fields-definition-trace-pairing

Let $L/K$ be a finite extension of fields. The {\it trace pairing} for $L/K$ is the symmetric $K$-bilinear form $$ Q_{L/K} : L \times L \longrightarrow K,\quad (\alpha, \beta) \longmapsto \text{Trace}_{L/K}(\alpha\beta) $$


fields-definition-discriminant

Let $L/K$ be a finite extension of fields. The {\it discriminant of $L/K$} is the discriminant of the trace pairing $Q_{L/K}$.


fields-definition-galois

A field extension $E/F$ is called {\it Galois} if it is algebraic, separable, and normal.


fields-definition-galois-group

If $E/F$ is a Galois extension, then the group $\text{Aut}(E/F)$ is called the {\it Galois group} and it is denoted $\text{Gal}(E/F)$.


fields-definition-transcendence

Let $k \subset K$ be a field extension. \begin{enumerate} \item A collection of elements $\{x_i\}_{i \in I}$ of $K$ is called {\it algebraically independent} over $k$ if the map $$ k[X_i; i\in I] \longrightarrow K $$ which maps $X_i$ to $x_i$ is injective. \item The field of fractions of a polynomial ring $k[x_i; i \in I]$ is denoted $k(x_i; i\in I)$. \item A {\it purely transcendental extension} of $k$ is any field extension $k \subset K$ isomorphic to the field of fractions of a polynomial ring over $k$. \item A {\it transcendence basis} of $K/k$ is a collection of elements $\{x_i\}_{i \in I}$ which are algebraically independent over $k$ and such that the extension $k(x_i; i\in I) \subset K$ is algebraic. \end{enumerate}


fields-definition-transcendence-degree

Let $k \subset K$ be a field extension. The {\it transcendence degree} of $K$ over $k$ is the cardinality of a transcendence basis of $K$ over $k$. It is denoted $\text{trdeg}_k(K)$.


fields-definition-algebraically-closed-in

Let $k \subset K$ be a field extension. \begin{enumerate} \item The {\it algebraic closure of $k$ in $K$} is the subfield $k'$ of $K$ consisting of elements of $K$ which are algebraic over $k$. \item We say $k$ is {\it algebraically closed in $K$} if every element of $K$ which is algebraic over $k$ is contained in $k$. \end{enumerate}


fields-definition-compositum

Consider a diagram \begin{equation} \label{equation-inside-omega} \vcenter{ \xymatrix{ L \ar[r] & \Omega \\ k \ar[r] \ar[u] & K \ar[u] } } \end{equation} of field extensions. The {\it compositum of $K$ and $L$ in $\Omega$} written $KL$ is the smallest subfield of $\Omega$ containing both $L$ and $K$.


fields-definition-linearly-disjoint

Consider a diagram of fields as in (\ref{equation-inside-omega}). We say that $K$ and $L$ are {\it linearly disjoint over $k$ in $\Omega$} if the map $$ K \otimes_k L \longrightarrow KL,\quad \sum x_i \otimes y_i \longmapsto \sum x_i y_i $$ is injective.


fields-definition-separable-algebraic

Algebraic field extensions. \begin{enumerate} \item A field extension $k \subset K$ is called {\it algebraic} if every element of $K$ is algebraic over $k$. \item An algebraic extension $k \subset k'$ is called {\it separable} if every $\alpha \in k'$ is separable over $k$. \item An algebraic extension $k \subset k'$ is called {\it purely inseparable} if the characteristic of $k$ is $p > 0$ and for every element $\alpha \in k'$ there exists a power $q$ of $p$ such that $\alpha^q \in k$. \item An algebraic extension $k \subset k'$ is called {\it normal} if for every $\alpha \in k'$ the minimal polynomial $P(T) \in k[T]$ of $\alpha$ over $k$ splits completely into linear factors over $k'$. \item An algebraic extension $k \subset k'$ is called {\it Galois} if it is separable and normal. \end{enumerate}


spaces-limits-definition-locally-finite-presentation

Let $S$ be a scheme. \begin{enumerate} \item A functor $F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$ is said to be {\it limit preserving} or {\it locally of finite presentation} if for every affine scheme $T$ over $S$ which is a limit $T = \lim T_i$ of a directed inverse system of affine schemes $T_i$ over $S$, we have $$ F(T) = \colim F(T_i). $$ We sometimes say that $F$ is {\it locally of finite presentation over $S$}. \item Let $F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. A transformation of functors $a : F \to G$ is {\it limit preserving} or {\it locally of finite presentation} if for every scheme $T$ over $S$ and every $y \in G(T)$ the functor $$ F_y : (\Sch/T)_{fppf}^{opp} \longrightarrow \textit{Sets}, \quad T'/T \longmapsto \{x \in F(T') \mid a(x) = y|_{T'}\} $$ is locally of finite presentation over $T$\footnote{The characterization (2) in Lemma \ref{lemma-characterize-relative-limit-preserving} may be easier to parse.}. We sometimes say that $F$ is {\it relatively limit preserving} over $G$. \end{enumerate}


spaces-limits-definition-subsheaf-sections-annihilated-by-ideal

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. The subsheaf $\mathcal{F}' \subset \mathcal{F}$ defined in Lemma \ref{lemma-sections-annihilated-by-ideal} above is called the {\it subsheaf of sections annihilated by $\mathcal{I}$}.


spaces-limits-definition-subsheaf-sections-supported-on-closed

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \subset |X|$ be a closed subset whose complement corresponds to an open subspace $U \subset X$ with quasi-compact inclusion morphism $U \to X$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. The quasi-coherent subsheaf $\mathcal{F}' \subset \mathcal{F}$ defined in Lemma \ref{lemma-sections-supported-on-closed-subset} above is called the {\it subsheaf of sections supported on $T$}.


stacks-geometry-definition-versal-ring-at-x

In Situation \ref{situation-versal} let $x_0 : \Spec(k) \to \mathcal{X}$ be a morphism, where $k$ is a finite type field over $S$. A {\it versal ring to $\mathcal{X}$ at $x_0$} is a complete Noetherian local $S$-algebra $A$ with residue field $k$ such that there exists a versal formal object $(A, \xi_n, f_n)$ as in Artin's Axioms, Definition \ref{artin-definition-versal-formal-object} with $\xi_1 \cong x_0$ (a $2$-isomorphism).


stacks-geometry-definition-multiplicity

Let $\mathcal{X}$ be a locally Noetherian algebraic stack. Let $T \subset |\mathcal{X}|$ be an irreducible component. The {\it multiplicity} of $T$ in $\mathcal{X}$ is defined as $m_{T, \mathcal{X}} = m_{T', U}$ where $f : U \to \mathcal{X}$ is a smooth morphism from a scheme and $T' \subset |U|$ is an irreducible component with $f(T') \subset T$.


stacks-geometry-definition-formal-branches

Let $\mathcal{X}$ be an algebraic stack locally of finite type over a locally Noetherian scheme $S$. Let $x_0 : \Spec(k) \to \mathcal{X}$ is a morphism where $k$ is a field of finite type over $S$. The {\it formal branches of $\mathcal{X}$ through $x_0$} is the set of irreducible components of $\Spec(A)$ for any choice of versal ring to $\mathcal{X}$ at $x_0$ identified for different choices of $A$ by the procedure described above.


stacks-geometry-definition-multiplicity-formal-branches

Let $\mathcal{X}$ be an algebraic stack locally of finite type over a locally Noetherian scheme $S$. Let $x_0 : \Spec(k) \to \mathcal{X}$ is a morphism where $k$ is a field of finite type over $S$. The {\it multiplicity of a formal branch of $\mathcal{X}$ through $x_0$} is the multiplicity of the corresponding irreducible component of $\Spec(A)$ for any choice of versal ring to $\mathcal{X}$ at $x_0$ (see discussion above).


stacks-geometry-definition-relative-dimension

If $f : T \to \mathcal{X}$ is a locally of finite type morphism from an algebraic space to an algebraic stack, and if $t \in |T|$ is a point with image $x \in | \mathcal{X}|$, then we define {\it the relative dimension} of $f$ at $t$, denoted $\dim_t(T_x),$ as follows: choose a morphism $\Spec k \to \mathcal{X}$, with source the spectrum of a field, which represents $x$, and choose a point $t' \in |T \times_{\mathcal{X}} \Spec k|$ mapping to $t$ under the projection to $|T|$ (such a point $t'$ exists, by Properties of Stacks, Lemma \ref{stacks-properties-lemma-points-cartesian}); then $$ \dim_t(T_x) = \dim_{t'}(T \times_{\mathcal{X}} \Spec k ). $$


stacks-geometry-definition-relative-dimension-for-stacks

If $f : \mathcal{T} \to \mathcal{X}$ is a locally of finite type morphism between locally Noetherian algebraic stacks, and if $t \in |\mathcal{T}|$ is a point with image $x \in |\mathcal{X}|$, then we define the {\it relative dimension} of $f$ at $t$, denoted $\dim_t(\mathcal{T}_x),$ as follows: choose a morphism $\Spec k \to \mathcal{X}$, with source the spectrum of a field, which represents $x$, and choose a point $t' \in |\mathcal{T} \times_{\mathcal{X}} \Spec k|$ mapping to $t$ under the projection to $|\mathcal{T}|$ (such a point $t'$ exists, by Properties of Stacks, Lemma \ref{stacks-properties-lemma-points-cartesian}; then $$ \dim_t(\mathcal{T}_x) = \dim_{t'}(\mathcal{T} \times_{\mathcal{X}} \Spec k ). $$


stacks-geometry-definition-pseudo-catenary

We say that a locally Noetherian algebraic stack $\mathcal{X}$ is {\it pseudo-catenary} if there exists a smooth and surjective morphism $U \to \mathcal{X}$ whose source is a universally catenary scheme.


stacks-geometry-definition-dimension-local-ring

Let $\mathcal{X}$ be a locally Noetherian algebraic stack. Let $x \in |\mathcal{X}|$ be a finite type point. The {\it dimension of the local ring of $\mathcal{X}$ at $x$} is $d \in \mathbf{Z}$ if the equivalent conditions of Lemma \ref{lemma-dimension-local-ring} are satisfied.


spaces-morphisms-definition-separated

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\Delta_{X/Y} : X \to X \times_Y X$ be the diagonal morphism. \begin{enumerate} \item We say $f$ is {\it separated} if $\Delta_{X/Y}$ is a closed immersion. \item We say $f$ is {\it locally separated}\footnote{In the literature this term often refers to quasi-separated and locally separated morphisms.} if $\Delta_{X/Y}$ is an immersion. \item We say $f$ is {\it quasi-separated} if $\Delta_{X/Y}$ is quasi-compact. \end{enumerate}


spaces-morphisms-definition-surjective

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


spaces-morphisms-definition-open

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


spaces-morphisms-definition-submersive

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \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 $|X| \to |Y|$ is submersive, see Topology, Definition \ref{topology-definition-submersive}. \item We say $f$ is {\it universally submersive} if for every morphism of algebraic spaces $Y' \to Y$ the base change $Y' \times_Y X \to Y'$ is submersive. \end{enumerate}


spaces-morphisms-definition-quasi-compact

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. We say $f$ is {\it quasi-compact} if for every quasi-compact algebraic space $Z$ and morphism $Z \to Y$ the fibre product $Z \times_Y X$ is quasi-compact.


spaces-morphisms-definition-closed

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


spaces-morphisms-definition-monomorphism

Let $S$ be a scheme. A morphism of algebraic spaces over $S$ is called a {\it monomorphism} if it is an injective map of sheaves, i.e., a monomorphism in the category of sheaves on $(\Sch/S)_{fppf}$.


spaces-morphisms-definition-inverse-image-closed-subspace

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Let $Z \subset X$ be a closed subspace. The {\it inverse image $f^{-1}(Z)$ of the closed subspace $Z$} is the closed subspace $Z \times_X Y$ of $Y$.


spaces-morphisms-definition-scheme-theoretic-intersection-union

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z, Y \subset X$ be closed subspaces corresponding to quasi-coherent ideal sheaves $\mathcal{I}, \mathcal{J} \subset \mathcal{O}_X$. The {\it scheme theoretic intersection} of $Z$ and $Y$ is the closed subspace of $X$ cut out by $\mathcal{I} + \mathcal{J}$. Then {\it scheme theoretic union} of $Z$ and $Y$ is the closed subspace of $X$ cut out by $\mathcal{I} \cap \mathcal{J}$.


spaces-morphisms-definition-scheme-theoretic-support

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. The {\it scheme theoretic support of $\mathcal{F}$} is the closed subspace $Z \subset X$ constructed in Lemma \ref{lemma-scheme-theoretic-support}.


spaces-morphisms-definition-scheme-theoretic-image

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The {\it scheme theoretic image} of $f$ is the smallest closed subspace $Z \subset Y$ through which $f$ factors, see Lemma \ref{lemma-scheme-theoretic-image} above.


spaces-morphisms-definition-scheme-theoretically-dense

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U \subset X$ be an open subspace. \begin{enumerate} \item The scheme theoretic image of the morphism $U \to X$ is called the {\it scheme theoretic closure of $U$ in $X$}. \item We say $U$ is {\it scheme theoretically dense in $X$} if the equivalent conditions of Lemma \ref{lemma-scheme-theoretically-dense} are satisfied. \end{enumerate}


spaces-morphisms-definition-dominant

Let $S$ be a scheme. A morphism $f : X \to Y$ of algebraic spaces over $S$ is called {\it dominant} if the image of $|f| : |X| \to |Y|$ is dense in $|Y|$.


spaces-morphisms-definition-universally-injective

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. We say $f$ is {\it universally injective} if for every morphism $Y' \to Y$ the induced map $|Y' \times_Y X| \to |Y'|$ is injective.


spaces-morphisms-definition-affine

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. We say $f$ is {\it affine} if for every affine scheme $Z$ and morphism $Z \to Y$ the algebraic space $X \times_Y Z$ is representable by an affine scheme.


spaces-morphisms-definition-relative-spec

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-algebras. The {\it relative spectrum of $\mathcal{A}$ over $X$}, or simply the {\it spectrum of $\mathcal{A}$ over $X$} is the affine morphism $\underline{\Spec}(\mathcal{A}) \to X$ corresponding to $\mathcal{A}$ under the equivalence of categories of Lemma \ref{lemma-affine-equivalence-algebras}.


spaces-morphisms-definition-quasi-affine

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. We say $f$ is {\it quasi-affine} if for every affine scheme $Z$ and morphism $Z \to Y$ the algebraic space $X \times_Y Z$ is representable by a quasi-affine scheme.


spaces-morphisms-definition-P

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


spaces-morphisms-definition-P-at-point

Let $\mathcal{Q}$ be a property of morphisms of germs of schemes which is \'etale local on the source-and-target. Let $S$ be a scheme. Given a morphism $f : X \to Y$ of algebraic spaces over $S$ and a point $x \in |X|$ we say that $f$ {\it has property $\mathcal{Q}$ at $x$} if the equivalent conditions of Lemma \ref{lemma-local-source-target-at-point} hold.


spaces-morphisms-definition-locally-finite-type

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \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 Let $x \in |X|$. We say $f$ is of {\it finite type at $x$} if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is 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}


spaces-morphisms-definition-finite-type-point

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We say a point $x \in |X|$ 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 $X_{\text{ft-pts}}$ the set of finite type points of $X$.


spaces-morphisms-definition-locally-quasi-finite

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item We say $f$ is {\it locally quasi-finite} if the equivalent conditions of Lemma \ref{lemma-local-source-target} hold with $\mathcal{P} = \text{locally quasi-finite}$. \item Let $x \in |X|$. We say $f$ is {\it quasi-finite at $x$} if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is locally quasi-finite. \item A morphism of algebraic spaces $f : X \to Y$ is {\it quasi-finite} if it is locally quasi-finite and quasi-compact. \end{enumerate}


spaces-morphisms-definition-locally-finite-presentation

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item We say $f$ is {\it locally of finite presentation} if the equivalent conditions of Lemma \ref{lemma-local-source-target} hold with $\mathcal{P} =$``locally of finite presentation''. \item Let $x \in |X|$. We say $f$ is of {\it finite presentation at $x$} if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is locally of finite presentation\footnote{It seems awkward to use ``locally of finite presentation at $x$'', but the current terminology may be misleading in the sense that ``of finite presentation at $x$'' does {\bf not} mean that there is an open neighbourhood $X' \subset X$ such that $f|_{X'}$ is of finite presentation.}. \item A morphism of algebraic spaces $f : X \to Y$ is {\it of finite presentation} if it is locally of finite presentation, quasi-compact and quasi-separated. \end{enumerate}


spaces-morphisms-definition-flat

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item We say $f$ is {\it flat} if the equivalent conditions of Lemma \ref{lemma-local-source-target} with $\mathcal{P} =$``flat''. \item Let $x \in |X|$. We say $f$ is {\it flat at $x$} if the equivalent conditions of Lemma \ref{lemma-local-source-target-at-point} holds with $\mathcal{Q} =$``induced map local rings is flat''. \end{enumerate} Note that the second part makes sense by Descent, Lemma \ref{descent-lemma-flat-at-point}.


spaces-morphisms-definition-flat-module

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. \begin{enumerate} \item Let $x \in |X|$. We say $\mathcal{F}$ is {\it flat at $x$ over $Y$} if the equivalent conditions of Lemma \ref{lemma-flat-at-point} hold. \item We say $\mathcal{F}$ is {\it flat over $Y$} if $\mathcal{F}$ is flat over $Y$ at all $x \in |X|$. \end{enumerate}


spaces-morphisms-definition-dimension-fibre

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$. Let $d, r \in \{0, 1, 2, \ldots, \infty\}$. \begin{enumerate} \item We say the {\it dimension of the local ring of the fibre of $f$ at $x$} is $d$ if the equivalent conditions of Lemma \ref{lemma-local-source-target-at-point} hold for the property $\mathcal{P}_d$ described in Descent, Lemma \ref{descent-lemma-dimension-local-ring-fibre}. \item We say the {\it transcendence degree of $x/f(x)$} is $r$ if the equivalent conditions of Lemma \ref{lemma-local-source-target-at-point} hold for the property $\mathcal{P}_r$ described in Descent, Lemma \ref{descent-lemma-transcendence-degree-at-point}. \item We say {\it $f$ has relative dimension $d$ at $x$} if the equivalent conditions of Lemma \ref{lemma-local-source-target-at-point} hold for the property $\mathcal{P}_d$ described in Descent, Lemma \ref{descent-lemma-dimension-at-point}. \end{enumerate}


spaces-morphisms-definition-relative-dimension

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $d \in \{0, 1, 2, \ldots\}$. \begin{enumerate} \item We say $f$ has {\it relative dimension $\leq d$} if $f$ has relative dimension $\leq d$ at all $x \in |X|$. \item We say $f$ has {\it relative dimension $d$} if $f$ has relative dimension $d$ at all $x \in |X|$. \end{enumerate}


spaces-morphisms-definition-syntomic

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item We say $f$ is {\it syntomic} if the equivalent conditions of Lemma \ref{lemma-local-source-target} hold with $\mathcal{P} =$``syntomic''. \item Let $x \in |X|$. We say $f$ is {\it syntomic at $x$} if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is syntomic. \end{enumerate}


spaces-morphisms-definition-smooth

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item We say $f$ is {\it smooth} if the equivalent conditions of Lemma \ref{lemma-local-source-target} hold with $\mathcal{P} =$``smooth''. \item Let $x \in |X|$. We say $f$ is {\it smooth at $x$} if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is smooth. \end{enumerate}


spaces-morphisms-definition-unramified

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item We say $f$ is {\it unramified} if the equivalent conditions of Lemma \ref{lemma-local-source-target} hold with $\mathcal{P} = \text{unramified}$. \item Let $x \in |X|$. We say $f$ is {\it unramified at $x$} if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is unramified. \item We say $f$ is {\it G-unramified} if the equivalent conditions of Lemma \ref{lemma-local-source-target} hold with $\mathcal{P} = \text{G-unramified}$. \item Let $x \in |X|$. We say $f$ is {\it G-unramified at $x$} if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is G-unramified. \end{enumerate}


spaces-morphisms-definition-etale

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$. We say $f$ is {\it \'etale at $x$} if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is \'etale.


spaces-morphisms-definition-proper

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


spaces-morphisms-definition-valuative-criterion

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. We say $f$ {\it satisfies the uniqueness part of the valuative criterion} if given any commutative solid diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & X \ar[d] \\ \Spec(A) \ar[r] \ar@{-->}[ru] & Y } $$ where $A$ is a valuation ring with field of fractions $K$, there exists at most one dotted arrow (without requiring existence). We say $f$ {\it satisfies the existence part of the valuative criterion} if given any solid diagram as above there exists an extension $K \subset K'$ of fields, a valuation ring $A' \subset K'$ dominating $A$ and a morphism $\Spec(A') \to X$ such that the following diagram commutes $$ \xymatrix{ \Spec(K') \ar[r] \ar[d] & \Spec(K) \ar[r] & X \ar[d] \\ \Spec(A') \ar[r] \ar[rru] & \Spec(A) \ar[r] & Y } $$ We say $f$ {\it satisfies the valuative criterion} if $f$ satisfies both the existence and uniqueness part.


spaces-morphisms-definition-integral

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item We say that $f$ is {\it integral} if for every affine scheme $Z$ and morphisms $Z \to Y$ the algebraic space $X \times_Y Z$ is representable by an affine scheme integral over $Z$. \item We say that $f$ is {\it finite} if for every affine scheme $Z$ and morphisms $Z \to Y$ the algebraic space $X \times_Y Z$ is representable by an affine scheme finite over $Z$. \end{enumerate}


spaces-morphisms-definition-finite-locally-free

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. We say that $f$ is {\it finite locally free} if $f$ is affine and $f_*\mathcal{O}_X$ is a finite locally free $\mathcal{O}_Y$-module. In this case we say $f$ is has {\it rank} or {\it degree} $d$ if the sheaf $f_*\mathcal{O}_X$ is finite locally free of rank $d$.


spaces-morphisms-definition-rational-map

Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. \begin{enumerate} \item Let $f : U \to Y$, $g : V \to Y$ be morphisms of algebraic spaces over $S$ defined on dense open subspaces $U$, $V$ of $X$. We say that $f$ is {\it equivalent} to $g$ if $f|_W = g|_W$ for some dense open subspace $W \subset U \cap V$. \item A {\it rational map from $X$ to $Y$} is an equivalence class for the equivalence relation defined in (1). \item Given morphisms $X \to B$ and $Y \to B$ of algebraic spaces over $S$ we say that a rational map from $X$ to $Y$ is a {\it $B$-rational map from $X$ to $Y$} if there exists a representative $f : U \to Y$ of the equivalence class which is a morphism over $B$. \end{enumerate}


spaces-morphisms-definition-rational-function

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A {\it rational function on $X$} is a rational map from $X$ to $\mathbf{A}^1_S$.


spaces-morphisms-definition-ring-of-rational-functions

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The {\it ring of rational functions on $X$} is the ring $R(X)$ whose elements are rational functions with addition and multiplication as just described.


spaces-morphisms-definition-domain-of-definition

Let $S$ be a scheme. Let $\varphi$ be a rational map between two algebraic spaces $X$ and $Y$ over $S$. We say $\varphi$ is {\it defined in a point $x \in |X|$} if there exists a representative $(U, f)$ of $\varphi$ with $x \in |U|$. The {\it domain of definition} of $\varphi$ is the set of all points where $\varphi$ is defined.


spaces-morphisms-definition-dominant-rational

Let $S$ be a scheme. Let $X$ and $Y$ be algebraic spaces over $S$. Assume $|X|$ and $|Y|$ are irreducible. A rational map from $X$ to $Y$ is called {\it dominant} if any representative $f : U \to Y$ is a dominant morphism in the sense of Definition \ref{definition-dominant}.


spaces-morphisms-definition-birational-spaces

Let $S$ be a scheme. Let $X$ and $Y$ be algebraic spaces over $S$ with $|X|$ and $|Y|$ irreducible. We say $X$ and $Y$ are {\it birational} if $X$ and $Y$ are isomorphic in the category of irreducible algebraic spaces over $S$ and dominant rational maps.


spaces-morphisms-definition-integral-closure

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-algebras. The {\it integral closure of $\mathcal{O}_X$ in $\mathcal{A}$} is the quasi-coherent $\mathcal{O}_X$-subalgebra $\mathcal{A}' \subset \mathcal{A}$ constructed in Lemma \ref{lemma-integral-closure} above.


spaces-morphisms-definition-normalization-X-in-Y

Let $S$ be a scheme. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $\mathcal{O}'$ be the integral closure of $\mathcal{O}_X$ in $f_*\mathcal{O}_Y$. The {\it normalization of $X$ in $Y$} is the morphism of algebraic spaces $$ \nu : X' = \underline{\Spec}_X(\mathcal{O}') \to X $$ over $S$. It comes equipped with a natural factorization $$ Y \xrightarrow{f'} X' \xrightarrow{\nu} X $$ of the initial morphism $f$.


spaces-morphisms-definition-normalization

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ satisfying the equivalent conditions of Lemma \ref{lemma-prepare-normalization}. We define the {\it normalization} of $X$ as the morphism $$ \nu : X^\nu \longrightarrow X $$ constructed in Lemma \ref{lemma-normalization}.


spaces-morphisms-definition-universal-homeomorphism

Let $S$ be a scheme. A morphism $f : X \to Y$ of algebraic spaces over $S$ is called a {\it universal homeomorphism} if and only if for every morphism of algebraic spaces $Z \to Y$ the base change $Z \times_Y X \to Z$ induces a homeomorphism $|Z \times_Y X| \to |Z|$.


weil-definition-chow-group-motives

Let $k$ be a base field. Let $M = (X, p, m)$ be a Chow motive over $k$. For $i \in \mathbf{Z}$ we define the {\it $i$th Chow group of $M$} by the formula $$ \CH^i(M) = p\left(\CH^{i + m}(X) \otimes \mathbf{Q}\right) $$


weil-definition-weil-cohomology-theory-classical

Let $k$ be an algebraically closed field. Let $F$ be a field of characteristic $0$. A {\it classical Weil cohomology theory} over $k$ with coefficients in $F$ is given by data (D1), (D2), and (D3) satisfying Poincar\'e duality, the K\"unneth formula, and compatibility with cycle classes, more precisely, satisfying (A), (B), and (C).


weil-definition-weil-cohomology-theory

Let $k$ be a field. Let $F$ be a field of characteristic $0$. A {\it Weil cohomology theory} over $k$ with coefficients in $F$ is given by data (D0), (D1), (D2), and (D3) satisfying Poincar\'e duality, the K\"unneth formula, and compatibility with cycle classes, more precisely, satisfying axioms (A), (B), and (C) of Section \ref{section-axioms} and in addition such that the equivalent conditions (1) and (2) of Lemma \ref{lemma-H-0-separable} hold for every smooth projective $X$ over $k$.


morphisms-definition-scheme-theoretic-intersection-union

Let $X$ be a scheme. Let $Z, Y \subset X$ be closed subschemes corresponding to quasi-coherent ideal sheaves $\mathcal{I}, \mathcal{J} \subset \mathcal{O}_X$. The {\it scheme theoretic intersection} of $Z$ and $Y$ is the closed subscheme of $X$ cut out by $\mathcal{I} + \mathcal{J}$. The {\it scheme theoretic union} of $Z$ and $Y$ is the closed subscheme of $X$ cut out by $\mathcal{I} \cap \mathcal{J}$.


morphisms-definition-scheme-theoretic-support

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. The {\it scheme theoretic support of $\mathcal{F}$} is the closed subscheme $Z \subset X$ constructed in Lemma \ref{lemma-scheme-theoretic-support}.


morphisms-definition-scheme-theoretic-image

Let $f : X \to Y$ be a morphism of schemes. The {\it scheme theoretic image} of $f$ is the smallest closed subscheme $Z \subset Y$ through which $f$ factors, see Lemma \ref{lemma-scheme-theoretic-image} above.


morphisms-definition-scheme-theoretically-dense

Let $X$ be a scheme. Let $U \subset X$ be an open subscheme. \begin{enumerate} \item The scheme theoretic image of the morphism $U \to X$ is called the {\it scheme theoretic closure of $U$ in $X$}. \item We say $U$ is {\it scheme theoretically dense in $X$} if for every open $V \subset X$ the scheme theoretic closure of $U \cap V$ in $V$ is equal to $V$. \end{enumerate}


morphisms-definition-dominant

A morphism $f : X \to S$ of schemes is called {\it dominant} if the image of $f$ is a dense subset of $S$.


morphisms-definition-surjective

A morphism of schemes is said to be {\it surjective} if it is surjective on underlying topological spaces.


morphisms-definition-universally-injective

Let $f : X \to S$ be a morphism. \begin{enumerate} \item We say that $f$ is {\it universally injective} if and only if for any morphism of schemes $S' \to S$ the base change $f' : X_{S'} \to S'$ is injective (on underlying topological spaces). \item We say $f$ is {\it radicial} if $f$ is injective as a map of topological spaces, and for every $x \in X$ the field extension $\kappa(x) \supset \kappa(f(x))$ is purely inseparable. \end{enumerate}


morphisms-definition-affine

A morphism of schemes $f : X \to S$ is called {\it affine} if the inverse image of every affine open of $S$ is an affine open of $X$.


morphisms-definition-family-ample-invertible-modules

\begin{reference} \cite[II Definition 2.2.4]{SGA6} \end{reference} Let $X$ be a scheme. Let $\{\mathcal{L}_i\}_{i \in I}$ be a family of invertible $\mathcal{O}_X$-modules. We say $\{\mathcal{L}_i\}_{i \in I}$ is an {\it ample family of invertible modules on $X$} if \begin{enumerate} \item $X$ is quasi-compact, and \item for every $x \in X$ there exists an $i \in I$, an $n \geq 1$, and $s \in \Gamma(X, \mathcal{L}_i^{\otimes n})$ such that $x \in X_s$ and $X_s$ is affine. \end{enumerate}


morphisms-definition-quasi-affine

A morphism of schemes $f : X \to S$ is called {\it quasi-affine} if the inverse image of every affine open of $S$ is a quasi-affine scheme.


morphisms-definition-property-local

Let $P$ be a property of ring maps. \begin{enumerate} \item We say that $P$ is {\it local} if the following hold: \begin{enumerate} \item For any ring map $R \to A$, and any $f \in R$ we have $P(R \to A) \Rightarrow P(R_f \to A_f)$. \item For any rings $R$, $A$, any $f \in R$, $a\in A$, and any ring map $R_f \to A$ we have $P(R_f \to A) \Rightarrow P(R \to A_a)$. \item For any ring map $R \to A$, and $a_i \in A$ such that $(a_1, \ldots, a_n) = A$ then $\forall i, P(R \to A_{a_i}) \Rightarrow P(R \to A)$. \end{enumerate} \item We say that $P$ is {\it stable under base change} if for any ring maps $R \to A$, $R \to R'$ we have $P(R \to A) \Rightarrow P(R' \to R' \otimes_R A)$. \item We say that $P$ is {\it stable under composition} if for any ring maps $A \to B$, $B \to C$ we have $P(A \to B) \wedge P(B \to C) \Rightarrow P(A \to C)$. \end{enumerate}


morphisms-definition-locally-P

Let $P$ be a property of ring maps. Let $f : X \to S$ be a morphism of schemes. We say $f$ is {\it locally of type $P$} if for any $x \in X$ there exists an affine open neighbourhood $U$ of $x$ in $X$ which maps into an affine open $V \subset S$ such that the induced ring map $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ has property $P$.


morphisms-definition-finite-type

Let $f : X \to S$ be a morphism of schemes. \begin{enumerate} \item We say that $f$ is of {\it finite type at $x \in X$} if there exists an affine open neighbourhood $\Spec(A) = U \subset X$ of $x$ and an affine open $\Spec(R) = V \subset S$ with $f(U) \subset V$ such that the induced ring map $R \to A$ is of finite type. \item We say that $f$ is {\it locally of finite type} if it is of finite type at every point of $X$. \item We say that $f$ is of {\it finite type} if it is locally of finite type and quasi-compact. \end{enumerate}


morphisms-definition-finite-type-point

Let $S$ be a scheme. Let us say that a point $s$ of $S$ is a {\it finite type point} if the canonical morphism $\Spec(\kappa(s)) \to S$ is of finite type. We denote $S_{\text{ft-pts}}$ the set of finite type points of $S$.


morphisms-definition-universally-catenary

Let $S$ be a scheme. Assume $S$ is locally Noetherian. We say $S$ is {\it universally catenary} if for every morphism $X \to S$ locally of finite type the scheme $X$ is catenary.


morphisms-definition-J

Let $X$ be a locally Noetherian scheme. We say $X$ is {\it J-2} if for every morphism $Y \to X$ which is locally of finite type the regular locus $\text{Reg}(Y)$ is open in $Y$.


morphisms-definition-quasi-finite

\begin{reference} \cite[II Definition 6.2.3]{EGA} \end{reference} Let $f : X \to S$ be a morphism of schemes. \begin{enumerate} \item We say that $f$ is {\it quasi-finite at a point $x \in X$} if there exist an affine neighbourhood $\Spec(A) = U \subset X$ of $x$ and an affine open $\Spec(R) = V \subset S$ such that $f(U) \subset V$, the ring map $R \to A$ is of finite type, and $R \to A$ is quasi-finite at the prime of $A$ corresponding to $x$ (see above). \item We say $f$ is {\it locally quasi-finite} if $f$ is quasi-finite at every point $x$ of $X$. \item We say that $f$ is {\it quasi-finite} if $f$ is of finite type and every point $x$ is an isolated point of its fibre. \end{enumerate}


morphisms-definition-finite-presentation

Let $f : X \to S$ be a morphism of schemes. \begin{enumerate} \item We say that $f$ is of {\it finite presentation at $x \in X$} if there exists an affine open neighbourhood $\Spec(A) = U \subset X$ of $x$ and affine open $\Spec(R) = V \subset S$ with $f(U) \subset V$ such that the induced ring map $R \to A$ is of finite presentation. \item We say that $f$ is {\it locally of finite presentation} if it is of finite presentation at every point of $X$. \item We say that $f$ is of {\it finite presentation} if it is locally of finite presentation, quasi-compact and quasi-separated. \end{enumerate}


morphisms-definition-open

Let $f : X \to S$ be a morphism. \begin{enumerate} \item We say $f$ is {\it open} if the map on underlying topological spaces is open. \item We say $f$ is {\it universally open} if for any morphism of schemes $S' \to S$ the base change $f' : X_{S'} \to S'$ is open. \end{enumerate}


morphisms-definition-submersive

Let $f : X \to Y$ be a morphism of schemes. \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 of underlying topological spaces is submersive, see Topology, Definition \ref{topology-definition-submersive}. \item We say $f$ is {\it universally submersive} if for every morphism of schemes $Y' \to Y$ the base change $Y' \times_Y X \to Y'$ is submersive. \end{enumerate}


morphisms-definition-flat

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


morphisms-definition-scheme-structure-connected-component

Let $X$ be a scheme. Let $T \subset X$ be a connected component. The {\it canonical scheme structure on $T$} is the unique scheme structure on $T$ such that the closed immersion $T \to X$ is flat, see Lemma \ref{lemma-characterize-flat-closed-immersions}.


morphisms-definition-relative-dimension-d

Let $f : X \to S$ be a morphism of schemes. Assume $f$ is locally of finite type. \begin{enumerate} \item We say $f$ is of {\it relative dimension $\leq d$ at $x$} if $\dim_x(X_{f(x)}) \leq d$. \item We say $f$ is of {\it relative dimension $\leq d$} if $\dim_x(X_{f(x)}) \leq d$ for all $x \in X$. \item We say $f$ is of {\it relative dimension $d$} if all nonempty fibres $X_s$ are equidimensional of dimension $d$. \end{enumerate}


morphisms-definition-syntomic

Let $f : X \to S$ be a morphism of schemes. \begin{enumerate} \item We say that $f$ is {\it syntomic at $x \in X$} if there exists an affine open neighbourhood $\Spec(A) = U \subset X$ of $x$ and affine open $\Spec(R) = V \subset S$ with $f(U) \subset V$ such that the induced ring map $R \to A$ is syntomic. \item We say that $f$ is {\it syntomic} if it is syntomic at every point of $X$. \item If $S = \Spec(k)$ and $f$ is syntomic, then we say that $X$ is a {\it local complete intersection over $k$}. \item A morphism of affine schemes $f : X \to S$ is called {\it standard syntomic} if there exists a global relative complete intersection $R \to R[x_1, \ldots, x_n]/(f_1, \ldots, f_c)$ (see Algebra, Definition \ref{algebra-definition-relative-global-complete-intersection}) such that $X \to S$ is isomorphic to $$ \Spec(R[x_1, \ldots, x_n]/(f_1, \ldots, f_c)) \to \Spec(R). $$ \end{enumerate}


morphisms-definition-syntomic-relative-dimension

Let $d \geq 0$ be an integer. We say a morphism of schemes $f : X \to S$ is {\it syntomic of relative dimension $d$} if $f$ is syntomic and the function $\dim_x(X_{f(x)}) = d$ for all $x \in X$.


morphisms-definition-conormal-sheaf

Let $i : Z \to X$ be an immersion. The {\it conormal sheaf $\mathcal{C}_{Z/X}$ of $Z$ in $X$} or the {\it conormal sheaf of $i$} is the quasi-coherent $\mathcal{O}_Z$-module $\mathcal{I}/\mathcal{I}^2$ described above.


morphisms-definition-sheaf-differentials

Let $f : X \to S$ be a morphism of schemes. The {\it sheaf of differentials $\Omega_{X/S}$ of $X$ over $S$} is the sheaf of differentials of $f$ viewed as a morphism of ringed spaces (Modules, Definition \ref{modules-definition-differentials}) equipped with its {\it universal $S$-derivation} $$ \text{d}_{X/S} : \mathcal{O}_X \longrightarrow \Omega_{X/S}. $$


morphisms-definition-smooth

Let $f : X \to S$ be a morphism of schemes. \begin{enumerate} \item We say that $f$ is {\it smooth at $x \in X$} if there exists an affine open neighbourhood $\Spec(A) = U \subset X$ of $x$ and affine open $\Spec(R) = V \subset S$ with $f(U) \subset V$ such that the induced ring map $R \to A$ is smooth. \item We say that $f$ is {\it smooth} if it is smooth at every point of $X$. \item A morphism of affine schemes $f : X \to S$ is called {\it standard smooth} if there exists a standard smooth ring map $R \to R[x_1, \ldots, x_n]/(f_1, \ldots, f_c)$ (see Algebra, Definition \ref{algebra-definition-standard-smooth}) such that $X \to S$ is isomorphic to $$ \Spec(R[x_1, \ldots, x_n]/(f_1, \ldots, f_c)) \to \Spec(R). $$ \end{enumerate}


morphisms-definition-smooth-relative-dimension

Let $d \geq 0$ be an integer. We say a morphism of schemes $f : X \to S$ is {\it smooth of relative dimension $d$} if $f$ is smooth and $\Omega_{X/S}$ is finite locally free of constant rank $d$.


morphisms-definition-unramified

Let $f : X \to S$ be a morphism of schemes. \begin{enumerate} \item We say that $f$ is {\it unramified at $x \in X$} if there exists an affine open neighbourhood $\Spec(A) = U \subset X$ of $x$ and affine open $\Spec(R) = V \subset S$ with $f(U) \subset V$ such that the induced ring map $R \to A$ is unramified. \item We say that $f$ is {\it G-unramified at $x \in X$} if there exists an affine open neighbourhood $\Spec(A) = U \subset X$ of $x$ and affine open $\Spec(R) = V \subset S$ with $f(U) \subset V$ such that the induced ring map $R \to A$ is G-unramified. \item We say that $f$ is {\it unramified} if it is unramified at every point of $X$. \item We say that $f$ is {\it G-unramified} if it is G-unramified at every point of $X$. \end{enumerate}


morphisms-definition-etale

Let $f : X \to S$ be a morphism of schemes. \begin{enumerate} \item We say that $f$ is {\it \'etale at $x \in X$} if there exists an affine open neighbourhood $\Spec(A) = U \subset X$ of $x$ and affine open $\Spec(R) = V \subset S$ with $f(U) \subset V$ such that the induced ring map $R \to A$ is \'etale. \item We say that $f$ is {\it \'etale} if it is \'etale at every point of $X$. \item A morphism of affine schemes $f : X \to S$ is called {\it standard \'etale} if $X \to S$ is isomorphic to $$ \Spec(R[x]_h/(g)) \to \Spec(R) $$ where $R \to R[x]_h/(g)$ is a standard \'etale ring map, see Algebra, Definition \ref{algebra-definition-standard-etale}, i.e., $g$ is monic and $g'$ invertible in $R[x]_h/(g)$. \end{enumerate}


morphisms-definition-relatively-ample

\begin{reference} \cite[II Definition 4.6.1]{EGA} \end{reference} Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. We say $\mathcal{L}$ is {\it relatively ample}, or {\it $f$-relatively ample}, or {\it ample on $X/S$}, or {\it $f$-ample} if $f : X \to S$ is quasi-compact, and if for every affine open $V \subset S$ the restriction of $\mathcal{L}$ to the open subscheme $f^{-1}(V)$ of $X$ is ample.


morphisms-definition-very-ample

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. We say $\mathcal{L}$ is {\it relatively very ample} or more precisely {\it $f$-relatively very ample}, or {\it very ample on $X/S$}, or {\it $f$-very ample} if there exist a quasi-coherent $\mathcal{O}_S$-module $\mathcal{E}$ and an immersion $i : X \to \mathbf{P}(\mathcal{E})$ over $S$ such that $\mathcal{L} \cong i^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$.


morphisms-definition-quasi-projective

\begin{reference} \cite[II, Definition 5.3.1]{EGA} and \cite[page 103]{H} \end{reference} Let $f : X \to S$ be a morphism of schemes. \begin{enumerate} \item We say $f$ is {\it quasi-projective} if $f$ is of finite type and there exists an $f$-relatively ample invertible $\mathcal{O}_X$-module. \item We say $f$ is {\it H-quasi-projective} if there exists a quasi-compact immersion $X \to \mathbf{P}^n_S$ over $S$ for some $n$.\footnote{This is not exactly the same as the definition in Hartshorne. Namely, the definition in Hartshorne (8th corrected printing, 1997) is that $f$ should be the composition of an open immersion followed by a H-projective morphism (see Definition \ref{definition-projective}), which does not imply $f$ is quasi-compact. See Lemma \ref{lemma-H-quasi-projective-open-H-projective} for the implication in the other direction.} \item We say $f$ is {\it locally quasi-projective} if there exists an open covering $S = \bigcup V_j$ such that each $f^{-1}(V_j) \to V_j$ is quasi-projective. \end{enumerate}


morphisms-definition-proper

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


morphisms-definition-projective

Let $f : X \to S$ be a morphism of schemes. \begin{enumerate} \item We say $f$ is {\it projective} if $X$ is isomorphic as an $S$-scheme to a closed subscheme of a projective bundle $\mathbf{P}(\mathcal{E})$ for some quasi-coherent, finite type $\mathcal{O}_S$-module $\mathcal{E}$. \item We say $f$ is {\it H-projective} if there exists an integer $n$ and a closed immersion $X \to \mathbf{P}^n_S$ over $S$. \item We say $f$ is {\it locally projective} if there exists an open covering $S = \bigcup U_i$ such that each $f^{-1}(U_i) \to U_i$ is projective. \end{enumerate}


morphisms-definition-integral

Let $f : X \to S$ be a morphism of schemes. \begin{enumerate} \item We say that $f$ is {\it integral} if $f$ is affine and if for every affine open $\Spec(R) = V \subset S$ with inverse image $\Spec(A) = f^{-1}(V) \subset X$ the associated ring map $R \to A$ is integral. \item We say that $f$ is {\it finite} if $f$ is affine and if for every affine open $\Spec(R) = V \subset S$ with inverse image $\Spec(A) = f^{-1}(V) \subset X$ the associated ring map $R \to A$ is finite. \end{enumerate}


morphisms-definition-universal-homeomorphism

A morphisms $f : X \to Y$ of schemes is called a {\it universal homeomorphism} if the base change $f' : Y' \times_Y X \to Y'$ is a homeomorphism for every morphism $Y' \to Y$.


morphisms-definition-seminormal-ring

Let $A$ be a ring. \begin{enumerate} \item We say $A$ is {\it seminormal} if for all $x, y \in A$ with $x^3 = y^2$ there is a unique $a \in A$ with $x = a^2$ and $y = a^3$. \item We say $A$ is {\it absolutely weakly normal} if (a) $A$ is seminormal and (b) for any prime number $p$ and $x, y \in A$ with $p^px = y^p$ there is a unique $a \in A$ with $x = a^p$ and $y = pa$. \end{enumerate}


morphisms-definition-seminormal

Let $X$ be a scheme. \begin{enumerate} \item We say $X$ is {\it seminormal} if every $x \in X$ has an affine open neighbourhood $\Spec(R) = U \subset X$ such that the ring $R$ is seminormal. \item We say $X$ is {\it absolutely weakly normal} if every $x \in X$ has an affine open neighbourhood $\Spec(R) = U \subset X$ such that the ring $R$ is absolutely weakly normal. \end{enumerate}


morphisms-definition-seminormalization

Let $X$ be a scheme. \begin{enumerate} \item The morphism $X^{sn} \to X$ constructed in Lemma \ref{lemma-seminormalization} is the {\it seminormalization} of $X$. \item The morphism $X^{awn} \to X$ constructed in Lemma \ref{lemma-seminormalization} is the {\it absolute weak normalization} of $X$. \end{enumerate}


morphisms-definition-finite-locally-free

Let $f : X \to S$ be a morphism of schemes. We say $f$ is {\it finite locally free} if $f$ is affine and $f_*\mathcal{O}_X$ is a finite locally free $\mathcal{O}_S$-module. In this case we say $f$ is has {\it rank} or {\it degree} $d$ if the sheaf $f_*\mathcal{O}_X$ is finite locally free of degree $d$.


morphisms-definition-rational-map

Let $X$, $Y$ be schemes. \begin{enumerate} \item Let $f : U \to Y$, $g : V \to Y$ be morphisms of schemes defined on dense open subsets $U$, $V$ of $X$. We say that $f$ is {\it equivalent} to $g$ if $f|_W = g|_W$ for some $W \subset U \cap V$ dense open in $X$. \item A {\it rational map from $X$ to $Y$} is an equivalence class for the equivalence relation defined in (1). \item If $X$, $Y$ are schemes over a base scheme $S$ we say that a rational map from $X$ to $Y$ is an {\it $S$-rational map from $X$ to $Y$} if there exists a representative $f : U \to Y$ of the equivalence class which is an $S$-morphism. \end{enumerate}


morphisms-definition-rational-function

Let $X$ be a scheme. A {\it rational function on $X$} is a rational map from $X$ to $\mathbf{A}^1_{\mathbf{Z}}$.


morphisms-definition-ring-of-rational-functions

Let $X$ be a scheme. The {\it ring of rational functions on $X$} is the ring $R(X)$ whose elements are rational functions with addition and multiplication as just described.


morphisms-definition-function-field

Let $X$ be an integral scheme. The {\it function field}, or the {\it field of rational functions} of $X$ is the field $R(X)$.


morphisms-definition-domain-of-definition

Let $\varphi$ be a rational map between two schemes $X$ and $Y$. We say $\varphi$ is {\it defined in a point $x \in X$} if there exists a representative $(U, f)$ of $\varphi$ with $x \in U$. The {\it domain of definition} of $\varphi$ is the set of all points where $\varphi$ is defined.


morphisms-definition-dominant-rational

Let $X$ and $Y$ be irreducible schemes. A rational map from $X$ to $Y$ is called {\it dominant} if any representative $f : U \to Y$ is a dominant morphism of schemes.


morphisms-definition-birational-schemes

Let $X$ and $Y$ be irreducible schemes. \begin{enumerate} \item We say $X$ and $Y$ are {\it birational} if $X$ and $Y$ are isomorphic in the category of irreducible schemes and dominant rational maps. \item Assume $X$ and $Y$ are schemes over a base scheme $S$. We say $X$ and $Y$ are {\it $S$-birational} if $X$ and $Y$ are isomorphic in the category of irreducible schemes over $S$ and dominant $S$-rational maps. \end{enumerate}


morphisms-definition-birational

\begin{reference} \cite[(2.2.9)]{EGA1} \end{reference} Let $X$, $Y$ be schemes. Assume $X$ and $Y$ have finitely many irreducible components. We say a morphism $f : X \to Y$ is {\it birational} if \begin{enumerate} \item $f$ induces a bijection between the set of generic points of irreducible components of $X$ and the set of generic points of the irreducible components of $Y$, and \item for every generic point $\eta \in X$ of an irreducible component of $X$ the local ring map $\mathcal{O}_{Y, f(\eta)} \to \mathcal{O}_{X, \eta}$ is an isomorphism. \end{enumerate}


morphisms-definition-degree

Let $X$ and $Y$ be integral schemes. Let $f : X \to Y$ be locally of finite type and dominant. Assume $[R(X) : R(Y)] < \infty$, or any other of the equivalent conditions (1) -- (4) of Lemma \ref{lemma-finite-degree}. Then the positive integer $$ \text{deg}(X/Y) = [R(X) : R(Y)] $$ is called the {\it degree of $X$ over $Y$}.


morphisms-definition-modification

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


morphisms-definition-alteration

\begin{reference} \cite[Definition 2.20]{alterations} \end{reference} Let $X$ be an integral scheme. An {\it alteration of $X$} is a proper dominant morphism $f : Y \to X$ with $Y$ integral such that $f^{-1}(U) \to U$ is finite for some nonempty open $U \subset X$.


morphisms-definition-integral-closure

Let $X$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-algebras. The {\it integral closure of $\mathcal{O}_X$ in $\mathcal{A}$} is the quasi-coherent $\mathcal{O}_X$-subalgebra $\mathcal{A}' \subset \mathcal{A}$ constructed in Lemma \ref{lemma-integral-closure} above.


morphisms-definition-normalization-X-in-Y

Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of schemes. Let $\mathcal{O}'$ be the integral closure of $\mathcal{O}_X$ in $f_*\mathcal{O}_Y$. The {\it normalization of $X$ in $Y$} is the scheme\footnote{The scheme $X'$ need not be normal, for example if $Y = X$ and $f = \text{id}_X$, then $X' = X$.} $$ \nu : X' = \underline{\Spec}_X(\mathcal{O}') \to X $$ over $X$. It comes equipped with a natural factorization $$ Y \xrightarrow{f'} X' \xrightarrow{\nu} X $$ of the initial morphism $f$.


morphisms-definition-normalization

Let $X$ be a scheme such that every quasi-compact open has finitely many irreducible components. We define the {\it normalization} of $X$ as the morphism $$ \nu : X^\nu \longrightarrow X $$ which is the normalization of $X$ in the morphism $f : Y \to X$ (\ref{equation-generic-points}) constructed above.


morphisms-definition-universally-bounded

Let $f : X \to Y$ be a morphism of schemes. \begin{enumerate} \item We say the integer $n$ {\it bounds the degrees of the fibres of $f$} if for all $y \in Y$ the fibre $X_y$ is a finite scheme over $\kappa(y)$ whose degree over $\kappa(y)$ is $\leq n$. \item We say the {\it fibres of $f$ are universally bounded}\footnote{This is probably nonstandard notation.} if there exists an integer $n$ which bounds the degrees of the fibres of $f$. \end{enumerate}


smoothing-definition-singular-ideal

Let $R \to A$ be a ring map. The {\it singular ideal of $A$ over $R$}, denoted $H_{A/R}$ is the unique radical ideal $H_{A/R} \subset A$ with $$ V(H_{A/R}) = \{\mathfrak q \in \Spec(A) \mid R \to A \text{ not smooth at }\mathfrak q\} $$


smoothing-definition-strictly-standard

Let $R \to A$ be a ring map of finite presentation. We say an element $a \in A$ is {\it elementary standard in $A$ over $R$} if there exists a presentation $A = R[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$ and $0 \leq c \leq \min(n, m)$ such that \begin{equation} \label{equation-elementary-standard-one} a = a' \det(\partial f_j/\partial x_i)_{i, j = 1, \ldots, c} \end{equation} for some $a' \in A$ and \begin{equation} \label{equation-elementary-standard-two} a f_{c + j} \in (f_1, \ldots, f_c) + (f_1, \ldots, f_m)^2 \end{equation} for $j = 1, \ldots, m - c$. We say $a \in A$ is {\it strictly standard in $A$ over $R$} if there exists a presentation $A = R[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$ and $0 \leq c \leq \min(n, m)$ such that \begin{equation} \label{equation-strictly-standard-one} a = \sum\nolimits_{I \subset \{1, \ldots, n\},\ |I| = c} a_I \det(\partial f_j/\partial x_i)_{j = 1, \ldots, c,\ i \in I} \end{equation} for some $a_I \in A$ and \begin{equation} \label{equation-strictly-standard-two} a f_{c + j} \in (f_1, \ldots, f_c) + (f_1, \ldots, f_m)^2 \end{equation} for $j = 1, \ldots, m - c$.


chow-definition-periodic-complex

Let $R$ be a ring. \begin{enumerate} \item A {\it $2$-periodic complex} over $R$ is given by a quadruple $(M, N, \varphi, \psi)$ consisting of $R$-modules $M$, $N$ and $R$-module maps $\varphi : M \to N$, $\psi : N \to M$ such that $$ \xymatrix{ \ldots \ar[r] & M \ar[r]^\varphi & N \ar[r]^\psi & M \ar[r]^\varphi & N \ar[r] & \ldots } $$ is a complex. In this setting we define the {\it cohomology modules} of the complex to be the $R$-modules $$ H^0(M, N, \varphi, \psi) = \Ker(\varphi)/\Im(\psi) \quad\text{and}\quad H^1(M, N, \varphi, \psi) = \Ker(\psi)/\Im(\varphi). $$ We say the $2$-periodic complex is {\it exact} if the cohomology groups are zero. \item A {\it $(2, 1)$-periodic complex} over $R$ is given by a triple $(M, \varphi, \psi)$ consisting of an $R$-module $M$ and $R$-module maps $\varphi : M \to M$, $\psi : M \to M$ such that $$ \xymatrix{ \ldots \ar[r] & M \ar[r]^\varphi & M \ar[r]^\psi & M \ar[r]^\varphi & M \ar[r] & \ldots } $$ is a complex. Since this is a special case of a $2$-periodic complex we have its {\it cohomology modules} $H^0(M, \varphi, \psi)$, $H^1(M, \varphi, \psi)$ and a notion of exactness. \end{enumerate}


chow-definition-periodic-length

Let $(M, N, \varphi, \psi)$ be a $2$-periodic complex over a ring $R$ whose cohomology modules have finite length. In this case we define the {\it multiplicity} of $(M, N, \varphi, \psi)$ to be the integer $$ e_R(M, N, \varphi, \psi) = \text{length}_R(H^0(M, N, \varphi, \psi)) - \text{length}_R(H^1(M, N, \varphi, \psi)) $$ In the case of a $(2, 1)$-periodic complex $(M, \varphi, \psi)$, we denote this by $e_R(M, \varphi, \psi)$ and we will sometimes call this the {\it (additive) Herbrand quotient}.


chow-definition-delta-dimension

Let $(S, \delta)$ as in Situation \ref{situation-setup}. For any scheme $X$ locally of finite type over $S$ and any irreducible closed subset $Z \subset X$ we define $$ \dim_\delta(Z) = \delta(\xi) $$ where $\xi \in Z$ is the generic point of $Z$. We will call this the {\it $\delta$-dimension of $Z$}. If $Z$ is a closed subscheme of $X$, then we define $\dim_\delta(Z)$ as the supremum of the $\delta$-dimensions of its irreducible components.


chow-definition-cycles

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. 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 subschemes $Z \subset X$, each $n_Z \in \mathbf{Z}$, and the collection $\{Z; n_Z \not = 0\}$ is locally finite (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}


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

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $Z \subset X$ be a closed subscheme. \begin{enumerate} \item For any irreducible component $Z' \subset Z$ with generic point $\xi$ the integer $m_{Z', Z} = \text{length}_{\mathcal{O}_{X, \xi}} \mathcal{O}_{Z, \xi}$ (Lemma \ref{lemma-multiplicity-finite}) is called the {\it multiplicity of $Z'$ in $Z$}. \item Assume $\dim_\delta(Z) \leq k$. The {\it $k$-cycle associated to $Z$} is $$ [Z]_k = \sum m_{Z', Z}[Z'] $$ where the sum is over the irreducible components of $Z$ of $\delta$-dimension $k$. (This is a $k$-cycle by Divisors, Lemma \ref{divisors-lemma-components-locally-finite}.) \end{enumerate}


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

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. \begin{enumerate} \item For any irreducible component $Z' \subset \text{Supp}(\mathcal{F})$ with generic point $\xi$ the integer $m_{Z', \mathcal{F}} = \text{length}_{\mathcal{O}_{X, \xi}} \mathcal{F}_\xi$ (Lemma \ref{lemma-length-finite}) is called the {\it multiplicity of $Z'$ in $\mathcal{F}$}. \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 irreducible components of $\text{Supp}(\mathcal{F})$ of $\delta$-dimension $k$. (This is a $k$-cycle by Lemma \ref{lemma-length-finite}.) \end{enumerate}


chow-definition-proper-pushforward

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a morphism. Assume $f$ is proper. \begin{enumerate} \item Let $Z \subset X$ be an integral closed subscheme with $\dim_\delta(Z) = k$. We define $$ f_*[Z] = \left\{ \begin{matrix} 0 & \text{if} & \dim_\delta(f(Z))< k, \\ \deg(Z/f(Z)) [f(Z)] & \text{if} & \dim_\delta(f(Z)) = k. \end{matrix} \right. $$ Here we think of $f(Z) \subset Y$ as an integral closed subscheme. The degree of $Z$ over $f(Z)$ is finite if $\dim_\delta(f(Z)) = \dim_\delta(Z)$ by Lemma \ref{lemma-equal-dimension}. \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}


chow-definition-flat-pullback

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a morphism. Assume $f$ is flat of relative dimension $r$. \begin{enumerate} \item Let $Z \subset Y$ be an integral closed subscheme of $\delta$-dimension $k$. We define $f^*[Z]$ to be the $(k+r)$-cycle on $X$ 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}


chow-definition-principal-divisor

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. 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 Divisors, Definition \ref{divisors-definition-principal-divisor}. This makes sense because prime divisors have $\delta$-dimension $n - 1$ by Lemma \ref{lemma-divisor-delta-dimension}.


chow-definition-rational-equivalence

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $k \in \mathbf{Z}$. \begin{enumerate} \item Given any locally finite collection $\{W_j \subset X\}$ of integral closed subschemes 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}


chow-definition-divisor-invertible-sheaf

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. 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 Divisors, Definition \ref{divisors-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 Divisors, Lemma \ref{divisors-lemma-divisor-meromorphic-well-defined}. \end{enumerate}


chow-definition-cap-c1

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. 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 subscheme $i : W \to X$ 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}


chow-definition-gysin-homomorphism

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. 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 zero scheme of $s$, and denote $i : D \to X$ the closed immersion. We define, for every integer $k$, a {\it Gysin homomorphism} $$ i^* : Z_{k + 1}(X) \to \CH_k(D). $$ by the following rules: \begin{enumerate} \item Given a integral closed subscheme $W \subset X$ 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 $i^*\alpha$ to a class on $X$. \end{enumerate}


chow-definition-bivariant-class

\begin{reference} Similar to \cite[Definition 17.1]{F} \end{reference} Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $f : X \to Y$ be a morphism of schemes locally of finite type over $S$. Let $p \in \mathbf{Z}$. A {\it bivariant class $c$ of degree $p$ for $f$} is given by a rule which assigns to every locally of finite type morphism $Y' \to Y$ 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, then $c \cap (Y'' \to Y')_*\alpha'' = (X'' \to X')_*(c \cap \alpha'')$ for all $\alpha''$ on $Y''$ where $X'' = Y'' \times_Y X$, \item if $Y'' \to Y'$ is flat locally of finite type of fixed relative dimension, then $c \cap (Y'' \to Y')^*\alpha' = (X'' \to X')^*(c \cap \alpha')$ for all $\alpha'$ on $Y'$, and \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 $A^p(X \to Y)$.


chow-definition-chow-cohomology

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. The {\it Chow cohomology} of $X$ is the graded $\mathbf{Z}$-algebra $A^*(X)$ whose degree $p$ component is $A^p(X \to X)$.


chow-definition-first-chern-class

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. The {\it first Chern class} $c_1(\mathcal{L}) \in A^1(X)$ of $\mathcal{L}$ is the bivariant class of Lemma \ref{lemma-cap-c1-bivariant}.


chow-definition-chern-classes

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ is integral and $n = \dim_\delta(X)$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $(\pi : P \to X, \mathcal{O}_P(1))$ be the projective space bundle associated to $\mathcal{E}$. \begin{enumerate} \item By Lemma \ref{lemma-chow-ring-projective-bundle} there are elements $c_i \in \CH_{n - i}(X)$, $i = 0, \ldots, r$ such that $c_0 = [X]$, and \begin{equation} \label{equation-chern-classes} \sum\nolimits_{i = 0}^r (-1)^i c_1(\mathcal{O}_P(1))^i \cap \pi^*c_{r - i} = 0. \end{equation} \item With notation as above we set $c_i(\mathcal{E}) \cap [X] = c_i$ as an element of $\CH_{n - i}(X)$. We call these the {\it Chern classes of $\mathcal{E}$ on $X$}. \item The {\it total Chern class of $\mathcal{E}$ on $X$} is the combination $$ c({\mathcal E}) \cap [X] = c_0({\mathcal E}) \cap [X] + c_1({\mathcal E}) \cap [X] + \ldots + c_r({\mathcal E}) \cap [X] $$ which is an element of $\CH_*(X) = \bigoplus_{k \in \mathbf{Z}} \CH_k(X)$. \end{enumerate}


chow-definition-cap-chern-classes

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. We define, for every integer $k$ and any $0 \leq j \leq r$, an operation $$ c_j(\mathcal{E}) \cap - : Z_k(X) \to \CH_{k - j}(X) $$ called {\it intersection with the $j$th Chern class of $\mathcal{E}$}. \begin{enumerate} \item Given an integral closed subscheme $i : W \to X$ of $\delta$-dimension $k$ we define $$ c_j(\mathcal{E}) \cap [W] = i_*(c_j({i^*\mathcal{E}}) \cap [W]) \in \CH_{k - j}(X) $$ where $c_j({i^*\mathcal{E}}) \cap [W]$ is as defined in Definition \ref{definition-chern-classes}. \item For a general $k$-cycle $\alpha = \sum n_i [W_i]$ we set $$ c_j(\mathcal{E}) \cap \alpha = \sum n_i c_j(\mathcal{E}) \cap [W_i] $$ \end{enumerate}


chow-definition-chern-classes-final

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_X$-module of rank $r$. 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-cap-cp-bivariant}. 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 $X$.


chow-definition-degree-zero-cycle

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


chow-definition-defined-on-perfect

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $E \in D(\mathcal{O}_X)$ be a perfect object. If $E$ is isomorphic in $D(\mathcal{O}_X)$ to a finite complex $\mathcal{E}^\bullet$ of finite locally free $\mathcal{O}_X$-modules, then we say {\it Chern classes of $E$ are defined}. If this is the case, then we define $c(E) = c(\mathcal{E}^\bullet) \in A^*(X)$, $ch(E) = ch(\mathcal{E}^\bullet) \in A^*(X) \otimes \mathbf{Q}$, and $P_p(E) = P_p(\mathcal{E}^\bullet) \in A^p(X)$.


chow-definition-localized-chern

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $i : Z \to X$ be a closed immersion. Let $E \in D(\mathcal{O}_X)$ be a perfect object whose Chern classes are defined. \begin{enumerate} \item If the restriction $E|_{X \setminus Z}$ is zero, then for all $p \geq 0$ we define $$ P_p(Z \to X, E) \in A^p(Z \to X) $$ by the construction given above and we define the {\it localized Chern character} by the formula $$ ch(Z \to X, E) = \sum\nolimits_{p = 0, 1, 2, \ldots} \frac{P_p(Z \to X, E)}{p!} \quad\text{in}\quad A^*(Z \to X) \otimes \mathbf{Q} $$ \item If the restriction $E|_{X \setminus Z}$ is isomorphic to a finite locally free $\mathcal{O}_{X \setminus Z}$-module of rank $< p$ sitting in cohomological degree $0$, then we define the {\it localized $p$th Chern class} $c_p(Z \to X, E)$ by the construction above. \end{enumerate}


chow-definition-lci-gysin

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $f : X \to Y$ be a local complete intersection morphism of schemes locally of finite type over $S$. We say {\it the gysin map for $f$ exists} if we can write $f = g \circ i$ with $g$ smooth and $i$ an immersion. In this case we define the {\it gysin map} $f^! = i^! \circ g^* \in A^*(X \to Y)$ as above.


chow-definition-determinant

Let $R$ be a local ring with maximal ideal $\mathfrak m$ and residue field $\kappa$. Let $M$ be a finite length $R$-module. Say $l = \text{length}_R(M)$. \begin{enumerate} \item Given elements $x_1, \ldots, x_r \in M$ we denote $\langle x_1, \ldots, x_r \rangle = Rx_1 + \ldots + Rx_r$ the $R$-submodule of $M$ generated by $x_1, \ldots, x_r$. \item We will say an $l$-tuple of elements $(e_1, \ldots, e_l)$ of $M$ is {\it admissible} if $\mathfrak m e_i \subset \langle e_1, \ldots, e_{i - 1} \rangle$ for $i = 1, \ldots, l$. \item A {\it symbol} $[e_1, \ldots, e_l]$ will mean $(e_1, \ldots, e_l)$ is an admissible $l$-tuple. \item An {\it admissible relation} between symbols is one of the following: \begin{enumerate} \item if $(e_1, \ldots, e_l)$ is an admissible sequence and for some $1 \leq a \leq l$ we have $e_a \in \langle e_1, \ldots, e_{a - 1}\rangle$, then $[e_1, \ldots, e_l] = 0$, \item if $(e_1, \ldots, e_l)$ is an admissible sequence and for some $1 \leq a \leq l$ we have $e_a = \lambda e'_a + x$ with $\lambda \in R^*$, and $x \in \langle e_1, \ldots, e_{a - 1}\rangle$, then $$ [e_1, \ldots, e_l] = \overline{\lambda} [e_1, \ldots, e_{a - 1}, e'_a, e_{a + 1}, \ldots, e_l] $$ where $\overline{\lambda} \in \kappa^*$ is the image of $\lambda$ in the residue field, and \item if $(e_1, \ldots, e_l)$ is an admissible sequence and $\mathfrak m e_a \subset \langle e_1, \ldots, e_{a - 2}\rangle$ then $$ [e_1, \ldots, e_l] = - [e_1, \ldots, e_{a - 2}, e_a, e_{a - 1}, e_{a + 1}, \ldots, e_l]. $$ \end{enumerate} \item We define the {\it determinant of the finite length $R$-module $M$} to be $$ \det\nolimits_\kappa(M) = \left\{ \frac{\kappa\text{-vector space generated by symbols}} {\kappa\text{-linear combinations of admissible relations}} \right\} $$ \end{enumerate}


chow-definition-periodic-determinant

Let $R$ be a local ring with residue field $\kappa$. Let $(M, \varphi, \psi)$ be a $(2, 1)$-periodic complex over $R$. Assume that $M$ has finite length and that $(M, \varphi, \psi)$ is exact. The {\it determinant of $(M, \varphi, \psi)$} is the element $$ \det\nolimits_\kappa(M, \varphi, \psi) \in \kappa^* $$ such that the composition $$ \det\nolimits_\kappa(M) \xrightarrow{\gamma_\psi \circ \sigma \circ \gamma_\varphi^{-1}} \det\nolimits_\kappa(M) $$ is multiplication by $(-1)^{\text{length}_R(I_\varphi)\text{length}_R(I_\psi)} \det\nolimits_\kappa(M, \varphi, \psi)$.


chow-definition-symbol-M

Let $A$ be a Noetherian local ring with residue field $\kappa$. Let $a, b \in A$. Let $M$ be a finite $A$-module of dimension $1$ such that $a, b$ are nonzerodivisors on $M$. We define the {\it symbol associated to $M, a, b$} to be the element $$ d_M(a, b) = \det\nolimits_\kappa(M/abM, a, b) \in \kappa^* $$


chow-definition-tame-symbol

Let $A$ be a Noetherian local domain of dimension $1$ with residue field $\kappa$. Let $K$ be the fraction field of $A$. We define the {\it tame symbol} of $A$ to be the map $$ K^* \times K^* \longrightarrow \kappa^*, \quad (x, y) \longmapsto d_A(x, y) $$ where $d_A(x, y)$ is extended to $K^* \times K^*$ by the multiplicativity of Lemma \ref{lemma-multiplicativity-symbol}.


flat-definition-one-step-devissage

Let $S$ be a scheme. Let $X$ be locally of finite type over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. Let $s \in S$ be a point. A {\it one step d\'evissage of $\mathcal{F}/X/S$ over $s$} is given by morphisms of schemes over $S$ $$ \xymatrix{ X & Z \ar[l]_i \ar[r]^\pi & Y } $$ and a quasi-coherent $\mathcal{O}_Z$-module $\mathcal{G}$ of finite type such that \begin{enumerate} \item $X$, $S$, $Z$ and $Y$ are affine, \item $i$ is a closed immersion of finite presentation, \item $\mathcal{F} \cong i_*\mathcal{G}$, \item $\pi$ is finite, and \item the structure morphism $Y \to S$ is smooth with geometrically irreducible fibres of dimension $\dim(\text{Supp}(\mathcal{F}_s))$. \end{enumerate} In this case we say $(Z, Y, i, \pi, \mathcal{G})$ is a one step d\'evissage of $\mathcal{F}/X/S$ over $s$.


flat-definition-one-step-devissage-at-x

Let $S$ be a scheme. Let $X$ be locally of finite type over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. Let $x \in X$ be a point with image $s$ in $S$. A {\it one step d\'evissage of $\mathcal{F}/X/S$ at $x$} is a system $(Z, Y, i, \pi, \mathcal{G}, z, y)$, where $(Z, Y, i, \pi, \mathcal{G})$ is a one step d\'evissage of $\mathcal{F}/X/S$ over $s$ and \begin{enumerate} \item $\dim_x(\text{Supp}(\mathcal{F}_s)) = \dim(\text{Supp}(\mathcal{F}_s))$, \item $z \in Z$ is a point with $i(z) = x$ and $\pi(z) = y$, \item we have $\pi^{-1}(\{y\}) = \{z\}$, \item the extension $\kappa(s) \subset \kappa(y)$ is purely transcendental. \end{enumerate}


flat-definition-shrink

Let $S$, $X$, $\mathcal{F}$, $x$, $s$ be as in Definition \ref{definition-one-step-devissage-at-x}. Let $(Z, Y, i, \pi, \mathcal{G}, z, y)$ be a one step d\'evissage of $\mathcal{F}/X/S$ at $x$. Let us define a {\it standard shrinking} of this situation to be given by standard opens $S' \subset S$, $X' \subset X$, $Z' \subset Z$, and $Y' \subset Y$ such that $s \in S'$, $x \in X'$, $z \in Z'$, and $y \in Y'$ and such that $$ (Z', Y', i|_{Z'}, \pi|_{Z'}, \mathcal{G}|_{Z'}, z, y) $$ is a one step d\'evissage of $\mathcal{F}|_{X'}/X'/S'$ at $x$.


flat-definition-complete-devissage

Let $S$ be a scheme. Let $X$ be locally of finite type over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. Let $s \in S$ be a point. A {\it complete d\'evissage of $\mathcal{F}/X/S$ over $s$} is given by a diagram $$ \xymatrix{ X & Z_1 \ar[l]^{i_1} \ar[d]^{\pi_1} \\ & Y_1 & Z_2 \ar[l]^{i_2} \ar[d]^{\pi_2} \\ & & Y_2 & Z_3 \ar[l] \ar[d] \\ & & & ... & ... \ar[l] \ar[d] \\ & & & & Y_n } $$ of schemes over $S$, finite type quasi-coherent $\mathcal{O}_{Z_k}$-modules $\mathcal{G}_k$, and $\mathcal{O}_{Y_k}$-module maps $$ \alpha_k : \mathcal{O}_{Y_k}^{\oplus r_k} \longrightarrow \pi_{k, *}\mathcal{G}_k, \quad k = 1, \ldots, n $$ satisfying the following properties: \begin{enumerate} \item $(Z_1, Y_1, i_1, \pi_1, \mathcal{G}_1)$ is a one step d\'evissage of $\mathcal{F}/X/S$ over $s$, \item the map $\alpha_k$ induces an isomorphism $$ \kappa(\xi_k)^{\oplus r_k} \longrightarrow (\pi_{k, *}\mathcal{G}_k)_{\xi_k} \otimes_{\mathcal{O}_{Y_k, \xi_k}} \kappa(\xi_k) $$ where $\xi_k \in (Y_k)_s$ is the unique generic point, \item for $k = 2, \ldots, n$ the system $(Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k)$ is a one step d\'evissage of $\Coker(\alpha_{k - 1})/Y_{k - 1}/S$ over $s$, \item $\Coker(\alpha_n) = 0$. \end{enumerate} In this case we say that $(Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k)_{k = 1, \ldots, n}$ is a complete d\'evissage of $\mathcal{F}/X/S$ over $s$.


flat-definition-complete-devissage-at-x

Let $S$ be a scheme. Let $X$ be locally of finite type over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. Let $x \in X$ be a point with image $s \in S$. A {\it complete d\'evissage of $\mathcal{F}/X/S$ at $x$} is given by a system $$ (Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k, z_k, y_k)_{k = 1, \ldots, n} $$ such that $(Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k)$ is a complete d\'evissage of $\mathcal{F}/X/S$ over $s$, and such that \begin{enumerate} \item $(Z_1, Y_1, i_1, \pi_1, \mathcal{G}_1, z_1, y_1)$ is a one step d\'evissage of $\mathcal{F}/X/S$ at $x$, \item for $k = 2, \ldots, n$ the system $(Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, z_k, y_k)$ is a one step d\'evissage of $\Coker(\alpha_{k - 1})/Y_{k - 1}/S$ at $y_{k - 1}$. \end{enumerate}


flat-definition-shrink-complete

Let $S$, $X$, $\mathcal{F}$, $x$, $s$ be as in Definition \ref{definition-complete-devissage-at-x}. Consider a complete d\'evissage $(Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k, z_k, y_k)_{k = 1, \ldots, n}$ of $\mathcal{F}/X/S$ at $x$. Let us define a {\it standard shrinking} of this situation to be given by standard opens $S' \subset S$, $X' \subset X$, $Z'_k \subset Z_k$, and $Y'_k \subset Y_k$ such that $s_k \in S'$, $x_k \in X'$, $z_k \in Z'$, and $y_k \in Y'$ and such that $$ (Z'_k, Y'_k, i'_k, \pi'_k, \mathcal{G}'_k, \alpha'_k, z_k, y_k)_{k = 1, \ldots, n} $$ is a one step d\'evissage of $\mathcal{F}'/X'/S'$ at $x$ where $\mathcal{G}'_k = \mathcal{G}_k|_{Z'_k}$ and $\mathcal{F}' = \mathcal{F}|_{X'}$.


flat-definition-elementary-etale-neighbourhood

Let $R \to S$ be a ring map. Let $\mathfrak q$ be a prime of $S$ lying over the prime $\mathfrak p$ of $R$. A {\it elementary \'etale localization of the ring map $R \to S$ at $\mathfrak q$} is given by a commutative diagram of rings and accompanying primes $$ \xymatrix{ S \ar[r] & S' \\ R \ar[u] \ar[r] & R' \ar[u] } \quad\quad \xymatrix{ \mathfrak q \ar@{-}[r] & \mathfrak q' \\ \mathfrak p \ar@{-}[u] \ar@{-}[r] & \mathfrak p' \ar@{-}[u] } $$ such that $R \to R'$ and $S \to S'$ are \'etale ring maps and $\kappa(\mathfrak p) = \kappa(\mathfrak p')$ and $\kappa(\mathfrak q) = \kappa(\mathfrak q')$.


flat-definition-complete-devissage-algebra

Let $R \to S$ be a finite type ring map. Let $\mathfrak r$ be a prime of $R$. Let $N$ be a finite $S$-module. A {\it complete d\'evissage of $N/S/R$ over $\mathfrak r$} is given by $R$-algebra maps $$ \xymatrix{ & A_1 & & A_2 & & ... & & A_n \\ S \ar[ru] & & B_1 \ar[lu] \ar[ru] & & ... \ar[lu] \ar[ru] & & ... \ar[lu] \ar[ru] & & B_n \ar[lu] } $$ finite $A_i$-modules $M_i$ and $B_i$-module maps $\alpha_i : B_i^{\oplus r_i} \to M_i$ such that \begin{enumerate} \item $S \to A_1$ is surjective and of finite presentation, \item $B_i \to A_{i + 1}$ is surjective and of finite presentation, \item $B_i \to A_i$ is finite, \item $R \to B_i$ is smooth with geometrically irreducible fibres, \item $N \cong M_1$ as $S$-modules, \item $\Coker(\alpha_i) \cong M_{i + 1}$ as $B_i$-modules, \item $\alpha_i : \kappa(\mathfrak p_i)^{\oplus r_i} \to M_i \otimes_{B_i} \kappa(\mathfrak p_i)$ is an isomorphism where $\mathfrak p_i = \mathfrak rB_i$, and \item $\Coker(\alpha_n) = 0$. \end{enumerate} In this situation we say that $(A_i, B_i, M_i, \alpha_i)_{i = 1, \ldots, n}$ is a complete d\'evissage of $N/S/R$ over $\mathfrak r$.


flat-definition-complete-devissage-at-x-algebra

Let $R \to S$ be a finite type ring map. Let $\mathfrak q$ be a prime of $S$ lying over the prime $\mathfrak r$ of $R$. Let $N$ be a finite $S$-module. A {\it complete d\'evissage of $N/S/R$ at $\mathfrak q$} is given by a complete d\'evissage $(A_i, B_i, M_i, \alpha_i)_{i = 1, \ldots, n}$ of $N/S/R$ over $\mathfrak r$ and prime ideals $\mathfrak q_i \subset B_i$ lying over $\mathfrak r$ such that \begin{enumerate} \item $\kappa(\mathfrak r) \subset \kappa(\mathfrak q_i)$ is purely transcendental, \item there is a unique prime $\mathfrak q'_i \subset A_i$ lying over $\mathfrak q_i \subset B_i$, \item $\mathfrak q = \mathfrak q'_1 \cap S$ and $\mathfrak q_i = \mathfrak q'_{i + 1} \cap A_i$, \item $R \to B_i$ has relative dimension $\dim_{\mathfrak q_i}(\text{Supp}(M_i \otimes_R \kappa(\mathfrak r)))$. \end{enumerate}


flat-definition-impurity

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


flat-definition-pure

Let $f : X \to S$ be a morphism of schemes which is of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. \begin{enumerate} \item Let $s \in S$. We say $\mathcal{F}$ is {\it pure along $X_s$} if there is no impurity $(g : T \to S, t' \leadsto t, \xi)$ of $\mathcal{F}$ above $s$ with $(T, t) \to (S, s)$ an elementary \'etale neighbourhood. \item We say $\mathcal{F}$ is {\it universally pure along $X_s$} if there does not exist any impurity of $\mathcal{F}$ above $s$. \item We say that $X$ is {\it pure along $X_s$} if $\mathcal{O}_X$ is pure along $X_s$. \item We say $\mathcal{F}$ is {\it universally $S$-pure}, or {\it universally pure relative to $S$} if $\mathcal{F}$ is universally pure along $X_s$ for every $s \in S$. \item We say $\mathcal{F}$ is {\it $S$-pure}, or {\it pure relative to $S$} if $\mathcal{F}$ is pure along $X_s$ for every $s \in S$. \item We say that $X$ is {\it $S$-pure} or {\it pure relative to $S$} if $\mathcal{O}_X$ is pure relative to $S$. \end{enumerate}


flat-definition-flat-dimension-n

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


flat-definition-flattening

Let $X \to S$ be a morphism of schemes. 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 representable by a scheme $S'$ over $S$. We say that the {\it universal flattening of $X$ exists} if the universal flattening of $\mathcal{O}_X$ exists.


flat-definition-flattening-stratification

Let $X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. We say that $\mathcal{F}$ has a {\it flattening stratification} if the functor $F_{flat}$ defined in Situation \ref{situation-flat} is representable by a monomorphism $S' \to S$ associated to a stratification of $S$ by locally closed subschemes. We say that $X$ has a {\it flattening stratification} if $\mathcal{O}_X$ has a flattening stratification.


flat-definition-h-covering

Let $T$ be a scheme. A {\it h covering of $T$} is a family of morphisms $\{f_i : T_i \to T\}_{i \in I}$ such that each $f_i$ is locally of finite presentation and one of the equivalent conditions of Lemma \ref{lemma-equivalence-h-v-locally-finite-presentation} is satisfied.


flat-definition-big-h-site

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


flat-definition-standard-h

Let $T$ be an affine scheme. A {\it standard h covering} of $T$ is a family $\{f_i : T_i \to T\}_{i = 1, \ldots, n}$ with each $T_i$ affine, with $f_i$ of finite presentation satisfying either of the following equivalent conditions: (1) $\{U_i \to U\}$ can be refined by a standard ph covering or (2) $\{U_i \to U\}$ is a V covering.


flat-definition-big-small-h

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


curves-definition-normal-projective-model

Let $k$ be a field. Let $X$ be a curve. A {\it nonsingular projective model of $X$} is a pair $(Y, \varphi)$ where $Y$ is a nonsingular projective curve and $\varphi : k(X) \to k(Y)$ is an isomorphism of function fields.


curves-definition-linear-series

Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Let $d \geq 0$ and $r \geq 0$. A {\it linear series of degree $d$ and dimension $r$} is a pair $(\mathcal{L}, V)$ where $\mathcal{L}$ in an invertible $\mathcal{O}_X$-module of degree $d$ (Varieties, Definition \ref{varieties-definition-degree-invertible-sheaf}) and $V \subset H^0(X, \mathcal{L})$ is a $k$-subvector space of dimension $r + 1$. We will abbreviate this by saying $(\mathcal{L}, V)$ is a {\it $\mathfrak g^r_d$} on $X$.


curves-definition-genus

Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_X) = k$. Then the {\it genus} of $X$ is $g = \dim_k H^1(X, \mathcal{O}_X)$.


curves-definition-geometric-genus

Let $k$ be a field. Let $X$ be a geometrically irreducible curve over $k$. The {\it geometric genus} of $X$ is the genus of a smooth projective model of $X$ possibly defined over an extension field of $k$ as in Lemma \ref{lemma-smooth-models}.


curves-definition-multicross

Let $k$ be an algebraically closed field. Let $X$ be an algebraic $1$-dimensional $k$-scheme. Let $x \in X$ be a closed point. We say $x$ defines a {\it multicross singularity} if the completion $\mathcal{O}_{X, x}^\wedge$ is isomorphic to (\ref{equation-multicross}) for some $n \geq 2$. We say $x$ is a {\it node}, or an {\it ordinary double point}, or {\it defines a nodal singularity} if $n = 2$.


curves-definition-nodal

Let $k$ be a field. Let $X$ be a $1$-dimensional locally algebraic $k$-scheme. \begin{enumerate} \item We say a closed point $x \in X$ is a {\it node}, or an {\it ordinary double point}, or {\it defines a nodal singularity} if there exists an ordinary double point $\overline{x} \in X_{\overline{k}}$ mapping to $x$. \item We say the {\it singularities of $X$ are at-worst-nodal} if all closed points of $X$ are either in the smooth locus of the structure morphism $X \to \Spec(k)$ or are ordinary double points. \end{enumerate}


curves-definition-split-node

Let $k$ be a field. Let $X$ be a $1$-dimensional algebraic $k$-scheme. Let $x \in X$ be a closed point. We say $x$ is a {\it split node} if $x$ is a node, $\kappa(x) = k$, and the equivalent assertions of Remark \ref{remark-trivial-quadratic-extension} hold for $A = \mathcal{O}_{X, x}$.


curves-definition-nodal-family

Let $f : X \to S$ be a morphism of schemes. We say $f$ is {\it at-worst-nodal of relative dimension $1$} if $f$ satisfies the equivalent conditions of Lemma \ref{lemma-nodal-family}.


etale-cohomology-definition-etale-covering-initial

A family of morphisms $\{ \varphi_i : U_i \to X\}_{i \in I}$ is called an {\it \'etale covering} if each $\varphi_i$ is an \'etale morphism and their images cover $X$, i.e., $X = \bigcup_{i \in I} \varphi_i(U_i)$.


etale-cohomology-definition-presheaf

Let $\mathcal{C}$ be a category. A {\it presheaf of sets} (respectively, an {\it abelian presheaf}) on $\mathcal{C}$ is a functor $\mathcal{C}^{opp} \to \textit{Sets}$ (resp.\ $\textit{Ab}$).


etale-cohomology-definition-family-morphisms-fixed-target

Let $\mathcal{C}$ be a category. A {\it family of morphisms with fixed target} $\mathcal{U} = \{\varphi_i : U_i \to U\}_{i\in I}$ is the data of \begin{enumerate} \item an object $U \in \mathcal{C}$, \item a set $I$ (possibly empty), and \item for all $i\in I$, a morphism $\varphi_i : U_i \to U$ of $\mathcal{C}$ with target $U$. \end{enumerate}


etale-cohomology-definition-site

A {\it site}\footnote{What we call a site is a called a category endowed with a pretopology in \cite[Expos\'e II, D\'efinition 1.3]{SGA4}. In \cite{ArtinTopologies} it is called a category with a Grothendieck topology.} consists of a category $\mathcal{C}$ and a set $\text{Cov}(\mathcal{C})$ consisting of families of morphisms with fixed target called {\it coverings}, such that \begin{enumerate} \item (isomorphism) if $\varphi : V \to U$ is an isomorphism in $\mathcal{C}$, then $\{\varphi : V \to U\}$ is a covering, \item (locality) if $\{\varphi_i : U_i \to U\}_{i\in I}$ is a covering and for all $i \in I$ we are given a covering $\{\psi_{ij} : U_{ij} \to U_i \}_{j\in I_i}$, then $$ \{ \varphi_i \circ \psi_{ij} : U_{ij} \to U \}_{(i, j)\in \prod_{i\in I} \{i\} \times I_i} $$ is also a covering, and \item (base change) if $\{U_i \to U\}_{i\in I}$ is a covering and $V \to U$ is a morphism in $\mathcal{C}$, then \begin{enumerate} \item for all $i \in I$ the fibre product $U_i \times_U V$ exists in $\mathcal{C}$, and \item $\{U_i \times_U V \to V\}_{i\in I}$ is a covering. \end{enumerate} \end{enumerate}


etale-cohomology-definition-sheaf

A presheaf $\mathcal{F}$ of sets (resp. abelian presheaf) on a site $\mathcal{C}$ is said to be a {\it separated presheaf} if for all coverings $\{\varphi_i : U_i \to U\}_{i\in I} \in \text{Cov} (\mathcal{C})$ the map $$ \mathcal{F}(U) \longrightarrow \prod\nolimits_{i\in I} \mathcal{F}(U_i) $$ is injective. Here the map is $s \mapsto (s|_{U_i})_{i\in I}$. The presheaf $\mathcal{F}$ is a {\it sheaf} if for all coverings $\{\varphi_i : U_i \to U\}_{i\in I} \in \text{Cov} (\mathcal{C})$, the diagram \begin{equation} \label{equation-sheaf-axiom} \xymatrix{ \mathcal{F}(U) \ar[r] & \prod_{i\in I} \mathcal{F}(U_i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod_{i, j \in I} \mathcal{F}(U_i \times_U U_j), } \end{equation} where the first map is $s \mapsto (s|_{U_i})_{i\in I}$ and the two maps on the right are $(s_i)_{i\in I} \mapsto (s_i |_{U_i \times_U U_j})$ and $(s_i)_{i\in I} \mapsto (s_j |_{U_i \times_U U_j})$, is an equalizer diagram in the category of sets (resp.\ abelian groups).


etale-cohomology-definition-category-sheaves

We denote $\Sh(\mathcal{C})$ (resp.\ $\textit{Ab}(\mathcal{C})$) the full subcategory of $\textit{PSh}(\mathcal{C})$ (resp.\ $\textit{PAb}(\mathcal{C})$) whose objects are sheaves. This is the {\it category of sheaves of sets} (resp.\ {\it abelian sheaves}) on $\mathcal{C}$.


etale-cohomology-definition-0-cech

Let $\mathcal{F}$ be a presheaf on the site $\mathcal{C}$ and $\mathcal{U} = \{U_i \to U\} \in \text{Cov} (\mathcal{C})$. We define the {\it zeroth {\v C}ech cohomology group} of $\mathcal{F}$ with respect to $\mathcal{U}$ by $$ \check H^0 (\mathcal{U}, \mathcal{F}) = \left\{ (s_i)_{i\in I} \in \prod\nolimits_{i\in I }\mathcal{F}(U_i) \text{ such that } s_i|_{U_i \times_U U_j} = s_j |_{U_i \times_U U_j} \right\}. $$


etale-cohomology-definition-fpqc-covering

Let $T$ be a scheme. An {\it fpqc covering} of $T$ is a family $\{ \varphi_i : T_i \to T\}_{i \in I}$ such that \begin{enumerate} \item each $\varphi_i$ is a flat morphism and $\bigcup_{i\in I} \varphi_i(T_i) = T$, and \item for each affine open $U \subset T$ there exists a finite set $K$, a map $\mathbf{i} : K \to I$ and affine opens $U_{\mathbf{i}(k)} \subset T_{\mathbf{i}(k)}$ such that $U = \bigcup_{k \in K} \varphi_{\mathbf{i}(k)}(U_{\mathbf{i}(k)})$. \end{enumerate}


etale-cohomology-definition-sheaf-property-fpqc

Let $S$ be a scheme. The category of schemes over $S$ is denoted $\Sch/S$. Consider a functor $\mathcal{F} : (\Sch/S)^{opp} \to \textit{Sets}$, in other words a presheaf of sets. We say $\mathcal{F}$ {\it satisfies the sheaf property for the fpqc topology} if for every fpqc covering $\{U_i \to U\}_{i \in I}$ of schemes over $S$ the diagram (\ref{equation-sheaf-axiom}) is an equalizer diagram.


etale-cohomology-definition-descent-datum

Let $\mathcal{U} = \{ t_i : T_i \to T\}_{i \in I}$ be a family of morphisms of schemes with fixed target. A {\it descent datum} for quasi-coherent sheaves with respect to $\mathcal{U}$ is a collection $((\mathcal{F}_i)_{i \in I}, (\varphi_{ij})_{i, j \in I})$ where \begin{enumerate} \item $\mathcal{F}_i$ is a quasi-coherent sheaf on $T_i$, and \item $\varphi_{ij} : \text{pr}_0^* \mathcal{F}_i \to \text{pr}_1^* \mathcal{F}_j$ is an isomorphism of modules on $T_i \times_T T_j$, \end{enumerate} such that the {\it cocycle condition} holds: the diagrams $$ \xymatrix{ \text{pr}_0^*\mathcal{F}_i \ar[dr]_{\text{pr}_{02}^*\varphi_{ik}} \ar[rr]^{\text{pr}_{01}^*\varphi_{ij}} & & \text{pr}_1^*\mathcal{F}_j \ar[dl]^{\text{pr}_{12}^*\varphi_{jk}} \\ & \text{pr}_2^*\mathcal{F}_k } $$ commute on $T_i \times_T T_j \times_T T_k$. This descent datum is called {\it effective} if there exist a quasi-coherent sheaf $\mathcal{F}$ over $T$ and $\mathcal{O}_{T_i}$-module isomorphisms $\varphi_i : t_i^* \mathcal{F} \cong \mathcal{F}_i$ compatible with the maps $\varphi_{ij}$, namely $$ \varphi_{ij} = \text{pr}_1^* (\varphi_j) \circ \text{pr}_0^* (\varphi_i)^{-1}. $$


etale-cohomology-definition-descent-datum-modules

Let $A \to B$ be a ring map and $N$ a $B$-module. A {\it descent datum} for $N$ with respect to $A \to B$ is an isomorphism $\varphi : N \otimes_A B \cong B \otimes_A N$ of $B \otimes_A B$-modules such that the diagram of $B \otimes_A B \otimes_A B$-modules $$ \xymatrix{ {N \otimes_A B \otimes_A B} \ar[dr]_{\varphi_{02}} \ar[rr]^{\varphi_{01}} & & {B \otimes_A N \otimes_A B} \ar[dl]^{\varphi_{12}} \\ & {B \otimes_A B \otimes_A N} } $$ commutes where $\varphi_{01} = \varphi \otimes \text{id}_B$ and similarly for $\varphi_{12}$ and $\varphi_{02}$.


etale-cohomology-definition-effective-modules

A descent datum $(N, \varphi)$ is called {\it effective} if there exists an $A$-module $M$ such that $(N, \varphi) \cong (B \otimes_A M, \varphi_\text{can})$, with the obvious notion of isomorphism of descent data.