divisors-definition-regular-ideal-sheaf

Let $X$ be a ringed space. Let $\mathcal{J} \subset \mathcal{O}_X$ be a sheaf of ideals. \begin{enumerate} \item We say $\mathcal{J}$ is {\it regular} if for every $x \in \text{Supp}(\mathcal{O}_X/\mathcal{J})$ there exists an open neighbourhood $x \in U \subset X$ and a regular sequence $f_1, \ldots, f_r \in \mathcal{O}_X(U)$ such that $\mathcal{J}|_U$ is generated by $f_1, \ldots, f_r$. \item We say $\mathcal{J}$ is {\it Koszul-regular} if for every $x \in \text{Supp}(\mathcal{O}_X/\mathcal{J})$ there exists an open neighbourhood $x \in U \subset X$ and a Koszul-regular sequence $f_1, \ldots, f_r \in \mathcal{O}_X(U)$ such that $\mathcal{J}|_U$ is generated by $f_1, \ldots, f_r$. \item We say $\mathcal{J}$ is {\it $H_1$-regular} if for every $x \in \text{Supp}(\mathcal{O}_X/\mathcal{J})$ there exists an open neighbourhood $x \in U \subset X$ and a $H_1$-regular sequence $f_1, \ldots, f_r \in \mathcal{O}_X(U)$ such that $\mathcal{J}|_U$ is generated by $f_1, \ldots, f_r$. \item We say $\mathcal{J}$ is {\it quasi-regular} if for every $x \in \text{Supp}(\mathcal{O}_X/\mathcal{J})$ there exists an open neighbourhood $x \in U \subset X$ and a quasi-regular sequence $f_1, \ldots, f_r \in \mathcal{O}_X(U)$ such that $\mathcal{J}|_U$ is generated by $f_1, \ldots, f_r$. \end{enumerate}


divisors-definition-regular-immersion

Let $i : Z \to X$ be an immersion of schemes. Choose an open subscheme $U \subset X$ such that $i$ identifies $Z$ with a closed subscheme of $U$ and denote $\mathcal{I} \subset \mathcal{O}_U$ the corresponding quasi-coherent sheaf of ideals. \begin{enumerate} \item We say $i$ is a {\it regular immersion} if $\mathcal{I}$ is regular. \item We say $i$ is a {\it Koszul-regular immersion} if $\mathcal{I}$ is Koszul-regular. \item We say $i$ is a {\it $H_1$-regular immersion} if $\mathcal{I}$ is $H_1$-regular. \item We say $i$ is a {\it quasi-regular immersion} if $\mathcal{I}$ is quasi-regular. \end{enumerate}


divisors-definition-relative-H1-regular-immersion

Let $f : X \to S$ be a morphism of schemes. Let $i : Z \to X$ be an immersion. \begin{enumerate} \item We say $i$ is a {\it relative quasi-regular immersion} if $Z \to S$ is flat and $i$ is a quasi-regular immersion. \item We say $i$ is a {\it relative $H_1$-regular immersion} if $Z \to S$ is flat and $i$ is an $H_1$-regular immersion. \end{enumerate}


divisors-definition-sheaf-meromorphic-functions

Let $(X, \mathcal{O}_X)$ be a locally ringed space. The {\it sheaf of meromorphic functions on $X$} is the sheaf {\it $\mathcal{K}_X$} associated to the presheaf displayed above. A {\it meromorphic function} on $X$ is a global section of $\mathcal{K}_X$.


divisors-definition-meromorphic-section

Let $X$ be a locally ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. \begin{enumerate} \item We denote $\mathcal{K}_X(\mathcal{F})$ the sheaf of $\mathcal{K}_X$-modules which is the sheafification of the presheaf $U \mapsto \mathcal{S}(U)^{-1}\mathcal{F}(U)$. Equivalently $\mathcal{K}_X(\mathcal{F}) = \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{K}_X$ (see above). \item A {\it meromorphic section of $\mathcal{F}$} is a global section of $\mathcal{K}_X(\mathcal{F})$. \end{enumerate}


divisors-definition-pullback-meromorphic-sections

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of locally ringed spaces. We say that {\it pullbacks of meromorphic functions are defined for $f$} if for every pair of open $U \subset X$, $V \subset Y$ such that $f(U) \subset V$, and any section $s \in \Gamma(V, \mathcal{S}_Y)$ the pullback $f^\sharp(s) \in \Gamma(U, \mathcal{O}_X)$ is an element of $\Gamma(U, \mathcal{S}_X)$.


divisors-definition-regular-meromorphic-section

Let $X$ be a locally ringed space. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. A meromorphic section $s$ of $\mathcal{L}$ is said to be {\it regular} if the induced map $\mathcal{K}_X \to \mathcal{K}_X(\mathcal{L})$ is injective. In other words, $s$ is a regular section of the invertible $\mathcal{K}_X$-module $\mathcal{K}_X(\mathcal{L})$, see Definition \ref{definition-regular-section}.


divisors-definition-regular-meromorphic-ideal-denominators

Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s$ be a regular meromorphic section of $\mathcal{L}$. The sheaf of ideals $\mathcal{I}$ constructed in Lemma \ref{lemma-regular-meromorphic-ideal-denominators} is called the {\it ideal sheaf of denominators of $s$}.


divisors-definition-Weil-divisor

Let $X$ be a locally Noetherian integral scheme. \begin{enumerate} \item A {\it prime divisor} is an integral closed subscheme $Z \subset X$ of codimension $1$. \item A {\it Weil divisor} is a formal sum $D = \sum n_Z Z$ where the sum is over prime divisors of $X$ and the collection $\{Z \mid n_Z \not = 0\}$ is locally finite (Topology, Definition \ref{topology-definition-locally-finite}). \end{enumerate} The group of all Weil divisors on $X$ is denoted $\text{Div}(X)$.


divisors-definition-order-vanishing

Let $X$ be a locally Noetherian integral scheme. Let $f \in R(X)^*$. For every prime divisor $Z \subset X$ we define the {\it order of vanishing of $f$ along $Z$} as the integer $$ \text{ord}_Z(f) = \text{ord}_{\mathcal{O}_{X, \xi}}(f) $$ where the right hand side is the notion of Algebra, Definition \ref{algebra-definition-ord} and $\xi$ is the generic point of $Z$.


divisors-definition-principal-divisor

Let $X$ be a locally Noetherian integral scheme. Let $f \in R(X)^*$. The {\it principal Weil divisor associated to $f$} is the Weil divisor $$ \text{div}(f) = \text{div}_X(f) = \sum \text{ord}_Z(f) [Z] $$ where the sum is over prime divisors and $\text{ord}_Z(f)$ is as in Definition \ref{definition-order-vanishing}. This makes sense by Lemma \ref{lemma-divisor-locally-finite}.


divisors-definition-class-group

Let $X$ be a locally Noetherian integral scheme. The {\it Weil divisor class group} of $X$ is the quotient of the group of Weil divisors by the subgroup of principal Weil divisors. Notation: $\text{Cl}(X)$.


divisors-definition-order-vanishing-meromorphic

Let $X$ be a locally Noetherian integral scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{K}_X(\mathcal{L}))$ be a regular meromorphic section of $\mathcal{L}$. For every prime divisor $Z \subset X$ we define the {\it order of vanishing of $s$ along $Z$} as the integer $$ \text{ord}_{Z, \mathcal{L}}(s) = \text{ord}_{\mathcal{O}_{X, \xi}}(s/s_\xi) $$ where the right hand side is the notion of Algebra, Definition \ref{algebra-definition-ord}, $\xi \in Z$ is the generic point, and $s_\xi \in \mathcal{L}_\xi$ is a generator.


divisors-definition-divisor-invertible-sheaf

Let $X$ be a locally Noetherian integral scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. \begin{enumerate} \item For any nonzero meromorphic section $s$ of $\mathcal{L}$ we define the {\it Weil divisor associated to $s$} as $$ \text{div}_\mathcal{L}(s) = \sum \text{ord}_{Z, \mathcal{L}}(s) [Z] \in \text{Div}(X) $$ where the sum is over prime divisors. \item We define {\it Weil divisor class associated to $\mathcal{L}$} as the image of $\text{div}_\mathcal{L}(s)$ in $\text{Cl}(X)$ where $s$ is any nonzero meromorphic section of $\mathcal{L}$ over $X$. This is well defined by Lemma \ref{lemma-divisor-meromorphic-well-defined}. \end{enumerate}


divisors-definition-blow-up

Let $X$ be a scheme. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals, and let $Z \subset X$ be the closed subscheme corresponding to $\mathcal{I}$, see Schemes, Definition \ref{schemes-definition-immersion}. The {\it blowing up of $X$ along $Z$}, or the {\it blowing up of $X$ in the ideal sheaf $\mathcal{I}$} is the morphism $$ b : \underline{\text{Proj}}_X \left(\bigoplus\nolimits_{n \geq 0} \mathcal{I}^n\right) \longrightarrow X $$ The {\it exceptional divisor} of the blowup is the inverse image $b^{-1}(Z)$. Sometimes $Z$ is called the {\it center} of the blowup.


divisors-definition-strict-transform

With $Z \subset S$ and $f : X \to S$ as above. \begin{enumerate} \item Given a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ the {\it strict transform} of $\mathcal{F}$ with respect to the blowup of $S$ in $Z$ is the quotient $\mathcal{F}'$ of $\text{pr}_X^*\mathcal{F}$ by the submodule of sections supported on $\text{pr}_{S'}^{-1}E$. \item The {\it strict transform} of $X$ is the closed subscheme $X' \subset X \times_S S'$ cut out by the quasi-coherent ideal of sections of $\mathcal{O}_{X \times_S S'}$ supported on $\text{pr}_{S'}^{-1}E$. \end{enumerate}


divisors-definition-admissible-blowup

Let $X$ be a scheme. Let $U \subset X$ be an open subscheme. A morphism $X' \to X$ is called a {\it $U$-admissible blowup} if there exists a closed immersion $Z \to X$ of finite presentation with $Z$ disjoint from $U$ such that $X'$ is isomorphic to the blowup of $X$ in $Z$.


spaces-definition-relative-representable-property

With $S$, and $a : F \to G$ representable as above. Let $\mathcal{P}$ be a property of morphisms of schemes which \begin{enumerate} \item is preserved under any base change, see Schemes, Definition \ref{schemes-definition-preserved-by-base-change}, and \item is fppf local on the base, see Descent, Definition \ref{descent-definition-property-morphisms-local}. \end{enumerate} In this case we say that $a$ has {\it property $\mathcal{P}$} if for every $U \in \Ob((\Sch/S)_{fppf})$ and any $\xi \in G(U)$ the resulting morphism of schemes $V_\xi \to U$ has property $\mathcal{P}$.


spaces-definition-algebraic-space

Let $S$ be a scheme contained in $\Sch_{fppf}$. An {\it algebraic space over $S$} is a presheaf $$ F : (\Sch/S)^{opp}_{fppf} \longrightarrow \textit{Sets} $$ with the following properties \begin{enumerate} \item The presheaf $F$ is a sheaf. \item The diagonal morphism $F  \to F \times F$ is representable. \item There exists a scheme $U \in \Ob((\Sch/S)_{fppf})$ and a map $h_U \to F$ which is surjective, and \'etale. \end{enumerate}


spaces-definition-morphism-algebraic-spaces

Let $F$, $F'$ be algebraic spaces over $S$. A {\it morphism $f : F \to F'$ of algebraic spaces over $S$} is a transformation of functors from $F$ to $F'$.


spaces-definition-etale-equivalence-relation

Let $S$ be a scheme. Let $U$ be a scheme over $S$. An {\it \'etale equivalence relation} on $U$ over $S$ is an equivalence relation $j : R \to U \times_S U$ such that $s, t : R \to U$ are \'etale morphisms of schemes.


spaces-definition-presentation

Let $F$ be an algebraic space over $S$. A {\it presentation} of $F$ is given by a scheme $U$ over $S$ and an \'etale equivalence relation $R$ on $U$ over $S$, and a surjective \'etale morphism $U \to F$ such that $R = U \times_F U$.


spaces-definition-immersion

Let $S \in \Ob(\Sch_{fppf})$ be a scheme. Let $F$ be an algebraic space over $S$. \begin{enumerate} \item A morphism of algebraic spaces over $S$ is called an {\it open immersion} if it is representable, and an open immersion in the sense of Definition \ref{definition-relative-representable-property}. \item An {\it open subspace} of $F$ is a subfunctor $F' \subset F$ such that $F'$ is an algebraic space and $F' \to F$ is an open immersion. \item A morphism of algebraic spaces over $S$ is called a {\it closed immersion} if it is representable, and a closed immersion in the sense of Definition \ref{definition-relative-representable-property}. \item A {\it closed subspace} of $F$ is a subfunctor $F' \subset F$ such that $F'$ is an algebraic space and $F' \to F$ is a closed immersion. \item A morphism of algebraic spaces over $S$ is called an {\it immersion} if it is representable, and an immersion in the sense of Definition \ref{definition-relative-representable-property}. \item A {\it locally closed subspace} of $F$ is a subfunctor $F' \subset F$ such that $F'$ is an algebraic space and $F' \to F$ is an immersion. \end{enumerate}


spaces-definition-Zariski-open-covering

Let $S \in \Ob(\Sch_{fppf})$ be a scheme. Let $F$ be an algebraic space over $S$. A {\it Zariski covering} $\{F_i \subset F\}_{i \in I}$ of $F$ is given by a set $I$ and a collection of open subspaces $F_i \subset F$ such that $\coprod F_i \to F$ is a surjective map of sheaves.


spaces-definition-small-Zariski-site

Let $S \in \Ob(\Sch_{fppf})$ be a scheme. Let $F$ be an algebraic space over $S$. A {\it small Zariski site $F_{Zar}$} of an algebraic space $F$ is one of the sites described above.


spaces-definition-separated

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F$ be an algebraic space over $S$. Let $\Delta : F \to F \times F$ be the diagonal morphism. \begin{enumerate} \item We say $F$ is {\it separated over $S$} if $\Delta$ is a closed immersion. \item We say $F$ is {\it locally separated over $S$}\footnote{In the literature this often refers to quasi-separated and locally separated algebraic spaces.} if $\Delta$ is an immersion. \item We say $F$ is {\it quasi-separated over $S$} if $\Delta$ is quasi-compact. \item We say $F$ is {\it Zariski locally quasi-separated over $S$}\footnote{This definition was suggested by B.\ Conrad.} if there exists a Zariski covering $F = \bigcup_{i \in I} F_i$ such that each $F_i$ is quasi-separated. \end{enumerate}


spaces-definition-quotient

Notation $U \to S$, $G$, $R$ as in Lemma \ref{lemma-quotient}. If the action of $G$ on $U$ satisfies $(*)$ we say $G$ {\it acts freely} on the scheme $U$. In this case the algebraic space $U/R$ is denoted $U/G$ and is called the {\it quotient of $U$ by $G$}.


spaces-definition-base-change

Let $\Sch_{fppf}$ be a big fppf site. Let $S \to S'$ be a morphism of this site. \begin{enumerate} \item If $F'$ is an algebraic space over $S'$, then the {\it base change of $F'$ to $S$} is the algebraic space $j^{-1}F'$ described in Lemma \ref{lemma-change-base-scheme}. We denote it $F'_S$. \item If $F$ is an algebraic space over $S$, then $F$ {\it viewed as an algebraic space over $S'$} is the algebraic space $j_!F$ over $S'$ described in Lemma \ref{lemma-change-base-scheme}. We often simply denote this $F$; if not then we will write $j_!F$. \end{enumerate}


topology-definition-separated

A continuous map $f : X \to Y$ of topological spaces is called {\it separated} if and only if the diagonal $\Delta : X \to X \times_Y X$ is a closed map.


topology-definition-base

Let $X$ be a topological space. A collection of subsets $\mathcal{B}$ of $X$ is called a {\it base for the topology on $X$} or a {\it basis for the topology on $X$} if the following conditions hold: \begin{enumerate} \item Every element $B \in \mathcal{B}$ is open in $X$. \item For every open $U \subset X$ and every $x \in U$, there exists an element $B \in \mathcal{B}$ such that $x \in B \subset U$. \end{enumerate}


topology-definition-subbase

Let $X$ be a topological space. A collection of subsets $\mathcal{B}$ of $X$ is called a {\it subbase for the topology on $X$} or a {\it subbasis for the topology on $X$} if the finite intersections of elements of $\mathcal{B}$ form a basis for the topology on $X$.


topology-definition-submersive

Let $f : X \to Y$ be a continuous map of topological spaces. \begin{enumerate} \item We say $f$ is a {\it strict map of topological spaces} if the induced topology and the quotient topology on $f(X)$ agree (see discussion above). \item We say $f$ is {\it submersive}\footnote{This is very different from the notion of a submersion between differential manifolds! It is probably a good idea to use ``strict and surjective'' in stead of ``submersive''.} if $f$ is surjective and strict. \end{enumerate}


topology-definition-connected-components

Let $X$ be a topological space. \begin{enumerate} \item We say $X$ is {\it connected} if $X$ is not empty and whenever $X = T_1 \amalg T_2$ with $T_i \subset X$ open and closed, then either $T_1 = \emptyset$ or $T_2 = \emptyset$. \item We say $T \subset X$ is a {\it connected component} of $X$ if $T$ is a maximal connected subset of $X$. \end{enumerate}


topology-definition-totally-disconnected

A topological space is {\it totally disconnected} if the connected components are all singletons.


topology-definition-locally-connected

A topological space $X$ is called {\it locally connected} if every point $x \in X$ has a fundamental system of connected neighbourhoods.


topology-definition-irreducible-components

Let $X$ be a topological space. \begin{enumerate} \item We say $X$ is {\it irreducible}, if $X$ is not empty, and whenever $X = Z_1 \cup Z_2$ with $Z_i$ closed, we have $X = Z_1$ or $X = Z_2$. \item We say $Z \subset X$ is an {\it irreducible component} of $X$ if $Z$ is a maximal irreducible subset of $X$. \end{enumerate}


topology-definition-generic-point

Let $X$ be a topological space. \begin{enumerate} \item Let $Z \subset X$ be an irreducible closed subset. A {\it generic point} of $Z$ is a point $\xi \in Z$ such that $Z = \overline{\{\xi\}}$. \item The space $X$ is called {\it Kolmogorov}, if for every $x, x' \in X$, $x \not = x'$ there exists a closed subset of $X$ which contains exactly one of the two points. \item The space $X$ is called {\it quasi-sober} if every irreducible closed subset has a generic point. \item The space $X$ is called {\it sober} if every irreducible closed subset has a unique generic point. \end{enumerate}


topology-definition-noetherian

A topological space is called {\it Noetherian} if the descending chain condition holds for closed subsets of $X$. A topological space is called {\it locally Noetherian} if every point has a neighbourhood which is Noetherian.


topology-definition-Krull

Let $X$ be a topological space. \begin{enumerate} \item A {\it chain of irreducible closed subsets} of $X$ is a sequence $Z_0 \subset Z_1 \subset \ldots \subset Z_n \subset X$ with $Z_i$ closed irreducible and $Z_i \not = Z_{i + 1}$ for $i = 0, \ldots, n - 1$. \item The {\it length} of a chain $Z_0 \subset Z_1 \subset \ldots \subset Z_n \subset X$ of irreducible closed subsets of $X$ is the integer $n$. \item The {\it dimension} or more precisely the {\it Krull dimension} $\dim(X)$ of $X$ is the element of $\{-\infty, 0, 1, 2, 3, \ldots, \infty\}$ defined by the formula: $$ \dim(X) = \sup \{\text{lengths of chains of irreducible closed subsets}\} $$ Thus $\dim(X) = -\infty$ if and only if $X$ is the empty space. \item Let $x \in X$. The {\it Krull dimension of $X$ at $x$} is defined as $$ \dim_x(X) = \min \{\dim(U), x\in U\subset X\text{ open}\} $$ the minimum of $\dim(U)$ where $U$ runs over the open neighbourhoods of $x$ in $X$. \end{enumerate}


topology-definition-equidimensional

Let $X$ be a topological space. We say that $X$ is {\it equidimensional} if every irreducible component of $X$ has the same dimension.


topology-definition-codimension

Let $X$ be a topological space. Let $Y \subset X$ be an irreducible closed subset. The {\it codimension} of $Y$ in $X$ is the supremum of the lengths $e$ of chains $$ Y = Y_0 \subset Y_1 \subset \ldots \subset Y_e \subset X $$ of irreducible closed subsets in $X$ starting with $Y$. We will denote this $\text{codim}(Y, X)$.


topology-definition-catenary

Let $X$ be a topological space. We say $X$ is {\it catenary} if for every pair of irreducible closed subsets $T \subset T'$ we have $\text{codim}(T, T') < \infty$ and every maximal chain of irreducible closed subsets $$ T = T_0 \subset T_1 \subset \ldots \subset T_e = T' $$ has the same length (equal to the codimension).


topology-definition-quasi-compact

Quasi-compactness. \begin{enumerate} \item We say that a topological space $X$ is {\it quasi-compact} if every open covering of $X$ has a finite refinement. \item We say that a continuous map $f : X \to Y$ is {\it quasi-compact} if the inverse image $f^{-1}(V)$ of every quasi-compact open $V \subset Y$ is quasi-compact. \item We say a subset $Z \subset X$ is {\it retrocompact} if the inclusion map $Z \to X$ is quasi-compact. \end{enumerate}


topology-definition-locally-quasi-compact

A topological space $X$ is called {\it locally quasi-compact}\footnote{This may not be standard notation. Alternative notions used in the literature are: (1) Every point has some quasi-compact neighbourhood, and (2) Every point has a closed quasi-compact neighbourhood. A scheme has the property that every point has a fundamental system of open quasi-compact neighbourhoods.} if every point has a fundamental system of quasi-compact neighbourhoods.


topology-definition-constructible

Let $X$ be a topological space. Let $E \subset X$ be a subset of $X$. \begin{enumerate} \item We say $E$ is {\it constructible}\footnote{In the second edition of EGA I \cite{EGA1-second} this was called a ``globally constructible'' set and a the terminology ``constructible'' was used for what we call a locally constructible set.} in $X$ if $E$ is a finite union of subsets of the form $U \cap V^c$ where $U, V \subset X$ are open and retrocompact in $X$. \item We say $E$ is {\it locally constructible} in $X$ if there exists an open covering $X = \bigcup V_i$ such that each $E \cap V_i$ is constructible in $V_i$. \end{enumerate}


topology-definition-proper-map

Let $f : X\to Y$ be a continuous map between topological spaces. \begin{enumerate} \item We say that the map $f$ is {\it closed} iff the image of every closed subset is closed. \item We say that the map $f$ is {\it proper}\footnote{This is the terminology used in \cite{Bourbaki}. Usually this is what is called ``universally closed'' in the literature. Thus our notion of proper does not involve any separation conditions.} iff the map $Z \times X\to Z \times Y$ is closed for any topological space $Z$. \item We say that the map $f$ is {\it quasi-proper} iff the inverse image $f^{-1}(V)$ of every quasi-compact subset $V \subset Y$ is quasi-compact. \item We say that $f$ is {\it universally closed} iff the map $f': Z \times_Y X \to Z$ is closed for any map $g: Z \to Y$. \end{enumerate}


topology-definition-space-jacobson

Let $X$ be a topological space. Let $X_0$ be the set of closed points of $X$. We say that $X$ is {\it Jacobson} if every closed subset $Z \subset X$ is the closure of $Z \cap X_0$.


topology-definition-specialization

Let $X$ be a topological space. \begin{enumerate} \item If $x, x' \in X$ then we say $x$ is a {\it specialization} of $x'$, or $x'$ is a {\it generalization} of $x$ if $x \in \overline{\{x'\}}$. Notation: $x' \leadsto x$. \item A subset $T \subset X$ is {\it stable under specialization} if for all $x' \in T$ and every specialization $x' \leadsto x$ we have $x \in T$. \item A subset $T \subset X$ is {\it stable under generalization} if for all $x \in T$ and every generalization $x' \leadsto x$ we have $x' \in T$. \end{enumerate}


topology-definition-lift-specializations

Let $f : X \to Y$ be a continuous map of topological spaces. \begin{enumerate} \item We say that {\it specializations lift along $f$} or that $f$ is {\it specializing} if given $y' \leadsto y$ in $Y$ and any $x'\in X$ with $f(x') = y'$ there exists a specialization $x' \leadsto x$ of $x'$ in $X$ such that $f(x) = y$. \item We say that {\it generalizations lift along $f$} or that $f$ is {\it generalizing} if given $y' \leadsto y$ in $Y$ and any $x\in X$ with $f(x) = y$ there exists a generalization $x' \leadsto x$ of $x$ in $X$ such that $f(x') = y'$. \end{enumerate}


topology-definition-dimension-function

Let $X$ be a topological space. \begin{enumerate} \item  Let $x, y \in X$, $x \not = y$. Suppose $x \leadsto y$, that is $y$ is a specialization of $x$. We say $y$ is an {\it immediate specialization} of $x$ if there is no $z \in X \setminus \{x, y\}$ with $x \leadsto z$ and $z \leadsto y$. \item A map $\delta : X \to \mathbf{Z}$ is called a {\it dimension function}\footnote{This is likely nonstandard notation. This notion is usually introduced only for (locally) Noetherian schemes, in which case condition (a) is implied by (b).} if \begin{enumerate} \item whenever $x \leadsto y$ and $x \not = y$ we have $\delta(x) > \delta(y)$, and \item for every immediate specialization $x \leadsto y$ in $X$ we have $\delta(x) = \delta(y) + 1$. \end{enumerate} \end{enumerate}


topology-definition-nowhere-dense

Let $X$ be a topological space. \begin{enumerate} \item Given a subset $T \subset X$ the {\it interior} of $T$ is the largest open subset of $X$ contained in $T$. \item A subset $T \subset X$ is called {\it nowhere dense} if the closure of $T$ has empty interior. \end{enumerate}


topology-definition-profinite

A topological space is {\it profinite} if it is homeomorphic to a limit of a diagram of finite discrete spaces.


topology-definition-spectral-space

A topological space $X$ is called {\it spectral} if it is sober, quasi-compact, the intersection of two quasi-compact opens is quasi-compact, and the collection of quasi-compact opens forms a basis for the topology. A continuous map $f : X \to Y$ of spectral spaces is called {\it spectral} if the inverse image of a quasi-compact open is quasi-compact.


topology-definition-extremally-disconnected

A topological space $X$ is called {\it extremally disconnected} if the closure of every open subset of $X$ is open.


topology-definition-isolated-point

Let $X$ be a topological space. We say $x \in X$ is an {\it isolated point} of $X$ if $\{x\}$ is open in $X$.


topology-definition-paritition

Let $X$ be a topological space. A {\it partition} of $X$ is a decomposition $X = \coprod X_i$ into locally closed subsets $X_i$. The $X_i$ are called the {\it parts} of the partition. Given two partitions of $X$ we say one {\it refines} the other if the parts of one are unions of parts of the other.


topology-definition-good-stratification

Let $X$ be a topological space. A {\it good stratification} of $X$ is a partition $X = \coprod X_i$ such that for all $i, j \in I$ we have $$ X_i \cap \overline{X_j} \not = \emptyset \Rightarrow X_i \subset \overline{X_j}. $$


topology-definition-stratification

Let $X$ be a topological space. A {\it stratification} of $X$ is given by a partition $X = \coprod_{i \in I} X_i$ and a partial ordering on $I$ such that for each $j \in I$ we have $$ \overline{X_j} \subset \bigcup\nolimits_{i \leq j} X_i $$ The parts $X_i$ are called the {\it strata} of the stratification.


topology-definition-locally-finite

Let $X$ be a topological space. Let $I$ be a set and for $i \in I$ let $E_i \subset X$ be a subset. We say the collection $\{E_i\}_{i \in I}$ is {\it locally finite} if for all $x \in X$ there exists an open neighbourhood $U$ of $x$ such that $\{i \in I | E_i \cap U \not = \emptyset\}$ is finite.


topology-definition-topological-group

A {\it topological group} is a group $G$ endowed with a topology such that multiplication $G \times G \to G$, $(x, y) \mapsto xy$ and inverse $G \to G$, $x \mapsto x^{-1}$ are continuous. A {\it homomorphism of topological groups} is a homomorphism of groups which is continuous.


topology-definition-profinite-group

A topological group is called a {\it profinite group} if it satisfies the equivalent conditions of Lemma \ref{lemma-profinite-group}.


topology-definition-topological-ring

A {\it topological ring} is a ring $R$ endowed with a topology such that addition $R \times R \to R$, $(x, y) \mapsto x + y$ and multiplication $R \times R \to R$, $(x, y) \mapsto xy$ are continuous. A {\it homomorphism of topological rings} is a homomorphism of rings which is continuous.


topology-definition-topological-module

Let $R$ be a topological ring. A {\it topological module} is an $R$-module $M$ endowed with a topology such that addition $M \times M \to M$ and scalar multiplication $R \times M \to M$ are continuous. A {\it homomorphism of topological modules} is a homomorphism of modules which is continuous.


hypercovering-definition-SR

Let $\mathcal{C}$ be a category. We denote $\text{SR}(\mathcal{C})$ the category of {\it semi-representable objects} defined as follows \begin{enumerate} \item objects are families of objects $\{U_i\}_{i \in I}$, and \item morphisms $\{U_i\}_{i \in I} \to \{V_j\}_{j \in J}$ are given by a map $\alpha : I \to J$ and for each $i \in I$ a morphism $f_i : U_i \to V_{\alpha(i)}$ of $\mathcal{C}$. \end{enumerate} Let $X \in \Ob(\mathcal{C})$ be an object of $\mathcal{C}$. The category of {\it semi-representable objects over $X$} is the category $\text{SR}(\mathcal{C}, X) = \text{SR}(\mathcal{C}/X)$.


hypercovering-definition-SR-F

Let $\mathcal{C}$ be a category. We denote $F$ the functor {\it which associates a presheaf to a semi-representable object}. In a formula \begin{eqnarray*} F : \text{SR}(\mathcal{C}) & \longrightarrow & \textit{PSh}(\mathcal{C}) \\ \{U_i\}_{i \in I} & \longmapsto & \amalg_{i\in I} h_{U_i} \end{eqnarray*} where $h_U$ denotes the representable presheaf associated to the object $U$.


hypercovering-definition-covering-SR

Let $\mathcal{C}$ be a site. Let $f = (\alpha, f_i) : \{U_i\}_{i \in I} \to \{V_j\}_{j \in J}$ be a morphism in the category $\text{SR}(\mathcal{C})$. We say that $f$ is a {\it covering} if for every $j \in J$ the family of morphisms $\{U_i \to V_j\}_{i \in I, \alpha(i) = j}$ is a covering for the site $\mathcal{C}$. Let $X$ be an object of $\mathcal{C}$. A morphism $K \to L$ in $\text{SR}(\mathcal{C}, X)$ is a {\it covering} if its image in $\text{SR}(\mathcal{C})$ is a covering.


hypercovering-definition-hypercovering

Let $\mathcal{C}$ be a site. Assume $\mathcal{C}$ has fibre products. Let $X \in \Ob(\mathcal{C})$ be an object of $\mathcal{C}$. A {\it hypercovering of $X$} is a simplicial object $K$ of $\text{SR}(\mathcal{C}, X)$ such that \begin{enumerate} \item The object $K_0$ is a covering of $X$ for the site $\mathcal{C}$. \item For every $n \geq 0$ the canonical morphism $$ K_{n + 1} \longrightarrow (\text{cosk}_n \text{sk}_n K)_{n + 1} $$ is a covering in the sense defined above. \end{enumerate}


hypercovering-definition-homology

Let $\mathcal{C}$ be a site. Let $K$ be a simplicial object of $\textit{PSh}(\mathcal{C})$. By the above we get a simplicial object $\mathbf{Z}_K^\#$ of $\textit{Ab}(\mathcal{C})$. We can take its associated complex of abelian presheaves $s(\mathbf{Z}_K^\#)$, see Simplicial, Section \ref{simplicial-section-complexes}. The {\it homology of $K$} is the homology of the complex of abelian sheaves $s(\mathbf{Z}_K^\#)$.


hypercovering-definition-hypercovering-variant

Let $\mathcal{C}$ be a site. Assume $\mathcal{C}$ has equalizers and fibre products. Let $\mathcal{G}$ be a presheaf of sets. A {\it hypercovering of $\mathcal{G}$} is a simplicial object $K$ of $\text{SR}(\mathcal{C})$ endowed with an augmentation $F(K) \to \mathcal{G}$ such that \begin{enumerate} \item $F(K_0) \to \mathcal{G}$ becomes surjective after sheafification, \item $F(K_1) \to F(K_0) \times_\mathcal{G} F(K_0)$ becomes surjective after sheafification, and \item $F(K_{n + 1}) \longrightarrow F((\text{cosk}_n \text{sk}_n K)_{n + 1})$ for $n \geq 1$ becomes surjective after sheafification. \end{enumerate} We say that a simplicial object $K$ of $\text{SR}(\mathcal{C})$ is a {\it hypercovering} if $K$ is a hypercovering of the final object $*$ of $\textit{PSh}(\mathcal{C})$.


algebraic-definition-representable-by-algebraic-space

Let $S$ be a scheme contained in $\Sch_{fppf}$. A category fibred in groupoids $p : \mathcal{X} \to (\Sch/S)_{fppf}$ is called {\it representable by an algebraic space over $S$} if there exists an algebraic space $F$ over $S$ and an equivalence $j : \mathcal{X} \to \mathcal{S}_F$ of categories over $(\Sch/S)_{fppf}$.


algebraic-definition-representable-by-algebraic-spaces

Let $S$ be a scheme contained in $\Sch_{fppf}$. A $1$-morphism $f : \mathcal{X} \to \mathcal{Y}$ of categories fibred in groupoids over $(\Sch/S)_{fppf}$ is called {\it representable by algebraic spaces} if for any $U \in \Ob((\Sch/S)_{fppf})$ and any $y : (\Sch/U)_{fppf} \to \mathcal{Y}$ the category fibred in groupoids $$ (\Sch/U)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X} $$ over $(\Sch/U)_{fppf}$ is representable by an algebraic space over $U$.


algebraic-definition-relative-representable-property

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Assume $f$ is representable by algebraic spaces. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which \begin{enumerate} \item is preserved under any base change, and \item is fppf local on the base, see Descent on Spaces, Definition \ref{spaces-descent-definition-property-morphisms-local}. \end{enumerate} In this case we say that $f$ has {\it property $\mathcal{P}$} if for every $U \in \Ob((\Sch/S)_{fppf})$ and any $y \in \mathcal{Y}_U$ the resulting morphism of algebraic spaces $f_y : F_y \to U$, see diagram (\ref{equation-representable-by-algebraic-spaces}), has property $\mathcal{P}$.


algebraic-definition-algebraic-stack

Let $S$ be a base scheme contained in $\Sch_{fppf}$. An {\it algebraic stack over $S$} is a category $$ p : \mathcal{X} \to (\Sch/S)_{fppf} $$ over $(\Sch/S)_{fppf}$ with the following properties: \begin{enumerate} \item The category $\mathcal{X}$ is a stack in groupoids over $(\Sch/S)_{fppf}$. \item The diagonal $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces. \item There exists a scheme $U \in \Ob((\Sch/S)_{fppf})$ and a $1$-morphism $(\Sch/U)_{fppf} \to \mathcal{X}$ which is surjective and smooth\footnote{In future chapters we will denote this simply $U \to \mathcal{X}$ as is customary in the literature. Another good alternative would be to formulate this condition as the existence of a representable category fibred in groupoids $\mathcal{U}$ and a surjective smooth $1$-morphism $\mathcal{U} \to \mathcal{X}$.}. \end{enumerate}


algebraic-definition-deligne-mumford

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X}$ be an algebraic stack over $S$. We say $\mathcal{X}$ is a {\it Deligne-Mumford stack} if there exists a scheme $U$ and a surjective \'etale morphism $(\Sch/U)_{fppf} \to \mathcal{X}$.


algebraic-definition-morphism-algebraic-stacks

Let $S$ be a scheme contained in $\Sch_{fppf}$. The {\it $2$-category of algebraic stacks over $S$} is the sub $2$-category of the $2$-category of categories fibred in groupoids over $(\Sch/S)_{fppf}$ (see Categories, Definition \ref{categories-definition-categories-fibred-in-groupoids-over-C}) defined as follows: \begin{enumerate} \item Its objects are those categories fibred in groupoids over $(\Sch/S)_{fppf}$ which are algebraic stacks over $S$. \item Its $1$-morphisms $f : \mathcal{X} \to \mathcal{Y}$ are any functors of categories over $(\Sch/S)_{fppf}$, as in Categories, Definition \ref{categories-definition-categories-over-C}. \item Its $2$-morphisms are transformations between functors over $(\Sch/S)_{fppf}$, as in Categories, Definition \ref{categories-definition-categories-over-C}. \end{enumerate}


algebraic-definition-smooth-groupoid

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. We say $(U, R, s, t, c)$ is a {\it smooth groupoid}\footnote{This terminology might be a bit confusing: it does not imply that $[U/R]$ is smooth over anything.} if $s, t : R \to U$ are smooth morphisms of algebraic spaces.


algebraic-definition-presentation

Let $\mathcal{X}$ be an algebraic stack over $S$. A {\it presentation} of $\mathcal{X}$ is given by a smooth groupoid $(U, R, s, t, c)$ in algebraic spaces over $S$, and an equivalence $f : [U/R] \to \mathcal{X}$.


algebraic-definition-viewed-as

Let $\Sch_{fppf}$ be a big fppf site. Let $S \to S'$ be a morphism of this site. If $p : \mathcal{X} \to (\Sch/S)_{fppf}$ is an algebraic stack over $S$, then $\mathcal{X}$ {\it viewed as an algebraic stack over $S'$} is the algebraic stack $$ \mathcal{X} \longrightarrow (\Sch/S')_{fppf} $$ gotten by applying construction A of Lemma \ref{lemma-category-of-spaces-over-smaller-base-scheme} to $\mathcal{X}$.


algebraic-definition-change-of-base

Let $\Sch_{fppf}$ be a big fppf site. Let $S \to S'$ be a morphism of this site. Let $\mathcal{X}'$ be an algebraic stack over $S'$. The {\it change of base of $\mathcal{X}'$} is the algebraic space $\mathcal{X}'_S$ over $S$ described above.


sites-definition-presheaves-sets

A {\it presheaf of sets} on $\mathcal{C}$ is a contravariant functor from $\mathcal{C}$ to $\textit{Sets}$. {\it Morphisms of presheaves} are transformations of functors. The category of presheaves of sets is denoted $\textit{PSh}(\mathcal{C})$.


sites-definition-presheaf

Let $\mathcal{C}$, $\mathcal{A}$ be categories. A {\it presheaf} $\mathcal{F}$ on $\mathcal{C}$ with values in $\mathcal{A}$ is a contravariant functor from $\mathcal{C}$ to $\mathcal{A}$, i.e., $\mathcal{F} : \mathcal{C}^{opp} \to \mathcal{A}$. A {\it morphism} of presheaves $\mathcal{F} \to \mathcal{G}$ on $\mathcal{C}$ with values in $\mathcal{A}$ is a transformation of functors from $\mathcal{F}$ to $\mathcal{G}$.


sites-definition-presheaves-injective-surjective

Let $\mathcal{C}$ be a category, and let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of presheaves of sets. \begin{enumerate} \item We say that $\varphi$ is {\it injective} if for every object $U$ of $\mathcal{C}$ the map $\varphi_U : \mathcal{F}(U) \to \mathcal{G}(U)$ is injective. \item We say that $\varphi$ is {\it surjective} if for every object $U$ of $\mathcal{C}$ the map $\varphi_U : \mathcal{F}(U) \to \mathcal{G}(U)$ is surjective. \end{enumerate}


sites-definition-sub-presheaf

We say $\mathcal{F}$ is a {\it subpresheaf} of $\mathcal{G}$ if for every object $U \in \Ob(\mathcal{C})$ the set $\mathcal{F}(U)$ is a subset of $\mathcal{G}(U)$, compatibly with the restriction mappings.


sites-definition-image

Notation as in Lemma \ref{lemma-image}. We say that $\mathcal{G}'$ is the {\it image of $\varphi$}.


sites-definition-family-morphisms-fixed-target

Let $\mathcal{C}$ be a category, see Conventions, Section \ref{conventions-section-categories}. A {\it family of morphisms with fixed target} in $\mathcal{C}$ is given by an object $U \in \Ob(\mathcal{C})$, a set $I$ and for each $i\in I$ a morphism $U_i \to U$ of $\mathcal{C}$ with target $U$. We use the notation $\{U_i \to U\}_{i\in I}$ to indicate this.


sites-definition-site

A {\it site}\footnote{This notation differs from that of \cite{SGA4}, as explained in the introduction.} is given by a category $\mathcal{C}$ and a set $\text{Cov}(\mathcal{C})$ of families of morphisms with fixed target $\{U_i \to U\}_{i \in I}$, called {\it coverings of $\mathcal{C}$}, satisfying the following axioms \begin{enumerate} \item If $V \to U$ is an isomorphism then $\{V \to U\} \in \text{Cov}(\mathcal{C})$. \item If $\{U_i \to U\}_{i\in I} \in \text{Cov}(\mathcal{C})$ and for each $i$ we have $\{V_{ij} \to U_i\}_{j\in J_i} \in \text{Cov}(\mathcal{C})$, then $\{V_{ij} \to U\}_{i \in I, j\in J_i} \in \text{Cov}(\mathcal{C})$. \item If $\{U_i \to U\}_{i\in I}\in \text{Cov}(\mathcal{C})$ and $V \to U$ is a morphism of $\mathcal{C}$ then $U_i \times_U V$ exists for all $i$ and $\{U_i \times_U V \to V \}_{i\in I} \in \text{Cov}(\mathcal{C})$. \end{enumerate}


sites-definition-sheaf-sets

Let $\mathcal{C}$ be a site, and let $\mathcal{F}$ be a presheaf of sets on $\mathcal{C}$. We say $\mathcal{F}$ is a {\it sheaf} if for every covering $\{U_i \to U\}_{i \in I} \in \text{Cov}(\mathcal{C})$ the diagram \begin{equation} \label{equation-sheaf-condition} \xymatrix{ \mathcal{F}(U) \ar[r] & \prod\nolimits_{i\in I} \mathcal{F}(U_i) \ar@<1ex>[r]^-{\text{pr}_0^*} \ar@<-1ex>[r]_-{\text{pr}_1^*} & \prod\nolimits_{(i_0, i_1) \in I \times I} \mathcal{F}(U_{i_0} \times_U U_{i_1}) } \end{equation} represents the first arrow as the equalizer of $\text{pr}_0^*$ and $\text{pr}_1^*$.


sites-definition-category-sheaves-sets

The category {\it $\Sh(\mathcal{C})$} of sheaves of sets is the full subcategory of the category $\textit{PSh}(\mathcal{C})$ whose objects are the sheaves of sets.


sites-definition-sheaf

Let $\mathcal{C}$ be a site, let $\mathcal{A}$ be a category and let $\mathcal{F}$ be a presheaf on $\mathcal{C}$ with values in $\mathcal{A}$. We say that $\mathcal{F}$ is a {\it sheaf} if for all objects $X$ of $\mathcal{A}$ the presheaf of sets $\mathcal{F}_X$ (defined above) is a sheaf.


sites-definition-morphism-coverings

Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{U_i \to U\}_{i\in I}$ be a family of morphisms of $\mathcal{C}$ with fixed target. Let $\mathcal{V} = \{V_j \to V\}_{j\in J}$ be another. \begin{enumerate} \item A {\it morphism of families of maps with fixed target of $\mathcal{C}$ from  $\mathcal{U}$ to $\mathcal{V}$}, or simply a {\it morphism from $\mathcal{U}$ to $\mathcal{V}$} is given by a morphism $U \to V$, a map of sets $\alpha : I \to J$ and for each $i\in I$ a morphism $U_i \to V_{\alpha(i)}$ such that the diagram $$ \xymatrix{ U_i \ar[r] \ar[d] & V_{\alpha(i)} \ar[d] \\ U \ar[r] & V } $$ is commutative. \item In the special case that $U = V$ and $U \to V$ is the identity we call $\mathcal{U}$ a {\it refinement} of the family $\mathcal{V}$. \end{enumerate}


sites-definition-combinatorial-tautological

Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{\varphi_i : U_i \to U\}_{i\in I}$, and $\mathcal{V} = \{\psi_j : V_j \to U\}_{j\in J}$ be two families of morphisms with fixed target. \begin{enumerate} \item  We say $\mathcal{U}$ and $\mathcal{V}$ are {\it combinatorially equivalent} if there exist maps $\alpha : I \to J$ and $\beta : J\to I$ such that $\varphi_i = \psi_{\alpha(i)}$ and $\psi_j = \varphi_{\beta(j)}$. \item We say $\mathcal{U}$ and $\mathcal{V}$ are {\it tautologically equivalent} if there exist maps $\alpha : I \to J$ and $\beta : J\to I$ and for all $i\in I$ and $j \in J$ commutative diagrams $$ \xymatrix{ U_i \ar[rd] \ar[rr] & & V_{\alpha(i)} \ar[ld] & & V_j \ar[rd] \ar[rr] & & U_{\beta(j)} \ar[ld] \\ & U & & & & U & } $$ with isomorphisms as horizontal arrows. \end{enumerate}


sites-definition-separated

We say that a presheaf of sets $\mathcal{F}$ on a site $\mathcal{C}$ is {\it separated} if, for all coverings of $\{U_i \rightarrow U\}$, the map $\mathcal{F}(U) \to \prod \mathcal{F}(U_i)$ is injective.


sites-definition-associated-sheaf

Let $\mathcal{C}$ be a site and let $\mathcal{F}$ be a presheaf of sets on $\mathcal{C}$. The sheaf $\mathcal{F}^\# := \mathcal{F}^{++}$ together with the canonical map $\mathcal{F} \to \mathcal{F}^\#$ is called the {\it sheaf associated to $\mathcal{F}$}.


sites-definition-sheaves-injective-surjective

Let $\mathcal{C}$ be a site, and let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of sheaves of sets. \begin{enumerate} \item We say that $\varphi$ is {\it injective} if for every object $U$ of $\mathcal{C}$ the map $\varphi : \mathcal{F}(U) \to \mathcal{G}(U)$ is injective. \item We say that $\varphi$ is {\it surjective} if for every object $U$ of $\mathcal{C}$ and every section $s\in \mathcal{G}(U)$ there exists a covering $\{U_i \to U\}$ such that for all $i$ the restriction $s|_{U_i}$ is in the image of $\varphi : \mathcal{F}(U_i) \to \mathcal{G}(U_i)$. \end{enumerate}


sites-definition-universal-effective-epimorphisms

Let $\mathcal{C}$ be a category. We say that a family $\{U_i \to U\}_{i \in I}$ is an {\it effective epimorphism} if all the morphisms $U_i \to U$ are representable (see Categories, Definition \ref{categories-definition-representable-morphism}), and for any $X\in \Ob(\mathcal{C})$ the sequence $$ \xymatrix{ \Mor_\mathcal{C}(U, X) \ar[r] & \prod\nolimits_{i \in I} \Mor_\mathcal{C}(U_i, X) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod\nolimits_{(i, j) \in I^2} \Mor_\mathcal{C}(U_i \times_U U_j, X) } $$ is an equalizer diagram. We say that a family $\{U_i \to U\}$ is a {\it universal effective epimorphism} if for any morphism $V \to U$ the base change $\{U_i \times_U V \to V\}$ is an effective epimorphism.


sites-definition-weaker-than-canonical

We say that the topology on a site $\mathcal{C}$ is {\it weaker than the canonical topology}, or that the topology is {\it subcanonical} if all the coverings of $\mathcal{C}$ are universal effective epimorphisms.


sites-definition-representable-sheaf

Let $\mathcal{C}$ be a site whose topology is subcanonical. The Yoneda embedding $h$ (see Categories, Section \ref{categories-section-opposite}) presents $\mathcal{C}$ as a full subcategory of the category of sheaves of $\mathcal{C}$. In this case we call sheaves of the form $h_U$ with $U \in \Ob(\mathcal{C})$ {\it representable sheaves} on $\mathcal{C}$. Notation: Sometimes, the representable sheaf $h_U$ associated to $U$ is denoted {\it $\underline{U}$}.


sites-definition-continuous

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. A functor $u : \mathcal{C} \to \mathcal{D}$ is called {\it continuous} if for every $\{V_i \to V\}_{i\in I} \in \text{Cov}(\mathcal{C})$ we have the following \begin{enumerate} \item $\{u(V_i) \to u(V)\}_{i\in I}$ is in $\text{Cov}(\mathcal{D})$, and \item for any morphism $T \to V$ in $\mathcal{C}$ the morphism $u(T \times_V V_i) \to u(T) \times_{u(V)} u(V_i)$ is an isomorphism. \end{enumerate}


sites-definition-morphism-sites

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. A {\it morphism of sites} $f : \mathcal{D} \to \mathcal{C}$ is given by a continuous functor $u : \mathcal{C} \to \mathcal{D}$ such that the functor $u_s$ is exact.


sites-definition-composition-morphisms-sites

Let $\mathcal{C}_i$, $i = 1, 2, 3$ be sites. Let $f : \mathcal{C}_1 \to \mathcal{C}_2$ and $g : \mathcal{C}_2 \to \mathcal{C}_3$ be morphisms of sites given by continuous functors $u : \mathcal{C}_2 \to \mathcal{C}_1$ and $v : \mathcal{C}_3 \to \mathcal{C}_2$. The {\it composition} $g \circ f$ is the morphism of sites corresponding to the functor $u \circ v$.


sites-definition-topos

A {\it topos} is the category $\Sh(\mathcal{C})$ of sheaves on a site $\mathcal{C}$. \begin{enumerate} \item Let $\mathcal{C}$, $\mathcal{D}$ be sites. A {\it morphism of topoi} $f$ from $\Sh(\mathcal{D})$ to $\Sh(\mathcal{C})$ is given by a pair of functors $f_* : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$ and $f^{-1} : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ such that \begin{enumerate} \item we have $$ \Mor_{\Sh(\mathcal{D})}(f^{-1}\mathcal{G}, \mathcal{F}) = \Mor_{\Sh(\mathcal{C})}(\mathcal{G}, f_*\mathcal{F}) $$ bifunctorially, and \item the functor $f^{-1}$ commutes with finite limits, i.e., is left exact. \end{enumerate} \item Let $\mathcal{C}$, $\mathcal{D}$, $\mathcal{E}$ be sites. Given morphisms of topoi $f :\Sh(\mathcal{D}) \to \Sh(\mathcal{C})$ and $g :\Sh(\mathcal{E}) \to \Sh(\mathcal{D})$ the {\it composition $f\circ g$} is the morphism of topoi defined by the functors $(f \circ g)_* = f_* \circ g_*$ and $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$. \end{enumerate}


sites-definition-quasi-compact

Let $\mathcal{C}$ be a site. An object $U$ of $\mathcal{C}$ is {\it quasi-compact} if given a covering $\mathcal{U} = \{U_i \to U\}_{i \in I}$ in $\mathcal{C}$ there exists another covering $\mathcal{V} = \{V_j \to U\}_{j \in J}$ and a morphism $\mathcal{V} \to \mathcal{U}$ of families of maps with fixed target given by $\text{id} : U \to U$, $\alpha : J \to I$, and $V_j \to U_{\alpha(j)}$ (see Definition \ref{definition-morphism-coverings}) such that the image of $\alpha$ is a finite subset of $I$.


sites-definition-quasi-compact-topos

An object $\mathcal{F}$ of a topos $\Sh(\mathcal{C})$ is {\it quasi-compact} if for any surjective map $\coprod_{i \in I} \mathcal{F}_i \to \mathcal{F}$ of $\Sh(\mathcal{C})$ there exists a finite subset $J \subset I$ such that $\coprod_{i \in J} \mathcal{F}_i \to \mathcal{F}$ is surjective. A topos $\Sh(\mathcal{C})$ is said to be {\it quasi-compact} if its final object $*$ is a quasi-compact object.


sites-definition-cocontinuous

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. The functor $u$ is called {\it cocontinuous} if for every $U \in \Ob(\mathcal{C})$ and every covering $\{V_j \to u(U)\}_{j \in J}$ of $\mathcal{D}$ there exists a covering $\{U_i \to U\}_{i\in I}$ of $\mathcal{C}$ such that the family of maps $\{u(U_i) \to u(U)\}_{i \in I}$ refines the covering $\{V_j \to u(U)\}_{j \in J}$.


sites-definition-localize

Let $\mathcal{C}$ be a site. Let $U \in \Ob(\mathcal{C})$. \begin{enumerate} \item The site $\mathcal{C}/U$ is called the {\it localization of the site $\mathcal{C}$ at the object $U$}. \item The morphism of topoi $j_U : \Sh(\mathcal{C}/U) \to \Sh(\mathcal{C})$ is called the {\it localization morphism}. \item The functor $j_{U*}$ is called the {\it direct image functor}. \item For a sheaf $\mathcal{F}$ on $\mathcal{C}$ the sheaf $j_U^{-1}\mathcal{F}$ is called the {\it restriction of $\mathcal{F}$ to $\mathcal{C}/U$}. \item For a sheaf $\mathcal{G}$ on $\mathcal{C}/U$ the sheaf $j_{U!}\mathcal{G}$ is called the {\it extension of $\mathcal{G}$ by the empty set}. \end{enumerate}


sites-definition-special-cocontinuous-functor

Let $\mathcal{C}$, $\mathcal{D}$ be sites. A {\it special cocontinuous functor $u$ from $\mathcal{C}$ to $\mathcal{D}$} is a cocontinuous functor $u : \mathcal{C} \to \mathcal{D}$ satisfying the assumptions and conclusions of Lemma \ref{lemma-equivalence}.


sites-definition-localize-topos

Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. \begin{enumerate} \item The topos $\Sh(\mathcal{C})/\mathcal{F}$ is called the {\it localization of the topos $\Sh(\mathcal{C})$ at $\mathcal{F}$}. \item The morphism of topoi $j_\mathcal{F} : \Sh(\mathcal{C})/\mathcal{F} \to \Sh(\mathcal{C})$ of Lemma \ref{lemma-localize-topos} is called the {\it localization morphism}. \end{enumerate}


sites-definition-point-topos

Let $\mathcal{C}$ be a site. A {\it point of the topos $\Sh(\mathcal{C})$} is a morphism of topoi $p$ from $\Sh(pt)$ to $\Sh(\mathcal{C})$.


sites-definition-point

Let $\mathcal{C}$ be a site. A {\it point $p$ of the site $\mathcal{C}$} is given by a functor $u : \mathcal{C} \to \textit{Sets}$ such that \begin{enumerate} \item For every covering $\{U_i \to U\}$ of $\mathcal{C}$ the map $\coprod u(U_i) \to u(U)$ is surjective. \item For every covering $\{U_i \to U\}$ of $\mathcal{C}$ and every morphism $V \to U$ the maps $u(U_i \times_U V) \to u(U_i) \times_{u(U)} u(V)$ are bijective. \item The stalk functor $\Sh(\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}_p$ is left exact. \end{enumerate}


sites-definition-pushforward-point

Let $p$ be a point of the site $\mathcal{C}$ given by the functor $u$. For a set $E$ we define $p_*E = u^sE$ the sheaf described in Lemma \ref{lemma-point-pushforward-sheaf} above. We sometimes call this a {\it skyscraper sheaf}.


sites-definition-2-morphism-topoi

Let $f, g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be two morphisms of topoi. A {\it 2-morphism from $f$ to $g$} is given by a transformation of functors $t : f_* \to g_*$.


sites-definition-morphism-points

Let $\mathcal{C}$ be a site. Let $p, p'$ be points of $\mathcal{C}$ given by functors $u, u' : \mathcal{C} \to \textit{Sets}$. A {\it morphism $f : p \to p'$} is given by a transformation of functors $$ f_u : u' \to u. $$


sites-definition-enough-points

Let $\mathcal{C}$ be a site. \begin{enumerate} \item A family of points $\{p_i\}_{i\in I}$ is called {\it conservative} if every map of sheaves $\phi : \mathcal{F} \to \mathcal{G}$ which is an isomorphism on all the fibres $\mathcal{F}_{p_i} \to \mathcal{G}_{p_i}$ is an isomorphism. \item  We say that $\mathcal{C}$ {\it has enough points} if there exists a conservative family of points. \end{enumerate}


sites-definition-w-contractible

Let $\mathcal{C}$ be a site. \begin{enumerate} \item We say an object $U$ of $\mathcal{C}$ is {\it weakly contractible} if the equivalent conditions of Lemma \ref{lemma-w-contractible} hold. \item We say a site has {\it enough weakly contractible objects} if every object $U$ of $\mathcal{C}$ has a covering $\{U_i \to U\}$ with $U_i$ weakly contractible for all $i$. \item More generally, if $P$ is a property of objects of $\mathcal{C}$ we say that $\mathcal{C}$ has {\it enough $P$ objects} if every object $U$ of $\mathcal{C}$ has a covering $\{U_i \to U\}$ such that $U_i$ has $P$ for all $i$. \end{enumerate}


sites-definition-empty

Let $\mathcal{C}$ be a site. We say an object $U$ of $\mathcal{C}$ is {\it sheaf theoretically empty} if $\emptyset^\# \to h_U^\#$ is an isomorphism of sheaves.


sites-definition-almost-cocontinuous

Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. We say $u$ is {\it almost cocontinuous} if for every object $U$ of $\mathcal{C}$ and every covering $\{V_j \to u(U)\}_{j \in J}$ there exists a covering $\{U_i \to U\}_{i \in I}$ in $\mathcal{C}$ such that for each $i$ in $I$ we have at least one of the following two conditions \begin{enumerate} \item $u(U_i)$ is sheaf theoretically empty, or \item the morphism $u(U_i) \to u(U)$ factors through $V_j$ for some $j \in J$. \end{enumerate}


sites-definition-embedding

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. A morphism of topoi $f : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$ is called an {\it embedding} if $f_*$ is fully faithful.


sites-definition-subtopos

Let $\mathcal{C}$ be a site. A strictly full subcategory $E \subset \Sh(\mathcal{C})$ is a {\it subtopos} if there exists an embedding of topoi $f : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$ such that $E$ is equal to the essential image of the functor $f_*$.


sites-definition-open-subtopos

Let $\mathcal{C}$ be a site. A strictly full subcategory $E \subset \Sh(\mathcal{C})$ is an {\it open subtopos} if there exists a subsheaf $\mathcal{F}$ of the final object of $\Sh(\mathcal{C})$ such that $E$ is the subtopos $\Sh(\mathcal{C})/\mathcal{F}$ described in Lemma \ref{lemma-open-subtopos}.


sites-definition-closed-subtopos

Let $\mathcal{C}$ be a site. A strictly full subcategory $E \subset \Sh(\mathcal{C})$ is an {\it closed subtopos} if there exists a subsheaf $\mathcal{F}$ of the final object of $\Sh(\mathcal{C})$ such that $E$ is the subtopos described in Lemma \ref{lemma-closed-subtopos}.


sites-definition-immersion-topoi

Let $f : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$ be a morphism of topoi. \begin{enumerate} \item We say $f$ is an {\it open immersion} if $f$ is an embedding and the essential image of $f_*$ is an open subtopos. \item We say $f$ is a {\it closed immersion} if $f$ is an embedding and the essential image of $f_*$ is a closed subtopos. \end{enumerate}


sites-definition-pushforward-algebraic-structures

Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites given by a functor $u : \mathcal{C} \to \mathcal{D}$. We define the {\it pushforward} functor for presheaves of algebraic structures by the rule $u^p\mathcal{F}(U) = \mathcal{F}(uU)$, and for sheaves of algebraic structures by the same rule, namely $f_*\mathcal{F}(U) = \mathcal{F}(uU)$.


sites-definition-global-sections

The {\it global sections} of a presheaf of sets $\mathcal{F}$ over a site $\mathcal{C}$ is the set $$ \Gamma(\mathcal{C}, \mathcal{F}) = \Mor_{\textit{PSh}(\mathcal{C})}(*, \mathcal{F}) $$ where $*$ is the final object in the category of presheaves on $\mathcal{C}$, i.e., the presheaf which associates to every object a singleton.


sites-definition-sieve

Let $\mathcal{C}$ be a category. Let $U \in \Ob(\mathcal{C})$. A {\it sieve $S$ on $U$} is a subpresheaf $S \subset h_U$.


sites-definition-sieve-generated

Let $\mathcal{C}$ be a category. Given a family of morphisms $\{f_i : U_i \to U\}_{i\in I}$ of $\mathcal{C}$ with target $U$ we say the sieve $S$ on $U$ described in Lemma \ref{lemma-sieves-set} part (\ref{item-sieve-generated}) is the {\it sieve  on $U$ generated by the morphisms $f_i$}.


sites-definition-pullback-sieve

Let $\mathcal{C}$ be a category. Let $f : V \to U$ be a morphism of $\mathcal{C}$. Let $S \subset h_U$ be a sieve. We define the {\it pullback of $S$ by $f$} to be the sieve $S \times_U V$ of $V$ defined by the rule $$ (\alpha : T \to V) \in (S \times_U V)(T) \Leftrightarrow (f \circ \alpha : T \to U) \in S(T) $$


sites-definition-topology

Let $\mathcal{C}$ be a category. A {\it topology on $\mathcal{C}$} is given by a rule which assigns to every $U \in \Ob(\mathcal{C})$ a subset $J(U)$ of the set of all sieves on $U$ satisfying the following conditions \begin{enumerate} \item For every morphism $f : V \to U$ in $\mathcal{C}$, and every element $S \in J(U)$ the pullback $S \times_U V$ is an element of $J(V)$. \item If $S$ and $S'$ are sieves on $U \in \Ob(\mathcal{C})$, if $S \in J(U)$, and if for all $f \in S(V)$ the pullback $S' \times_U V$ belongs to $J(V)$, then $S'$ belongs to $J(U)$. \item For every $U \in \Ob(\mathcal{C})$ the maximal sieve $S = h_U$ belongs to $J(U)$. \end{enumerate}


sites-definition-finer

Let $\mathcal{C}$ be a category. Let $J$, $J'$ be two topologies on $\mathcal{C}$. We say that $J$ is {\it finer} or {\it stronger} than $J'$ if and only if for every object $U$ of $\mathcal{C}$ we have $J'(U) \subset J(U)$. In this case we also say that $J'$ is {\it coarser} or {\it weaker} than $J$.


sites-definition-sheaf-sets-topology

Let $\mathcal{C}$ be a category endowed with a topology $J$. Let $\mathcal{F}$ be a presheaf of sets on $\mathcal{C}$. We say that $\mathcal{F}$ is a {\it sheaf} on $\mathcal{C}$ if for every $U \in \Ob(\mathcal{C})$ and for every covering sieve $S$ of $U$ the canonical map $$ \Mor_{\textit{PSh}(\mathcal{C})}(h_U, \mathcal{F}) \longrightarrow \Mor_{\textit{PSh}(\mathcal{C})}(S, \mathcal{F}) $$ is bijective.


sites-definition-canonical-topology

Let $\mathcal{C}$ be a category. The finest topology on $\mathcal{C}$ such that all representable presheaves are sheaves, see Lemma \ref{lemma-topology-presheaves-sheaves}, is called the {\it canonical topology} of $\mathcal{C}$.


sites-definition-topology-associated-site

Let $\mathcal{C}$ be a site with coverings $\text{Cov}(\mathcal{C})$. The {\it topology associated to $\mathcal{C}$} is the topology $J$ constructed in Lemma \ref{lemma-site-gives-topology} above.


sites-definition-presheaf-separated-topology

Let $\mathcal{C}$ be a category. Let $J$ be a topology on $\mathcal{C}$. We say that a presheaf of sets $\mathcal{F}$ is {\it separated} if for every object $U$ and every covering sieve $S$ on $U$ the canonical map $\mathcal{F}(U) \to \Mor_{\textit{PSh}(\mathcal{C})}(S, \mathcal{F})$ is injective.


sites-definition-associated-sheaf-topology

Let $\mathcal{C}$ be a category endowed with a topology $J$. Let $\mathcal{F}$ be a presheaf of sets on $\mathcal{C}$. The sheaf $\mathcal{F}^\# := LL\mathcal{F}$ together with the canonical map $\mathcal{F} \to \mathcal{F}^\#$ is called the {\it sheaf associated to $\mathcal{F}$}.


sites-definition-point-topology

Let $\mathcal{C}$ be a category. Let $J$ be a topology on $\mathcal{C}$. A {\it point $p$} of the topology is given by a functor $u : \mathcal{C} \to \textit{Sets}$ such that \begin{enumerate} \item For every covering sieve $S$ on $U$ the map $S_p \to (h_U)_p$ is surjective. \item The stalk functor $\Sh(\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \to \mathcal{F}_p$ is exact. \end{enumerate}


more-etale-definition-f-shriek-separated

Let $f : X \to Y$ be a morphism of schemes which is separated (!) and locally of finite type. Let $\mathcal{F}$ be an abelian sheaf on $X_\etale$. The subsheaf $f_!\mathcal{F} \subset f_*\mathcal{F}$ constructed in Lemma \ref{lemma-f-shriek-separated} is called the {\it direct image with compact support}.


more-etale-definition-compact-support

Let $X$ be a separated scheme locally of finite type over a field $k$. Let $\mathcal{F}$ be an abelian sheaf on $X_\etale$. We let $H^0_c(X, \mathcal{F}) \subset H^0(X, \mathcal{F})$ be the set of sections whose support is proper over $k$. Elements of $H^0_c(X, \mathcal{F})$ are called {\it sections with compact support}.


more-etale-definition-f-shriek-lqf

Let $f : X \to Y$ be a locally quasi-finite morphism of schemes. We define the {\it direct image with compact support} to be the functor $$ f_! : \textit{Ab}(X_\etale) \longrightarrow \textit{Ab}(Y_\etale) $$ defined by the formula $f_!\mathcal{F} = (f_{p!}\mathcal{F})^\#$, i.e., $f_!\mathcal{F}$ is the sheafification of the presheaf $f_{p!}\mathcal{F}$ constructed above.


stacks-properties-definition-points

Let $\mathcal{X}$ be an algebraic stack. A {\it point} of $\mathcal{X}$ is an equivalence class of morphisms from spectra of fields into $\mathcal{X}$. The set of points of $\mathcal{X}$ is denoted $|\mathcal{X}|$.


stacks-properties-definition-topological-space

Let $\mathcal{X}$ be an algebraic stack. The underlying {\it topological space} of $\mathcal{X}$ is the set of points $|\mathcal{X}|$ endowed with the topology constructed in Lemma \ref{lemma-topology-points}.


stacks-properties-definition-surjective

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is {\it surjective} if the map $|f| : |\mathcal{X}| \to |\mathcal{Y}|$ of associated topological spaces is surjective.


stacks-properties-definition-quasi-compact

Let $\mathcal{X}$ be an algebraic stack. We say $\mathcal{X}$ is {\it quasi-compact} if and only if $|\mathcal{X}|$ is quasi-compact.


stacks-properties-definition-type-property

Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{P}$ be a property of schemes which is local in the smooth topology. We say $\mathcal{X}$ {\it has property $\mathcal{P}$} if any of the equivalent conditions of Lemma \ref{lemma-type-property} hold.


stacks-properties-definition-property-at-point

Let $\mathcal{P}$ be a property of germs of schemes which is smooth local. Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$. We say $\mathcal{X}$ {\it has property $\mathcal{P}$ at $x$} if any of the equivalent conditions of Lemma \ref{lemma-local-source-target-at-point} holds.


stacks-properties-definition-monomorphism

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is a {\it monomorphism} if it is representable by algebraic spaces and a monomorphism in the sense of Section \ref{section-properties-morphisms}.


stacks-properties-definition-immersion

Immersions. \begin{enumerate} \item A morphism of algebraic stacks is called an {\it open immersion} if it is representable, and an open immersion in the sense of Section \ref{section-properties-morphisms}. \item A morphism of algebraic stacks is called a {\it closed immersion} if it is representable, and a closed immersion in the sense of Section \ref{section-properties-morphisms}. \item A morphism of algebraic stacks is called an {\it immersion} if it is representable, and an immersion in the sense of Section \ref{section-properties-morphisms}. \end{enumerate}


stacks-properties-definition-substacks

Let $\mathcal{X}$ be an algebraic stack. \begin{enumerate} \item An {\it open substack} of $\mathcal{X}$ is a strictly full subcategory $\mathcal{X}' \subset \mathcal{X}$ such that $\mathcal{X}'$ is an algebraic stack and $\mathcal{X}' \to \mathcal{X}$ is an open immersion. \item A {\it closed substack} of $\mathcal{X}$ is a strictly full subcategory $\mathcal{X}' \subset \mathcal{X}$ such that $\mathcal{X}'$ is an algebraic stack and $\mathcal{X}' \to \mathcal{X}$ is a closed immersion. \item A {\it locally closed substack} of $\mathcal{X}$ is a strictly full subcategory $\mathcal{X}' \subset \mathcal{X}$ such that $\mathcal{X}'$ is an algebraic stack and $\mathcal{X}' \to \mathcal{X}$ is an immersion. \end{enumerate}


stacks-properties-definition-reduced-induced-stack

Let $\mathcal{X}$ be an algebraic stack. Let $Z \subset |\mathcal{X}|$ be a closed subset. An {\it algebraic stack structure on $Z$} is given by a closed substack $\mathcal{Z}$ of $\mathcal{X}$ with $|\mathcal{Z}|$ equal to $Z$. The {\it reduced induced algebraic stack structure} on $Z$ is the one constructed in Lemma \ref{lemma-reduced-closed-substack}. The {\it reduction $\mathcal{X}_{red}$ of $\mathcal{X}$} is the reduced induced algebraic stack structure on $|\mathcal{X}|$.


stacks-properties-definition-residual-gerbe

Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$. \begin{enumerate} \item We say the {\it residual gerbe of $\mathcal{X}$ at $x$ exists} if the equivalent conditions (1), (2), and (3) of Lemma \ref{lemma-residual-gerbe} hold. \item If the residual gerbe of $\mathcal{X}$ at $x$ exists, then the {\it residual gerbe of $\mathcal{X}$ at $x$}\footnote{This clashes with \cite{LM-B} in spirit, but not in fact. Namely, in Chapter 11 they associate to any point on any quasi-separated algebraic stack a gerbe (not necessarily algebraic) which they call the residual gerbe. We will see in Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-every-point-residual-gerbe} that on a quasi-separated algebraic stack every point has a residual gerbe in our sense which is then equivalent to theirs. For more information on this topic see \cite[Appendix B]{rydh_etale_devissage}.} is the strictly full subcategory $\mathcal{Z}_x \subset \mathcal{X}$ constructed in Lemma \ref{lemma-residual-gerbe}. \end{enumerate}


stacks-properties-definition-dimension-at-point

Let $\mathcal{X}$ be a locally Noetherian algebraic stack over a scheme $S$. Let $x \in |\mathcal{X}|$ be a point of $\mathcal{X}$. Let $[U/R] \to \mathcal{X}$ be a presentation (Algebraic Stacks, Definition \ref{algebraic-definition-presentation}) where $U$ is a scheme and let $u \in U$ be a point that maps to $x$. We define the {\it dimension of $\mathcal{X}$ at $x$} to be the element $\dim_x(\mathcal{X}) \in \mathbf{Z} \cup \infty$ such that  $$ \dim_x(\mathcal{X}) = \dim_u(U)-\dim_{e(u)}(R_u). $$ with notation as in Lemma \ref{lemma-dimension-at-point-well-defined}.


stacks-properties-definition-dimension

Let $S$ be a scheme. Let $\mathcal{X}$ be a locally Noetherian algebraic stack over $S$. The {\it dimension} $\dim(\mathcal{X})$ of $\mathcal{X}$ is defined to be $$ \dim(\mathcal{X}) = \sup\nolimits_{x \in |\mathcal{X}|} \dim_x(\mathcal{X}) $$


stacks-properties-definition-number-of-geometric-branches

Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$. \begin{enumerate} \item The {\it number of geometric branches of $\mathcal{X}$ at $x$} is either $n \in \mathbf{N}$ if the equivalent conditions of Lemma \ref{lemma-local-source-target-at-point} hold for $\mathcal{P}_n$ defined above, or else $\infty$. \item We say $\mathcal{X}$ is {\it geometrically unibranch at $x$} if the number of geometric branches of $\mathcal{X}$ at $x$ is $1$. \end{enumerate}


stacks-definition-mor-presheaf

Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category, see Categories, Section \ref{categories-section-fibred-categories}. Given an object $U$ of $\mathcal{C}$ and objects $x$, $y$ of the fibre category, the {\it presheaf of morphisms from $x$ to $y$} is the presheaf $$ (f : V \to U) \longmapsto \Mor_{\mathcal{S}_V}(f^*x, f^*y) $$ described above. It is denoted $\mathit{Mor}(x, y)$. The subpresheaf $\mathit{Isom}(x, y)$ whose values over $V$ is the set of isomorphisms $f^*x \to f^*y$ in the fibre category $\mathcal{S}_V$ is called the {\it presheaf of isomorphisms from $x$ to $y$}.


stacks-definition-descent-data

Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. Make a choice of pullbacks as in Categories, Definition \ref{categories-definition-pullback-functor-fibred-category}. Let $\mathcal{U} = \{f_i : U_i \to U\}_{i \in I}$ be a family of morphisms of $\mathcal{C}$. Assume all the fibre products $U_i \times_U U_j$, and $U_i \times_U U_j \times_U U_k$ exist. \begin{enumerate} \item A {\it descent datum $(X_i, \varphi_{ij})$ in $\mathcal{S}$ relative to the family $\{f_i : U_i \to U\}$} is given by an object $X_i$ of $\mathcal{S}_{U_i}$ for each $i \in I$, an isomorphism $\varphi_{ij} : \text{pr}_0^*X_i \to \text{pr}_1^*X_j$ in $\mathcal{S}_{U_i \times_U U_j}$ for each pair $(i, j) \in I^2$ such that for every triple of indices $(i, j, k) \in I^3$ the diagram $$ \xymatrix{ \text{pr}_0^*X_i \ar[rd]_{\text{pr}_{01}^*\varphi_{ij}} \ar[rr]_{\text{pr}_{02}^*\varphi_{ik}} & & \text{pr}_2^*X_k \\ & \text{pr}_1^*X_j \ar[ru]_{\text{pr}_{12}^*\varphi_{jk}} & } $$ in the category $\mathcal{S}_{U_i \times_U U_j \times_U U_k}$ commutes. This is called the {\it cocycle condition}. \item A {\it morphism $\psi : (X_i, \varphi_{ij}) \to (X'_i, \varphi'_{ij})$ of descent data} is given by a family $\psi = (\psi_i)_{i\in I}$ of morphisms $\psi_i : X_i \to X'_i$ in $\mathcal{S}_{U_i}$ such that all the diagrams $$ \xymatrix{ \text{pr}_0^*X_i \ar[r]_{\varphi_{ij}} \ar[d]_{\text{pr}_0^*\psi_i} & \text{pr}_1^*X_j \ar[d]^{\text{pr}_1^*\psi_j} \\ \text{pr}_0^*X'_i \ar[r]^{\varphi'_{ij}} & \text{pr}_1^*X'_j \\ } $$ in the categories $\mathcal{S}_{U_i \times_U U_j}$ commute. \item The category of descent data relative to $\mathcal{U}$ is denoted $DD(\mathcal{U})$. \end{enumerate}


stacks-definition-pullback-functor

With $\mathcal{U} = \{U_i \to U\}_{i \in I}$, $\mathcal{V} = \{V_j \to V\}_{j \in J}$, $\alpha : I \to J$, $h : U \to V$, and $g_i : U_i \to V_{\alpha(i)}$ as in Lemma \ref{lemma-pullback} 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.


stacks-definition-effective-descent-datum

Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. Make a choice of pullbacks as in Categories, Definition \ref{categories-definition-pullback-functor-fibred-category}. Let $\mathcal{U} = \{f_i : U_i \to U\}_{i \in I}$ be a family of morphisms with target $U$. Assume all the fibre products $U_i \times_U U_j$ and $U_i \times_U U_j \times_U U_k$ exist. \begin{enumerate} \item Given an object $X$ of $\mathcal{S}_U$ the {\it trivial descent datum} is the descent datum $(X, \text{id}_X)$ with respect to the family $\{\text{id}_U : U \to U\}$. \item Given an object $X$ of $\mathcal{S}_U$ we have a {\it canonical descent datum} on the family of objects $f_i^*X$ by pulling back the trivial descent datum $(X, \text{id}_X)$ via the obvious map $\{f_i : U_i \to U\} \to \{\text{id}_U : U \to U\}$. We denote this descent datum $(f_i^*X, can)$. \item A descent datum $(X_i, \varphi_{ij})$ relative to $\{f_i : U_i \to U\}$ is called {\it effective} if there exists an object $X$ of $\mathcal{S}_U$ such that $(X_i, \varphi_{ij})$ is isomorphic to $(f_i^*X, can)$. \end{enumerate}


stacks-definition-stack

Let $\mathcal{C}$ be a site. A {\it stack} over $\mathcal{C}$ is a category $p : \mathcal{S} \to \mathcal{C}$ over $\mathcal{C}$ which satisfies the following conditions: \begin{enumerate} \item $p : \mathcal{S} \to \mathcal{C}$ is a fibred category, see Categories, Definition \ref{categories-definition-fibred-category}, \item for any $U \in \Ob(\mathcal{C})$ and any $x, y \in \mathcal{S}_U$ the presheaf $\mathit{Mor}(x, y)$ (see Definition \ref{definition-mor-presheaf}) is a sheaf on the site $\mathcal{C}/U$, and \item for any covering $\mathcal{U} = \{f_i : U_i \to U\}_{i \in I}$ of the site $\mathcal{C}$, any descent datum in $\mathcal{S}$ relative to $\mathcal{U}$ is effective. \end{enumerate}


stacks-definition-stacks-over-C

Let $\mathcal{C}$ be a site. The {\it $2$-category of stacks over $\mathcal{C}$} is the sub $2$-category of the $2$-category of fibred categories over $\mathcal{C}$ (see Categories, Definition \ref{categories-definition-fibred-categories-over-C}) defined as follows: \begin{enumerate} \item Its objects will be stacks $p : \mathcal{S} \to \mathcal{C}$. \item Its $1$-morphisms $(\mathcal{S}, p) \to (\mathcal{S}', p')$ will be functors $G : \mathcal{S} \to \mathcal{S}'$ such that $p' \circ G = p$ and such that $G$ maps strongly cartesian morphisms to strongly cartesian morphisms. \item Its $2$-morphisms $t : G \to H$ for $G, H : (\mathcal{S}, p) \to (\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_x) = \text{id}_{p(x)}$ for all $x \in \Ob(\mathcal{S})$. \end{enumerate}


stacks-definition-stack-in-groupoids

A {\it stack in groupoids} over a site $\mathcal{C}$ is a category $p : \mathcal{S} \to \mathcal{C}$ over $\mathcal{C}$ such that \begin{enumerate} \item $p : \mathcal{S} \to \mathcal{C}$ is fibred in groupoids over $\mathcal{C}$ (see Categories, Definition \ref{categories-definition-fibred-groupoids}), \item for all $U \in \Ob(\mathcal{C})$, for all $x, y\in \Ob(\mathcal{S}_U)$ the presheaf $\mathit{Isom}(x, y)$ is a sheaf on the site $\mathcal{C}/U$, and \item for all coverings $\mathcal{U} = \{U_i \to U\}$ in $\mathcal{C}$, all descent data $(x_i, \phi_{ij})$ for $\mathcal{U}$ are effective. \end{enumerate}


stacks-definition-stacks-in-groupoids-over-C

Let $\mathcal{C}$ be a site. The {\it $2$-category of stacks in groupoids over $\mathcal{C}$} is the sub $2$-category of the $2$-category of stacks over $\mathcal{C}$ (see Definition \ref{definition-stacks-over-C}) defined as follows: \begin{enumerate} \item Its objects will be stacks in groupoids $p : \mathcal{S} \to \mathcal{C}$. \item Its $1$-morphisms $(\mathcal{S}, p) \to (\mathcal{S}', p')$ will be functors $G : \mathcal{S} \to \mathcal{S}'$ such that $p' \circ G = p$. (Since every morphism is strongly cartesian every functor preserves them.) \item Its $2$-morphisms $t : G \to H$ for $G, H : (\mathcal{S}, p) \to (\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_x) = \text{id}_{p(x)}$ for all $x \in \Ob(\mathcal{S})$. \end{enumerate}


stacks-definition-stack-in-sets

Let $\mathcal{C}$ be a site. \begin{enumerate} \item A {\it stack in setoids} over $\mathcal{C}$ is a stack over $\mathcal{C}$ all of whose fibre categories are setoids. \item A {\it stack in sets}, or a {\it stack in discrete categories} is a stack over $\mathcal{C}$ all of whose fibre categories are discrete. \end{enumerate}


stacks-definition-stacks-in-setoids-over-C

Let $\mathcal{C}$ be a site. The {\it $2$-category of stacks in setoids over $\mathcal{C}$} is the sub $2$-category of the $2$-category of stacks over $\mathcal{C}$ (see Definition \ref{definition-stacks-over-C}) defined as follows: \begin{enumerate} \item Its objects will be stacks in setoids $p : \mathcal{S} \to \mathcal{C}$. \item Its $1$-morphisms $(\mathcal{S}, p) \to (\mathcal{S}', p')$ will be functors $G : \mathcal{S} \to \mathcal{S}'$ such that $p' \circ G = p$. (Since every morphism is strongly cartesian every functor preserves them.) \item Its $2$-morphisms $t : G \to H$ for $G, H : (\mathcal{S}, p) \to (\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_x) = \text{id}_{p(x)}$ for all $x \in \Ob(\mathcal{S})$. \end{enumerate}


stacks-definition-topology-inherited

Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. We say $(\mathcal{S}, \text{Cov}(\mathcal{S}))$ as in Lemma \ref{lemma-topology-inherited} is the {\it structure of site on $\mathcal{S}$ inherited from $\mathcal{C}$}. We sometimes indicate this by saying that {\it $\mathcal{S}$ is endowed with the topology inherited from $\mathcal{C}$}.


stacks-definition-gerbe

A {\it gerbe} over a site $\mathcal{C}$ is a category $p : \mathcal{S} \to \mathcal{C}$ over $\mathcal{C}$ such that \begin{enumerate} \item $p : \mathcal{S} \to \mathcal{C}$ is a stack in groupoids over $\mathcal{C}$ (see Definition \ref{definition-stack-in-groupoids}), \item for $U \in \Ob(\mathcal{C})$ there exists a covering $\{U_i \to U\}$ in $\mathcal{C}$ such that $\mathcal{S}_{U_i}$ is nonempty, and \item for $U \in \Ob(\mathcal{C})$ and $x, y \in \Ob(\mathcal{S}_U)$ there exists a covering $\{U_i \to U\}$ in $\mathcal{C}$ such that $x|_{U_i} \cong y|_{U_i}$ in $\mathcal{S}_{U_i}$. \end{enumerate}


stacks-definition-gerbe-over-stack-in-groupoids

Let $\mathcal{C}$ be a site. Let $\mathcal{X}$ and $\mathcal{Y}$ be stacks in groupoids over $\mathcal{C}$. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories over $\mathcal{C}$. We say $\mathcal{X}$ is a {\it gerbe over} $\mathcal{Y}$ if the equivalent conditions of Lemma \ref{lemma-when-gerbe} are satisfied.


stacks-definition-pushforward-stack

Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites given by the continuous functor $u : \mathcal{C} \to \mathcal{D}$. Let $\mathcal{S}$ be a fibred category over $\mathcal{D}$. In this setting we write {\it $f_*\mathcal{S}$} for the fibred category $u^p\mathcal{S}$ defined above. We say that $f_*\mathcal{S}$ is the {\it pushforward of $\mathcal{S}$ along $f$}.


stacks-definition-pullback-stack

Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites given by a continuous functor $u : \mathcal{C} \to \mathcal{D}$ satisfying the hypotheses and conclusions of Sites, Proposition \ref{sites-proposition-get-morphism}. Let $\mathcal{S}$ be a stack over $\mathcal{C}$. In this setting we write {\it $f^{-1}\mathcal{S}$} for the stackification of the fibred category $u_p\mathcal{S}$ over $\mathcal{D}$ constructed above. We say that $f^{-1}\mathcal{S}$ is the {\it pullback of $\mathcal{S}$ along $f$}.


spaces-simplicial-definition-cartesian-sheaf

In Situation \ref{situation-simplicial-site}. \begin{enumerate} \item A sheaf $\mathcal{F}$ of sets or of abelian groups on $\mathcal{C}$ is {\it cartesian} if the maps $\mathcal{F}(\varphi) : f_\varphi^{-1}\mathcal{F}_m \to \mathcal{F}_n$ are isomorphisms for all $\varphi : [m] \to [n]$. \item If $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}_{total}$, then a sheaf $\mathcal{F}$ of $\mathcal{O}$-modules is {\it cartesian} if  the maps $f_\varphi^*\mathcal{F}_m \to \mathcal{F}_n$ are isomorphisms for all $\varphi : [m] \to [n]$. \item An object $K$ of $D(\mathcal{C}_{total})$ is {\it cartesian} if the maps $f_\varphi^{-1}K_m \to K_n$ are isomorphisms for all $\varphi : [m] \to [n]$. \item If $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}_{total}$, then an object $K$ of $D(\mathcal{O})$ is {\it cartesian} if the maps $Lf_\varphi^*K_m \to K_n$ are isomorphisms for all $\varphi : [m] \to [n]$. \end{enumerate}


spaces-simplicial-definition-cartesian-derived

In Situation \ref{situation-simplicial-site}. A {\it simplicial system of the derived category} consists of the following data \begin{enumerate} \item for every $n$ an object $K_n$ of $D(\mathcal{C}_n)$, \item for every $\varphi : [m] \to [n]$ a map $K_\varphi : f_\varphi^{-1}K_m \to K_n$ in $D(\mathcal{C}_n)$ \end{enumerate} subject to the condition that $$ K_{\varphi \circ \psi} = K_\varphi \circ f_\varphi^{-1}K_\psi : f_{\varphi \circ \psi}^{-1}K_l = f_\varphi^{-1} f_\psi^{-1}K_l \longrightarrow K_n $$ for any morphisms $\varphi : [m] \to [n]$ and $\psi : [l] \to [m]$ of $\Delta$. We say the simplicial system is {\it cartesian} if the maps $K_\varphi$ are isomorphisms for all $\varphi$. Given two simplicial systems of the derived category there is an obvious notion of a {\it morphism of simplicial systems of the derived category}.


spaces-simplicial-definition-cartesian-derived-modules

In Situation \ref{situation-simplicial-site}. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. A {\it simplicial system of the derived category of modules} consists of the following data \begin{enumerate} \item for every $n$ an object $K_n$ of $D(\mathcal{O}_n)$, \item for every $\varphi : [m] \to [n]$ a map $K_\varphi : Lf_\varphi^*K_m \to K_n$ in $D(\mathcal{O}_n)$ \end{enumerate} subject to the condition that $$ K_{\varphi \circ \psi} = K_\varphi \circ Lf_\varphi^*K_\psi : Lf_{\varphi \circ \psi}^*K_l = Lf_\varphi^* Lf_\psi^*K_l \longrightarrow K_n $$ for any morphisms $\varphi : [m] \to [n]$ and $\psi : [l] \to [m]$ of $\Delta$. We say the simplicial system is {\it cartesian} if the maps $K_\varphi$ are isomorphisms for all $\varphi$. Given two simplicial systems of the derived category there is an obvious notion of a {\it morphism of simplicial systems of the derived category of modules}.


spaces-simplicial-definition-cartesian-morphism

Let $a : Y \to X$ be a morphism of simplicial schemes. We say $a$ is {\it cartesian}, or that {\it $Y$ is cartesian over $X$}, if for every morphism $\varphi : [n] \to [m]$ of $\Delta$ the corresponding diagram $$ \xymatrix{ Y_m \ar[r]_a \ar[d]_{Y(\varphi)} & X_m \ar[d]^{X(\varphi)}\\ Y_n \ar[r]^{a} & X_n } $$ is a fibre square in the category of schemes.


spaces-simplicial-definition-fibre-products-simplicial-scheme

Let $f : X \to S$ be a morphism of schemes. The {\it simplicial scheme associated to $f$}, denoted $(X/S)_\bullet$, is the functor $\Delta^{opp} \to \Sch$, $[n] \mapsto X \times_S \ldots \times_S X$ described in Simplicial, Example \ref{simplicial-example-fibre-products-simplicial-object}.


examples-stacks-definition-hilbert-d-stack

We will denote $\mathcal{H}_d(\mathcal{X}/\mathcal{Y})$ the {\it degree $d$ finite Hilbert stack of $\mathcal{X}$ over $\mathcal{Y}$} constructed above. If $\mathcal{Y} = S$ we write $\mathcal{H}_d(\mathcal{X}) = \mathcal{H}_d(\mathcal{X}/\mathcal{Y})$. If $\mathcal{X} = \mathcal{Y} = S$ we denote it $\mathcal{H}_d$.


models-definition-type

A {\it numerical type} $T$ is given by $$ n, m_i, a_{ij}, w_i, g_i $$ where $n \geq 1$ is an integer and $m_i$, $a_{ij}$, $w_i$, $g_i$ are integers for $1 \leq i, j \leq n$ subject to the following conditions \begin{enumerate} \item $m_i > 0$, $w_i > 0$, $g_i \geq 0$, \item the matrix $A = (a_{ij})$ is symmetric and $a_{ij} \geq 0$ for $i \not = j$, \item there is no proper nonempty subset $I \subset \{1, \ldots, n\}$ such that $a_{ij} = 0$ for $i \in I$, $j \not \in I$, \item for each $i$ we have $\sum_j a_{ij}m_j = 0$, and \item $w_i | a_{ij}$. \end{enumerate}


models-definition-type-equivalent

We say two numerical types $n, m_i, a_{ij}, w_i, g_i$ and $n', m'_i, a'_{ij}, w'_i, g'_i$ are {\it equivalent types} if there exists a permutation $\sigma$ of $\{1, \ldots, n\}$ such that $m_i = m'_{\sigma(i)}$, $a_{ij} = a'_{\sigma(i)\sigma(j)}$, $w_i = w'_{\sigma(i)}$, and $g_i = g'_{\sigma(i)}$.


models-definition-genus

We say $n, m_i, a_{ij}, w_i, g_i$ is a {\it numerical type of genus $g$} if $g = 1 + \sum m_i(w_i(g_i - 1) - \frac{1}{2} a_{ii})$ is the integer from Lemma \ref{lemma-genus}.


models-definition-type-minus-one

Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type. We say $i$ is a {\it $(-1)$-index} if $g_i = 0$ and $a_{ii} = -w_i$.


models-definition-top-genus

Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$. The {\it topological genus of $T$} is the nonnegative integer $g_{top} = 1 - n + e$ from Lemma \ref{lemma-top-genus}.


models-definition-type-minimal

We say the numerical type $n, m_i, a_{ij}, w_i, g_i$ of genus $g$ is {\it minimal} if there does not exist an $i$ with $g_i = 0$ and $a_{ii} = -w_i$, in other words, if there does not exist a $(-1)$-index.


models-definition-type-minus-two

Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$. We say $i$ is a {\it $(-2)$-index} if $g_i = 0$ and $a_{ii} = -2w_i$.


models-definition-picard-group

Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$. The {\it Picard group of $T$} is the cokernel of the matrix $(a_{ij}/w_i)$, more precisely $$ \Pic(T) = \Coker\left( \mathbf{Z}^{\oplus n} \to \mathbf{Z}^{\oplus n},\quad e_i \mapsto \sum \frac{a_{ij}}{w_j}e_j \right) $$ where $e_i$ denotes the $i$th standard basis vector for $\mathbf{Z}^{\oplus n}$.


models-definition-minimal-model

Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_C) = K$. A {\it minimal model} will be a regular, proper model $X$ for $C$ such that $X$ does not contain an exceptional curve of the first kind (Resolution of Surfaces, Section \ref{resolve-section-minus-one}).


models-definition-numerical-type-model

In Situation \ref{situation-regular-model} the {\it numerical type associated to $X$} is the numerical type described in Lemma \ref{lemma-numerical-type-of-model}.


models-definition-semistable

Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_C) = K$. We say that $C$ has {\it semistable reduction} if the equivalent conditions of Lemma \ref{lemma-semistable} are satisfied.


models-definition-good

Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_C) = K$. We say that $C$ has {\it good reduction} if the equivalent conditions of Lemma \ref{lemma-good} are satisfied.


spaces-groupoids-definition-equivalence-relation

Let $B \to S$ as in Section \ref{section-notation}. Let $U$ be an algebraic space over $B$. \begin{enumerate} \item A {\it pre-relation} on $U$ over $B$ is any morphism $j : R \to U \times_B U$ of algebraic spaces over $B$. In this case we set $t = \text{pr}_0 \circ j$ and $s = \text{pr}_1 \circ j$, so that $j = (t, s)$. \item A {\it relation} on $U$ over $B$ is a monomorphism $j : R \to U \times_B U$ of algebraic spaces over $B$. \item A {\it pre-equivalence relation} is a pre-relation $j : R \to U \times_B U$ such that the image of $j : R(T) \to U(T) \times U(T)$ is an equivalence relation for all schemes $T$ over $B$. \item We say a morphism $R \to U \times_B U$ of algebraic spaces over $B$ is an {\it equivalence relation on $U$ over $B$} if and only if for every $T$ over $B$ the $T$-valued points of $R$ define an equivalence relation on the set of $T$-valued points of $U$. \end{enumerate}


spaces-groupoids-definition-restrict-relation

Let $B \to S$ as in Section \ref{section-notation}. Let $U$ be an algebraic space over $B$. Let $j : R \to U \times_B U$ be a pre-relation. Let $g : U' \to U$ be a morphism of algebraic spaces over $B$. The pre-relation $j' : R' \to U' \times_B U'$ of Lemma \ref{lemma-restrict-relation} is called the {\it restriction}, or {\it pullback} of the pre-relation $j$ to $U'$. In this situation we sometimes write $R' = R|_{U'}$.


spaces-groupoids-definition-group-space

Let $B \to S$ as in Section \ref{section-notation}. \begin{enumerate} \item A {\it group algebraic space over $B$} is a pair $(G, m)$, where $G$ is an algebraic space over $B$ and $m : G \times_B G \to G$ is a morphism of algebraic spaces over $B$ with the following property: For every scheme $T$ over $B$ the pair $(G(T), m)$ is a group. \item A {\it morphism $\psi : (G, m) \to (G', m')$ of group algebraic spaces over $B$} is a morphism $\psi : G \to G'$ of algebraic spaces over $B$ such that for every $T/B$ the induced map $\psi : G(T) \to G'(T)$ is a homomorphism of groups. \end{enumerate}


spaces-groupoids-definition-action-group-space

Let $B \to S$ as in Section \ref{section-notation}. Let $(G, m)$ be a group algebraic space over $B$. Let $X$ be an algebraic space over $B$. \begin{enumerate} \item An {\it action of $G$ on the algebraic space $X/B$} is a morphism $a : G \times_B X \to X$ over $B$ such that for every scheme $T$ over $B$ the map $a : G(T) \times X(T) \to X(T)$ defines the structure of a $G(T)$-set on $X(T)$. \item Suppose that $X$, $Y$ are algebraic spaces over $B$ each endowed with an action of $G$. An {\it equivariant} or more precisely a {\it $G$-equivariant} morphism $\psi : X \to Y$ is a morphism of algebraic spaces over $B$ such that for every $T$ over $B$ the map $\psi : X(T) \to Y(T)$ is a morphism of $G(T)$-sets. \end{enumerate}


spaces-groupoids-definition-free-action

Let $B \to S$, $G \to B$, and $X \to B$ as in Definition \ref{definition-action-group-space}. Let $a : G \times_B X \to X$ be an action of $G$ on $X/B$. We say the action is {\it free} if for every scheme $T$ over $B$ the action $a : G(T) \times X(T) \to X(T)$ is a free action of the group $G(T)$ on the set $X(T)$.


spaces-groupoids-definition-pseudo-torsor

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(G, m)$ be a group algebraic space over $B$. Let $X$ be an algebraic space over $B$, and let $a : G \times_B X \to X$ be an action of $G$ on $X$. \begin{enumerate} \item We say $X$ is a {\it pseudo $G$-torsor} or that $X$ is {\it formally principally homogeneous under $G$} if the induced morphism $G \times_B X \to X \times_B X$, $(g, x) \mapsto (a(g, x), x)$ is an isomorphism. \item A pseudo $G$-torsor $X$ is called {\it trivial} if there exists an $G$-equivariant isomorphism $G \to X$ over $B$ where $G$ acts on $G$ by left multiplication. \end{enumerate}


spaces-groupoids-definition-principal-homogeneous-space

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(G, m)$ be a group algebraic space over $B$. Let $X$ be a pseudo $G$-torsor over $B$. \begin{enumerate} \item We say $X$ is a {\it principal homogeneous space}, or more precisely a {\it principal homogeneous $G$-space over $B$} if there exists a fpqc covering\footnote{The default type of torsor in Groupoids, Definition \ref{groupoids-definition-principal-homogeneous-space} is a pseudo torsor which is trivial on an fpqc covering. Since $G$, as an algebraic space, can be seen a sheaf of groups there already is a notion of a $G$-torsor which corresponds to fppf-torsor, see Lemma \ref{lemma-torsor}. Hence we use ``principal homogeneous space'' for a pseudo torsor which is fpqc locally trivial, and we try to avoid using the word torsor in this situation.} $\{B_i \to B\}_{i \in I}$ such that each $X_{B_i} \to B_i$ has a section (i.e., is a trivial pseudo $G_{B_i}$-torsor). \item Let $\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}$. We say $X$ is a {\it $G$-torsor in the $\tau$ topology}, or a {\it $\tau$ $G$-torsor}, or simply a {\it $\tau$ torsor} if there exists a $\tau$ covering $\{B_i \to B\}_{i \in I}$ such that each $X_{B_i} \to B_i$ has a section. \item If $X$ is a principal homogeneous $G$-space over $B$, then we say that it is {\it quasi-isotrivial} if it is a torsor for the \'etale topology. \item If $X$ is a principal homogeneous $G$-space over $B$, then we say that it is {\it locally trivial} if it is a torsor for the Zariski topology. \end{enumerate}


spaces-groupoids-definition-equivariant-module

Let $B \to S$ as in Section \ref{section-notation}. Let $(G, m)$ be a group algebraic space over $B$, and let $a : G \times_B X \to X$ be an action of $G$ on the algebraic space $X$ over $B$. An {\it $G$-equivariant quasi-coherent $\mathcal{O}_X$-module}, or simply a {\it equivariant quasi-coherent $\mathcal{O}_X$-module}, is a pair $(\mathcal{F}, \alpha)$, where $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-module, and $\alpha$ is a $\mathcal{O}_{G \times_B X}$-module map $$ \alpha : a^*\mathcal{F} \longrightarrow \text{pr}_1^*\mathcal{F} $$ where $\text{pr}_1 : G \times_B X \to X$ is the projection such that \begin{enumerate} \item the diagram $$ \xymatrix{ (1_G \times a)^*\text{pr}_2^*\mathcal{F} \ar[r]_-{\text{pr}_{12}^*\alpha} & \text{pr}_2^*\mathcal{F} \\ (1_G \times a)^*a^*\mathcal{F} \ar[u]^{(1_G \times a)^*\alpha} \ar@{=}[r] & (m \times 1_X)^*a^*\mathcal{F} \ar[u]_{(m \times 1_X)^*\alpha} } $$ is a commutative in the category of $\mathcal{O}_{G \times_B G \times_B X}$-modules, and \item the pullback $$ (e \times 1_X)^*\alpha : \mathcal{F} \longrightarrow \mathcal{F} $$ is the identity map. \end{enumerate} For explanation compare with the relevant diagrams of Equation (\ref{equation-action}).


spaces-groupoids-definition-groupoid

Let $B \to S$ as in Section \ref{section-notation}. \begin{enumerate} \item A {\it groupoid in algebraic spaces over $B$} is a quintuple $(U, R, s, t, c)$ where $U$ and $R$ are algebraic spaces over $B$, and $s, t : R \to U$ and $c : R \times_{s, U, t} R \to R$ are morphisms of algebraic spaces over $B$ with the following property: For any scheme $T$ over $B$ the quintuple $$ (U(T), R(T), s, t, c) $$ is a groupoid category. \item A {\it morphism $f : (U, R, s, t, c) \to (U', R', s', t', c')$ of groupoids in algebraic spaces over $B$} is given by morphisms of algebraic spaces $f : U \to U'$ and $f : R \to R'$ over $B$ with the following property:  For any scheme $T$ over $B$ the maps $f$ define a functor from the groupoid category $(U(T), R(T), s, t, c)$ to the groupoid category $(U'(T), R'(T), s', t', c')$. \end{enumerate}


spaces-groupoids-definition-groupoid-module

Let $B \to S$ as in Section \ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. A {\it quasi-coherent module on $(U, R, s, t, c)$} is a pair $(\mathcal{F}, \alpha)$, where $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_U$-module, and $\alpha$ is a $\mathcal{O}_R$-module map $$ \alpha : t^*\mathcal{F} \longrightarrow s^*\mathcal{F} $$ such that \begin{enumerate} \item the diagram $$ \xymatrix{ & \text{pr}_1^*t^*\mathcal{F} \ar[r]_-{\text{pr}_1^*\alpha} & \text{pr}_1^*s^*\mathcal{F} \ar@{=}[rd] & \\ \text{pr}_0^*s^*\mathcal{F} \ar@{=}[ru] & & & c^*s^*\mathcal{F} \\ & \text{pr}_0^*t^*\mathcal{F} \ar[lu]^{\text{pr}_0^*\alpha} \ar@{=}[r] & c^*t^*\mathcal{F} \ar[ru]_{c^*\alpha} } $$ is a commutative in the category of $\mathcal{O}_{R \times_{s, U, t} R}$-modules, and \item the pullback $$ e^*\alpha : \mathcal{F} \longrightarrow \mathcal{F} $$ is the identity map. \end{enumerate} Compare with the commutative diagrams of Lemma \ref{lemma-diagram}.


spaces-groupoids-definition-stabilizer-groupoid

Let $B \to S$ as in Section \ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. The group algebraic space $j^{-1}(\Delta_{U/B}) \to U$ is called the {\it stabilizer of the groupoid in algebraic spaces $(U, R, s, t, c)$}.


spaces-groupoids-definition-restrict-groupoid

Let $B \to S$ as in Section \ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $g : U' \to U$ be a morphism of algebraic spaces over $B$. The morphism of groupoids in algebraic spaces $(U', R', s', t', c') \to (U, R, s, t, c)$ constructed in Lemma \ref{lemma-restrict-groupoid} is called the {\it restriction of $(U, R, s, t, c)$ to $U'$}. We sometime use the notation $R' = R|_{U'}$ in this case.


spaces-groupoids-definition-invariant-open

Let $B \to S$ as in Section \ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over the base $B$. \begin{enumerate} \item We say an open subspace $W \subset U$ is {\it $R$-invariant} if $t(s^{-1}(W)) \subset W$. \item A locally closed subspace $Z \subset U$ is called {\it $R$-invariant} if $t^{-1}(Z) = s^{-1}(Z)$ as locally closed subspaces of $R$. \item A monomorphism of algebraic spaces $T \to U$ is {\it $R$-invariant} if $T \times_{U, t} R = R \times_{s, U} T$ as algebraic spaces over $R$. \end{enumerate}


spaces-groupoids-definition-quotient-sheaf

Let $B \to S$ and the pre-relation $j : R \to U \times_B U$ be as above. In this setting the {\it quotient sheaf $U/R$} associated to $j$ is the sheafification of the presheaf (\ref{equation-quotient-presheaf}) on $(\Sch/S)_{fppf}$. If $j : R \to U \times_B U$ comes from the action of a group algebraic space $G$ over $B$ on $U$ as in Lemma \ref{lemma-groupoid-from-action} then we denote the quotient sheaf $U/G$.


spaces-groupoids-definition-representable-quotient

In the situation of Definition \ref{definition-quotient-sheaf}. We say that the pre-relation $j$ has a {\it quotient representable by an algebraic space} if the sheaf $U/R$ is an algebraic space. We say that the pre-relation $j$ has a {\it representable quotient} if the sheaf $U/R$ is representable by a scheme. We will say a groupoid in algebraic spaces $(U, R, s, t, c)$ over $B$ has a {\it representable quotient} (resp.\ {\it quotient representable by an algebraic space} if the quotient $U/R$ with $j = (t, s)$ is representable (resp.\ an algebraic space).


spaces-groupoids-definition-quotient-stack

Quotient stacks. Let $B \to S$ be as above. \begin{enumerate} \item Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. The {\it quotient stack} $$ p : [U/R] \longrightarrow (\Sch/S)_{fppf} $$ of $(U, R, s, t, c)$ is the stackification (see Stacks, Lemma \ref{stacks-lemma-stackify-groupoids}) of the category fibred in groupoids $[U/_{\!p}R]$ over $(\Sch/S)_{fppf}$ associated to (\ref{equation-quotient-stack}). \item Let $(G, m)$ be a group algebraic space over $B$. Let $a : G \times_B X \to X$ be an action of $G$ on an algebraic space over $B$. The {\it quotient stack} $$ p : [X/G] \longrightarrow (\Sch/S)_{fppf} $$ is the quotient stack associated to the groupoid in algebraic spaces $(X, G \times_B X, s, t, c)$ over $B$ of Lemma \ref{lemma-groupoid-from-action}. \end{enumerate}


spaces-descent-definition-descent-datum-quasi-coherent

Let $S$ be a scheme. Let $\{f_i : X_i \to X\}_{i \in I}$ be a family of morphisms of algebraic spaces over $S$ with fixed target $X$. \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 $X_i$ for each $i \in I$, an isomorphism of quasi-coherent $\mathcal{O}_{X_i \times_X X_j}$-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 $\mathcal{O}_{X_i \times_X X_j \times_X X_k}$-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 $\mathcal{O}_{X_i}$-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}


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

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


spaces-descent-definition-property-morphisms-local

Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. Let $\tau \in \{fpqc, fppf, syntomic, smooth, \etale\}$. 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 $\{Y_i \to Y\}_{i \in I}$ of algebraic spaces and any morphism of algebraic spaces $f : X \to Y$ we have $$ f \text{ has }\mathcal{P} \Leftrightarrow \text{each }Y_i \times_Y X \to Y_i\text{ has }\mathcal{P}. $$