categories-definition-functor-symmetric-monoidal-categories

Let $\mathcal{C}$ and $\mathcal{C}'$ be symmetric monoidal categories. A {\it functor of symmetric monoidal categories} $F : \mathcal{C} \to \mathcal{C}'$ is given by a functor $F$ as indicated and a natural transformation $$ F(X) \otimes F(Y) \to F(X \otimes Y) $$ such that $F$ is a functor of monoidal categories and such that for all objects $X, Y$ the diagram $$ \xymatrix{ F(X) \otimes F(Y) \ar[r] \ar[d] & F(X \otimes Y) \ar[d] \\ F(Y) \otimes F(X) \ar[r] & F(Y \otimes X) } $$ commutes.


topologies-definition-zariski-covering

Let $T$ be a scheme. A {\it Zariski covering of $T$} is a family of morphisms $\{f_i : T_i \to T\}_{i \in I}$ of schemes such that each $f_i$ is an open immersion and such that $T = \bigcup f_i(T_i)$.


topologies-definition-standard-Zariski

Compare Schemes, Definition \ref{schemes-definition-standard-covering}. Let $T$ be an affine scheme. A {\it standard Zariski covering} of $T$ is a Zariski covering $\{U_j \to T\}_{j = 1, \ldots, m}$ with each $U_j \to T$ inducing an isomorphism with a standard affine open of $T$.


topologies-definition-big-zariski-site

A {\it big Zariski site} is any site $\Sch_{Zar}$ as in Sites, Definition \ref{sites-definition-site} constructed as follows: \begin{enumerate} \item Choose any set of schemes $S_0$, and any set of Zariski coverings $\text{Cov}_0$ among these schemes. \item As underlying category of $\Sch_{Zar}$ take any category $\Sch_\alpha$ constructed as in Sets, Lemma \ref{sets-lemma-construct-category} starting with the set $S_0$. \item As coverings of $\Sch_{Zar}$ choose any set of coverings as in Sets, Lemma \ref{sets-lemma-coverings-site} starting with the category $\Sch_\alpha$ and the class of Zariski coverings, and the set $\text{Cov}_0$ chosen above. \end{enumerate}


topologies-definition-big-small-Zariski

Let $S$ be a scheme. Let $\Sch_{Zar}$ be a big Zariski site containing $S$. \begin{enumerate} \item The {\it big Zariski site of $S$}, denoted $(\Sch/S)_{Zar}$, is the site $\Sch_{Zar}/S$ introduced in Sites, Section \ref{sites-section-localize}. \item The {\it small Zariski site of $S$}, which we denote $S_{Zar}$, is the full subcategory of $(\Sch/S)_{Zar}$ whose objects are those $U/S$ such that $U \to S$ is an open immersion. A covering of $S_{Zar}$ is any covering $\{U_i \to U\}$ of $(\Sch/S)_{Zar}$ with $U \in \Ob(S_{Zar})$. \item The {\it big affine Zariski site of $S$}, denoted $(\textit{Aff}/S)_{Zar}$, is the full subcategory of $(\Sch/S)_{Zar}$ whose objects are affine $U/S$. A covering of $(\textit{Aff}/S)_{Zar}$ is any covering $\{U_i \to U\}$ of $(\Sch/S)_{Zar}$ which is a standard Zariski covering. \end{enumerate}


topologies-definition-restriction-small-zariski

In the situation of Lemma \ref{lemma-at-the-bottom} the functor $i_S^{-1} = \pi_{S, *}$ is often called the {\it restriction to the small Zariski site}, and for a sheaf $\mathcal{F}$ on the big Zariski site we denote $\mathcal{F}|_{S_{Zar}}$ this restriction.


topologies-definition-etale-covering

Let $T$ be a scheme. An {\it \'etale covering of $T$} is a family of morphisms $\{f_i : T_i \to T\}_{i \in I}$ of schemes such that each $f_i$ is \'etale and such that $T = \bigcup f_i(T_i)$.


topologies-definition-standard-etale

Let $T$ be an affine scheme. A {\it standard \'etale covering} of $T$ is a family $\{f_j : U_j \to T\}_{j = 1, \ldots, m}$ with each $U_j$ is affine and \'etale over $T$ and $T = \bigcup f_j(U_j)$.


topologies-definition-big-etale-site

A {\it big \'etale site} is any site $\Sch_\etale$ as in Sites, Definition \ref{sites-definition-site} constructed as follows: \begin{enumerate} \item Choose any set of schemes $S_0$, and any set of \'etale coverings $\text{Cov}_0$ among these schemes. \item As underlying category take any category $\Sch_\alpha$ constructed as in Sets, Lemma \ref{sets-lemma-construct-category} starting with the set $S_0$. \item Choose any set of coverings as in Sets, Lemma \ref{sets-lemma-coverings-site} starting with the category $\Sch_\alpha$ and the class of \'etale coverings, and the set $\text{Cov}_0$ chosen above. \end{enumerate}


topologies-definition-big-small-etale

Let $S$ be a scheme. Let $\Sch_\etale$ be a big \'etale site containing $S$. \begin{enumerate} \item The {\it big \'etale site of $S$}, denoted $(\Sch/S)_\etale$, is the site $\Sch_\etale/S$ introduced in Sites, Section \ref{sites-section-localize}. \item The {\it small \'etale site of $S$}, which we denote $S_\etale$, is the full subcategory of $(\Sch/S)_\etale$ whose objects are those $U/S$ such that $U \to S$ is \'etale. A covering of $S_\etale$ is any covering $\{U_i \to U\}$ of $(\Sch/S)_\etale$ with $U \in \Ob(S_\etale)$. \item The {\it big affine \'etale site of $S$}, denoted $(\textit{Aff}/S)_\etale$, is the full subcategory of $(\Sch/S)_\etale$ whose objects are affine $U/S$. A covering of $(\textit{Aff}/S)_\etale$ is any covering $\{U_i \to U\}$ of $(\Sch/S)_\etale$ which is a standard \'etale covering. \end{enumerate}


topologies-definition-restriction-small-etale

In the situation of Lemma \ref{lemma-at-the-bottom-etale} the functor $i_S^{-1} = \pi_{S, *}$ is often called the {\it restriction to the small \'etale site}, and for a sheaf $\mathcal{F}$ on the big \'etale site we denote $\mathcal{F}|_{S_\etale}$ this restriction.


topologies-definition-smooth-covering

Let $T$ be a scheme. A {\it smooth covering of $T$} is a family of morphisms $\{f_i : T_i \to T\}_{i \in I}$ of schemes such that each $f_i$ is smooth and such that $T = \bigcup f_i(T_i)$.


topologies-definition-standard-smooth

Let $T$ be an affine scheme. A {\it standard smooth covering} of $T$ is a family $\{f_j : U_j \to T\}_{j = 1, \ldots, m}$ with each $U_j$ is affine, $U_j \to T$ standard smooth and $T = \bigcup f_j(U_j)$.


topologies-definition-big-smooth-site

A {\it big smooth site} is any site $\Sch_{smooth}$ as in Sites, Definition \ref{sites-definition-site} constructed as follows: \begin{enumerate} \item Choose any set of schemes $S_0$, and any set of smooth coverings $\text{Cov}_0$ among these schemes. \item As underlying category take any category $\Sch_\alpha$ constructed as in Sets, Lemma \ref{sets-lemma-construct-category} starting with the set $S_0$. \item Choose any set of coverings as in Sets, Lemma \ref{sets-lemma-coverings-site} starting with the category $\Sch_\alpha$ and the class of smooth coverings, and the set $\text{Cov}_0$ chosen above. \end{enumerate}


topologies-definition-big-small-smooth

Let $S$ be a scheme. Let $\Sch_{smooth}$ be a big smooth site containing $S$. \begin{enumerate} \item The {\it big smooth site of $S$}, denoted $(\Sch/S)_{smooth}$, is the site $\Sch_{smooth}/S$ introduced in Sites, Section \ref{sites-section-localize}. \item The {\it big affine smooth site of $S$}, denoted $(\textit{Aff}/S)_{smooth}$, is the full subcategory of $(\Sch/S)_{smooth}$ whose objects are affine $U/S$. A covering of $(\textit{Aff}/S)_{smooth}$ is any covering $\{U_i \to U\}$ of $(\Sch/S)_{smooth}$ which is a standard smooth covering. \end{enumerate}


topologies-definition-syntomic-covering

Let $T$ be a scheme. An {\it syntomic covering of $T$} is a family of morphisms $\{f_i : T_i \to T\}_{i \in I}$ of schemes such that each $f_i$ is syntomic and such that $T = \bigcup f_i(T_i)$.


topologies-definition-standard-syntomic

Let $T$ be an affine scheme. A {\it standard syntomic covering} of $T$ is a family $\{f_j : U_j \to T\}_{j = 1, \ldots, m}$ with each $U_j$ is affine, $U_j \to T$ standard syntomic and $T = \bigcup f_j(U_j)$.


topologies-definition-big-syntomic-site

A {\it big syntomic site} is any site $\Sch_{syntomic}$ as in Sites, Definition \ref{sites-definition-site} constructed as follows: \begin{enumerate} \item Choose any set of schemes $S_0$, and any set of syntomic coverings $\text{Cov}_0$ among these schemes. \item As underlying category take any category $\Sch_\alpha$ constructed as in Sets, Lemma \ref{sets-lemma-construct-category} starting with the set $S_0$. \item Choose any set of coverings as in Sets, Lemma \ref{sets-lemma-coverings-site} starting with the category $\Sch_\alpha$ and the class of syntomic coverings, and the set $\text{Cov}_0$ chosen above. \end{enumerate}


topologies-definition-big-small-syntomic

Let $S$ be a scheme. Let $\Sch_{syntomic}$ be a big syntomic site containing $S$. \begin{enumerate} \item The {\it big syntomic site of $S$}, denoted $(\Sch/S)_{syntomic}$, is the site $\Sch_{syntomic}/S$ introduced in Sites, Section \ref{sites-section-localize}. \item The {\it big affine syntomic site of $S$}, denoted $(\textit{Aff}/S)_{syntomic}$, is the full subcategory of $(\Sch/S)_{syntomic}$ whose objects are affine $U/S$. A covering of $(\textit{Aff}/S)_{syntomic}$ is any covering $\{U_i \to U\}$ of $(\Sch/S)_{syntomic}$ which is a standard syntomic covering. \end{enumerate}


topologies-definition-fppf-covering

Let $T$ be a scheme. An {\it fppf covering of $T$} is a family of morphisms $\{f_i : T_i \to T\}_{i \in I}$ of schemes such that each $f_i$ is flat, locally of finite presentation and such that $T = \bigcup f_i(T_i)$.


topologies-definition-standard-fppf

Let $T$ be an affine scheme. A {\it standard fppf covering} of $T$ is a family $\{f_j : U_j \to T\}_{j = 1, \ldots, m}$ with each $U_j$ is affine, flat and of finite presentation over $T$ and $T = \bigcup f_j(U_j)$.


topologies-definition-big-fppf-site

A {\it big fppf site} is any site $\Sch_{fppf}$ as in Sites, Definition \ref{sites-definition-site} constructed as follows: \begin{enumerate} \item Choose any set of schemes $S_0$, and any set of fppf coverings $\text{Cov}_0$ among these schemes. \item As underlying category take any category $\Sch_\alpha$ constructed as in Sets, Lemma \ref{sets-lemma-construct-category} starting with the set $S_0$. \item Choose any set of coverings as in Sets, Lemma \ref{sets-lemma-coverings-site} starting with the category $\Sch_\alpha$ and the class of fppf coverings, and the set $\text{Cov}_0$ chosen above. \end{enumerate}


topologies-definition-big-small-fppf

Let $S$ be a scheme. Let $\Sch_{fppf}$ be a big fppf site containing $S$. \begin{enumerate} \item The {\it big fppf site of $S$}, denoted $(\Sch/S)_{fppf}$, is the site $\Sch_{fppf}/S$ introduced in Sites, Section \ref{sites-section-localize}. \item The {\it big affine fppf site of $S$}, denoted $(\textit{Aff}/S)_{fppf}$, is the full subcategory of $(\Sch/S)_{fppf}$ whose objects are affine $U/S$. A covering of $(\textit{Aff}/S)_{fppf}$ is any covering $\{U_i \to U\}$ of $(\Sch/S)_{fppf}$ which is a standard fppf covering. \end{enumerate}


topologies-definition-standard-ph-covering

Let $T$ be an affine scheme. A {\it standard ph covering} is a family $\{f_j : U_j \to T\}_{j = 1, \ldots, m}$ constructed from a proper surjective morphism $f : U \to T$ and an affine open covering $U = \bigcup_{j = 1, \ldots, m} U_j$ by setting $f_j = f|_{U_j}$.


topologies-definition-ph-covering

Let $T$ be a scheme. A {\it ph covering of $T$} is a family of morphisms $\{f_i : T_i \to T\}_{i \in I}$ of schemes such that $f_i$ is locally of finite type and such that for every affine open $U \subset T$ there exists a standard ph covering $\{U_j \to U\}_{j = 1, \ldots, m}$ refining the family $\{T_i \times_T U \to U\}_{i \in I}$.


topologies-definition-big-ph-site

A {\it big ph site} is any site $\Sch_{ph}$ as in Sites, Definition \ref{sites-definition-site} constructed as follows: \begin{enumerate} \item Choose any set of schemes $S_0$, and any set of ph coverings $\text{Cov}_0$ among these schemes. \item As underlying category take any category $\Sch_\alpha$ constructed as in Sets, Lemma \ref{sets-lemma-construct-category} starting with the set $S_0$. \item Choose any set of coverings as in Sets, Lemma \ref{sets-lemma-coverings-site} starting with the category $\Sch_\alpha$ and the class of ph coverings, and the set $\text{Cov}_0$ chosen above. \end{enumerate}


topologies-definition-big-small-ph

Let $S$ be a scheme. Let $\Sch_{ph}$ be a big ph site containing $S$. \begin{enumerate} \item The {\it big ph site of $S$}, denoted $(\Sch/S)_{ph}$, is the site $\Sch_{ph}/S$ introduced in Sites, Section \ref{sites-section-localize}. \item The {\it big affine ph site of $S$}, denoted $(\textit{Aff}/S)_{ph}$, is the full subcategory of $(\Sch/S)_{ph}$ whose objects are affine $U/S$. A covering of $(\textit{Aff}/S)_{ph}$ is any finite covering $\{U_i \to U\}$ of $(\Sch/S)_{ph}$ with $U_i$ and $U$ affine. \end{enumerate}


topologies-definition-fpqc-covering

Let $T$ be a scheme. An {\it fpqc covering of $T$} is a family of morphisms $\{f_i : T_i \to T\}_{i \in I}$ of schemes such that each $f_i$ is flat and such that for every affine open $U \subset T$ there exists $n \geq 0$, a map $a : \{1, \ldots, n\} \to I$ and affine opens $V_j \subset T_{a(j)}$, $j = 1, \ldots, n$ with $\bigcup_{j = 1}^n f_{a(j)}(V_j) = U$.


topologies-definition-standard-fpqc

Let $T$ be an affine scheme. A {\it standard fpqc covering} of $T$ is a family $\{f_j : U_j \to T\}_{j = 1, \ldots, n}$ with each $U_j$ is affine, flat over $T$ and $T = \bigcup f_j(U_j)$.


topologies-definition-sheaf-property-fpqc

Let $F$ be a contravariant functor on the category of schemes with values in sets. \begin{enumerate} \item Let $\{U_i \to T\}_{i \in I}$ be a family of morphisms of schemes with fixed target. We say that $F$ {\it satisfies the sheaf property for the given family} if for any collection of elements $\xi_i \in F(U_i)$ such that $\xi_i|_{U_i \times_T U_j} = \xi_j|_{U_i \times_T U_j}$ there exists a unique element $\xi \in F(T)$ such that $\xi_i = \xi|_{U_i}$ in $F(U_i)$. \item We say that $F$ {\it satisfies the sheaf property for the fpqc topology} if it satisfies the sheaf property for any fpqc covering. \end{enumerate}


topologies-definition-standard-V-covering

Let $T$ be an affine scheme. A {\it standard V covering} is a finite family $\{T_j \to T\}_{j = 1, \ldots, m}$ with $T_j$ affine such that for every morphism $g : \Spec(V) \to T$ where $V$ is a valuation ring, there is an extension $V \subset W$ of valuation rings (More on Algebra, Definition \ref{more-algebra-definition-extension-valuation-rings}), an index $1 \leq j \leq m$, and a commutative diagram $$ \xymatrix{ \Spec(W) \ar[r] \ar[d] & T_j \ar[d] \\ \Spec(V) \ar[r]^g & T } $$


topologies-definition-V-covering

Let $T$ be a scheme. A {\it V covering of $T$} is a family of morphisms $\{T_i \to T\}_{i \in I}$ of schemes such that for every affine open $U \subset T$ there exists a standard V covering $\{U_j \to U\}_{j = 1, \ldots, m}$ refining the family $\{T_i \times_T U \to U\}_{i \in I}$.


topologies-definition-sheaf-property-V

Let $F$ be a contravariant functor on the category of schemes with values in sets. We say that $F$ {\it satisfies the sheaf property for the V topology} if it satisfies the sheaf property for any V covering (see Definition \ref{definition-sheaf-property-fpqc}).


pic-definition-picard-functor

Let $\Sch_{fppf}$ be a big site as in Topologies, Definition \ref{topologies-definition-big-small-fppf}. Let $f : X \to S$ be a morphism of this site. The {\it Picard functor} $\Picardfunctor_{X/S}$ is the fppf sheafification of the functor $$ (\Sch/S)_{fppf} \longrightarrow \textit{Sets},\quad T \longmapsto \Pic(X_T) $$ If this functor is representable, then we denote $\underline{\Picardfunctor}_{X/S}$ a scheme representing it.


pic-definition-genus

Let $k$ be a field. Let $X$ be a smooth projective geometrically irreducible curve over $k$. The {\it genus} of $X$ is $g = \dim_k H^1(X, \mathcal{O}_X)$.


constructions-definition-relative-spec

Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_S$-algebras. The {\it relative spectrum of $\mathcal{A}$ over $S$}, or simply the {\it spectrum of $\mathcal{A}$ over $S$} is the scheme constructed in Lemma \ref{lemma-glue-relative-spec} which represents the functor $F$ (\ref{equation-spec}), see Lemma \ref{lemma-glueing-gives-functor-spec}. We denote it $\pi : \underline{\Spec}_S(\mathcal{A}) \to S$. The ``universal family'' is a morphism of $\mathcal{O}_S$-algebras $$ \mathcal{A} \longrightarrow \pi_*\mathcal{O}_{\underline{\Spec}_S(\mathcal{A})} $$


constructions-definition-affine-n-space

Let $S$ be a scheme and $n \geq 0$. The scheme $$ \mathbf{A}^n_S = \underline{\Spec}_S(\mathcal{O}_S[T_1, \ldots, T_n]) $$ over $S$ is called {\it affine $n$-space over $S$}. If $S = \Spec(R)$ is affine then we also call this {\it affine $n$-space over $R$} and we denote it $\mathbf{A}^n_R$.


constructions-definition-vector-bundle

Let $S$ be a scheme. Let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_S$-module\footnote{The reader may expect here the condition that $\mathcal{E}$ is finite locally free. We do not do so in order to be consistent with \cite[II, Definition 1.7.8]{EGA}.}. The {\it vector bundle associated to $\mathcal{E}$} is $$ \mathbf{V}(\mathcal{E}) = \underline{\Spec}_S(\text{Sym}(\mathcal{E})). $$


constructions-definition-abstract-vector-bundle

Let $S$ be a scheme. A {\it vector bundle $\pi : V \to S$ over $S$} is an affine morphism of schemes such that $\pi_*\mathcal{O}_V$ is endowed with the structure of a graded $\mathcal{O}_S$-algebra $\pi_*\mathcal{O}_V = \bigoplus\nolimits_{n \geq 0} \mathcal{E}_n$ such that $\mathcal{E}_0 = \mathcal{O}_S$ and such that the maps $$ \text{Sym}^n(\mathcal{E}_1) \longrightarrow \mathcal{E}_n $$ are isomorphisms for all $n \geq 0$. A {\it morphism of vector bundles over $S$} is a morphism $f : V \to V'$ such that the induced map $$ f^* : \pi'_*\mathcal{O}_{V'} \longrightarrow \pi_*\mathcal{O}_V $$ is compatible with the given gradings.


constructions-definition-cone

Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_S$-algebra. Assume that $\mathcal{O}_S \to \mathcal{A}_0$ is an isomorphism\footnote{Often one imposes the assumption that $\mathcal{A}$ is generated by $\mathcal{A}_1$ over $\mathcal{O}_S$. We do not assume this in order to be consistent with \cite[II, (8.3.1)]{EGA}.}. The {\it cone associated to $\mathcal{A}$} or the {\it affine cone associated to $\mathcal{A}$} is $$ C(\mathcal{A}) = \underline{\Spec}_S(\mathcal{A}). $$


constructions-definition-abstract-cone

Let $S$ be a scheme. A {\it cone $\pi : C \to S$ over $S$} is an affine morphism of schemes such that $\pi_*\mathcal{O}_C$ is endowed with the structure of a graded $\mathcal{O}_S$-algebra $\pi_*\mathcal{O}_C = \bigoplus\nolimits_{n \geq 0} \mathcal{A}_n$ such that $\mathcal{A}_0 = \mathcal{O}_S$. A {\it morphism of cones} from $\pi : C \to S$ to $\pi' : C' \to S$ is a morphism $f : C \to C'$ such that the induced map $$ f^* : \pi'_*\mathcal{O}_{C'} \longrightarrow \pi_*\mathcal{O}_C $$ is compatible with the given gradings.


constructions-definition-standard-covering

Let $S$ be a graded ring. Suppose that $D_{+}(f) \subset \text{Proj}(S)$ is a standard open. A {\it standard open covering} of $D_{+}(f)$ is a covering $D_{+}(f) = \bigcup_{i = 1}^n D_{+}(g_i)$, where $g_1, \ldots, g_n \in S$ are homogeneous of positive degree.


constructions-definition-structure-sheaf

Let $S$ be a graded ring. \begin{enumerate} \item The {\it structure sheaf $\mathcal{O}_{\text{Proj}(S)}$ of the homogeneous spectrum of $S$} is the unique sheaf of rings $\mathcal{O}_{\text{Proj}(S)}$ which agrees with $\widetilde S$ on the basis of standard opens. \item The locally ringed space $(\text{Proj}(S), \mathcal{O}_{\text{Proj}(S)})$ is called the {\it homogeneous spectrum} of $S$ and denoted $\text{Proj}(S)$. \item The sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules extending $\widetilde M$ to all opens of $\text{Proj}(S)$ is called the sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules associated to $M$. This sheaf is denoted $\widetilde M$ as well. \end{enumerate}


constructions-definition-twist

Let $S$ be a graded ring. Let $X = \text{Proj}(S)$. \begin{enumerate} \item We define $\mathcal{O}_X(n) = \widetilde{S(n)}$. This is called the $n$th {\it twist of the structure sheaf of $\text{Proj}(S)$}. \item For any sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ we set $\mathcal{F}(n) = \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{O}_X(n)$. \end{enumerate}


constructions-definition-projective-space

The scheme $\mathbf{P}^n_{\mathbf{Z}} = \text{Proj}(\mathbf{Z}[T_0, \ldots, T_n])$ is called {\it projective $n$-space over $\mathbf{Z}$}. Its base change $\mathbf{P}^n_S$ to a scheme $S$ is called {\it projective $n$-space over $S$}. If $R$ is a ring the base change to $\Spec(R)$ is denoted $\mathbf{P}^n_R$ and called {\it projective $n$-space over $R$}.


constructions-definition-relative-proj

Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent sheaf of graded $\mathcal{O}_S$-algebras. The {\it relative homogeneous spectrum of $\mathcal{A}$ over $S$}, or the {\it homogeneous spectrum of $\mathcal{A}$ over $S$}, or the {\it relative Proj of $\mathcal{A}$ over $S$} is the scheme constructed in Lemma \ref{lemma-glue-relative-proj} which represents the functor $F$ (\ref{equation-proj}), see Lemma \ref{lemma-glueing-gives-functor-proj}. We denote it $\pi : \underline{\text{Proj}}_S(\mathcal{A}) \to S$.


constructions-definition-projective-bundle

Let $S$ be a scheme. Let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_S$-module\footnote{The reader may expect here the condition that $\mathcal{E}$ is finite locally free. We do not do so in order to be consistent with \cite[II, Definition 4.1.1]{EGA}.}. We denote $$ \pi : \mathbf{P}(\mathcal{E}) = \underline{\text{Proj}}_S(\text{Sym}(\mathcal{E})) \longrightarrow S $$ and we call it the {\it projective bundle associated to $\mathcal{E}$}. The symbol $\mathcal{O}_{\mathbf{P}(\mathcal{E})}(n)$ indicates the invertible $\mathcal{O}_{\mathbf{P}(\mathcal{E})}$-module of Lemma \ref{lemma-apply-relative} and is called the $n$th {\it twist of the structure sheaf}.


constructions-definition-grassmannian

Let $0 < k < n$. The scheme $\mathbf{G}(k, n)$ representing the functor $G(k, n)$ is called {\it Grassmannian over $\mathbf{Z}$}. Its base change $\mathbf{G}(k, n)_S$ to a scheme $S$ is called {\it Grassmannian over $S$}. If $R$ is a ring the base change to $\Spec(R)$ is denoted $\mathbf{G}(k, n)_R$ and called {\it Grassmannian over $R$}.


algebraization-definition-derived-complete

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a sheaf of ideals. Let $K \in D(\mathcal{O})$. We say that $K$ is {\it derived complete with respect to $\mathcal{I}$} if for every object $U$ of $\mathcal{C}$ and $f \in \mathcal{I}(U)$ the object $T(K|_U, f)$ of $D(\mathcal{O}_U)$ is zero.


algebraization-definition-algebraizable

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. We say {\it $(\mathcal{F}_n)$ extends to $X$} if there exists an object $(\mathcal{G}_n)$ of $\textit{Coh}(X, I\mathcal{O}_X)$ whose restriction to $U$ is isomorphic to $(\mathcal{F}_n)$.


algebraization-definition-canonically-algebraizable

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. We say {\it $(\mathcal{F}_n)$ canonically extends to $X$} if the the inverse system $$ \{\widetilde{H^0(U, \mathcal{F}_n)}\}_{n \geq 1} $$ in $\QCoh(\mathcal{O}_X)$ is pro-isomorphic to an object $(\mathcal{G}_n)$ of $\textit{Coh}(X, I\mathcal{O}_X)$.


algebraization-definition-s-d-inequalities

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. Let $a, b$ be integers. Let $\delta^Y_Z$ be as in (\ref{equation-delta-Z}). We say {\it $(\mathcal{F}_n)$ satisfies the $(a, b)$-inequalities} if for $y \in U \cap Y$ and a prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge$ with $\mathfrak p \not \in V(I\mathcal{O}_{X, y}^\wedge)$ \begin{enumerate} \item if $V(\mathfrak p) \cap V(I\mathcal{O}_{X, y}^\wedge) \not = \{\mathfrak m_y^\wedge\}$, then $$ \text{depth}((\mathcal{F}^\wedge_y)_\mathfrak p) + \delta^Y_Z(y) \geq a \quad\text{or}\quad \text{depth}((\mathcal{F}^\wedge_y)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) + \delta^Y_Z(y) > b $$ \item if $V(\mathfrak p) \cap V(I\mathcal{O}_{X, y}^\wedge) = \{\mathfrak m_y^\wedge\}$, then $$ \text{depth}((\mathcal{F}^\wedge_y)_\mathfrak p) + \delta^Y_Z(y) > a $$ \end{enumerate} We say {\it $(\mathcal{F}_n)$ satisfies the strict $(a, b)$-inequalities} if for $y \in U \cap Y$ and a prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge$ with $\mathfrak p \not \in V(I\mathcal{O}_{X, y}^\wedge)$ we have $$ \text{depth}((\mathcal{F}^\wedge_y)_\mathfrak p) + \delta^Y_Z(y) > a \quad\text{or}\quad \text{depth}((\mathcal{F}^\wedge_y)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) + \delta^Y_Z(y) > b $$


spaces-over-fields-definition-integral-algebraic-space

Let $S$ be a scheme. We say an algebraic space $X$ over $S$ is {\it integral} if it is reduced, decent, and $|X|$ is irreducible.


spaces-over-fields-definition-function-field

Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$. The {\it function field}, or the {\it field of rational functions} of $X$ is the field $R(X)$ of Lemma \ref{lemma-integral-algebraic-space-rational-functions}.


spaces-over-fields-definition-degree

Let $S$ be a scheme. Let $X$ and $Y$ be integral algebraic spaces over $S$. Let $f : X \to Y$ be locally of finite type and dominant. Assume any of the equivalent conditions (1) -- (5) of Lemma \ref{lemma-finite-degree}. Let $x \in |X|$ and $y \in |Y|$ be the generic points. Then the positive integer $$ \text{deg}(X/Y) = [\kappa(x) : \kappa(y)] $$ is called the {\it degree of $X$ over $Y$}.


spaces-over-fields-definition-Weil-divisor

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


spaces-over-fields-definition-order-vanishing

Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. Let $f \in R(X)^*$. For every prime divisor $Z \subset X$ we define the {\it order of vanishing of $f$ along $Z$} as the integer $$ \text{ord}_Z(f) = \text{length}_{\mathcal{O}_{X, \xi}^h} (\mathcal{O}_{X, \xi}^h/a \mathcal{O}_{X, \xi}^h) - \text{length}_{\mathcal{O}_{X, \xi}^h} (\mathcal{O}_{X, \xi}^h/b \mathcal{O}_{X, \xi}^h) $$ where $a, b \in \mathcal{O}_{X, \xi}^h$ are nonzerodivisors such that the image of $f$ in $Q(\mathcal{O}_{X, \xi}^h)$ (Lemma \ref{lemma-order-vanishing}) is equal to $a/b$. This is well defined by Algebra, Lemma \ref{algebra-lemma-ord-additive}.


spaces-over-fields-definition-principal-divisor

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


spaces-over-fields-definition-class-group

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


spaces-over-fields-definition-order-vanishing-meromorphic

Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic algebraic space over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{K}_X(\mathcal{L}))$ be a regular meromorphic section of $\mathcal{L}$. For every prime divisor $Z \subset X$ with generic point $\xi \in |Z|$ we define the {\it order of vanishing of $s$ along $Z$} as the integer $$ \text{ord}_{Z, \mathcal{L}}(s) = \text{length}_{\mathcal{O}_{X, \xi}^h} (\mathcal{O}_{X, \xi}^h/a \mathcal{O}_{X, \xi}^h) - \text{length}_{\mathcal{O}_{X, \xi}^h} (\mathcal{O}_{X, \xi}^h/b \mathcal{O}_{X, \xi}^h) $$ where $a, b \in \mathcal{O}_{X, \xi}^h$ are nonzerodivisors such that the element $s/s_\xi$ of $Q(\mathcal{O}_{X, \xi}^h)$ constructed above is equal to $a/b$. This is well defined by the above and Algebra, Lemma \ref{algebra-lemma-ord-additive}.


spaces-over-fields-definition-divisor-invertible-sheaf

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


spaces-over-fields-definition-modification

Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$. A {\it modification of $X$} is a birational proper morphism $f : X' \to X$ of algebraic spaces over $S$ with $X'$ integral.


spaces-over-fields-definition-alteration

Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$. An {\it alteration of $X$} is a proper dominant morphism $f : Y \to X$ of algebraic spaces over $S$ with $Y$ integral such that $f^{-1}(U) \to U$ is finite for some nonempty open $U \subset X$.


spaces-over-fields-definition-geometrically-reduced

Let $k$ be a field. Let $X$ be an algebraic space over $k$. \begin{enumerate} \item Let $x \in |X|$ be a point. We say $X$ is {\it geometrically reduced at $x$} if $\mathcal{O}_{X, \overline{x}}$ is geometrically reduced over $k$. \item We say $X$ is {\it geometrically reduced} over $k$ if $X$ is geometrically reduced at every point of $X$. \end{enumerate}


spaces-over-fields-definition-geometrically-connected

Let $X$ be an algebraic space over the field $k$. We say $X$ is {\it geometrically connected} over $k$ if the base change $X_{k'}$ is connected for every field extension $k'$ of $k$.


spaces-over-fields-definition-geometrically-irreducible

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


spaces-over-fields-definition-geometrically-integral

Let $X$ be an algebraic space over the field $k$. We say $X$ is {\it geometrically integral} over $k$ if the algebraic space $X_{k'}$ is integral (Definition \ref{definition-integral-algebraic-space}) for every field extension $k'$ of $k$.


spaces-over-fields-definition-euler-characteristic

Let $k$ be a field. Let $X$ be a proper algebraic 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.


spaces-over-fields-definition-intersection-number

Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. Let $i : Z \to X$ be a closed subspace 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 $(\mathcal{L}^d \cdot Z)$.


spaces-divisors-definition-weakly-associated

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \in |X|$. \begin{enumerate} \item We say $x$ is {\it weakly associated} to $\mathcal{F}$ if the equivalent conditions (1), (2), and (3) of Lemma \ref{lemma-associated} are satisfied. \item We denote $\text{WeakAss}(\mathcal{F})$ the set of weakly associated points of $\mathcal{F}$. \item The {\it weakly associated points of $X$} are the weakly associated points of $\mathcal{O}_X$. \end{enumerate} If $X$ is locally Noetherian we will say {\it $x$ is associated to $\mathcal{F}$} if and only if $x$ is weakly associated to $\mathcal{F}$ and we set $\text{Ass}(\mathcal{F}) = \text{WeakAss}(\mathcal{F})$. Finally (still assuming $X$ is locally Noetherian), we will say {\it $x$ is an associated point of $X$} if and only if $x$ is a weakly associated point of $X$.


spaces-divisors-definition-locally-Noetherian-fibre

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $y \in |Y|$. We say {\it the fibre of $f$ over $y$ is locally Noetherian} if the equivalent conditions (1), (2), and (3) of Lemma \ref{lemma-locally-noetherian-fibre} are satisfied. We say {\it the fibres of $f$ are locally Noetherian} if this holds for every $y \in |Y|$.


spaces-divisors-definition-relative-weak-assassin

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. The {\it relative weak assassin of $\mathcal{F}$ in $X$ over $Y$} is the set $\text{WeakAss}_{X/Y}(\mathcal{F}) \subset |X|$ consisting of those $x \in |X|$ such that the equivalent conditions of Lemma \ref{lemma-relative-assassin} are satisfied. If the fibres of $f$ are locally Noetherian (Definition \ref{definition-locally-Noetherian-fibre}) then we use the notation $\text{Ass}_{X/Y}(\mathcal{F})$.


spaces-divisors-definition-effective-Cartier-divisor

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item A {\it locally principal closed subspace} of $X$ is a closed subspace whose sheaf of ideals is locally generated by $1$ element. \item An {\it effective Cartier divisor} on $X$ is a closed subspace $D \subset X$ such that the ideal sheaf $\mathcal{I}_D \subset \mathcal{O}_X$ is an invertible $\mathcal{O}_X$-module. \end{enumerate}


spaces-divisors-definition-sum-effective-Cartier-divisors

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Given effective Cartier divisors $D_1$, $D_2$ on $X$ we set $D = D_1 + D_2$ equal to the closed subspace of $X$ corresponding to the quasi-coherent sheaf of ideals $\mathcal{I}_{D_1}\mathcal{I}_{D_2} \subset \mathcal{O}_S$. We call this the {\it sum of the effective Cartier divisors $D_1$ and $D_2$}.


spaces-divisors-definition-pullback-effective-Cartier-divisor

Let $S$ be a scheme. Let $f : X' \to X$ be a morphism of algebraic spaces over $S$. Let $D \subset X$ be an effective Cartier divisor. We say the {\it pullback of $D$ by $f$ is defined} if the closed subspace $f^{-1}(D) \subset X'$ is an effective Cartier divisor. In this case we denote it either $f^*D$ or $f^{-1}(D)$ and we call it the {\it pullback of the effective Cartier divisor}.


spaces-divisors-definition-invertible-sheaf-effective-Cartier-divisor

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ and let $D \subset X$ be an effective Cartier divisor with ideal sheaf $\mathcal{I}_D$. \begin{enumerate} \item The {\it invertible sheaf $\mathcal{O}_X(D)$ associated to $D$} is defined by $$ \mathcal{O}_X(D) = \SheafHom_{\mathcal{O}_X}(\mathcal{I}_D, \mathcal{O}_X) = \mathcal{I}_D^{\otimes -1}. $$ \item The canonical section, usually denoted $1$ or $1_D$, is the global section of $\mathcal{O}_X(D)$ corresponding to the inclusion mapping $\mathcal{I}_D \to \mathcal{O}_X$. \item We write $\mathcal{O}_X(-D) = \mathcal{O}_X(D)^{\otimes -1} = \mathcal{I}_D$. \item Given a second effective Cartier divisor $D' \subset X$ we define $\mathcal{O}_X(D - D') = \mathcal{O}_X(D) \otimes_{\mathcal{O}_X} \mathcal{O}_X(-D')$. \end{enumerate}


spaces-divisors-definition-regular-section

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{L}$ be an invertible sheaf on $X$. A global section $s \in \Gamma(X, \mathcal{L})$ is called a {\it regular section} if the map $\mathcal{O}_X \to \mathcal{L}$, $f \mapsto fs$ is injective.


spaces-divisors-definition-zero-scheme-s

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{L}$ be an invertible sheaf. Let $s \in \Gamma(X, \mathcal{L})$. The {\it zero scheme} of $s$ is the closed subspace $Z(s) \subset X$ defined by the quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_X$ which is the image of the map $s : \mathcal{L}^{\otimes -1} \to \mathcal{O}_X$.


spaces-divisors-definition-relative-effective-Cartier-divisor

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. A {\it relative effective Cartier divisor} on $X/Y$ is an effective Cartier divisor $D \subset X$ such that $D \to Y$ is a flat morphism of algebraic spaces.


spaces-divisors-definition-sheaf-meromorphic-functions

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The {\it sheaf of meromorphic functions on $X$} is the sheaf {\it $\mathcal{K}_X$} on $X_\etale$ associated to the presheaf displayed above. A {\it meromorphic function} on $X$ is a global section of $\mathcal{K}_X$.


spaces-divisors-definition-meromorphic-section

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


spaces-divisors-definition-pullback-meromorphic-sections

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. We say that {\it pullbacks of meromorphic functions are defined for $f$} if for every commutative diagram $$ \xymatrix{ U \ar[r] \ar[d] & X \ar[d] \\ V \ar[r] & Y } $$ with $U \in X_\etale$ and $V \in Y_\etale$ and any section $s \in \mathcal{S}_Y(V)$ the pullback $f^\sharp(s) \in \mathcal{O}_X(U)$ is an element of $\mathcal{S}_X(U)$.


spaces-divisors-definition-regular-meromorphic-section

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. A meromorphic section $s$ of $\mathcal{L}$ is said to be {\it regular} if the induced map $\mathcal{K}_X \to \mathcal{K}_X(\mathcal{L})$ is injective.


spaces-divisors-definition-relative-proj

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent sheaf of graded $\mathcal{O}_X$-algebras. The {\it relative homogeneous spectrum of $\mathcal{A}$ over $X$}, or the {\it homogeneous spectrum of $\mathcal{A}$ over $X$}, or the {\it relative Proj of $\mathcal{A}$ over $X$} is the algebraic space $F$ over $X$ of Lemma \ref{lemma-relative-proj}. We denote it $\pi : \underline{\text{Proj}}_X(\mathcal{A}) \to X$.


spaces-divisors-definition-relatively-ample

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. We say $\mathcal{L}$ is {\it relatively ample}, or {\it $f$-relatively ample}, or {\it ample on $X/Y$}, or {\it $f$-ample} if $f : X \to Y$ is representable and for every morphism $Z \to Y$ where $Z$ is a scheme, the pullback $\mathcal{L}_Z$ of $\mathcal{L}$ to $X_Z = Z \times_Y X$ is ample on $X_Z/Z$ as in Morphisms, Definition \ref{morphisms-definition-relatively-ample}.


spaces-divisors-definition-blow-up

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals, and let $Z \subset X$ be the closed subspace corresponding to $\mathcal{I}$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-ideals}). The {\it blowing up of $X$ along $Z$}, or the {\it blowing up of $X$ in the ideal sheaf $\mathcal{I}$} is the morphism $$ b : \underline{\text{Proj}}_X \left(\bigoplus\nolimits_{n \geq 0} \mathcal{I}^n\right) \longrightarrow X $$ The {\it exceptional divisor} of the blowup is the inverse image $b^{-1}(Z)$. Sometimes $Z$ is called the {\it center} of the blowup.


spaces-divisors-definition-strict-transform

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


spaces-divisors-definition-admissible-blowup

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


dga-definition-dga

Let $R$ be a commutative ring. A {\it differential graded algebra over $R$} is either \begin{enumerate} \item a chain complex $A_\bullet$ of $R$-modules endowed with $R$-bilinear maps $A_n \times A_m \to A_{n + m}$, $(a, b) \mapsto ab$ such that $$ \text{d}_{n + m}(ab) = \text{d}_n(a)b + (-1)^n a\text{d}_m(b) $$ and such that $\bigoplus A_n$ becomes an associative and unital $R$-algebra, or \item a cochain complex $A^\bullet$ of $R$-modules endowed with $R$-bilinear maps $A^n \times A^m \to A^{n + m}$, $(a, b) \mapsto ab$ such that $$ \text{d}^{n + m}(ab) = \text{d}^n(a)b + (-1)^n a\text{d}^m(b) $$ and such that $\bigoplus A^n$ becomes an associative and unital $R$-algebra. \end{enumerate}


dga-definition-homomorphism-dga

A {\it homomorphism of differential graded algebras} $f : (A, \text{d}) \to (B, \text{d})$ is an algebra map $f : A \to B$ compatible with the gradings and $\text{d}$.


dga-definition-cdga

A differential graded algebra $(A, \text{d})$ is {\it commutative} if $ab = (-1)^{nm}ba$ for $a$ in degree $n$ and $b$ in degree $m$. We say $A$ is {\it strictly commutative} if in addition $a^2 = 0$ for $\deg(a)$ odd.


dga-definition-tensor-product

Let $R$ be a ring. Let $(A, \text{d})$, $(B, \text{d})$ be differential graded algebras over $R$. The {\it tensor product differential graded algebra} of $A$ and $B$ is the algebra $A \otimes_R B$ with multiplication defined by $$ (a \otimes b)(a' \otimes b') = (-1)^{\deg(a')\deg(b)} aa' \otimes bb' $$ endowed with differential $\text{d}$ defined by the rule $\text{d}(a \otimes b) = \text{d}(a) \otimes b + (-1)^m a \otimes \text{d}(b)$ where $m = \deg(a)$.


dga-definition-dgm

Let $R$ be a ring. Let $(A, \text{d})$ be a differential graded algebra over $R$. A (right) {\it differential graded module} $M$ over $A$ is a right $A$-module $M$ which has a grading $M = \bigoplus M^n$ and a differential $\text{d}$ such that $M^n A^m \subset M^{n + m}$, such that $\text{d}(M^n) \subset M^{n + 1}$, and such that $$ \text{d}(ma) = \text{d}(m)a + (-1)^n m\text{d}(a) $$ for $a \in A$ and $m \in M^n$. A {\it homomorphism of differential graded modules} $f : M \to N$ is an $A$-module map compatible with gradings and differentials. The category of (right) differential graded $A$-modules is denoted $\text{Mod}_{(A, \text{d})}$.


dga-definition-shift

Let $(A, \text{d})$ be a differential graded algebra. Let $M$ be a differential graded module. For any $k \in \mathbf{Z}$ we define the {\it $k$-shifted module} $M[k]$ as follows \begin{enumerate} \item as $A$-module $M[k] = M$, \item $M[k]^n = M^{n + k}$, \item $\text{d}_{M[k]} = (-1)^k\text{d}_M$. \end{enumerate} For a morphism $f : M \to N$ of differential graded $A$-modules we let $f[k] : M[k] \to N[k]$ be the map equal to $f$ on underlying $A$-modules. This defines a functor $[k] : \text{Mod}_{(A, \text{d})} \to \text{Mod}_{(A, \text{d})}$.


dga-definition-homotopy

Let $(A, \text{d})$ be a differential graded algebra. Let $f, g : M \to N$ be homomorphisms of differential graded $A$-modules. A {\it homotopy between $f$ and $g$} is an $A$-module map $h : M \to N$ such that \begin{enumerate} \item $h(M^n) \subset N^{n - 1}$ for all $n$, and \item $f(x) - g(x) = \text{d}_N(h(x)) + h(\text{d}_M(x))$ for all $x \in M$. \end{enumerate} If a homotopy exists, then we say $f$ and $g$ are {\it homotopic}.


dga-definition-complexes-notation

Let $(A, \text{d})$ be a differential graded algebra. The {\it homotopy category}, denoted $K(\text{Mod}_{(A, \text{d})})$, is the category whose objects are the objects of $\text{Mod}_{(A, \text{d})}$ and whose morphisms are homotopy classes of homomorphisms of differential graded $A$-modules.


dga-definition-cone

Let $(A, \text{d})$ be a differential graded algebra. Let $f : K \to L$ be a homomorphism of differential graded $A$-modules. The {\it cone} of $f$ is the differential graded $A$-module $C(f)$ given by $C(f) = L \oplus K$ with grading $C(f)^n = L^n \oplus K^{n + 1}$ and differential $$ d_{C(f)} = \left( \begin{matrix} \text{d}_L & f \\ 0 & -\text{d}_K \end{matrix} \right) $$ It comes equipped with canonical morphisms of complexes $i : L \to C(f)$ and $p : C(f) \to K[1]$ induced by the obvious maps $L \to C(f)$ and $C(f) \to K$.


dga-definition-admissible-ses

Let $(A, \text{d})$ be a differential graded algebra. \begin{enumerate} \item A homomorphism $K \to L$ of differential graded $A$-modules is an {\it admissible monomorphism} if there exists a graded $A$-module map $L \to K$ which is left inverse to $K \to L$. \item A homomorphism $L \to M$ of differential graded $A$-modules is an {\it admissible epimorphism} if there exists a graded $A$-module map $M \to L$ which is right inverse to $L \to M$. \item A short exact sequence $0 \to K \to L \to M \to 0$ of differential graded $A$-modules is an {\it admissible short exact sequence} if it is split as a sequence of graded $A$-modules. \end{enumerate}


dga-definition-distinguished-triangle

Let $(A, \text{d})$ be a differential graded algebra. \begin{enumerate} \item If $0 \to K \to L \to M \to 0$ is an admissible short exact sequence of differential graded $A$-modules, then the {\it triangle associated to $0 \to K \to L \to M \to 0$} is the triangle  (\ref{equation-triangle-associated-to-admissible-ses}) of $K(\text{Mod}_{(A, \text{d})})$. \item A triangle of $K(\text{Mod}_{(A, \text{d})})$ is called a {\it distinguished triangle} if it is isomorphic to a triangle associated to an admissible short exact sequence of differential graded $A$-modules. \end{enumerate}


dga-definition-opposite-dga

Let $R$ be a ring. Let $(A, \text{d})$ be a differential graded algebra over $R$. The {\it opposite differential graded algebra} is the differential graded algebra $(A^{opp}, \text{d})$ over $R$ where $A^{opp} = A$ as a graded $R$-module, $\text{d} = \text{d}$, and multiplication is given by $$ a \cdot_{opp} b = (-1)^{\deg(a)\deg(b)} b a $$ for homogeneous elements $a, b \in A$.


dga-definition-shift-graded-module

Let $R$ be a ring. Let $A$ be a $\mathbf{Z}$-graded $R$-algebra. \begin{enumerate} \item Given a right graded $A$-module $M$ we define the {\it $k$th shifted $A$-module} $M[k]$ as the same as a right $A$-module but with grading $(M[k])^n = M^{n + k}$. \item Given a left graded $A$-module $M$ we define the {\it $k$th shifted $A$-module} $M[k]$ as the module with grading $(M[k])^n = M^{n + k}$ and multiplication $A^n \times (M[k])^m \to (M[k])^{n + m}$ equal to $(-1)^{nk}$ times the given multiplication $A^n \times M^{m + k} \to M^{n + m + k}$. \end{enumerate}


dga-definition-unbounded-derived-category

Let $(A, \text{d})$ be a differential graded algebra. Let $\text{Ac}$ and $\text{Qis}$ be as in Lemma \ref{lemma-acyclic}. The {\it derived category of $(A, \text{d})$} is the triangulated category $$ D(A, \text{d}) = K(\text{Mod}_{(A, \text{d})})/\text{Ac} = \text{Qis}^{-1}K(\text{Mod}_{(A, \text{d})}). $$ We denote $H^0 : D(A, \text{d}) \to \text{Mod}_R$ the unique functor whose composition with the quotient functor gives back the functor $H^0$ defined above.


dga-definition-linear-category

Let $R$ be a ring. An {\it $R$-linear category $\mathcal{A}$} is a category where every morphism set is given the structure of an $R$-module and where for $x, y, z \in \Ob(\mathcal{A})$ composition law $$ \Hom_\mathcal{A}(y, z) \times \Hom_\mathcal{A}(x, y) \longrightarrow \Hom_\mathcal{A}(x, z) $$ is $R$-bilinear.


dga-definition-functor-linear-categories

Let $R$ be a ring. A {\it functor of $R$-linear categories}, or an {\it $R$-linear} is a functor $F : \mathcal{A} \to \mathcal{B}$ where for all objects $x, y$ of $\mathcal{A}$ the map $F : \Hom_\mathcal{A}(x, y) \to \Hom_\mathcal{A}(F(x), F(y))$ is a homomorphism of $R$-modules.


dga-definition-graded-category

Let $R$ be a ring. A {\it graded category $\mathcal{A}$ over $R$} is a category where every morphism set is given the structure of a graded $R$-module and where for $x, y, z \in \Ob(\mathcal{A})$ composition is $R$-bilinear and induces a homomorphism $$ \Hom_\mathcal{A}(y, z) \otimes_R \Hom_\mathcal{A}(x, y) \longrightarrow \Hom_\mathcal{A}(x, z) $$ of graded $R$-modules (i.e., preserving degrees).


dga-definition-functor-graded-categories

Let $R$ be a ring. A {\it functor of graded categories over $R$}, or a {\it graded functor} is a functor $F : \mathcal{A} \to \mathcal{B}$ where for all objects $x, y$ of $\mathcal{A}$ the map $F : \Hom_\mathcal{A}(x, y) \to \Hom_\mathcal{A}(F(x), F(y))$ is a homomorphism of graded $R$-modules.


dga-definition-H0-of-graded-category

Let $R$ be a ring. Let $\mathcal{A}$ be a graded category over $R$. We let {\it $\mathcal{A}^0$} be the category with the same objects as $\mathcal{A}$ and with $$ \Hom_{\mathcal{A}^0}(x, y) = \Hom^0_\mathcal{A}(x, y) $$ the degree $0$ graded piece of the graded module of morphisms of $\mathcal{A}$.


dga-definition-graded-direct-sum

Let $R$ be a ring. Let $\mathcal{A}$ be a graded category over $R$. A direct sum $(x, y, z, i, j, p, q)$ in $\mathcal{A}$ (notation as in Homology, Remark \ref{homology-remark-direct-sum}) is a {\it graded direct sum} if $i, j, p, q$ are homogeneous of degree $0$.


dga-definition-dga-category

Let $R$ be a ring. A {\it differential graded category $\mathcal{A}$ over $R$} is a category where every morphism set is given the structure of a differential graded $R$-module and where for $x, y, z \in \Ob(\mathcal{A})$ composition is $R$-bilinear and induces a homomorphism $$ \Hom_\mathcal{A}(y, z) \otimes_R \Hom_\mathcal{A}(x, y) \longrightarrow \Hom_\mathcal{A}(x, z) $$ of differential graded $R$-modules.


dga-definition-functor-dga-categories

Let $R$ be a ring. A {\it functor of differential graded categories over $R$} is a functor $F : \mathcal{A} \to \mathcal{B}$ where for all objects $x, y$ of $\mathcal{A}$ the map $F : \Hom_\mathcal{A}(x, y) \to \Hom_\mathcal{A}(F(x), F(y))$ is a homomorphism of differential graded $R$-modules.


dga-definition-homotopy-category-of-dga-category

Let $R$ be a ring. Let $\mathcal{A}$ be a differential graded category over $R$. Then we let \begin{enumerate} \item the {\it category of complexes of $\mathcal{A}$}\footnote{This may be nonstandard terminology.} be the category $\text{Comp}(\mathcal{A})$ whose objects are the same as the objects of $\mathcal{A}$ and with $$ \Hom_{\text{Comp}(\mathcal{A})}(x, y) = \Ker(d : \Hom^0_\mathcal{A}(x, y) \to \Hom^1_\mathcal{A}(x, y)) $$ \item the {\it homotopy category of $\mathcal{A}$} be the category $K(\mathcal{A})$ whose objects are the same as the objects of $\mathcal{A}$ and with $$ \Hom_{K(\mathcal{A})}(x, y) = H^0(\Hom_\mathcal{A}(x, y)) $$ \end{enumerate}


dga-definition-dg-direct-sum

Let $R$ be a ring. Let $\mathcal{A}$ be a differential graded category over $R$. A direct sum $(x, y, z, i, j, p, q)$ in $\mathcal{A}$ (notation as in Homology, Remark \ref{homology-remark-direct-sum}) is a {\it differential graded direct sum} if $i, j, p, q$ are homogeneous of degree $0$ and closed, i.e., $\text{d}(i) = 0$, etc.


dga-definition-bimodule

Bimodules. Let $R$ be a ring. \begin{enumerate} \item Let $A$ and $B$ be $R$-algebras. An {\it $(A, B)$-bimodule} is an $R$-module $M$ equippend with $R$-bilinear maps $$ A \times M \to M, (a, x) \mapsto ax \quad\text{and}\quad M \times B \to M, (x, b) \mapsto xb $$ such that the following hold \begin{enumerate} \item $a'(ax) = (a'a)x$ and $(xb)b' = x(bb')$, \item $a(xb) = (ax)b$, and \item $1 x = x = x 1$. \end{enumerate} \item Let $A$ and $B$ be $\mathbf{Z}$-graded $R$-algebras. A {\it graded $(A, B)$-bimodule} is an $(A, B)$-bimodule $M$ which has a grading $M = \bigoplus M^n$ such that $A^n M^m \subset M^{n + m}$ and $M^n B^m \subset M^{n + m}$. \item Let $A$ and $B$ be differential graded $R$-algebras. A {\it differential graded $(A, B)$-bimodule} is a graded $(A, B)$-bimodule which comes equipped with a differential $\text{d} : M \to M$ homogeneous of degree $1$ such that $\text{d}(ax) = \text{d}(a)x + (-1)^{\deg(a)}a\text{d}(x)$ and $\text{d}(xb) = \text{d}(x)b + (-1)^{\deg(x)}x\text{d}(b)$ for homogeneous elements $a \in A$, $x \in M$, $b \in B$. \end{enumerate}


spaces-more-groupoids-definition-split-at-point

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$ Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $u \in |U|$ be a point. \begin{enumerate} \item We say $R$ is {\it strongly split over $u$} if there exists an open subspace $P \subset R$ such that \begin{enumerate} \item $(U, P, s|_P, t|_P, c|_{P \times_{s, U, t} P})$ is a groupoid in algebraic spaces over $B$, \item $s|_P$, $t|_P$ are finite, and \item $\{r \in |R| : s(r) = u, t(r) = u\} \subset |P|$. \end{enumerate} The choice of such a $P$ will be called a {\it strong splitting of $R$ over $u$}. \item We say $R$ is {\it split over $u$} if there exists an open subspace $P \subset R$ such that \begin{enumerate} \item $(U, P, s|_P, t|_P, c|_{P \times_{s, U, t} P})$ is a groupoid in algebraic spaces over $B$, \item $s|_P$, $t|_P$ are finite, and \item $\{g \in |G| : g\text{ maps to }u\} \subset |P|$ where $G \to U$ is the stabilizer. \end{enumerate} The choice of such a $P$ will be called a {\it splitting of $R$ over $u$}. \item We say $R$ is {\it quasi-split over $u$} if there exists an open subspace $P \subset R$ such that \begin{enumerate} \item $(U, P, s|_P, t|_P, c|_{P \times_{s, U, t} P})$ is a groupoid in algebraic spaces over $B$, \item $s|_P$, $t|_P$ are finite, and \item $e(u) \in |P|$\footnote{This condition is implied by (a).}. \end{enumerate} The choice of such a $P$ will be called a {\it quasi-splitting of $R$ over $u$}. \end{enumerate}


modules-definition-globally-generated

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. We say that $\mathcal{F}$ is {\it generated by global sections} if there exist a set $I$, and global sections $s_i \in \Gamma(X, \mathcal{F})$, $i \in I$ such that the map $$ \bigoplus\nolimits_{i \in I} \mathcal{O}_X \longrightarrow \mathcal{F} $$ which is the map associated to $s_i$ on the summand corresponding to $i$, is surjective. In this case we say that the sections $s_i$ {\it generate} $\mathcal{F}$.


modules-definition-generated-by-local-sections

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. Given a set $I$, and local sections $s_i$, $i \in I$ of $\mathcal{F}$ we say that the subsheaf $\mathcal{G}$ of Lemma \ref{lemma-generated-by-local-sections} above is the {\it subsheaf generated by the $s_i$}.


modules-definition-support

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. \begin{enumerate} \item The {\it support of $\mathcal{F}$} is the set of points $x \in X$ such that $\mathcal{F}_x \not = 0$. \item We denote $\text{Supp}(\mathcal{F})$ the support of $\mathcal{F}$. \item Let $s \in \Gamma(X, \mathcal{F})$ be a global section. The {\it support of $s$} is the set of points $x \in X$ such that the image $s_x \in \mathcal{F}_x$ of $s$ is not zero. \end{enumerate}


modules-definition-locally-generated

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. We say that $\mathcal{F}$ is {\it locally generated by sections} if for every $x \in X$ there exists an open neighbourhood $U$ such that $\mathcal{F}|_U$ is globally generated as a sheaf of $\mathcal{O}_U$-modules.


modules-definition-finite-type

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. We say that $\mathcal{F}$ is of {\it finite type} if for every $x \in X$ there exists an open neighbourhood $U$ such that $\mathcal{F}|_U$ is generated by finitely many sections.


modules-definition-quasi-coherent

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. We say that $\mathcal{F}$ is a {\it quasi-coherent sheaf of $\mathcal{O}_X$-modules} if for every point $x \in X$ there exists an open neighbourhood $x\in U \subset X$ such that $\mathcal{F}|_U$ is isomorphic to the cokernel of a map $$ \bigoplus\nolimits_{j \in J} \mathcal{O}_U \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O}_U $$ The category of quasi-coherent $\mathcal{O}_X$-modules is denoted $\QCoh(\mathcal{O}_X)$.


modules-definition-sheaf-associated

In the situation of Lemma \ref{lemma-construct-quasi-coherent-sheaves} we say $\mathcal{F}_M$ is the {\it sheaf associated to the module $M$ and the ring map $\alpha$}. If $R = \Gamma(X, \mathcal{O}_X)$ and $\alpha = \text{id}_R$ we simply say $\mathcal{F}_M$ is the {\it sheaf associated to the module $M$}.


modules-definition-finite-presentation

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. We say that $\mathcal{F}$ is of {\it finite presentation} if for every point $x \in X$ there exists an open neighbourhood $x\in U \subset X$, and  $n, m \in \mathbf{N}$ such that $\mathcal{F}|_U$ is isomorphic to the cokernel of a map $$ \bigoplus\nolimits_{j = 1, \ldots, m} \mathcal{O}_U \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} \mathcal{O}_U $$


modules-definition-coherent

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. We say that $\mathcal{F}$ is a {\it coherent $\mathcal{O}_X$-module} if the following two conditions hold: \begin{enumerate} \item $\mathcal{F}$ is of finite type, and \item for every open $U \subset X$ and every finite collection $s_i \in \mathcal{F}(U)$, $i = 1, \ldots, n$ the kernel of the associated map $\bigoplus_{i = 1, \ldots, n} \mathcal{O}_U \to \mathcal{F}|_U$ is of finite type. \end{enumerate} The category of coherent $\mathcal{O}_X$-modules is denoted $\textit{Coh}(\mathcal{O}_X)$.


modules-definition-closed-immersion

A {\it closed immersion of ringed spaces}\footnote{This is nonstandard notation; see discussion above.} is a morphism $i : (Z, \mathcal{O}_Z) \to (X, \mathcal{O}_X)$ with the following properties: \begin{enumerate} \item The map $i$ is a closed immersion of topological spaces. \item The associated map $\mathcal{O}_X \to i_*\mathcal{O}_Z$ is surjective. Denote the kernel by $\mathcal{I}$. \item The $\mathcal{O}_X$-module $\mathcal{I}$ is locally generated by sections. \end{enumerate}


modules-definition-locally-free

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. \begin{enumerate} \item We say $\mathcal{F}$ is {\it locally free} if for every point $x \in X$ there exists a set $I$ and an open neighbourhood $x \in U \subset X$ such that $\mathcal{F}|_U$ is isomorphic to $\bigoplus_{i \in I} \mathcal{O}_X|_U$ as an $\mathcal{O}_X|_U$-module. \item We say $\mathcal{F}$ is {\it finite locally free} if we may choose the index sets $I$ to be finite. \item We say $\mathcal{F}$ is {\it finite locally free of rank $r$} if we may choose the index sets $I$ to have cardinality $r$. \end{enumerate}


modules-definition-flat

Let $(X, \mathcal{O}_X)$ be a ringed space. An $\mathcal{O}_X$-module $\mathcal{F}$ is {\it flat} if the functor $$ \textit{Mod}(\mathcal{O}_X) \longrightarrow \textit{Mod}(\mathcal{O}_X), \quad \mathcal{G} \mapsto \mathcal{G} \otimes_\mathcal{O} \mathcal{F} $$ is exact.


modules-definition-flat-at-point

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $x \in X$. An $\mathcal{O}_X$-module $\mathcal{F}$ is {\it flat at $x$} if $\mathcal{F}_x$ is a flat $\mathcal{O}_{X, x}$-module.


modules-definition-flat-morphism

Let $f : X \to Y$ be a morphism of ringed spaces. Let $x \in X$. We say $f$ is {\it flat at $x$} if the map of rings $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is flat. We say $f$ is {\it flat} if $f$ is flat at every $x \in X$.


modules-definition-flat-module

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. \begin{enumerate} \item We say that $\mathcal{F}$ is {\it flat over $Y$ at a point $x \in X$} if the stalk $\mathcal{F}_x$ is a flat $\mathcal{O}_{Y, f(x)}$-module. \item We say that $\mathcal{F}$ is {\it flat over $Y$} if $\mathcal{F}$ is flat over $Y$ at every point $x$ of $X$. \end{enumerate}


modules-definition-koszul

Let $X$ be a ringed space. Let $\varphi : \mathcal{E} \to \mathcal{O}_X$ be an $\mathcal{O}_X$-module map. The {\it Koszul complex} $K_\bullet(\varphi)$ associated to $\varphi$ is the sheaf of commutative differential graded algebras defined as follows: \begin{enumerate} \item the underlying graded algebra is the exterior algebra $K_\bullet(\varphi) = \wedge(\mathcal{E})$, \item the differential $d : K_\bullet(\varphi) \to K_\bullet(\varphi)$ is the unique derivation such that $d(e) = \varphi(e)$ for all local sections $e$ of $\mathcal{E} = K_1(\varphi)$. \end{enumerate}


modules-definition-koszul-complex

Let $X$ be a ringed space and let $f_1, \ldots, f_n \in \Gamma(X, \mathcal{O}_X)$. The {\it Koszul complex on $f_1, \ldots, f_r$} is the Koszul complex associated to the map $(f_1, \ldots, f_n) : \mathcal{O}_X^{\oplus n} \to \mathcal{O}_X$. Notation $K_\bullet(\mathcal{O}_X, f_1, \ldots, f_n)$, or $K_\bullet(\mathcal{O}_X, f_\bullet)$.


modules-definition-invertible

Let $(X, \mathcal{O}_X)$ be a ringed space. An {\it invertible $\mathcal{O}_X$-module} is a sheaf of $\mathcal{O}_X$-modules $\mathcal{L}$ such that the functor $$ \textit{Mod}(\mathcal{O}_X) \longrightarrow \textit{Mod}(\mathcal{O}_X),\quad \mathcal{F} \longmapsto \mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{F} $$ is an equivalence of categories. We say that $\mathcal{L}$ is {\it trivial} if it is isomorphic as an $\mathcal{O}_X$-module to $\mathcal{O}_X$.


modules-definition-powers

Let $(X, \mathcal{O}_X)$ be a ringed space. Given an invertible sheaf $\mathcal{L}$ on $X$ and $n \in \mathbf{Z}$ we define the $n$th {\it tensor power} $\mathcal{L}^{\otimes n}$ of $\mathcal{L}$ as the image of $\mathcal{O}_X$ under applying the equivalence $\mathcal{F} \mapsto\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}$ exactly $n$ times.


modules-definition-gamma-star

Let $(X, \mathcal{O}_X)$ be a ringed space. Given an invertible sheaf $\mathcal{L}$ on $X$ we define the {\it associated graded ring} to be $$ \Gamma_*(X, \mathcal{L}) = \bigoplus\nolimits_{n \geq 0} \Gamma(X, \mathcal{L}^{\otimes n}) $$ Given a sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ we set $$ \Gamma_*(X, \mathcal{L}, \mathcal{F}) = \bigoplus\nolimits_{n \in \mathbf{Z}} \Gamma(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) $$ which we think of as a graded $\Gamma_*(X, \mathcal{L})$-module.


modules-definition-pic

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


modules-definition-derivation

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


modules-definition-module-differentials

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


modules-definition-differentials

Let $(f, f^\sharp) : (X, \mathcal{O}_X) \to (S, \mathcal{O}_S)$ be a morphism of ringed spaces. \begin{enumerate} \item Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. An {\it $S$-derivation} into $\mathcal{F}$ is a $f^{-1}\mathcal{O}_S$-derivation, or more precisely a $f^\sharp$-derivation in the sense of Definition \ref{definition-derivation}. We denote $\text{Der}_S(\mathcal{O}_X, \mathcal{F})$ the set of $S$-derivations into $\mathcal{F}$. \item The {\it sheaf of differentials $\Omega_{X/S}$ of $X$ over $S$} is the module of differentials $\Omega_{\mathcal{O}_X/f^{-1}\mathcal{O}_S}$ endowed with its universal $S$-derivation $\text{d}_{X/S} : \mathcal{O}_X \to \Omega_{X/S}$. \end{enumerate}


modules-definition-differential-operators

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


modules-definition-module-principal-parts

Let $X$ be a topoological space. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a map of sheaves of rings on $X$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_2$-modules. The module $\mathcal{P}^k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F})$ constructed in Lemma \ref{lemma-module-principal-parts} is called the {\it module of principal parts of order $k$} of $\mathcal{F}$.


modules-definition-relative-differential-operators

Let $(f, f^\sharp) : (X, \mathcal{O}_X) \to (S, \mathcal{O}_S)$ be a morphism of ringed spaces. Let $\mathcal{F}$ and $\mathcal{G}$ be $\mathcal{O}_X$-modules. Let $k \geq 0$ be an integer. A {\it differential operator of order $k$ on $X/S$} is a differential operator $D : \mathcal{F} \to \mathcal{G}$ with respect to $f^\sharp : f^{-1}\mathcal{O}_S \to \mathcal{O}_X$ We denote $\text{Diff}^k_{X/S}(\mathcal{F}, \mathcal{G})$ the set of these differential operators.


modules-definition-de-rham-complex

In the situation above, the {\it de Rham complex of $\mathcal{B}$ over $\mathcal{A}$} is the unique complex $$ \Omega_{\mathcal{B}/\mathcal{A}}^0 \to \Omega_{\mathcal{B}/\mathcal{A}}^1 \to \Omega_{\mathcal{B}/\mathcal{A}}^2 \to \ldots $$ of sheaves of $\mathcal{A}$-modules whose differential in degree $0$ is given by $\text{d} : \mathcal{B} \to \Omega_{\mathcal{B}/\mathcal{A}}$ and whose differentials in higher degrees have the following property \begin{equation} \label{equation-rule} \text{d}\left(b_0\text{d}b_1 \wedge \ldots \wedge \text{d}b_p\right) = \text{d}b_0 \wedge \text{d}b_1 \wedge \ldots \wedge \text{d}b_p \end{equation} where $b_0, \ldots, b_p \in \mathcal{B}(U)$ are sections over a common open $U \subset X$.


modules-definition-de-rham-complex-morphism-ringed-spaces

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. The {\it de Rham complex} of $f$ or of $X$ over $Y$ is the complex $$ \Omega^\bullet_{X/Y} = \Omega^\bullet_{\mathcal{O}_X/f^{-1}\mathcal{O}_Y} $$


modules-definition-naive-cotangent-complex

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


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

The {\it naive cotangent complex} $\NL_f = \NL_{X/Y}$ of a morphism of ringed spaces $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ is $\NL_{\mathcal{O}_X/f^{-1}\mathcal{O}_Y}$.


defos-definition-strict-morphism-thickenings

In Situation \ref{situation-morphism-thickenings} we say that $(f, f')$ is a {\it strict morphism of thickenings} if the map $(f')^*\mathcal{J} \longrightarrow \mathcal{I}$ is surjective.


defos-definition-strict-morphism-thickenings-ringed-topoi

In Situation \ref{situation-morphism-thickenings-ringed-topoi} we say that $(f, f')$ is a {\it strict morphism of thickenings} if the map $(f')^*\mathcal{J} \longrightarrow \mathcal{I}$ is surjective.


groupoids-quotients-definition-invariant

Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j = (t, s) : R \to U \times_B U$ be a pre-relation of algebraic spaces over $B$. We say a morphism $\phi : U \to X$ of algebraic spaces over $B$ is {\it $R$-invariant} if the diagram $$ \xymatrix{ R \ar[r]_s \ar[d]_t & U \ar[d]^\phi \\ U \ar[r]^\phi & X } $$ is commutative. If $j : R \to U \times_B U$ comes from the action of a group algebraic space $G$ on $U$ over $B$ as in Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-groupoid-from-action}, then we say that $\phi$ is {\it $G$-invariant}.


groupoids-quotients-definition-base-change

In the situation of Lemma \ref{lemma-base-change-on-invariant} we call $j' : R' \to U' \times_B U'$ the {\it base change} of the pre-relation $j$ to $X'$. We say it is a {\it flat base change} if $X' \to X$ is a flat morphism of algebraic spaces.


groupoids-quotients-definition-categorical

Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j = (t, s) : R \to U \times_B U$ be pre-relation in algebraic spaces over $B$. \begin{enumerate} \item We say a morphism $\phi : U \to X$ of algebraic spaces over $B$ is a {\it categorical quotient} if it is $R$-invariant, and for every $R$-invariant morphism $\psi : U \to Y$ of algebraic spaces over $B$ there exists a unique morphism $\chi : X \to Y$ such that $\psi = \phi \circ \chi$. \item Let $\mathcal{C}$ be a full subcategory of the category of algebraic spaces over $B$. Assume $U$, $R$ are objects of $\mathcal{C}$. In this situation we say a morphism $\phi : U \to X$ of algebraic spaces over $B$ is a {\it categorical quotient in $\mathcal{C}$} if $X \in \Ob(\mathcal{C})$, and $\phi$ is $R$-invariant, and for every $R$-invariant morphism $\psi : U \to Y$ with $Y \in \Ob(\mathcal{C})$ there exists a unique morphism $\chi : X \to Y$ such that $\psi = \phi \circ \chi$. \item If $B = S$ and $\mathcal{C}$ is the category of schemes over $S$, then we say $U \to X$ is a {\it categorical quotient in the category of schemes}, or simply a {\it categorical quotient in schemes}. \end{enumerate}


groupoids-quotients-definition-universal-categorical

Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $\mathcal{C}$ be a full subcategory of the category of algebraic spaces over $B$ closed under fibre products. Let $j = (t, s) : R \to U \times_B U$ be pre-relation in $\mathcal{C}$, and let $U \to X$ be an $R$-invariant morphism with $X \in \Ob(\mathcal{C})$. \begin{enumerate} \item We say $U \to X$ is a {\it universal categorical quotient} in $\mathcal{C}$ if for every morphism $X' \to X$ in $\mathcal{C}$ the morphism $U' = X' \times_X U \to X'$ is the categorical quotient in $\mathcal{C}$ of the base change $j' : R' \to U'$ of $j$. \item We say $U \to X$ is a {\it uniform categorical quotient} in $\mathcal{C}$ if for every flat morphism $X' \to X$ in $\mathcal{C}$ the morphism $U' = X' \times_X U \to X'$ is the categorical quotient in $\mathcal{C}$ of the base change $j' : R' \to U'$ of $j$. \end{enumerate}


groupoids-quotients-definition-orbit

Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j : R \to U \times_B U$ be a pre-relation over $B$. If $u \in |U|$, then the {\it orbit}, or more precisely the {\it $R$-orbit} of $u$ is $$ O_u = \left\{ u' \in |U|\ : \begin{matrix} \exists n \geq 1, \ \exists u_0, \ldots, u_n \in |U|\text{ such that } u_0 = u \text{ and } u_n = u' \\ \text{and for all }i \in \{0, \ldots, n - 1\}\text{ either } u_i = u_{i + 1}\text{ or } \\ \exists r \in |R|, \ s(r) = u_i, t(r) = u_{i + 1} \text{ or } \\ \exists r \in |R|, \ t(r) = u_i, s(r) = u_{i + 1} \end{matrix} \right\} $$


groupoids-quotients-definition-geometric-orbits

Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j : R \to U \times_B U$ be a pre-relation over $B$. Let $\Spec(k) \to B$ be a geometric point of $B$. \begin{enumerate} \item We say $\overline{u}, \overline{u}' \in U(k)$ are {\it weakly $R$-equivalent} if they are in the same equivalence class for the equivalence relation generated by the relation $j(R(k)) \subset U(k) \times U(k)$. \item We say $\overline{u}, \overline{u}' \in U(k)$ are {\it $R$-equivalent} if for some overfield $k \subset \Omega$ the images in $U(\Omega)$ are weakly $R$-equivalent. \item The {\it weak orbit}, or more precisely the {\it weak $R$-orbit} of $\overline{u} \in U(k)$ is set of all elements of $U(k)$ which are weakly $R$-equivalent to $\overline{u}$. \item The {\it orbit}, or more precisely the {\it $R$-orbit} of $\overline{u} \in U(k)$ is set of all elements of $U(k)$ which are $R$-equivalent to $\overline{u}$. \end{enumerate}


groupoids-quotients-definition-set-theoretically-invariant

Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j : R \to U \times_B U$ be a pre-relation over $B$. \begin{enumerate} \item We say $\phi : U \to X$ is {\it set-theoretically $R$-invariant} if and only if the map $U(k) \to X(k)$ equalizes the two maps $s, t : R(k) \to U(k)$ for every algebraically closed field $k$ over $B$. \item We say $\phi : U \to X$ {\it separates orbits}, or {\it separates $R$-orbits} if it is set-theoretically $R$-invariant and $\phi(\overline{u}) = \phi(\overline{u}')$ in $X(k)$ implies that $\overline{u}, \overline{u}' \in U(k)$ are in the same orbit for every algebraically closed field $k$ over $B$. \end{enumerate}


groupoids-quotients-definition-set-theoretic-equivalence

Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j : R \to U \times_B U$ be a pre-relation over $B$. \begin{enumerate} \item We say $j$ is a {\it set-theoretic pre-equivalence relation} if for all algebraically closed fields $k$ over $B$ the relation $\sim_R$ on $U(k)$ defined by $$ \overline{u} \sim_R \overline{u}' \Leftrightarrow \begin{matrix} \exists\text{ field extension }K/k, \ \exists\ r \in R(K), \\ s(r) = \overline{u}, \ t(r) = \overline{u}' \end{matrix} $$ is an equivalence relation. \item We say $j$ is a {\it set-theoretic equivalence relation} if $j$ is universally injective and a set-theoretic pre-equivalence relation. \end{enumerate}


groupoids-quotients-definition-orbit-space

Let $B \to S$ as in Section \ref{section-conventions-notation}. Let $j : R \to U \times_B U$ be a pre-relation. We say $\phi : U \to X$ is an {\it orbit space for $R$} if \begin{enumerate} \item $\phi$ is $R$-invariant, \item $\phi$ separates $R$-orbits, and \item $\phi$ is surjective. \end{enumerate}


groupoids-quotients-definition-coarse

Let $S$ be a scheme and $B$ an algebraic space over $S$. Let $j : R \to U \times_B U$ be a pre-relation. A morphism $\phi : U \to X$ of algebraic spaces over $B$ is called a {\it coarse quotient} if \begin{enumerate} \item $\phi$ is a categorical quotient, and \item $\phi$ is an orbit space. \end{enumerate} If $S = B$, $U$, $R$ are all schemes, then we say a morphism of schemes $\phi : U \to X$ is a {\it coarse quotient in schemes} if \begin{enumerate} \item $\phi$ is a categorical quotient in schemes, and \item $\phi$ is an orbit space. \end{enumerate}


groupoids-quotients-definition-topological

Let $S$ be a scheme and $B$ an algebraic space over $S$. Let $j : R \to U \times_B U$ be a pre-relation. Let $\phi : U \to X$ be an $R$-invariant morphism of algebraic spaces over $B$. \begin{enumerate} \item \label{item-submersive} The morphism $\phi$ is submersive. \item \label{item-invariant-closed} For any $R$-invariant closed subset $Z \subset |U|$ the image $\phi(Z)$ is closed in $|X|$. \item \label{item-intersect-invariant-closed} Condition (\ref{item-invariant-closed}) holds and for any pair of $R$-invariant closed subsets $Z_1, Z_2 \subset |U|$ we have $$ \phi(Z_1 \cap Z_2) = \phi(Z_1) \cap \phi(Z_2) $$ \item The morphism $(t, s) : R \to U \times_X U$ is universally submersive. \label{item-strong} \end{enumerate} For each of these properties we can also require them to hold after any flat base change, or after any base change, see Definition \ref{definition-base-change}. In this case we say condition (\ref{item-submersive}), (\ref{item-invariant-closed}), (\ref{item-intersect-invariant-closed}), or (\ref{item-strong}) holds {\it uniformly} or {\it universally}.


groupoids-quotients-definition-functions

Let $S$ be a scheme and $B$ an algebraic space over $S$. Let $j : R \to U \times_B U$ be a pre-relation. Let $\phi : U \to X$ be an $R$-invariant morphism. Denote $\phi' = \phi \circ s = \phi \circ t : R \to X$. \begin{enumerate} \item We denote $(\phi_*\mathcal{O}_U)^R$ the $\mathcal{O}_X$-sub-algebra of $\phi_*\mathcal{O}_U$ which is the equalizer of the two maps $$ \xymatrix{ \phi_*\mathcal{O}_U \ar@<1ex>[rr]^{\phi_*s^\sharp} \ar@<-1ex>[rr]_{\phi_*t^\sharp} & & \phi'_*\mathcal{O}_R } $$ on $X_\etale$. We sometimes call this the {\it sheaf of $R$-invariant functions on $X$}. \item We say {\it the functions on $X$ are the $R$-invariant functions on $U$} if the natural map $\mathcal{O}_X \to (\phi_*\mathcal{O}_U)^R$ is an isomorphism. \end{enumerate}


groupoids-quotients-definition-good

Let $S$ be a scheme and $B$ an algebraic space over $S$. Let $j : R \to U \times_B U$ be a pre-relation. A morphism $\phi : U \to X$ of algebraic spaces over $B$ is called a {\it good quotient} if \begin{enumerate} \item $\phi$ is invariant, \item $\phi$ is affine, \item $\phi$ is surjective, \item condition (\ref{item-intersect-invariant-closed}) holds universally, and \item the functions on $X$ are the $R$-invariant functions on $U$. \end{enumerate}


groupoids-quotients-definition-geometric

Let $S$ be a scheme and $B$ an algebraic space over $S$. Let $j : R \to U \times_B U$ be a pre-relation. A morphism $\phi : U \to X$ of algebraic spaces over $B$ is called a {\it geometric quotient} if \begin{enumerate} \item $\phi$ is an orbit space, \item condition (\ref{item-submersive}) holds universally, i.e., $\phi$ is universally submersive, and \item the functions on $X$ are the $R$-invariant functions on $U$. \end{enumerate}


spaces-resolve-definition-blowup-at-point

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$ be a closed point. By Decent Spaces, Lemma \ref{decent-spaces-lemma-decent-space-closed-point} we can represent $x$ by a closed immersion $i : \Spec(k) \to X$. The {\it blowing up $X' \to X$ of $X$ at $x$} means the blowing up of $X$ in the closed subspace $Z = i(\Spec(k)) \subset X$.


spaces-resolve-definition-normalized-blowup

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$ satisfying the equivalent conditions of Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-prepare-normalization}. 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 $X$ at $x$ (Definition \ref{definition-blowup-at-point}) and $X'' \to X'$ is the normalization of $X'$.


spaces-resolve-definition-resolution

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


spaces-resolve-definition-resolution-surface

Let $S$ be a scheme. Let $Y$ be a $2$-dimensional Noetherian integral algebraic space over $S$. We say $Y$ has a {\it resolution of singularities by normalized blowups} if there exists a sequence $$ Y_n \to X_{n - 1} \to \ldots \to Y_1 \to Y_0 \to Y $$ 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}


duality-definition-dualizing-scheme

Let $X$ be a locally Noetherian scheme. An object $K$ of $D(\mathcal{O}_X)$ is called a {\it dualizing complex} if $K$ satisfies the equivalent conditions of Lemma \ref{lemma-equivalent-definitions}.


duality-definition-good-dualizing

Let $S$ be a Noetherian scheme and let $\omega_S^\bullet$ be a dualizing complex on $S$. Let $X$ be a scheme of finite type over $S$. The complex $K$ constructed above is called the {\it dualizing complex normalized relative to $\omega_S^\bullet$} and is denoted $\omega_X^\bullet$.


duality-definition-gorenstein

Let $X$ be a scheme. We say $X$ is {\it Gorenstein} if $X$ is locally Noetherian and $\mathcal{O}_{X, x}$ is Gorenstein for all $x \in X$.


duality-definition-gorenstein-morphism

Let $f : X \to Y$ be a morphism of schemes. Assume that all the fibres $X_y$ are locally Noetherian schemes. \begin{enumerate} \item Let $x \in X$, and $y = f(x)$. We say that $f$ is {\it Gorenstein at $x$} if $f$ is flat at $x$, and the local ring of the scheme $X_y$ at $x$ is Gorenstein. \item We say $f$ is a {\it Gorenstein morphism} if $f$ is Gorenstein at every point of $X$. \end{enumerate}


duality-definition-relative-dualizing-complex

Let $X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $W \subset X \times_S X$ be any open such that the diagonal $\Delta_{X/S} : X \to X \times_S X$ factors through a closed immersion $\Delta : X \to W$. A {\it relative dualizing complex} is a pair $(K, \xi)$ consisting of an object $K \in D(\mathcal{O}_X)$ and a map $$ \xi : \Delta_*\mathcal{O}_X \longrightarrow L\text{pr}_1^*K|_W $$ in $D(\mathcal{O}_W)$ such that \begin{enumerate} \item $K$ is $S$-perfect (Derived Categories of Schemes, Definition \ref{perfect-definition-relatively-perfect}), and \item $\xi$ defines an isomorphism of $\Delta_*\mathcal{O}_X$ with $R\SheafHom_{\mathcal{O}_W}( \Delta_*\mathcal{O}_X, L\text{pr}_1^*K|_W)$. \end{enumerate}


more-morphisms-definition-thickening

Thickenings. \begin{enumerate} \item We say a scheme $X'$ is a {\it thickening} of a scheme $X$ if $X$ is a closed subscheme of $X'$ and the underlying topological spaces are equal. \item We say a scheme $X'$ is a {\it first order thickening} of a scheme $X$ if $X$ is a closed subscheme of $X'$ and the quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_{X'}$ defining $X$ has square zero. \item Given two thickenings $X \subset X'$ and $Y \subset Y'$ a {\it morphism of thickenings} is a morphism $f' : X' \to Y'$ such that $f'(X) \subset Y$, i.e., such that $f'|_X$ factors through the closed subscheme $Y$. In this situation we set $f = f'|_X : X \to Y$ and we say that $(f, f') : (X \subset X') \to (Y \subset Y')$ is a morphism of thickenings. \item Let $S$ be a scheme. We similarly define {\it thickenings over $S$}, and {\it morphisms of thickenings over $S$}. This means that the schemes $X, X', Y, Y'$ above are schemes over $S$, and that the morphisms $X \to X'$, $Y \to Y'$ and $f' : X' \to Y'$ are morphisms over $S$. \end{enumerate}


more-morphisms-definition-first-order-infinitesimal-neighbourhood

Let $i : Z \to X$ be an immersion of schemes. The {\it first order infinitesimal neighbourhood} of $Z$ in $X$ is the first order thickening $Z \subset Z'$ over $X$ described above.


more-morphisms-definition-formally-unramified

Let $f : X \to S$ be a morphism of schemes. We say $f$ is {\it formally unramified} if given any solid commutative diagram $$ \xymatrix{ X \ar[d]_f & T \ar[d]^i \ar[l] \\ S & T' \ar[l] \ar@{-->}[lu] } $$ where $T \subset T'$ is a first order thickening of affine schemes over $S$ there exists at most one dotted arrow making the diagram commute.


more-morphisms-definition-universal-thickening

Let $h : Z \to X$ be a formally unramified morphism of schemes. \begin{enumerate} \item The {\it universal first order thickening} of $Z$ over $X$ is the thickening $Z \subset Z'$ constructed in Lemma \ref{lemma-universal-thickening}. \item The {\it conormal sheaf of $Z$ over $X$} is the conormal sheaf of $Z$ in its universal first order thickening $Z'$ over $X$. \end{enumerate} We often denote the conormal sheaf $\mathcal{C}_{Z/X}$ in this situation.


more-morphisms-definition-formally-etale

Let $f : X \to S$ be a morphism of schemes. We say $f$ is {\it formally \'etale} if given any solid commutative diagram $$ \xymatrix{ X \ar[d]_f & T \ar[d]^i \ar[l] \\ S & T' \ar[l] \ar@{-->}[lu] } $$ where $T \subset T'$ is a first order thickening of affine schemes over $S$ there exists exactly one dotted arrow making the diagram commute.


more-morphisms-definition-formally-smooth

Let $f : X \to S$ be a morphism of schemes. We say $f$ is {\it formally smooth} if given any solid commutative diagram $$ \xymatrix{ X \ar[d]_f & T \ar[d]^i \ar[l] \\ S & T' \ar[l] \ar@{-->}[lu] } $$ where $T \subset T'$ is a first order thickening of affine schemes over $S$ there exists a dotted arrow making the diagram commute.


more-morphisms-definition-netherlander

Let $f : X \to Y$ be a morphism of schemes. The {\it naive cotangent complex of $f$} is the complex defined in Modules, Definition \ref{modules-definition-cotangent-complex-morphism-ringed-topoi}. Notation: $\NL_f$ or $\NL_{X/Y}$.


more-morphisms-definition-normal

Let $f : X \to Y$ be a morphism of schemes. Assume that all the fibres $X_y$ are locally Noetherian schemes. \begin{enumerate} \item Let $x \in X$, and $y = f(x)$. We say that $f$ is {\it normal at $x$} if $f$ is flat at $x$, and the scheme $X_y$ is geometrically normal at $x$ over $\kappa(y)$ (see Varieties, Definition \ref{varieties-definition-geometrically-normal}). \item We say $f$ is a {\it normal morphism} if $f$ is normal at every point of $X$. \end{enumerate}


more-morphisms-definition-regular

Let $f : X \to Y$ be a morphism of schemes. Assume that all the fibres $X_y$ are locally Noetherian schemes. \begin{enumerate} \item Let $x \in X$, and $y = f(x)$. We say that $f$ is {\it regular at $x$} if $f$ is flat at $x$, and the scheme $X_y$ is geometrically regular at $x$ over $\kappa(y)$ (see Varieties, Definition \ref{varieties-definition-geometrically-regular}). \item We say $f$ is a {\it regular morphism} if $f$ is regular at every point of $X$. \end{enumerate}


more-morphisms-definition-CM

Let $f : X \to Y$ be a morphism of schemes. Assume that all the fibres $X_y$ are locally Noetherian schemes. \begin{enumerate} \item Let $x \in X$, and $y = f(x)$. We say that $f$ is {\it Cohen-Macaulay at $x$} if $f$ is flat at $x$, and the local ring of the scheme $X_y$ at $x$ is Cohen-Macaulay. \item We say $f$ is a {\it Cohen-Macaulay morphism} if $f$ is Cohen-Macaulay at every point of $X$. \end{enumerate}


more-morphisms-definition-etale-neighbourhood

Let $S$ be a scheme. Let $s \in S$ be a point. \begin{enumerate} \item An {\it \'etale neighbourhood of $(S, s)$} is a pair $(U, u)$ together with an \'etale morphism of schemes $\varphi : U \to S$ such that $\varphi(u) = s$. \item A {\it morphism of \'etale neighbourhoods} $f : (V, v) \to (U, u)$ of $(S, s)$ is simply a morphism of $S$-schemes $f : V \to U$ such that $f(v) = u$. \item An {\it elementary \'etale neighbourhood} is an \'etale neighbourhood $\varphi : (U, u) \to (S, s)$ such that $\kappa(s) = \kappa(u)$. \end{enumerate}


more-morphisms-definition-relatively-finitely-presented-sheaf

Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. We say $\mathcal{F}$ is {\it finitely presented relative to $S$} or {\it of finite presentation relative to $S$} if there exists an affine open covering $S = \bigcup V_i$ and for every $i$ an affine open covering $f^{-1}(V_i) = \bigcup_j U_{ij}$ such that $\mathcal{F}(U_{ij})$ is a $\mathcal{O}_X(U_{ij})$-module of finite presentation relative to $\mathcal{O}_S(V_i)$.


more-morphisms-definition-relative-pseudo-coherence

Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E$ be an object of $D(\mathcal{O}_X)$. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Fix $m \in \mathbf{Z}$. \begin{enumerate} \item We say $E$ is {\it $m$-pseudo-coherent relative to $S$} if there exists an affine open covering $S = \bigcup V_i$ and for each $i$ an affine open covering $f^{-1}(V_i) = \bigcup U_{ij}$ such that the equivalent conditions of Lemma \ref{lemma-relatively-pseudo-coherent} are satisfied for each of the pairs $(U_{ij} \to V_i, E|_{U_{ij}})$. \item We say $E$ is {\it pseudo-coherent relative to $S$} if $E$ is $m$-pseudo-coherent relative to $S$ for all $m \in \mathbf{Z}$. \item We say $\mathcal{F}$ is {\it $m$-pseudo-coherent relative to $S$} if $\mathcal{F}$ viewed as an object of $D(\mathcal{O}_X)$ is $m$-pseudo-coherent relative to $S$. \item We say $\mathcal{F}$ is {\it pseudo-coherent relative to $S$} if $\mathcal{F}$ viewed as an object of $D(\mathcal{O}_X)$ is pseudo-coherent relative to $S$. \end{enumerate}


more-morphisms-definition-pseudo-coherent

A morphism of schemes $f : X \to S$ is called {\it pseudo-coherent} if the equivalent conditions of Lemma \ref{lemma-pseudo-coherent} are satisfied. In this case we also say that $X$ is pseudo-coherent over $S$.


more-morphisms-definition-perfect

A morphism of schemes $f : X \to S$ is called {\it perfect} if the equivalent conditions of Lemma \ref{lemma-perfect} are satisfied. In this case we also say that $X$ is perfect over $S$.


more-morphisms-definition-lci

Let $f : X \to S$ be a morphism of schemes. \begin{enumerate} \item Let $x \in X$. We say that $f$ is {\it Koszul at $x$} if $f$ is of finite type at $x$ and there exists an open neighbourhood and a factorization of $f|_U$ as $\pi \circ i$ where $i : U \to P$ is a Koszul-regular immersion and $\pi : P \to S$ is smooth. \item We say $f$ is a {\it Koszul morphism}, or that $f$ is a {\it local complete intersection morphism} if $f$ is Koszul at every point. \end{enumerate}


more-morphisms-definition-weakly-etale

A morphism of schemes $X \to Y$ is {\it weakly \'etale} or {\it absolutely flat} if both $X \to Y$ and the diagonal morphism $X \to X \times_Y X$ are flat.


more-morphisms-definition-ind-quasi-affine

A scheme $X$ is {\it ind-quasi-affine} if every quasi-compact open of $X$ is quasi-affine. Similarly, a morphism of schemes $X \to Y$ is {\it ind-quasi-affine} if $f^{-1}(V)$ is ind-quasi-affine for each affine open $V$ in $Y$.


more-morphisms-definition-affine-stratification

Let $X$ be a scheme. An {\it affine stratification} is a locally finite stratification $X = \coprod_{i \in I} X_i$ whose strata $X_i$ are affine and such that the inclusion morphisms $X_i \to X$ are affine.


more-morphisms-definition-affine-stratification-number

Let $X$ be a nonempty quasi-compact and quasi-separated scheme. The {\it affine stratification number} is the smallest integer $n \geq 0$ such that the following equivalent conditions are satisfied \begin{enumerate} \item there exists a finite affine stratification $X = \coprod_{i \in I} X_i$ where $I$ has length $n$, \item there exists an affine stratification $X = X_0 \amalg X_1 \amalg \ldots \amalg X_n$ with index set $\{0, \ldots, n\}$. \end{enumerate}


more-morphisms-definition-weighting

Let $f : X \to Y$ be a locally quasi-finite morphism. A {\it weighting} or a {\it pond\'eration} of $f$ is a map $w : X \to \mathbf{Z}$ such that for any diagram $$ \xymatrix{ X \ar[d]_f & U \ar[l]^h \ar[d]^\pi \\ Y & V \ar[l]_g } $$ where $V \to Y$ is \'etale, $U \subset X_V$ is open, and $U \to V$ finite, the function $\int_\pi (w \circ h)$ is locally constant.


sites-modules-definition-free-abelian-presheaf-on

Let $\mathcal{C}$ be a category. Let $\mathcal{G}$ be a presheaf of sets. The {\it free abelian presheaf} $\mathbf{Z}_\mathcal{G}$ on $\mathcal{G}$ is the abelian presheaf defined by the rule $$ U \longmapsto \mathbf{Z}[\mathcal{G}(U)]. $$ In the special case $\mathcal{G} = h_X$ of a representable presheaf associated to an object $X$ of $\mathcal{C}$ we use the notation $\mathbf{Z}_X = \mathbf{Z}_{h_X}$. In other words $$ \mathbf{Z}_X(U) = \mathbf{Z}[\Mor_\mathcal{C}(U, X)]. $$


sites-modules-definition-free-abelian-sheaf-on

Let $\mathcal{C}$ be a site. Let $\mathcal{G}$ be a presheaf of sets. The {\it free abelian sheaf} $\mathbf{Z}_\mathcal{G}^\#$ on $\mathcal{G}$ is the abelian sheaf $\mathbf{Z}_\mathcal{G}^\#$ which is the sheafification of the free abelian presheaf on $\mathcal{G}$. In the special case $\mathcal{G} = h_X$ of a representable presheaf associated to an object $X$ of $\mathcal{C}$ we use the notation $\mathbf{Z}_X^\#$.


sites-modules-definition-ringed-site

Ringed sites. \begin{enumerate} \item A {\it ringed site} is a pair $(\mathcal{C}, \mathcal{O})$ where $\mathcal{C}$ is a site and $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}$. The sheaf $\mathcal{O}$ is called the {\it structure sheaf} of the ringed site. \item Let $(\mathcal{C}, \mathcal{O})$, $(\mathcal{C}', \mathcal{O}')$ be ringed sites. A {\it morphism of ringed sites} $$ (f, f^\sharp) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{C}', \mathcal{O}') $$ is given by a morphism of sites $f : \mathcal{C} \to \mathcal{C}'$ (see Sites, Definition \ref{sites-definition-morphism-sites}) together with a map of sheaves of rings $f^\sharp : f^{-1}\mathcal{O}' \to \mathcal{O}$, which by adjunction is the same thing as a map of sheaves of rings $f^\sharp : \mathcal{O}' \to f_*\mathcal{O}$. \item Let $(f, f^\sharp) : (\mathcal{C}_1, \mathcal{O}_1) \to (\mathcal{C}_2, \mathcal{O}_2)$ and $(g, g^\sharp) : (\mathcal{C}_2, \mathcal{O}_2) \to (\mathcal{C}_3, \mathcal{O}_3)$ be morphisms of ringed sites. Then we define the {\it composition of morphisms of ringed sites} by the rule $$ (g, g^\sharp) \circ (f, f^\sharp) = (g \circ f, f^\sharp \circ g^\sharp). $$ Here we use composition of morphisms of sites defined in Sites, Definition \ref{sites-definition-composition-morphisms-sites} and $f^\sharp \circ g^\sharp$ indicates the morphism of sheaves of rings $$ \mathcal{O}_3 \xrightarrow{g^\sharp} g_*\mathcal{O}_2 \xrightarrow{g_*f^\sharp} g_*f_*\mathcal{O}_1 = (g \circ f)_*\mathcal{O}_1 $$ \end{enumerate}


sites-modules-definition-ringed-topos

Ringed topoi. \begin{enumerate} \item A {\it ringed topos} is a pair $(\Sh(\mathcal{C}), \mathcal{O})$ where $\mathcal{C}$ is a site and $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}$. The sheaf $\mathcal{O}$ is called the {\it structure sheaf} of the ringed topos. \item Let $(\Sh(\mathcal{C}), \mathcal{O})$, $(\Sh(\mathcal{C}'), \mathcal{O}')$ be ringed topoi. A {\it morphism of ringed topoi} $$ (f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}) \longrightarrow (\Sh(\mathcal{C}'), \mathcal{O}') $$ is given by a morphism of topoi $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{C}')$ (see Sites, Definition \ref{sites-definition-topos}) together with a map of sheaves of rings $f^\sharp : f^{-1}\mathcal{O}' \to \mathcal{O}$, which by adjunction is the same thing as a map of sheaves of rings $f^\sharp : \mathcal{O}' \to f_*\mathcal{O}$. \item Let $(f, f^\sharp) : (\Sh(\mathcal{C}_1), \mathcal{O}_1) \to (\Sh(\mathcal{C}_2), \mathcal{O}_2)$ and $(g, g^\sharp) : (\Sh(\mathcal{C}_2), \mathcal{O}_2) \to (\Sh(\mathcal{C}_3), \mathcal{O}_3)$ be morphisms of ringed topoi. Then we define the {\it composition of morphisms of ringed topoi} by the rule $$ (g, g^\sharp) \circ (f, f^\sharp) = (g \circ f, f^\sharp \circ g^\sharp). $$ Here we use composition of morphisms of topoi defined in Sites, Definition \ref{sites-definition-topos} and $f^\sharp \circ g^\sharp$ indicates the morphism of sheaves of rings $$ \mathcal{O}_3 \xrightarrow{g^\sharp} g_*\mathcal{O}_2 \xrightarrow{g_*f^\sharp} g_*f_*\mathcal{O}_1 = (g \circ f)_*\mathcal{O}_1 $$ \end{enumerate}


sites-modules-definition-2-morphism-ringed-topoi

Let $f, g : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be two morphisms of ringed topoi. A {\it 2-morphism from $f$ to $g$} is given by a transformation of functors $t : f_* \to g_*$ such that $$ \xymatrix{ & \mathcal{O}_\mathcal{D} \ar[ld]_{f^\sharp} \ar[rd]^{g^\sharp} \\ f_*\mathcal{O}_\mathcal{C} \ar[rr]^t & & g_*\mathcal{O}_\mathcal{C} } $$ is commutative.