathcal{C}', \mathcal{O}')$ is locally ringed. \end{enumerate}


sites-modules-lemma-invertible-is-locally-free-rank-1

Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos. Any locally free $\mathcal{O}$-module of rank $1$ is invertible. If $(\mathcal{C}, \mathcal{O})$ is locally ringed, then the converse holds as well (but in general this is not the case).


sites-modules-lemma-locally-ringed-morphism

Let $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. Consider the following conditions \begin{enumerate} \item The diagram of sheaves $$ \xymatrix{ f^{-1}(\mathcal{O}^*_\mathcal{D}) \ar[r]_-{f^\sharp} \ar[d] & \mathcal{O}^*_\mathcal{C} \ar[d] \\ f^{-1}(\mathcal{O}_\mathcal{D}) \ar[r]^-{f^\sharp} & \mathcal{O}_\mathcal{C} } $$ is cartesian. \item For any point $p$ of $\mathcal{C}$, setting $q = f \circ p$, the diagram $$ \xymatrix{ \mathcal{O}^*_{\mathcal{D}, q} \ar[r] \ar[d] & \mathcal{O}^*_{\mathcal{C}, p} \ar[d] \\ \mathcal{O}_{\mathcal{D}, q} \ar[r] & \mathcal{O}_{\mathcal{C}, p} } $$ of sets is cartesian. \end{enumerate} We always have (1) $\Rightarrow$ (2). If $\mathcal{C}$ has enough points then (1) and (2) are equivalent. If $(\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})$ and $(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ are locally ringed topoi then (2) is equivalent to \begin{enumerate} \item[(3)] For any point $p$ of $\mathcal{C}$, setting $q = f \circ p$, the ring map $\mathcal{O}_{\mathcal{D}, q} \to \mathcal{O}_{\mathcal{C}, p}$ is a local ring map. \end{enumerate} In fact, properties (2), or (3) for a conservative family of points implies (1).


sites-modules-lemma-composition-morphisms-locally-ringed-topoi

Let $(f, f^\sharp) : (\Sh(\mathcal{C}_1), \mathcal{O}_1) \to (\Sh(\mathcal{C}_2), \mathcal{O}_2)$ and $(g, g^\sharp) : (\Sh(\mathcal{C}_2), \mathcal{O}_2) \to (\Sh(\mathcal{C}_3), \mathcal{O}_3)$ be morphisms of locally ringed topoi. Then the composition $(g, g^\sharp) \circ (f, f^\sharp)$ (see Definition \ref{definition-ringed-topos}) is also a morphism of locally ringed topoi.


sites-modules-lemma-locally-ringed-intrinsic-morphism

If $f : \Sh(\mathcal{C}') \to \Sh(\mathcal{C})$ is a morphism of topoi. If $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}$, then $$ f^{-1}(\mathcal{O}^*) = (f^{-1}\mathcal{O})^*. $$ In particular, if $\mathcal{O}$ turns $\mathcal{C}$ into a locally ringed site, then setting $f^\sharp = \text{id}$ the morphism of ringed topoi $$ (f, f^\sharp) : (\Sh(\mathcal{C}'), f^{-1}\mathcal{O}) \to (\Sh(\mathcal{C}, \mathcal{O}) $$ is a morphism of locally ringed topoi.


sites-modules-lemma-localize-morphism-locally-ringed-topoi

Localization of locally ringed sites and topoi. \begin{enumerate} \item Let $(\mathcal{C}, \mathcal{O})$ be a locally ringed site. Let $U$ be an object of $\mathcal{C}$. Then the localization $(\mathcal{C}/U, \mathcal{O}_U)$ is a locally ringed site, and the localization morphism $$ (j_U, j_U^\sharp) : (\Sh(\mathcal{C}/U), \mathcal{O}_U) \to (\Sh(\mathcal{C}), \mathcal{O}) $$ is a morphism of locally ringed topoi. \item Let $(\mathcal{C}, \mathcal{O})$ be a locally ringed site. Let $f : V \to U$ be a morphism of $\mathcal{C}$. Then the morphism $$ (j, j^\sharp) : (\Sh(\mathcal{C}/V), \mathcal{O}_V) \to (\Sh(\mathcal{C}/U), \mathcal{O}_U) $$ of Lemma \ref{lemma-relocalize} is a morphism of locally ringed topoi. \item Let $(f, f^\sharp) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{D}, \mathcal{O}')$ be a morphism of locally ringed sites where $f$ is given by the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V$ be an object of $\mathcal{D}$ and let $U = u(V)$. Then the morphism $$ (f', (f')^\sharp) : (\Sh(\mathcal{C}/U), \mathcal{O}_U) \to (\Sh(\mathcal{D}/V), \mathcal{O}'_V) $$ of Lemma \ref{lemma-localize-morphism-ringed-sites} is a morphism of locally ringed sites. \item Let $(f, f^\sharp) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{D}, \mathcal{O}')$ be a morphism of locally ringed sites where $f$ is given by the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V \in \Ob(\mathcal{D})$, $U \in \Ob(\mathcal{C})$, and $c : U \to u(V)$. Then the morphism $$ (f_c, (f_c)^\sharp) : (\Sh(\mathcal{C}/U), \mathcal{O}_U) \to (\Sh(\mathcal{D}/V), \mathcal{O}'_V) $$ of Lemma \ref{lemma-relocalize-morphism-ringed-sites} is a morphism of locally ringed topoi. \item Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a locally ringed topos. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Then the localization $(\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F})$ is a locally ringed topos and the localization morphism $$ (j_\mathcal{F}, j_\mathcal{F}^\sharp) : (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) \to (\Sh(\mathcal{C}), \mathcal{O}) $$ is a morphism of locally ringed topoi. \item  Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a locally ringed topos. Let $s : \mathcal{G} \to \mathcal{F}$ be a map of sheaves on $\mathcal{C}$. Then the morphism $$ (j, j^\sharp) : (\Sh(\mathcal{C})/\mathcal{G}, \mathcal{O}_\mathcal{G}) \longrightarrow (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) $$ of Lemma \ref{lemma-relocalize-ringed-topos} is a morphism of locally ringed topoi. \item Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \longrightarrow (\Sh(\mathcal{D}), \mathcal{O}')$ be a morphism of locally ringed topoi. Let $\mathcal{G}$ be a sheaf on $\mathcal{D}$. Set $\mathcal{F} = f^{-1}\mathcal{G}$. Then the morphism $$ (f', (f')^\sharp) : (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) \longrightarrow (\Sh(\mathcal{D})/\mathcal{G}, \mathcal{O}'_\mathcal{G}) $$ of Lemma \ref{lemma-localize-morphism-ringed-topoi} is a morphism of locally ringed topoi. \item  Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \longrightarrow (\Sh(\mathcal{D}), \mathcal{O}')$ be a morphism of locally ringed topoi. Let $\mathcal{G}$ be a sheaf on $\mathcal{D}$, let $\mathcal{F}$ be a sheaf on $\mathcal{C}$, and let $s : \mathcal{F} \to f^{-1}\mathcal{G}$ be a morphism of sheaves. Then the morphism $$ (f_s, (f_s)^\sharp) : (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) \longrightarrow (\Sh(\mathcal{D})/\mathcal{G}, \mathcal{O}'_\mathcal{G}) $$ of Lemma \ref{lemma-relocalize-morphism-ringed-topoi} is a morphism of locally ringed topoi. \end{enumerate}


sites-modules-lemma-lower-shriek-modules

Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous and cocontinuous functor between sites. Denote $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ the associated morphism of topoi. Let $\mathcal{O}_\mathcal{D}$ be a sheaf of rings on $\mathcal{D}$. Set $\mathcal{O}_\mathcal{C} = g^{-1}\mathcal{O}_\mathcal{D}$. Hence $g$ becomes a morphism of ringed topoi with $g^* = g^{-1}$. In this case there exists a functor $$ g_! : \textit{Mod}(\mathcal{O}_\mathcal{C}) \longrightarrow \textit{Mod}(\mathcal{O}_\mathcal{D}) $$ which is left adjoint to $g^*$.


sites-modules-lemma-special-square-cocontinuous

Assume given a commutative diagram $$ \xymatrix{ (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_{(g', (g')^\sharp)} \ar[d]_{(f', (f')^\sharp)} & (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^{(f, f^\sharp)} \\ (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^{(g, g^\sharp)} & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) } $$ of ringed topoi. Assume \begin{enumerate} \item $f$, $f'$, $g$, and $g'$ correspond to cocontinuous functors $u$, $u'$, $v$, and $v'$ as in Sites, Lemma \ref{sites-lemma-cocontinuous-morphism-topoi}, \item $v \circ u' = u \circ v'$, \item $v$ and $v'$ are continuous as well as cocontinuous, \item for any object $V'$ of $\mathcal{D}'$ the functor ${}^{u'}_{V'}\mathcal{I} \to {}^{\ \ \ u}_{v(V')}\mathcal{I}$ given by $v$ is cofinal, and \item $g^{-1}\mathcal{O}_{\mathcal{D}} = \mathcal{O}_{\mathcal{D}'}$ and $(g')^{-1}\mathcal{O}_{\mathcal{C}} = \mathcal{O}_{\mathcal{C}'}$. \end{enumerate} Then we have $f'_* \circ (g')^* = g^* \circ f_*$ and $g'_! \circ (f')^{-1} = f^{-1} \circ g_!$ on modules.


sites-modules-lemma-special-square-continuous

Consider a commutative diagram $$ \xymatrix{ (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_{(g', (g')^\sharp)} \ar[d]_{(f', (f')^\sharp)} & (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^{(f, f^\sharp)} \\ (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^{(g, g^\sharp)} & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) } $$ of ringed topoi and suppose we have functors $$ \xymatrix{ \mathcal{C}' \ar[r]_{v'} & \mathcal{C} \\ \mathcal{D}' \ar[r]^v \ar[u]^{u'} & \mathcal{D} \ar[u]_u } $$ such that (with notation as in Sites, Sections \ref{sites-section-morphism-sites} and \ref{sites-section-cocontinuous-morphism-topoi}) we have \begin{enumerate} \item $u$ and $u'$ are continuous and give rise to the morphisms $f$ and $f'$, \item $v$ and $v'$ are cocontinuous giving rise to the morphisms $g$ and $g'$, \item $u \circ v = v' \circ u'$, \item $v$ and $v'$ are continuous as well as cocontinuous, and \item $g^{-1}\mathcal{O}_{\mathcal{D}} = \mathcal{O}_{\mathcal{D}'}$ and $(g')^{-1}\mathcal{O}_{\mathcal{C}} = \mathcal{O}_{\mathcal{C}'}$. \end{enumerate} Then $f'_* \circ (g')^* = g^* \circ f_*$ and $g'_! \circ (f')^{-1} = f^{-1} \circ g_!$ on modules.


sites-modules-lemma-constant-exact

Let $\mathcal{C}$ be a site. If $0 \to A \to B \to C \to 0$ is a short exact sequence of abelian groups, then $0 \to \underline{A} \to \underline{B} \to \underline{C} \to 0$ is an exact sequence of abelian sheaves and in fact it is even exact as a sequence of abelian presheaves.


sites-modules-lemma-tensor-with-finitely-presented

Let $\mathcal{C}$ be a site. Let $\Lambda$ be a ring and let $M$ and $Q$ be $\Lambda$-modules. If $Q$ is a finitely presented $\Lambda$-module, then we have $\underline{M \otimes_\Lambda Q}(U) = \underline{M}(U) \otimes_\Lambda Q$ for all $U \in \Ob(\mathcal{C})$.


sites-modules-lemma-flat-sections

Let $\mathcal{C}$ be a site. Let $\Lambda$ be a coherent ring. Let $M$ be a flat $\Lambda$-module. For $U \in \Ob(\mathcal{C})$ the module $\underline{M}(U)$ is a flat $\Lambda$-module.


sites-modules-lemma-completion-flat

Let $\mathcal{C}$ be a site. Let $\Lambda$ be a Noetherian ring. Let $I \subset \Lambda$ be an ideal. The sheaf $\underline{\Lambda}^\wedge = \lim \underline{\Lambda/I^n}$ is a flat $\underline{\Lambda}$-algebra. Moreover we have canonical identifications $$ \underline{\Lambda}/I\underline{\Lambda} = \underline{\Lambda}/\underline{I} = \underline{\Lambda}^\wedge/I\underline{\Lambda}^\wedge = \underline{\Lambda}^\wedge/\underline{I} \cdot \underline{\Lambda}^\wedge = \underline{\Lambda}^\wedge/\underline{I}^\wedge = \underline{\Lambda/I} $$ where $\underline{I}^\wedge = \lim \underline{I/I^n}$.


sites-modules-lemma-locally-constant-finite-type

Let $\mathcal{C}$ be a site. Let $\Lambda$ be a ring and let $M$ be a $\Lambda$-module. Assume $\Sh(\mathcal{C})$ is not the empty topos. Then \begin{enumerate} \item $\underline{M}$ is a finite type sheaf of $\underline{\Lambda}$-modules if and only if $M$ is a finite $\Lambda$-module, and \item $\underline{M}$ is a finitely presented sheaf of $\underline{\Lambda}$-modules if and only if $M$ is a  finitely presented $\Lambda$-module. \end{enumerate}


sites-modules-lemma-pullback-locally-constant

Let $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be a morphism of topoi. If $\mathcal{G}$ is a locally constant sheaf of sets, groups, abelian groups, rings, modules over a fixed ring $\Lambda$, etc on $\mathcal{D}$, the same is true for $f^{-1}\mathcal{G}$ on $\mathcal{C}$.


sites-modules-lemma-morphism-locally-constant

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


sites-modules-lemma-locally-constant

Let $\mathcal{C}$ be a site. Let $\Lambda$ be a ring. Let $M$, $N$ be $\Lambda$-modules. Let $\mathcal{F}, \mathcal{G}$ be a locally constant sheaves of $\Lambda$-modules. \begin{enumerate} \item If $M$ is of finite presentation, then $$ \underline{\Hom_\Lambda(M, N)} = \SheafHom_{\underline{\Lambda}}(\underline{M}, \underline{N}) $$ \item If $M$ and $N$ are both of finite presentation, then $$ \underline{\text{Isom}_\Lambda(M, N)} = \mathit{Isom}_{\underline{\Lambda}}(\underline{M}, \underline{N}) $$ \item If $\mathcal{F}$ is of finite presentation, then $\SheafHom_{\underline{\Lambda}}(\mathcal{F}, \mathcal{G})$ is a locally constant sheaf of $\Lambda$-modules. \item If $\mathcal{F}$ and $\mathcal{G}$ are both of finite presentation, then $\mathit{Isom}_{\underline{\Lambda}}(\mathcal{F}, \mathcal{G})$ is a locally constant sheaf of sets. \end{enumerate}


sites-modules-lemma-kernel-finite-locally-constant

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


sites-modules-lemma-tensor-product-locally-constant

Let $\mathcal{C}$ be a site. Let $\Lambda$ be a ring. The tensor product of two locally constant sheaves of $\Lambda$-modules on $\mathcal{C}$ is a locally constant sheaf of $\Lambda$-modules.


sites-modules-lemma-simple-invert

In the situation above the map to the sheafification $$ \mathcal{O} \longrightarrow (\mathcal{S}^{-1}\mathcal{O})^\# $$ is a homomorphism of sheaves of rings with the following universal property: for any homomorphism of sheaves of rings $\mathcal{O} \to \mathcal{A}$ such that each local section of $\mathcal{S}$ maps to an invertible section of $\mathcal{A}$ there exists a unique factorization $(\mathcal{S}^{-1}\mathcal{O})^\# \to \mathcal{A}$.


sites-modules-lemma-simple-invert-module

In the situation above the map to the sheafification $$ \mathcal{F} \longrightarrow (\mathcal{S}^{-1}\mathcal{F})^\# $$ has the following universal property: for any homomorphism of $\mathcal{O}$-modules $\mathcal{F} \to \mathcal{G}$ such that each local section of $\mathcal{S}$ acts invertibly on $\mathcal{G}$ there exists a unique factorization $(\mathcal{S}^{-1}\mathcal{F})^\# \to \mathcal{G}$. Moreover we have $$ (\mathcal{S}^{-1}\mathcal{F})^\# = (\mathcal{S}^{-1}\mathcal{O})^\# \otimes_\mathcal{O} \mathcal{F} $$ as sheaves of $(\mathcal{S}^{-1}\mathcal{O})^\#$-modules.


derham-lemma-base-change-de-rham

Let $$ \xymatrix{ X' \ar[r]_f \ar[d] & X \ar[d] \\ S' \ar[r] & S } $$ be a cartesian diagram of schemes. Then the maps discussed above induce isomorphisms $f^*\Omega^p_{X/S} \to \Omega^p_{X'/S'}$.


derham-lemma-etale

Consider a commutative diagram of schemes $$ \xymatrix{ X' \ar[r]_f \ar[d] & X \ar[d] \\ S' \ar[r] & S } $$ If $X' \to X$ and $S' \to S$ are \'etale, then the maps discussed above induce isomorphisms $f^*\Omega^p_{X/S} \to \Omega^p_{X'/S'}$.


derham-lemma-de-rham-affine

Let $X \to S$ be a morphism of affine schemes given by the ring map $R \to A$. Then $R\Gamma(X, \Omega^\bullet_{X/S}) = \Omega^\bullet_{A/R}$ in $D(R)$ and $H^i_{dR}(X/S) = H^i(\Omega^\bullet_{A/R})$.


derham-lemma-quasi-coherence-relative

Let $p : X \to S$ be a morphism of schemes. If $p$ is quasi-compact and quasi-separated, then $Rp_*\Omega^\bullet_{X/S}$ is an object of $D_\QCoh(\mathcal{O}_S)$.


derham-lemma-coherence-relative

Let $p : X \to S$ be a proper morphism of schemes with $S$ locally Noetherian. Then $Rp_*\Omega^\bullet_{X/S}$ is an object of $D_{\textit{Coh}}(\mathcal{O}_S)$.


derham-lemma-proper-smooth-de-Rham

Let $f : X \to S$ be a proper smooth morphism of schemes. Then $Rf_*\Omega^p_{X/S}$, $p \geq 0$ and $Rf_*\Omega^\bullet_{X/S}$ are perfect objects of $D(\mathcal{O}_S)$ whose formation commutes with arbitrary change of base.


derham-lemma-cup-product-graded-commutative

Let $p : X \to S$ be a morphism of schemes. The cup product on $H^*_{dR}(X/S)$ is associative and graded commutative.


derham-lemma-de-rham-complex-product

In the situation above there is a canonical isomorphism $$ \text{Tot}(\Omega^\bullet_{X/S} \boxtimes \Omega^\bullet_{Y/S}) \longrightarrow \Omega^\bullet_{X \times_S Y/S} $$ of complexes of $f^{-1}\mathcal{O}_S$-modules.


derham-lemma-kunneth-de-rham

Assume $X$ and $Y$ are smooth, quasi-compact, with affine diagonal over $S = \Spec(A)$. Then the map $$ R\Gamma(X, \Omega^\bullet_{X/S}) \otimes_A^\mathbf{L} R\Gamma(Y, \Omega^\bullet_{Y/S}) \longrightarrow R\Gamma(X \times_S Y, \Omega^\bullet_{X \times_S Y/S}) $$ is an isomorphism in $D(A)$.


derham-lemma-pullback-c1

Given a commutative diagram $$ \xymatrix{ X' \ar[r]_f \ar[d] & X \ar[d] \\ S' \ar[r] & S } $$ of schemes the diagrams $$ \xymatrix{ \Pic(X') \ar[d]_{c_1^{dR}} & \Pic(X) \ar[d]^{c_1^{dR}} \ar[l]^{f^*} \\ H^2_{dR}(X'/S') & H^2_{dR}(X/S) \ar[l]_{f^*} } \quad \xymatrix{ \Pic(X') \ar[d]_{c_1^{Hodge}} & \Pic(X) \ar[d]^{c_1^{Hodge}} \ar[l]^{f^*} \\ H^1(X', \Omega^1_{X'/S'}) & H^1(X, \Omega^1_{X/S}) \ar[l]_{f^*} } $$ commute.


derham-lemma-the-complex-for-L-star

With notation as above, there is a short exact sequence of complexes $$ 0 \to \Omega^\bullet_{X/S} \to \Omega^\bullet_{L^\star/S, 0} \to \Omega^\bullet_{X/S}[-1] \to 0 $$


derham-lemma-the-complex-for-L-star-gives-chern-class

The ``boundary'' map $\delta : \Omega^\bullet_{X/S} \to \Omega^\bullet_{X/S}[2]$ in $D(X, f^{-1}\mathcal{O}_S)$ coming from the short exact sequence in Lemma \ref{lemma-the-complex-for-L-star} is the map of Remark \ref{remark-cup-product-as-a-map} for $\xi = c_1^{dR}(\mathcal{L})$.


derham-lemma-push-omega-a

With notation as above we have \begin{enumerate} \item $\Omega^p_{L^\star/S, n} = \Omega^p_{L^\star/S, 0} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}$ for all $n \in \mathbf{Z}$ as quasi-coherent $\mathcal{O}_X$-modules, \item $\Omega^\bullet_{X/S} = \Omega^\bullet_{L/X, 0}$ as complexes, and \item for $n > 0$ and $p \geq 0$ we have $\Omega^p_{L/X, n} = \Omega^p_{L^\star/S, n}$. \end{enumerate}


derham-lemma-line-bundle-characteristic-zero

In the situation above, assume there is a morphism $S \to \Spec(\mathbf{Q})$. Then $\Omega^\bullet_{X/S} \to \pi_*\Omega^\bullet_{L/S}$ is a quasi-isomorphism and $H_{dR}^*(X/S) = H_{dR}^*(L/S)$.


derham-lemma-euler-sequence

There exists a short exact sequence $$ 0 \to \Omega \to \mathcal{O}(-1)^{\oplus n + 1} \to \mathcal{O} \to 0 $$


derham-lemma-twisted-hodge-cohomology-projective-space

In the situation above we have the following cohomology groups \begin{enumerate} \item $H^q(\mathbf{P}^n_A, \Omega^p) = 0$ unless $0 \leq p = q \leq n$, \item for $0 \leq p \leq n$ the $A$-module $H^p(\mathbf{P}^n_A, \Omega^p)$ free of rank $1$. \item for $q > 0$, $k > 0$, and $p$ arbitrary we have $H^q(\mathbf{P}^n_A, \Omega^p(k)) = 0$, and \item add more here. \end{enumerate}


derham-lemma-hodge-cohomology-projective-space

We have $H^q(\mathbf{P}^n_A, \Omega^p) = 0$ unless $0 \leq p = q \leq n$. For $0 \leq p \leq n$ the $A$-module $H^p(\mathbf{P}^n_A, \Omega^p)$ free of rank $1$ with basis element $c_1^{Hodge}(\mathcal{O}(1))^p$.


derham-lemma-de-rham-cohomology-projective-space

For $0 \leq i \leq n$ the de Rham cohomology $H^{2i}_{dR}(\mathbf{P}^n_A/A)$ is a free $A$-module of rank $1$ with basis element $c_1^{dR}(\mathcal{O}(1))^i$. In all other degrees the de Rham cohomology of $\mathbf{P}^n_A$ over $A$ is zero.


derham-lemma-spectral-sequence-smooth

Let $f : X \to Y$ be a quasi-compact, quasi-separated, and smooth morphism of schemes over a base scheme $S$. There is a bounded spectral sequence with first page $$ E_1^{p, q} = H^q(\Omega^p_{Y/S} \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*\Omega^\bullet_{X/Y}) $$ converging to $R^{p + q}f_*\Omega^\bullet_{X/S}$.


derham-lemma-relative-global-generation-on-fibres

Let $f : X \to Y$ be a smooth proper morphism of schemes. Let $N$ and $n_1, \ldots, n_N \geq 0$ be integers and let $\xi_i \in H^{n_i}_{dR}(X/Y)$, $1 \leq i \leq N$. Assume for all points $y \in Y$ the images of $\xi_1, \ldots, \xi_N$ in $H^*_{dR}(X_y/y)$ form a basis over $\kappa(y)$. Then the map $$ \bigoplus\nolimits_{i = 1}^N \mathcal{O}_Y[-n_i] \longrightarrow Rf_*\Omega^\bullet_{X/Y} $$ associated to $\xi_1, \ldots, \xi_N$ is an isomorphism.


derham-lemma-global-generation-on-fibres

Let $f : X \to Y$ be a smooth proper morphism of schemes over a base $S$. Assume \begin{enumerate} \item $Y$ and $S$ are affine, and \item there exist integers $N$ and $n_1, \ldots, n_N \geq 0$ and $\xi_i \in H^{n_i}_{dR}(X/S)$, $1 \leq i \leq N$ such that for all points $y \in Y$ the images of $\xi_1, \ldots, \xi_N$ in $H^*_{dR}(X_y/y)$ form a basis over $\kappa(y)$. \end{enumerate} Then the map $$ \bigoplus\nolimits_{i = 1}^N H^*_{dR}(Y/S) \longrightarrow H^*_{dR}(X/S), \quad (a_1, \ldots, a_N) \longmapsto  \sum \xi_i \cup f^*a_i $$ is an isomorphism.


derham-lemma-log-complex

Let $X \to S$ be a morphism of schemes. Let $Y \subset X$ be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for $Y \subset X$ over $S$. There is a canonical short exact sequence of complexes $$ 0 \to \Omega^\bullet_{X/S} \to \Omega^\bullet_{X/S}(\log Y) \to \Omega^\bullet_{Y/S}[-1] \to 0 $$


derham-lemma-gysin-via-log-complex

Let $f : X \to S$ be a morphism of schemes. Let $Y \subset X$ be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for $Y \subset X$ over $S$. Denote $$ \delta : \Omega^\bullet_{Y/S} \to \Omega^\bullet_{X/S}[2] $$ in $D(X, f^{-1}\mathcal{O}_S)$ the ``boundary'' map coming from the short exact sequence in Lemma \ref{lemma-log-complex}. Denote $$ \xi' : \Omega^\bullet_{X/S} \to \Omega^\bullet_{X/S}[2] $$ in $D(X, f^{-1}\mathcal{O}_S)$ the map of Remark \ref{remark-cup-product-as-a-map} corresponding to $\xi = c_1^{dR}(\mathcal{O}_X(-Y))$. Denote $$ \zeta' : \Omega^\bullet_{Y/S} \to \Omega^\bullet_{Y/S}[2] $$ in $D(Y, f|_Y^{-1}\mathcal{O}_S)$ the map of Remark \ref{remark-cup-product-as-a-map} corresponding to $\zeta = c_1^{dR}(\mathcal{O}_X(-Y)|_Y)$. Then the diagram $$ \xymatrix{ \Omega^\bullet_{X/S} \ar[d]_{\xi'} \ar[r] & \Omega^\bullet_{Y/S} \ar[d]^{\zeta'} \ar[ld]_\delta \\ \Omega^\bullet_{X/S}[2] \ar[r] & \Omega^\bullet_{Y/S}[2] } $$ is commutative in $D(X, f^{-1}\mathcal{O}_S)$.


derham-lemma-log-complex-consequence

Let $X \to S$ be a morphism of schemes. Let $Y \subset X$ be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for $Y \subset X$ over $S$. Let $b \in H^m_{dR}(X/S)$ be a de Rham cohomology class whose restriction to $Y$ is zero. Then $c_1^{dR}(\mathcal{O}_X(Y)) \cup b = 0$ in $H^{m + 2}_{dR}(X/S)$.


derham-lemma-check-log-smooth

Let $X \to T \to S$ be morphisms of schemes. Let $Y \subset X$ be an effective Cartier divisor. If both $X \to T$ and $Y \to T$ are smooth, then the de Rham complex of log poles is defined for $Y \subset X$ over $S$.


derham-lemma-comparison

For $a \geq 0$ we have \begin{enumerate} \item the map $\Omega^a_{X/S} \to b_*\Omega^a_{L/S}$ is an isomorphism, \item the map $\Omega^a_{Z/S} \to p_*\Omega^a_{P/S}$ is an isomorphism, and \item the map $Rb_*\Omega^a_{L/S} \to i_*Rp_*\Omega^a_{P/S}$ is an isomorphism on cohomology sheaves in degree $\geq 1$. \end{enumerate}


derham-lemma-comparison-bis

For $a \geq 0$ there are canonical maps $$ b^*\Omega^a_{X/S} \longrightarrow \Omega^a_{L/S} \longrightarrow b^*\Omega^a_{X/S} \otimes_{\mathcal{O}_L} \mathcal{O}_L((n - 1)E) $$ whose composition is induced by the inclusion $\mathcal{O}_L \subset \mathcal{O}_L((n - 1)E)$.


derham-lemma-blowup-twist-same-cohomology

Let $E = 0(P)$ be the exceptional divisor of the blowing up $b$. For any locally free $\mathcal{O}_X$-module $\mathcal{E}$ and $0 \leq i \leq n - 1$ the map $$ \mathcal{E} \longrightarrow Rb_*(b^*\mathcal{E} \otimes_{\mathcal{O}_L} \mathcal{O}_L(iE)) $$ is an isomorphism in $D(\mathcal{O}_X)$.


derham-lemma-blowup

Let $S$ be a scheme. Let $Z \to X$ be a closed immersion of schemes smooth over $S$. Let $b : X' \to X$ be the blowing up of $Z$ with exceptional divisor $E \subset X'$. Then $X'$ and $E$ are smooth over $S$. The morphism $p : E \to Z$ is canonically isomorphic to the projective space bundle $$ \mathbf{P}(\mathcal{I}/\mathcal{I}^2) \longrightarrow Z $$ where $\mathcal{I} \subset \mathcal{O}_X$ is the ideal sheaf of $Z$. The relative $\mathcal{O}_E(1)$ coming from the projective space bundle structure is isomorphic to the restriction of $\mathcal{O}_{X'}(-E)$ to $E$.


derham-lemma-comparison-general

With notation as in Lemma \ref{lemma-blowup} for $a \geq 0$ we have \begin{enumerate} \item the map $\Omega^a_{X/S} \to b_*\Omega^a_{X'/S}$ is an isomorphism, \item the map $\Omega^a_{Z/S} \to p_*\Omega^a_{E/S}$ is an isomorphism, \item the map $Rb_*\Omega^a_{X'/S} \to i_*Rp_*\Omega^a_{E/S}$ is an isomorphism on cohomology sheaves in degree $\geq 1$. \end{enumerate}


derham-lemma-distinguished-triangle-blowup

With notation as in Lemma \ref{lemma-blowup} and denoting $f : X \to S$ the structure morphism there is a canonical distinguished triangle $$ \Omega^\bullet_{X/S} \to Rb_*(\Omega^\bullet_{X'/S}) \oplus i_*\Omega^\bullet_{Z/S} \to i_*Rp_*(\Omega^\bullet_{E/S}) \to \Omega^\bullet_{X/S}[1] $$ in $D(X, f^{-1}\mathcal{O}_S)$ where the four maps $$ \begin{matrix} \Omega^\bullet_{X/S} & \to & Rb_*(\Omega^\bullet_{X'/S}), \\ \Omega^\bullet_{X/S} & \to & i_*\Omega^\bullet_{Z/S}, \\ Rb_*(\Omega^\bullet_{X'/S}) & \to & i_*Rp_*(\Omega^\bullet_{E/S}), \\ i_*\Omega^\bullet_{Z/S} & \to & i_*Rp_*(\Omega^\bullet_{E/S}) \end{matrix} $$ are the canonical ones (Section \ref{section-de-rham-complex}), except with sign reversed for one of them.


derham-lemma-ext-zero

Let $i : Z \to X$ be a closed immersion of schemes which is regular of codimension $c$. Then $\Ext^q_{\mathcal{O}_X}(i_*\mathcal{F}, \mathcal{E}) = 0$ for $q < c$ for $\mathcal{E}$ locally free on $X$ and $\mathcal{F}$ any $\mathcal{O}_Z$-module.


derham-lemma-funny-map

Let $R$ be a ring and consider a commutative diagram $$ \xymatrix{ 0 \ar[r] & K^0 \ar[r] & L^0 \ar[r] & M^0 \ar[r] & 0 \\ & & L^{-1} \ar[u]_\partial \ar@{=}[r] & M^{-1} \ar[u] } $$ of $R$-modules with exact top row and $M^0$ and $M^{-1}$ finite free of the same rank. Then there are canonical maps $$ \wedge^i(H^0(L^\bullet)) \longrightarrow \wedge^i(K^0) \otimes_R \det(M^\bullet) $$ whose composition with $\wedge^i(K^0) \to \wedge^i(H^0(L^\bullet))$ is equal to multiplication with $\delta(M^\bullet)$.


derham-lemma-Garel-upstairs

There exists a unique rule that to every locally quasi-finite syntomic morphism of schemes $f : Y \to X$ assigns $\mathcal{O}_Y$-module maps $$ c^p_{Y/X} : \Omega^p_{Y/\mathbf{Z}} \longrightarrow f^*\Omega^p_{X/\mathbf{Z}} \otimes_{\mathcal{O}_Y} \det(\NL_{Y/X}) $$ satisfying the following two properties \begin{enumerate} \item the composition with $f^*\Omega^p_{X/\mathbf{Z}} \to \Omega^p_{Y/\mathbf{Z}}$ is multiplication by $\delta(\NL_{Y/X})$, and \item the rule is compatible with restriction to opens and with base change. \end{enumerate}


derham-lemma-Garel

There exists a unique rule that to every finite syntomic morphism of schemes $f : Y \to X$ assigns $\mathcal{O}_X$-module maps $$ \Theta^p_{Y/X} : f_*\Omega^p_{Y/\mathbf{Z}} \longrightarrow \Omega^p_{X/\mathbf{Z}} $$ satisfying the following properties \begin{enumerate} \item the composition with $\Omega^p_{X/\mathbf{Z}} \otimes_{\mathcal{O}_X} f_*\mathcal{O}_Y \to f_*\Omega^p_{Y/\mathbf{Z}}$ is equal to $\text{id} \otimes \text{Trace}_f$ where $\text{Trace}_f : f_*\mathcal{O}_Y \to \mathcal{O}_X$ is the map from Discriminants, Section \ref{discriminant-section-discriminant}, \item the rule is compatible with base change. \end{enumerate}


derham-lemma-Garel-map-frobenius-smooth-char-p

Let $p$ be a prime number. Let $X \to S$ be a smooth morphism of relative dimension $d$ of schemes in characteristic $p$. The relative Frobenius $F_{X/S} : X \to X^{(p)}$ of $X/S$ (Varieties, Definition \ref{varieties-definition-relative-frobenius}) is finite syntomic and the corresponding map $$ \Theta_{X/X^{(p)}} : F_{X/S, *}\Omega^\bullet_{X/S} \to \Omega^\bullet_{X^{(p)}/S} $$ is zero in all degrees except in degree $d$ where it defines a surjection.


derham-lemma-duality-hodge

Let $k$ be a field. Let $X$ be a nonempty smooth proper scheme over $k$ equidimensional of dimension $d$. There exists a $k$-linear map $$ t : H^d(X, \Omega^d_{X/k}) \longrightarrow k $$ unique up to precomposing by multiplication by a unit of $H^0(X, \mathcal{O}_X)$ with the following property: for all $p, q$ the pairing $$ H^q(X, \Omega^p_{X/k}) \times H^{d - q}(X, \Omega^{d - p}_{X/k}) \longrightarrow k, \quad (\xi, \xi') \longmapsto t(\xi \cup \xi') $$ is perfect.


derham-lemma-bottom-part-degenerates

Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$. The map $$ \text{d} : H^0(X, \mathcal{O}_X) \to H^0(X, \Omega^1_{X/k}) $$ is zero.


derham-lemma-top-part-degenerates

Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$ equidimensional of dimension $d$. The map $$ \text{d} : H^d(X, \Omega^{d - 1}_{X/k}) \to H^d(X, \Omega^d_{X/k}) $$ is zero.


derham-lemma-chern-classes

There is a unique rule which assigns to every quasi-compact and quasi-separated scheme $X$ a total Chern class $$ c^{dR} : K_0(\textit{Vect}(X)) \longrightarrow \prod\nolimits_{i \geq 0} H^{2i}_{dR}(X/\mathbf{Z}) $$ with the following properties \begin{enumerate} \item we have $c^{dR}(\alpha + \beta) = c^{dR}(\alpha) c^{dR}(\beta)$ for $\alpha, \beta \in K_0(\textit{Vect}(X))$, \item if $f : X \to X'$ is a morphism of quasi-compact and quasi-separated schemes, then $c^{dR}(f^*\alpha) = f^*c^{dR}(\alpha)$, \item given $\mathcal{L} \in \Pic(X)$ we have $c^{dR}([\mathcal{L}]) = 1 + c_1^{dR}(\mathcal{L})$ \end{enumerate}


derham-lemma-gysin-differential

The gysin map (\ref{equation-gysin}) is compatible with the de Rham differentials on $\Omega^\bullet_{X/S}$ and $\Omega^\bullet_{Z/S}$.


derham-lemma-gysin-global

Let $X \to S$ be a morphism of schemes. Let $Z \to X$ be a closed immersion of finite presentation whose conormal sheaf $\mathcal{C}_{Z/X}$ is locally free of rank $c$. Then there is a canonical map $$ \gamma^p : \Omega^p_{Z/S} \to \mathcal{H}^c_Z(\Omega^{p + c}_{X/S}) $$ which is locally given by the maps $\gamma^p_{f_1, \ldots, f_c}$ of Remark \ref{remark-gysin-equations}.


derham-lemma-gysin-differential-global

Let $X \to S$ and $i : Z \to X$ be as in Lemma \ref{lemma-gysin-global}. The gysin map $\gamma^p$ is compatible with the de Rham differentials on $\Omega^\bullet_{X/S}$ and $\Omega^\bullet_{Z/S}$.


derham-lemma-gysin-projection

Let $X \to S$ and $i : Z \to X$ be as in Lemma \ref{lemma-gysin-global}. Given $\alpha \in H^q(X, \Omega^p_{X/S})$ we have $\gamma^p(\alpha|_Z) = i^{-1}\alpha \wedge \gamma^0(1)$ in $H^q(Z, \mathcal{H}^c_Z(\Omega^{p + c}_{X/S}))$. Please see proof for notation.


derham-lemma-gysin-transverse

Let $c \geq 0$ be a integer. Let $$ \xymatrix{ Z' \ar[d]_h \ar[r] & X' \ar[d]_g \ar[r] & S' \ar[d] \\ Z \ar[r] & X \ar[r] & S } $$ be a commutative diagram of schemes. Assume \begin{enumerate} \item $Z \to X$ and $Z' \to X'$ satisfy the assumptions of Lemma \ref{lemma-gysin-global}, \item the left square in the diagram is cartesian, and \item $h^*\mathcal{C}_{Z/X} \to \mathcal{C}_{Z'/X'}$ (Morphisms, Lemma \ref{morphisms-lemma-conormal-functorial}) is an isomorphism. \end{enumerate} Then the diagram $$ \xymatrix{ h^*\Omega^p_{Z/S} \ar[rr]_-{h^{-1}\gamma^p} \ar[d] & & \mathcal{O}_{X'}|_{Z'} \otimes_{h^{-1}\mathcal{O}_X|_Z} h^{-1}\mathcal{H}^c_Z(\Omega^{p + c}_{X/S}) \ar[d] \\ \Omega^p_{Z'/S'} \ar[rr]^{\gamma^p} & & \mathcal{H}^c_{Z'}(\Omega^{p + c}_{X'/S'}) } $$ is commutative. The left vertical arrow is functoriality of modules of differentials and the right vertical arrow uses Cohomology, Remark \ref{cohomology-remark-support-functorial}.


derham-lemma-relative-bottom-part-degenerates

In Situation \ref{situation-relative-duality} the psuhforward $f_*\mathcal{O}_X$ is a finite \'etale $\mathcal{O}_S$-algebra and locally on $S$ we have $Rf_*\mathcal{O}_X = f_*\mathcal{O}_X \oplus P$ in $D(\mathcal{O}_S)$ with $P$ perfect of tor amplitude in $[1, \infty)$. The map $\text{d} : f_*\mathcal{O}_X \to f_*\Omega_{X/S}$ is zero.


derham-lemma-relative-duality-hodge

In Situation \ref{situation-relative-duality} there exists an $\mathcal{O}_S$-module map $$ t : Rf_*\Omega^d_{X/S}[d] \longrightarrow \mathcal{O}_S $$ unique up to precomposing by multiplication by a unit of $H^0(X, \mathcal{O}_X)$ with the following property: for all $p$ the pairing $$ Rf_*\Omega^p_{X/S} \otimes_{\mathcal{O}_S}^\mathbf{L} Rf_*\Omega^{d - p}_{X/S}[d] \longrightarrow \mathcal{O}_S $$ given by the relative cup product composed with $t$ is a perfect pairing of perfect complexes on $S$.


derham-lemma-relative-top-part-degenerates

In Situation \ref{situation-relative-duality} the map $\text{d} : R^df_*\Omega^{d - 1}_{X/S} \to R^df_*\Omega^d_{X/S}$ is zero.


derham-proposition-global-generation-on-fibres

Let $f : X \to Y$ be a smooth proper morphism of schemes over a base $S$. Let $N$ and $n_1, \ldots, n_N \geq 0$ be integers and let $\xi_i \in H^{n_i}_{dR}(X/S)$, $1 \leq i \leq N$. Assume for all points $y \in Y$ the images of $\xi_1, \ldots, \xi_N$ in $H^*_{dR}(X_y/y)$ form a basis over $\kappa(y)$. The map $$ \tilde \xi = \bigoplus \tilde \xi_i[-n_i] : \bigoplus \Omega^\bullet_{Y/S}[-n_i] \longrightarrow Rf_*\Omega^\bullet_{X/S} $$ (see proof) is an isomorphism in $D(Y, (Y \to S)^{-1}\mathcal{O}_S)$ and correspondingly the map $$ \bigoplus\nolimits_{i = 1}^N H^*_{dR}(Y/S) \longrightarrow H^*_{dR}(X/S), \quad (a_1, \ldots, a_N) \longmapsto  \sum \xi_i \cup f^*a_i $$ is an isomorphism.


derham-proposition-projective-space-bundle-formula

Let $X \to S$ be a morphism of schemes. Let $\mathcal{E}$ be a locally free $\mathcal{O}_X$-module of constant rank $r$. Consider the morphism $p : P = \mathbf{P}(\mathcal{E}) \to X$. Then the map $$ \bigoplus\nolimits_{i = 0, \ldots, r - 1} H^*_{dR}(X/S) \longrightarrow H^*_{dR}(P/S) $$ given by the rule $$ (a_0, \ldots, a_{r - 1}) \longmapsto \sum\nolimits_{i = 0, \ldots, r - 1} c_1^{dR}(\mathcal{O}_P(1))^i \cup p^*(a_i) $$ is an isomorphism.


derham-proposition-blowup-split

With notation as in Lemma \ref{lemma-blowup} the map $\Omega^\bullet_{X/S} \to Rb_*\Omega^\bullet_{X'/S}$ has a splitting in $D(X, (X \to S)^{-1}\mathcal{O}_S)$.


derham-proposition-Garel

\begin{reference} \cite{Garel} \end{reference} Let $f : Y \to X$ be a finite syntomic morphism of schemes. The maps $\Theta^p_{Y/X}$ of Lemma \ref{lemma-Garel} define a map of complexes $$ \Theta_{Y/X} : f_*\Omega^\bullet_{Y/\mathbf{Z}} \longrightarrow \Omega^\bullet_{X/\mathbf{Z}} $$ with the following properties \begin{enumerate} \item in degree $0$ we get $\text{Trace}_f : f_*\mathcal{O}_Y \to \mathcal{O}_X$, see Discriminants, Section \ref{discriminant-section-discriminant}, \item we have $\Theta_{Y/X}(\omega \wedge \eta) = \omega \wedge \Theta_{Y/X}(\eta)$ for $\omega$ in $\Omega^\bullet_{X/\mathbf{Z}}$ and $\eta$ in $f_*\Omega^\bullet_{Y/\mathbf{Z}}$, \item if $f$ is a morphism over a base scheme $S$, then $\Theta_{Y/X}$ induces a map of complexes $f_*\Omega^\bullet_{Y/S} \to \Omega^\bullet_{X/S}$. \end{enumerate}


derham-proposition-poincare-duality

Let $k$ be a field. Let $X$ be a nonempty smooth proper scheme over $k$ equidimensional of dimension $d$. There exists a $k$-linear map $$ t : H^{2d}_{dR}(X/k) \longrightarrow k $$ unique up to precomposing by multiplication by a unit of $H^0(X, \mathcal{O}_X)$ with the following property: for all $i$ the pairing $$ H^i_{dR}(X/k) \times H_{dR}^{2d - i}(X/k) \longrightarrow k, \quad (\xi, \xi') \longmapsto t(\xi \cup \xi') $$ is perfect.


trace-theorem-baffling

Let $X$ be a scheme in characteristic $p > 0$. Then the absolute frobenius induces (by pullback) the trivial map on cohomology, i.e., for all integers $j\geq 0$, $$ F_X^* : H^j (X, \underline{\mathbf{Z}/n\mathbf{Z}}) \longrightarrow H^j (X, \underline{\mathbf{Z}/n\mathbf{Z}}) $$ is the identity.


trace-theorem-geometric-arithmetic-inverse

Let $\mathcal{F}$ be an abelian sheaf on $X_\etale$. Then for all $j\geq 0$, $\text{frob}_k$ acts on the cohomology group $H^j(X_{\bar k}, \mathcal{F}|_{X_{\bar k}})$ as the inverse of the map $\pi_X^*$.


trace-theorem-trace

Let $X$ be a projective curve over a finite field $k$, $\Lambda$ a finite ring and $K \in D_{ctf}(X, \Lambda)$. Then the global and local Lefschetz numbers of $K$ are equal, i.e., \begin{equation} \label{equation-trace-formula} \text{Tr}(\pi^*_X |_{R\Gamma(X_{\bar k}, K)}) = \sum\nolimits_{x\in X(k)} \text{Tr}(\pi_X |_{K_{\bar x}}) \end{equation} in $\Lambda^\natural$.


trace-theorem-weil-trace-formula

Let $C$ be a nonsingular projective curve over an algebraically closed field $k$, and $\varphi : C \to C$ a $k$-endomorphism of $C$ distinct from the identity. Let $V(\varphi) = \Delta_C \cdot \Gamma_\varphi$, where $\Delta_C$ is the diagonal, $\Gamma_\varphi$ is the graph of $\varphi$, and the intersection number is taken on $C \times C$. Let $J = \underline{\Picardfunctor}^0_{C/k}$ be the jacobian of $C$ and denote $\varphi^* : J \to J$ the action induced by $\varphi$ by taking pullbacks. Then $$ V(\varphi) = 1 - \text{Tr}_J(\varphi^*) + \deg \varphi. $$


trace-theorem-trace-formula-again

Let $k$ be a finite field and $X$ a finite type, separated scheme of dimension at most 1 over $k$. Let $\Lambda$ be a finite ring whose cardinality is prime to that of $k$, and $K\in D_{ctf}(X, \Lambda)$. Then \begin{equation} \label{equation-trace-formula-again} \text{Tr}(\pi_X^* |_{R\Gamma_c(X_{\bar k}, K)}) = \sum\nolimits_{x\in X(k)} \text{Tr}(\pi_x |_{K_{\bar x}}) \end{equation} in $\Lambda^{\natural}$.


trace-theorem-A

Let $X$ be a scheme of finite type over a finite field $k$. Let $\Lambda$ be a finite ring of order prime to the characteristic of $k$ and $\mathcal{F}$ a constructible flat $\Lambda$-module on $X_\etale$. Then $$ L(X, \mathcal{F}) = \det(1 - \pi_X^*\ T |_{R\Gamma_c(X_{\bar k}, \mathcal{F})})^{-1} \in \Lambda[[T]]. $$


trace-theorem-B

Let $X$ be a scheme of finite type over a finite field $k$, and $\mathcal{F}$ a $\mathbf{Q}_\ell$-sheaf on $X$. Then $$ L(X, \mathcal{F}) = \prod\nolimits_i \det(1 - \pi_X^*T |_{H_c^i(X_{\bar k} , \mathcal{F})})^{(-1)^{i + 1}} \in \mathbf{Q}_\ell[[T]]. $$


trace-theorem-D

Let $X/k$ be as above, let $\Lambda$ be a finite ring with $\#\Lambda \in k^*$ and $K\in D_{ctf}(X, \Lambda)$. Then $R\Gamma_c(X_{\bar k}, K)\in D_{perf}(\Lambda)$ and $$ \sum_{x\in X(k)}\text{Tr}\left(\pi_x |_{K_{\bar x}}\right) = \text{Tr}\left(\pi_X^* |_{R\Gamma_c(X_{\bar k}, K )}\right). $$


trace-theorem-C

Let $X$ be a separated scheme of finite type over a finite field $k$ and $\mathcal{F}$ be a $\mathbf{Q}_\ell$-sheaf on $X$. Then $\dim_{\mathbf{Q}_\ell}H_c^i(X_{\bar k}, \mathcal{F})$ is finite for all $i$, and is nonzero for $0\leq i \leq 2 \dim X$ only. Furthermore, we have $$ \sum_{x\in X(k)} \text{Tr}\left(\pi_x |_{\mathcal{F}_{\bar x}}\right) = \sum_i (-1)^i\text{Tr}\left(\pi_X^* |_{H_c^i(X_{\bar k}, \mathcal{F})}\right). $$


trace-theorem-fundamental-group

Let $X$ be a connected scheme. \begin{enumerate} \item There is a topology on $\pi_1(X, \overline{x})$ such that the open subgroups form a fundamental system of open nbhds of $e\in \pi_1(X, \overline x)$. \item With topology of (1) the group $\pi_1(X, \overline{x})$ is a profinite group. \item The functor $$ \begin{matrix} \text{ schemes finite } \atop \text{ \'etale over }X & \to & \text{ finite discrete continuous } \atop \pi_1(X, \overline{x})\text{-sets}\\ Y / X& \mapsto & F_{\overline{x}}(Y) \text{ with its natural action} \end{matrix} $$ is an equivalence of categories. \end{enumerate}


trace-theorem-weil-II

For a sheaf $\mathcal{F}_\rho$ with $\rho$ satisfying the conclusions of the conjecture above then the eigenvalues of $\pi_X^*$ on $H_c^i(X_{\overline{k}}, \mathcal{F}_{\rho})$ are algebraic numbers $\alpha$ with absolute values $$ |\alpha|=q^{w/2}, \text{ for }w\in \mathbf{Z},\ w\leq i $$ Moreover, if $X$ smooth and proj. then $w = i$.


trace-theorem-drinfeld-make-rho

Given an eigenform $f$ with values in $\overline{\mathbf{Q}}_l$ and eigenvalues $u_v\in \overline{\mathbf{Z}}_l^*$ then there exists $$ \rho : \pi_1(X)\to \text{GL}_2(E) $$ continuous, absolutely irreducible where $E$ is a finite extension of $\mathbf{Q}_\ell$ contained in $\overline{\mathbf{Q}}_l$ such that $t_v = \text{Tr}(\rho(F_v))$, and $u_v = q_v^{-1}\det\left(\rho(F_v)\right)$ for all places $v$.


trace-theorem-drinfeld-make-f

Suppose $\mathbf{Q}_l \subset E$ finite, and $$ \rho : \pi_1(X)\to \text{GL}_2(E) $$ absolutely irreducible, continuous. Then there exists an eigenform $f$ with values in $\overline{\mathbf{Q}}_l$ whose eigenvalues $t_v$, $u_v$ satisfy the equalities $t_v = \text{Tr}(\rho(F_v))$ and $u_v = q_v^{-1}\det(\rho(F_v))$.


trace-theorem-conjecture-n-2

The Conjecture holds if $n\leq 2$.


trace-theorem-conjecture-l-bigger-2n

Conjecture holds if $l > 2n$ modulo some unproven things.