etale-cohomology-definition-ringed-site

Let $\mathcal{C}$ be a {\it ringed site}, i.e., a site endowed with a sheaf of rings $\mathcal{O}$. A sheaf of $\mathcal{O}$-modules $\mathcal{F}$ on $\mathcal{C}$ is called {\it quasi-coherent} if for all $U \in \Ob(\mathcal{C})$ there exists a covering $\{U_i \to U\}_{i\in I}$ of $\mathcal{C}$ such that the restriction $\mathcal{F}|_{\mathcal{C}/U_i}$ is isomorphic to the cokernel of an $\mathcal{O}$-linear map of free $\mathcal{O}$-modules $$ \bigoplus\nolimits_{k \in K} \mathcal{O}|_{\mathcal{C}/U_i} \longrightarrow \bigoplus\nolimits_{l \in L} \mathcal{O}|_{\mathcal{C}/U_i}. $$ The direct sum over $K$ is the sheaf associated to the presheaf $V \mapsto \bigoplus_{k \in K} \mathcal{O}(V)$ and similarly for the other.


etale-cohomology-definition-cech-complex

Let $\mathcal{C}$ be a category, $\mathcal{U} = \{U_i \to U\}_{i \in I}$ a family of morphisms of $\mathcal{C}$ with fixed target, and $\mathcal{F} \in \textit{PAb}(\mathcal{C})$ an abelian presheaf. We define the {\it {\v C}ech complex} $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$ by $$ \prod_{i_0\in I} \mathcal{F}(U_{i_0}) \to \prod_{i_0, i_1\in I} \mathcal{F}(U_{i_0} \times_U U_{i_1}) \to \prod_{i_0, i_1, i_2 \in I} \mathcal{F}(U_{i_0} \times_U U_{i_1} \times_U U_{i_2}) \to \ldots $$ where the first term is in degree 0, and the maps are the usual ones. Again, it is essential to allow the case $i_0 = i_1$ etc. The {\it {\v C}ech cohomology groups} are defined by $$ \check{H}^p(\mathcal{U}, \mathcal{F}) = H^p(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})). $$


etale-cohomology-definition-free-abelian-presheaf

Let $\mathcal{C}$ be a category. Given a presheaf of sets $\mathcal{G}$, we define the {\it free abelian presheaf on $\mathcal{G}$}, denoted $\mathbf{Z}_\mathcal{G}$, by the rule $$ \mathbf{Z}_\mathcal{G}(U) = \mathbf{Z}[\mathcal{G}(U)] $$ for $U \in \Ob(\mathcal{C})$ with restriction maps induced by the restriction maps of $\mathcal{G}$. In the special case $\mathcal{G} = h_U$ we write simply $\mathbf{Z}_U = \mathbf{Z}_{h_U}$.


etale-cohomology-definition-tau-covering

(See Topologies, Definitions \ref{topologies-definition-fppf-covering}, \ref{topologies-definition-syntomic-covering}, \ref{topologies-definition-smooth-covering}, \ref{topologies-definition-etale-covering}, and \ref{topologies-definition-zariski-covering}.) Let $\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}$. A family of morphisms of schemes $\{f_i : T_i \to T\}_{i \in I}$ with fixed target is called a {\it $\tau$-covering} if and only if each $f_i$ is flat of finite presentation, syntomic, smooth, \'etale, resp.\ an open immersion, and we have $\bigcup f_i(T_i) = T$.


etale-cohomology-definition-standard-tau

(See Topologies, Definitions \ref{topologies-definition-standard-fppf}, \ref{topologies-definition-standard-syntomic}, \ref{topologies-definition-standard-smooth}, \ref{topologies-definition-standard-etale}, and \ref{topologies-definition-standard-Zariski}.) Let $\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}$. Let $T$ be an affine scheme. A {\it standard $\tau$-covering} of $T$ is a family $\{f_j : U_j \to T\}_{j = 1, \ldots, m}$ with each $U_j$ is affine, and each $f_j$ flat and of finite presentation, standard syntomic, standard smooth, \'etale, resp.\ the immersion of a standard principal open in $T$ and $T = \bigcup f_j(U_j)$.


etale-cohomology-definition-tau-site

Let $S$ be a scheme. Let $\tau \in \{fppf, syntomic, smooth, \etale, \linebreak[0] Zariski\}$. \begin{enumerate} \item A {\it big $\tau$-site of $S$} is any of the sites $(\Sch/S)_\tau$ constructed as explained above and in more detail in Topologies, Definitions \ref{topologies-definition-big-small-fppf}, \ref{topologies-definition-big-small-syntomic}, \ref{topologies-definition-big-small-smooth}, \ref{topologies-definition-big-small-etale}, and \ref{topologies-definition-big-small-Zariski}. \item If $\tau \in \{\etale, Zariski\}$, then the {\it small $\tau$-site of $S$} is the full subcategory $S_\tau$ of $(\Sch/S)_\tau$ whose objects are schemes $T$ over $S$ whose structure morphism $T \to S$ is \'etale, resp.\ an open immersion. A covering in $S_\tau$ is a covering $\{U_i \to U\}$ in $(\Sch/S)_\tau$ such that $U$ is an object of $S_\tau$. \end{enumerate}


etale-cohomology-definition-etale-topos

Let $S$ be a scheme. \begin{enumerate} \item The {\it \'etale topos}, or the {\it small \'etale topos} of $S$ is the category $\Sh(S_\etale)$ of sheaves of sets on the small \'etale site of $S$. \item The {\it Zariski topos}, or the {\it small Zariski topos} of $S$ is the category $\Sh(S_{Zar})$ of sheaves of sets on the small Zariski site of $S$. \item For $\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}$ a {\it big $\tau$-topos} is the category of sheaves of set on a big $\tau$-topos of $S$. \end{enumerate}


etale-cohomology-definition-additive-sheaf

On any of the sites $(\Sch/S)_\tau$ or $S_\tau$ of Section \ref{section-big-small}. \begin{enumerate} \item The sheaf $T \mapsto \Gamma(T, \mathcal{O}_T)$ is denoted $\mathcal{O}_S$, or $\mathbf{G}_a$, or $\mathbf{G}_{a, S}$ if we want to indicate the base scheme. \item Similarly, the sheaf $T \mapsto \Gamma(T, \mathcal{O}^*_T)$ is denoted $\mathcal{O}_S^*$, or $\mathbf{G}_m$, or $\mathbf{G}_{m, S}$ if we want to indicate the base scheme. \item The {\it constant sheaf} $\underline{\mathbf{Z}/n\mathbf{Z}}$ on any site is the sheafification of the constant presheaf $U \mapsto \mathbf{Z}/n\mathbf{Z}$. \end{enumerate}


etale-cohomology-definition-structure-sheaf

Let $S$ be a scheme. The {\it structure sheaf} of $S$ is the sheaf of rings $\mathcal{O}_S$ on any of the sites $S_{Zar}$, $S_\etale$, or $(\Sch/S)_\tau$ discussed above.


etale-cohomology-definition-etale-morphism

A morphism of schemes is {\it \'etale} if it is smooth of relative dimension 0.


etale-cohomology-definition-standard-etale

A ring map $A \to B$ is called {\it standard \'etale} if $B \cong \left(A[t]/(f)\right)_g$ with $f, g \in A[t]$, with $f$ monic, and $\text{d}f/\text{d}t$ invertible in $B$.


etale-cohomology-definition-etale-covering

An {\it \'etale covering} of a scheme $U$ is a family of morphisms of schemes $\{\varphi_i : U_i \to U\}_{i \in I}$ such that \begin{enumerate} \item each $\varphi_i$ is an \'etale morphism, \item the $U_i$ cover $U$, i.e., $U = \bigcup_{i\in I}\varphi_i(U_i)$. \end{enumerate}


etale-cohomology-definition-big-etale-site

(For more details see Section \ref{section-big-small}, or Topologies, Section \ref{topologies-section-etale}.) Let $S$ be a scheme. The {\it big \'etale site over $S$} is the site $(\Sch/S)_\etale$, see Definition \ref{definition-tau-site}. The {\it small \'etale site over $S$} is the site $S_\etale$, see Definition \ref{definition-tau-site}. We define similarly the {\it big} and {\it small Zariski sites} on $S$, denoted $(\Sch/S)_{Zar}$ and $S_{Zar}$.


etale-cohomology-definition-geometric-point

Let $S$ be a scheme. \begin{enumerate} \item A {\it geometric point} of $S$ is a morphism $\Spec(k) \to S$ where $k$ is algebraically closed. Such a point is usually denoted $\overline{s}$, i.e., by an overlined small case letter. We often use $\overline{s}$ to denote the scheme $\Spec(k)$ as well as the morphism, and we use $\kappa(\overline{s})$ to denote $k$. \item We say $\overline{s}$ {\it lies over} $s$ to indicate that $s \in S$ is the image of $\overline{s}$. \item An {\it \'etale neighborhood} of a geometric point $\overline{s}$ of $S$ is a commutative diagram $$ \xymatrix{ & U \ar[d]^\varphi \\ {\overline{s}} \ar[r]^{\overline{s}} \ar[ur]^{\bar u} & S } $$ where $\varphi$ is an \'etale morphism of schemes. We write $(U, \overline{u}) \to (S, \overline{s})$. \item A {\it morphism of \'etale neighborhoods} $(U, \overline{u}) \to (U', \overline{u}')$ is an $S$-morphism $h: U \to U'$ such that $\overline{u}' = h \circ \overline{u}$. \end{enumerate}


etale-cohomology-definition-stalk

Let $S$ be a scheme. Let $\mathcal{F}$ be a presheaf on $S_\etale$. Let $\overline{s}$ be a geometric point of $S$. The {\it stalk} of $\mathcal{F}$ at $\overline{s}$ is $$ \mathcal{F}_{\overline{s}} = \colim_{(U, \overline{u})} \mathcal{F}(U) $$ where $(U, \overline{u})$ runs over all \'etale neighborhoods of $\overline{s}$ in $S$.


etale-cohomology-definition-support

Let $S$ be a scheme. Let $\mathcal{F}$ be an abelian sheaf on $S_\etale$. \begin{enumerate} \item The {\it support of $\mathcal{F}$} is the set of points $s \in S$ such that $\mathcal{F}_{\overline{s}} \not = 0$ for any (some) geometric point $\overline{s}$ lying over $s$. \item Let $\sigma \in \mathcal{F}(U)$ be a section. The {\it support of $\sigma$} is the closed subset $U \setminus W$, where $W \subset U$ is the largest open subset of $U$ on which $\sigma$ restricts to zero (see Lemma \ref{lemma-zero-over-image}). \end{enumerate}


etale-cohomology-definition-henselian

(See Algebra, Definition \ref{algebra-definition-henselian}.) A local ring $(R, \mathfrak m, \kappa)$ is called {\it henselian} if for all $f \in R[T]$ monic, for all $a_0 \in \kappa$ such that $\bar f(a_0) = 0$ and $\bar f'(a_0) \neq 0$, there exists an $a \in R$ such that $f(a) = 0$ and $a \bmod \mathfrak m = a_0$.


etale-cohomology-definition-strictly-henselian

A local ring $R$ is called {\it strictly henselian} if it is henselian and its residue field is separably closed.


etale-cohomology-definition-etale-local-rings

Let $S$ be a scheme. Let $\overline{s}$ be a geometric point of $S$ lying over the point $s \in S$. \begin{enumerate} \item The {\it \'etale local ring of $S$ at $\overline{s}$} is the stalk of the structure sheaf $\mathcal{O}_S$ on $S_\etale$ at $\overline{s}$. We sometimes call this the {\it strict henselization of $\mathcal{O}_{S, s}$} relative to the geometric point $\overline{s}$. Notation used: $\mathcal{O}_{S, \overline{s}} = \mathcal{O}_{S, s}^{sh}$. \item The {\it henselization of $\mathcal{O}_{S, s}$} is the henselization of the local ring of $S$ at $s$. See Algebra, Definition \ref{algebra-definition-henselization}, and Theorem \ref{theorem-henselization}. Notation: $\mathcal{O}_{S, s}^h$. \item The {\it strict henselization of $S$ at $\overline{s}$} is the scheme $\Spec(\mathcal{O}_{S, s}^{sh})$. \item The {\it henselization of $S$ at $s$} is the scheme $\Spec(\mathcal{O}_{S, s}^h)$. \end{enumerate}


etale-cohomology-definition-direct-image-presheaf

Let $f: X\to Y$ be a morphism of schemes. Let $\mathcal{F} $ a presheaf of sets on $X_\etale$. The {\it direct image}, or {\it pushforward} of $\mathcal{F}$ (under $f$) is $$ f_*\mathcal{F} : Y_\etale^{opp} \longrightarrow \textit{Sets}, \quad (V/Y) \longmapsto \mathcal{F}(X \times_Y V/X). $$ We sometimes write $f_* = f_{small, *}$ to distinguish from other direct image functors (such as usual Zariski pushforward or $f_{big, *}$).


etale-cohomology-definition-direct-image-sheaf

Let $f: X\to Y$ be a morphism of schemes. Let $\mathcal{F} $ a sheaf of sets on $X_\etale$. The {\it direct image}, or {\it pushforward} of $\mathcal{F}$ (under $f$) is $$ f_*\mathcal{F} : Y_\etale^{opp} \longrightarrow \textit{Sets}, \quad (V/Y) \longmapsto \mathcal{F}(X \times_Y V/X) $$ which is a sheaf by Remark \ref{remark-direct-image-sheaf}. We sometimes write $f_* = f_{small, *}$ to distinguish from other direct image functors (such as usual Zariski pushforward or $f_{big, *}$).


etale-cohomology-definition-higher-direct-images

Let $f: X \to Y$ be a morphism of schemes. The right derived functors $\{R^pf_*\}_{p \geq 1}$ of $f_* : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)$ are called {\it higher direct images}.


etale-cohomology-definition-inverse-image

Let $f: X\to Y$ be a morphism of schemes. The {\it inverse image}, or {\it pullback}\footnote{We use the notation $f^{-1}$ for pullbacks of sheaves of sets or sheaves of abelian groups, and we reserve $f^*$ for pullbacks of sheaves of modules via a morphism of ringed sites/topoi.} functors are the functors $$ f^{-1} = f_{small}^{-1} : \Sh(Y_\etale) \longrightarrow \Sh(X_\etale) $$ and $$ f^{-1} = f_{small}^{-1} : \textit{Ab}(Y_\etale) \longrightarrow \textit{Ab}(X_\etale) $$ which are left adjoint to $f_* = f_{small, *}$. Thus $f^{-1}$ thus characterized by the fact that $$ \Hom_{{\Sh(X_\etale)}} (f^{-1}\mathcal{G}, \mathcal{F}) = \Hom_{\Sh(Y_\etale)} (\mathcal{G}, f_*\mathcal{F}) $$ functorially, for any $\mathcal{F} \in \Sh(X_\etale)$ and $\mathcal{G} \in \Sh(Y_\etale)$. We similarly have $$ \Hom_{{\textit{Ab}(X_\etale)}} (f^{-1}\mathcal{G}, \mathcal{F}) = \Hom_{\textit{Ab}(Y_\etale)} (\mathcal{G}, f_*\mathcal{F}) $$ for $\mathcal{F} \in \textit{Ab}(X_\etale)$ and $\mathcal{G} \in \textit{Ab}(Y_\etale)$.


etale-cohomology-definition-inverse-system-sheaves

Let $I$ be a preordered set. Let $(X_i, f_{i'i})$ be an inverse system of schemes over $I$. A {\it system $(\mathcal{F}_i, \varphi_{i'i})$ of sheaves on $(X_i, f_{i'i})$} is given by \begin{enumerate} \item a sheaf $\mathcal{F}_i$ on $(X_i)_\etale$ for all $i \in I$, \item for $i' \geq i$ a map $\varphi_{i'i} : f_{i'i}^{-1}\mathcal{F}_i \to \mathcal{F}_{i'}$ of sheaves on $(X_{i'})_\etale$ \end{enumerate} such that $\varphi_{i''i} = \varphi_{i''i'} \circ f_{i'' i'}^{-1}\varphi_{i'i}$ whenever $i'' \geq i' \geq i$.


etale-cohomology-definition-algebraic-geometric-point

Let $S$ be a scheme. Let $\overline{s}$ be a geometric point lying over the point $s$ of $S$. Let $\kappa(s) \subset \kappa(s)^{sep} \subset \kappa(\overline{s})$ denote the separable algebraic closure of $\kappa(s)$ in the algebraically closed field $\kappa(\overline{s})$. \begin{enumerate} \item In this situation the {\it absolute Galois group} of $\kappa(s)$ is $\text{Gal}(\kappa(s)^{sep}/\kappa(s))$. It is sometimes denoted $\text{Gal}_{\kappa(s)}$. \item The geometric point $\overline{s}$ is called {\it algebraic} if $\kappa(s) \subset \kappa(\overline{s})$ is an algebraic closure of $\kappa(s)$. \end{enumerate}


etale-cohomology-definition-G-module-continuous

Let $G$ be a topological group. \begin{enumerate} \item A {\it $G$-module}, sometimes called a {\it discrete $G$-module}, is an abelian group $M$ endowed with a left action $a : G \times M \to M$ by group homomorphisms such that $a$ is continuous when $M$ is given the discrete topology. \item A {\it morphism of $G$-modules} $f : M \to N$ is a $G$-equivariant homomorphism from $M$ to $N$. \item The category of $G$-modules is denoted $\text{Mod}_G$. \end{enumerate} Let $R$ be a ring. \begin{enumerate} \item An {\it $R\text{-}G$-module} is an $R$-module $M$ endowed with a left action $a : G \times M \to M$ by $R$-linear maps such that $a$ is continuous when $M$ is given the discrete topology. \item A {\it morphism of $R\text{-}G$-modules} $f : M \to N$ is a $G$-equivariant $R$-module map from $M$ to $N$. \item The category of $R\text{-}G$-modules is denoted $\text{Mod}_{R, G}$. \end{enumerate}


etale-cohomology-definition-galois-cohomology

Let $G$ be a topological group. Let $M$ be a discrete $G$-module with continuous $G$-action. In other words, $M$ is an object of the category $\text{Mod}_G$ introduced in Definition \ref{definition-G-module-continuous}. \begin{enumerate} \item The right derived functors $H^i(G, M)$ of $H^0(G, M)$ on the category $\text{Mod}_G$ are called the {\it continuous group cohomology groups} of $M$. \item If $G$ is an abstract group endowed with the discrete topology then the $H^i(G, M)$ are called the {\it group cohomology groups} of $M$. \item If $G$ is a Galois group, then the groups $H^i(G, M)$ are called the {\it Galois cohomology groups} of $M$. \item If $G$ is the absolute Galois group of a field $K$, then the groups $H^i(G, M)$ are sometimes called the {\it Galois cohomology groups of $K$ with coefficients in $M$}. In this case we sometimes write $H^i(K, M)$ instead of $H^i(G, M)$. \end{enumerate}


etale-cohomology-definition-brauer-equivalent

Two finite central simple algebras $A_1$ and $A_2$ over $K$ are called {\it similar}, or {\it equivalent} if there exist $m, n \geq 1$ such that $\text{Mat}(n \times n, A_1) \cong \text{Mat}(m \times m, A_2)$. We write $A_1 \sim A_2$.


etale-cohomology-definition-brauer-group

Let $K$ be a field. The {\it Brauer group} of $K$ is the set $\text{Br} (K)$ of similarity classes of finite central simple algebras over $K$, endowed with the group law induced by tensor product (over $K$). The class of $A$ in $\text{Br}(K)$ is denoted by $[A]$. The neutral element is $[K] = [\text{Mat}(d \times d, K)]$ for any $d \geq 1$.


etale-cohomology-definition-finite-locally-constant

Let $X$ be a scheme. Let $\mathcal{F}$ be a sheaf of sets on $X_\etale$. \begin{enumerate} \item Let $E$ be a set. We say $\mathcal{F}$ is the {\it constant sheaf with value $E$} if $\mathcal{F}$ is the sheafification of the presheaf $U \mapsto E$. Notation: $\underline{E}_X$ or $\underline{E}$. \item We say $\mathcal{F}$ is a {\it constant sheaf} if it is isomorphic to a sheaf as in (1). \item We say $\mathcal{F}$ is {\it locally constant} if there exists a covering $\{U_i \to X\}$ such that $\mathcal{F}|_{U_i}$ is a constant sheaf. \item We say that $\mathcal{F}$ is {\it finite locally constant} if it is locally constant and the values are finite sets. \end{enumerate} Let $\mathcal{F}$ be a sheaf of abelian groups on $X_\etale$. \begin{enumerate} \item Let $A$ be an abelian group. We say $\mathcal{F}$ is the {\it constant sheaf with value $A$} if $\mathcal{F}$ is the sheafification of the presheaf $U \mapsto A$. Notation: $\underline{A}_X$ or $\underline{A}$. \item We say $\mathcal{F}$ is a {\it constant sheaf} if it is isomorphic as an abelian sheaf to a sheaf as in (1). \item We say $\mathcal{F}$ is {\it locally constant} if there exists a covering $\{U_i \to X\}$ such that $\mathcal{F}|_{U_i}$ is a constant sheaf. \item We say that $\mathcal{F}$ is {\it finite locally constant} if it is locally constant and the values are finite abelian groups. \end{enumerate} Let $\Lambda$ be a ring. Let $\mathcal{F}$ be a sheaf of $\Lambda$-modules on $X_\etale$. \begin{enumerate} \item Let $M$ be a $\Lambda$-module. We say $\mathcal{F}$ is the {\it constant sheaf with value $M$} if $\mathcal{F}$ is the sheafification of the presheaf $U \mapsto M$. Notation: $\underline{M}_X$ or $\underline{M}$. \item We say $\mathcal{F}$ is a {\it constant sheaf} if it is isomorphic as a sheaf of $\Lambda$-modules to a sheaf as in (1). \item We say $\mathcal{F}$ is {\it locally constant} if there exists a covering $\{U_i \to X\}$ such that $\mathcal{F}|_{U_i}$ is a constant sheaf. \end{enumerate}


etale-cohomology-definition-trace-map

Let $f : Y \to X$ be a finite \'etale morphism of schemes. The map $f_* f^{-1} \to \text{id}$ described above and explicitly below is called the {\it trace}.


etale-cohomology-definition-Cr

A field $K$ is called {\it $C_r$} if for every $0 < d^r < n$ and every $f \in K[T_1, \ldots, T_n]$ homogeneous of degree $d$, there exist $\alpha = (\alpha_1, \ldots, \alpha_n)$, $\alpha_i \in K$ not all zero, such that $f(\alpha) = 0$. Such an $\alpha$ is called a {\it nontrivial solution} of $f$.


etale-cohomology-definition-variety

Let $k$ be a field. A {\it variety} is separated, integral scheme of finite type over $k$. A {\it curve} is a variety of dimension $1$.


etale-cohomology-definition-extension-zero

Let $j : U \to X$ be an \'etale morphism of schemes. \begin{enumerate} \item The restriction functor $j^{-1} : \Sh(X_\etale) \to \Sh(U_\etale)$ has a left adjoint $j_!^{Sh} : \Sh(U_\etale) \to \Sh(X_\etale)$. \item The restriction functor $j^{-1} : \textit{Ab}(X_\etale) \to \textit{Ab}(U_\etale)$ has a left adjoint which is denoted $j_! : \textit{Ab}(U_\etale) \to \textit{Ab}(X_\etale)$ and called {\it extension by zero}. \item Let $\Lambda$ be a ring. The restriction functor $j^{-1} : \textit{Mod}(X_\etale, \Lambda) \to \textit{Mod}(U_\etale, \Lambda)$ has a left adjoint which is denoted $j_! : \textit{Mod}(U_\etale, \Lambda) \to \textit{Mod}(X_\etale, \Lambda)$ and called {\it extension by zero}. \end{enumerate}


etale-cohomology-definition-constructible

Let $X$ be a scheme. \begin{enumerate} \item A sheaf of sets on $X_\etale$ is {\it constructible} if for every affine open $U \subset X$ there exists a finite decomposition of $U$ into constructible locally closed subschemes $U = \coprod_i U_i$ such that $\mathcal{F}|_{U_i}$ is finite locally constant for all $i$. \item A sheaf of abelian groups on $X_\etale$ is {\it constructible} if for every affine open $U \subset X$ there exists a finite decomposition of $U$ into constructible locally closed subschemes $U = \coprod_i U_i$ such that $\mathcal{F}|_{U_i}$ is finite locally constant for all $i$. \item Let $\Lambda$ be a Noetherian ring. A sheaf of $\Lambda$-modules on $X_\etale$ is {\it constructible} if for every affine open $U \subset X$ there exists a finite decomposition of $U$ into constructible locally closed subschemes $U = \coprod_i U_i$ such that $\mathcal{F}|_{U_i}$ is of finite type and locally constant for all $i$. \end{enumerate}


etale-cohomology-definition-c

Let $X$ be a scheme. Let $\Lambda$ be a Noetherian ring. We denote {\it $D_c(X_\etale, \Lambda)$} the full subcategory of $D(X_\etale, \Lambda)$ of complexes whose cohomology sheaves are constructible sheaves of $\Lambda$-modules.


etale-cohomology-definition-ctf

Let $X$ be a scheme. Let $\Lambda$ be a Noetherian ring. We denote {\it $D_{ctf}(X_\etale, \Lambda)$} the full subcategory of $D_c(X_\etale, \Lambda)$ consisting of objects having locally finite tor dimension.


etale-cohomology-definition-cd

Let $X$ be a quasi-compact and quasi-separated scheme. The {\it cohomological dimension of $X$} is the smallest element $$ \text{cd}(X) \in \{0, 1, 2, \ldots\} \cup \{\infty\} $$ such that for any abelian torsion sheaf $\mathcal{F}$ on $X_\etale$ we have $H^i_\etale(X, \mathcal{F}) = 0$ for $i > \text{cd}(X)$. If $X = \Spec(A)$ we sometimes call this the cohomological dimension of $A$.


etale-cohomology-definition-cd-f

Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of schemes. The {\it cohomological dimension of $f$} is the smallest element $$ \text{cd}(f) \in \{0, 1, 2, \ldots\} \cup \{\infty\} $$ such that for any abelian torsion sheaf $\mathcal{F}$ on $X_\etale$ we have $R^if_*\mathcal{F} = 0$ for $i > \text{cd}(f)$.


equiv-definition-Serre-functor

Let $k$ be a field. Let $\mathcal{T}$ be a $k$-linear triangulated category such that $\dim_k \Hom_\mathcal{T}(X, Y) < \infty$ for all $X, Y \in \Ob(\mathcal{T})$. We say {\it a Serre functor exists} if the equivalent conditions of Lemma \ref{lemma-Serre-functor-exists} are satisfied. In this case a {\it Serre functor} is a $k$-linear equivalence $S : \mathcal{T} \to \mathcal{T}$ endowed with $k$-linear isomorphisms $c_{X, Y} : \Hom_\mathcal{T}(X, Y) \to \Hom_\mathcal{T}(Y, S(X))^\vee$ functorial in $X, Y \in \Ob(\mathcal{T})$.


equiv-definition-fourier-mukai-functor

Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$. Let $K \in D(\mathcal{O}_{X \times_S Y})$. The exact functor $$ \Phi_K : D(\mathcal{O}_X) \longrightarrow D(\mathcal{O}_Y),\quad M \longmapsto R\text{pr}_{2, *}( L\text{pr}_1^*M \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K) $$ of triangulated categories is called a {\it Fourier-Mukai functor} and $K$ is called a {\it Fourier-Mukai kernel} for this functor. Moreover, \begin{enumerate} \item if $\Phi_K$ sends $D_\QCoh(\mathcal{O}_X)$ into $D_\QCoh(\mathcal{O}_Y)$ then the resulting exact functor $\Phi_K : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ is called a Fourier-Mukai functor, \item if $\Phi_K$ sends $D_{perf}(\mathcal{O}_X)$ into $D_{perf}(\mathcal{O}_Y)$ then the resulting exact functor $\Phi_K : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ is called a Fourier-Mukai functor, and \item if $X$ and $Y$ are Noetherian and $\Phi_K$ sends $D^b_{\textit{Coh}}(\mathcal{O}_X)$ into $D^b_{\textit{Coh}}(\mathcal{O}_Y)$ then the resulting exact functor $\Phi_K : D^b_{\textit{Coh}}(\mathcal{O}_X) \to D^b_{\textit{Coh}}(\mathcal{O}_Y)$ is called a Fourier-Mukai functor. Similarly for $D_{\textit{Coh}}$, $D^+_{\textit{Coh}}$, $D^-_{\textit{Coh}}$. \end{enumerate}


equiv-definition-siblings

Let $\mathcal{A}$ be an abelian category. Let $\mathcal{D}$ be a triangulated category. We say two exact functors of triangulated categories $$ F, F' : D^b(\mathcal{A}) \longrightarrow \mathcal{D} $$ are {\it siblings}, or we say $F'$ is a {\it sibling} of $F$, if the following two conditions are satisfied \begin{enumerate} \item the functors $F \circ i$ and $F' \circ i$ are isomorphic where $i : \mathcal{A} \to D^b(\mathcal{A})$ is the inclusion functor, and \item $F(K) \cong F'(K)$ for any $K$ in $D^b(\mathcal{A})$. \end{enumerate}


equiv-definition-siblings-geometric

Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$. Recall that $D^b_{\textit{Coh}}(\mathcal{O}_X) = D^b(\textit{Coh}(\mathcal{O}_X))$ by Derived Categories of Schemes, Proposition \ref{perfect-proposition-DCoh}. We say two $k$-linear exact functors $$ F, F' : D^b_{\textit{Coh}}(\mathcal{O}_X) = D^b(\textit{Coh}(\mathcal{O}_X)) \longrightarrow D^b_{\textit{Coh}}(\mathcal{O}_Y) $$ are {\it siblings}, or we say $F'$ is a {\it sibling} of $F$ if $F$ and $F'$ are siblings in the sense of Definition \ref{definition-siblings} with abelian category being $\textit{Coh}(\mathcal{O}_X)$. If $X$ is regular then $D_{perf}(\mathcal{O}_X) = D^b_{\textit{Coh}}(\mathcal{O}_X)$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-perfect-on-noetherian} and we use the same terminology for $k$-linear exact functors $F, F' : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$.


equiv-definition-relative-equivalence-kernel

Let $S$ be a scheme. Let $X \to S$ and $Y \to S$ be smooth proper morphisms. An object $K \in D_{perf}(\mathcal{O}_{X \times_S Y})$ is said to be {\it the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$} if there exist an object $K' \in D_{perf}(\mathcal{O}_{X \times_S Y})$ such that $$ \Delta_{X/S, *}\mathcal{O}_X \cong R\text{pr}_{13, *}(L\text{pr}_{12}^*K \otimes_{\mathcal{O}_{X \times_S Y \times_S X}}^\mathbf{L} L\text{pr}_{23}^*K') $$ in $D(\mathcal{O}_{X \times_S X})$ and $$ \Delta_{Y/S, *}\mathcal{O}_Y \cong R\text{pr}_{13, *}(L\text{pr}_{12}^*K' \otimes_{\mathcal{O}_{Y \times_S X \times_S Y}}^\mathbf{L} L\text{pr}_{23}^*K) $$ in $D(\mathcal{O}_{Y \times_S Y})$. In other words, the isomorphism class of $K$ defines an invertible arrow in the category defined in Section \ref{section-category-Fourier-Mukai-kernels}.


equiv-definition-derived-equivalent

Let $k$ be a field. Let $X$ and $Y$ be smooth projective schemes over $k$. We say $X$ and $Y$ are {\it derived equivalent} if there exists a $k$-linear exact equivalence $D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$.


stacks-more-morphisms-definition-thickening

Thickenings. \begin{enumerate} \item We say an algebraic stack $\mathcal{X}'$ is a {\it thickening} of an algebraic stack $\mathcal{X}$ if $\mathcal{X}$ is a closed substack of $\mathcal{X}'$ and the associated topological spaces are equal. \item Given two thickenings $\mathcal{X} \subset \mathcal{X}'$ and $\mathcal{Y} \subset \mathcal{Y}'$ a {\it morphism of thickenings} is a morphism $f' : \mathcal{X}' \to \mathcal{Y}'$ of algebraic stacks such that $f'|_\mathcal{X}$ factors through the closed substack $\mathcal{Y}$. In this situation we set $f = f'|_\mathcal{X} : \mathcal{X} \to \mathcal{Y}$ and we say that $(f, f') : (\mathcal{X} \subset \mathcal{X}') \to (\mathcal{Y} \subset \mathcal{Y}')$ is a morphism of thickenings. \item Let $\mathcal{Z}$ be an algebraic stack. We similarly define {\it thickenings over $\mathcal{Z}$} and {\it morphisms of thickenings over $\mathcal{Z}$}. This means that the algebraic stacks $\mathcal{X}'$ and $\mathcal{Y}'$ are algebraic stack endowed with a structure morphism to $\mathcal{Z}$ and that $f'$ fits into a suitable $2$-commutative diagram of algebraic stacks. \end{enumerate}


stacks-more-morphisms-definition-first-order-thickening

We say an algebraic stack $\mathcal{X}'$ is a {\it first order thickening} of an algebraic stack $\mathcal{X}$ if $\mathcal{X}$ is a closed substack of $\mathcal{X}'$ and $\mathcal{X} \to \mathcal{X}'$ is a first order thickening in the sense of Properties of Stacks, Section \ref{stacks-properties-section-properties-morphisms}.


stacks-more-morphisms-definition-formally-smooth

A morphism $f : \mathcal{X} \to \mathcal{Y}$ of algebraic stacks is said to be {\it formally smooth} if it is formally smooth on objects as a $1$-morphism in categories fibred in groupoids as explained in Criteria for Representability, Section \ref{criteria-section-formally-smooth}.


stacks-more-morphisms-definition-categorical-quotient

Let $\mathcal{X}$ be an algebraic stack. Let $f : \mathcal{X} \to Y$ be a morphism to an algebraic space $Y$. \begin{enumerate} \item We say $f$ is a {\it categorical moduli space} if any morphism $\mathcal{X} \to W$ to an algebraic space $W$ factors uniquely through $f$. \item We say $f$ is a {\it uniform categorical moduli space} if for any flat morphism $Y' \to Y$ of algebraic spaces the base change $f' : Y' \times_Y \mathcal{X} \to Y'$ is a categorical moduli space. \end{enumerate} Let $\mathcal{C}$ be a full subcategory of the category of algebraic spaces. \begin{enumerate} \item[(3)] We say $f$ is a {\it categorical moduli space in $\mathcal{C}$} if $Y \in \Ob(\mathcal{C})$ and any morphism $\mathcal{X} \to W$ with $W \in \Ob(\mathcal{C})$ factors uniquely through $f$. \item[(4)] We say is a {\it uniform categorical moduli space in $\mathcal{C}$} if $Y \in \Ob(\mathcal{C})$ and for every flat morphism $Y' \to Y$ in $\mathcal{C}$ the base change $f' : Y' \times_Y \mathcal{X} \to Y'$ is a categorical moduli space in $\mathcal{C}$. \end{enumerate}


stacks-more-morphisms-definition-well-nigh-affine

Let $\mathcal{X}$ be an algebraic stack. We say $\mathcal{X}$ is {\it well-nigh affine} if there exists an affine scheme $U$ and a surjective, flat, finite, and finitely presented morphism $U \to \mathcal{X}$.


perfect-definition-supported-on

Let $X$ be a scheme. Let $E$ be an object of $D(\mathcal{O}_X)$. Let $T \subset X$ be a closed subset. We say $E$ is {\it supported on $T$} if the cohomology sheaves $H^i(E)$ are supported on $T$.


perfect-definition-approximation-holds

Let $X$ be a scheme. Consider triples $(T, E, m)$ where \begin{enumerate} \item $T \subset X$ is a closed subset, \item $E$ is an object of $D_\QCoh(\mathcal{O}_X)$, and \item $m \in \mathbf{Z}$. \end{enumerate} We say {\it approximation holds for the triple} $(T, E, m)$ if there exists a perfect object $P$ of $D(\mathcal{O}_X)$ supported on $T$ and a map $\alpha : P \to E$ which induces isomorphisms $H^i(P) \to H^i(E)$ for $i > m$ and a surjection $H^m(P) \to H^m(E)$.


perfect-definition-approximation

Let $X$ be a scheme. We say {\it approximation by perfect complexes holds} on $X$ if for any closed subset $T \subset X$ with $X \setminus T$ retro-compact in $X$ there exists an integer $r$ such that for every triple $(T, E, m)$ as in Definition \ref{definition-approximation-holds} with \begin{enumerate} \item $E$ is $(m - r)$-pseudo-coherent, and \item $H^i(E)$ is supported on $T$ for $i \geq m - r$ \end{enumerate} approximation holds.


perfect-definition-tor-independent

Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$. We say $X$ and $Y$ are {\it Tor independent over $S$} if for every $x \in X$ and $y \in Y$ mapping to the same point $s \in S$ the rings $\mathcal{O}_{X, x}$ and $\mathcal{O}_{Y, y}$ are Tor independent over $\mathcal{O}_{S, s}$ (see More on Algebra, Definition \ref{more-algebra-definition-tor-independent}).


perfect-definition-relatively-perfect

Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. An object $E$ of $D(\mathcal{O}_X)$ is {\it perfect relative to $S$} or {\it $S$-perfect} if $E$ is pseudo-coherent (Cohomology, Definition \ref{cohomology-definition-pseudo-coherent}) and $E$ locally has finite tor dimension as an object of $D(f^{-1}\mathcal{O}_S)$ (Cohomology, Definition \ref{cohomology-definition-tor-amplitude}).


perfect-definition-resolution-property

Let $X$ be a scheme. We say $X$ has the {\it resolution property} if every quasi-coherent $\mathcal{O}_X$-module of finite type is the quotient of a finite locally free $\mathcal{O}_X$-module.


perfect-definition-K-group

Let $X$ be a scheme. \begin{enumerate} \item We denote $K_0(X)$ the {\it Grothendieck group of $X$}. It is the zeroth K-group of the strictly full, saturated, triangulated subcategory $D_{perf}(\mathcal{O}_X)$ of $D(\mathcal{O}_X)$ consisting of perfect objects. In a formula $$ K_0(X) = K_0(D_{perf}(\mathcal{O}_X)) $$ \item If $X$ is locally Noetherian, then we denote $K'_0(X)$ the {\it Grothendieck group of coherent sheaves on $X$}. It is the is the zeroth $K$-group of the abelian category of coherent $\mathcal{O}_X$-modules. In a formula $$ K'_0(X) = K_0(\textit{Coh}(\mathcal{O}_X)) $$ \end{enumerate}


spaces-flat-definition-impurity

In Situation \ref{situation-pre-pure} we say a diagram (\ref{equation-impurity}) defines an {\it impurity of $\mathcal{F}$ above $y$} if $\xi \in \text{Ass}_{X_T/T}(\mathcal{F}_T)$ and $t \not \in f_T(\overline{\{\xi\}})$. We will indicate this by saying ``let $(g : T \to Y, t' \leadsto t, \xi)$ be an impurity of $\mathcal{F}$ above $y$''.


spaces-flat-definition-pure

In Situation \ref{situation-pre-pure}. \begin{enumerate} \item We say $\mathcal{F}$ is {\it pure above $y$} if {\bf none} of the equivalent conditions of Lemma \ref{lemma-pure-along-X-y} hold. \item We say $\mathcal{F}$ is {\it universally pure above $y$} if there does not exist any impurity of $\mathcal{F}$ above $y$. \item We say that $X$ is {\it pure above $y$} if $\mathcal{O}_X$ is pure above $y$. \item We say $\mathcal{F}$ is {\it universally $Y$-pure}, or {\it universally pure relative to $Y$} if $\mathcal{F}$ is universally pure above $y$ for every $y \in |Y|$. \item We say $\mathcal{F}$ is {\it $Y$-pure}, or {\it pure relative to $Y$} if $\mathcal{F}$ is pure above $y$ for every $y \in |Y|$. \item We say that $X$ is {\it $Y$-pure} or {\it pure relative to $Y$} if $\mathcal{O}_X$ is pure relative to $Y$. \end{enumerate}


spaces-flat-definition-flattening

Let $S$ be a scheme. Let $X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. We say that the {\it universal flattening of $\mathcal{F}$ exists} if the functor $F_{flat}$ defined in Situation \ref{situation-flat} is an algebraic space. We say that the {\it universal flattening of $X$ exists} if the universal flattening of $\mathcal{O}_X$ exists.


spaces-flat-definition-flat-dimension-n

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. Let $n \geq 0$. We say {\it $\mathcal{F}$ is flat over $Y$ in dimensions $\geq n$} if the equivalent conditions of Lemma \ref{lemma-pre-flat-dimension-n} are satisfied.


spaces-chow-definition-delta-dimension

In Situation \ref{situation-setup} for any good $X/B$ and any irreducible closed subset $T \subset |X|$ we define $$ \dim_\delta(T) = \delta(\xi) $$ where $\xi \in T$ is the generic point of $T$. We will call this the {\it $\delta$-dimension of $T$}. If $T \subset |X|$ is any closed subset, then we define $\dim_\delta(T)$ as the supremum of the $\delta$-dimensions of the irreducible components of $T$. If $Z$ is a closed subspace of $X$, then we set $\dim_\delta(Z) = \dim_\delta(|Z|)$.


spaces-chow-definition-cycles

In Situation \ref{situation-setup} let $X/B$ be good. Let $k \in \mathbf{Z}$. \begin{enumerate} \item A {\it cycle on $X$} is a formal sum $$ \alpha = \sum n_Z [Z] $$ where the sum is over integral closed subspaces $Z \subset X$, each $n_Z \in \mathbf{Z}$, and $\{|Z|; n_Z \not = 0\}$ is a locally finite collection of subsets of $|X|$ (Topology, Definition \ref{topology-definition-locally-finite}). \item A {\it $k$-cycle} on $X$ is a cycle $$ \alpha = \sum n_Z [Z] $$ where $n_Z \not = 0 \Rightarrow \dim_\delta(Z) = k$. \item The abelian group of all $k$-cycles on $X$ is denoted $Z_k(X)$. \end{enumerate}


spaces-chow-definition-length-at-x

Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $x \in |X|$. Let $d \in \{0, 1, 2, \ldots, \infty\}$. We say {\it $\mathcal{F}$ has length $d$ at $x$} if the equivalent conditions of Lemma \ref{lemma-length} are satisfied.


spaces-chow-definition-cycle-associated-to-closed-subscheme

In Situation \ref{situation-setup} let $X/B$ be good. Let $Y \subset X$ be a closed subspace. \begin{enumerate} \item For an irreducible component $Z \subset Y$ with generic point $\xi$ the length of $\mathcal{O}_Y$ at $\xi$ (Definition \ref{definition-length-at-x}) is called the {\it multiplicity of $Z$ in $Y$}. By Lemma \ref{lemma-length-finite} applied to $\mathcal{O}_Y$ on $Y$ this is a positive integer. \item Assume $\dim_\delta(Y) \leq k$. The {\it $k$-cycle associated to $Y$} is $$ [Y]_k = \sum m_{Z, Y}[Z] $$ where the sum is over the irreducible components $Z$ of $Y$ of $\delta$-dimension $k$ and $m_{Z, Y}$ is the multiplicity of $Z$ in $Y$. This is a $k$-cycle by Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-components-locally-finite}. \end{enumerate}


spaces-chow-definition-cycle-associated-to-coherent-sheaf

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. \begin{enumerate} \item For an integral closed subspace $Z \subset X$ with generic point $\xi$ such that $|Z|$ is an irreducible component of $\text{Supp}(\mathcal{F})$ the length of $\mathcal{F}$ at $\xi$ (Definition \ref{definition-length-at-x}) is called the {\it multiplicity of $Z$ in $\mathcal{F}$}. By Lemma \ref{lemma-length-finite} this is a positive integer. \item Assume $\dim_\delta(\text{Supp}(\mathcal{F})) \leq k$. The {\it $k$-cycle associated to $\mathcal{F}$} is $$ [\mathcal{F}]_k = \sum m_{Z, \mathcal{F}}[Z] $$ where the sum is over the integral closed subspaces $Z \subset X$ corresponding to irreducible components of $\text{Supp}(\mathcal{F})$ of $\delta$-dimension $k$ and $m_{Z, \mathcal{F}}$ is the multiplicity of $Z$ in $\mathcal{F}$. This is a $k$-cycle by Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-components-locally-finite}. \end{enumerate}


spaces-chow-definition-proper-pushforward

In Situation \ref{situation-setup} let $X, Y/B$ be good. Let $f : X \to Y$ be a morphism over $B$. Assume $f$ is proper. \begin{enumerate} \item Let $Z \subset X$ be an integral closed subspace with $\dim_\delta(Z) = k$. Let $Z' \subset Y$ be the image of $Z$ as in Lemma \ref{lemma-proper-image}. We define $$ f_*[Z] = \left\{ \begin{matrix} 0 & \text{if} & \dim_\delta(Z')< k, \\ \deg(Z/Z') [Z'] & \text{if} & \dim_\delta(Z') = k. \end{matrix} \right. $$ The degree of $Z$ over $Z'$ is defined and finite if $\dim_\delta(Z') = \dim_\delta(Z)$ by Lemma \ref{lemma-equal-dimension} and Spaces over Fields, Definition \ref{spaces-over-fields-definition-degree}. \item Let $\alpha = \sum n_Z [Z]$ be a $k$-cycle on $X$. The {\it pushforward} of $\alpha$ as the sum $$ f_* \alpha = \sum n_Z f_*[Z] $$ where each $f_*[Z]$ is defined as above. The sum is locally finite by Lemma \ref{lemma-quasi-compact-locally-finite} above. \end{enumerate}


spaces-chow-definition-flat-pullback

In Situation \ref{situation-setup} let $X, Y/B$ be good. Let $f : X \to Y$ be a morphism over $B$. Assume $f$ is flat of relative dimension $r$. \begin{enumerate} \item Let $Z \subset Y$ be an integral closed subspace of $\delta$-dimension $k$. We define $f^*[Z]$ to be the $(k+r)$-cycle on $X$ associated to the scheme theoretic inverse image $$ f^*[Z] = [f^{-1}(Z)]_{k+r}. $$ This makes sense since $\dim_\delta(f^{-1}(Z)) = k + r$ by Lemma \ref{lemma-flat-inverse-image-dimension}. \item Let $\alpha = \sum n_i [Z_i]$ be a $k$-cycle on $Y$. The {\it flat pullback of $\alpha$ by $f$} is the sum $$ f^* \alpha = \sum n_i f^*[Z_i] $$ where each $f^*[Z_i]$ is defined as above. The sum is locally finite by Lemma \ref{lemma-inverse-image-locally-finite}. \item We denote $f^* : Z_k(Y) \to Z_{k + r}(X)$ the map of abelian groups so obtained. \end{enumerate}


spaces-chow-definition-principal-divisor

In Situation \ref{situation-setup} let $X/B$ be good. Assume $X$ is integral with $\dim_\delta(X) = n$. Let $f \in R(X)^*$. The {\it principal divisor associated to $f$} is the $(n - 1)$-cycle $$ \text{div}(f) = \text{div}_X(f) = \sum \text{ord}_Z(f) [Z] $$ defined in Spaces over Fields, Definition \ref{spaces-over-fields-definition-principal-divisor}. This makes sense because prime divisors have $\delta$-dimension $n - 1$ by Lemma \ref{lemma-divisor-delta-dimension}.


spaces-chow-definition-rational-equivalence

In Situation \ref{situation-setup} let $X/B$ be good. Let $k \in \mathbf{Z}$. \begin{enumerate} \item Given any locally finite collection $\{W_j \subset X\}$ of integral closed subspaces with $\dim_\delta(W_j) = k + 1$, and any $f_j \in R(W_j)^*$ we may consider $$ \sum (i_j)_*\text{div}(f_j) \in Z_k(X) $$ where $i_j : W_j \to X$ is the inclusion morphism. This makes sense as the morphism $\coprod i_j : \coprod W_j \to X$ is proper. \item We say that $\alpha \in Z_k(X)$ is {\it rationally equivalent to zero} if $\alpha$ is a cycle of the form displayed above. \item We say $\alpha, \beta \in Z_k(X)$ are {\it rationally equivalent} and we write $\alpha \sim_{rat} \beta$ if $\alpha - \beta$ is rationally equivalent to zero. \item We define $$ \CH_k(X) = Z_k(X) / \sim_{rat} $$ to be the {\it Chow group of $k$-cycles on $X$}. This is sometimes called the {\it Chow group of $k$-cycles modulo rational equivalence on $X$}. \end{enumerate}


spaces-chow-definition-divisor-invertible-sheaf

In Situation \ref{situation-setup} let $X/B$ be good. Assume $X$ is integral and $n = \dim_\delta(X)$. 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 $s$} is the $(n - 1)$-cycle $$ \text{div}_\mathcal{L}(s) = \sum \text{ord}_{Z, \mathcal{L}}(s) [Z] $$ defined in Spaces over Fields, Definition \ref{spaces-over-fields-definition-divisor-invertible-sheaf}. This makes sense because Weil divisors have $\delta$-dimension $n - 1$ by Lemma \ref{lemma-divisor-delta-dimension}. \item We define {\it Weil divisor associated to $\mathcal{L}$} as $$ c_1(\mathcal{L}) \cap [X] = \text{class of }\text{div}_\mathcal{L}(s) \in \CH_{n - 1}(X) $$ where $s$ is any nonzero meromorphic section of $\mathcal{L}$ over $X$. This is well defined by Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-divisor-meromorphic-well-defined}. \end{enumerate}


spaces-chow-definition-cap-c1

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. We define, for every integer $k$, an operation $$ c_1(\mathcal{L}) \cap - : Z_{k + 1}(X) \to \CH_k(X) $$ called {\it intersection with the first Chern class of $\mathcal{L}$}. \begin{enumerate} \item Given an integral closed subspace $i : W \to X$ with $\dim_\delta(W) = k + 1$ we define $$ c_1(\mathcal{L}) \cap [W] = i_*(c_1({i^*\mathcal{L}}) \cap [W]) $$ where the right hand side is defined in Definition \ref{definition-divisor-invertible-sheaf}. \item For a general $(k + 1)$-cycle $\alpha = \sum n_i [W_i]$ we set $$ c_1(\mathcal{L}) \cap \alpha = \sum n_i c_1(\mathcal{L}) \cap [W_i] $$ \end{enumerate}


spaces-chow-definition-gysin-homomorphism

In Situation \ref{situation-setup} let $X/B$ be good. Let $(\mathcal{L}, s)$ be a pair consisting of an invertible sheaf and a global section $s \in \Gamma(X, \mathcal{L})$. Let $D = Z(s)$ be the vanishing locus of $s$, and denote $i : D \to X$ the closed immersion. We define, for every integer $k$, a (refined) {\it Gysin homomorphism} $$ i^* : Z_{k + 1}(X) \to \CH_k(D). $$ by the following rules: \begin{enumerate} \item Given a integral closed subspace $W \subset X$ with $\dim_\delta(W) = k + 1$ we define \begin{enumerate} \item if $W \not \subset D$, then $i^*[W] = [D \cap W]_k$ as a $k$-cycle on $D$, and \item if $W \subset D$, then $i^*[W] = i'_*(c_1(\mathcal{L}|_W) \cap [W])$, where $i' : W \to D$ is the induced closed immersion. \end{enumerate} \item For a general $(k + 1)$-cycle $\alpha = \sum n_j[W_j]$ we set $$ i^*\alpha = \sum n_j i^*[W_j] $$ \item If $D$ is an effective Cartier divisor, then we denote $D \cdot \alpha = i_*i^*\alpha$ the pushforward of the class to a class on $X$. \end{enumerate}


spaces-chow-definition-bivariant-class

\begin{reference} Similar to \cite[Definition 17.1]{F} \end{reference} In Situation \ref{situation-setup} let $f : X \to Y$ be a morphism of good algebraic spaces over $B$. Let $p \in \mathbf{Z}$. A {\it bivariant class $c$ of degree $p$ for $f$} is given by a rule which assigns to every morphism $Y' \to Y$ of good algebraic spaces over $B$ and every $k$ a map $$ c \cap - : \CH_k(Y') \longrightarrow \CH_{k - p}(X') $$ where $X' = Y' \times_Y X$, satisfying the following conditions \begin{enumerate} \item if $Y'' \to Y'$ is a proper morphism, then $c \cap (Y'' \to Y')_*\alpha'' = (X'' \to X')_*(c \cap \alpha'')$ for all $\alpha''$ on $Y''$, \item if $Y'' \to Y'$ a morphism of good algebraic spaces over $B$ which is flat of relative dimension $r$, then $c \cap (Y'' \to Y')^*\alpha' = (X'' \to X')^*(c \cap \alpha')$ for all $\alpha'$ on $Y'$, \item if $(\mathcal{L}', s', i' : D' \to Y')$ is as in Definition \ref{definition-gysin-homomorphism} with pullback $(\mathcal{N}', t', j' : E' \to X')$ to $X'$, then we have $c \cap (i')^*\alpha' = (j')^*(c \cap \alpha')$ for all $\alpha'$ on $Y'$. \end{enumerate} The collection of all bivariant classes of degree $p$ for $f$ is denoted $A^p(X \to Y)$.


spaces-chow-definition-chow-cohomology

In Situation \ref{situation-setup} let $X/B$ be good. The {\it Chow cohomology} of $X$ is the graded $\mathbf{Z}$-algebra $A^*(X)$ whose degree $p$ component is $A^p(X \to X)$.


spaces-chow-definition-chern-classes

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. For $i = 0, \ldots, r$ the {\it $i$th Chern class of $\mathcal{E}$} is the bivariant class $c_i(\mathcal{E}) \in A^i(X)$ of degree $i$ constructed in Lemma \ref{lemma-segre-classes}. The {\it total Chern class of $\mathcal{E}$} is the formal sum $$ c(\mathcal{E}) = c_0(\mathcal{E}) + c_1(\mathcal{E}) + \ldots + c_r(\mathcal{E}) $$ which is viewed as a nonhomogeneous bivariant class on $X$.


spaces-chow-definition-degree-zero-cycle

Let $k$ be a field. Let $p : X \to \Spec(k)$ be a proper morphism of algebraic spaces. The {\it degree of a zero cycle} on $X$ is given by proper pushforward $$ p_* : \CH_0(X) \longrightarrow \CH_0(\Spec(k)) \longrightarrow \mathbf{Z} $$ (Lemma \ref{lemma-proper-pushforward-rational-equivalence}) composed with the natural isomorphism $\CH_0(\Spec(k)) \to \mathbf{Z}$ which maps $[\Spec(k)]$ to $1$. Notation: $\deg(\alpha)$.


sdga-definition-ga

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. A {\it sheaf of graded $\mathcal{O}$-algebras} or a {\it sheaf of graded algebras} on $(\mathcal{C}, \mathcal{O})$ is given by a family $\mathcal{A}^n$ indexed by $n \in \mathbf{Z}$ of $\mathcal{O}$-modules endowed with $\mathcal{O}$-bilinear maps $$ \mathcal{A}^n \times \mathcal{A}^m \to \mathcal{A}^{n + m},\quad (a, b) \longmapsto ab $$ called the multiplication maps with the following properties \begin{enumerate} \item multiplication is associative, and \item there is a global section $1$ of $\mathcal{A}^0$ which is a two-sided identity for multiplication. \end{enumerate} We often denote such a structure $\mathcal{A}$. A {\it homomorphism of graded $\mathcal{O}$-algebras} $f : \mathcal{A} \to \mathcal{B}$ is a family of maps $f^n : \mathcal{A}^n \to \mathcal{B}^n$ of $\mathcal{O}$-modules compatible with the multiplication maps.


sdga-definition-gm

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of graded algebras on $(\mathcal{C}, \mathcal{O})$. A (right) {\it graded $\mathcal{A}$-module} or (right) {\it graded module} over $\mathcal{A}$ is given by a family $\mathcal{M}^n$ indexed by $n \in \mathbf{Z}$ of $\mathcal{O}$-modules endowed with $\mathcal{O}$-bilinear maps $$ \mathcal{M}^n \times \mathcal{A}^m \to \mathcal{M}^{n + m},\quad (x, a) \longmapsto xa $$ called the multiplication maps with the following properties \begin{enumerate} \item multiplication satisfies $(xa)a' = x(aa')$, \item the identity section $1$ of $\mathcal{A}^0$ acts as the identity on $\mathcal{M}^n$ for all $n$. \end{enumerate} We often say ``let $\mathcal{M}$ be a graded $\mathcal{A}$-module'' to indicate this situation. A {\it homomorphism of graded $\mathcal{A}$-modules} $f : \mathcal{M} \to \mathcal{N}$ is a family of maps $f^n : \mathcal{M}^n \to \mathcal{N}^n$ of $\mathcal{O}$-modules compatible with the multiplication maps. The category of (right) graded $\mathcal{A}$-modules is denoted $\text{Mod}_\mathcal{A}$.


sdga-definition-bimodule

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of graded algebras on $(\mathcal{C}, \mathcal{O})$. A {\it graded $(\mathcal{A}, \mathcal{B})$-bimodule} is given by a family $\mathcal{M}^n$ indexed by $n \in \mathbf{Z}$ of $\mathcal{O}$-modules endowed with $\mathcal{O}$-bilinear maps $$ \mathcal{M}^n \times \mathcal{B}^m \to \mathcal{M}^{n + m},\quad (x, b) \longmapsto xb $$ and $$ \mathcal{A}^n \times \mathcal{M}^m \to \mathcal{M}^{n + m},\quad (a, x) \longmapsto ax $$ called the multiplication maps with the following properties \begin{enumerate} \item multiplication satisfies $a(a'x) = (aa')x$ and $(xb)b' = x(bb')$, \item $(ax)b = a(xb)$, \item the identity section $1$ of $\mathcal{A}^0$ acts as the identity by multiplication, and \item the identity section $1$ of $\mathcal{B}^0$ acts as the identity by multiplication. \end{enumerate} We often denote such a structure $\mathcal{M}$. A {\it homomorphism of graded $(\mathcal{A}, \mathcal{B})$-bimodules} $f : \mathcal{M} \to \mathcal{N}$ is a family of maps $f^n : \mathcal{M}^n \to \mathcal{N}^n$ of $\mathcal{O}$-modules compatible with the multiplication maps.


sdga-definition-dga

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. A {\it sheaf of differential graded $\mathcal{O}$-algebras} or a {\it sheaf of differential graded algebras} on $(\mathcal{C}, \mathcal{O})$ is a cochain complex $\mathcal{A}^\bullet$ of $\mathcal{O}$-modules endowed with $\mathcal{O}$-bilinear maps $$ \mathcal{A}^n \times \mathcal{A}^m \to \mathcal{A}^{n + m},\quad (a, b) \longmapsto ab $$ called the multiplication maps with the following properties \begin{enumerate} \item multiplication is associative, \item there is a global section $1$ of $\mathcal{A}^0$ which is a two-sided identity for multiplication, \item for $U \in \Ob(\mathcal{C})$, $a \in \mathcal{A}^n(U)$, and $b \in \mathcal{A}^m(U)$ we have $$ \text{d}^{n + m}(ab) = \text{d}^n(a)b + (-1)^n a\text{d}^m(b) $$ \end{enumerate} We often denote such a structure $(\mathcal{A}, \text{d})$. A {\it homomorphism of differential graded $\mathcal{O}$-algebras} from $(\mathcal{A}, \text{d})$ to $(\mathcal{B}, \text{d})$ is a map $f : \mathcal{A}^\bullet \to \mathcal{B}^\bullet$ of complexes of $\mathcal{O}$-modules compatible with the multiplication maps.


sdga-definition-dgm

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. A (right) {\it differential graded $\mathcal{A}$-module} or (right) {\it differential graded module} over $\mathcal{A}$ is a cochain complex $\mathcal{M}^\bullet$ endowed with $\mathcal{O}$-bilinear maps $$ \mathcal{M}^n \times \mathcal{A}^m \to \mathcal{M}^{n + m},\quad (x, a) \longmapsto xa $$ called the multiplication maps with the following properties \begin{enumerate} \item multiplication satisfies $(xa)a' = x(aa')$, \item the identity section $1$ of $\mathcal{A}^0$ acts as the identity on $\mathcal{M}^n$ for all $n$, \item for $U \in \Ob(\mathcal{C})$, $x \in \mathcal{M}^n(U)$, and $a \in \mathcal{A}^m(U)$ we have $$ \text{d}^{n + m}(xa) = \text{d}^n(x)a + (-1)^n x\text{d}^m(a) $$ \end{enumerate} We often say ``let $\mathcal{M}$ be a differential graded $\mathcal{A}$-module'' to indicate this situation. A {\it homomorphism of differential graded $\mathcal{A}$-modules} from $\mathcal{M}$ to $\mathcal{N}$ is a map $f : \mathcal{M}^\bullet \to \mathcal{N}^\bullet$ of complexes of $\mathcal{O}$-modules compatible with the multiplication maps. The category of (right) differential graded $\mathcal{A}$-modules is denoted $\text{Mod}_{(\mathcal{A}, \text{d})}$.


sdga-definition-dg-bimodule

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. A {\it differential graded $(\mathcal{A}, \mathcal{B})$-bimodule} is given by a complex $\mathcal{M}^\bullet$ of $\mathcal{O}$-modules endowed with $\mathcal{O}$-bilinear maps $$ \mathcal{M}^n \times \mathcal{B}^m \to \mathcal{M}^{n + m},\quad (x, b) \longmapsto xb $$ and $$ \mathcal{A}^n \times \mathcal{M}^m \to \mathcal{M}^{n + m},\quad (a, x) \longmapsto ax $$ called the multiplication maps with the following properties \begin{enumerate} \item multiplication satisfies $a(a'x) = (aa')x$ and $(xb)b' = x(bb')$, \item $(ax)b = a(xb)$, \item $\text{d}(ax) = \text{d}(a) x + (-1)^{\deg(a)}a \text{d}(x)$ and $\text{d}(xb) = \text{d}(x) b + (-1)^{\deg(x)}x \text{d}(b)$, \item the identity section $1$ of $\mathcal{A}^0$ acts as the identity by multiplication, and \item the identity section $1$ of $\mathcal{B}^0$ acts as the identity by multiplication. \end{enumerate} We often denote such a structure $\mathcal{M}$ and sometimes we write ${}_\mathcal{A}\mathcal{M}_\mathcal{B}$. A {\it homomorphism of differential graded $(\mathcal{A}, \mathcal{B})$-bimodules} $f : \mathcal{M} \to \mathcal{N}$ is a map of complexes $f : \mathcal{M}^\bullet \to \mathcal{N}^\bullet$ of $\mathcal{O}$-modules compatible with the multiplication maps.


sdga-definition-homotopy

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $f, g : \mathcal{M} \to \mathcal{N}$ be homomorphisms of differential graded $\mathcal{A}$-modules. A {\it homotopy between $f$ and $g$} is a graded $\mathcal{A}$-module map $h : \mathcal{M} \to \mathcal{N}$ homogeneous of degree $-1$ such that $$ f - g = \text{d}_\mathcal{N} \circ h + h \circ \text{d}_\mathcal{M} $$ If a homotopy exists, then we say $f$ and $g$ are {\it homotopic}.


sdga-definition-complexes-notation

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


sdga-definition-cone

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $f : \mathcal{K} \to \mathcal{L}$ be a homomorphism of differential graded $\mathcal{A}$-modules. The {\it cone} of $f$ is the differential graded $\mathcal{A}$-module $C(f)$ defined as follows: \begin{enumerate} \item the underlying complex of $\mathcal{O}$-modules is the cone of the corresponding map $f : \mathcal{K}^\bullet \to \mathcal{L}^\bullet$ of complexes of $\mathcal{A}$-modules, i.e., we have $C(f)^n = \mathcal{L}^n \oplus \mathcal{K}^{n + 1}$ and differential $$ d_{C(f)} = \left( \begin{matrix} \text{d}_\mathcal{L} & f \\ 0 & -\text{d}_\mathcal{K} \end{matrix} \right) $$ \item the multiplication map $$ C(f)^n \times \mathcal{A}^m \to C(f)^{n + m} $$ is the direct sum of the multiplication map $\mathcal{L}^n \times \mathcal{A}^m \to \mathcal{L}^{n + m}$ and the multiplication map $\mathcal{K}^{n + 1} \times \mathcal{A}^m \to \mathcal{K}^{n + 1 + m}$. \end{enumerate} It comes equipped with canonical hommorphisms of differential graded $\mathcal{A}$-modules $i : \mathcal{L} \to C(f)$ and $p : C(f) \to \mathcal{K}[1]$ induced by the obvious maps.


sdga-definition-graded-injective

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. A diffential graded $\mathcal{A}$-module $\mathcal{I}$ is said to be {\it graded injective}\footnote{This may be nonstandard terminology.} if $\mathcal{M}$ viewed as a graded $\mathcal{A}$-module is an injective object of the category $\text{Mod}_\mathcal{A}$ of graded $\mathcal{A}$-modules.


sdga-definition-K-injective

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. A diffential graded $\mathcal{A}$-module $\mathcal{I}$ is {\it K-injective} if for every acyclic differential graded $\mathcal{M}$ we have  $$ \Hom_{K(\text{Mod}_{(\mathcal{A}, \text{d})})}(\mathcal{M}, \mathcal{I}) = 0 $$


sdga-definition-derived-category

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\text{Qis}$ be as in Lemma \ref{lemma-qis}. The {\it derived category of $(\mathcal{A}, \text{d})$} is the triangulated category $$ D(\mathcal{A}, \text{d}) = \text{Qis}^{-1}K(\text{Mod}_{(A, \text{d})}) $$ discussed in more detail above.


sdga-definition-pullback

Derived tensor product and derived pullback. \begin{enumerate} \item Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$, $\mathcal{B}$ be differential graded $\mathcal{O}$-algebras. Let $\mathcal{N}$ be a  differential graded $(\mathcal{A}, \mathcal{B})$-bimodule. The functor $D(\mathcal{A}, \text{d}) \to D(\mathcal{B}, \text{d})$ constructed in Lemma \ref{lemma-derived-tensor-product} is called the {\it derived tensor product} and denoted $- \otimes_\mathcal{A}^\mathbf{L} \mathcal{N}$. \item Let $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. Let $\mathcal{A}$ be a differential graded $\mathcal{O}_\mathcal{C}$-algebra. Let $\mathcal{B}$ be a differential graded $\mathcal{O}_\mathcal{D}$-algebra. Let $\varphi : \mathcal{B} \to f_*\mathcal{A}$ be a homomorphism of differential graded $\mathcal{O}_\mathcal{D}$-algebras. The functor $D(\mathcal{B}, \text{d}) \to D(\mathcal{A}, \text{d})$ constructed in Lemma \ref{lemma-derived-tensor-product} is called {\it derived pullback} and denote $Lf^*$. \end{enumerate}


sdga-definition-pushforward

Derived internal hom and derived pushforward. \begin{enumerate} \item Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$, $\mathcal{B}$ be differential graded $\mathcal{O}$-algebras. Let $\mathcal{N}$ be a  differential graded $(\mathcal{A}, \mathcal{B})$-bimodule. The right derived extension $$ R\SheafHom_\mathcal{B}(\mathcal{N}, -) : D(\mathcal{B}, \text{d}) \longrightarrow D(\mathcal{A}, \text{d}) $$ of the internal hom functor $\SheafHom_\mathcal{B}^{dg}(\mathcal{N}, -)$ is called {\it derived internal hom}. \item Let $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. Let $\mathcal{A}$ be a differential graded $\mathcal{O}_\mathcal{C}$-algebra. Let $\mathcal{B}$ be a differential graded $\mathcal{O}_\mathcal{D}$-algebra. Let $\varphi : \mathcal{B} \to f_*\mathcal{A}$ be a homomorphism of differential graded $\mathcal{O}_\mathcal{D}$-algebras. The right derived extension $$ Rf_* : D(\mathcal{A}, \text{d}) \longrightarrow D(\mathcal{B}, \text{d}) $$ of the pushforward $f_*$ is called {\it derived pushforward}. \end{enumerate}


stacks-morphisms-definition-separated

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. \begin{enumerate} \item We say $f$ is {\it DM} if $\Delta_f$ is unramified\footnote{The letters DM stand for Deligne-Mumford. If $f$ is DM then given any scheme $T$ and any morphism $T \to \mathcal{Y}$ the fibre product $\mathcal{X}_T = \mathcal{X} \times_\mathcal{Y} T$ is an algebraic stack over $T$ whose diagonal is unramified, i.e., $\mathcal{X}_T$ is DM. This implies $\mathcal{X}_T$ is a Deligne-Mumford stack, see Theorem \ref{theorem-DM}. In other words a DM morphism is one whose ``fibres'' are Deligne-Mumford stacks. This hopefully at least motivates the terminology.}. \item We say $f$ is {\it quasi-DM} if $\Delta_f$ is locally quasi-finite\footnote{If $f$ is quasi-DM, then the ``fibres'' $\mathcal{X}_T$ of $\mathcal{X} \to \mathcal{Y}$ are quasi-DM. An algebraic stack $\mathcal{X}$ is quasi-DM exactly if there exists a scheme $U$ and a surjective flat morphism $U \to \mathcal{X}$ of finite presentation which is locally quasi-finite, see Theorem \ref{theorem-quasi-DM}. Note the similarity to being Deligne-Mumford, which is defined in terms of having an \'etale covering by a scheme.}. \item We say $f$ is {\it separated} if $\Delta_f$ is proper. \item We say $f$ is {\it quasi-separated} if $\Delta_f$ is quasi-compact and quasi-separated. \end{enumerate}


stacks-morphisms-definition-absolute-separated

Let $\mathcal{X}$ be an algebraic stack over the base scheme $S$. Denote $p : \mathcal{X} \to S$ the structure morphism. \begin{enumerate} \item We say $\mathcal{X}$ is {\it DM over $S$} if $p : \mathcal{X} \to S$ is DM. \item We say $\mathcal{X}$ is {\it quasi-DM over $S$} if $p : \mathcal{X} \to S$ is quasi-DM. \item We say $\mathcal{X}$ is {\it separated over $S$} if $p : \mathcal{X} \to S$ is separated. \item We say $\mathcal{X}$ is {\it quasi-separated over $S$} if $p : \mathcal{X} \to S$ is quasi-separated. \item We say $\mathcal{X}$ is {\it DM} if $\mathcal{X}$ is DM\footnote{Theorem \ref{theorem-DM} shows that this is equivalent to $\mathcal{X}$ being a Deligne-Mumford stack.} over $\Spec(\mathbf{Z})$. \item We say $\mathcal{X}$ is {\it quasi-DM} if $\mathcal{X}$ is quasi-DM over $\Spec(\mathbf{Z})$. \item We say $\mathcal{X}$ is {\it separated} if $\mathcal{X}$ is separated over $\Spec(\mathbf{Z})$. \item We say $\mathcal{X}$ is {\it quasi-separated} if $\mathcal{X}$ is quasi-separated over $\Spec(\mathbf{Z})$. \end{enumerate} In the last 4 definitions we view $\mathcal{X}$ as an algebraic stack over $\Spec(\mathbf{Z})$ via Algebraic Stacks, Definition \ref{algebraic-definition-viewed-as}.


stacks-morphisms-definition-isom

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $Z$ be an algebraic space. \begin{enumerate} \item Let $x : Z \to \mathcal{X}$ be a morphism. We set $$ \mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x, x) = Z \times_{x, \mathcal{X}} \mathcal{I}_{\mathcal{X}/\mathcal{Y}} $$ We endow it with the structure of a group algebraic space over $Z$ by pulling back the composition law discussed in Remark \ref{remark-inertia-is-group-in-spaces}. We will sometimes refer to $\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x, x)$ as the {\it relative sheaf of automorphisms of $x$}. \item Let $x_1, x_2 : Z \to \mathcal{X}$ be morphisms. Set $y_i = f \circ x_i$. Let $\alpha : y_1 \to y_2$ be a $2$-morphism. Then $\alpha$ determines a morphism $\Delta^\alpha : Z \to Z \times_{y_1, \mathcal{Y}, y_2} Z$ and we set $$ \mathit{Isom}_{\mathcal{X}/\mathcal{Y}}^\alpha(x_1, x_2) = (Z \times_{x_1, \mathcal{X}, x_2} Z) \times_{Z \times_{y_1, \mathcal{Y}, y_2} Z, \Delta^\alpha} Z. $$ We will sometimes refer to $\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}^\alpha(x_1, x_2)$ as the {\it relative sheaf of isomorphisms from $x_1$ to $x_2$}. \end{enumerate} If $\mathcal{Y} = \Spec(\mathbf{Z})$ or more generally when $\mathcal{Y}$ is an algebraic space, then we use the notation $\mathit{Isom}_\mathcal{X}(x, x)$ and $\mathit{Isom}_\mathcal{X}(x_1, x_2)$ and we use the terminology {\it sheaf of automorphisms of $x$} and {\it sheaf of isomorphisms from $x_1$ to $x_2$}.


stacks-morphisms-definition-quasi-compact

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is {\it quasi-compact} if for every quasi-compact algebraic stack $\mathcal{Z}$ and morphism $\mathcal{Z} \to \mathcal{Y}$ the fibre product $\mathcal{Z} \times_\mathcal{Y} \mathcal{X}$ is quasi-compact.


stacks-morphisms-definition-noetherian

Let $\mathcal{X}$ be an algebraic stack. We say $\mathcal{X}$ is {\it Noetherian} if $\mathcal{X}$ is quasi-compact, quasi-separated and locally Noetherian.


stacks-morphisms-definition-affine

A morphism of algebraic stacks is said to be {\it affine} if it is representable and affine in the sense of Properties of Stacks, Section \ref{stacks-properties-section-properties-morphisms}.


stacks-morphisms-definition-integral

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. \begin{enumerate} \item We say $f$ is {\it integral} if $f$ is representable and integral in the sense of Properties of Stacks, Section \ref{stacks-properties-section-properties-morphisms}. \item We say $f$ is {\it finite} if $f$ is representable and finite in the sense of Properties of Stacks, Section \ref{stacks-properties-section-properties-morphisms}. \end{enumerate}


stacks-morphisms-definition-open

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. \begin{enumerate} \item We say $f$ is {\it open} if the map of topological spaces $|\mathcal{X}| \to |\mathcal{Y}|$ is open. \item We say $f$ is {\it universally open} if for every morphism of algebraic stacks $\mathcal{Z} \to \mathcal{Y}$ the morphism of topological spaces $$ |\mathcal{Z} \times_\mathcal{Y} \mathcal{X}| \to |\mathcal{Z}| $$ is open, i.e., the base change $\mathcal{Z} \times_\mathcal{Y} \mathcal{X} \to \mathcal{Z}$ is open. \end{enumerate}


stacks-morphisms-definition-submersive

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. \begin{enumerate} \item We say $f$ is {\it submersive}\footnote{This is very different from the notion of a submersion of differential manifolds.} if the continuous map $|\mathcal{X}| \to |\mathcal{Y}|$ is submersive, see Topology, Definition \ref{topology-definition-submersive}. \item We say $f$ is {\it universally submersive} if for every morphism of algebraic stacks $\mathcal{Y}' \to \mathcal{Y}$ the base change $\mathcal{Y}' \times_\mathcal{Y} \mathcal{X} \to \mathcal{Y}'$ is submersive. \end{enumerate}


stacks-morphisms-definition-closed

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. \begin{enumerate} \item We say $f$ is {\it closed} if the map of topological spaces $|\mathcal{X}| \to |\mathcal{Y}|$ is closed. \item We say $f$ is {\it universally closed} if for every morphism of algebraic stacks $\mathcal{Z} \to \mathcal{Y}$ the morphism of topological spaces $$ |\mathcal{Z} \times_\mathcal{Y} \mathcal{X}| \to |\mathcal{Z}| $$ is closed, i.e., the base change $\mathcal{Z} \times_\mathcal{Y} \mathcal{X} \to \mathcal{Z}$ is closed. \end{enumerate}


stacks-morphisms-definition-universally-injective

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is {\it universally injective} if for every morphism of algebraic stacks $\mathcal{Z} \to \mathcal{Y}$ the map $$ |\mathcal{Z} \times_\mathcal{Y} \mathcal{X}| \to |\mathcal{Z}| $$ is injective.


stacks-morphisms-definition-universal-homeomorphism

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is a {\it universal homeomorphism} if for every morphism of algebraic stacks $\mathcal{Z} \to \mathcal{Y}$ the map of topological spaces $$ |\mathcal{Z} \times_\mathcal{Y} \mathcal{X}| \to |\mathcal{Z}| $$ is a homeomorphism.


stacks-morphisms-definition-P

Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is smooth local on the source-and-target. We say a morphism $f : \mathcal{X} \to \mathcal{Y}$ of algebraic stacks {\it has property $\mathcal{P}$} if the equivalent conditions of Lemma \ref{lemma-local-source-target} hold.


stacks-morphisms-definition-locally-finite-type

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. \begin{enumerate} \item We say $f$ {\it locally of finite type} if the equivalent conditions of Lemma \ref{lemma-local-source-target} hold with $\mathcal{P} = \text{locally of finite type}$. \item We say $f$ is {\it of finite type} if it is locally of finite type and quasi-compact. \end{enumerate}


stacks-morphisms-definition-finite-type-point

Let $\mathcal{X}$ be an algebraic stack. We say a point $x \in |\mathcal{X}|$ is a {\it finite type point}\footnote{This is a slight abuse of language as it would perhaps be more correct to say ``locally finite type point''.} if the equivalent conditions of Lemma \ref{lemma-point-finite-type} are satisfied. We denote $\mathcal{X}_{\text{ft-pts}}$ the set of finite type points of $\mathcal{X}$.


stacks-morphisms-definition-locally-quasi-finite

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is {\it locally quasi-finite} if $f$ is quasi-DM, locally of finite type, and for every morphism $\Spec(k) \to \mathcal{Y}$ where $k$ is a field the space $|\mathcal{X}_k|$ is discrete.


stacks-morphisms-definition-quasi-finite

\begin{reference} \cite{rydh_approx} \end{reference} Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is {\it quasi-finite} if $f$ is locally quasi-finite (Definition \ref{definition-locally-quasi-finite}) and quasi-compact (Definition \ref{definition-quasi-compact}).


stacks-morphisms-definition-flat

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is {\it flat} if the equivalent conditions of Lemma \ref{lemma-local-source-target} hold with $\mathcal{P} = \text{flat}$.


stacks-morphisms-definition-flat-at-point

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $x \in |\mathcal{X}|$. We say $f$ is {\it flat at $x$} if the equivalent conditions of Lemma \ref{lemma-flat-at-point} hold.


stacks-morphisms-definition-locally-finite-presentation

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. \begin{enumerate} \item We say $f$ {\it locally of finite presentation} if the equivalent conditions of Lemma \ref{lemma-local-source-target} hold with $\mathcal{P} = \text{locally of finite presentation}$. \item We say $f$ is {\it of finite presentation} if it is locally of finite presentation, quasi-compact, and quasi-separated. \end{enumerate}


stacks-morphisms-definition-gerbe

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $\mathcal{X}$ is a {\it gerbe over} $\mathcal{Y}$ if $\mathcal{X}$ is a gerbe over $\mathcal{Y}$ as stacks in groupoids over $(\Sch/S)_{fppf}$, see Stacks, Definition \ref{stacks-definition-gerbe-over-stack-in-groupoids}. We say an algebraic stack $\mathcal{X}$ is a {\it gerbe} if there exists a morphism $\mathcal{X} \to X$ where $X$ is an algebraic space which turns $\mathcal{X}$ into a gerbe over $X$.


stacks-morphisms-definition-smooth

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is {\it smooth} if the equivalent conditions of Lemma \ref{lemma-local-source-target} hold with $\mathcal{P} = \text{smooth}$.


stacks-morphisms-definition-etale-smooth-P

Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is \'etale-smooth local on the source-and-target. We say a DM morphism $f : \mathcal{X} \to \mathcal{Y}$ of algebraic stacks {\it has property $\mathcal{P}$} if the equivalent conditions of Lemma \ref{lemma-local-source-target} hold.


stacks-morphisms-definition-etale

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is {\it \'etale} if $f$ is DM and the equivalent conditions of Lemma \ref{lemma-etale-smooth-local-source-target} hold with $\mathcal{P} = \etale$.


stacks-morphisms-definition-unramified

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is {\it unramified} if $f$ is DM and the equivalent conditions of Lemma \ref{lemma-etale-smooth-local-source-target} hold with $\mathcal{P} =$``unramified''.


stacks-morphisms-definition-proper

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is {\it proper} if $f$ is separated, finite type, and universally closed.


stacks-morphisms-definition-scheme-theoretic-image

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The {\it scheme theoretic image} of $f$ is the smallest closed substack $\mathcal{Z} \subset \mathcal{Y}$ through which $f$ factors\footnote{We will see in Lemma \ref{lemma-scheme-theoretic-image-existence} that the scheme theoretic image always exists.}.


stacks-morphisms-definition-fill-in-diagram

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Consider a $2$-commutative solid diagram \begin{equation} \label{equation-diagram} \vcenter{ \xymatrix{ \Spec(K) \ar[r]_-x \ar[d]_j & \mathcal{X} \ar[d]^f \\ \Spec(A) \ar[r]^-y \ar@{..>}[ru] & \mathcal{Y} } } \end{equation} where $A$ is a valuation ring with field of fractions $K$. Let $$ \gamma : y \circ j \longrightarrow f \circ x $$ be a $2$-morphism witnessing the $2$-commutativity of the diagram. (Notation as in Categories, Sections \ref{categories-section-formal-cat-cat} and \ref{categories-section-2-categories}.) Given (\ref{equation-diagram}) and $\gamma$ a {\it dotted arrow} is a triple $(a, \alpha, \beta)$ consisting of a morphism $a : \Spec(A) \to \mathcal{X}$ and $2$-arrows $\alpha : a \circ j \to x$, $\beta : y \to f \circ a$ such that $\gamma = (\text{id}_f \star \alpha) \circ (\beta \star \text{id}_j)$, in other words such that $$ \xymatrix{ & f \circ a \circ j \ar[rd]^{\text{id}_f \star \alpha} \\ y \circ j \ar[ru]^{\beta \star \text{id}_j} \ar[rr]^\gamma & & f \circ x } $$ is commutative. A {\it morphism of dotted arrows} $(a, \alpha, \beta) \to (a', \alpha', \beta')$ is a $2$-arrow $\theta : a \to a'$ such that $\alpha = \alpha' \circ (\theta \star \text{id}_j)$ and $\beta' = (\text{id}_f \star \theta) \circ \beta$.


stacks-morphisms-definition-uniqueness

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ satisfies the {\it uniqueness part of the valuative criterion} if for every diagram (\ref{equation-diagram}) and $\gamma$ as in Definition \ref{definition-fill-in-diagram} the category of dotted arrows is either empty or a setoid with exactly one isomorphism class.


stacks-morphisms-definition-existence

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ satisfies the {\it existence part of the valuative criterion} if for every diagram (\ref{equation-diagram}) and $\gamma$ as in Definition \ref{definition-fill-in-diagram} there exists an extension $K'/K$ of fields, a valuation ring $A' \subset K'$ dominating $A$ such that the category of dotted arrows for the outer rectangle of the diagram $$ \xymatrix{ \Spec(K') \ar[r] \ar@/^2em/[rr]_{x'} \ar[d]_{j'} & \Spec(K) \ar[d]_j \ar[r]_-x & \mathcal{X} \ar[d]^f \\ \Spec(A') \ar[r] \ar@/_2em/[rr]^{y'} & \Spec(A) \ar[r]^-y & \mathcal{Y} } $$ with induced $2$-arrow $\gamma' : y' \circ j' \to f \circ x'$ is nonempty.


stacks-morphisms-definition-lci

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is a {\it local complete intersection morphism} or {\it Koszul} if the equivalent conditions of Lemma \ref{lemma-local-source-target} hold with $\mathcal{P} = \text{local complete intersection}$.


schemes-definition-locally-ringed-space

Locally ringed spaces. \begin{enumerate} \item A {\it locally ringed space $(X, \mathcal{O}_X)$} is a pair consisting of a topological space $X$ and a sheaf of rings $\mathcal{O}_X$ all of whose stalks are local rings. \item Given a locally ringed space $(X, \mathcal{O}_X)$ we say that $\mathcal{O}_{X, x}$ is the {\it local ring of $X$ at $x$}. We denote $\mathfrak{m}_{X, x}$ or simply $\mathfrak{m}_x$ the maximal ideal of $\mathcal{O}_{X, x}$. Moreover, the {\it residue field of $X$ at $x$} is the residue field $\kappa(x) = \mathcal{O}_{X, x}/\mathfrak{m}_x$. \item A {\it morphism of locally ringed spaces} $(f, f^\sharp) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ is a morphism of ringed spaces such that for all $x \in X$ the induced ring map $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is a local ring map. \end{enumerate}


schemes-definition-immersion-locally-ringed-spaces

Let $f : X \to Y$ be a morphism of locally ringed spaces. We say that $f$ is an {\it open immersion} if $f$ is a homeomorphism of $X$ onto an open subset of $Y$, and the map $f^{-1}\mathcal{O}_Y \to \mathcal{O}_X$ is an isomorphism.


schemes-definition-open-subspace

Let $X$ be a locally ringed space. Let $U \subset X$ be an open subset. The locally ringed space $(U, \mathcal{O}_U)$ of Example \ref{example-open-subspace} above is the {\it open subspace of $X$ associated to $U$}.


schemes-definition-closed-immersion-locally-ringed-spaces

Let $i : Z \to X$ be a morphism of locally ringed spaces. We say that $i$ is a {\it closed immersion} if: \begin{enumerate} \item The map $i$ is a homeomorphism of $Z$ onto a closed subset of $X$. \item The map $\mathcal{O}_X \to i_*\mathcal{O}_Z$ is surjective; let $\mathcal{I}$ denote the kernel. \item The $\mathcal{O}_X$-module $\mathcal{I}$ is locally generated by sections. \end{enumerate}


schemes-definition-closed-subspace

Let $X$ be a locally ringed space. Let $\mathcal{I}$ be a sheaf of ideals on $X$ which is locally generated by sections. The locally ringed space $(Z, \mathcal{O}_Z)$ of Example \ref{example-closed-subspace} above is the {\it closed subspace of $X$ associated to the sheaf of ideals $\mathcal{I}$}.


schemes-definition-standard-covering

Let $R$ be a ring. \begin{enumerate} \item A {\it standard open covering} of $\Spec(R)$ is a covering $\Spec(R) = \bigcup_{i = 1}^n D(f_i)$, where $f_1, \ldots, f_n \in R$. \item Suppose that $D(f) \subset \Spec(R)$ 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 $g_1, \ldots, g_n \in R$. \end{enumerate}


schemes-definition-structure-sheaf

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


schemes-definition-affine-scheme

An {\it affine scheme} is a locally ringed space isomorphic as a locally ringed space to $\Spec(R)$ for some ring $R$. A {\it morphism of affine schemes} is a morphism in the category of locally ringed spaces.


schemes-definition-scheme

\begin{history} In \cite{EGA1} what we call a scheme was called a ``pre-sch\'ema'' and the name ``sch\'ema'' was reserved for what is a separated scheme in the Stacks project. In the second edition \cite{EGA1-second} the terminology was changed to the terminology that is now standard. However, one may occasionally encounter the terminology ``prescheme'', for example in \cite{Murre-lectures}. \end{history} A {\it scheme} is a locally ringed space with the property that every point has an open neighbourhood which is an affine scheme. A {\it morphism of schemes} is a morphism of locally ringed spaces. The category of schemes will be denoted $\Sch$.


schemes-definition-immersion

Let $X$ be a scheme. \begin{enumerate} \item A morphism of schemes is called an {\it open immersion} if it is an open immersion of locally ringed spaces (see Definition \ref{definition-immersion-locally-ringed-spaces}). \item An {\it open subscheme} of $X$ is an open subspace of $X$ in the sense of Definition \ref{definition-open-subspace}; an open subscheme of $X$ is a scheme by Lemma \ref{lemma-open-subspace-scheme}. \item A morphism of schemes is called a {\it closed immersion} if it is a closed immersion of locally ringed spaces (see Definition \ref{definition-closed-immersion-locally-ringed-spaces}). \item A {\it closed subscheme} of $X$ is a closed subspace of $X$ in the sense of Definition \ref{definition-closed-subspace}; a closed subscheme is a scheme by Lemma \ref{lemma-closed-subspace-scheme}. \item A morphism of schemes $f : X \to Y$ is called an {\it immersion}, or a {\it locally closed immersion} if it can be factored as $j \circ i$ where $i$ is a closed immersion and $j$ is an open immersion. \end{enumerate}


schemes-definition-reduced

Let $X$ be a scheme. We say $X$ is {\it reduced} if every local ring $\mathcal{O}_{X, x}$ is reduced.


schemes-definition-reduced-induced-scheme

Let $X$ be a scheme. Let $Z \subset X$ be a closed subset. A {\it scheme structure on $Z$} is given by a closed subscheme $Z'$ of $X$ whose underlying set is equal to $Z$. We often say ``let $(Z, \mathcal{O}_Z)$ be a scheme structure on $Z$'' to indicate this. The {\it reduced induced scheme structure} on $Z$ is the one constructed in Lemma \ref{lemma-reduced-closed-subscheme}. The {\it reduction $X_{red}$ of $X$} is the reduced induced scheme structure on $X$ itself.


schemes-definition-representable-functor

(See Categories, Definition \ref{categories-definition-representable-functor}.) Let $F$ be a contravariant functor from the category of schemes to the category of sets (as above). We say that $F$ is {\it representable by a scheme} or {\it representable} if there exists a scheme $X$ such that $h_X \cong F$.


schemes-definition-representable-by-open-immersions

Let $F$ be a contravariant functor on the category of schemes with values in sets. \begin{enumerate} \item We say that $F$ {\it satisfies the sheaf property for the Zariski topology} if for every scheme $T$ and every open covering $T = \bigcup_{i \in I} U_i$, and for any collection of elements $\xi_i \in F(U_i)$ such that $\xi_i|_{U_i \cap U_j} = \xi_j|_{U_i \cap U_j}$ there exists a unique element $\xi \in F(T)$ such that $\xi_i = \xi|_{U_i}$ in $F(U_i)$. \item A {\it subfunctor $H \subset F$} is a rule that associates to every scheme $T$ a subset $H(T) \subset F(T)$ such that the maps $F(f) : F(T) \to F(T')$ maps $H(T)$ into $H(T')$ for all morphisms of schemes $f : T' \to T$. \item Let $H \subset F$ be a subfunctor. We say that $H \subset F$ is {\it representable by open immersions} if for all pairs $(T, \xi)$, where $T$ is a scheme and $\xi \in F(T)$ there exists an open subscheme $U_\xi \subset T$ with the following property: \begin{itemize} \item[(*)] A morphism $f : T' \to T$ factors through $U_\xi$ if and only if $f^*\xi \in H(T')$. \end{itemize} \item Let $I$ be a set. For each $i \in I$ let $H_i \subset F$ be a subfunctor. We say that the collection $(H_i)_{i \in I}$ {\it covers $F$} if and only if for every $\xi \in F(T)$ there exists an open covering $T = \bigcup U_i$ such that $\xi|_{U_i} \in H_i(U_i)$. \end{enumerate}


schemes-definition-fibre-product

Given morphisms of schemes $f : X \to S$ and $g : Y \to S$ the {\it fibre product} is a scheme $X \times_S Y$ together with projection morphisms $p : X \times_S Y \to X$ and $q : X \times_S Y \to Y$ sitting into the following commutative diagram $$ \xymatrix{ X \times_S Y \ar[r]_q \ar[d]_p & Y \ar[d]^g \\ X \ar[r]^f & S } $$ which is universal among all diagrams of this sort, see Categories, Definition \ref{categories-definition-fibre-products}.


schemes-definition-inverse-image-closed-subscheme

Let $f : X \to Y$ be a morphism of schemes. Let $Z \subset Y$ be a closed subscheme of $Y$. The {\it inverse image $f^{-1}(Z)$ of the closed subscheme $Z$} is the closed subscheme $Z \times_Y X$ of $X$. See Lemma \ref{lemma-fibre-product-immersion} above.


schemes-definition-base-change

Let $S$ be a scheme. \begin{enumerate} \item We say $X$ is a {\it scheme over $S$} to mean that $X$ comes equipped with a morphism of schemes $X \to S$. The morphism $X \to S$ is sometimes called the {\it structure morphism}. \item If $R$ is a ring we say $X$ is a {\it scheme over $R$} instead of $X$ is a scheme over $\Spec(R)$. \item A {\it morphism $f : X \to Y$ of schemes over $S$} is a morphism of schemes such that the composition $X \to Y \to S$ of $f$ with the structure morphism of $Y$ is equal to the structure morphism of $X$. \item We denote $\Mor_S(X, Y)$ the set of all morphisms from $X$ to $Y$ over $S$. \item Let $X$ be a scheme over $S$. Let $S' \to S$ be a morphism of schemes. The {\it base change} of $X$ is the scheme $X_{S'} = S' \times_S X$ over $S'$. \item Let $f : X \to Y$ be a morphism of schemes over $S$. Let $S' \to S$ be a morphism of schemes. The {\it base change} of $f$ is the induced morphism $f' : X_{S'} \to Y_{S'}$ (namely the morphism $\text{id}_{S'} \times_{\text{id}_S} f$). \item Let $R$ be a ring. Let $X$ be a scheme over $R$. Let $R \to R'$ be a ring map. The {\it base change} $X_{R'}$ is the scheme $\Spec(R') \times_{\Spec(R)} X$ over $R'$. \end{enumerate}


schemes-definition-preserved-by-base-change

Properties and base change. \begin{enumerate} \item Let $\mathcal{P}$ be a property of schemes over a base. We say that $\mathcal{P}$ is {\it preserved under arbitrary base change}, or simply that $\mathcal{P}$ is {\it preserved under base change} if whenever $X/S$ has $\mathcal{P}$, any base change $X_{S'}/S'$ has $\mathcal{P}$. \item Let $\mathcal{P}$ be a property of morphisms of schemes over a base. We say that $\mathcal{P}$ is {\it preserved under arbitrary base change}, or simply that {\it preserved under base change} if whenever $f : X \to Y$ over $S$ has $\mathcal{P}$, any base change $f' : X_{S'} \to Y_{S'}$ over $S'$ has $\mathcal{P}$. \end{enumerate}


schemes-definition-fibre

Let $f : X \to S$ be a morphism of schemes. Let $s \in S$ be a point. The {\it scheme theoretic fibre $X_s$ of $f$ over $s$}, or simply the {\it fibre of $f$ over $s$}, is the scheme fitting in the following fibre product diagram $$ \xymatrix{ X_s = \Spec(\kappa(s)) \times_S X \ar[r] \ar[d] & X \ar[d] \\ \Spec(\kappa(s)) \ar[r] & S } $$ We think of the fibre $X_s$ always as a scheme over $\kappa(s)$.


schemes-definition-quasi-compact

A morphism of schemes is called {\it quasi-compact} if the underlying map of topological spaces is quasi-compact, see Topology, Definition \ref{topology-definition-quasi-compact}.


schemes-definition-universally-closed

A morphism of schemes $f : X \to S$ is said to be {\it universally closed} if every base change $f' : X_{S'} \to S'$ is closed.


schemes-definition-valuative-criterion

Let $f : X \to S$ be a morphism of schemes. We say $f$ {\it satisfies the existence part of the valuative criterion} if given any commutative solid diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & X \ar[d] \\ \Spec(A) \ar[r] \ar@{-->}[ru] & S } $$ where $A$ is a valuation ring with field of fractions $K$, the dotted arrow exists. We say $f$ {\it satisfies the uniqueness part of the valuative criterion} if there is at most one dotted arrow given any diagram as above (without requiring existence of course).


schemes-definition-separated

Let $f : X \to S$ be a morphism of schemes. \begin{enumerate} \item We say $f$ is {\it separated} if the diagonal morphism $\Delta_{X/S}$ is a closed immersion. \item We say $f$ is {\it quasi-separated} if the diagonal morphism $\Delta_{X/S}$ is a quasi-compact morphism. \item We say a scheme $Y$ is {\it separated} if the morphism $Y \to \Spec(\mathbf{Z})$ is separated. \item We say a scheme $Y$ is {\it quasi-separated} if the morphism $Y \to \Spec(\mathbf{Z})$ is quasi-separated. \end{enumerate}


schemes-definition-monomorphism

A morphism of schemes is called a {\it monomorphism} if it is a monomorphism in the category of schemes, see Categories, Definition \ref{categories-definition-mono-epi}.


injectives-definition-small

Let $\mathcal{C}$ be a category, let $I \subset \text{Arrows}(\mathcal{C})$, and let $\alpha$ be an ordinal. An object $A$ of $\mathcal{C}$ is said to be {\it $\alpha$-small with respect to $I$} if whenever $\{B_\beta\}$ is a system over $\alpha$ with transition maps in $I$, then the map (\ref{equation-compare}) is an isomorphism.


injectives-definition-grothendieck-conditions

Let $\mathcal{A}$ be an abelian category. We name some conditions \begin{enumerate} \item[AB3] $\mathcal{A}$ has direct sums, \item[AB4] $\mathcal{A}$ has AB3 and direct sums are exact, \item[AB5] $\mathcal{A}$ has AB3 and filtered colimits are exact. \end{enumerate} Here are the dual notions \begin{enumerate} \item[AB3*] $\mathcal{A}$ has products, \item[AB4*] $\mathcal{A}$ has AB3* and products are exact, \item[AB5*] $\mathcal{A}$ has AB3* and filtered limits are exact. \end{enumerate} We say an object $U$ of $\mathcal{A}$ is a {\it generator} if for every $N \subset M$, $N \not = M$ in $\mathcal{A}$ there exists a morphism $U \to M$ which does not factor through $N$. We say $\mathcal{A}$ is a {\it Grothendieck abelian category} if it has AB5 and a generator.


injectives-definition-size

Let $\mathcal{A}$ be a Grothendieck abelian category. Let $M$ be an object of $\mathcal{A}$. The {\it size} $|M|$ of $M$ is the cardinality of the set of subobjects of $M$.


brauer-definition-finite

Let $A$ be a $k$-algebra. We say $A$ is {\it finite} if $\dim_k(A) < \infty$. In this case we write $[A : k] = \dim_k(A)$.


brauer-definition-skew-field

A {\it skew field} is a possibly noncommutative ring with an identity element $1$, with $1 \not = 0$, in which every nonzero element has a multiplicative inverse.


brauer-definition-simple

Let $A$ be a $k$-algebra. We say an $A$-module $M$ is {\it simple} if it is nonzero and the only $A$-submodules are $0$ and $M$. We say $A$ is {\it simple} if the only two-sided ideals of $A$ are $0$ and $A$.


brauer-definition-central

A $k$-algebra $A$ is {\it central} if the center of $A$ is the image of $k \to A$.


brauer-definition-opposite

Given a $k$-algebra $A$ we denote $A^{op}$ the $k$-algebra we get by reversing the order of multiplication in $A$. This is called the {\it opposite algebra}.


brauer-definition-brauer-group

Let $k$ be a field. The {\it Brauer group} of $k$ is the abelian group of similarity classes of finite central simple $k$-algebras defined above. Notation $\text{Br}(k)$.


brauer-definition-splitting

Let $A$ be a finite central simple $k$-algebra. We say a field extension $k \subset k'$ {\it splits} $A$, or $k'$ is a {\it splitting field} for $A$ if $A \otimes_k k'$ is a matrix algebra over $k'$.


divisors-definition-associated

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. \begin{enumerate} \item We say $x \in X$ is {\it associated} to $\mathcal{F}$ if the maximal ideal $\mathfrak m_x$ is associated to the $\mathcal{O}_{X, x}$-module $\mathcal{F}_x$. \item We denote $\text{Ass}(\mathcal{F})$ or $\text{Ass}_X(\mathcal{F})$ the set of associated points of $\mathcal{F}$. \item The {\it associated points of $X$} are the associated points of $\mathcal{O}_X$. \end{enumerate}


divisors-definition-embedded

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. \begin{enumerate} \item An {\it embedded associated point} of $\mathcal{F}$ is an associated point which is not maximal among the associated points of $\mathcal{F}$, i.e., it is the specialization of another associated point of $\mathcal{F}$. \item A point $x$ of $X$ is called an {\it embedded point} if $x$ is an embedded associated point of $\mathcal{O}_X$. \item An {\it embedded component} of $X$ is an irreducible closed subset $Z = \overline{\{x\}}$ where $x$ is an embedded point of $X$. \end{enumerate}


divisors-definition-weakly-associated

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. \begin{enumerate} \item We say $x \in X$ is {\it weakly associated} to $\mathcal{F}$ if the maximal ideal $\mathfrak m_x$ is weakly associated to the $\mathcal{O}_{X, x}$-module $\mathcal{F}_x$. \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}


divisors-definition-relative-assassin

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. The {\it relative assassin of $\mathcal{F}$ in $X$ over $S$} is the set $$ \text{Ass}_{X/S}(\mathcal{F}) = \bigcup\nolimits_{s \in S} \text{Ass}_{X_s}(\mathcal{F}_s) $$ where $\mathcal{F}_s = (X_s \to X)^*\mathcal{F}$ is the restriction of $\mathcal{F}$ to the fibre of $f$ at $s$.


divisors-definition-relative-weak-assassin

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. The {\it relative weak assassin of $\mathcal{F}$ in $X$ over $S$} is the set $$ \text{WeakAss}_{X/S}(\mathcal{F}) = \bigcup\nolimits_{s \in S} \text{WeakAss}(\mathcal{F}_s) $$ where $\mathcal{F}_s = (X_s \to X)^*\mathcal{F}$ is the restriction of $\mathcal{F}$ to the fibre of $f$ at $s$.


divisors-definition-torsion

Let $X$ be an integral scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. \begin{enumerate} \item We say a local section of $\mathcal{F}$ is {\it torsion} if it satisfies the equivalent conditions of Lemma \ref{lemma-torsion-sections}. \item We say $\mathcal{F}$ is {\it torsion free} if every torsion section of $\mathcal{F}$ is $0$. \end{enumerate}


divisors-definition-reflexive

Let $X$ be an integral locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. The {\it reflexive hull} of $\mathcal{F}$ is the $\mathcal{O}_X$-module $$ \mathcal{F}^{**} = \SheafHom_{\mathcal{O}_X}( \SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{O}_X), \mathcal{O}_X) $$ We say $\mathcal{F}$ is {\it reflexive} if the natural map $j : \mathcal{F} \longrightarrow \mathcal{F}^{**}$ is an isomorphism.


divisors-definition-effective-Cartier-divisor

Let $S$ be a scheme. \begin{enumerate} \item A {\it locally principal closed subscheme} of $S$ is a closed subscheme whose sheaf of ideals is locally generated by a single element. \item An {\it effective Cartier divisor} on $S$ is a closed subscheme $D \subset S$ whose ideal sheaf $\mathcal{I}_D \subset \mathcal{O}_S$ is an invertible $\mathcal{O}_S$-module. \end{enumerate}


divisors-definition-sum-effective-Cartier-divisors

Let $S$ be a scheme. Given effective Cartier divisors $D_1$, $D_2$ on $S$ we set $D = D_1 + D_2$ equal to the closed subscheme of $S$ 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 $D_2$}.


divisors-definition-pullback-effective-Cartier-divisor

Let $f : S' \to S$ be a morphism of schemes. Let $D \subset S$ be an effective Cartier divisor. We say the {\it pullback of $D$ by $f$ is defined} if the closed subscheme $f^{-1}(D) \subset S'$ 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}.


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

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


divisors-definition-regular-section

Let $(X, \mathcal{O}_X)$ be a locally ringed space. 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.


divisors-definition-zero-scheme-s

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


divisors-definition-relative-effective-Cartier-divisor

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


divisors-definition-conormal-sheaf

Let $f : Z \to X$ be an immersion. The {\it conormal algebra $\mathcal{C}_{Z/X, *}$ of $Z$ in $X$} or the {\it conormal algebra of $f$} is the quasi-coherent sheaf of graded $\mathcal{O}_Z$-algebras $\bigoplus_{n \geq 0} \mathcal{I}^n/\mathcal{I}^{n + 1}$ described above.


divisors-definition-normal-cone

Let $i : Z \to X$ be an immersion of schemes. The {\it normal cone $C_ZX$} of $Z$ in $X$ is $$ C_ZX = \underline{\Spec}_Z(\mathcal{C}_{Z/X, *}) $$ see Constructions, Definitions \ref{constructions-definition-cone} and \ref{constructions-definition-abstract-cone}. The {\it normal bundle} of $Z$ in $X$ is the vector bundle $$ N_ZX = \underline{\Spec}_Z(\text{Sym}(\mathcal{C}_{Z/X})) $$ see Constructions, Definitions \ref{constructions-definition-vector-bundle} and \ref{constructions-definition-abstract-vector-bundle}.