sites-modules-definition-presheaf-modules

Let $\mathcal{C}$ be a category, and let $\mathcal{O}$ be a presheaf of rings on $\mathcal{C}$. \begin{enumerate} \item A {\it presheaf of $\mathcal{O}$-modules} is given by an abelian presheaf $\mathcal{F}$ together with a map of presheaves of sets $$ \mathcal{O} \times \mathcal{F} \longrightarrow \mathcal{F} $$ such that for every object $U$ of $\mathcal{C}$ the map $\mathcal{O}(U) \times \mathcal{F}(U) \to \mathcal{F}(U)$ defines the structure of an $\mathcal{O}(U)$-module structure on the abelian group $\mathcal{F}(U)$. \item A {\it morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of presheaves of $\mathcal{O}$-modules} is a morphism of abelian presheaves $\varphi : \mathcal{F} \to \mathcal{G}$ such that the diagram $$ \xymatrix{ \mathcal{O} \times \mathcal{F} \ar[r] \ar[d]_{\text{id} \times \varphi} & \mathcal{F} \ar[d]^{\varphi} \\ \mathcal{O} \times \mathcal{G} \ar[r] & \mathcal{G} } $$ commutes. \item The set of $\mathcal{O}$-module morphisms as above is denoted $\Hom_\mathcal{O}(\mathcal{F}, \mathcal{G})$. \item The category of presheaves of $\mathcal{O}$-modules is denoted $\textit{PMod}(\mathcal{O})$. \end{enumerate}


sites-modules-definition-sheaf-modules

Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. \begin{enumerate} \item A {\it sheaf of $\mathcal{O}$-modules} is a presheaf of $\mathcal{O}$-modules $\mathcal{F}$, see Definition \ref{definition-presheaf-modules}, such that the underlying presheaf of abelian groups $\mathcal{F}$ is a sheaf. \item A {\it morphism of sheaves of $\mathcal{O}$-modules} is a morphism of presheaves of $\mathcal{O}$-modules. \item Given sheaves of $\mathcal{O}$-modules $\mathcal{F}$ and $\mathcal{G}$ we denote $\Hom_\mathcal{O}(\mathcal{F}, \mathcal{G})$ the set of morphism of sheaves of $\mathcal{O}$-modules. \item The category of sheaves of $\mathcal{O}$-modules is denoted $\textit{Mod}(\mathcal{O})$. \end{enumerate}


sites-modules-definition-pushforward

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. \begin{enumerate} \item Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{C}$-modules. We define the {\it pushforward} of $\mathcal{F}$ as the sheaf of $\mathcal{O}_\mathcal{D}$-modules which as a sheaf of abelian groups equals $f_*\mathcal{F}$ and with module structure given by the restriction via $f^\sharp : \mathcal{O}_\mathcal{D} \to f_*\mathcal{O}_\mathcal{C}$ of the module structure $$ f_*\mathcal{O}_\mathcal{C} \times f_*\mathcal{F} \longrightarrow f_*\mathcal{F} $$ from Lemma \ref{lemma-pushforward-module}. \item Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_\mathcal{D}$-modules. We define the {\it pullback} $f^*\mathcal{G}$ to be the sheaf of $\mathcal{O}_\mathcal{C}$-modules defined by the formula $$ f^*\mathcal{G} = \mathcal{O}_\mathcal{C} \otimes_{f^{-1}\mathcal{O}_\mathcal{D}} f^{-1}\mathcal{G} $$ where the ring map $f^{-1}\mathcal{O}_\mathcal{D} \to \mathcal{O}_\mathcal{C}$ is $f^\sharp$, and where the  module structure is given by Lemma \ref{lemma-pullback-module}. \end{enumerate}


sites-modules-definition-g-shriek

With $u : \mathcal{C} \to \mathcal{D}$ satisfying (a), (b) above. For $\mathcal{F} \in \textit{PAb}(\mathcal{C})$ we define {\it $g_{p!}\mathcal{F}$} as the presheaf $$ V \longmapsto \colim_{V \to u(U)} \mathcal{F}(U) $$ with colimits over $(\mathcal{I}_V^u)^{opp}$ taken in $\textit{Ab}$. For $\mathcal{F} \in \textit{PAb}(\mathcal{C})$ we set {\it $g_!\mathcal{F} = (g_{p!}\mathcal{F})^\#$}.


sites-modules-definition-global

Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. \begin{enumerate} \item We say $\mathcal{F}$ is a {\it free $\mathcal{O}$-module} if $\mathcal{F}$ is isomorphic as an $\mathcal{O}$-module to a sheaf of the form $\bigoplus_{i \in I} \mathcal{O}$. \item We say $\mathcal{F}$ is {\it finite free} if $\mathcal{F}$ is isomorphic as an $\mathcal{O}$-module to a sheaf of the form $\bigoplus_{i \in I} \mathcal{O}$ with a finite index set $I$. \item We say $\mathcal{F}$ is {\it generated by global sections} if there exists a surjection $$ \bigoplus\nolimits_{i \in I} \mathcal{O} \longrightarrow \mathcal{F} $$ from a free $\mathcal{O}$-module onto $\mathcal{F}$. \item Given $r \geq 0$ we say $\mathcal{F}$ is {\it generated by $r$ global sections} if there exists a surjection $\mathcal{O}^{\oplus r} \to \mathcal{F}$. \item We say $\mathcal{F}$ is {\it generated by finitely many global sections} if it is generated by $r$ global sections for some $r \geq 0$. \item We say $\mathcal{F}$ has a {\it global presentation} if there exists an exact sequence $$ \bigoplus\nolimits_{j \in J} \mathcal{O} \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O} \longrightarrow \mathcal{F} \longrightarrow 0 $$ of $\mathcal{O}$-modules. \item We say $\mathcal{F}$ has a {\it global finite presentation} if there exists an exact sequence $$ \bigoplus\nolimits_{j \in J} \mathcal{O} \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O} \longrightarrow \mathcal{F} \longrightarrow 0 $$ of $\mathcal{O}$-modules with $I$ and $J$ finite sets. \end{enumerate}


sites-modules-definition-localize-ringed-site

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \Ob(\mathcal{C})$. \begin{enumerate} \item The ringed site $(\mathcal{C}/U, \mathcal{O}_U)$ is called the {\it localization of the ringed site $(\mathcal{C}, \mathcal{O})$ at the object $U$}. \item The morphism of ringed topoi $(j_U, j_U^\sharp) : (\Sh(\mathcal{C}/U), \mathcal{O}_U) \to (\Sh(\mathcal{C}), \mathcal{O})$ is called the {\it localization morphism}. \item The functor $j_{U*} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})$ is called the {\it direct image functor}. \item For a sheaf of $\mathcal{O}$-modules $\mathcal{F}$ on $\mathcal{C}$ the sheaf $j_U^*\mathcal{F}$ is called the {\it restriction of $\mathcal{F}$ to $\mathcal{C}/U$}. We will sometimes denote it by $\mathcal{F}|_{\mathcal{C}/U}$ or even $\mathcal{F}|_U$. It is described by the simple rule $j_U^*(\mathcal{F})(X/U) = \mathcal{F}(X)$. \item The left adjoint $j_{U!} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})$ of restriction is called {\it extension by zero}. It exists and is exact by Lemmas \ref{lemma-extension-by-zero} and \ref{lemma-extension-by-zero-exact}. \end{enumerate}


sites-modules-definition-localize-ringed-topos

Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $\mathcal{F} \in \Sh(\mathcal{C})$. \begin{enumerate} \item The ringed topos $(\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F})$ is called the {\it localization of the ringed topos $(\Sh(\mathcal{C}), \mathcal{O})$ at $\mathcal{F}$}. \item The morphism of ringed topoi $(j_\mathcal{F}, j_\mathcal{F}^\sharp) : (\Sh(\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal{F}) \to (\Sh(\mathcal{C}), \mathcal{O})$ of Lemma \ref{lemma-localize-ringed-topos} is called the {\it localization morphism}. \end{enumerate}


sites-modules-definition-site-local

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. We will freely use the notions defined in Definition \ref{definition-global}. \begin{enumerate} \item We say $\mathcal{F}$ is {\it locally free} if for every object $U$ of $\mathcal{C}$ there exists a covering $\{U_i \to U\}_{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_i}$ is a free $\mathcal{O}_{U_i}$-module. \item We say $\mathcal{F}$ is {\it finite locally free} if for every object $U$ of $\mathcal{C}$ there exists a covering $\{U_i \to U\}_{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_i}$ is a finite free $\mathcal{O}_{U_i}$-module. \item We say $\mathcal{F}$ is {\it locally generated by sections} if for every object $U$ of $\mathcal{C}$ there exists a covering $\{U_i \to U\}_{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_i}$ is an $\mathcal{O}_{U_i}$-module generated by global sections. \item Given $r \geq 0$ we sat $\mathcal{F}$ is {\it locally generated by $r$ sections} if for every object $U$ of $\mathcal{C}$ there exists a covering $\{U_i \to U\}_{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_i}$ is an $\mathcal{O}_{U_i}$-module generated by $r$ global sections. \item We say $\mathcal{F}$ is {\it of finite type} if for every object $U$ of $\mathcal{C}$ there exists a covering $\{U_i \to U\}_{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_i}$ is an $\mathcal{O}_{U_i}$-module generated by finitely many global sections. \item We say $\mathcal{F}$ is {\it quasi-coherent} if for every object $U$ of $\mathcal{C}$ there exists a covering $\{U_i \to U\}_{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_i}$ is an $\mathcal{O}_{U_i}$-module which has a global presentation. \item We say $\mathcal{F}$ is {\it of finite presentation} if for every object $U$ of $\mathcal{C}$ there exists a covering $\{U_i \to U\}_{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_i}$ is an $\mathcal{O}_{U_i}$-module which has a finite global presentation. \item We say $\mathcal{F}$ is {\it coherent} if and only if $\mathcal{F}$ is of finite type, and for every object $U$ of $\mathcal{C}$ and any $s_1, \ldots, s_n \in \mathcal{F}(U)$ the kernel of the map $\bigoplus_{i = 1, \ldots, n} \mathcal{O}_U \to \mathcal{F}|_U$ is of finite type on $(\mathcal{C}/U, \mathcal{O}_U)$. \end{enumerate}


sites-modules-definition-flat

Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings. \begin{enumerate} \item A presheaf $\mathcal{F}$ of $\mathcal{O}$-modules is called {\it flat} if the functor $$ \textit{PMod}(\mathcal{O}) \longrightarrow \textit{PMod}(\mathcal{O}), \quad \mathcal{G} \mapsto \mathcal{G} \otimes_{p, \mathcal{O}} \mathcal{F} $$ is exact. \item A map $\mathcal{O} \to \mathcal{O}'$ of presheaves of rings is called {\it flat} if $\mathcal{O}'$ is flat as a presheaf of $\mathcal{O}$-modules. \item If $\mathcal{C}$ is a site, $\mathcal{O}$ is a sheaf of rings and $\mathcal{F}$ is a sheaf of $\mathcal{O}$-modules, then we say $\mathcal{F}$ is {\it flat} if the functor $$ \textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}), \quad \mathcal{G} \mapsto \mathcal{G} \otimes_\mathcal{O} \mathcal{F} $$ is exact. \item A map $\mathcal{O} \to \mathcal{O}'$ of sheaves of rings on a site is called {\it flat} if $\mathcal{O}'$ is flat as a sheaf of $\mathcal{O}$-modules. \end{enumerate}


sites-modules-definition-flat-morphism

Let $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}) \longrightarrow (\Sh(\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. We say $(f, f^\sharp)$ is {\it flat} if the ring map $f^\sharp : f^{-1}\mathcal{O}' \to \mathcal{O}$ is flat. We say a morphism of ringed sites is {\it flat} if the associated morphism of ringed topoi is flat.


sites-modules-definition-flat-module

Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$ be a morphism of ringed topoi. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. We say that $\mathcal{F}$ is {\it flat over $(\Sh(\mathcal{D}), \mathcal{O}')$} if $\mathcal{F}$ is flat as an $f^{-1}\mathcal{O}'$-module.


sites-modules-definition-invertible-sheaf

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. \begin{enumerate} \item A finite locally free $\mathcal{O}$-module $\mathcal{F}$ is said to have {\it rank $r$} if for every object $U$ of $\mathcal{C}$ there exists a covering $\{U_i \to U\}$ of $U$ such that $\mathcal{F}|_{U_i}$ is isomorphic to $\mathcal{O}_{U_i}^{\oplus r}$ as an $\mathcal{O}_{U_i}$-module. \item An $\mathcal{O}$-module $\mathcal{L}$ is {\it invertible} if the functor $$ \textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}),\quad \mathcal{F}  \longmapsto \mathcal{F} \otimes_\mathcal{O} \mathcal{L} $$ is an equivalence. \item The sheaf {\it $\mathcal{O}^*$} is the subsheaf of $\mathcal{O}$ defined by the rule $$ U \longmapsto \mathcal{O}^*(U) = \{f \in \mathcal{O}(U) \mid \exists g \in \mathcal{O}(U)\text{ such that }fg = 1\} $$ It is a sheaf of abelian groups with multiplication as the group law. \end{enumerate}


sites-modules-definition-pic

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. The {\it Picard group} $\Pic(\mathcal{O})$ of the ringed site is the abelian group whose elements are isomorphism classes of invertible $\mathcal{O}$-modules, with addition corresponding to tensor product.


sites-modules-definition-derivation

Let $\mathcal{C}$ be a site. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. Let $\mathcal{F}$ be an $\mathcal{O}_2$-module. A {\it $\mathcal{O}_1$-derivation} or more precisely a {\it $\varphi$-derivation} into $\mathcal{F}$ is a map $D : \mathcal{O}_2 \to \mathcal{F}$ which is additive, annihilates the image of $\mathcal{O}_1 \to \mathcal{O}_2$, and satisfies the {\it Leibniz rule} $$ D(ab) = aD(b) + D(a)b $$ for all $a, b$ local sections of $\mathcal{O}_2$ (wherever they are both defined). We denote $\text{Der}_{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{F})$ the set of $\varphi$-derivations into $\mathcal{F}$.


sites-modules-definition-module-differentials

Let $\mathcal{C}$ be a site. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. The {\it module of differentials} of the ring map $\varphi$ is the object representing the functor $\mathcal{F} \mapsto \text{Der}_{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{F})$ which exists by Lemma \ref{lemma-universal-module}. It is denoted $\Omega_{\mathcal{O}_2/\mathcal{O}_1}$, and the {\it universal $\varphi$-derivation} is denoted $\text{d} : \mathcal{O}_2 \to \Omega_{\mathcal{O}_2/\mathcal{O}_1}$.


sites-modules-definition-sheaf-differentials

Let $X = (\Sh(\mathcal{C}), \mathcal{O})$ and $Y = (\Sh(\mathcal{C}'), \mathcal{O}')$ be ringed topoi. Let $(f, f^\sharp) : X \to Y$ be a morphism of ringed topoi. In this situation \begin{enumerate} \item for a sheaf $\mathcal{F}$ of $\mathcal{O}$-modules a {\it $Y$-derivation} $D : \mathcal{O} \to \mathcal{F}$ is just a $f^\sharp$-derivation, and \item the {\it sheaf of differentials $\Omega_{X/Y}$ of $X$ over $Y$} is the module of differentials of $f^\sharp : f^{-1}\mathcal{O}' \to \mathcal{O}$, see Definition \ref{definition-module-differentials}. \end{enumerate} Thus $\Omega_{X/Y}$ comes equipped with a {\it universal $Y$-derivation} $\text{d}_{X/Y} : \mathcal{O} \longrightarrow \Omega_{X/Y}$. We sometimes write $\Omega_{X/Y} = \Omega_f$.


sites-modules-definition-differential-operators

Let $\mathcal{C}$ be a site. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. Let $k \geq 0$ be an integer. Let $\mathcal{F}$, $\mathcal{G}$ be sheaves of $\mathcal{O}_2$-modules. A {\it differential operator $D : \mathcal{F} \to \mathcal{G}$ of order $k$} is an is an $\mathcal{O}_1$-linear map such that for all local sections $g$ of $\mathcal{O}_2$ the map $s \mapsto D(gs) - gD(s)$ is a differential operator of order $k - 1$. For the base case $k = 0$ we define a differential operator of order $0$ to be an $\mathcal{O}_2$-linear map.


sites-modules-definition-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. The module $\mathcal{P}^k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F})$ constructed in Lemma \ref{lemma-module-principal-parts} is called the {\it module of principal parts of order $k$} of $\mathcal{F}$.


sites-modules-definition-naive-cotangent-complex

Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $\mathcal{C}$. The {\it naive cotangent complex} $\NL_{\mathcal{B}/\mathcal{A}}$ is the chain complex (\ref{equation-naive-cotangent-complex}) $$ \NL_{\mathcal{B}/\mathcal{A}} = \left(\mathcal{I}/\mathcal{I}^2 \longrightarrow \Omega_{\mathcal{A}[\mathcal{B}]/\mathcal{A}} \otimes_{\mathcal{A}[\mathcal{B}]} \mathcal{B}\right) $$ with $\mathcal{I}/\mathcal{I}^2$ placed in degree $-1$ and $\Omega_{\mathcal{A}[\mathcal{B}]/\mathcal{A}} \otimes_{\mathcal{A}[\mathcal{B}]} \mathcal{B}$ placed in degree $0$.


sites-modules-definition-cotangent-complex-morphism-ringed-topoi

Let $X = (\Sh(\mathcal{C}), \mathcal{O})$ and $Y = (\Sh(\mathcal{C}'), \mathcal{O}')$ be ringed topoi. Let $(f, f^\sharp) : X \to Y$ be a morphism of ringed topoi. The {\it naive cotangent complex} $\NL_f = \NL_{X/Y}$ of the given morphism of ringed topoi is $\NL_{\mathcal{O}_X/f^{-1}\mathcal{O}_Y}$. We sometimes write $\NL_{X/Y} = \NL_{\mathcal{O}_X/\mathcal{O}_Y}$.


sites-modules-definition-locally-ringed

A ringed site $(\mathcal{C}, \mathcal{O})$ is said to be {\it locally ringed site} if (\ref{equation-one-is-never-zero}) is an isomorphism, and the equivalent properties of Lemma \ref{lemma-locally-ringed} are satisfied.


sites-modules-definition-locally-ringed-topos

A ringed topos $(\Sh(\mathcal{C}), \mathcal{O})$ is said to be {\it locally ringed} if the underlying ringed site $(\mathcal{C}, \mathcal{O})$ is locally ringed.


sites-modules-definition-morphism-locally-ringed-topoi

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. Assume $(\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})$ and $(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ are locally ringed topoi. We say that $(f, f^\sharp)$ is a {\it morphism of locally ringed topoi} if and only if the diagram of sheaves $$ \xymatrix{ f^{-1}(\mathcal{O}^*_\mathcal{D}) \ar[r]_-{f^\sharp} \ar[d] & \mathcal{O}^*_\mathcal{C} \ar[d] \\ f^{-1}(\mathcal{O}_\mathcal{D}) \ar[r]^-{f^\sharp} & \mathcal{O}_\mathcal{C} } $$ (see Lemma \ref{lemma-locally-ringed-morphism}) is cartesian. If $(f, f^\sharp)$ is a morphism of ringed sites, then we say that it is a {\it morphism of locally ringed sites} if the associated morphism of ringed topoi is a morphism of locally ringed topoi.


sites-modules-definition-locally-constant

Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf of sets, groups, abelian groups, rings, modules over a fixed ring $\Lambda$, etc. \begin{enumerate} \item We say $\mathcal{F}$ is a {\it constant sheaf} of sets, groups, abelian groups, rings, modules over a fixed ring $\Lambda$, etc if it is isomorphic as a sheaf of sets, groups, abelian groups, rings, modules over a fixed ring $\Lambda$, etc to a constant sheaf $\underline{E}$ as in Section \ref{section-constant}. \item We say $\mathcal{F}$ is {\it locally constant} if for every object $U$ of $\mathcal{C}$ there exists a covering $\{U_i \to U\}$ such that $\mathcal{F}|_{U_i}$ is a constant sheaf. \item If $\mathcal{F}$ is a sheaf of sets or groups, then we say $\mathcal{F}$ is {\it finite locally constant} if the constant values are finite sets or finite groups. \end{enumerate}


derham-definition-hodge-filtration

Let $X \to S$ be a morphism of schemes. The {\it Hodge filtration} on $H^n_{dR}(X/S)$ is the filtration with terms $$ F^pH^n_{dR}(X/S) = \Im\left(H^n(X, \sigma_{\geq p}\Omega^\bullet_{X/S}) \longrightarrow H^n_{dR}(X/S)\right) $$ where $\sigma_{\geq p}\Omega^\bullet_{X/S}$ is as in Homology, Section \ref{homology-section-truncations}.


derham-definition-local-product

Let $X \to S$ be a morphism of schemes. Let $Y \subset X$ be an effective Cartier divisor. We say the {\it de Rham complex of log poles is defined for $Y \subset X$ over $S$} if for all $y \in Y$ and local equation $f \in \mathcal{O}_{X, y}$ of $Y$ we have \begin{enumerate} \item $\mathcal{O}_{X, y} \to \Omega_{X/S, y}$, $g \mapsto g \text{d}f$ is a split injection, and \item $\Omega^p_{X/S, y}$ is $f$-torsion free for all $p$. \end{enumerate}


derham-definition-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 $S$. Then the complex $$ \Omega^\bullet_{X/S}(\log Y) $$ constructed in Lemma \ref{lemma-log-complex} is the {\it de Rham complex of log poles for $Y \subset X$ over $S$}.


trace-definition-geometric-frobenius

Let $k$ be a finite field with $q = p^f$ elements. Let $X$ be a scheme over $k$. The {\it geometric frobenius} of $X$ is the morphism $\pi_X : X \to X$ over $\Spec(k)$ which equals $F_X^f$.


trace-definition-arithmetic-frobenius

The {\it arithmetic frobenius} is the map $\text{frob}_k : \bar k \to \bar k$, $x \mapsto x^q$ of $G_k$.


trace-definition-geometric-frobenius-on-stalk

If $x \in X(k)$ is a rational point and $\bar x : \Spec(\bar k) \to X$ the geometric point lying over $x$, we let $\pi_x : \mathcal{F}_{\bar x} \to \mathcal{F}_{\bar x}$ denote the action by $\text{frob}_k^{-1}$ and call it the {\it geometric frobenius}\footnote{This notation is not standard. This operator is denoted $F_x$ in \cite{SGA4.5}. We will likely change this notation in the future.}


trace-definition-trace

The {\it trace} of the endomorphism $a$ is the sum of the diagonal entries of a matrix representing it. This defines an additive map $\text{Tr} : \text{End}_\Lambda(\Lambda^{\oplus m}) \to \Lambda^\natural$.


trace-definition-derived-functor

Let $F: \mathcal{A} \to \mathcal{B}$ be a left exact functor and assume that $\mathcal{A}$ has enough injectives. We define the {\it total right derived functor of $F$} as the functor $RF: D^+(\mathcal{A}) \to D^+(\mathcal{B})$ fitting into the diagram $$ \xymatrix{ D^+(\mathcal{A}) \ar[r]^{RF} & D^+(\mathcal{B}) \\ K^+(\mathcal I) \ar[u] \ar[r]^F & K^+(\mathcal{B}). \ar[u] } $$ This is possible since the left vertical arrow is invertible by the previous lemma. Similarly, let $G: \mathcal{A} \to \mathcal{B}$ be a right exact functor and assume that $\mathcal{A}$ has enough projectives. We define the {\it total left derived functor of $G$} as the functor $LG: D^-(\mathcal{A}) \to D^-(\mathcal{B})$ fitting into the diagram $$ \xymatrix{ D^-(\mathcal{A}) \ar[r]^{LG} & D^-(\mathcal{B}) \\ K^-(\mathcal{P}) \ar[u] \ar[r]^G & K^-(\mathcal{B}). \ar[u] } $$ This is possible since the left vertical arrow is invertible by the previous lemma.


trace-definition-filtered

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item Let $\text{Fil}(\mathcal{A})$ be the category of filtered objects $(A, F)$ of $\mathcal{A}$, where $F$ is a filtration of the form $$ A \supset \ldots \supset F^n A \supset F^{n+1}A \supset \ldots \supset 0. $$ This is an additive category. \item We denote $\text{Fil}^f(\mathcal{A})$ the full subcategory of $\text{Fil}(\mathcal{A})$ whose objects $(A, F)$ have finite filtration. This is also an additive category. \item An object $I \in \text{Fil}^f(\mathcal{A})$ is called {\it filtered injective} (respectively {\it projective}) provided that $\text{gr}^p(I) = \text{gr}_F^p(I) = F^pI/F^{p+1}I$ is injective (resp. projective) in $\mathcal{A}$ for all $p$. \item The category of complexes $\text{Comp}(\text{Fil}^f(\mathcal{A})) \supset \text{Comp}^+(\text{Fil}^f(\mathcal{A}))$ and its homotopy category $K(\text{Fil}^f(\mathcal{A})) \supset K^+(\text{Fil}^f(\mathcal A))$ are defined as before. \item A morphism $\alpha : K^\bullet \to L^\bullet$ of complexes in $\text{Comp}(\text{Fil}^f(\mathcal{A}))$ is called a {\it filtered quasi-isomorphism} provided that $$ \text{gr}^p(\alpha): \text{gr}^p(K^\bullet) \to \text{gr}^p(L^\bullet) $$ is a quasi-isomorphism for all $p \in \mathbf{Z}$. \item We define $DF(\mathcal{A})$ (resp. $DF^+(\mathcal{A})$) by inverting the filtered quasi-isomorphisms in $K(\text{Fil}^f(\mathcal{A}))$ (resp. $K^+(\text{Fil}^f(\mathcal{A}))$). \end{enumerate}


trace-definition-filtered-derived-functors

Let $T: \mathcal{A} \to \mathcal{B}$ be a left exact functor and assume that $\mathcal{A}$ has enough injectives. Define $RT: DF^+(\mathcal{A}) \to D F^+(\mathcal{B})$ to fit in the diagram $$ \xymatrix{ DF^+(\mathcal{A}) \ar[r]^{RT} & DF^+(\mathcal{B}) \\ K^+(\mathcal{I}) \ar[u] \ar[r]^{T \quad} & K^+(\text{Fil}^f(\mathcal{B})). \ar[u]} $$ This is well-defined by the previous lemma. Let $G: \mathcal{A} \to \mathcal{B}$ be a right exact functor and assume that $\mathcal{A}$ has enough projectives. Define $LG: DF^+(\mathcal{A}) \to DF^+(\mathcal{B})$ to fit in the diagram $$ \xymatrix{ DF^-(\mathcal{A}) \ar[r]^{LG} & DF^-(\mathcal{B}) \\ K^-(\mathcal{P}) \ar[u] \ar[r]^{G \quad} & K^-(\text{Fil}^f(\mathcal{B})). \ar[u]} $$ Again, this is well-defined by the previous lemma. The functors $RT$, resp.\ $LG$, are called the {\it filtered derived functor} of $T$, resp.\ $G$.


trace-definition-perfect

We denote by $K_{perf}(\Lambda)$ the category whose objects are bounded complexes of finite projective $\Lambda$-modules, and whose morphisms are morphisms of complexes up to homotopy. The functor $K_{perf}(\Lambda)\to D(\Lambda)$ is fully faithful (Derived Categories, Lemma \ref{derived-lemma-morphisms-from-projective-complex}). Denote $D_{perf}(\Lambda)$ its essential image. An object of $D(\Lambda)$ is called {\it perfect} if it is in $D_{perf}(\Lambda)$.


trace-definition-finite-tor-dimension

Let $\Lambda$ be a (possibly noncommutative) ring. An object $K\in D(\Lambda)$ has {\it finite $\text{Tor}$-dimension} if there exist $a, b \in \mathbf{Z}$ such that for any right $\Lambda$-module $N$, we have $H^i(N \otimes_{\Lambda}^\mathbf{L} K) = 0$ for all $i \not \in [a, b]$.


trace-definition-global-lefschetz-number

Let $\Lambda$ be a finite ring, $X$ a projective curve over a finite field $k$ and $K \in D_{ctf}(X, \Lambda)$ (for instance $K = \underline\Lambda$). There is a canonical map $c_K : \pi_X^{-1}K \to K$, and its base change $c_K|_{X_{\bar k}}$ induces an action denoted $\pi_X^*$ on the perfect complex $R\Gamma(X_{\bar k}, K|_{X_{\bar k}})$. The {\it global Lefschetz number} of $K$ is the trace $\text{Tr}(\pi_X^* |_{R\Gamma(X_{\bar k}, K)})$ of that action. It is an element of $\Lambda^\natural$.


trace-definition-local-lefschetz-number

With $\Lambda, X, k, K$ as in Definition \ref{definition-global-lefschetz-number}. Since $K\in D_{ctf}(X, \Lambda)$, for any geometric point $\bar x$ of $X$, the complex $K_{\bar x}$ is a perfect complex (in $D_{perf}(\Lambda)$). As we have seen in Section \ref{section-frobenii}, the Frobenius $\pi_X$ acts on $K_{\bar x}$. The {\it local Lefschetz number} of $K$ is the sum $$ \sum\nolimits_{x\in X(k)} \text{Tr}(\pi_X |_{K_{\overline{x}}}) $$ which is again an element of $\Lambda^\natural$.


trace-definition-trace-G

Let $f : P\to P$ be an endomorphism of a finite projective $\Lambda[G]$-module $P$. We define $$ \text{Tr}_{\Lambda}^G(f; P) := \varepsilon\left(\text{Tr}_{\Lambda[G]}(f; P)\right) $$ to be the {\it $G$-trace of $f$ on $P$}.


trace-definition-l-adic-sheaf

Let $X$ be a Noetherian scheme. A {\it $\mathbf{Z}_\ell$-sheaf} on $X$, or simply an {\it $\ell$-adic sheaf} $\mathcal{F}$ is an inverse system $\left\{\mathcal{F}_n\right\}_{n\geq 1}$ where \begin{enumerate} \item $\mathcal{F}_n$ is a constructible $\mathbf{Z}/\ell^n\mathbf{Z}$-module on $X_\etale$, and \item the transition maps $\mathcal{F}_{n+1}\to \mathcal{F}_n$ induce isomorphisms $\mathcal{F}_{n+1} \otimes_{\mathbf{Z}/\ell^{n+1}\mathbf{Z}} \mathbf{Z}/\ell^n\mathbf{Z} \cong \mathcal{F}_n$. \end{enumerate} We say that $\mathcal{F}$ is {\it lisse} if each $\mathcal{F}_n$ is locally constant. A {\it morphism} of such is merely a morphism of inverse systems.


trace-definition-torsion-l-adic-sheaf

A $\mathbf{Z}_\ell$-sheaf $\mathcal{F}$ is {\it torsion} if $\ell^n : \mathcal{F} \to \mathcal{F}$ is the zero map for some $n$. The abelian category of $\mathbf{Q}_\ell$-sheaves on $X$ is the quotient of the abelian category of $\mathbf{Z}_\ell$-sheaves by the Serre subcategory of torsion sheaves. In other words, its objects are $\mathbf{Z}_\ell$-sheaves on $X$, and if $\mathcal{F}, \mathcal{G}$ are two such, then $$ \Hom_{\mathbf{Q}_\ell} \left(\mathcal{F}, \mathcal{G} \right) = \Hom_{\mathbf{Z}_\ell} \left(\mathcal{F}, \mathcal{G}\right) \otimes_{\mathbf{Z}_\ell} \mathbf{Q}_\ell. $$ We denote by $\mathcal{F} \mapsto \mathcal{F} \otimes \mathbf{Q}_\ell$ the quotient functor (right adjoint to the inclusion). If $\mathcal{F} = \mathcal{F}' \otimes \mathbf{Q}_\ell$ where $\mathcal{F}'$ is a $\mathbf{Z}_\ell$-sheaf and $\bar x$ is a geometric point, then the {\it stalk} of $\mathcal{F}$ at $\bar x$ is $\mathcal{F}_{\bar x} = \mathcal{F}'_{\bar x} \otimes \mathbf{Q}_\ell$.


trace-definition-cohomology-l-adic

If $X$ is a separated scheme of finite type over an algebraically closed field $k$ and $\mathcal{F} = \left\{\mathcal{F}_n\right\}_{n\geq 1}$ is a $\mathbf{Z}_\ell$-sheaf on $X$, then we define $$ H^i(X, \mathcal{F}) := \lim_n H^i(X, \mathcal{F}_n) \quad\text{and}\quad H_c^i(X, \mathcal{F}) := \lim_n H_c^i(X, \mathcal{F}_n). $$ If $\mathcal{F} = \mathcal{F}'\otimes \mathbf{Q}_\ell$ for a $\mathbf{Z}_\ell$-sheaf $\mathcal{F}'$ then we set $$ H_c^i(X , \mathcal{F}) := H_c^i(X, \mathcal{F}')\otimes_{\mathbf{Z}_\ell}\mathbf{Q}_\ell. $$ We call these the {\it $\ell$-adic cohomology} of $X$ with coefficients $\mathcal{F}$.


trace-definition-L-function-finite-ring

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 we set $$ L(X, \mathcal{F}) := \prod\nolimits_{x\in |X|} \det(1 - \pi_x^*T^{\deg x} |_{\mathcal{F}_{\bar x}})^{-1} \in \Lambda [[ T ]] $$ where $|X|$ is the set of closed points of $X$, $\deg x = [\kappa(x): k]$ and $\bar x$ is a geometric point lying over $x$. This definition clearly generalizes to the case where $\mathcal{F}$ is replaced by a $K \in D_{ctf}(X, \Lambda)$. We call this the {\it $L$-function of $\mathcal{F}$}.


trace-definition-L-function-l-adic

Now assume that $\mathcal{F}$ is a $\mathbf{Q}_\ell$-sheaf on $X$. In this case we define $$ L(X, \mathcal{F}) := \prod\nolimits_{x \in |X|} \det(1 - \pi_x^*T^{\deg x} |_{\mathcal{F}_{\bar x}})^{-1} \in \mathbf{Q}_\ell[[T]]. $$ Note that this product converges since there are finitely many points of a given degree. We call this the {\it $L$-function of $\mathcal{F}$}.


trace-definition-open

A subgroup of the form $\text{Stab}(\overline y\in F_{\overline{x}}(Y))\subset \pi_1(X, \overline{x})$ is called {\it open}.


trace-definition-unramified

An {\it unramified cusp form on $\text{GL}_2(\mathbf{A})$ with values in $\Lambda$}\footnote{This is likely nonstandard notation.} is a function $$ f : \text{GL}_2(\mathbf{A}) \to \Lambda $$ such that \begin{enumerate} \item $f(x\gamma) = f(x)$ for all $x\in \text{GL}_2(\mathbf{A})$ and all $\gamma\in \text{GL}_2(K)$ \item $f(ux) = f(x)$ for all $x\in \text{GL}_2(\mathbf{A})$ and all $u\in \text{GL}_2(O)$ \item for all $x\in \text{GL}_2(\mathbf{A})$, $$ \int_{\mathbf{A} \mod K} f \left(x \left( \begin{matrix} 1 & z \\ 0 & 1 \end{matrix} \right) \right) dz = 0 $$ see \cite[Section 4.1]{dJ-conjecture} for an explanation of how to make sense out of this for a general ring $\Lambda$ in which $p$ is invertible. \end{enumerate}


sheaves-definition-presheaf

Let $X$ be a topological space. \begin{enumerate} \item A {\it presheaf $\mathcal{F}$ of sets on $X$} is a rule which assigns to each open $U \subset X$ a set $\mathcal{F}(U)$ and to each inclusion $V \subset U$ a map $\rho^U_V : \mathcal{F}(U) \to \mathcal{F}(V)$ such that $\rho^U_U = \text{id}_{\mathcal{F}(U)}$ and whenever $W \subset V \subset U$ we have $\rho^U_W = \rho^V_W \circ \rho ^U_V$. \item A {\it morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of presheaves of sets on $X$} is a rule which assigns to each open $U \subset X$ a map of sets $\varphi : \mathcal{F}(U) \to \mathcal{G}(U)$ compatible with restriction maps, i.e., whenever $V \subset U \subset X$ are open the diagram $$ \xymatrix{ \mathcal{F}(U) \ar[r]^\varphi \ar[d]^{\rho^U_V} & \mathcal{G}(U) \ar[d]^{\rho^U_V} \\ \mathcal{F}(V) \ar[r]^\varphi & \mathcal{G}(V) } $$ commutes. \item The category of presheaves of sets on $X$ will be denoted $\textit{PSh}(X)$. \end{enumerate}


sheaves-definition-constant-presheaf

Let $X$ be a topological space. Let $A$ be a set. The {\it constant presheaf with value $A$} is the presheaf that assigns the set $A$ to every open $U \subset X$, and such that all restriction mappings are $\text{id}_A$.


sheaves-definition-abelian-presheaves

Let $X$ be a topological space. \begin{enumerate} \item A {\it presheaf of abelian groups on $X$} or an {\it abelian presheaf over $X$} is a presheaf of sets $\mathcal{F}$ such that for each open $U \subset X$ the set $\mathcal{F}(U)$ is endowed with the structure of an abelian group, and such that all restriction maps $\rho^U_V$ are homomorphisms of abelian groups, see Lemma \ref{lemma-abelian-presheaves} above. \item A {\it morphism of abelian presheaves over $X$} $\varphi : \mathcal{F} \to \mathcal{G}$ is a morphism of presheaves of sets which induces a homomorphism of abelian groups $\mathcal{F}(U) \to \mathcal{G}(U)$ for every open $U \subset X$. \item The category of presheaves of abelian groups on $X$ is denoted $\textit{PAb}(X)$. \end{enumerate}


sheaves-definition-presheaf-values-in-category

Let $X$ be a topological space. Let $\mathcal{C}$ be a category. \begin{enumerate} \item A {\it presheaf $\mathcal{F}$ on $X$ with values in $\mathcal{C}$} is given by a rule which assigns to every open $U \subset X$ an object $\mathcal{F}(U)$ of $\mathcal{C}$ and to each inclusion $V \subset U$ a morphism $\rho_V^U : \mathcal{F}(U) \to \mathcal{F}(V)$ in $\mathcal{C}$ such that whenever $W \subset V \subset U$ we have $\rho_W^U = \rho_W^V \circ \rho_V^U$. \item A {\it morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of presheaves with value in $\mathcal{C}$} is given by a morphism $\varphi : \mathcal{F}(U) \to \mathcal{G}(U)$ in $\mathcal{C}$ compatible with restriction morphisms. \end{enumerate}


sheaves-definition-underlying-presheaf-sets

Let $X$ be a topological space. Let $\mathcal{C}$ be a category. Let $F : \mathcal{C} \to \textit{Sets}$ be a faithful functor. Let $\mathcal{F}$ be a presheaf on $X$ with values in $\mathcal{C}$. The presheaf of sets $U \mapsto F(\mathcal{F}(U))$ is called the {\it underlying presheaf of sets of $\mathcal{F}$}.


sheaves-definition-presheaf-modules

Let $X$ be a topological space, and let $\mathcal{O}$ be a presheaf of rings on $X$. \begin{enumerate} \item A {\it presheaf of $\mathcal{O}$-modules} is given by an abelian presheaf $\mathcal{F}$ together with a map of presheaves of sets $$ \mathcal{O} \times \mathcal{F} \longrightarrow \mathcal{F} $$ such that for every open $U \subset X$ the map $\mathcal{O}(U) \times \mathcal{F}(U) \to \mathcal{F}(U)$ defines the structure of an $\mathcal{O}(U)$-module structure on the abelian group $\mathcal{F}(U)$. \item A {\it morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of presheaves of $\mathcal{O}$-modules} is a morphism of abelian presheaves $\varphi : \mathcal{F} \to \mathcal{G}$ such that the diagram $$ \xymatrix{ \mathcal{O} \times \mathcal{F} \ar[r] \ar[d]_{\text{id} \times \varphi} & \mathcal{F} \ar[d]^{\varphi} \\ \mathcal{O} \times \mathcal{G} \ar[r] & \mathcal{G} } $$ commutes. \item The set of $\mathcal{O}$-module morphisms as above is denoted $\Hom_\mathcal{O}(\mathcal{F}, \mathcal{G})$. \item The category of presheaves of $\mathcal{O}$-modules is denoted $\textit{PMod}(\mathcal{O})$. \end{enumerate}


sheaves-definition-sheaf

Let $X$ be a topological space. \begin{enumerate} \item A {\it sheaf $\mathcal{F}$ of sets on $X$} is a presheaf of sets which satisfies the following additional property: Given any open covering $U = \bigcup_{i \in I} U_i$ and any collection of sections $s_i \in \mathcal{F}(U_i)$, $i \in I$ such that $\forall i, j\in I$ $$ s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j} $$ there exists a unique section $s \in \mathcal{F}(U)$ such that $s_i = s|_{U_i}$ for all $i \in I$. \item A {\it morphism of sheaves of sets} is simply a morphism of presheaves of sets. \item The category of sheaves of sets on $X$ is denoted $\Sh(X)$. \end{enumerate}


sheaves-definition-constant-sheaf

Let $X$ be a topological space. Let $A$ be a set. The {\it constant sheaf with value $A$} denoted $\underline{A}$, or $\underline{A}_X$ is the sheaf that assigns to an open $U \subset X$ the set of all locally constant maps $U \to A$ with restriction mappings given by restrictions of functions.


sheaves-definition-abelian-sheaf

Let $X$ be a topological space. \begin{enumerate} \item An {\it abelian sheaf on $X$} or {\it sheaf of abelian groups on $X$} is an abelian presheaf on $X$ such that the underlying presheaf of sets is a sheaf. \item The category of sheaves of abelian groups is denoted $\textit{Ab}(X)$. \end{enumerate}


sheaves-definition-sheaf-values-in-category

Let $X$ be a topological space. Let $\mathcal{C}$ be a category with products. A presheaf $\mathcal{F}$ with values in $\mathcal{C}$ on $X$ is a {\it sheaf} if for every open covering the diagram $$ \xymatrix{ \mathcal{F}(U) \ar[r] & \prod\nolimits_{i\in I} \mathcal{F}(U_i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod\nolimits_{(i_0, i_1) \in I \times I} \mathcal{F}(U_{i_0} \cap U_{i_1}) } $$ is an equalizer diagram in the category $\mathcal{C}$.


sheaves-definition-sheaf-modules

Let $X$ be a topological space. Let $\mathcal{O}$ be a sheaf of rings on $X$. \begin{enumerate} \item A {\it sheaf of $\mathcal{O}$-modules} is a presheaf of $\mathcal{O}$-modules $\mathcal{F}$, see Definition \ref{definition-presheaf-modules}, such that the underlying presheaf of abelian groups $\mathcal{F}$ is a sheaf. \item A {\it morphism of sheaves of $\mathcal{O}$-modules} is a morphism of presheaves of $\mathcal{O}$-modules. \item Given sheaves of $\mathcal{O}$-modules $\mathcal{F}$ and $\mathcal{G}$ we denote $\Hom_\mathcal{O}(\mathcal{F}, \mathcal{G})$ the set of morphism of sheaves of $\mathcal{O}$-modules. \item The category of sheaves of $\mathcal{O}$-modules is denoted $\textit{Mod}(\mathcal{O})$. \end{enumerate}


sheaves-definition-separated

Let $X$ be a topological space. A presheaf of sets $\mathcal{F}$ on $X$ is {\it separated} if for every open $U \subset X$ the map $\mathcal{F}(U) \to \prod_{x \in U} \mathcal{F}_x$ is injective.


sheaves-definition-algebraic-structure

A {\it type of algebraic structure} is given by a category $\mathcal{C}$ and a functor $F : \mathcal{C} \to \textit{Sets}$ with the following properties \begin{enumerate} \item $F$ is faithful, \item $\mathcal{C}$ has limits and $F$ commutes with limits, \item $\mathcal{C}$ has filtered colimits and $F$ commutes with them, and \item $F$ reflects isomorphisms. \end{enumerate}


sheaves-definition-injective-surjective

Let $X$ be a topological space. \begin{enumerate} \item A presheaf $\mathcal{F}$ is called a {\it subpresheaf} of a presheaf $\mathcal{G}$ if $\mathcal{F}(U) \subset \mathcal{G}(U)$ for all open $U \subset X$ such that the restriction maps of $\mathcal{G}$ induce the restriction maps of $\mathcal{F}$. If $\mathcal{F}$ and $\mathcal{G}$ are sheaves, then $\mathcal{F}$ is called a {\it subsheaf} of $\mathcal{G}$. We sometimes indicate this by the notation $\mathcal{F} \subset \mathcal{G}$. \item A morphism of presheaves of sets $\varphi : \mathcal{F} \to \mathcal{G}$ on $X$ is called {\it injective} if and only if $\mathcal{F}(U) \to \mathcal{G}(U)$ is injective for all $U$ open in $X$. \item A morphism of presheaves of sets $\varphi : \mathcal{F} \to \mathcal{G}$ on $X$ is called {\it surjective} if and only if $\mathcal{F}(U) \to \mathcal{G}(U)$ is surjective for all $U$ open in $X$. \item A morphism of sheaves of sets $\varphi : \mathcal{F} \to \mathcal{G}$ on $X$ is called {\it injective} if and only if $\mathcal{F}(U) \to \mathcal{G}(U)$ is injective for all $U$ open in $X$. \item A morphism of sheaves of sets $\varphi : \mathcal{F} \to \mathcal{G}$ on $X$ is called {\it surjective} if and only if for every open $U$ of $X$ and every section $s$ of $\mathcal{G}(U)$ there exists an open covering $U = \bigcup U_i$ such that $s|_{U_i}$ is in the image of $\mathcal{F}(U_i) \to \mathcal{G}(U_i)$ for all $i$. \end{enumerate}


sheaves-definition-f-map

Let $f : X \to Y$ be a continuous map. Let $\mathcal{F}$ be a sheaf of sets on $X$ and let $\mathcal{G}$ be a sheaf of sets on $Y$. An {\it $f$-map $\xi : \mathcal{G} \to \mathcal{F}$} is a collection of maps $\xi_V : \mathcal{G}(V) \to \mathcal{F}(f^{-1}(V))$ indexed by open subsets $V \subset Y$ such that $$ \xymatrix{ \mathcal{G}(V) \ar[r]_{\xi_V} \ar[d]_{\text{restriction of }\mathcal{G}} & \mathcal{F}(f^{-1}V) \ar[d]^{\text{restriction of }\mathcal{F}} \\ \mathcal{G}(V') \ar[r]^{\xi_{V'}} & \mathcal{F}(f^{-1}V') } $$ commutes for all $V' \subset V \subset Y$ open.


sheaves-definition-composition-f-maps

Suppose that $f : X \to Y$ and $g : Y \to Z$ are continuous maps of topological spaces. Suppose that $\mathcal{F}$ is a sheaf on $X$, $\mathcal{G}$ is a sheaf on $Y$, and $\mathcal{H}$ is a sheaf on $Z$. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be an $f$-map. Let $\psi : \mathcal{H} \to \mathcal{G}$ be an $g$-map. The {\it composition of $\varphi$ and $\psi$} is the $(g \circ f)$-map $\varphi \circ \psi$ defined by the commutativity of the diagrams $$ \xymatrix{ \mathcal{H}(W) \ar[rr]_{(\varphi \circ \psi)_W} \ar[rd]_{\psi_W} & & \mathcal{F}(f^{-1}g^{-1}W) \\ & \mathcal{G}(g^{-1}W) \ar[ru]_{\varphi_{g^{-1}W}} } $$


sheaves-definition-ringed-space

A {\it ringed space} is a pair $(X, \mathcal{O}_X)$ consisting of a topological space $X$ and a sheaf of rings $\mathcal{O}_X$ on $X$. A {\it morphism of ringed spaces} $(X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ is a pair consisting of a continuous map $f : X \to Y$ and an $f$-map of sheaves of rings $f^\sharp : \mathcal{O}_Y \to \mathcal{O}_X$.


sheaves-definition-composition-maps-ringed-spaces

Let $(f, f^\sharp) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ and $(g, g^\sharp) : (Y, \mathcal{O}_Y) \to (Z, \mathcal{O}_Z)$ be morphisms of ringed spaces. Then we define the {\it composition of morphisms of ringed spaces} by the rule $$ (g, g^\sharp) \circ (f, f^\sharp) = (g \circ f, f^\sharp \circ g^\sharp). $$ Here we use composition of $f$-maps defined in Definition \ref{definition-composition-f-maps}.


sheaves-definition-pushforward

Let $(f, f^\sharp) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. \begin{enumerate} \item Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. We define the {\it pushforward} of $\mathcal{F}$ as the sheaf of $\mathcal{O}_Y$-modules which as a sheaf of abelian groups equals $f_*\mathcal{F}$ and with module structure given by the restriction via $f^\sharp : \mathcal{O}_Y \to f_*\mathcal{O}_X$ of the module structure given in Lemma \ref{lemma-pushforward-module}. \item Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_Y$-modules. We define the {\it pullback} $f^*\mathcal{G}$ to be the sheaf of $\mathcal{O}_X$-modules defined by the formula $$ f^*\mathcal{G} = \mathcal{O}_X \otimes_{f^{-1}\mathcal{O}_Y} f^{-1}\mathcal{G} $$ where the ring map $f^{-1}\mathcal{O}_Y \to \mathcal{O}_X$ is the map corresponding to $f^\sharp$, and where the  module structure is given by Lemma \ref{lemma-pullback-module}. \end{enumerate}


sheaves-definition-skyscraper-sheaf

Let $X$ be a topological space. \begin{enumerate} \item Let $x \in X$ be a point. Denote $i_x : \{x\} \to X$ the inclusion map. Let $A$ be a set and think of $A$ as a sheaf on the one point space $\{x\}$. We call $i_{x, *}A$ the {\it skyscraper sheaf at $x$ with value $A$}. \item If in (1) above $A$ is an abelian group then we think of $i_{x, *}A$ as a sheaf of abelian groups on $X$. \item If in (1) above $A$ is an algebraic structure then we think of $i_{x, *}A$ as a sheaf of algebraic structures. \item If $(X, \mathcal{O}_X)$ is a ringed space, then we think of $i_x : \{x\} \to X$ as a morphism of ringed spaces $(\{x\}, \mathcal{O}_{X, x}) \to (X, \mathcal{O}_X)$ and if $A$ is a $\mathcal{O}_{X, x}$-module, then we think of $i_{x, *}A$ as a sheaf of $\mathcal{O}_X$-modules. \item We say a sheaf of sets $\mathcal{F}$ is a {\it skyscraper sheaf} if there exists an point $x$ of $X$ and a set $A$ such that $\mathcal{F} \cong i_{x, *}A$. \item We say a sheaf of abelian groups $\mathcal{F}$ is a {\it skyscraper sheaf} if there exists an point $x$ of $X$ and an abelian group $A$ such that $\mathcal{F} \cong i_{x, *}A$ as sheaves of abelian groups. \item We say a sheaf of algebraic structures $\mathcal{F}$ is a {\it skyscraper sheaf} if there exists an point $x$ of $X$ and an algebraic structure $A$ such that $\mathcal{F} \cong i_{x, *}A$ as sheaves of algebraic structures. \item If $(X, \mathcal{O}_X)$ is a ringed space and $\mathcal{F}$ is a sheaf of $\mathcal{O}_X$-modules, then we say $\mathcal{F}$ is a {\it skyscraper sheaf} if there exists a point $x \in X$ and a $\mathcal{O}_{X, x}$-module $A$ such that $\mathcal{F} \cong i_{x, *}A$ as sheaves of $\mathcal{O}_X$-modules. \end{enumerate}


sheaves-definition-presheaf-basis

Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology on $X$. \begin{enumerate} \item A {\it presheaf $\mathcal{F}$ of sets on $\mathcal{B}$} is a rule which assigns to each $U \in \mathcal{B}$ a set $\mathcal{F}(U)$ and to each inclusion $V \subset U$ of elements of $\mathcal{B}$ a map $\rho^U_V : \mathcal{F}(U) \to \mathcal{F}(V)$ such that $\rho^U_U = \text{id}_{\mathcal{F}(U)}$ for all $U \in \mathcal{B}$ whenever $W \subset V \subset U$ in $\mathcal{B}$ we have $\rho^U_W = \rho^V_W \circ \rho ^U_V$. \item A {\it morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of presheaves of sets on $\mathcal{B}$} is a rule which assigns to each element $U \in \mathcal{B}$ a map of sets $\varphi : \mathcal{F}(U) \to \mathcal{G}(U)$ compatible with restriction maps. \end{enumerate}


sheaves-definition-sheaf-basis

Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology on $X$. \begin{enumerate} \item A {\it sheaf $\mathcal{F}$ of sets on $\mathcal{B}$} is a presheaf of sets on $\mathcal{B}$ which satisfies the following additional property: Given any $U \in \mathcal{B}$, and any covering $U = \bigcup_{i \in I} U_i$ with $U_i \in \mathcal{B}$, and any coverings $U_i \cap U_j = \bigcup_{k \in I_{ij}} U_{ijk}$ with $U_{ijk} \in \mathcal{B}$ the sheaf condition holds: \begin{itemize} \item[(**)] For any collection of sections $s_i \in \mathcal{F}(U_i)$, $i \in I$ such that $\forall i, j\in I$, $\forall k\in I_{ij}$ $$ s_i|_{U_{ijk}} = s_j|_{U_{ijk}} $$ there exists a unique section $s \in \mathcal{F}(U)$ such that $s_i = s|_{U_i}$ for all $i \in I$. \end{itemize} \item A {\it morphism of sheaves of sets on $\mathcal{B}$} is simply a morphism of presheaves of sets. \end{enumerate}


sheaves-definition-sheaf-structures-basis

Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology on $X$. Let $(\mathcal{C}, F)$ be a type of algebraic structure. \begin{enumerate} \item A {\it presheaf $\mathcal{F}$ with values in $\mathcal{C}$ on $\mathcal{B}$} is a rule which assigns to each $U \in \mathcal{B}$ an object $\mathcal{F}(U)$ of $\mathcal{C}$ and to each inclusion $V \subset U$ of elements of $\mathcal{B}$ a morphism $\rho^U_V : \mathcal{F}(U) \to \mathcal{F}(V)$ in $\mathcal{C}$ such that $\rho^U_U = \text{id}_{\mathcal{F}(U)}$ for all $U \in \mathcal{B}$ and whenever $W \subset V \subset U$ in $\mathcal{B}$ we have $\rho^U_W = \rho^V_W \circ \rho ^U_V$. \item A {\it morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of presheaves with values in $\mathcal{C}$ on $\mathcal{B}$} is a rule which assigns to each element $U \in \mathcal{B}$ a morphism of algebraic structures $\varphi : \mathcal{F}(U) \to \mathcal{G}(U)$ compatible with restriction maps. \item Given a presheaf $\mathcal{F}$ with values in $\mathcal{C}$ on $\mathcal{B}$ we say that $U \mapsto F(\mathcal{F}(U))$ is the underlying presheaf of sets. \item A {\it sheaf $\mathcal{F}$ with values in $\mathcal{C}$ on $\mathcal{B}$} is a presheaf with values in $\mathcal{C}$ on $\mathcal{B}$ whose underlying presheaf of sets is a sheaf. \end{enumerate}


sheaves-definition-sheaf-modules-basis

Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology on $X$. Let $\mathcal{O}$ be a presheaf of rings on $\mathcal{B}$. \begin{enumerate} \item A {\it presheaf of $\mathcal{O}$-modules $\mathcal{F}$ on $\mathcal{B}$} is a presheaf of abelian groups on $\mathcal{B}$ together with a morphism of presheaves of sets $\mathcal{O} \times \mathcal{F} \to \mathcal{F}$ such that for all $U \in \mathcal{B}$ the map $\mathcal{O}(U) \times \mathcal{F}(U) \to \mathcal{F}(U)$ turns the group $\mathcal{F}(U)$ into an $\mathcal{O}(U)$-module. \item A {\it morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of presheaves of $\mathcal{O}$-modules on $\mathcal{B}$} is a morphism of abelian presheaves on $\mathcal{B}$ which induces an $\mathcal{O}(U)$-module homomorphism $\mathcal{F}(U) \to \mathcal{G}(U)$ for every $U \in \mathcal{B}$. \item Suppose that $\mathcal{O}$ is a sheaf of rings on $\mathcal{B}$. A {\it sheaf $\mathcal{F}$ of $\mathcal{O}$-modules on $\mathcal{B}$} is a presheaf of $\mathcal{O}$-modules on $\mathcal{B}$ whose underlying presheaf of abelian groups is a sheaf. \end{enumerate}


sheaves-definition-restriction

Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset. \begin{enumerate} \item Let $\mathcal{G}$ be a presheaf of sets, abelian groups or algebraic structures on $X$. The presheaf $j_p\mathcal{G}$ described in Lemma \ref{lemma-j-pullback} is called the {\it restriction of $\mathcal{G}$ to $U$} and denoted $\mathcal{G}|_U$. \item Let $\mathcal{G}$ be a sheaf of sets on $X$, abelian groups or algebraic structures on $X$. The sheaf $j^{-1}\mathcal{G}$ is called the {\it restriction of $\mathcal{G}$ to $U$} and denoted $\mathcal{G}|_U$. \item If $(X, \mathcal{O})$ is a ringed space, then the pair $(U, \mathcal{O}|_U)$ is called the {\it open subspace of $(X, \mathcal{O})$ associated to $U$}. \item If $\mathcal{G}$ is a presheaf of $\mathcal{O}$-modules then $\mathcal{G}|_U$ together with the multiplication map $\mathcal{O}|_U \times \mathcal{G}|_U \to \mathcal{G}|_U$ (see Lemma \ref{lemma-pullback-module}) is called the {\it restriction of $\mathcal{G}$ to $U$}. \end{enumerate}


sheaves-definition-j-shriek

Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset. \begin{enumerate} \item Let $\mathcal{F}$ be a presheaf of sets on $U$. We define the {\it extension of $\mathcal{F}$ by the empty set $j_{p!}\mathcal{F}$} to be the presheaf of sets on $X$ defined by the rule $$ j_{p!}\mathcal{F}(V) = \left\{ \begin{matrix} \emptyset & \text{if} & V \not \subset U \\ \mathcal{F}(V) & \text{if} & V \subset U \end{matrix} \right. $$ with obvious restriction mappings. \item Let $\mathcal{F}$ be a sheaf of sets on $U$. We define the {\it extension of $\mathcal{F}$ by the empty set $j_!\mathcal{F}$} to be the sheafification of the presheaf $j_{p!}\mathcal{F}$. \end{enumerate}


sheaves-definition-j-shriek-structures

Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset. \begin{enumerate} \item Let $\mathcal{F}$ be an abelian presheaf on $U$. We define the {\it extension $j_{p!}\mathcal{F}$ of $\mathcal{F}$ by $0$} to be the abelian presheaf on $X$ defined by the rule $$ j_{p!}\mathcal{F}(V) = \left\{ \begin{matrix} 0 & \text{if} & V \not \subset U \\ \mathcal{F}(V) & \text{if} & V \subset U \end{matrix} \right. $$ with obvious restriction mappings. \item Let $\mathcal{F}$ be an abelian sheaf on $U$. We define the {\it extension $j_!\mathcal{F}$ of $\mathcal{F}$ by $0$} to be the sheafification of the abelian presheaf $j_{p!}\mathcal{F}$. \item Let $\mathcal{C}$ be a category having an initial object $e$. Let $\mathcal{F}$ be a presheaf on $U$ with values in $\mathcal{C}$. We define the {\it extension $j_{p!}\mathcal{F}$ of $\mathcal{F}$ by $e$} to be the presheaf on $X$ with values in $\mathcal{C}$ defined by the rule $$ j_{p!}\mathcal{F}(V) = \left\{ \begin{matrix} e & \text{if} & V \not \subset U \\ \mathcal{F}(V) & \text{if} & V \subset U \end{matrix} \right. $$ with obvious restriction mappings. \item Let $(\mathcal{C}, F)$ be a type of algebraic structure such that $\mathcal{C}$ has an initial object $e$. Let $\mathcal{F}$ be a sheaf of algebraic structures on $U$ (of the give type). We define the {\it extension $j_!\mathcal{F}$ of $\mathcal{F}$ by $e$} to be the sheafification of the presheaf $j_{p!}\mathcal{F}$ defined above. \item Let $\mathcal{O}$ be a presheaf of rings on $X$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}|_U$-modules. In this case we define the {\it extension by $0$} to be the presheaf of $\mathcal{O}$-modules which is equal to $j_{p!}\mathcal{F}$ as an abelian presheaf endowed with the multiplication map $\mathcal{O} \times j_{p!}\mathcal{F} \to j_{p!}\mathcal{F}$. \item Let $\mathcal{O}$ be a sheaf of rings on $X$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}|_U$-modules. In this case we define the {\it extension by $0$} to be the $\mathcal{O}$-module which is equal to $j_!\mathcal{F}$ as an abelian sheaf endowed with the multiplication map $\mathcal{O} \times j_!\mathcal{F} \to j_!\mathcal{F}$. \end{enumerate}


descent-definition-descent-datum-quasi-coherent

Let $S$ be a scheme. Let $\{f_i : S_i \to S\}_{i \in I}$ be a family of morphisms with target $S$. \begin{enumerate} \item A {\it descent datum $(\mathcal{F}_i, \varphi_{ij})$ for quasi-coherent sheaves} with respect to the given family is given by a quasi-coherent sheaf $\mathcal{F}_i$ on $S_i$ for each $i \in I$, an isomorphism of quasi-coherent $\mathcal{O}_{S_i \times_S S_j}$-modules $\varphi_{ij} : \text{pr}_0^*\mathcal{F}_i \to \text{pr}_1^*\mathcal{F}_j$ for each pair $(i, j) \in I^2$ such that for every triple of indices $(i, j, k) \in I^3$ the diagram $$ \xymatrix{ \text{pr}_0^*\mathcal{F}_i \ar[rd]_{\text{pr}_{01}^*\varphi_{ij}} \ar[rr]_{\text{pr}_{02}^*\varphi_{ik}} & & \text{pr}_2^*\mathcal{F}_k \\ & \text{pr}_1^*\mathcal{F}_j \ar[ru]_{\text{pr}_{12}^*\varphi_{jk}} & } $$ of $\mathcal{O}_{S_i \times_S S_j \times_S S_k}$-modules commutes. This is called the {\it cocycle condition}. \item A {\it morphism $\psi : (\mathcal{F}_i, \varphi_{ij}) \to (\mathcal{F}'_i, \varphi'_{ij})$ of descent data} is given by a family $\psi = (\psi_i)_{i\in I}$ of morphisms of $\mathcal{O}_{S_i}$-modules $\psi_i : \mathcal{F}_i \to \mathcal{F}'_i$ such that all the diagrams $$ \xymatrix{ \text{pr}_0^*\mathcal{F}_i \ar[r]_{\varphi_{ij}} \ar[d]_{\text{pr}_0^*\psi_i} & \text{pr}_1^*\mathcal{F}_j \ar[d]^{\text{pr}_1^*\psi_j} \\ \text{pr}_0^*\mathcal{F}'_i \ar[r]^{\varphi'_{ij}} & \text{pr}_1^*\mathcal{F}'_j \\ } $$ commute. \end{enumerate}


descent-definition-descent-datum-effective-quasi-coherent

Let $S$ be a scheme. Let $\{S_i \to S\}_{i \in I}$ be a family of morphisms with target $S$. \begin{enumerate} \item Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_S$-module. We call the unique descent on $\mathcal{F}$ datum with respect to the covering $\{S \to S\}$ the {\it trivial descent datum}. \item The pullback of the trivial descent datum to $\{S_i \to S\}$ is called the {\it canonical descent datum}. Notation: $(\mathcal{F}|_{S_i}, can)$. \item A descent datum $(\mathcal{F}_i, \varphi_{ij})$ for quasi-coherent sheaves with respect to the given covering is said to be {\it effective} if there exists a quasi-coherent sheaf $\mathcal{F}$ on $S$ such that $(\mathcal{F}_i, \varphi_{ij})$ is isomorphic to $(\mathcal{F}|_{S_i}, can)$. \end{enumerate}


descent-definition-descent-datum-modules

Let $R \to A$ be a ring map. \begin{enumerate} \item A {\it descent datum $(N, \varphi)$ for modules with respect to $R \to A$} is given by an $A$-module $N$ and an isomorphism of $A \otimes_R A$-modules $$ \varphi : N \otimes_R A \to A \otimes_R N $$ such that the {\it cocycle condition} holds: the diagram of $A \otimes_R A \otimes_R A$-module maps $$ \xymatrix{ N \otimes_R A \otimes_R A \ar[rr]_{\varphi_{02}} \ar[rd]_{\varphi_{01}} & & A \otimes_R A \otimes_R N \\ & A \otimes_R N \otimes_R A \ar[ru]_{\varphi_{12}} & } $$ commutes (see below for notation). \item A {\it morphism $(N, \varphi) \to (N', \varphi')$ of descent data} is a morphism of $A$-modules $\psi : N \to N'$ such that the diagram $$ \xymatrix{ N \otimes_R A \ar[r]_\varphi \ar[d]_{\psi \otimes \text{id}_A} & A \otimes_R N \ar[d]^{\text{id}_A \otimes \psi} \\ N' \otimes_R A \ar[r]^{\varphi'} & A \otimes_R N' } $$ is commutative. \end{enumerate}


descent-definition-descent-datum-effective-module

Let $R \to A$ be a ring map. We say a descent datum $(N, \varphi)$ is {\it effective} if there exists an $R$-module $M$ and an isomorphism of descent data from $(M \otimes_R A, can)$ to $(N, \varphi)$.


descent-definition-split-equalizer

A {\it split equalizer} is a diagram (\ref{equation-equalizer}) with $g_1 \circ f = g_2 \circ f$ for which there exist auxiliary morphisms $h : B \to A$ and $i : C \to B$ such that \begin{equation} \label{equation-split-equalizer-conditions} h \circ f = 1_A, \quad f \circ h = i \circ g_1, \quad i \circ g_2 = 1_B. \end{equation}


descent-definition-universally-injective

A ring map $f: R \to S$ is {\it universally injective} if it is universally injective as a morphism in $\text{Mod}_R$.


descent-definition-C

Let $R$ be a ring. Define the contravariant functor {\it $C$} $ : \text{Mod}_R \to \text{Mod}_R$ by setting $$ C(M) = \Hom_{\textit{Ab}}(M, \mathbf{Q}/\mathbf{Z}), $$ with the $R$-action on $C(M)$ given by $rf(s) = f(rs)$.


descent-definition-effective-descent

The functor $f^*: \text{Mod}_R \to DD_{S/R}$ is called {\it base extension along $f$}. We say that $f$ is a {\it descent morphism for modules} if $f^*$ is fully faithful. We say that $f$ is an {\it effective descent morphism for modules} if $f^*$ is an equivalence of categories.


descent-definition-pushforward

Define the functor {\it $f_*$} $: DD_{S/R} \to \text{Mod}_R$ by taking $f_*(M, \theta)$ to be the $R$-submodule of $M$ for which the diagram \begin{equation} \label{equation-equalizer-f} \xymatrix@C=8pc{f_*(M,\theta) \ar[r] & M \ar@<1ex>^{\theta \circ (1_M \otimes  \delta_0^1)}[r] \ar@<-1ex>_{1_M \otimes \delta_1^1}[r] &  M \otimes_{S, \delta_1^1} S_2  } \end{equation} is an equalizer.


descent-definition-structure-sheaf

Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$. Let $S$ be a scheme. Let $\Sch_\tau$ be a big site containing $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_S$-module. \begin{enumerate} \item The {\it structure sheaf of the big site $(\Sch/S)_\tau$} is the sheaf of rings $T/S \mapsto \Gamma(T, \mathcal{O}_T)$ which is denoted $\mathcal{O}$ or $\mathcal{O}_S$. \item If $\tau = \etale$ the structure sheaf of the small site $S_\etale$ is the sheaf of rings $T/S \mapsto \Gamma(T, \mathcal{O}_T)$ which is denoted $\mathcal{O}$ or $\mathcal{O}_S$. \item The {\it sheaf of $\mathcal{O}$-modules associated to $\mathcal{F}$} on the big site $(\Sch/S)_\tau$ is the sheaf of $\mathcal{O}$-modules $(f : T \to S) \mapsto \Gamma(T, f^*\mathcal{F})$ which is denoted $\mathcal{F}^a$ (and often simply $\mathcal{F}$). \item Let $\tau = \etale$ (resp.\ $\tau = Zariski$). The {\it sheaf of $\mathcal{O}$-modules associated to $\mathcal{F}$} on the small site $S_\etale$ (resp.\ $S_{Zar}$) is the sheaf of $\mathcal{O}$-modules $(f : T \to S) \mapsto \Gamma(T, f^*\mathcal{F})$ which is denoted $\mathcal{F}^a$ (and often simply $\mathcal{F}$). \end{enumerate}


descent-definition-parasitic

Let $S$ be a scheme. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}$-modules on $(\Sch/S)_\tau$. \begin{enumerate} \item $\mathcal{F}$ is called {\it parasitic}\footnote{This may be nonstandard notation.} if for every flat morphism $U \to S$ we have $\mathcal{F}(U) = 0$. \item $\mathcal{F}$ is called {\it parasitic for the $\tau$-topology} if for every $\tau$-covering $\{U_i \to S\}_{i \in I}$ we have $\mathcal{F}(U_i) = 0$ for all $i$. \end{enumerate}


descent-definition-property-local

Let $\mathcal{P}$ be a property of schemes. Let $\tau \in \{fpqc, \linebreak[0] fppf, \linebreak[0] syntomic, \linebreak[0] smooth, \linebreak[0] \etale, \linebreak[0] Zariski\}$. We say $\mathcal{P}$ is {\it local in the $\tau$-topology} if for any $\tau$-covering $\{S_i \to S\}_{i \in I}$ (see Topologies, Section \ref{topologies-section-procedure}) we have $$ S \text{ has }\mathcal{P} \Leftrightarrow \text{each }S_i \text{ has }\mathcal{P}. $$


descent-definition-germs

Germs of schemes. \begin{enumerate} \item A pair $(X, x)$ consisting of a scheme $X$ and a point $x \in X$ is called the {\it germ of $X$ at $x$}. \item A {\it morphism of germs} $f : (X, x) \to (S, s)$ is an equivalence class of morphisms of schemes $f : U \to S$ with $f(x) = s$ where $U \subset X$ is an open neighbourhood of $x$. Two such $f$, $f'$ are said to be equivalent if and only if $f$ and $f'$ agree in some open neighbourhood of $x$. \item We define the {\it composition of morphisms of germs} by composing representatives (this is well defined). \end{enumerate}


descent-definition-etale-morphism-germs

Let $f : (X, x) \to (S, s)$ be a morphism of germs. We say $f$ is {\it \'etale} (resp.\ {\it smooth}) if there exists a representative $f : U \to S$ of $f$ which is an \'etale morphism (resp.\ a smooth morphism) of schemes.


descent-definition-local-at-point

Let $\mathcal{P}$ be a property of germs of schemes. We say that $\mathcal{P}$ is {\it \'etale local} (resp.\ {\it smooth local}) if for any \'etale (resp.\ smooth) morphism of germs $(U', u') \to (U, u)$ we have $\mathcal{P}(U, u) \Leftrightarrow \mathcal{P}(U', u')$.


descent-definition-property-morphisms-local

Let $\mathcal{P}$ be a property of morphisms of schemes over a base. Let $\tau \in \{fpqc, fppf, syntomic, smooth, \etale, Zariski\}$. We say $\mathcal{P}$ is {\it $\tau$ local on the base}, or {\it $\tau$ local on the target}, or {\it local on the base for the $\tau$-topology} if for any $\tau$-covering $\{Y_i \to Y\}_{i \in I}$ (see Topologies, Section \ref{topologies-section-procedure}) and any morphism of schemes $f : X \to Y$ over $S$ we have $$ f \text{ has }\mathcal{P} \Leftrightarrow \text{each }Y_i \times_Y X \to Y_i\text{ has }\mathcal{P}. $$


descent-definition-property-morphisms-local-source

Let $\mathcal{P}$ be a property of morphisms of schemes. Let $\tau \in \{Zariski, \linebreak[0] fpqc, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$. We say $\mathcal{P}$ is {\it $\tau$ local on the source}, or {\it local on the source for the $\tau$-topology} if for any morphism of schemes $f : X \to Y$ over $S$, and any $\tau$-covering $\{X_i \to X\}_{i \in I}$ we have $$ f \text{ has }\mathcal{P} \Leftrightarrow \text{each }X_i \to Y\text{ has }\mathcal{P}. $$


descent-definition-local-source-target

Let $\mathcal{P}$ be a property of morphisms of schemes. We say $\mathcal{P}$ is {\it \'etale local on source-and-target} if \begin{enumerate} \item (stable under precomposing with \'etale maps) if $f : X \to Y$ is \'etale and $g : Y \to Z$ has $\mathcal{P}$, then $g \circ f$ has $\mathcal{P}$, \item (stable under \'etale base change) if $f : X \to Y$ has $\mathcal{P}$ and $Y' \to Y$ is \'etale, then the base change $f' : Y' \times_Y X \to Y'$ has $\mathcal{P}$, and \item (locality) given a morphism $f : X \to Y$ the following are equivalent \begin{enumerate} \item $f$ has $\mathcal{P}$, \item for every $x \in X$ there exists a commutative diagram $$ \xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y } $$ with \'etale vertical arrows and $u \in U$ with $a(u) = x$ such that $h$ has $\mathcal{P}$. \end{enumerate} \end{enumerate}


descent-definition-local-source-target-at-point

Let $\mathcal{Q}$ be a property of morphisms of germs of schemes. We say $\mathcal{Q}$ is  {\it \'etale local on the source-and-target} if for any commutative diagram $$ \xymatrix{ (U', u') \ar[d]_a \ar[r]_{h'} & (V', v') \ar[d]^b \\ (U, u) \ar[r]^h & (V, v) } $$ of germs with \'etale vertical arrows we have $\mathcal{Q}(h) \Leftrightarrow \mathcal{Q}(h')$.


descent-definition-descent-datum

Let $f : X \to S$ be a morphism of schemes. \begin{enumerate} \item Let $V \to X$ be a scheme over $X$. A {\it descent datum for $V/X/S$} is an isomorphism $\varphi : V \times_S X \to X \times_S V$ of schemes over $X \times_S X$ satisfying the {\it cocycle condition} that the diagram $$ \xymatrix{ V \times_S X \times_S X \ar[rd]^{\varphi_{01}} \ar[rr]_{\varphi_{02}} & & X \times_S X \times_S V\\ & X \times_S V \times_S X \ar[ru]^{\varphi_{12}} } $$ commutes (with obvious notation). \item We also say that the pair $(V/X, \varphi)$ is a {\it descent datum relative to $X \to S$}. \item A {\it morphism $f : (V/X, \varphi) \to (V'/X, \varphi')$ of descent data relative to $X \to S$} is a morphism $f : V \to V'$ of schemes over $X$ such that the diagram $$ \xymatrix{ V \times_S X \ar[r]_{\varphi} \ar[d]_{f \times \text{id}_X} & X \times_S V \ar[d]^{\text{id}_X \times f} \\ V' \times_S X \ar[r]^{\varphi'} & X \times_S V' } $$ commutes. \end{enumerate}


descent-definition-descent-datum-for-family-of-morphisms

Let $S$ be a scheme. Let $\{X_i \to S\}_{i \in I}$ be a family of morphisms with target $S$. \begin{enumerate} \item A {\it descent datum $(V_i, \varphi_{ij})$ relative to the family $\{X_i \to S\}$} is given by a scheme $V_i$ over $X_i$ for each $i \in I$, an isomorphism $\varphi_{ij} : V_i \times_S X_j \to X_i \times_S V_j$ of schemes over $X_i \times_S X_j$ for each pair $(i, j) \in I^2$ such that for every triple of indices $(i, j, k) \in I^3$ the diagram $$ \xymatrix{ V_i \times_S X_j \times_S X_k \ar[rd]^{\text{pr}_{01}^*\varphi_{ij}} \ar[rr]_{\text{pr}_{02}^*\varphi_{ik}} & & X_i \times_S X_j \times_S V_k\\ & X_i \times_S V_j \times_S X_k \ar[ru]^{\text{pr}_{12}^*\varphi_{jk}} } $$ of schemes over $X_i \times_S X_j \times_S X_k$ commutes (with obvious notation). \item A {\it morphism $\psi : (V_i, \varphi_{ij}) \to (V'_i, \varphi'_{ij})$ of descent data} is given by a family $\psi = (\psi_i)_{i \in I}$ of morphisms of $X_i$-schemes $\psi_i : V_i \to V'_i$ such that all the diagrams $$ \xymatrix{ V_i \times_S X_j \ar[r]_{\varphi_{ij}} \ar[d]_{\psi_i \times \text{id}} & X_i \times_S V_j \ar[d]^{\text{id} \times \psi_j} \\ V'_i \times_S X_j \ar[r]^{\varphi'_{ij}} & X_i \times_S V'_j } $$ commute. \end{enumerate}


descent-definition-pullback-functor

With $S, S', X, X', f, a, a', h$ as in Lemma \ref{lemma-pullback} the functor $$ (V, \varphi) \longmapsto f^*(V, \varphi) $$ constructed in that lemma is called the {\it pullback functor} on descent data.


descent-definition-pullback-functor-family

With $\mathcal{U} = \{U_i \to S'\}_{i \in I}$, $\mathcal{V} = \{V_j \to S\}_{j \in J}$, $\alpha : I \to J$, $h : S' \to S$, and $g_i : U_i \to V_{\alpha(i)}$ as in Lemma \ref{lemma-pullback-family} the functor $$ (Y_j, \varphi_{jj'}) \longmapsto (g_i^*Y_{\alpha(i)}, (g_i \times g_{i'})^*\varphi_{\alpha(i)\alpha(i')}) $$ constructed in that lemma is called the {\it pullback functor} on descent data.


descent-definition-effective

Let $S$ be a scheme. Let $f : X \to S$ be a morphism of schemes. \begin{enumerate} \item  Given a scheme $U$ over $S$ we have the {\it trivial descent datum} of $U$ relative to $\text{id} : S \to S$, namely the identity morphism on $U$. \item By Lemma \ref{lemma-pullback} we get a {\it canonical descent datum} on $X \times_S U$ relative to $X \to S$ by pulling back the trivial descent datum via $f$. We often denote $(X \times_S U, can)$ this descent datum. \item A descent datum $(V, \varphi)$ relative to $X/S$ is called {\it effective} if $(V, \varphi)$ is isomorphic to the canonical descent datum $(X \times_S U, can)$ for some scheme $U$ over $S$. \end{enumerate}


descent-definition-effective-family

Let $S$ be a scheme. Let $\{X_i \to S\}$ be a family of morphisms with target $S$. \begin{enumerate} \item  Given a scheme $U$ over $S$ we have a {\it canonical descent datum} on the family of schemes $X_i \times_S U$ by pulling back the trivial descent datum for $U$ relative to $\{\text{id} : S \to S\}$. We denote this descent datum $(X_i \times_S U, can)$. \item A descent datum $(V_i, \varphi_{ij})$ relative to $\{X_i \to S\}$ is called {\it effective} if there exists a scheme $U$ over $S$ such that $(V_i, \varphi_{ij})$ is isomorphic to $(X_i \times_S U, can)$. \end{enumerate}


descent-definition-descending-types-morphisms

Let $\mathcal{P}$ be a property of morphisms of schemes over a base. Let $\tau \in \{Zariski, fpqc, fppf, \etale, smooth, syntomic\}$. We say {\it morphisms of type $\mathcal{P}$ satisfy descent for $\tau$-coverings} if for any $\tau$-covering $\mathcal{U} : \{U_i \to S\}_{i \in I}$ (see Topologies, Section \ref{topologies-section-procedure}), any descent datum $(X_i, \varphi_{ij})$ relative to $\mathcal{U}$ such that each morphism $X_i \to U_i$ has property $\mathcal{P}$ is effective.


simplicial-definition-face-degeneracy

For any integer $n\geq 1$, and any $0\leq j \leq n$ we let {\it $\delta^n_j : [n-1] \to [n]$} denote the injective order preserving map skipping $j$. For any integer $n\geq 0$, and any $0\leq j \leq n$ we denote {\it $\sigma^n_j : [n + 1]  \to [n]$} the surjective order preserving map with $(\sigma^n_j)^{-1}(\{j\}) = \{j, j + 1\}$.


simplicial-definition-simplicial-object

Let $\mathcal{C}$ be a category. \begin{enumerate} \item A {\it simplicial object $U$ of $\mathcal{C}$} is a contravariant functor $U$ from $\Delta$ to $\mathcal{C}$, in a formula: $$ U : \Delta^{opp} \longrightarrow \mathcal{C} $$ \item If $\mathcal{C}$ is the category of sets, then we call $U$ a {\it simplicial set}. \item If $\mathcal{C}$ is the category of abelian groups, then we call $U$ a {\it simplicial abelian group}. \item A {\it morphism of simplicial objects $U \to U'$} is a transformation of functors. \item The {\it category of simplicial objects of $\mathcal{C}$} is denoted $\text{Simp}(\mathcal{C})$. \end{enumerate}


simplicial-definition-cosimplicial-object

Let $\mathcal{C}$ be a category. \begin{enumerate} \item A {\it cosimplicial object $U$ of $\mathcal{C}$} is a covariant functor $U$ from $\Delta$ to $\mathcal{C}$, in a formula: $$ U : \Delta \longrightarrow \mathcal{C} $$ \item If $\mathcal{C}$ is the category of sets, then we call $U$ a {\it cosimplicial set}. \item If $\mathcal{C}$ is the category of abelian groups, then we call $U$ a {\it cosimplicial abelian group}. \item A {\it morphism of cosimplicial objects $U \to U'$} is a transformation of functors. \item The {\it category of cosimplicial objects of $\mathcal{C}$} is denoted $\text{CoSimp}(\mathcal{C})$. \end{enumerate}


simplicial-definition-product

Let $\mathcal{C}$ be a category. Let $U$ and $V$ be simplicial objects of $\mathcal{C}$. Assume the products $U_n \times V_n$ exist in $\mathcal{C}$. The {\it product of $U$ and $V$} is the simplicial object $U \times V$ defined as follows: \begin{enumerate} \item $(U \times V)_n = U_n \times V_n$, \item $d^n_i = (d^n_i, d^n_i)$, and \item $s^n_i = (s^n_i, s^n_i)$. \end{enumerate} In other words, $U \times V$ is the product of the presheaves $U$ and $V$ on $\Delta$.


simplicial-definition-fibre-product

Let $\mathcal{C}$ be a category. Let $U, V, W$ be simplicial objects of $\mathcal{C}$. Let $a : V \to U$, $b : W \to U$ be morphisms. Assume the fibre products $V_n \times_{U_n} W_n$ exist in $\mathcal{C}$. The {\it fibre product of $V$ and $W$ over $U$} is the simplicial object $V \times_U W$ defined as follows: \begin{enumerate} \item $(V \times_U W)_n = V_n \times_{U_n} W_n$, \item $d^n_i = (d^n_i, d^n_i)$, and \item $s^n_i = (s^n_i, s^n_i)$. \end{enumerate} In other words, $V \times_U W$ is the fibre product of the presheaves $V$ and $W$ over the presheaf $U$ on $\Delta$.


simplicial-definition-push-out

Let $\mathcal{C}$ be a category. Let $U, V, W$ be simplicial objects of $\mathcal{C}$. Let $a : U \to V$, $b : U \to W$ be morphisms. Assume the pushouts $V_n \amalg_{U_n} W_n$ exist in $\mathcal{C}$. The {\it pushout of $V$ and $W$ over $U$} is the simplicial object $V\amalg_U W$ defined as follows: \begin{enumerate} \item $(V \amalg_U W)_n = V_n \amalg_{U_n} W_n$, \item $d^n_i = (d^n_i, d^n_i)$, and \item $s^n_i = (s^n_i, s^n_i)$. \end{enumerate} In other words, $V\amalg_U W$ is the pushout of the presheaves $V$ and $W$ over the presheaf $U$ on $\Delta$.


simplicial-definition-product-cosimplicial-objects

Let $\mathcal{C}$ be a category. Let $U$ and $V$ be cosimplicial objects of $\mathcal{C}$. Assume the products $U_n \times V_n$ exist in $\mathcal{C}$. The {\it product of $U$ and $V$} is the cosimplicial object $U \times V$ defined as follows: \begin{enumerate} \item $(U \times V)_n = U_n \times V_n$, \item for any $\varphi : [n] \to [m]$ the map $(U \times V)(\varphi) : U_n \times V_n \to U_m \times V_m$ is the product $U(\varphi) \times V(\varphi)$. \end{enumerate}


simplicial-definition-fibre-product-cosimplicial-objects

Let $\mathcal{C}$ be a category. Let $U, V, W$ be cosimplicial objects of $\mathcal{C}$. Let $a : V \to U$ and $b : W \to U$ be morphisms. Assume the fibre products $V_n \times_{U_n} W_n$ exist in $\mathcal{C}$. The {\it fibre product of $V$ and $W$ over $U$} is the cosimplicial object $V \times_U W$ defined as follows: \begin{enumerate} \item $(V \times_U W)_n = V_n \times_{U_n} W_n$, \item for any $\varphi : [n] \to [m]$ the map $(V \times_U W)(\varphi) : V_n \times_{U_n} W_n \to V_m \times_{U_m} W_m$ is the product $V(\varphi) \times_{U(\varphi)} W(\varphi)$. \end{enumerate}


simplicial-definition-terminology-simplicial-sets

Let $U$ be a simplicial set. We say $x$ is an {\it $n$-simplex of $U$} to signify that $x$ is an element of $U_n$. We say that $y$ is the $j$the {\it face of $x$} to signify that $d^n_jx = y$. We say that $z$ is the $j$th {\it degeneracy of $x$} if $z = s^n_jx$. A simplex is called {\it degenerate} if it is the degeneracy of another simplex.


simplicial-definition-truncated-simplicial-object

An {\it $n$-truncated simplicial object of $\mathcal{C}$} is a contravariant functor from $\Delta_{\leq n}$ to $\mathcal{C}$. A {\it morphism of $n$-truncated simplicial objects} is a transformation of functors. We denote the category of $n$-truncated simplicial objects of $\mathcal{C}$ by the symbol $\text{Simp}_n(\mathcal{C})$.


simplicial-definition-product-with-simplicial-set

Let $\mathcal{C}$ be a category such that the coproduct of any two objects of $\mathcal{C}$ exists. Let $U$ be a simplicial set. Let $V$ be a simplicial object of $\mathcal{C}$. Assume that each $U_n$ is finite nonempty. In this case we define the {\it product $U \times V$ of $U$ and $V$} to be the simplicial object of $\mathcal{C}$ whose $n$th term is the object $$ (U \times V)_n = \coprod\nolimits_{u\in U_n} V_n $$ with maps for $\varphi : [m] \to [n]$ given by the morphism $$ \coprod\nolimits_{u\in U_n} V_n \longrightarrow \coprod\nolimits_{u'\in U_m} V_m $$ which maps the component $V_n$ corresponding to $u$ to the component $V_m$ corresponding to $u' = U(\varphi)(u)$ via the morphism $V(\varphi)$. More loosely, if all of the coproducts displayed above exist (without assuming anything about $\mathcal{C}$) we will say that the {\it product $U \times V$ exists}.


simplicial-definition-hom-deltak-cosimplicial

Let $\mathcal{C}$ be a category with finite products. Let $V$ be a cosimplicial object of $\mathcal{C}$. Let $U$ be a simplicial set such that each $U_n$ is finite nonempty. We define {\it $\Hom(U, V)$} to be the cosimplicial object of $\mathcal{C}$ defined as follows: \begin{enumerate} \item we set $\Hom(U, V)_n = \prod_{u \in U_n} V_n$, in other words the unique object of $\mathcal{C}$ such that its $X$-valued points satisfy $$ \Mor_\mathcal{C}(X, \Hom(U, V)_n) = \text{Map}(U_n, \Mor_\mathcal{C}(X, V_n)) $$ and \item for $\varphi : [m] \to [n]$ we take the map $\Hom(U, V)_m \to \Hom(U, V)_n$ given by $f \mapsto V(\varphi) \circ f \circ U(\varphi)$ on $X$-valued points as above. \end{enumerate}


simplicial-definition-hom-deltak-simplicial

Let $\mathcal{C}$ be a category with finite products. Let $V$ be a simplicial object of $\mathcal{C}$. Let $U$ be a cosimplicial set such that each $U_n$ is finite nonempty. We define {\it $\Hom(U, V)$} to be the simplicial object of $\mathcal{C}$ defined as follows: \begin{enumerate} \item we set $\Hom(U, V)_n = \prod_{u \in U_n} V_n$, in other words the unique object of $\mathcal{C}$ such that its $X$-valued points satisfy $$ \Mor_\mathcal{C}(X, \Hom(U, V)_n) = \text{Map}(U_n, \Mor_\mathcal{C}(X, V_n)) $$ and \item for $\varphi : [m] \to [n]$ we take the map $\Hom(U, V)_n \to \Hom(U, V)_m$ given by $f \mapsto V(\varphi) \circ f \circ U(\varphi)$ on $X$-valued points as above. \end{enumerate}


simplicial-definition-hom-from-simplicial-set

Let $\mathcal{C}$ be a category such that the coproduct of any two objects exists. Let $U$ be a simplicial set, with $U_n$ finite nonempty for all $n \geq 0$. Let $V$ be a simplicial object of $\mathcal{C}$. We denote {\it $\Hom(U, V)$} any simplicial object of $\mathcal{C}$ such that $$ \Mor_{\text{Simp}(\mathcal{C})}(W, \Hom(U, V)) = \Mor_{\text{Simp}(\mathcal{C})}(W \times U, V) $$ functorially in the simplicial object $W$ of $\mathcal{C}$.


simplicial-definition-split

Let $\mathcal{C}$ be a category which admits finite nonempty coproducts. We say a simplicial object $U$ of $\mathcal{C}$ is {\it split} if there exist subobjects $N(U_m)$ of $U_m$, $m \geq 0$ with the property that \begin{equation} \label{equation-splitting} \coprod\nolimits_{\varphi : [n] \to [m]\text{ surjective}} N(U_m) \longrightarrow U_n \end{equation} is an isomorphism for all $n \geq 0$. If $U$ is an $r$-truncated simplicial object of $\mathcal{C}$ then we say $U$ is {\it split} if there exist subobjects $N(U_m)$ of $U_m$, $r \geq m \geq 0$ with the property that (\ref{equation-splitting}) is an isomorphism for $r \geq n \geq 0$.


simplicial-definition-augmentation

Let $\mathcal{C}$ be a category. Let $U$ be a simplicial object of $\mathcal{C}$. An {\it augmentation $\epsilon : U \to X$ of $U$ towards an object $X$ of $\mathcal{C}$} is a morphism from $U$ into the constant simplicial object $X$.


simplicial-definition-eilenberg-maclane

Let $\mathcal{A}$ be an abelian category. Let $A$ be an object of $\mathcal{A}$ and let $k$ be an integer $\geq 0$. The {\it Eilenberg-Maclane object $K(A, k)$} is given by the object $K(A, k) = i_{k!}U$ which is described in Lemma \ref{lemma-eilenberg-maclane-object} above.


simplicial-definition-homotopy

Let $\mathcal{C}$ be a category having finite coproducts. Suppose that $U$ and $V$ are two simplicial objects of $\mathcal{C}$. Let $a, b : U \to V$ be two morphisms. \begin{enumerate} \item We say a morphism $$ h : U \times \Delta[1] \longrightarrow V $$ is a {\it homotopy from $a$ to $b$} if $a = h \circ e_0$ and $b = h \circ e_1$. \item We say the morphisms $a$ and $b$ are {\it homotopic} or are {\it in the same homotopy class} if there exists a sequence of morphisms $a = a_0, a_1, \ldots, a_n = b$ from $U$ to $V$ such that for each $i = 1, \ldots, n$ there either exists a homotopy from $a_{i - 1}$ to $a_i$ or there exists a homotopy from $a_i$ to $a_{i - 1}$. \end{enumerate}


simplicial-definition-homotopy-equivalent

Let $U$ and $V$ be two simplicial objects of a category $\mathcal{C}$. We say a morphism $a : U \to V$ is a {\it homotopy equivalence} if there exists a morphism $b : V \to U$ such that $a \circ b$ is homotopic to $\text{id}_V$ and $b \circ a$ is homotopic to $\text{id}_U$. We say $U$ and $V$ are {\it homotopy equivalent} if there exists a homotopy equivalence $a : U \to V$.


simplicial-definition-homotopy-cosimplicial

Let $\mathcal{C}$ be a category having finite products. Let $U$ and $V$ be two cosimplicial objects of $\mathcal{C}$. Let $a, b : U \to V$ be two morphisms of cosimplicial objects of $\mathcal{C}$. \begin{enumerate} \item We say a morphism $$ h : U \longrightarrow \Hom(\Delta[1], V) $$ such that $a = e_0 \circ h$ and $b = e_1 \circ h$ is a {\it homotopy from $a$ to $b$}. \item We say $a$ and $b$ are {\it homotopic} or are {\it in the same homotopy class} if there exists a sequence $a = a_0, a_1, \ldots, a_n = b$ of morphisms from $U$ to $V$ such that for each $i = 1, \ldots, n$ there either exists a homotopy from $a_i$ to $a_{i - 1}$ or there exists a homotopy from $a_{i - 1}$ to $a_i$. \end{enumerate}


simplicial-definition-trivial-kan

A map $X \to Y$ of simplicial sets is called a {\it trivial Kan fibration} if $X_0 \to Y_0$ is surjective and for all $n \geq 1$ and any commutative solid diagram $$ \xymatrix{ \partial \Delta[n] \ar[r] \ar[d] & X \ar[d] \\ \Delta[n] \ar[r] \ar@{-->}[ru] & Y } $$ a dotted arrow exists making the diagram commute.


simplicial-definition-kan

A map $X \to Y$ of simplicial sets is called a {\it Kan fibration} if for all $k, n$ with $1 \leq n$, $0 \leq k \leq n$ and any commutative solid diagram $$ \xymatrix{ \Lambda_k[n] \ar[r] \ar[d] & X \ar[d] \\ \Delta[n] \ar[r] \ar@{-->}[ru] & Y } $$ a dotted arrow exists making the diagram commute. A {\it Kan complex} is a simplicial set $X$ such that $X \to *$ is a Kan fibration, where $*$ is the constant simplicial set on a singleton.


discriminant-definition-trace-element

Let $A \to B$ be a flat quasi-finite map of Noetherian rings. The {\it trace element} is the unique\footnote{Uniqueness and existence will be justified in Lemmas \ref{lemma-trace-unique} and \ref{lemma-dualizing-tau}.} element $\tau_{B/A} \in \omega_{B/A}$ with the following property: for any Noetherian $A$-algebra $A_1$ such that $B_1 = B \otimes_A A_1$ comes with a product decomposition $B_1 = C \times D$ with $A_1 \to C$ finite the image of $\tau_{B/A}$ in $\omega_{C/A_1}$ is $\text{Trace}_{C/A_1}$. Here we use the base change map (\ref{equation-bc-dualizing}) and Lemma \ref{lemma-dualizing-product} to get $\omega_{B/A} \to \omega_{B_1/A_1} \to \omega_{C/A_1}$.


discriminant-definition-kahler-different

Let $f : Y \to X$ be a morphism of schemes which is locally of finite type. The {\it K\"ahler different} is the $0$th fitting ideal of $\Omega_{Y/X}$.


discriminant-definition-different

Let $f : Y \to X$ be a flat quasi-finite morphism of Noetherian schemes. Let $\omega_{Y/X}$ be the relative dualizing module and let $\tau_{Y/X} \in \Gamma(Y, \omega_{Y/X})$ be the trace element (Remarks \ref{remark-relative-dualizing-for-quasi-finite} and \ref{remark-relative-dualizing-for-flat-quasi-finite}). The annihilator of $$ \Coker(\mathcal{O}_Y \xrightarrow{\tau_{Y/X}} \omega_{Y/X}) $$ is the {\it different} of $Y/X$. It is a coherent ideal $\mathfrak{D}_f \subset \mathcal{O}_Y$.


stacks-limits-definition-limit-preserving

Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. We say $f$ is {\it limit preserving} if for every directed limit $U = \lim U_i$ of affine schemes over $S$ the diagram $$ \xymatrix{ \colim \mathcal{X}_{U_i} \ar[r] \ar[d]_f & \mathcal{X}_U \ar[d]^f \\ \colim \mathcal{Y}_{U_i} \ar[r] & \mathcal{Y}_U } $$ of fibre categories is $2$-cartesian.


spaces-more-morphisms-remark-weakly-radicial

Let $X \to Y$ be a morphism of algebraic spaces. For some applications (of radicial morphisms) it is enough to require that for every $\Spec(K) \to Y$ where $K$ is a field \begin{enumerate} \item the space $|\Spec(K) \times_Y X|$ is a singleton, \item there exists a monomorphism $\Spec(L) \to \Spec(K) \times_Y X$, and \item $K \subset L$ is purely inseparable. \end{enumerate} If needed later we will may call such a morphism {\it weakly radicial}. For example if $X \to Y$ is a surjective weakly radicial morphism then $X(k) \to Y(k)$ is surjective for every algebraically closed field $k$. Note that the base change $X_{\overline{\mathbf{Q}}} \to \Spec(\overline{\mathbf{Q}})$ of the morphism in Example \ref{example-universally-injective-not-radicial} is weakly radicial, but not radicial. The analogue of Lemma \ref{lemma-when-universally-injective-radicial} is that if $X \to Y$ has property ($\beta$) and is universally injective, then it is weakly radicial (proof omitted).


spaces-more-morphisms-remark-alternative

Now that we know that $\Omega_{X/Y}$ is quasi-coherent we can attempt to construct it in another manner. For example we can use the result of Properties of Spaces, Section \ref{spaces-properties-section-quasi-coherent-presentation} to construct the sheaf of differentials by glueing. For example if $Y$ is a scheme and if $U \to X$ is a surjective \'etale morphism from a scheme towards $X$, then we see that $\Omega_{U/Y}$ is a quasi-coherent $\mathcal{O}_U$-module, and since $s, t : R \to U$ are \'etale we get an isomorphism $$ \alpha : s^*\Omega_{U/Y} \to \Omega_{R/Y} \to t^*\Omega_{U/Y} $$ by using Morphisms, Lemma \ref{morphisms-lemma-triangle-differentials-smooth}. You check that this satisfies the cocycle condition and you're done. If $Y$ is not a scheme, then you define $\Omega_{U/Y}$ as the cokernel of the map $(U \to Y)^*\Omega_{Y/S} \to \Omega_{U/S}$, and proceed as before. This two step process is a little bit ugly. Another possibility is to glue the sheaves $\Omega_{U/V}$ for any diagram as in Lemma \ref{lemma-localize-differentials} but this is not very elegant either. Both approaches will work however, and will give a slightly more elementary construction of the sheaf of differentials.


spaces-more-morphisms-remark-topological-invariance-etale-site

\begin{reference} Email by Lenny Taelman dated May 1, 2016. \end{reference} A universal homeomorphism of algebraic spaces need not be representable, see Morphisms of Spaces, Example \ref{spaces-morphisms-example-universal-homeomorphism}. In fact Theorem \ref{theorem-topological-invariance} does not hold for universal homeomorphisms. To see this, let $k$ be an algebraically closed field of characteristic $0$ and let $$ \mathbf{A}^1 \to X \to \mathbf{A}^1 $$ be as in Morphisms of Spaces, Example \ref{spaces-morphisms-example-universal-homeomorphism}. Recall that the first morphism is \'etale and identifies $t$ with $-t$ for $t \in \mathbf{A}^1_k \setminus \{0\}$ and that the second morphism is our universal homeomorphism. Since $\mathbf{A}^1_k$ has no nontrivial connected finite \'etale coverings (because $k$ is algebraically closed of characteristic zero; details omitted), it suffices to construct a nontrivial connected finite \'etale covering $Y \to X$. To do this, let $Y$ be the affine line with zero doubled (Schemes, Example \ref{schemes-example-affine-space-zero-doubled}). Then $Y = Y_1 \cup Y_2$ with $Y_i = \mathbf{A}^1_k$ glued along $\mathbf{A}^1_k \setminus \{0\}$. To define the morphism $Y \to X$ we use the morphisms $$ Y_1 \xrightarrow{1} \mathbf{A}^1_k \to X \quad\text{and}\quad Y_2 \xrightarrow{-1} \mathbf{A}^1_k \to X. $$ These glue over $Y_1 \cap Y_2$ by the construction of $X$ and hence define a morphism $Y \to X$. In fact, we claim that $$ \xymatrix{ Y \ar[d] & Y_1 \amalg Y_2 \ar[l] \ar[d] \\ X & \mathbf{A}^1_k \ar[l] } $$ is a cartesian square. We omit the details; you can use for example Groupoids, Lemma \ref{groupoids-lemma-criterion-fibre-product}. Since $\mathbf{A}^1_k \to X$ is \'etale and surjective, this proves that $Y \to X$ is finite \'etale of degree $2$ which gives the desired example. \medskip\noindent More simply, you can argue as follows. The scheme $Y$ has a free action of the group $G = \{+1, -1\}$ where $-1$ acts by swapping $Y_1$ and $Y_2$ and changing the sign of the coordinate. Then $X = Y/G$ (see Spaces, Definition \ref{spaces-definition-quotient}) and hence $Y \to X$ is finite \'etale. You can also show directly that there exists a universal homeomorphism $X \to \mathbf{A}^1_k$ by using $t \mapsto t^2$ on affine spaces. In fact, this $X$ is the same as the $X$ above.


spaces-more-morphisms-remark-tempting

It is tempting to think that in the situation of Lemma \ref{lemma-etale-on-top} we have ``$b$ formally smooth'' $\Leftrightarrow$ ``$b \circ a$ formally smooth''. However, this is likely not true in general.


spaces-more-morphisms-remark-action-by-derivations

Assumptions and notation as in Lemma \ref{lemma-action-by-derivations}. The action of a local section $\theta$ on $a'$ is sometimes indicated by $\theta \cdot a'$. Note that this means nothing else than the fact that $(a')^\sharp$ and $(\theta \cdot a')^\sharp$ differ by a derivation $D$ which is related to $\theta$ by Equation (\ref{equation-D}).


spaces-more-morphisms-remark-special-case

A special case of Lemmas \ref{lemma-difference-derivation}, \ref{lemma-action-by-derivations}, \ref{lemma-sheaf}, and \ref{lemma-action-sheaf} is where $Y = Y'$. In this case the map $A$ is always zero. The sheaf of Lemma \ref{lemma-sheaf} is just given by the rule $$ U' \mapsto \{a' : U' \to Y\text{ over }B\text{ with } a'|_U = a|_U\} $$ and we act on this by the sheaf $\SheafHom_{\mathcal{O}_X}(a^*\Omega_{Y/B}, \mathcal{C}_{X/X'})$.


spaces-more-morphisms-remark-another-special-case

Another special case of Lemmas \ref{lemma-difference-derivation}, \ref{lemma-action-by-derivations}, \ref{lemma-sheaf}, and \ref{lemma-action-sheaf} is where $B$ itself is a thickening $Z \subset Z' = B$ and $Y = Z \times_{Z'} Y'$. Picture $$ \xymatrix{ (X \subset X') \ar@{..>}[rr]_{(a, ?)} \ar[rd]_{(g, g')} & & (Y \subset Y') \ar[ld]^{(h, h')} \\ & (Z \subset Z') } $$ In this case the map $A : a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$ is determined by $a$: the map $h^*\mathcal{C}_{Z/Z'} \to \mathcal{C}_{Y/Y'}$ is surjective (because we assumed $Y = Z \times_{Z'} Y'$), hence the pullback $g^*\mathcal{C}_{Z/Z'} = a^*h^*\mathcal{C}_{Z/Z'} \to a^*\mathcal{C}_{Y/Y'}$ is surjective, and the composition $g^*\mathcal{C}_{Z/Z'} \to a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$ has to be the canonical map induced by $g'$. Thus the sheaf of Lemma \ref{lemma-sheaf} is just given by the rule $$ U' \mapsto \{a' : U' \to Y'\text{ over }Z'\text{ with } a'|_U = a|_U\} $$ and we act on this by the sheaf $\SheafHom_{\mathcal{O}_X}(a^*\Omega_{Y/Z}, \mathcal{C}_{X/X'})$.


spaces-more-morphisms-remark-chow-Noetherian

In Lemmas \ref{lemma-blowup-to-find-embedding} and \ref{lemma-chow-noetherian} the morphism $g : Z' \to Y$ is a composition of projective morphisms. Presumably (by the analogue for algebraic spaces of Morphisms, Lemma \ref{morphisms-lemma-ample-composition}) there exists a $g$-ample invertible sheaf on $Z'$. If we ever need this, then we will state and prove this here.


spaces-more-morphisms-remark-inverse-systems-kernel-cokernel-annihilated-by

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\mathcal{I}, \mathcal{K} \subset \mathcal{O}_X$ be quasi-coherent sheaves of ideals. Let $\alpha : (\mathcal{F}_n) \to (\mathcal{G}_n)$ be a morphism of $\textit{Coh}(X, \mathcal{I})$. Given an affine scheme $U = \Spec(A)$ and a surjective \'etale morphism $U \to X$ denote $I, K \subset A$ the ideals corresponding to the restrictions $\mathcal{I}|_U, \mathcal{K}|_U$. Denote $\alpha_U : M \to N$ of finite $A^\wedge$-modules which corresponds to $\alpha|_U$ via Cohomology of Schemes, Lemma \ref{coherent-lemma-inverse-systems-affine}. We claim the following are equivalent \begin{enumerate} \item there exists an integer $t \geq 1$ such that $\Ker(\alpha_n)$ and $\Coker(\alpha_n)$ are annihilated by $\mathcal{K}^t$ for all $n \geq 1$, \item for any (or some) affine open $\Spec(A) = U \subset X$ as above the modules $\Ker(\alpha_U)$ and $\Coker(\alpha_U)$ are annihilated by $K^t$ for some integer $t \geq 1$. \end{enumerate} If these equivalent conditions hold we will say that $\alpha$ is a {\it map whose kernel and cokernel are annihilated by a power of $\mathcal{K}$}. To see the equivalence we refer to Cohomology of Schemes, Remark \ref{coherent-remark-inverse-systems-kernel-cokernel-annihilated-by}.


spaces-more-morphisms-remark-reformulate-existence-theorem

Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Write $S = \Spec(A)$ and $S_n = \Spec(A/I^n)$. Let $X \to S$ be a morphism of algebraic spaces that is separated and of finite type. For $n \geq 1$ we set $X_n = X \times_S S_n$. Picture: $$ \xymatrix{ X_1 \ar[r]_{i_1} \ar[d] & X_2 \ar[r]_{i_2} \ar[d] & X_3 \ar[r] \ar[d] & \ldots & X \ar[d] \\ S_1 \ar[r] & S_2 \ar[r] & S_3 \ar[r] & \ldots & S } $$ In this situation we consider systems $(\mathcal{F}_n, \varphi_n)$ where \begin{enumerate} \item $\mathcal{F}_n$ is a coherent $\mathcal{O}_{X_n}$-module, \item $\varphi_n : i_n^*\mathcal{F}_{n + 1} \to \mathcal{F}_n$ is an isomorphism, and \item $\text{Supp}(\mathcal{F}_1)$ is proper over $S_1$. \end{enumerate} Theorem \ref{theorem-grothendieck-existence} says that the completion functor $$ \begin{matrix} \text{coherent }\mathcal{O}_X\text{-modules }\mathcal{F} \\ \text{with support proper over }A \end{matrix} \quad \longrightarrow \quad \begin{matrix} \text{systems }(\mathcal{F}_n) \\ \text{as above} \end{matrix} $$ is an equivalence of categories. In the special case that $X$ is proper over $A$ we can omit the conditions on the supports.


spaces-more-morphisms-remark-weaken-separation-axioms-question

We can ask if in Grothendieck's algebraization theorem (in the form of Lemma \ref{lemma-algebraize-morphism}), we can get by with weaker separation axioms on the target. Let us be more precise. Let $A$, $I$, $S$, $S_n$, $X$, $Y$, $X_n$, $Y_n$, and $g_n$ be as in the statement of Lemma \ref{lemma-algebraize-morphism} and assume that \begin{enumerate} \item $X \to S$ is proper, and \item $Y \to S$ is locally of finite type. \end{enumerate} Does there exist a morphism of algebraic spaces $g : X \to Y$ over $S$ such that $g_n$ is the base change of $g$ to $S_n$? We don't know the answer in general; if you do please email \href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}. If $Y \to S$ is separated, then the result holds by the lemma (there is an immediate reduction to the case where $X$ is finite type over $S$, by choosing a quasi-compact open containing the image of $g_1$). If we only assume $Y \to S$ is quasi-separated, then the result is true as well. First, as before we may assume $Y$ is quasi-compact as well as quasi-separated. Then we can use either \cite{Bhatt-Algebraize} or from \cite{Hall-Rydh-coherent} to algebraize $(g_n)$. Namely, to apply the first reference, we use $$ D_{perf}(X) \to \lim D_{perf}(X_n) \xrightarrow{\lim Lg_n^*} \lim D_{perf}(Y_n) = D_{perf}(Y) $$ where the last step uses a Grothendieck existence result for the derived category of the proper algebraic space $Y$ over $R$ (compare with Flatness on Spaces, Remark \ref{spaces-flat-remark-correct-generality}). The paper cited shows that this arrow determines a morphism $Y \to X$ as desired. To apply the second reference we use the same argument with coherent modules: $$ \textit{Coh}(\mathcal{O}_X) \to \lim \textit{Coh}(\mathcal{O}_{X_n}) \xrightarrow{\lim g_n^*} \lim \textit{Coh}(\mathcal{O}_{Y_n}) = \textit{Coh}(\mathcal{O}_Y) $$ where the final equality is a consequence of Grothendieck's existence theorem (Theorem \ref{theorem-grothendieck-existence}). The second reference tells us that this functor corresponds to a morphism $Y \to X$ over $R$. If we ever need this generalization we will precisely state and carefully prove the result here.


spaces-more-morphisms-remark-match-relative-pseudo-coherence

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of representable algebraic spaces over $S$ which is locally of finite type. Let $f_0 : X_0 \to Y_0$ be a morphism of schemes representing $f$ (awkward but temporary notation). Then $f_0$ is locally of finite type. If $E$ is an object of $D_\QCoh(\mathcal{O}_X)$, then $E$ is the pullback of a unique object $E_0$ in $D_\QCoh(\mathcal{O}_{X_0})$, see Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}. In this situation the phrase ``$E$ is $m$-pseudo-coherent relative to $Y$'' will be taken to mean ``$E_0$ is $m$-pseudo-coherent relative to $Y_0$'' as defined in More on Morphisms, Section \ref{more-morphisms-section-relative-pseudo-coherence}.


spaces-more-morphisms-remark-compare-L

The reader may have noticed the similarity between Lemma \ref{lemma-compute-ext-rel-perfect} and Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-compute-ext}. Indeed, the pseudo-coherent complex $L$ of Lemma \ref{lemma-compute-ext-rel-perfect} may be characterized as the unique pseudo-coherent complex on $Y$ such that there are functorial isomorphisms $$ \Ext^i_{\mathcal{O}_Y}(L, \mathcal{F}) \longrightarrow \Ext^i_{\mathcal{O}_X}(K, E \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*\mathcal{F}) $$ compatible with boundary maps for $\mathcal{F}$ ranging over $\QCoh(\mathcal{O}_Y)$. If we ever need this we will formulate a precise result here and give a detailed proof.


algebra-remark-tensor-product-not-exact

However, tensor product does NOT preserve exact sequences in general. In other words, if $M_1 \to M_2 \to M_3$ is exact, then it is not necessarily true that $M_1 \otimes N \to M_2 \otimes N \to M_3 \otimes N$ is exact for arbitrary $R$-module $N$.


algebra-remark-flat-module

For $R$-modules $N$, if the functor $-\otimes_R N$ is exact, i.e. tensoring with $N$ preserves all exact sequences, then $N$ is said to be {\it flat} $R$-module. We will discuss this later in Section \ref{section-flat}.


algebra-remark-fundamental-diagram

A fundamental commutative diagram associated to a ring map $\varphi : R \to S$, a prime $\mathfrak q \subset S$ and the corresponding prime $\mathfrak p = \varphi^{-1}(\mathfrak q)$ of $R$ is the following $$ \xymatrix{ \kappa(\mathfrak q) = S_{\mathfrak q}/{\mathfrak q}S_{\mathfrak q} & S_{\mathfrak q} \ar[l] & S \ar[r] \ar[l] & S/\mathfrak q \ar[r] & \kappa(\mathfrak q) \\ \kappa(\mathfrak p) \otimes_R S = S_{\mathfrak p}/{\mathfrak p}S_{\mathfrak p} \ar[u] & S_{\mathfrak p} \ar[u] \ar[l] & S \ar[u] \ar[r] \ar[l] & S/\mathfrak pS \ar[u] \ar[r] & (R \setminus \mathfrak p)^{-1}S/\mathfrak pS \ar[u] \\ \kappa(\mathfrak p) = R_{\mathfrak p}/{\mathfrak p}R_{\mathfrak p} \ar[u] & R_{\mathfrak p} \ar[u] \ar[l] & R \ar[u] \ar[r] \ar[l] & R/\mathfrak p \ar[u] \ar[r] & \kappa(\mathfrak p) \ar[u] } $$ In this diagram the arrows in the outer left and outer right columns are identical. The horizontal maps induce on the associated spectra always a homeomorphism onto the image. The lower two rows of the diagram make sense without assuming $\mathfrak q$ exists. The lower squares induce fibre squares of topological spaces. This diagram shows that $\mathfrak p$ is in the image of the map on Spec if and only if $S \otimes_R \kappa(\mathfrak p)$ is not the zero ring.