spaces-descent-definition-property-morphisms-local-source

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


spaces-descent-definition-local-source-target

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


spaces-descent-definition-etale-smooth-local-source-target

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


spaces-descent-definition-descent-datum

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


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

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


spaces-descent-definition-pullback-functor

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


spaces-descent-definition-pullback-functor-family

With $\mathcal{U}' = \{X'_i \to X'\}_{i \in I'}$, $\mathcal{U} = \{X_i \to X\}_{i \in I}$, $\alpha : I' \to I$, $g : X' \to X$, and $g_i : X'_i \to X_{\alpha(i)}$ as in Lemma \ref{lemma-pullback-family} the functor $$ (V_i, \varphi_{ij}) \longmapsto (g_i^*V_{\alpha(i)}, (g_i \times g_j)^*\varphi_{\alpha(i) \alpha(j)}) $$ constructed in that lemma is called the {\it pullback functor} on descent data.


spaces-descent-definition-effective

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


spaces-descent-definition-effective-family

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


decent-spaces-definition-universally-bounded

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$, and let $U$ be a scheme over $S$. Let $f : U \to X$ be a morphism over $S$. We say the {\it fibres of $f$ are universally bounded}\footnote{This is probably nonstandard notation.} if there exists an integer $n$ such that for all fields $k$ and all morphisms $\Spec(k) \to X$ the fibre product $\Spec(k) \times_X U$ is a finite scheme over $k$ whose degree over $k$ is $\leq n$.


decent-spaces-definition-very-reasonable

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item We say $X$ is {\it decent} if for every point $x \in X$ the equivalent conditions of Lemma \ref{lemma-UR-finite-above-x} hold, in other words property $(\gamma)$ of Lemma \ref{lemma-bounded-fibres} holds. \item We say $X$ is {\it reasonable} if the equivalent conditions of Lemma \ref{lemma-U-universally-bounded} hold, in other words property $(\delta)$ of Lemma \ref{lemma-bounded-fibres} holds. \item We say $X$ is {\it very reasonable} if the equivalent conditions of Lemma \ref{lemma-characterize-very-reasonable} hold, i.e., property $(\epsilon)$ of Lemma \ref{lemma-bounded-fibres} holds. \end{enumerate}


decent-spaces-definition-residue-field

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$. The {\it residue field of $X$ at $x$} is the unique field $\kappa(x)$ which comes equipped with a monomorphism $\Spec(\kappa(x)) \to X$ representing $x$.


decent-spaces-definition-elemenary-etale-neighbourhood

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in X$ be a point. An {\it elementary \'etale neighbourhood} is an \'etale morphism $(U, u) \to (X, x)$ where $U$ is a scheme, $u \in U$ is a point mapping to $x$, and $\kappa(x) \to \kappa(u)$ is an isomorphism. A {\it morphism of elementary \'etale neighbourhoods} $(U, u) \to (U', u')$ is defined as a morphism $U \to U'$ over $X$ mapping $u$ to $u'$.


decent-spaces-definition-henselian-local-ring

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$. The {\it henselian local ring of $X$ at $x$}, is $$ \mathcal{O}_{X, x}^h = \colim \Gamma(U, \mathcal{O}_U) $$ where the colimit is over the elementary \'etale neighbourhoods $(U, u) \to (X, x)$.


decent-spaces-definition-residual-space

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. The {\it residual space of $X$ at $x$}\footnote{This is nonstandard notation.} is the monomorphism $Z_x \to X$ constructed in Lemma \ref{lemma-find-singleton-from-point}.


decent-spaces-definition-relative-conditions

Let $S$ be a scheme. We say an algebraic space $X$ over $S$ {\it has property $(\beta)$} if $X$ has the corresponding property of Lemma \ref{lemma-bounded-fibres}. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item We say $f$ {\it has property $(\beta)$} if for any scheme $T$ and morphism $T \to Y$ the fibre product $T \times_Y X$ has property $(\beta)$. \item We say $f$ is {\it decent} if for any scheme $T$ and morphism $T \to Y$ the fibre product $T \times_Y X$ is a decent algebraic space. \item We say $f$ is {\it reasonable} if for any scheme $T$ and morphism $T \to Y$ the fibre product $T \times_Y X$ is a reasonable algebraic space. \item We say $f$ is {\it very reasonable} if for any scheme $T$ and morphism $T \to Y$ the fibre product $T \times_Y X$ is a very reasonable algebraic space. \end{enumerate}


decent-spaces-definition-birational

Let $S$ be a scheme. Let $X$ and $Y$ algebraic spaces over $S$. Assume $X$ and $Y$ are decent and that $|X|$ and $|Y|$ have finitely many irreducible components. We say a morphism $f : X \to Y$ is {\it birational} if \begin{enumerate} \item $|f|$ induces a bijection between the set of generic points of irreducible components of $|X|$ and the set of generic points of the irreducible components of $|Y|$, and \item for every generic point $x \in |X|$ of an irreducible component the local ring map $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is an isomorphism (see clarification below). \end{enumerate}


decent-spaces-definition-unibranch

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$. We say that $X$ is {\it unibranch at $x$} if the equivalent conditions of Lemma \ref{lemma-irreducible-local-ring} hold. We say that $X$ is {\it unibranch} if $X$ is unibranch at every $x \in |X|$.


decent-spaces-definition-number-of-geometric-branches

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$. The {\it number of branches of $X$ at $x$} is either $n \in \mathbf{N}$ if the equivalent conditions of Lemma \ref{lemma-nr-branches-local-ring} hold, or else $\infty$.


decent-spaces-definition-catenary

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. We say $X$ is {\it catenary} if $|X|$ is catenary (Topology, Definition \ref{topology-definition-catenary}).


decent-spaces-definition-universally-catenary

Let $S$ be a scheme. Let $X$ be a decent and locally Noetherian algebraic space over $S$. We say $X$ is {\it universally catenary} if for every morphism $Y \to X$ of algebraic spaces which is  locally of finite type and with $Y$ decent, the algebraic space $Y$ is catenary.


groupoids-definition-equivalence-relation

Let $S$ be a scheme. Let $U$ be a scheme over $S$. \begin{enumerate} \item A {\it pre-relation} on $U$ over $S$ is any morphism of schemes $j : R \to U \times_S U$. 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 $S$ is a monomorphism of schemes $j : R \to U \times_S U$. \item A {\it pre-equivalence relation} is a pre-relation $j : R \to U \times_S U$ such that the image of $j : R(T) \to U(T) \times U(T)$ is an equivalence relation for all $T/S$. \item We say a morphism $R \to U \times_S U$ of schemes is an {\it equivalence relation on $U$ over $S$} if and only if for every scheme $T$ over $S$ the $T$-valued points of $R$ define an equivalence relation on the set of $T$-valued points of $U$. \end{enumerate}


groupoids-definition-restrict-relation

Let $S$ be a scheme. Let $U$ be a scheme over $S$. Let $j : R \to U \times_S U$ be a pre-relation. Let $g : U' \to U$ be a morphism of schemes. The pre-relation $j' : R' \to U' \times_S U'$ is called the {\it restriction}, or {\it pullback} of the pre-relation $j$ to $U'$. In this situation we sometimes write $R' = R|_{U'}$.


groupoids-definition-group-scheme

Let $S$ be a scheme. \begin{enumerate} \item A {\it group scheme over $S$} is a pair $(G, m)$, where $G$ is a scheme over $S$ and $m : G \times_S G \to G$ is a morphism of schemes over $S$ with the following property: For every scheme $T$ over $S$ the pair $(G(T), m)$ is a group. \item A {\it morphism $\psi : (G, m) \to (G', m')$ of group schemes over $S$} is a morphism $\psi : G \to G'$ of schemes over $S$ such that for every $T/S$ the induced map $\psi : G(T) \to G'(T)$ is a homomorphism of groups. \end{enumerate}


groupoids-definition-closed-subgroup-scheme

Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$. \begin{enumerate} \item A {\it closed subgroup scheme} of $G$ is a closed subscheme $H \subset G$ such that $m|_{H \times_S H}$ factors through $H$ and induces a group scheme structure on $H$ over $S$. \item An {\it open subgroup scheme} of $G$ is an open subscheme $G' \subset G$ such that $m|_{G' \times_S G'}$ factors through $G'$ and induces a group scheme structure on $G'$ over $S$. \end{enumerate}


groupoids-definition-smooth-group-scheme

Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$. \begin{enumerate} \item We say $G$ is a {\it smooth group scheme} if the structure morphism $G \to S$ is smooth. \item We say $G$ is a {\it flat group scheme} if the structure morphism $G \to S$ is flat. \item We say $G$ is a {\it separated group scheme} if the structure morphism $G \to S$ is separated. \end{enumerate} Add more as needed.


groupoids-definition-abelian-variety

Let $k$ be a field. An {\it abelian variety} is a group scheme over $k$ which is also a proper, geometrically integral variety over $k$.


groupoids-definition-action-group-scheme

Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$. \begin{enumerate} \item An {\it action of $G$ on the scheme $X/S$} is a morphism $a : G \times_S X \to X$ over $S$ such that for every $T/S$ 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 schemes over $S$ 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 schemes over $S$ such that for every $T/S$ the map $\psi : X(T) \to Y(T)$ is a morphism of $G(T)$-sets. \end{enumerate}


groupoids-definition-free-action

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


groupoids-definition-pseudo-torsor

Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$. Let $X$ be a scheme over $S$, and let $a : G \times_S X \to X$ 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 of schemes $G \times_S X \to X \times_S X$, $(g, x) \mapsto (a(g, x), x)$ is an isomorphism of schemes over $S$. \item A pseudo $G$-torsor $X$ is called {\it trivial} if there exists an $G$-equivariant isomorphism $G \to X$ over $S$ where $G$ acts on $G$ by left multiplication. \end{enumerate}


groupoids-definition-principal-homogeneous-space

Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$. Let $X$ be a pseudo $G$-torsor over $S$. \begin{enumerate} \item We say $X$ is a {\it principal homogeneous space} or a {\it $G$-torsor} if there exists a fpqc covering\footnote{This means that the default type of torsor is a pseudo torsor which is trivial on an fpqc covering. This is the definition in \cite[Expos\'e IV, 6.5]{SGA3}. It is a little bit inconvenient for us as we most often work in the fppf topology.} $\{S_i \to S\}_{i \in I}$ such that each $X_{S_i} \to S_i$ has a section (i.e., is a trivial pseudo $G_{S_i}$-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 $\{S_i \to S\}_{i \in I}$ such that each $X_{S_i} \to S_i$ has a section. \item If $X$ is a $G$-torsor, then we say that it is {\it quasi-isotrivial} if it is a torsor for the \'etale topology. \item If $X$ is a $G$-torsor, then we say that it is {\it locally trivial} if it is a torsor for the Zariski topology. \end{enumerate}


groupoids-definition-equivariant-module

Let $S$ be a scheme, let $(G, m)$ be a group scheme over $S$, and let $a : G \times_S X \to X$ be an action of the group scheme $G$ on $X/S$. A {\it $G$-equivariant quasi-coherent $\mathcal{O}_X$-module}, or simply an {\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 $\mathcal{O}_{G \times_S X}$-module map $$ \alpha : a^*\mathcal{F} \longrightarrow \text{pr}_1^*\mathcal{F} $$ where $\text{pr}_1 : G \times_S X \to X$ is the projection such that \begin{enumerate} \item the diagram $$ \xymatrix{ (1_G \times a)^*\text{pr}_1^*\mathcal{F} \ar[r]_-{\text{pr}_{12}^*\alpha} & \text{pr}_2^*\mathcal{F} \\ (1_G \times a)^*a^*\mathcal{F} \ar[u]^{(1_G \times a)^*\alpha} \ar@{=}[r] & (m \times 1_X)^*a^*\mathcal{F} \ar[u]_{(m \times 1_X)^*\alpha} } $$ is a commutative in the category of $\mathcal{O}_{G \times_S G \times_S X}$-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}).


groupoids-definition-groupoid

Let $S$ be a scheme. \begin{enumerate} \item A {\it groupoid scheme over $S$}, or simply a {\it groupoid over $S$} is a quintuple $(U, R, s, t, c)$ where $U$ and $R$ are schemes over $S$, and $s, t : R \to U$ and $c : R \times_{s, U, t} R \to R$ are morphisms of schemes over $S$ with the following property: For any scheme $T$ over $S$ the quintuple $$ (U(T), R(T), s, t, c) $$ is a groupoid category in the sense described above. \item A {\it morphism $f : (U, R, s, t, c) \to (U', R', s', t', c')$ of groupoid schemes over $S$} is given by morphisms of schemes $f : U \to U'$ and $f : R \to R'$ with the following property:  For any scheme $T$ over $S$ 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}


groupoids-definition-groupoid-module

Let $S$ be a scheme, let $(U, R, s, t, c)$ be a groupoid scheme over $S$. 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}.


groupoids-definition-stabilizer-groupoid

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. The group scheme $j^{-1}(\Delta_{U/S})\to U$ is called the {\it stabilizer of the groupoid scheme $(U, R, s, t, c)$}.


groupoids-definition-restrict-groupoid

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \to U$ be a morphism of schemes. The morphism of groupoids $(U', R', s', t', c') \to (U, R, s, t, c)$ 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.


groupoids-definition-invariant-open

Let $(U, R, s, t, c)$ be a groupoid scheme over the base scheme $S$. \begin{enumerate} \item A subset $W \subset U$ is {\it set-theoretically $R$-invariant} if $t(s^{-1}(W)) \subset W$. \item An open $W \subset U$ is {\it $R$-invariant} if $t(s^{-1}(W)) \subset W$. \item A closed subscheme $Z \subset U$ is called {\it $R$-invariant} if $t^{-1}(Z) = s^{-1}(Z)$. Here we use the scheme theoretic inverse image, see Schemes, Definition \ref{schemes-definition-inverse-image-closed-subscheme}. \item A monomorphism of schemes $T \to U$ is {\it $R$-invariant} if $T \times_{U, t} R = R \times_{s, U} T$ as schemes over $R$. \end{enumerate}


groupoids-definition-quotient-sheaf

Let $\tau$, $S$, and the pre-relation $j : R \to U \times_S U$ 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}) in the $\tau$-topology. If $j : R \to U \times_S U$ comes from the action of a group scheme $G/S$ on $U$ as in Lemma \ref{lemma-groupoid-from-action} then we sometimes denote the quotient sheaf $U/G$.


groupoids-definition-representable-quotient

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


groupoids-definition-cartesian-morphism

Let $S$ be a scheme. Let $f : (U', R', s', t', c') \to (U, R, s, t, c)$ be a morphism of groupoid schemes over $S$. We say $f$ is {\it cartesian}, or that {\it $(U', R', s', t', c')$ is cartesian over $(U, R, s, t, c)$}, if the diagram $$ \xymatrix{ R' \ar[r]_f \ar[d]_{s'} & R \ar[d]^s \\ U' \ar[r]^f & U } $$ is a fibre square in the category of schemes. A {\it morphism of groupoid schemes cartesian over $(U, R, s, t, c)$} is a morphism of groupoid schemes compatible with the structure morphisms towards $(U, R, s, t, c)$.


local-cohomology-definition-cd

Let $I \subset A$ be a finitely generated ideal of a ring $A$. The smallest integer $d \geq -1$ satisfying the equivalent conditions of Lemma \ref{lemma-cd} is called the {\it cohomological dimension of $I$ in $A$} and is denoted $\text{cd}(A, I)$.


local-cohomology-definition-depth-complex

Let $I$ be an ideal of a Noetherian ring $A$. Let $K \in D^+_{\textit{Coh}}(A)$. We define the {\it $I$-depth} of $K$, denoted $\text{depth}_I(K)$, to be the maximal $m \in \mathbf{Z} \cup \{\infty\}$ such that $H^i_I(K) = 0$ for all $i < m$. If $A$ is local with maximal ideal $\mathfrak m$ then we call $\text{depth}_\mathfrak m(K)$ simply the {\it depth} of $K$.


more-algebra-definition-stably-free

Let $R$ be a ring.  \begin{enumerate} \item Two modules $M$, $N$ over $R$ are said to be {\it stably isomorphic} if there exist $n, m \geq 0$ such that $M \oplus R^{\oplus m} \cong N \oplus R^{\oplus n}$ as $R$-modules. \item A module $M$ is {\it stably free} if it is stably isomorphic to a free module. \end{enumerate}


more-algebra-definition-fitting-ideal

Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $k \geq 0$. The {\it $k$th Fitting ideal} of $M$ is the ideal $\text{Fit}_k(M)$ constructed in Lemma \ref{lemma-fitting-ideal}. Set $\text{Fit}_{-1}(M) = 0$.


more-algebra-definition-zariski-pair

A {\it Zariski pair} is a pair $(A, I)$ such that $I$ is contained in the Jacobson radical of $A$.


more-algebra-definition-henselian-pair

A {\it henselian pair} is a pair $(A, I)$ satisfying \begin{enumerate} \item $I$ is contained in the Jacobson radical of $A$, and \item for any monic polynomial $f \in A[T]$ and factorization $\overline{f} = g_0h_0$ with $g_0, h_0 \in A/I[T]$ monic generating the unit ideal in $A/I[T]$, there exists a factorization $f = gh$ in $A[T]$ with $g, h$ monic and $g_0 = \overline{g}$ and $h_0 = \overline{h}$. \end{enumerate}


more-algebra-definition-absolutely-integrally-closed

A ring $A$ is {\it absolutely integrally closed} if every monic $f \in A[T]$ is a product of linear factors.


more-algebra-definition-auto-ass

A ring $R$ is said to be {\it auto-associated} if $R$ is local and its maximal ideal $\mathfrak m$ is weakly associated to $R$.


more-algebra-definition-torsion

Let $R$ be a domain. Let $M$ be an $R$-module. \begin{enumerate} \item We say an element $x \in M$ is {\it torsion} if there exists a nonzero $f \in R$ such that $fx = 0$. \item We say $M$ is {\it torsion free} if the only torsion element of $M$ is $0$. \end{enumerate}


more-algebra-definition-reflexive

Let $R$ be a domain. We say an $R$-module $M$ is {\it reflexive} if the natural map $$ j : M \longrightarrow \Hom_R(\Hom_R(M, R), R) $$ which sends $m \in M$ to the map sending $\varphi \in \Hom_R(M, R)$ to $\varphi(m) \in R$ is an isomorphism.


more-algebra-definition-reflexive-hull

Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module. The module $M^{**} = \Hom_R(\Hom_R(M, R), R)$ is called the {\it reflexive hull} of $M$.


more-algebra-definition-content-ideal

Let $A$ be a ring. Let $M$ be a flat $A$-module. Let $x \in M$. If the set of ideals $I$ in $A$ such that $x \in IM$ has a smallest element, we call it the {\it content ideal of $x$}.


more-algebra-definition-strict-transform

Let $R$ be a ring. Let $I \subset R$ be an ideal and $a \in I$. Let $R[\frac{I}{a}]$ be the affine blowup algebra, see Algebra, Definition \ref{algebra-definition-blow-up}. Let $M$ be an $R$-module. The {\it strict transform of $M$ along $R \to R[\frac{I}{a}]$} is the $R[\frac{I}{a}]$-module $$ M' = \left(M \otimes_R R[\textstyle{\frac{I}{a}}]\right)/a\text{-power torsion} $$


more-algebra-definition-koszul

Let $R$ be a ring. Let $\varphi : E \to R$ be an $R$-module map. The {\it Koszul complex} $K_\bullet(\varphi)$ associated to $\varphi$ is the commutative differential graded algebra defined as follows: \begin{enumerate} \item the underlying graded algebra is the exterior algebra $K_\bullet(\varphi) = \wedge(E)$, \item the differential $d : K_\bullet(\varphi) \to K_\bullet(\varphi)$ is the unique derivation such that $d(e) = \varphi(e)$ for all $e \in E = K_1(\varphi)$. \end{enumerate}


more-algebra-definition-koszul-complex

Let $R$ be a ring and let $f_1, \ldots, f_r \in R$. The {\it Koszul complex on $f_1, \ldots, f_r$} is the Koszul complex associated to the map $(f_1, \ldots, f_r) : R^{\oplus r} \to R$. Notation $K_\bullet(f_\bullet)$, $K_\bullet(f_1, \ldots, f_r)$, $K_\bullet(R, f_1, \ldots, f_r)$, or $K_\bullet(R, f_\bullet)$.


more-algebra-definition-koszul-regular-sequence

Let $R$ be a ring. Let $r \geq 0$ and let $f_1, \ldots, f_r \in R$ be a sequence of elements. Let $M$ be an $R$-module. The sequence $f_1, \ldots, f_r$ is called \begin{enumerate} \item {\it $M$-Koszul-regular} if $H_i(K_\bullet(f_1, \ldots, f_r) \otimes_R M) = 0$ for all $i \not = 0$, \item {\it $M$-$H_1$-regular} if $H_1(K_\bullet(f_1, \ldots, f_r) \otimes_R M) = 0$, \item {\it Koszul-regular} if $H_i(K_\bullet(f_1, \ldots, f_r)) = 0$ for all $i \not = 0$, and \item {\it $H_1$-regular} if $H_1(K_\bullet(f_1, \ldots, f_r)) = 0$. \end{enumerate}


more-algebra-definition-regular-ideal

Let $R$ be a ring and let $I \subset R$ be an ideal. \begin{enumerate} \item We say $I$ is a {\it regular ideal} if for every $\mathfrak p \in V(I)$ there exists a $g \in R$, $g \not \in \mathfrak p$ and a regular sequence $f_1, \ldots, f_r \in R_g$ such that $I_g$ is generated by $f_1, \ldots, f_r$. \item We say $I$ is a {\it Koszul-regular ideal} if for every $\mathfrak p \in V(I)$ there exists a $g \in R$, $g \not \in \mathfrak p$ and a Koszul-regular sequence $f_1, \ldots, f_r \in R_g$ such that $I_g$ is generated by $f_1, \ldots, f_r$. \item We say $I$ is a {\it $H_1$-regular ideal} if for every $\mathfrak p \in V(I)$ there exists a $g \in R$, $g \not \in \mathfrak p$ and an $H_1$-regular sequence $f_1, \ldots, f_r \in R_g$ such that $I_g$ is generated by $f_1, \ldots, f_r$. \item We say $I$ is a {\it quasi-regular ideal} if for every $\mathfrak p \in V(I)$ there exists a $g \in R$, $g \not \in \mathfrak p$ and a quasi-regular sequence $f_1, \ldots, f_r \in R_g$ such that $I_g$ is generated by $f_1, \ldots, f_r$. \end{enumerate}


more-algebra-definition-local-complete-intersection

A ring map $A \to B$ is called a {\it local complete intersection} if it is of finite type and for some (equivalently any) presentation $B = A[x_1, \ldots, x_n]/I$ the ideal $I$ is Koszul-regular.


more-algebra-definition-topological-ring

\begin{reference} \cite[Sections 7.1 and 7.2]{EGA1} \end{reference} Let $R$ be a ring and let $M$ be an $R$-module. \begin{enumerate} \item We say $R$ is a {\it topological ring} if $R$ is endowed with a topology such that both addition and multiplication are continuous as maps $R \times R \to R$ where $R \times R$ has the product topology. In this case we say $M$ is a {\it topological module} if $M$ is endowed with a topology such that addition $M \times M \to M$ and scalar multiplication $R \times M \to M$ are continuous. \item A {\it homomorphism of topological modules} is just a continuous $R$-module map. A {\it homomorphism of topological rings} is a ring homomorphism which is continuous for the given topologies. \item We say $M$ is {\it linearly topologized} if $0$ has a fundamental system of neighbourhoods consisting of submodules. We say $R$ is {\it linearly topologized} if $0$ has a fundamental system of neighbourhoods consisting of ideals. \item If $R$ is linearly topologized, we say that $I \subset R$ is an {\it ideal of definition} if $I$ is open and if every neighbourhood of $0$ contains $I^n$ for some $n$. \item If $R$ is linearly topologized, we say that $R$ is {\it pre-admissible} if $R$ has an ideal of definition. \item If $R$ is linearly topologized, we say that $R$ is {\it admissible} if it is pre-admissible and complete\footnote{By our conventions this includes separated.}. \item If $R$ is linearly topologized, we say that $R$ is {\it pre-adic} if there exists an ideal of definition $I$ such that $\{I^n\}_{n \geq 0}$ forms a fundamental system of neighbourhoods of $0$. \item If $R$ is linearly topologized, we say that $R$ is {\it adic} if $R$ is pre-adic and complete. \end{enumerate} Note that a (pre)adic topological ring is the same thing as a (pre)admissible topological ring which has an ideal of definition $I$ such that $I^n$ is open for all $n \geq 1$.


more-algebra-definition-formally-smooth

Let $R \to S$ be a homomorphism of topological rings with $R$ and $S$ linearly topologized. We say $S$ is {\it formally smooth over $R$} if for every commutative solid diagram $$ \xymatrix{ S \ar[r] \ar@{-->}[rd] & A/J \\ R \ar[r] \ar[u] & A \ar[u] } $$ of homomorphisms of topological rings where $A$ is a discrete ring and $J \subset A$ is an ideal of square zero, a dotted arrow exists which makes the diagram commute.


more-algebra-definition-formally-smooth-adic

Let $R \to S$ be a ring map. Let $\mathfrak n \subset S$ be an ideal. If the equivalent conditions (2)(a) and (2)(b) of Lemma \ref{lemma-formally-smooth} hold, then we say $R \to S$ is {\it formally smooth for the $\mathfrak n$-adic topology}.


more-algebra-definition-regular

A ring map $R \to \Lambda$ is {\it regular} if it is flat and for every prime $\mathfrak p \subset R$ the fibre ring $$ \Lambda \otimes_R \kappa(\mathfrak p) = \Lambda_\mathfrak p/\mathfrak p\Lambda_\mathfrak p $$ is Noetherian and geometrically regular over $\kappa(\mathfrak p)$.


more-algebra-definition-p-basis

Let $p$ be a prime number. Let $k \to K$ be an extension of fields of characteristic $p$. Denote $kK^p$ the compositum of $k$ and $K^p$ in $K$. \begin{enumerate} \item A subset $\{x_i\} \subset K$ is called {\it p-independent over $k$} if the elements $x^E = \prod x_i^{e_i}$ where $0 \leq e_i < p$ are linearly independent over $kK^p$. \item A subset $\{x_i\}$ of $K$ is called a {\it p-basis of $K$ over $k$} if the elements $x^E$ form a basis of $K$ over $kK^p$. \end{enumerate}


more-algebra-definition-J

Let $R$ be a Noetherian ring. Let $X = \Spec(R)$. \begin{enumerate} \item We say $R$ is {\it J-0} if $\text{Reg}(X)$ contains a nonempty open. \item We say $R$ is {\it J-1} if $\text{Reg}(X)$ is open. \item We say $R$ is {\it J-2} if any finite type $R$-algebra is J-1. \end{enumerate}


more-algebra-definition-G-ring

A ring $R$ is called a {\it G-ring} if $R$ is Noetherian and for every prime $\mathfrak p$ of $R$ the ring map $R_\mathfrak p \to (R_\mathfrak p)^\wedge$ is regular.


more-algebra-definition-excellent

Let $R$ be a ring. \begin{enumerate} \item We say $R$ is {\it quasi-excellent} if $R$ is Noetherian, a G-ring, and J-2. \item We say $R$ is {\it excellent} if $R$ is quasi-excellent and universally catenary. \end{enumerate}


more-algebra-definition-projective

Let $R$ be a ring. An $R$-module $J$ is {\it injective} if and only if the functor $\Hom_R(-, J) : \text{Mod}_R \to \text{Mod}_R$ is an exact functor.


more-algebra-definition-simple-functors

Let $R$ be a ring. \begin{enumerate} \item For any $R$-module $M$ over $R$ we denote $M^\vee = \Hom(M, \mathbf{Q}/\mathbf{Z})$ with its natural $R$-module structure. We think of {\it $M \mapsto M^\vee$} as a contravariant functor from the category of $R$-modules to itself. \item For any $R$-module $M$ we denote $$ F(M) = \bigoplus\nolimits_{m \in M} R[m] $$ the {\it free module} with basis given by the elements $[m]$ with $m \in M$. We let $F(M)\to M$, $\sum f_i [m_i] \mapsto \sum f_i m_i$ be the natural surjection of $R$-modules. We think of $M \mapsto (F(M) \to M)$ as a functor from the category of $R$-modules to the category of arrows in $R$-modules. \end{enumerate}


more-algebra-definition-K-flat

Let $R$ be a ring. A complex $K^\bullet$ is called {\it K-flat} if for every acyclic complex $M^\bullet$ the total complex $\text{Tot}(M^\bullet \otimes_R K^\bullet)$ is acyclic.


more-algebra-definition-derived-tor

Let $R$ be a ring. Let $M^\bullet$ be an object of $D(R)$. The {\it derived tensor product} $$ - \otimes_R^{\mathbf{L}} M^\bullet : D(R) \longrightarrow D(R) $$ is the exact functor of triangulated categories described above.


more-algebra-definition-tor-independent

Let $R$ be a ring. Let $A$, $B$ be $R$-algebras. We say $A$ and $B$ are {\it Tor independent over $R$} if $\text{Tor}_p^R(A, B) = 0$ for all $p > 0$.


more-algebra-definition-pseudo-coherent

Let $R$ be a ring. Denote $D(R)$ its derived category. Let $m \in \mathbf{Z}$. \begin{enumerate} \item An object $K^\bullet$ of $D(R)$ is {\it $m$-pseudo-coherent} if there exists a bounded complex $E^\bullet$ of finite free $R$-modules and a morphism $\alpha : E^\bullet \to K^\bullet$ such that $H^i(\alpha)$ is an isomorphism for $i > m$ and $H^m(\alpha)$ is surjective. \item An object $K^\bullet$ of $D(R)$ is {\it pseudo-coherent} if it is quasi-isomorphic to a bounded above complex of finite free $R$-modules. \item An $R$-module $M$ is called {\it $m$-pseudo-coherent} if $M[0]$ is an $m$-pseudo-coherent object of $D(R)$. \item An $R$-module $M$ is called {\it pseudo-coherent}\footnote{This clashes with what is meant by a pseudo-coherent module in \cite{Bourbaki-CA}.} if $M[0]$ is a pseudo-coherent object of $D(R)$. \end{enumerate}


more-algebra-definition-tor-amplitude

Let $R$ be a ring. Denote $D(R)$ its derived category. Let $a, b \in \mathbf{Z}$. \begin{enumerate} \item An object $K^\bullet$ of $D(R)$ has {\it tor-amplitude in $[a, b]$} if $H^i(K^\bullet \otimes_R^\mathbf{L} M) = 0$ for all $R$-modules $M$ and all $i \not \in [a, b]$. \item An object $K^\bullet$ of $D(R)$ has {\it finite tor dimension} if it has tor-amplitude in $[a, b]$ for some $a, b$. \item An $R$-module $M$ has {\it tor dimension $\leq d$} if $M[0]$ as an object of $D(R)$ has tor-amplitude in $[-d, 0]$. \item An $R$-module $M$ has {\it finite tor dimension} if $M[0]$ as an object of $D(R)$ has finite tor dimension. \end{enumerate}


more-algebra-definition-projective-dimension

Let $R$ be a ring. Let $K$ be an object of $D(R)$. We say $K$ has {\it finite projective dimension} if $K$ can be represented by a bounded complex of projective modules. We say $K$ as {\it projective-amplitude in $[a, b]$} if  $K$ is quasi-isomorphic to a complex $$ \ldots \to 0 \to P^a \to P^{a + 1} \to \ldots \to P^{b - 1} \to P^b \to 0 \to \ldots $$ where $P^i$ is a projective $R$-module for all $i \in \mathbf{Z}$.


more-algebra-definition-injective-dimension

Let $R$ be a ring. Let $K$ be an object of $D(R)$. We say $K$ has {\it finite injective dimension} if $K$ can be represented by a finite complex of injective $R$-modules. We say $K$ has {\it injective-amplitude in $[a, b]$} if $K$ is isomorphic to a complex $$ \ldots \to 0 \to I^a \to I^{a + 1} \to \ldots \to I^{b - 1} \to I^b \to 0 \to \ldots $$ with $I^i$ an injective $R$-module for all $i \in \mathbf{Z}$.


more-algebra-definition-near-projective

Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $M$ be an $R$-module. We say $M$ is {\it $I$-projective}\footnote{This is nonstandard notation.} if the equivalent conditions of Lemma \ref{lemma-near-projective} hold.


more-algebra-definition-perfect

Let $R$ be a ring. Denote $D(R)$ the derived category of the abelian category of $R$-modules. \begin{enumerate} \item An object $K$ of $D(R)$ is {\it perfect} if it is quasi-isomorphic to a bounded complex of finite projective $R$-modules. \item An $R$-module $M$ is {\it perfect} if $M[0]$ is a perfect object in $D(R)$. \end{enumerate}


more-algebra-definition-relatively-finitely-presented

Let $R \to A$ be a finite type ring map. Let $M$ be an $A$-module. We say $M$ is an $A$-module {\it finitely presented relative to $R$} if the equivalent conditions of Lemma \ref{lemma-relatively-finitely-presented} hold.


more-algebra-definition-relatively-pseudo-coherent

Let $R \to A$ be a finite type ring map. Let $K^\bullet$ be a complex of $A$-modules. Let $M$ be an $A$-module. Let $m \in \mathbf{Z}$. \begin{enumerate} \item We say $K^\bullet$ is {\it $m$-pseudo-coherent relative to $R$} if the equivalent conditions of Lemma \ref{lemma-relatively-pseudo-coherent} hold. \item We say $K^\bullet$ is {\it pseudo-coherent relative to $R$} if $K^\bullet$ is $m$-pseudo-coherent relative to $R$ for all $m \in \mathbf{Z}$. \item We say $M$ is {\it $m$-pseudo-coherent relative to $R$} if $M[0]$ is $m$-pseudo-coherent relative to $R$. \item We say $M$ is {\it pseudo-coherent relative to $R$} if $M[0]$ is pseudo-coherent relative to $R$. \end{enumerate}


more-algebra-definition-pseudo-coherent-perfect

Let $A \to B$ be a ring map. \begin{enumerate} \item We say $A \to B$ is a {\it pseudo-coherent ring map} if it is of finite type and $B$, as a $B$-module, is pseudo-coherent relative to $A$. \item We say $A \to B$ is a {\it perfect ring map} if it is a pseudo-coherent ring map such that $B$ as an $A$-module has finite tor dimension. \end{enumerate}


more-algebra-definition-relatively-perfect

Let $R \to A$ be a flat ring map of finite presentation. An object $K$ of $D(A)$ is {\it $R$-perfect} or {\it perfect relative to $R$} if $K$ is pseudo-coherent (Definition \ref{definition-pseudo-coherent}) and has finite tor dimension over $R$ (Definition \ref{definition-tor-amplitude}).


more-algebra-definition-f-power-torsion

Let $R$ be a ring. Let $M$ be an $R$-module. \begin{enumerate} \item Let $I \subset R$ be an ideal. We say $M$ is an {\it $I$-power torsion module} if for every $m \in M$ there exists an $n > 0$ such that $I^n m = 0$. \item Let $f \in R$. We say $M$ is {\it an $f$-power torsion module} if for each $m \in M$, there exists an $n > 0$ such that $f^n m = 0$. \end{enumerate}


more-algebra-definition-derived-complete

Let $A$ be a ring. Let $K \in D(A)$. Let $I \subset A$ be an ideal. We say $K$ is {\it derived complete with respect to $I$} if for every $f \in I$ we have $T(K, f) = 0$. If $M$ is an $A$-module, then we say $M$ is {\it derived complete with respect to $I$} if $M[0] \in D(A)$ is derived complete with respect to $I$.


more-algebra-definition-weakly-etale

A ring $A$ is called {\it absolutely flat} if every $A$-module is flat over $A$. A ring map $A \to B$ is {\it weakly \'etale} or {\it absolutely flat} if both $A \to B$ and $B \otimes_A B \to B$ are flat.


more-algebra-definition-weak-dimension

Let $A$ be a ring. Let $d \geq 0$ be an integer. We say that $A$ has {\it weak dimension $\leq d$} if every $A$-module has tor dimension $\leq d$.


more-algebra-definition-unibranch

\begin{reference} \cite[Chapter 0 (23.2.1)]{EGA4} \end{reference} Let $A$ be a local ring. We say $A$ is {\it unibranch} if the reduction $A_{red}$ is a domain and if the integral closure $A'$ of $A_{red}$ in its field of fractions is local. We say $A$ is {\it geometrically unibranch} if $A$ is unibranch and moreover the residue field of $A'$ is purely inseparable over the residue field of $A$.


more-algebra-definition-number-of-branches

Let $A$ be a local ring with henselization $A^h$ and strict henselization $A^{sh}$. The {\it number of branches of $A$} is the number of minimal primes of $A^h$ if finite and $\infty$ otherwise. The {\it number of geometric branches of $A$} is the number of minimal primes of $A^{sh}$ if finite and $\infty$ otherwise.


more-algebra-definition-formally-catenary

A Noetherian local ring $A$ is {\it formally catenary} if for every minimal prime $\mathfrak p \subset A$ the spectrum of $A^\wedge/\mathfrak p A^\wedge$ is equidimensional.


more-algebra-definition-extension-discrete-valuation-rings

We say that $A \to B$ or $A \subset B$ is an {\it extension of discrete valuation rings} if $A$ and $B$ are discrete valuation rings and $A \to B$ is injective and local. In particular, if $\pi_A$ and $\pi_B$ are uniformizers of $A$ and $B$, then $\pi_A = u \pi_B^e$ for some $e \geq 1$ and unit $u$ of $B$. The integer $e$ does not depend on the choice of the uniformizers as it is also the unique integer $\geq 1$ such that $$ \mathfrak m_A B = \mathfrak m_B^e $$ The integer $e$ is called the {\it ramification index} of $B$ over $A$. We say that $B$ is {\it weakly unramified} over $A$ if $e = 1$. If the extension of residue fields $\kappa_A = A/\mathfrak m_A \subset \kappa_B = B/\mathfrak m_B$ is finite, then we set $f = [\kappa_B : \kappa_A]$ and we call it the {\it residual degree} or {\it residue degree} of the extension $A \subset B$.


more-algebra-definition-types-of-extensions

Let $A$ be a discrete valuation ring with fraction field $K$. Let $L \supset K$ be a finite separable extension. With $B$ and $\mathfrak m_i$, $i = 1, \ldots, n$ as in Remark \ref{remark-finite-separable-extension} we say the extension $L/K$ is \begin{enumerate} \item {\it unramified with respect to $A$} if $e_i = 1$ and the extension $\kappa_A \subset \kappa(\mathfrak m_i)$ is separable for all $i$, \item {\it tamely ramified with respect to $A$} if either the characteristic of $\kappa_A$ is $0$ or the characteristic of $\kappa_A$ is $p > 0$, the field extensions $\kappa_A \subset \kappa(\mathfrak m_i)$ are separable, and the ramification indices $e_i$ are prime to $p$, and \item {\it totally ramified with respect to $A$} if $n = 1$ and the residue field extension $\kappa_A \subset \kappa(\mathfrak m_1)$ is trivial. \end{enumerate} If the discrete valuation ring $A$ is clear from context, then we sometimes say $L/K$ is unramified, totally ramified, or tamely ramified for short.


more-algebra-definition-decomposition-inertia

Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$ be a finite Galois extension with Galois group $G$. Let $B$ be the integral closure of $A$ in $L$. Let $\mathfrak m \subset B$ be a maximal ideal. \begin{enumerate} \item The {\it decomposition group of $\mathfrak m$} is the subgroup $D = \{\sigma \in G \mid \sigma(\mathfrak m) = \mathfrak m\}$. \item The {\it inertia group of $\mathfrak m$} is the kernel $I$ of the map $D \to \text{Aut}(\kappa(\mathfrak m)/\kappa_A)$. \end{enumerate}


more-algebra-definition-wild-inertia

With assumptions and notation as in Lemma \ref{lemma-galois-inertia}. \begin{enumerate} \item The {\it wild inertia group of $\mathfrak m$} is the subgroup $P$. \item The {\it tame inertia group of $\mathfrak m$} is the quotient $I \to I_t$. \end{enumerate} We denote $\theta : I \to \mu_e(\kappa(\mathfrak m))$ the surjective map (\ref{equation-inertia-character}) whose kernel is $P$ and which induces the isomorphism $I_t \to \mu_e(\kappa(\mathfrak m))$.


more-algebra-definition-mixed

Let $A$ be a discrete valuation ring. We say $A$ has {\it mixed characteristic} if the characteristic of the residue field of $A$ is $p > 0$ and the characteristic of the fraction field of $A$ is $0$. In this case we obtain an extension of discrete valuation rings $\mathbf{Z}_{(p)} \subset A$ and the {\it absolute ramification index} of $A$ is the ramification index of this extension.


more-algebra-definition-solution

Let $A \to B$ be an extension of discrete valuation rings with fraction fields $K \subset L$. \begin{enumerate} \item We say a finite field extension $K \subset K_1$ is a {\it weak solution for $A \subset B$} if all the extensions $(A_1)_{\mathfrak m_i} \subset (B_1)_{\mathfrak m_{ij}}$ of Remark \ref{remark-construction} are weakly unramified. \item We say a finite field extension $K \subset K_1$ is a {\it solution for $A \subset B$} if each extension $(A_1)_{\mathfrak m_i} \subset (B_1)_{\mathfrak m_{ij}}$ of Remark \ref{remark-construction} is formally smooth in the $\mathfrak m_{ij}$-adic topology. \end{enumerate} We say a solution $K \subset K_1$ is a {\it separable solution} if $K \subset K_1$ is separable.


more-algebra-definition-invertible

Let $R$ be a ring. An $R$-module $M$ is {\it invertible} if the functor $$ \text{Mod}_R \longrightarrow \text{Mod}_R,\quad N \longmapsto M \otimes_R N $$ is an equivalence of categories. An invertible $R$-module is said to be {\it trivial} if it is isomorphic to $A$ as an $A$-module.


more-algebra-definition-extension-valuation-rings

We say that $A \to B$ or $A \subset B$ is an {\it extension of valuation rings} if $A$ and $B$ are valuation rings and $A \to B$ is injective and local. Such an extension induces a commutative diagram $$ \xymatrix{ A \setminus \{0\} \ar[r] \ar[d]_v & B \setminus \{0\} \ar[d]^v \\ \Gamma_A \ar[r] & \Gamma_B } $$ where $\Gamma_A$ and $\Gamma_B$ are the value groups. We say that $B$ is {\it weakly unramified} over $A$ if the lower horizontal arrow is a bijection. If the extension of residue fields $\kappa_A = A/\mathfrak m_A \subset \kappa_B = B/\mathfrak m_B$ is finite, then we set $f = [\kappa_B : \kappa_A]$ and we call it the {\it residual degree} or {\it residue degree} of the extension $A \subset B$.


more-algebra-definition-bezout

Let $R$ be a domain. \begin{enumerate} \item We say $R$ is a {\it B\'ezout domain} if every finitely generated ideal of $R$ is principal. \item We say $R$ is an {\it elementary divisor domain} if for all $n , m \geq 1$ and every $n \times m$ matrix $A$, there exist invertible matrices $U, V$ of size $n \times n, m \times m$ such that $$ U A V = \left( \begin{matrix} f_1 & 0 & 0 & \ldots \\ 0 & f_2 & 0 & \ldots \\ 0 & 0 & f_3 & \ldots \\ \ldots & \ldots & \ldots & \ldots \end{matrix} \right) $$ with $f_1, \ldots, f_{\min(n, m)} \in R$ and $f_1 | f_2 | \ldots$. \end{enumerate}


etale-definition-unramified-rings

Let $A$, $B$ be Noetherian local rings. A local homomorphism $A \to B$ is said to be {\it unramified homomorphism of local rings} if \begin{enumerate} \item $\mathfrak m_AB = \mathfrak m_B$, \item $\kappa(\mathfrak m_B)$ is a finite separable extension of $\kappa(\mathfrak m_A)$, and \item $B$ is essentially of finite type over $A$ (this means that $B$ is the localization of a finite type $A$-algebra at a prime). \end{enumerate}


etale-definition-unramified-schemes

(See Morphisms, Definition \ref{morphisms-definition-unramified} for the definition in the general case.) Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be locally of finite type. Let $x \in X$. \begin{enumerate} \item We say $f$ is {\it unramified at $x$} if $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is an unramified homomorphism of local rings. \item The morphism $f : X \to Y$ is said to be {\it unramified} if it is unramified at all points of $X$. \end{enumerate}


etale-definition-flat-rings

Flatness of modules and rings. \begin{enumerate} \item A module $N$ over a ring $A$ is said to be {\it flat} if the functor $M \mapsto M \otimes_A N$ is exact. \item If this functor is also faithful, we say that $N$ is {\it faithfully flat} over $A$. \item A morphism of rings $f : A \to B$ is said to be {\it flat (resp. faithfully flat)} if the functor $M \mapsto M \otimes_A B$ is exact (resp. faithful and exact). \end{enumerate}


etale-definition-flat-schemes

(See Morphisms, Definition \ref{morphisms-definition-flat}). Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. \begin{enumerate} \item Let $x \in X$. We say $\mathcal{F}$ is {\it flat over $Y$ at $x \in X$} if $\mathcal{F}_x$ is a flat $\mathcal{O}_{Y, f(x)}$-module. This uses the map $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ to think of $\mathcal{F}_x$ as a $\mathcal{O}_{Y, f(x)}$-module. \item Let $x \in X$. We say $f$ is {\it flat at $x \in X$} if $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is flat. \item We say $f$ is {\it flat} if it is flat at all points of $X$. \item A morphism $f : X \to Y$ that is flat and surjective is sometimes said to be {\it faithfully flat}. \end{enumerate}


etale-definition-etale-ring

Let $A$, $B$ be Noetherian local rings. A local homomorphism $f : A \to B$ is said to be an {\it \'etale homomorphism of local rings} if it is flat and an unramified homomorphism of local rings (please see Definition \ref{definition-unramified-rings}).


etale-definition-etale-schemes-1

(See Morphisms, Definition \ref{morphisms-definition-etale}.) Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be a morphism of schemes which is locally of finite type. \begin{enumerate} \item Let $x \in X$. We say $f$ is {\it \'etale at $x \in X$} if $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is an \'etale homomorphism of local rings. \item The morphism is said to be {\it \'etale} if it is \'etale at all its points. \end{enumerate}


etale-definition-strict-normal-crossings

Let $X$ be a locally Noetherian scheme. A {\it strict normal crossings divisor} on $X$ is an effective Cartier divisor $D \subset X$ such that for every $p \in D$ the local ring $\mathcal{O}_{X, p}$ is regular and there exists a regular system of parameters $x_1, \ldots, x_d \in \mathfrak m_p$ and $1 \leq r \leq d$ such that $D$ is cut out by $x_1 \ldots x_r$ in $\mathcal{O}_{X, p}$.


etale-definition-normal-crossings

Let $X$ be a locally Noetherian scheme. A {\it normal crossings divisor} on $X$ is an effective Cartier divisor $D \subset X$ such that for every $p \in D$ there exists an \'etale morphism $U \to X$ with $p$ in the image and $D \times_X U$ a strict normal crossings divisor on $U$.


crystalline-definition-divided-power-envelope

Let $(A, I, \gamma)$ be a divided power ring. Let $A \to B$ be a ring map. Let $J \subset B$ be an ideal with $IB \subset J$. The divided power algebra $(D, \bar J, \bar\gamma)$ constructed in Lemma \ref{lemma-divided-power-envelope} is called the {\it divided power envelope of $J$ in $B$ relative to $(A, I, \gamma)$} and is denoted $D_B(J)$ or $D_{B, \gamma}(J)$.


crystalline-definition-compatible

Let $(A, I, \gamma)$ and $(B, J, \delta)$ be divided power rings. Let $A \to B$ be a ring map. We say {\it $\delta$ is compatible with $\gamma$} if there exists a divided power structure $\bar\gamma$ on $J + IB$ such that $$ (A, I, \gamma) \to (B, J + IB, \bar \gamma)\quad\text{and}\quad (B, J, \delta) \to (B, J + IB, \bar \gamma) $$ are homomorphisms of divided power rings.


crystalline-definition-affine-thickening

In Situation \ref{situation-affine}. \begin{enumerate} \item A {\it divided power thickening} of $C$ over $(A, I, \gamma)$ is a homomorphism of divided power algebras $(A, I, \gamma) \to (B, J, \delta)$ such that $p$ is nilpotent in $B$ and a ring map $C \to B/J$ such that $$ \xymatrix{ B \ar[r] & B/J \\ & C \ar[u] \\ A \ar[uu] \ar[r] & A/I \ar[u] } $$ is commutative. \item A {\it homomorphism of divided power thickenings} $$ (B, J, \delta, C \to B/J) \longrightarrow (B', J', \delta', C \to B'/J') $$ is a homomorphism $\varphi : B \to B'$ of divided power $A$-algebras such that $C \to B/J \to B'/J'$ is the given map $C \to B'/J'$. \item We denote $\text{CRIS}(C/A, I, \gamma)$ or simply $\text{CRIS}(C/A)$ the category of divided power thickenings of $C$ over $(A, I, \gamma)$. \item We denote $\text{Cris}(C/A, I, \gamma)$ or simply $\text{Cris}(C/A)$ the full subcategory consisting of $(B, J, \delta, C \to B/J)$ such that $C \to B/J$ is an isomorphism. We often denote such an object $(B \to C, \delta)$ with $J = \Ker(B \to C)$ being understood. \end{enumerate}


crystalline-definition-derivation

Let $A$ be a ring. Let $(B, J, \delta)$ be a divided power ring. Let $A \to B$ be a ring map. Let $M$ be an $B$-module. A {\it divided power $A$-derivation} into $M$ is a map $\theta : B \to M$ which is additive, annihilates the elements of $A$, satisfies the Leibniz rule $\theta(bb') = b\theta(b') + b'\theta(b)$ and satisfies $$ \theta(\delta_n(x)) = \delta_{n - 1}(x)\theta(x) $$ for all $n \geq 1$ and all $x \in J$.


crystalline-definition-divided-power-structure

Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. Let $\mathcal{I} \subset \mathcal{O}$ be a sheaf of ideals. A {\it divided power structure $\gamma$} on $\mathcal{I}$ is a sequence of maps $\gamma_n : \mathcal{I} \to \mathcal{I}$, $n \geq 1$ such that for any object $U$ of $\mathcal{C}$ the triple $$ (\mathcal{O}(U), \mathcal{I}(U), \gamma) $$ is a divided power ring.


crystalline-definition-divided-power-scheme

A {\it divided power scheme} is a triple $(S, \mathcal{I}, \gamma)$ where $S$ is a scheme, $\mathcal{I}$ is a quasi-coherent sheaf of ideals, and $\gamma$ is a divided power structure on $\mathcal{I}$. A {\it morphism of divided power schemes} $(S, \mathcal{I}, \gamma) \to (S', \mathcal{I}', \gamma')$ is a morphism of schemes $f : S \to S'$ such that $f^{-1}\mathcal{I}'\mathcal{O}_S \subset \mathcal{I}$ and such that $$ (\mathcal{O}_{S'}(U'), \mathcal{I}'(U'), \gamma') \longrightarrow (\mathcal{O}_S(f^{-1}U'), \mathcal{I}(f^{-1}U'), \gamma) $$ is a homomorphism of divided power rings for all $U' \subset S'$ open.


crystalline-definition-divided-power-thickening

A triple $(U, T, \gamma)$ as above is called a {\it divided power thickening} if $U \to T$ is a thickening.


crystalline-definition-divided-power-thickening-X

In Situation \ref{situation-global}. \begin{enumerate} \item A {\it divided power thickening of $X$ relative to $(S, \mathcal{I}, \gamma)$} is given by a divided power thickening $(U, T, \delta)$ over $(S, \mathcal{I}, \gamma)$ and an $S$-morphism $U \to X$. \item A {\it morphism of divided power thickenings of $X$ relative to $(S, \mathcal{I}, \gamma)$} is defined in the obvious manner. \end{enumerate} The category of divided power thickenings of $X$ relative to $(S, \mathcal{I}, \gamma)$ is denoted $\text{CRIS}(X/S, \mathcal{I}, \gamma)$ or simply $\text{CRIS}(X/S)$.


crystalline-definition-big-crystalline-site

In Situation \ref{situation-global}. \begin{enumerate} \item A family of morphisms $\{(U_i, T_i, \delta_i) \to (U, T, \delta)\}$ of divided power thickenings of $X/S$ is a {\it Zariski, \'etale, smooth, syntomic, or fppf covering} if and only if \begin{enumerate} \item $U_i = U \times_T T_i$ for all $i$ and \item $\{T_i \to T\}$ is a Zariski, \'etale, smooth, syntomic, or fppf covering. \end{enumerate} \item The {\it big crystalline site} of $X$ over $(S, \mathcal{I}, \gamma)$, is the category $\text{CRIS}(X/S)$ endowed with the Zariski topology. \item The topos of sheaves on $\text{CRIS}(X/S)$ is denoted $(X/S)_{\text{CRIS}}$ or sometimes $(X/S, \mathcal{I}, \gamma)_{\text{CRIS}}$\footnote{This clashes with our convention to denote the topos associated to a site $\mathcal{C}$ by $\Sh(\mathcal{C})$.}. \end{enumerate}


crystalline-definition-crystalline-site

In Situation \ref{situation-global}. \begin{enumerate} \item The (small) {\it crystalline site} of $X$ over $(S, \mathcal{I}, \gamma)$, denoted $\text{Cris}(X/S, \mathcal{I}, \gamma)$ or simply $\text{Cris}(X/S)$ is the full subcategory of $\text{CRIS}(X/S)$ consisting of those $(U, T, \delta)$ in $\text{CRIS}(X/S)$ such that $U \to X$ is an open immersion. It comes endowed with the Zariski topology. \item The topos of sheaves on $\text{Cris}(X/S)$ is denoted $(X/S)_{\text{cris}}$ or sometimes $(X/S, \mathcal{I}, \gamma)_{\text{cris}}$\footnote{This clashes with our convention to denote the topos associated to a site $\mathcal{C}$ by $\Sh(\mathcal{C})$.}. \end{enumerate}


crystalline-definition-modules

In Situation \ref{situation-global}. Let $\mathcal{C} = \text{CRIS}(X/S)$ or $\mathcal{C} = \text{Cris}(X/S)$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_{X/S}$-modules on $\mathcal{C}$. \begin{enumerate} \item We say $\mathcal{F}$ is {\it locally quasi-coherent} if for every object $(U, T, \delta)$ of $\mathcal{C}$ the restriction $\mathcal{F}_T$ is a quasi-coherent $\mathcal{O}_T$-module. \item We say $\mathcal{F}$ is {\it quasi-coherent} if it is quasi-coherent in the sense of Modules on Sites, Definition \ref{sites-modules-definition-site-local}. \item We say $\mathcal{F}$ is a {\it crystal in $\mathcal{O}_{X/S}$-modules} if all the comparison maps (\ref{equation-comparison-modules}) are isomorphisms. \end{enumerate}


crystalline-definition-crystal-quasi-coherent-modules

If $\mathcal{F}$ satisfies the equivalent conditions of Lemma \ref{lemma-crystal-quasi-coherent-modules}, then we say that $\mathcal{F}$ is a {\it crystal in quasi-coherent modules}. We say that $\mathcal{F}$ is a {\it crystal in finite locally free modules} if, in addition, $\mathcal{F}$ is finite locally free.


crystalline-definition-global-derivation

In Situation \ref{situation-global} let $\mathcal{F}$ be a sheaf of $\mathcal{O}_{X/S}$-modules on $\text{Cris}(X/S)$. An {\it $S$-derivation $D : \mathcal{O}_{X/S} \to \mathcal{F}$} is a map of sheaves such that for every object $(U, T, \delta)$ of $\text{Cris}(X/S)$ the map $$ D : \Gamma(T, \mathcal{O}_T) \longrightarrow \Gamma(T, \mathcal{F}) $$ is a divided power $\Gamma(V, \mathcal{O}_V)$-derivation where $V \subset S$ is any open such that $T \to S$ factors through $V$.


crystalline-definition-F-crystal

In Situation \ref{situation-F-crystal} an {\it $F$-crystal on $X/S$ (relative to $\sigma$)} is a pair $(\mathcal{E}, F_\mathcal{E})$ given by a crystal in finite locally free $\mathcal{O}_{X/S}$-modules $\mathcal{E}$ together with a map $$ F_\mathcal{E} : (F_X)_{\text{cris}}^*\mathcal{E} \longrightarrow \mathcal{E} $$ An $F$-crystal is called {\it nondegenerate} if there exists an integer $i \geq 0$ a map $V : \mathcal{E} \to (F_X)_{\text{cris}}^*\mathcal{E}$ such that $V \circ F_{\mathcal{E}} = p^i \text{id}$.


varieties-definition-variety

Let $k$ be a field. A {\it variety} is a scheme $X$ over $k$ such that $X$ is integral and the structure morphism $X \to \Spec(k)$ is separated and of finite type.


varieties-definition-geometrically-reduced

Let $k$ be a field. Let $X$ be a scheme over $k$. \begin{enumerate} \item Let $x \in X$ be a point. We say $X$ is {\it geometrically reduced at $x$} if for any field extension $k \subset k'$ and any point $x' \in X_{k'}$ lying over $x$ the local ring $\mathcal{O}_{X_{k'}, x'}$ is reduced. \item We say $X$ is {\it geometrically reduced} over $k$ if $X$ is geometrically reduced at every point of $X$. \end{enumerate}


varieties-definition-geometrically-connected

Let $X$ be a scheme over the field $k$. We say $X$ is {\it geometrically connected} over $k$ if the scheme $X_{k'}$ is connected for every field extension $k'$ of $k$.


varieties-definition-geometrically-irreducible

Let $X$ be a scheme over the field $k$. We say $X$ is {\it geometrically irreducible} over $k$ if the scheme $X_{k'}$ is irreducible\footnote{An irreducible space is nonempty.} for any field extension $k'$ of $k$.


varieties-definition-geometrically-integral

Let $X$ be a scheme over the field $k$. \begin{enumerate} \item Let $x \in X$. We say $X$ is {\it geometrically pointwise integral at $x$} if for every field extension $k \subset k'$ and every $x' \in X_{k'}$ lying over $x$ the local ring $\mathcal{O}_{X_{k'}, x'}$ is integral. \item We say $X$ is {\it geometrically pointwise integral} if $X$ is geometrically pointwise integral at every point. \item We say $X$ is {\it geometrically integral} over $k$ if the scheme $X_{k'}$ is integral for every field extension $k'$ of $k$. \end{enumerate}


varieties-definition-geometrically-normal

Let $X$ be a scheme over the field $k$. \begin{enumerate} \item Let $x \in X$. We say $X$ is {\it geometrically normal at $x$} if for every field extension $k \subset k'$ and every $x' \in X_{k'}$ lying over $x$ the local ring $\mathcal{O}_{X_{k'}, x'}$ is normal. \item We say $X$ is {\it geometrically normal} over $k$ if $X$ is geometrically normal at every $x \in X$. \end{enumerate}


varieties-definition-geometrically-regular

Let $k$ be a field. Let $X$ be a locally Noetherian scheme over $k$. \begin{enumerate} \item Let $x \in X$. We say $X$ is {\it geometrically regular at $x$} over $k$ if for every finitely generated field extension $k \subset k'$ and any $x' \in X_{k'}$ lying over $x$ the local ring $\mathcal{O}_{X_{k'}, x'}$ is regular. \item We say $X$ is {\it geometrically regular over $k$} if $X$ is geometrically regular at all of its points. \end{enumerate}


varieties-definition-dual-numbers

For any ring $R$ the {\it dual numbers} over $R$ is the $R$-algebra denoted $R[\epsilon]$. As an $R$-module it is free with basis $1$, $\epsilon$ and the $R$-algebra structure comes from setting $\epsilon^2 = 0$.


varieties-definition-tangent-space

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. The set of dotted arrows making (\ref{equation-tangent-space}) commute with its canonical $\kappa(x)$-vector space structure is called the {\it tangent space of $X$ over $S$ at $x$} and we denote it $T_{X/S, x}$. An element of this space is called a {\it tangent vector} of $X/S$ at $x$.


varieties-definition-algebraic-scheme

Let $k$ be a field. An {\it algebraic $k$-scheme} is a scheme $X$ over $k$ such that the structure morphism $X \to \Spec(k)$ is of finite type. A {\it locally algebraic $k$-scheme} is a scheme $X$ over $k$ such that the structure morphism $X \to \Spec(k)$ is locally of finite type.


varieties-definition-variety-type

Let $k$ be a field. Let $X$ be a variety over $k$. \begin{enumerate} \item We say $X$ is an {\it affine variety} if $X$ is an affine scheme. This is equivalent to requiring $X$ to be isomorphic to a closed subscheme of $\mathbf{A}^n_k$ for some $n$. \item We say $X$ is a {\it projective variety} if the structure morphism $X \to \Spec(k)$ is projective. By Morphisms, Lemma \ref{morphisms-lemma-characterize-locally-projective} this is true if and only if $X$ is isomorphic to a closed subscheme of $\mathbf{P}^n_k$ for some $n$. \item We say $X$ is a {\it quasi-projective variety} if the structure morphism $X \to \Spec(k)$ is quasi-projective. By Morphisms, Lemma \ref{morphisms-lemma-characterize-locally-quasi-projective} this is true if and only if $X$ is isomorphic to a locally closed subscheme of $\mathbf{P}^n_k$ for some $n$. \item A {\it proper variety} is a variety such that the morphism $X \to \Spec(k)$ is proper. \item A {\it smooth variety} is a variety such that the morphism $X \to \Spec(k)$ is smooth. \end{enumerate}


varieties-definition-euler-characteristic

Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. In this situation the {\it Euler characteristic of $\mathcal{F}$} is the integer $$ \chi(X, \mathcal{F}) = \sum\nolimits_i (-1)^i \dim_k H^i(X, \mathcal{F}). $$ For justification of the formula see below.


varieties-definition-regularity

Let $k$ be a field. Let $n \geq 0$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbf{P}^n_k$. We say $\mathcal{F}$ is {\it $m$-regular} if $$ H^i(\mathbf{P}^n_k, \mathcal{F}(m - i)) = 0 $$ for $i = 1, \ldots, n$.


varieties-definition-hilbert-polynomial

Let $k$ be a field. Let $n \geq 0$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbf{P}^n_k$. The function $d \mapsto \chi(\mathbf{P}^n_k, \mathcal{F}(d))$ is called the {\it Hilbert polynomial} of $\mathcal{F}$.


varieties-definition-absolute-frobenius

Let $p$ be a prime number. Let $X$ be a scheme in characteristic $p$. The {\it absolute frobenius of $X$} is the morphism $F_X : X \to X$ given by the identity on the underlying topological space and with $F_X^\sharp : \mathcal{O}_X \to \mathcal{O}_X$ given by $g \mapsto g^p$.


varieties-definition-relative-frobenius

Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$. Let $X$ be a scheme over $S$. We define $$ X^{(p)} = X^{(p/S)} = X \times_{S, F_S} S $$ viewed as a scheme over $S$. Applying Lemma \ref{lemma-frobenius-endomorphism-identity} we see there is a unique morphism $F_{X/S} : X \longrightarrow X^{(p)}$ over $S$ fitting into the commutative diagram $$ \xymatrix{ X \ar[rr]_{F_{X/S}} \ar[rrd] \ar@/^1em/[rrrr]^{F_X} & & X^{(p)} \ar[rr] \ar[d] & & X \ar[d] \\ & & S \ar[rr]^{F_S} & & S } $$ where the right square is cartesian. The morphism $F_{X/S}$ is called the {\it relative Frobenius morphism of $X/S$}.


varieties-definition-delta-invariant-algebra

Let $A$ be a reduced Nagata local ring of dimension $1$. The {\it $\delta$-invariant of $A$} is $\text{length}_A(A'/A)$ where $A'$ is as in Lemma \ref{lemma-pre-delta-invariant}.


varieties-definition-delta-invariant

Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. Let $x \in X$ be a point such that $\mathcal{O}_{X, x}$ is reduced and $\dim(\mathcal{O}_{X, x}) = 1$. The {\it $\delta$-invariant of $X$ at $x$} is the $\delta$-invariant of $\mathcal{O}_{X, x}$ as defined in Definition \ref{definition-delta-invariant-algebra}.


varieties-definition-wedge

Let $A$ and $A_i$, $1 \leq i \leq n$ be local rings. We say {\it $A$ is a wedge of $A_1, \ldots, A_n$} if there exist isomorphisms $$ \kappa_{A_1} \to \kappa_{A_2} \to \ldots \to \kappa_{A_n} $$ and $A$ is isomorphic to the ring consisting of $n$-tuples $(a_1, \ldots, a_n) \in A_1 \times \ldots \times A_n$ which map to the same element of $\kappa_{A_n}$.


varieties-definition-curve

Let $k$ be a field. A {\it curve} is a variety of dimension $1$ over $k$.


varieties-definition-degree-invertible-sheaf

Let $k$ be a field, let $X$ be a proper scheme of dimension $\leq 1$ over $k$, and let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. The {\it degree} of $\mathcal{L}$ is defined by $$ \deg(\mathcal{L}) = \chi(X, \mathcal{L}) - \chi(X, \mathcal{O}_X) $$ More generally, if $\mathcal{E}$ is a locally free sheaf of rank $n$ we define the {\it degree} of $\mathcal{E}$ by $$ \deg(\mathcal{E}) = \chi(X, \mathcal{E}) - n\chi(X, \mathcal{O}_X) $$


varieties-definition-intersection-number

Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $i : Z \to X$ be a closed subscheme of dimension $d$. Let $\mathcal{L}_1, \ldots, \mathcal{L}_d$ be invertible $\mathcal{O}_X$-modules. We define the {\it intersection number} $(\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z)$ as the coefficient of $n_1 \ldots n_d$ in the numerical polynomial $$ \chi(X, i_*\mathcal{O}_Z \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_d^{\otimes n_d}) = \chi(Z, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_d^{\otimes n_d}|_Z) $$ In the special case that $\mathcal{L}_1 = \ldots = \mathcal{L}_d = \mathcal{L}$ we write $(\mathcal{L}^d \cdot Z)$.


varieties-definition-degree

Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_X$-module. For any closed subscheme the {\it degree of $Z$ with respect to $\mathcal{L}$}, denoted $\deg_\mathcal{L}(Z)$, is the intersection number $(\mathcal{L}^d \cdot Z)$ where $d = \dim(Z)$.


varieties-definition-embed-dim

Let $k$ be an algebraically closed field. Let $X$ be a locally algebraic $k$-scheme and let $x \in X$ be a closed point. The {\it embedding dimension of $X$ at $x$} is $\dim_k \mathfrak m_x/\mathfrak m_x^2$.


varieties-definition-embedding-dimension

Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. Let $x \in X$ be a point. The {\it embedding dimension of $X/k$ at $x$} is $\dim_{\kappa(x)}(T_{X/k, x})$.


cotangent-definition-standard-resolution

Let $A \to B$ be a ring map. The {\it standard resolution of $B$ over $A$} is the augmentation $\epsilon : P_\bullet \to B$ with terms $$ P_0 = A[B],\quad P_1 = A[A[B]],\quad \ldots $$ and maps as constructed above.


cotangent-definition-cotangent-complex-ring-map

The {\it cotangent complex} $L_{B/A}$ of a ring map $A \to B$ is the complex of $B$-modules associated to the simplicial $B$-module $$ \Omega_{P_\bullet/A} \otimes_{P_\bullet, \epsilon} B $$ where $\epsilon : P_\bullet \to B$ is the standard resolution of $B$ over $A$.


cotangent-definition-biderivation

Let $A \to B$ be a ring map. Let $M$ be a $(B, B)$-bimodule over $A$. An {\it $A$-biderivation} is an $A$-linear map $\lambda : B \to M$ such that $\lambda(xy) = x\lambda(y) + \lambda(x)y$.


cotangent-definition-atiyah-class

Let $A \to B$ be a ring map. Let $M$ be a $B$-module. The map $M \to L_{B/A} \otimes_B^\mathbf{L} M[1]$ in (\ref{equation-atiyah}) is called the {\it Atiyah class} of $M$.


cotangent-definition-standard-resolution-sheaves-rings

Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $\mathcal{C}$. The {\it standard resolution of $\mathcal{B}$ over $\mathcal{A}$} is the augmentation $\epsilon : \mathcal{P}_\bullet \to \mathcal{B}$ with terms $$ \mathcal{P}_0 = \mathcal{A}[\mathcal{B}],\quad \mathcal{P}_1 = \mathcal{A}[\mathcal{A}[\mathcal{B}]],\quad \ldots $$ and maps as constructed above.


cotangent-definition-cotangent-complex-morphism-sheaves-rings

Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $\mathcal{C}$. The {\it cotangent complex} $L_{\mathcal{B}/\mathcal{A}}$ is the complex of $\mathcal{B}$-modules associated to the simplicial module $$ \Omega_{\mathcal{P}_\bullet/\mathcal{A}} \otimes_{\mathcal{P}_\bullet, \epsilon} \mathcal{B} $$ where $\epsilon : \mathcal{P}_\bullet \to \mathcal{B}$ is the standard resolution of $\mathcal{B}$ over $\mathcal{A}$. We usually think of $L_{\mathcal{B}/\mathcal{A}}$ as an object of $D(\mathcal{B})$.


cotangent-definition-atiyah-class-general

Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings. Let $\mathcal{F}$ be a sheaf of $\mathcal{B}$-modules. The map $\mathcal{F} \to L_{\mathcal{B}/\mathcal{A}} \otimes_\mathcal{B}^\mathbf{L} \mathcal{F}[1]$ in (\ref{equation-atiyah-general}) is called the {\it Atiyah class} of $\mathcal{F}$.


cotangent-definition-cotangent-complex-morphism-ringed-spaces

Let $f : (X, \mathcal{O}_X) \to (S, \mathcal{O}_S)$ be a morphism of ringed spaces. The {\it cotangent complex} $L_f$ of $f$ is $L_f = L_{\mathcal{O}_X/f^{-1}\mathcal{O}_S}$. We will also use the notation $L_f = L_{X/S} = L_{\mathcal{O}_X/\mathcal{O}_S}$.


cotangent-definition-cotangent-complex-morphism-ringed-topoi

Let $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. The {\it cotangent complex} $L_f$ of $f$ is $L_f = L_{\mathcal{O}_\mathcal{C}/f^{-1}\mathcal{O}_\mathcal{D}}$. We sometimes write $L_f = L_{\mathcal{O}_\mathcal{C}/\mathcal{O}_\mathcal{D}}$.


cotangent-definition-cotangent-morphism-schemes

Let $f : X \to Y$ be a morphism of schemes. The {\it cotangent complex $L_{X/Y}$ of $X$ over $Y$} is the cotangent complex of $f$ as a morphism of ringed spaces (Definition \ref{definition-cotangent-complex-morphism-ringed-spaces}).


cotangent-definition-cotangent-morphism-spaces

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The {\it cotangent complex $L_{X/Y}$ of $X$ over $Y$} is the cotangent complex of the morphism of ringed topoi $f_{small}$ between the small \'etale sites of $X$ and $Y$ (see Properties of Spaces, Lemma \ref{spaces-properties-lemma-morphism-ringed-topoi} and Definition \ref{definition-cotangent-complex-morphism-ringed-topoi}).


spaces-cohomology-definition-alternating-cech-complex

Let $S$ be a scheme. Let $f : U \to X$ be a surjective \'etale morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be an object of $\textit{Ab}(X_\etale)$. The {\it alternating {\v C}ech complex}\footnote{This may be nonstandard notation} $\check{\mathcal{C}}^\bullet_{alt}(f, \mathcal{F})$ associated to $\mathcal{F}$ and $f$ is the complex $$ \Hom(K^0, \mathcal{F}) \to \Hom(K^1, \mathcal{F}) \to \Hom(K^2, \mathcal{F}) \to \ldots $$ with Hom groups computed in $\textit{Ab}(X_\etale)$.


spaces-cohomology-definition-coherent

Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. A quasi-coherent module $\mathcal{F}$ on $X$ is called {\it coherent} if $\mathcal{F}$ is a coherent $\mathcal{O}_X$-module on the site $X_\etale$ in the sense of Modules on Sites, Definition \ref{sites-modules-definition-site-local}.


artin-definition-RS

Let $S$ be a locally Noetherian scheme. Let $\mathcal{Z}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. We say $\mathcal{Z}$ satisfies {\it condition (RS)} if for every pushout $$ \xymatrix{ X \ar[r] \ar[d] & X' \ar[d] \\ Y \ar[r] & Y' = Y \amalg_X X' } $$ in the category of schemes over $S$ where \begin{enumerate} \item $X$, $X'$, $Y$, $Y'$ are spectra of local Artinian rings, \item $X$, $X'$, $Y$, $Y'$ are of finite type over $S$, and \item $X \to X'$ (and hence $Y \to Y'$) is a closed immersion \end{enumerate} the functor of fibre categories $$ \mathcal{Z}_{Y'} \longrightarrow \mathcal{Z}_Y \times_{\mathcal{Z}_X} \mathcal{Z}_{X'} $$ is an equivalence of categories.


artin-definition-formal-objects

Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. \begin{enumerate} \item A {\it formal object} $\xi = (R, \xi_n, f_n)$ of $\mathcal{X}$ consists of a Noetherian complete local $S$-algebra $R$, objects $\xi_n$ of $\mathcal{X}$ lying over $\Spec(R/\mathfrak m_R^n)$, and morphisms $f_n : \xi_n \to \xi_{n + 1}$ of $\mathcal{X}$ lying over $\Spec(R/\mathfrak m^n) \to \Spec(R/\mathfrak m^{n + 1})$ such that $R/\mathfrak m$ is a field of finite type over $S$. \item A {\it morphism of formal objects} $a : \xi = (R, \xi_n, f_n) \to \eta = (T, \eta_n, g_n)$ is given by morphisms $a_n : \xi_n \to \eta_n$ such that for every $n$ the diagram $$ \xymatrix{ \xi_n \ar[r]_{f_n} \ar[d]_{a_n} & \xi_{n + 1} \ar[d]^{a_{n + 1}} \\ \eta_n \ar[r]^{g_n} & \eta_{n + 1} } $$ is commutative. Applying the functor $p$ we obtain a compatible collection of morphisms $\Spec(R/\mathfrak m_R^n) \to \Spec(T/\mathfrak m_T^n)$ and hence a morphism $a_0 : \Spec(R) \to \Spec(T)$ over $S$. We say that $a$ {\it lies over} $a_0$. \end{enumerate}


artin-definition-effective

Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. A formal object $\xi = (R, \xi_n, f_n)$ of $\mathcal{X}$ is called {\it effective} if it is in the essential image of the functor (\ref{equation-approximation}).


artin-definition-limit-preserving

Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. We say $\mathcal{X}$ is {\it limit preserving} if for every affine scheme $T$ over $S$ which is a limit $T = \lim T_i$ of a directed inverse system of affine schemes $T_i$ over $S$, we have an equivalence $$ \colim \mathcal{X}_{T_i} \longrightarrow \mathcal{X}_T $$ of fibre categories.


artin-definition-versal-formal-object

Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\xi = (R, \xi_n, f_n)$ be a formal object. Set $k = R/\mathfrak m$ and $x_0 = \xi_1$. We will say that $\xi$ is {\it versal} if $\xi$ as a formal object of $\mathcal{F}_{\mathcal{X}, k, x_0}$ (Remark \ref{remark-formal-objects-match}) is versal in the sense of Formal Deformation Theory, Definition \ref{formal-defos-definition-versal}.


artin-definition-versal

Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be fibred in groupoids over $(\Sch/S)_{fppf}$. Let $U$ be a scheme locally of finite type over $S$. Let $x$ be an object of $\mathcal{X}$ lying over $U$. Let $u_0$ be finite type point of $U$. We say $x$ is {\it versal} at $u_0$ if the morphism $\hat x$ (\ref{equation-hat-x}) is smooth, see Formal Deformation Theory, Definition \ref{formal-defos-definition-smooth-morphism}.


artin-definition-openness-versality

Let $S$ be a locally Noetherian scheme. \begin{enumerate} \item Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. We say $\mathcal{X}$ satisfies {\it openness of versality} if given a scheme $U$ locally of finite type over $S$, an object $x$ of $\mathcal{X}$ over $U$, and a finite type point $u_0 \in U$ such that $x$ is versal at $u_0$, then there exists an open neighbourhood $u_0 \in U' \subset U$ such that $x$ is versal at every finite type point of $U'$. \item Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. We say $f$ satisfies {\it openness of versality} if given a scheme $U$ locally of finite type over $S$, an object $y$ of $\mathcal{Y}$ over $U$, openness of versality holds for $(\Sch/U)_{fppf} \times_\mathcal{Y} \mathcal{X}$. \end{enumerate}


artin-definition-RS-star

Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. We say $\mathcal{X}$ satisfies {\it condition (RS*)} if given a fibre product diagram $$ \xymatrix{ B' \ar[r] & B \\ A' = A \times_B B' \ar[u] \ar[r] & A \ar[u] } $$ of $S$-algebras, with $B' \to B$ surjective with square zero kernel, the functor of fibre categories $$ \mathcal{X}_{\Spec(A')} \longrightarrow \mathcal{X}_{\Spec(A)} \times_{\mathcal{X}_{\Spec(B)}} \mathcal{X}_{\Spec(B')} $$ is an equivalence of categories.


artin-definition-obstruction-theory

Let $S$ be a locally Noetherian base. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. An {\it obstruction theory} is  given by the following data \begin{enumerate} \item for every $S$-algebra $A$ such that $\Spec(A) \to S$ maps into an affine open and every object $x$ of $\mathcal{X}$ over $\Spec(A)$ an $A$-linear functor $$ \mathcal{O}_x : \text{Mod}_A \to \text{Mod}_A $$ of {\it obstruction modules}, \item for $(x, A)$ as in (1), a ring map $A \to B$, $M \in \text{Mod}_A$, $N \in \text{Mod}_B$, and an $A$-linear map $M \to N$ an induced $A$-linear map $\mathcal{O}_x(M) \to \mathcal{O}_y(N)$ where $y = x|_{\Spec(B)}$, and \item for every deformation situation $(x, A' \to A)$ an {\it obstruction} element $o_x(A') \in \mathcal{O}_x(I)$ where $I = \Ker(A' \to A)$. \end{enumerate} These data are subject to the following conditions \begin{enumerate} \item[(i)] the functoriality maps turn the obstruction modules into a functor from the category of triples $(x, A, M)$ to sets, \item[(ii)] for every morphism of deformation situations $(y, B' \to B) \to (x, A' \to A)$ the element $o_x(A')$ maps to $o_y(B')$, and \item[(iii)] we have $$ \text{Lift}(x, A') \not = \emptyset \Leftrightarrow o_x(A') = 0 $$ for every deformation situation $(x, A' \to A)$. \end{enumerate}


artin-definition-naive-obstruction-theory

Let $S$ be a locally Noetherian base. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. Assume that $\mathcal{X}$ satisfies (RS*). A {\it naive obstruction theory} is  given by the following data \begin{enumerate} \item \label{item-map} for every $S$-algebra $A$ such that $\Spec(A) \to S$ maps into an affine open $\Spec(\Lambda) \subset S$ and every object $x$ of $\mathcal{X}$ over $\Spec(A)$ we are given an object $E_x \in D^-(A)$ and a map $\xi_x : E \to \NL_{A/\Lambda}$, \item \label{item-inf} given $(x, A)$ as in (\ref{item-map}) there are transformations of functors $$ \text{Inf}_x( - ) \to \Ext^{-1}_A(E_x, -) \quad\text{and}\quad T_x(-) \to \Ext^0_A(E_x, -) $$ \item \label{item-functoriality} for $(x, A)$ as in (\ref{item-map}) and a ring map $A \to B$ setting $y = x|_{\Spec(B)}$ there is a functoriality map $E_x \to E_y$ in $D(A)$. \end{enumerate} These data are subject to the following conditions \begin{enumerate} \item[(i)] in the situation of (\ref{item-functoriality}) the diagram $$ \xymatrix{ E_y \ar[r]_{\xi_y} & \NL_{B/\Lambda} \\ E_x \ar[u] \ar[r]^{\xi_x} & \NL_{A/\Lambda} \ar[u] } $$ is commutative in $D(A)$, \item[(ii)] given $(x, A)$ as in (\ref{item-map}) and $A \to B \to C$ setting $y = x|_{\Spec(B)}$ and $z = x|_{\Spec(C)}$ the composition of the functoriality maps $E_x \to E_y$ and $E_y \to E_z$ is the functoriality map $E_x \to E_z$, \item[(iii)] the maps of (\ref{item-inf}) are isomorphisms compatible with the functoriality maps and the maps of Remark \ref{remark-functoriality}, \item[(iv)] the composition $E_x \to \NL_{A/\Lambda} \to \Omega_{A/\Lambda}$ corresponds to the canonical element of $T_x(\Omega_{A/\Lambda}) = \Ext^0(E_x, \Omega_{A/\Lambda})$, see Remark \ref{remark-canonical-element}, \item[(v)] given a deformation situation $(x, A' \to A)$ with $I = \Ker(A' \to A)$ the composition $E_x \to \NL_{A/\Lambda} \to \NL_{A/A'}$ is zero in $$ \Hom_A(E_x, \NL_{A/\Lambda}) = \Ext^0_A(E_x, \NL_{A/A'}) = \Ext^1_A(E_x, I) $$ if and only if $x$ lifts to $A'$. \end{enumerate}


obsolete-definition-epsilon

Let $(S, \delta)$ be as in Chow Homology, Situation \ref{chow-situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\dim_\delta(X) = n$. Let $D_1, D_2$ be two effective Cartier divisors in $X$. Let $Z \subset X$ be an integral closed subscheme with $\dim_\delta(Z) = n - 1$. The {\it $\epsilon$-invariant} of this situation is $$ \epsilon_Z(D_1, D_2) = n_Z \cdot m_Z $$ where $n_Z$, resp.\ $m_Z$ is the coefficient of $Z$ in the $(n - 1)$-cycle $[D_1]_{n - 1}$, resp.\ $[D_2]_{n - 1}$.


obsolete-definition-locally-finite-sum-effective-Cartier-divisors

Let $X$ be a scheme. Let $\{D_i\}_{i \in I}$ be a locally finite collection of effective Cartier divisors on $X$. Suppose given a function $I \to \mathbf{Z}_{\geq 0}$, $i \mapsto n_i$. The {\it sum of the effective Cartier divisors} $D = \sum n_i D_i$, is the unique effective Cartier divisor $D \subset X$ such that on any quasi-compact open $U \subset X$ we have $D|_U = \sum_{D_i \cap U \not = \emptyset} n_iD_i|_U$ is the sum as in Divisors, Definition \ref{divisors-definition-sum-effective-Cartier-divisors}.


stacks-sheaves-definition-presheaves

Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. \begin{enumerate} \item A {\it presheaf on $\mathcal{X}$} is a presheaf on the underlying category of $\mathcal{X}$. \item A {\it morphism of presheaves on $\mathcal{X}$} is a morphism of presheaves on the underlying category of $\mathcal{X}$. \end{enumerate} We denote $\textit{PSh}(\mathcal{X})$ the category of presheaves on $\mathcal{X}$.


stacks-sheaves-definition-inherited-topologies

Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. \begin{enumerate} \item The {\it associated Zariski site}, denoted $\mathcal{X}_{Zar}$, is the structure of site on $\mathcal{X}$ inherited from $(\Sch/S)_{Zar}$. \item The {\it associated \'etale site}, denoted $\mathcal{X}_\etale$, is the structure of site on $\mathcal{X}$ inherited from $(\Sch/S)_\etale$. \item The {\it associated smooth site}, denoted $\mathcal{X}_{smooth}$, is the structure of site on $\mathcal{X}$ inherited from $(\Sch/S)_{smooth}$. \item The {\it associated syntomic site}, denoted $\mathcal{X}_{syntomic}$, is the structure of site on $\mathcal{X}$ inherited from $(\Sch/S)_{syntomic}$. \item The {\it associated fppf site}, denoted $\mathcal{X}_{fppf}$, is the structure of site on $\mathcal{X}$ inherited from $(\Sch/S)_{fppf}$. \end{enumerate}


stacks-sheaves-definition-sheaves

Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$. \begin{enumerate} \item We say $\mathcal{F}$ is a {\it Zariski sheaf}, or a {\it sheaf for the Zariski topology} if $\mathcal{F}$ is a sheaf on the associated Zariski site $\mathcal{X}_{Zar}$. \item We say $\mathcal{F}$ is an {\it \'etale sheaf}, or a {\it sheaf for the \'etale topology} if $\mathcal{F}$ is a sheaf on the associated \'etale site $\mathcal{X}_\etale$. \item We say $\mathcal{F}$ is a {\it smooth sheaf}, or a {\it sheaf for the smooth topology} if $\mathcal{F}$ is a sheaf on the associated smooth site $\mathcal{X}_{smooth}$. \item We say $\mathcal{F}$ is a {\it syntomic sheaf}, or a {\it sheaf for the syntomic topology} if $\mathcal{F}$ is a sheaf on the associated syntomic site $\mathcal{X}_{syntomic}$. \item We say $\mathcal{F}$ is an {\it fppf sheaf}, or a {\it sheaf}, or a {\it sheaf for the fppf topology} if $\mathcal{F}$ is a sheaf on the associated fppf site $\mathcal{X}_{fppf}$. \end{enumerate} A morphism of sheaves is just a morphism of presheaves. We denote these categories of sheaves $\Sh(\mathcal{X}_{Zar})$, $\Sh(\mathcal{X}_\etale)$, $\Sh(\mathcal{X}_{smooth})$, $\Sh(\mathcal{X}_{syntomic})$, and $\Sh(\mathcal{X}_{fppf})$.


stacks-sheaves-definition-morphism

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. We denote $$ f = (f^{-1}, f_*) : \Sh(\mathcal{X}_{fppf}) \longrightarrow \Sh(\mathcal{Y}_{fppf}) $$ the {\it associated morphism of fppf topoi} constructed above. Similarly for the associated Zariski, \'etale, smooth, and syntomic topoi.


stacks-sheaves-definition-structure-sheaf

Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. The {\it structure sheaf of $\mathcal{X}$} is the sheaf of rings $\mathcal{O}_\mathcal{X} = p^{-1}\mathcal{O}$.


stacks-sheaves-definition-modules

Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. \begin{enumerate} \item A {\it presheaf of modules on $\mathcal{X}$} is a presheaf of $\mathcal{O}_\mathcal{X}$-modules. The category of presheaves of modules is denoted $\textit{PMod}(\mathcal{O}_\mathcal{X})$. \item We say a presheaf of modules $\mathcal{F}$ is an {\it $\mathcal{O}_\mathcal{X}$-module}, or more precisely a {\it sheaf of $\mathcal{O}_\mathcal{X}$-modules} if $\mathcal{F}$ is an fppf sheaf. The category of $\mathcal{O}_\mathcal{X}$-modules is denoted $\textit{Mod}(\mathcal{O}_\mathcal{X})$. \end{enumerate}


stacks-sheaves-definition-pullback

Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $x \in \Ob(\mathcal{X})$ lying over $U = p(x)$. Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$. \begin{enumerate} \item The {\it pullback $x^{-1}\mathcal{F}$ of $\mathcal{F}$} is the restriction $\mathcal{F}|_{(\mathcal{X}/x)}$ viewed as a presheaf on $(\Sch/U)_{fppf}$ via the equivalence $\mathcal{X}/x \to (\Sch/U)_{fppf}$ of Lemma \ref{lemma-localizing}. \item The {\it restriction of $\mathcal{F}$ to $U_\etale$} is $x^{-1}\mathcal{F}|_{U_\etale}$, abusively written $\mathcal{F}|_{U_\etale}$. \end{enumerate}


stacks-sheaves-definition-quasi-coherent

Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. A {\it quasi-coherent module on $\mathcal{X}$}, or a {\it quasi-coherent $\mathcal{O}_\mathcal{X}$-module} is a quasi-coherent module on the ringed site $(\mathcal{X}_{fppf}, \mathcal{O}_\mathcal{X})$ as in Modules on Sites, Definition \ref{sites-modules-definition-site-local}. The category of quasi-coherent sheaves on $\mathcal{X}$ is denoted $\QCoh(\mathcal{O}_\mathcal{X})$.


stacks-sheaves-definition-locally-quasi-coherent

Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_\mathcal{X}$-modules. We say $\mathcal{F}$ is {\it locally quasi-coherent}\footnote{This is nonstandard notation.} if $\mathcal{F}$ is a sheaf for the \'etale topology and for every object $x$ of $\mathcal{X}$ the restriction $x^*\mathcal{F}|_{U_\etale}$ is a quasi-coherent sheaf. Here $U = p(x)$.


resolve-definition-normalized-blowup

Let $X$ be a scheme such that every quasi-compact open has finitely many irreducible components. Let $x \in X$ be a closed point. The {\it normalized blowup of $X$ at $x$} is the composition $X'' \to X' \to X$ where $X' \to X$ is the blowup of $X$ in $x$ and $X'' \to X'$ is the normalization of $X'$.


resolve-definition-reduce-to-rational

Let $(A, \mathfrak m, \kappa)$ be a local normal Nagata domain of dimension $2$. \begin{enumerate} \item We say $A$ {\it defines a rational singularity} if for every normal modification $X \to \Spec(A)$ we have $H^1(X, \mathcal{O}_X) = 0$. \item We say that {\it reduction to rational singularities is possible for $A$} if the length of the $A$-modules $$ H^1(X, \mathcal{O}_X) $$ is bounded for all modifications $X \to \Spec(A)$ with $X$ normal. \end{enumerate}


resolve-definition-resolution

Let $Y$ be a Noetherian integral scheme. A {\it resolution of singularities} of $Y$ is a modification $f : X \to Y$ such that $X$ is regular.


resolve-definition-resolution-surface

Let $Y$ be a $2$-dimensional Noetherian integral scheme. We say $Y$ has a {\it resolution of singularities by normalized blowups} if there exists a sequence $$ Y_n \to Y_{n - 1} \to \ldots \to Y_1 \to Y_0 \to Y $$ where \begin{enumerate} \item $Y_i$ is proper over $Y$ for $i = 0, \ldots, n$, \item $Y_0 \to Y$ is the normalization, \item $Y_i \to Y_{i - 1}$ is a normalized blowup for $i = 1, \ldots, n$, and \item $Y_n$ is regular. \end{enumerate}


exercises-definition-directed-poset

A {\it directed set} is a nonempty set $I$ endowed with a preorder $\leq$ such that given any pair $i, j \in I$ there exists a $k \in I$ such that $i \leq k$ and $j \leq k$. A {\it system of rings} over $I$ is given by a ring $A_i$ for each $i \in I$ and a map of rings $\varphi_{ij} : A_i \to A_j$ whenever $i \leq j$ such that the composition $A_i \to A_j \to A_k$ is equal to $A_i \to A_k$ whenever $i \leq j \leq k$.


exercises-definition-colimit

The ring $A$ constructed in Exercise \ref{exercise-directed-colimit} is called the {\it colimit} of the system. Notation $\colim A_i$.


exercises-definition-finite-presentation

A module $M$ over $R$ is said to be of {\it finite presentation} over $R$ if it  is isomorphic to the cokernel of a map of finite free modules $ R^{\oplus n} \to R^{\oplus m}$.


exercises-definition-quasi-compact

A topological space $X$ is called {\it quasi-compact} if for any open covering $X = \bigcup_{i\in I} U_i$ there is a finite subset $\{i_1, \ldots, i_n\}\subset I$ such that $X = U_{i_1}\cup\ldots U_{i_n}$.


exercises-definition-Hausdorff

A topological space $X$ is said to verify the separation axiom $T_0$ if for any pair of points $x, y\in X$, $x\not = y$ there is an open subset of $X$ containing one but not the other. We say that $X$ is {\it Hausdorff} if for any pair $x, y\in X$, $x\not = y$ there are disjoint open subsets $U, V$ such that $x\in U$ and $y\in V$.


exercises-definition-irreducible

A topological space $X$ is called {\it irreducible} if $X$ is not empty and if $X = Z_1\cup Z_2$ with $Z_1, Z_2\subset X$ closed, then either $Z_1 = X$ or $Z_2 = X$. A subset $T\subset X$ of a topological space is called {\it irreducible} if it is an irreducible topological space with the topology induced from $X$. This definition implies $T$ is irreducible if and only if the closure $\bar T$ of $T$ in $X$ is irreducible.


exercises-definition-generic-point

A point $x$ of an irreducible topological space $X$ is called a {\it generic point} of $X$ if $X$ is equal to the closure of the subset $\{x\}$.


exercises-definition-Noetherian-space

A topological space $X$ is called {\it Noetherian} if any decreasing sequence $Z_1\supset Z_2 \supset Z_3\supset \ldots$ of closed subsets of $X$ stabilizes. (It is called {\it Artinian} if any increasing sequence of closed subsets stabilizes.)


exercises-definition-irreducible-component

A maximal irreducible subset $T\subset X$ is called an {\it irreducible component} of the space $X$. Such an irreducible component of $X$ is automatically a closed subset of $X$.


exercises-definition-closed

A point $x\in X$ is called {\it closed} if $\overline{\{x\}} = \{ x\}$. Let $x, y$ be points of $X$. We say that $x$ is a {\it specialization} of $y$, or that $y$ is a {\it generalization} of $x$ if $x\in \overline{\{y\}}$.


exercises-definition-connected-component

A topological space $X$ is called {\it connected} if it is nonempty and not the union of two nonempty disjoint open subsets. A {\it connected component} of $X$ is a maximal connected subset. Any point of $X$ is contained in a connected component of $X$ and any connected component of $X$ is closed in $X$. (But in general a connected component need not be open in $X$.)


exercises-definition-length

Let $A$ be a ring. Let $M$ be an $A$-module. The {\it length} of $M$ as an $R$-module is $$ \text{length}_A(M) = \sup \{ n \mid \exists\ 0 = M_0 \subset M_1 \subset \ldots \subset M_n = M, \text{ }M_i \not = M_{i + 1} \}. $$ In other words, the supremum of the lengths of chains of submodules.


exercises-definition-catenary

A Noetherian ring $A$ is said to be {\it catenary} if for any triple of prime ideals ${\mathfrak p}_1 \subset {\mathfrak p}_2 \subset {\mathfrak p}_3$ we have $$ ht({\mathfrak p}_3 / {\mathfrak p}_1) = ht({\mathfrak p}_3/{\mathfrak p}_2) + ht({\mathfrak p}_2/{\mathfrak p}_1). $$ Here $ht(\mathfrak p/\mathfrak q)$ means the height of $\mathfrak p/\mathfrak q$ in the ring $A/\mathfrak q$. In a formula $$ ht(\mathfrak p/\mathfrak q) = \dim(A_\mathfrak p/\mathfrak qA_\mathfrak p) = \dim((A/\mathfrak q)_\mathfrak p) = \dim((A/\mathfrak q)_{\mathfrak p/\mathfrak q}) $$ A topological space $X$ is {\it catenary}, if given $T \subset T' \subset X$ with $T$ and $T'$ closed and irreducible, then there exists a maximal chain of irreducible closed subsets $$ T = T_0 \subset T_1 \subset \ldots \subset T_n = T' $$ and every such chain has the same (finite) length.


exercises-definition-finite-locally-free

Let $A$ be a ring. Recall that a {\it finite locally free} $A$-module $M$ is a module such that for every ${\mathfrak p} \in \Spec(A)$ there exists an $f\in A$, $f \not \in {\mathfrak p}$ such that $M_f$ is a finite free $A_f$-module. We say $M$ is an {\it invertible module} if $M$ is finite locally free of rank $1$, i.e., for every ${\mathfrak p} \in \Spec(A)$ there exists an $f\in A$, $f \not \in \mathfrak p$ such that $M_f \cong A_f$ as an $A_f$-module.


exercises-definition-class-group

Let $A$ be a ring. The {\it class group of $A$}, sometimes called the {\it Picard group of $A$} is the set $\Pic(A)$ of isomorphism classes of invertible $A$-modules endowed with a group operation defined by tensor product (see Exercise \ref{exercise-tensor-finite-locally-free}).


exercises-definition-GU-GD

Let $\phi : A \to B$ be a homomorphism of rings. We say that the {\it going-up theorem} holds for $\phi$ if the following condition is satisfied: \begin{itemize} \item[(GU)] for any ${\mathfrak p}, {\mathfrak p}' \in \Spec(A)$ such that ${\mathfrak p} \subset {\mathfrak p}'$, and for any $P \in \Spec(B)$ lying over ${\mathfrak p}$, there exists $P'\in \Spec(B)$ lying over ${\mathfrak p}'$ such that $P \subset P'$. \end{itemize} Similarly, we say that the {\it going-down theorem} holds for $\phi$ if the following condition is satisfied: \begin{itemize} \item[(GD)] for any ${\mathfrak p}, {\mathfrak p}' \in \Spec(A)$ such that ${\mathfrak p} \subset {\mathfrak p}'$, and for any $P' \in \Spec(B)$ lying over ${\mathfrak p}'$, there exists $P\in \Spec(B)$ lying over ${\mathfrak p}$ such that $P \subset P'$. \end{itemize}


exercises-definition-numerical-polynomial

A {\it numerical polynomial} is a polynomial $f(x) \in {\mathbf Q}[x]$ such that $f(n) \in {\mathbf Z}$ for every integer $n$.


exercises-definition-graded-module

A {\it graded module} $M$ over a ring $A$ is an $A$-module $M$ endowed with a direct sum decomposition $ \bigoplus\nolimits_{n \in {\mathbf Z}} M_n $ into $A$-submodules. We will say that $M$ is {\it locally finite} if all of the $M_n$ are finite $A$-modules. Suppose that $A$ is a Noetherian ring and that $\varphi$ is a {\it Euler-Poincar\'e function} on finite $A$-modules. This means that for every finitely generated $A$-module $M$ we are given an integer $\varphi(M) \in {\mathbf Z}$ and for every short exact sequence $$ 0 \longrightarrow M' \longrightarrow M \longrightarrow M'' \longrightarrow 0 $$ we have $\varphi(M) = \varphi(M') + \varphi(M')$. The {\it Hilbert function} of a locally finite graded module $M$ (with respect to $\varphi$) is the function $\chi_\varphi(M, n) = \varphi(M_n)$. We say that $M$ has a {\it Hilbert polynomial} if there is some numerical polynomial $P_\varphi$ such that $\chi_\varphi(M, n) = P_\varphi(n)$ for all sufficiently large integers $n$.