spaces-morphisms-lemma-proper-local

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item $f$ is proper, \item for every scheme $Z$ and every morphism $Z \to Y$ the projection $Z \times_Y X \to Z$ is proper, \item for every affine scheme $Z$ and every morphism $Z \to Y$ the projection $Z \times_Y X \to Z$ is proper, \item there exists a scheme $V$ and a surjective \'etale morphism $V \to Y$ such that $V \times_Y X \to V$ is proper, and \item there exists a Zariski covering $Y = \bigcup Y_i$ such that each of the morphisms $f^{-1}(Y_i) \to Y_i$ is proper. \end{enumerate}


spaces-morphisms-lemma-base-change-proper

A base change of a proper morphism is proper.


spaces-morphisms-lemma-composition-proper

A composition of proper morphisms is proper.


spaces-morphisms-lemma-closed-immersion-proper

A closed immersion of algebraic spaces is a proper morphism of algebraic spaces.


spaces-morphisms-lemma-universally-closed-permanence

Let $S$ be a scheme. Consider a commutative diagram of algebraic spaces $$ \xymatrix{ X \ar[rr] \ar[rd] & & Y \ar[ld] \\ & B & } $$ over $S$. \begin{enumerate} \item If $X \to B$ is universally closed and $Y \to B$ is separated, then the morphism $X \to Y$ is universally closed. In particular, the image of $|X|$ in $|Y|$ is closed. \item If $X \to B$ is proper and $Y \to B$ is separated, then the morphism $X \to Y$ is proper. \end{enumerate}


spaces-morphisms-lemma-image-proper-is-proper

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $f : X \to Y$ be a morphism of algebraic spaces over $B$. If $X$ is universally closed over $B$ and $f$ is surjective then $Y$ is universally closed over $B$. In particular, if also $Y$ is separated and of finite type over $B$, then $Y$ is proper over $B$.


spaces-morphisms-lemma-scheme-theoretic-image-is-proper

Let $S$ be a scheme. Let $$ \xymatrix{ X \ar[rr]_h \ar[rd]_f & & Y \ar[ld]^g \\ & B } $$ be a commutative diagram of morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $X \to B$ is a proper morphism, \item $Y \to B$ is separated and locally of finite type, \end{enumerate} Then the scheme theoretic image $Z \subset Y$ of $h$ is proper over $B$ and $X \to Z$ is surjective.


spaces-morphisms-lemma-separated-diagonal-proper

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is separated, \item $\Delta_{X/Y} : X \to X \times_Y X$ is universally closed, and \item $\Delta_{X/Y} : X \to X \times_Y X$ is proper. \end{enumerate}


spaces-morphisms-lemma-push-down-solution

Let $S$ be a scheme. Let $f : X \to Y$ be a separated morphism of algebraic spaces over $S$. Suppose given a diagram $$ \xymatrix{ \Spec(K') \ar[r] \ar[d] & \Spec(K) \ar[r] & X \ar[d] \\ \Spec(A') \ar[r] \ar[rru] & \Spec(A) \ar[r] \ar@{-->}[ru] & Y } $$ as in Definition \ref{definition-valuative-criterion} with $K \subset K'$ arbitrary. Then the dotted arrow exists making the diagram commute.


spaces-morphisms-lemma-usual-enough

Let $S$ be a scheme. Let $f : X \to Y$ be a separated morphism of algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item $f$ satisfies the existence part of the valuative criterion as in Definition \ref{definition-valuative-criterion}, \item 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 a dotted arrow, i.e., $f$ satisfies the existence part of the valuative criterion as in Schemes, Definition \ref{schemes-definition-valuative-criterion}. \end{enumerate}


spaces-morphisms-lemma-base-change-valuative-criteria

The base change of a morphism of algebraic spaces which satisfies the existence part of (resp.\ uniqueness part of) the valuative criterion by any morphism of algebraic spaces satisfies the existence part of (resp.\ uniqueness part of) the valuative criterion.


spaces-morphisms-lemma-composition-valuative-criteria

The composition of two morphisms of algebraic spaces which satisfy the (existence part of, resp.\ uniqueness part of) the valuative criterion satisfies the (existence part of, resp.\ uniqueness part of) the valuative criterion.


spaces-morphisms-lemma-quasi-compact-existence-universally-closed

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $f$ is quasi-compact, and \item $f$ satisfies the existence part of the valuative criterion. \end{enumerate} Then $f$ is universally closed.


spaces-morphisms-lemma-characterize-universally-closed-quasi-separated

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item If $f$ is quasi-separated and universally closed, then $f$ satisfies the existence part of the valuative criterion. \item If $f$ is quasi-compact and quasi-separated, then $f$ is universally closed if and only if the existence part of the valuative criterion holds. \end{enumerate}


spaces-morphisms-lemma-characterize-universally-closed-separated

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-compact and separated. Then the following are equivalent \begin{enumerate} \item $f$ is universally closed, \item the existence part of the valuative criterion holds as in Definition \ref{definition-valuative-criterion}, and \item 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 a dotted arrow, i.e., $f$ satisfies the existence part of the valuative criterion as in Schemes, Definition \ref{schemes-definition-valuative-criterion}. \end{enumerate}


spaces-morphisms-lemma-lift-valuation-ring-through-flat-morphism

Let $S$ be a scheme. Let $f : X \to Y$ be a flat morphism of algebraic spaces over $S$. Let $\Spec(A) \to Y$ be a morphism where $A$ is a valuation ring. If the closed point of $\Spec(A)$ maps to a point of $|Y|$ in the image of $|X| \to |Y|$, then there exists a commutative diagram $$ \xymatrix{ \Spec(A') \ar[r] \ar[d] & X \ar[d] \\ \Spec(A) \ar[r] & Y } $$ where $A \to A'$ is an extension of valuation rings (More on Algebra, Definition \ref{more-algebra-definition-extension-valuation-rings}).


spaces-morphisms-lemma-refined-valuative-criterion-universally-closed

Let $S$ be a scheme. Let $f : X \to Y$ and $h : U \to X$ be morphisms of algebraic spaces over $S$. If \begin{enumerate} \item $f$ and $h$ are quasi-compact, \item $|h|(|U|)$ is dense in $|X|$, and \end{enumerate} given any commutative solid diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & U \ar[r] & X \ar[d] \\ \Spec(A) \ar[rr] \ar@{-->}[rru] & & Y } $$ where $A$ is a valuation ring with field of fractions $K$ \begin{enumerate} \item[(3)] there exists at most one dotted arrow making the diagram commute, and \item[(4)] 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 $$ \xymatrix{ \Spec(K') \ar[r] \ar[d] & \Spec(K) \ar[r] & U \ar[r] & X \ar[d] \\ \Spec(A') \ar[r] \ar[rrru] & \Spec(A) \ar[rr] & & Y } $$ \end{enumerate} then $f$ is universally closed. If moreover \begin{enumerate} \item[(5)] $f$ is quasi-separated \end{enumerate} then $f$ is separated and universally closed.


spaces-morphisms-lemma-separated-implies-valuative

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is separated, then $f$ satisfies the uniqueness part of the valuative criterion.


spaces-morphisms-lemma-valuative-criterion-separatedness

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item the morphism $f$ is quasi-separated, and \item the morphism $f$ satisfies the uniqueness part of the valuative criterion. \end{enumerate} Then $f$ is separated.


spaces-morphisms-lemma-characterize-separated-and-universally-closed

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-compact and quasi-separated. Then the following are equivalent \begin{enumerate} \item $f$ is separated and universally closed, \item the valuative criterion holds as in Definition \ref{definition-valuative-criterion}, \item 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 a unique dotted arrow, i.e., $f$ satisfies the valuative criterion as in Schemes, Definition \ref{schemes-definition-valuative-criterion}. \end{enumerate}


spaces-morphisms-lemma-characterize-proper

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is of finite type and quasi-separated. Then the following are equivalent \begin{enumerate} \item $f$ is proper, \item the valuative criterion holds as in Definition \ref{definition-valuative-criterion}, \item 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 a unique dotted arrow, i.e., $f$ satisfies the valuative criterion as in Schemes, Definition \ref{schemes-definition-valuative-criterion}. \end{enumerate}


spaces-morphisms-lemma-integral-local

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is representable and integral (resp.\ finite), \item $f$ is integral (resp.\ finite), \item there exists a scheme $V$ and a surjective \'etale morphism $V \to Y$ such that $V \times_Y X \to V$ is integral (resp. finite), and \item there exists a Zariski covering $Y = \bigcup Y_i$ such that each of the morphisms $f^{-1}(Y_i) \to Y_i$ is integral (resp.\ finite). \end{enumerate}


spaces-morphisms-lemma-composition-integral

The composition of integral (resp.\ finite) morphisms is integral (resp.\ finite).


spaces-morphisms-lemma-base-change-integral

The base change of an integral (resp.\ finite) morphism is integral (resp.\ finite).


spaces-morphisms-lemma-finite-integral

A finite morphism of algebraic spaces is integral. An integral morphism of algebraic spaces which is locally of finite type is finite.


spaces-morphisms-lemma-integral-universally-closed

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item $f$ is integral, and \item $f$ is affine and universally closed. \end{enumerate}


spaces-morphisms-lemma-finite-quasi-finite

A finite morphism of algebraic spaces is quasi-finite.


spaces-morphisms-lemma-finite-proper

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item $f$ is finite, and \item $f$ is affine and proper. \end{enumerate}


spaces-morphisms-lemma-closed-immersion-finite

A closed immersion is finite (and a fortiori integral).


spaces-morphisms-lemma-finite-union-finite

Let $S$ be a scheme. Let $X_i \to Y$, $i = 1, \ldots, n$ be finite morphisms of algebraic spaces over $S$. Then $X_1 \amalg \ldots \amalg X_n \to Y$ is finite too.


spaces-morphisms-lemma-finite-permanence

Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of algebraic spaces over $S$. \begin{enumerate} \item If $g \circ f$ is finite and $g$ separated then $f$ is finite. \item If $g \circ f$ is integral and $g$ separated then $f$ is integral. \end{enumerate}


spaces-morphisms-lemma-finite-locally-free-local

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is representable and finite locally free, \item $f$ is finite locally free, \item there exists a scheme $V$ and a surjective \'etale morphism $V \to Y$ such that $V \times_Y X \to V$ is finite locally free, and \item there exists a Zariski covering $Y = \bigcup Y_i$ such that each morphism $f^{-1}(Y_i) \to Y_i$ is finite locally free. \end{enumerate}


spaces-morphisms-lemma-composition-finite-locally-free

The composition of finite locally free morphisms is finite locally free.


spaces-morphisms-lemma-base-change-finite-locally-free

The base change of a finite locally free morphism is finite locally free.


spaces-morphisms-lemma-finite-flat

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is finite locally free, \item $f$ is finite, flat, and locally of finite presentation. \end{enumerate} If $Y$ is locally Noetherian these are also equivalent to \begin{enumerate} \item[(3)] $f$ is finite and flat. \end{enumerate}


spaces-morphisms-lemma-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. There exists a quasi-coherent sheaf of $\mathcal{O}_X$-algebras $\mathcal{A}' \subset \mathcal{A}$ such that for any affine object $U$ of $X_\etale$ the ring $\mathcal{A}'(U) \subset \mathcal{A}(U)$ is the integral closure of $\mathcal{O}_X(U)$ in $\mathcal{A}(U)$.


spaces-morphisms-lemma-properties-normalization

Let $S$ be a scheme. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $Y \to X' \to X$ be the normalization of $X$ in $Y$. \begin{enumerate} \item If $W \to X$ is an \'etale morphism of algebraic spaces over $S$, then $W \times_X X'$ is the normalization of $W$ in $W \times_X Y$. \item If $Y$ and $X$ are representable, then $Y'$ is representable and is canonically isomorphic to the normalization of the scheme $X$ in the scheme $Y$ as constructed in Morphisms, Section \ref{morphisms-section-normalization}. \end{enumerate}


spaces-morphisms-lemma-characterize-normalization

Let $S$ be a scheme. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. The factorization $f = \nu \circ f'$, where $\nu : X' \to X$ is the normalization of $X$ in $Y$ is characterized by the following two properties: \begin{enumerate} \item the morphism $\nu$ is integral, and \item for any factorization $f = \pi \circ g$, with $\pi : Z \to X$ integral, there exists a commutative diagram $$ \xymatrix{ Y \ar[d]_{f'} \ar[r]_g & Z \ar[d]^\pi \\ X' \ar[ru]^h \ar[r]^\nu & X } $$ for a unique morphism $h : X' \to Z$. \end{enumerate} Moreover, in (2) the morphism $h : X' \to Z$ is the normalization of $Z$ in $Y$.


spaces-morphisms-lemma-normalization-in-reduced

Let $S$ be a scheme. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $X' \to X$ be the normalization of $X$ in $Y$. If $Y$ is reduced, so is $X'$.


spaces-morphisms-lemma-normalization-generic

Let $S$ be a scheme. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of schemes. Let $X' \to X$ be the normalization of $X$ in $Y$. If $x' \in |X'|$ is a point of codimension $0$ (Properties of Spaces, Definition \ref{spaces-properties-definition-dimension-local-ring}), then $x'$ is the image of some $y \in |Y|$ of codimension $0$.


spaces-morphisms-lemma-normalization-in-disjoint-union

Let $S$ be a scheme. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Suppose that $Y = Y_1 \amalg Y_2$ is a disjoint union of two algebraic spaces. Write $f_i = f|_{Y_i}$. Let $X_i'$ be the normalization of $X$ in $Y_i$. Then $X_1' \amalg X_2'$ is the normalization of $X$ in $Y$.


spaces-morphisms-lemma-normalization-in-universally-closed

Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact, quasi-separated and universally closed morphisms of algebraic spaces over $S$. Then $f_*\mathcal{O}_X$ is integral over $\mathcal{O}_Y$. In other words, the normalization of $Y$ in $X$ is equal to the factorization $$ X \longrightarrow \underline{\Spec}_Y(f_*\mathcal{O}_X) \longrightarrow Y $$ of Remark \ref{remark-factorization-quasi-compact-quasi-separated}.


spaces-morphisms-lemma-normalization-in-integral

Let $S$ be a scheme. Let $f : Y \to X$ be an integral morphism of algebraic spaces over $S$. Then the integral closure of $X$ in $Y$ is equal to $Y$.


spaces-morphisms-lemma-prepare-normalization

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent \begin{enumerate} \item there is a surjective \'etale morphism $U \to X$ where $U$ is as scheme such that every quasi-compact open of $U$ has finitely many irreducible components, \item for every scheme $U$ and every \'etale morphism $U \to X$ every quasi-compact open of $U$ has finitely many irreducible components, and \item for every quasi-compact algebraic space $Y$ \'etale over $X$ the space $|Y|$ has finitely many irreducible components. \end{enumerate} If $X$ is representable this means that every quasi-compact open of $X$ has finitely many irreducible components.


spaces-morphisms-lemma-normalization

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ satisfying the equivalent conditions of Lemma \ref{lemma-prepare-normalization}. Then there exists an integral morphism of algebraic spaces $$ X^\nu \longrightarrow X $$ such that for every scheme $U$ and \'etale morphism $U \to X$ the fibre product $X^\nu \times_X U$ is the normalization of $U$.


spaces-morphisms-lemma-normalization-normal

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ satisfying the equivalent conditions of Lemma \ref{lemma-prepare-normalization}. \begin{enumerate} \item The normalization $X^\nu$ is normal. \item The morphism $\nu : X^\nu \to X$ is integral and surjective. \item The map $|\nu| : |X^\nu| \to |X|$ induces a bijection between the sets of points of codimension $0$ (Properties of Spaces, Definition \ref{spaces-properties-definition-dimension-local-ring}). \item Let $Z \to X$ be a morphism. Assume $Z$ is a normal algebraic space and that for $z \in |Z|$ we have: $z$ has codimension $0$ in $Z \Rightarrow f(z)$ has codimension $0$ in $X$. Then there exists a unique factorization $Z \to X^\nu \to X$. \end{enumerate}


spaces-morphisms-lemma-nagata-normalization

Let $S$ be a scheme. Let $X$ be a Nagata algebraic space over $S$. The normalization $\nu : X^\nu \to X$ is a finite morphism.


spaces-morphisms-lemma-neighbourhood-scheme

Let $S$ be a scheme. Consider a commutative diagram $$ \xymatrix{ V' \ar[r] \ar[rd] & T' \times_T X \ar[r] \ar[d] & X \ar[d] \\ & T' \ar[r] & T } $$ of algebraic spaces over $S$. Assume \begin{enumerate} \item $T' \to T$ is an \'etale morphism of affine schemes, \item $X \to T$ is a separated, locally quasi-finite morphism, \item $V'$ is an open subspace of $T' \times_T X$, and \item $V' \to T'$ is quasi-affine. \end{enumerate} In this situation the image $U$ of $V'$ in $X$ is a quasi-compact open subspace of $X$ which is representable.


spaces-morphisms-lemma-locally-quasi-finite-separated-representable

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally quasi-finite and separated, then $f$ is representable.


spaces-morphisms-lemma-etale-universally-injective-open

\begin{slogan} Universally injective \'etale maps are open immersions. \end{slogan} Let $S$ be a scheme. Let $f : X \to Y$ be an \'etale and universally injective morphism of algebraic spaces over $S$. Then $f$ is an open immersion.


spaces-morphisms-lemma-finite-type-separated

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is representable, of finite type, and separated. Let $Y'$ be the normalization of $Y$ in $X$. Picture: $$ \xymatrix{ X \ar[rd]_f \ar[rr]_{f'} & & Y' \ar[ld]^\nu \\ & Y & } $$ Then there exists an open subspace $U' \subset Y'$ such that \begin{enumerate} \item $(f')^{-1}(U') \to U'$ is an isomorphism, and \item $(f')^{-1}(U') \subset X$ is the set of points at which $f$ is quasi-finite. \end{enumerate}


spaces-morphisms-lemma-quasi-finite-separated-quasi-affine

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-finite and separated. Let $Y'$ be the normalization of $Y$ in $X$. Picture: $$ \xymatrix{ X \ar[rd]_f \ar[rr]_{f'} & & Y' \ar[ld]^\nu \\ & Y & } $$ Then $f'$ is a quasi-compact open immersion and $\nu$ is integral. In particular $f$ is quasi-affine.


spaces-morphisms-lemma-base-change-universal-homeomorphism

The base change of a universal homeomorphism of algebraic spaces by any morphism of algebraic spaces is a universal homeomorphism.


spaces-morphisms-lemma-composition-universal-homeomorphism

The composition of a pair of universal homeomorphisms of algebraic spaces is a universal homeomorphism.


spaces-morphisms-lemma-reduction-universal-homeomorphism

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The canonical closed immersion $X_{red} \to X$ (see Properties of Spaces, Definition \ref{spaces-properties-definition-reduced-induced-space}) is a universal homeomorphism.


spaces-morphisms-lemma-integral-universally-injective-push-pull

Let $S$ be a scheme. Let $f : Y \to X$ be a universally injective, integral morphism of algebraic spaces over $S$. \begin{enumerate} \item The functor $$ f_{small, *} : \Sh(Y_\etale) \longrightarrow \Sh(X_\etale) $$ is fully faithful and its essential image is those sheaves of sets $\mathcal{F}$ on $X_\etale$ whose restriction to $|X| \setminus f(|Y|)$ is isomorphic to $*$, and \item the functor $$ f_{small, *} : \textit{Ab}(Y_\etale) \longrightarrow \textit{Ab}(X_\etale) $$ is fully faithful and its essential image is those abelian sheaves on $Y_\etale$ whose support is contained in $f(|Y|)$. \end{enumerate} In both cases $f_{small}^{-1}$ is a left inverse to the functor $f_{small, *}$.


spaces-morphisms-proposition-generic-flatness-reduced

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 of $\mathcal{O}_X$-modules. Assume \begin{enumerate} \item $Y$ is reduced, \item $f$ is of finite type, and \item $\mathcal{F}$ is a finite type $\mathcal{O}_X$-module. \end{enumerate} Then there exists an open dense subspace $W \subset Y$ such that the base change $X_W \to W$ of $f$ is flat, locally of finite presentation, and quasi-compact and such that $\mathcal{F}|_{X_W}$ is flat over $W$ and of finite presentation over $\mathcal{O}_{X_W}$.


spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme

Let $S$ be a scheme. Let $f : X \to T$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $T$ is representable, \item $f$ is locally quasi-finite, and \item $f$ is separated. \end{enumerate} Then $X$ is representable.


weil-lemma-composition-correspondences

We have the following for correspondences: \begin{enumerate} \item composition of correspondences is $\mathbf{Q}$-bilinear and associative, \item there is a canonical isomorphism $$ \CH_{-r}(X) \otimes \mathbf{Q} = \text{Corr}^r(X, \Spec(k)) $$ such that pullback by correspondences corresponds to composition, \item there is a canonical isomorphism $$ \CH^r(X) \otimes \mathbf{Q} = \text{Corr}^r(\Spec(k), X) $$ such that pushforward by correspondences corresponds to composition, \item composition of correspondences is compatible with pushforward and pullback of cycles. \end{enumerate}


weil-lemma-category-correspondences

Smooth projective schemes over $k$ with correspondences and composition of correspondences as defined above form a graded category over $\mathbf{Q}$ (Differential Graded Algebra, Definition \ref{dga-definition-graded-category}).


weil-lemma-contravariant-functor

There is a contravariant functor from the category of smooth projective schemes over $k$ to the category of correspondences which is the identity on objects and sends $f : Y \to X$ to the element $[\Gamma_f] \in \text{Corr}^0(X, Y)$.


weil-lemma-functor-and-cycles

Let $f : Y \to X$ be a morphism of smooth projective schemes over $k$. Let $[\Gamma_f] \in \text{Corr}^0(X, Y)$ be as in Example \ref{example-graph-correspondence}. Then \begin{enumerate} \item pushforward of cycles by the correspondence $[\Gamma_f]$ agrees with the gysin map $f^! : \CH^*(Y) \to \CH^*(X)$, \item pullback of cycles by the correspondence $[\Gamma_f]$ agrees with the pushforward map $f_* : \CH_*(Y) \to \CH_*(X)$, \item if $X$ and $Y$ are equidimensional of dimensions $d$ and $e$, then \begin{enumerate} \item pushforward of cycles by the correspondence $[\Gamma_f^t]$ of Remark \ref{remark-transpose} corresponds to pushforward of cycles by $f$, and \item pullback of cycles by the correspondence $[\Gamma_f^t]$ of Remark \ref{remark-transpose} corresponds to the gysin map $f^!$. \end{enumerate} \end{enumerate}


weil-lemma-tensor-product

The tensor product of correspondences defined above turns the category of correspondences into a symmetric monoidal category with unit $\Spec(k)$.


weil-lemma-prep-dual

Let $f : Y \to X$ be a morphism of smooth projective schemes over $k$. Assume $X$ and $Y$ equidimensional of dimensions $d$ and $e$. Denote $a = [\Gamma_f] \in \text{Corr}^0(X, Y)$ and $a^t = [\Gamma_f^t] \in \text{Corr}^{d - e}(Y, X)$. Set $\eta_X = [\Gamma_{X \to X \times X}] \in \text{Corr}^0(X \times X, X)$, $\eta_Y = [\Gamma_{Y \to Y \times Y}] \in \text{Corr}^0(Y \times Y, Y)$, $[X] \in \text{Corr}^{-d}(X, \Spec(k))$, and $[Y] \in \text{Corr}^{-e}(Y, \Spec(k))$. The diagram $$ \xymatrix{ X \otimes Y \ar[r]_{a \otimes \text{id}} \ar[d]_{\text{id} \otimes a^t} & Y \otimes Y \ar[r]_{\eta_Y} & Y \ar[d]^{[Y]} \\ X \otimes X \ar[r]^{\eta_X} & X \ar[r]^{[X]} & \Spec(k) } $$ is commutative in the category of correspondences.


weil-lemma-inverse-h2

With notation as in Example \ref{example-decompose-P1} \begin{enumerate} \item the motive $(X, c_0, 0)$ is isomorphic to the motive $\mathbf{1} = (\Spec(k), 1, 0)$. \item the motive $(X, c_2, 0)$ is isomorphic to the motive $\mathbf{1}(-1) = (\Spec(k), 1, -1)$. \end{enumerate}


weil-lemma-additive

The category $M_k$ is additive.


weil-lemma-characterize-motives

Let $X$, $c_2$ be as in Example \ref{example-decompose-P1}. Let $\mathcal{C}$ be a $\mathbf{Q}$-linear Karoubian symmetric monoidal category. Any $\mathbf{Q}$-linear functor $$ F : \left\{ \begin{matrix} \text{smooth projective schemes over }k\\ \text{morphisms are correspondences of degree }0 \end{matrix} \right\} \longrightarrow \mathcal{C} $$ of symmetric monoidal categories such that the image of $F(c_2)$ on $F(X)$ is an invertible object, factors uniquely through a functor $F : M_k \to \mathcal{C}$ of symmetric monoidal categories.


weil-lemma-dual

Let $X$ be a smooth projective scheme over $k$ which is equidimensional of dimension $d$. Then $h(X)(d)$ is a left dual to $h(X)$ in $M_k$.


weil-lemma-dual-general

Every object of $M_k$ has a left dual.


weil-lemma-chow-groups-representable

Let $k$ be a base field. The functor $\CH^i(-)$ on the category of motives $M_k$ is representable by $\mathbf{1}(-i)$, i.e., we have $$ \CH^i(M) = \Hom_{M_k}(\mathbf{1}(-i), M) $$ functorially in $M$ in $M_k$.


weil-lemma-manin

Let $k$ be a base field. Let $c : M \to N$ be a morphism of motives. If for every smooth projective scheme $X$ over $k$ the map $c \otimes 1 : M \otimes h(X) \to N \otimes h(X)$ induces an isomorphism on Chow groups, then $c$ is an isomorphism.


weil-lemma-projective-space-bundle-formula

In the situation above, the map $$ \sum\nolimits_{i = 0, \ldots, r - 1} c_i : \bigoplus\nolimits_{i = 0, \ldots, r - 1} h(X)(i) \longrightarrow h(P) $$ is an isomorphism in the category of motives.


weil-lemma-diagonal-projective-bundle

Let $p : P \to X$ be as in Lemma \ref{lemma-projective-space-bundle-formula}. The class $[\Delta_P]$ of the diagonal of $P$ in $\CH^*(P \times P)$ can be written as $$ [\Delta_P] = \left(\sum\nolimits_{i = 0, \ldots, r - 1} {r - 1 \choose i} c_{r - 1 - i}(\text{pr}_1^*\mathcal{S}^\vee) \cap c_1(\text{pr}_2^*\mathcal{O}_P(1))^i\right) \cap (p \times p)^*[\Delta_X] $$ where $\mathcal{S}$ is the kernel of the canonical surjection $p^*\mathcal{E} \to \mathcal{O}_P(1)$.


weil-lemma-pushforward-classical

Assume given (D1) and (D3) satisfying (A). For $f : X \to Y$ a morphism of smooth projective varieties we have $f_*(f^*b \cup a) = b \cup f_*a$. If $g : Y \to Z$ is a second morphism of smooth projective varieties, then $g_* \circ f_* = (g \circ f)_*$.


weil-lemma-degrees-cycles-classical

Let $H^*$ be a classical Weil cohomology theory (Definition \ref{definition-weil-cohomology-theory-classical}). Let $X$ be a smooth projective variety of dimension $d$. The diagram $$ \xymatrix{ \CH^d(X) \ar[r]_-\gamma \ar@{=}[d] & H^{2d}(X) \ar[d]^{\int_X} \\ \CH_0(X) \ar[r]^\deg & F } $$ commutes where $\deg : \CH_0(X) \to \mathbf{Z}$ is the degree of zero cycles discussed in Chow Homology, Section \ref{chow-section-degree-zero-cycles}.


weil-lemma-trace-product-classical

Let $H^*$ be a classical Weil cohomology theory (Definition \ref{definition-weil-cohomology-theory-classical}). Let $X$ and $Y$ be smooth projective varieties. Then $\int_{X \times Y} = \int_X \otimes \int_Y$.


Let $H^*$ be a classical Weil cohomology theory (Definition \ref{definition-weil-cohomology-theory-classical}). Let $X$ and $Y$ be smooth projective varieties. Then $\text{pr}_{2, *} : H^*(X \times Y) \to H^*(Y)$ sends $a \otimes b$ to $(\int_X a) b$.


weil-lemma-class-diagonal-classical

Let $H^*$ be a classical Weil cohomology theory (Definition \ref{definition-weil-cohomology-theory-classical}). Let $X$ be a smooth projective variety of dimension $d$. Choose a basis $e_{i, j}, j = 1, \ldots, \beta_i$ of $H^i(X)$ over $F$. Using K\"unneth write $$ \gamma([\Delta]) = \sum\nolimits_{i = 0, \ldots, 2d} \sum\nolimits_j e_{i, j} \otimes e'_{2d - i , j} \quad\text{in}\quad \bigoplus\nolimits_i H^i(X) \otimes_F H^{2d - i}(X) $$ with $e'_{2d - i, j} \in H^{2d - i}(X)$. Then $\int_X e_{i, j} \cup e'_{2d - i, j'} = (-1)^i\delta_{jj'}$.


weil-lemma-square-diagonal-classical

Let $H^*$ be a classical Weil cohomology theory (Definition \ref{definition-weil-cohomology-theory-classical}). Let $X$ be a smooth projective variety. We have $$ \sum\nolimits_{i = 0, \ldots, 2\dim(X)} (-1)^i\dim_F H^i(X) = \deg([\Delta] \cdot [\Delta]) = \deg(c_d(\mathcal{T}_X) \cap [X]) $$


weil-lemma-from-functor-to-weil-classical

Let $k$ be an algebraically closed field. Let $F$ be a field of characteristic $0$. Consider a $\mathbf{Q}$-linear functor $$ G : M_k \longrightarrow \text{graded }F\text{-vector spaces} $$ of symmetric monoidal categories such that $G(\mathbf{1}(1))$ is nonzero only in degree $-2$. Then we obtain data (D1), (D2), (D3) satisfying all of (A), (B), (C) except for possibly (A)(c) and (A)(d).


weil-lemma-from-weil-to-functor-classical

Let $k$ be an algebraically closed field. Let $F$ be a field of characteristic $0$. Let $H^*$ be a classical Weil cohomology theory. Then we can construct a $\mathbf{Q}$-linear functor $$ G : M_k \longrightarrow \text{graded }F\text{-vector spaces} $$ of symmetric monoidal categories such that $H^*(X) = G(h(X))$.


weil-lemma-generated-by-separable

Let $k$ be a field. Let $X$ be a smooth projective scheme over $k$. Then $\CH_0(X)$ is generated by classes of closed points whose residue fields are separable over $k$.


weil-lemma-chow-limit

Let $K/k$ be an algebraic field extension. Let $X$ be a finite type scheme over $k$. Then $\CH_i(X_K) = \colim \CH_i(X_{k'})$ where the colimit is over the subextensions $K/k'/k$ with $k'/k$ finite.


weil-lemma-divide-difference-points

Let $k$ be a field. Let $X$ be a geometrically irreducible smooth projective scheme over $k$. Let $x, x' \in X$ be $k$-rational points. Let $n$ be an integer invertible in $k$. Then there exists a finite separable extension $k'/k$ such that the pullback of $[x] - [x']$ to $X_{k'}$ is divisible by $n$ in $\CH_0(X_{k'})$.


weil-lemma-kernel-to-closure

Let $K/k$ be an algebraic extension of fields. Let $X$ be a finite type scheme over $k$. The kernel of the map $\CH_i(X) \to \CH_i(X_K)$ constructed in Lemma \ref{lemma-chow-limit} is torsion.


weil-lemma-pushforward

Assume given (D0), (D1), and (D3) satisfying (A). For $f : X \to Y$ a morphism of nonempty equidimensional smooth projective schemes over $k$ we have $f_*(f^*b \cup a) = b \cup f_*a$. If $g : Y \to Z$ is a second morphism with $Z$ nonempty smooth projective and equidimensional, then $g_* \circ f_* = (g \circ f)_*$.


weil-lemma-pr2star

Assume given (D0), (D1), and (D3) satisfying (A) and (B). Let $X$ and $Y$ be nonempty smooth projective schemes over $k$ equidimensional of dimensions $d$ and $e$. Then $\text{pr}_{2, *} : H^*(X \times Y)(d + e) \to H^*(Y)(e)$ sends $a \otimes b$ to $(\int_X a) b$.


weil-lemma-base

Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Then $H^i(\Spec(k)) = 0$ for $i \not = 0$ and there is a unique $F$-algebra isomorphism $F = H^0(\Spec(k))$. We have $\gamma([\Spec(k)]) = 1$ and $\int_{\Spec(k)} 1 = 1$.


weil-lemma-unit

Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $X$ be a smooth projective scheme over $k$. If $X = \emptyset$, then $H^*(X) = 0$. If $X$ is nonempty, then $\gamma([X]) = 1$ and $1 \not = 0$ in $H^0(X)$.


weil-lemma-push-unit

Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $i : X \to Y$ be a closed immersion of nonempty smooth projective equidimensional schemes over $k$. Then $\gamma([X]) = i_*1$ in $H^{2c}(Y)(c)$ where $c = \dim(Y) - \dim(X)$.


weil-lemma-class-diagonal

Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $X$ be a nonempty smooth projective scheme over $k$ equidimensional of dimension $d$. Choose a basis $e_{i, j}, j = 1, \ldots, \beta_i$ of $H^i(X)$ over $F$. Using K\"unneth write $$ \gamma([\Delta]) = \sum\nolimits_i \sum\nolimits_j e_{i, j} \otimes e'_{2d - i , j} \quad\text{in}\quad \bigoplus\nolimits_i H^i(X) \otimes_F H^{2d - i}(X)(d) $$ with $e'_{2d - i, j} \in H^{2d - i}(X)(d)$. Then $\int_X e_{i, j} \cup e'_{2d - i, j'} = (-1)^i\delta_{jj'}$.


weil-lemma-cohomology-P1

Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Then $H^*(\mathbf{P}^1_k)$ is $1$-dimensional in dimensions $0$ and $2$ and zero in other degrees.


weil-lemma-weil-additive

Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). If $X$ and $Y$ are smooth projective schemes over $k$, then $H^*(X \amalg Y) \to H^*(X) \times H^*(Y)$, $a \mapsto (i^*a, j^*a)$ is an isomorphism where $i$, $j$ are the coprojections.


weil-lemma-from-functor-to-weil

Let $k$ be a field. Let $F$ be a field of characteristic $0$. Assume given a $\mathbf{Q}$-linear functor $$ G : M_k \longrightarrow \text{graded }F\text{-vector spaces} $$ of symmetric monoidal categories such that $G(\mathbf{1}(1))$ is nonzero only in degree $-2$. Then we obtain data (D0), (D1), (D2), and (D3) satisfying all of (A), (B), and (C) above.


weil-lemma-from-weil-to-functor

Let $k$ be a field. Let $F$ be a field of characteristic $0$. Given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C) we can construct a $\mathbf{Q}$-linear functor $$ G : M_k \longrightarrow \text{graded }F\text{-vector spaces} $$ of symmetric monoidal categories such that $H^*(X) = G(h(X))$.


weil-lemma-trace-disjoint-union

Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $X, Y$ be nonempty smooth projective schemes both equidimensional of dimension $d$ over $k$. Then $\int_{X \amalg Y} = \int_X + \int_Y$.


weil-lemma-dim-0

Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $X$ be a smooth projective scheme of dimension zero over $k$. Then \begin{enumerate} \item $H^i(X) = 0$ for $i \not = 0$, \item $H^0(X)$ is a finite separable algebra over $F$, \item $\dim_F H^0(X) = \deg(X \to \Spec(F))$, \item $\int_X : H^0(X) \to F$ is the trace map, \item $\gamma([X]) = 1$, and \item $\int_X \gamma([X]) = \deg(X \to \Spec(k))$. \end{enumerate}


weil-lemma-degrees-cycles

Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $X$ be a nonempty smooth projective scheme equidimensional of dimension $d$ over $k$. The diagram $$ \xymatrix{ \CH^d(X) \ar[r]_-\gamma \ar@{=}[d] & H^{2d}(X)(d) \ar[d]^{\int_X} \\ \CH_0(X) \ar[r]^\deg & F } $$ commutes where $\deg : \CH_0(X) \to \mathbf{Z}$ is the degree of zero cycles discussed in Chow Homology, Section \ref{chow-section-degree-zero-cycles}.


weil-lemma-algebra-relations

Let $F$ be a field of characteristic $0$. Let $F'$ and $F_i$, $i = 1, \ldots, r$ be finite separable $F$-algebras. Let $A$ be a finite $F$-algebra. Let $\sigma, \sigma' : A \to F'$ and $\sigma_i : A \to F_i$ be $F$-algebra maps. Assume $\sigma$ and $\sigma'$ surjective. If there is a relation $$ \text{Tr}_{F'/F} \circ \sigma - \text{Tr}_{F'/F} \circ \sigma' = n(\sum m_i \text{Tr}_{F_i/F} \circ \sigma_i) $$ where $n > 1$ and $m_i$ are integers, then $\sigma = \sigma'$.


weil-lemma-relations-classes-points

Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $k'/k$ be a finite separable extension. Let $X$ be a smooth projective scheme over $k'$. Let $x, x' \in X$ be $k'$-rational points. If $\gamma(x) \not = \gamma(x')$, then $[x] - [x']$ is not divisible by any integer $n > 1$ in $\CH_0(X)$.


weil-lemma-classes-points

Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $k'/k$ be a finite separable extension. Let $X$ be a geometrically irreducible smooth projective scheme over $k'$ of dimension $d$. Then $\gamma : \CH_0(X) \to H^{2d}(X)(d)$ factors through $\deg : \CH_0(X) \to \mathbf{Z}$.


weil-lemma-injective

Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $f : X \to Y$ be a dominant morphism of irreducible smooth projective schemes over $k$. Then $H^*(Y) \to H^*(X)$ is injective.


weil-lemma-otimes

Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $k''/k'/k$ be finite separable algebras and let $X$ be a smooth projective scheme over $k'$. Then $$ H^*(X) \otimes_{H^0(\Spec(k'))} H^0(\Spec(k'')) = H^*(X \times_{\Spec(k')} \Spec(k'')) $$


weil-lemma-H-0-separable

Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $X$ be a smooth projective scheme over $k$. Set $k' = \Gamma(X, \mathcal{O}_X)$. The following are equivalent \begin{enumerate} \item there exist finitely many closed points $x_1, \ldots, x_r \in X$ whose residue fields are separable over $k$ such that $H^0(X) \to H^0(x_1) \oplus \ldots \oplus H^0(x_r)$ is injective, \item the map $H^0(\Spec(k')) \to H^0(X)$ is an isomorphism. \end{enumerate} If $X$ is equidimensional of dimension $d$, these are also equivalent to \begin{enumerate} \item[(3)] the classes of closed points generate $H^{2d}(X)(d)$ as a module over $H^0(X)$. \end{enumerate} If this is true, then $H^0(X)$ is a finite separable algebra over $F$.


weil-lemma-splitting-principle

In the situation above. Let $X \in \Ob(\mathcal{C})$. Let $\mathcal{E}_i$ be a finite collection of locally free $\mathcal{O}_X$-modules of rank $r_i$. There exists a morphism $p : P \to X$ in $\mathcal{C}$ such that \begin{enumerate} \item $p^* : A(X) \to A(P)$ is injective, \item each $p^*\mathcal{E}_i$ has a filtration whose successive quotients $\mathcal{L}_{i, 1}, \ldots, \mathcal{L}_{i, r_i}$ are invertible $\mathcal{O}_P$-modules. \end{enumerate}


weil-lemma-chern-classes-E-tensor-L

Let $X \in \Ob(\mathcal{C})$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Then $$ c^A_i({\mathcal E} \otimes {\mathcal L}) = \sum\nolimits_{j = 0}^i \binom{r - i + j}{j} c^A_{i - j}({\mathcal E}) \cup c^A_1({\mathcal L})^j $$


weil-lemma-adams-and-chern

In the situation above let $X \in \Ob(\mathcal{C})$. If $\psi^2$ is as in Chow Homology, Lemma \ref{chow-lemma-second-adams-operator} and $c^A$ and $ch^A$ are as in Propositions \ref{proposition-chern-class} and \ref{proposition-chern-character} then we have $c^A_i(\psi^2(\alpha)) = 2^i c^A_i(\alpha)$ and $ch^A_i(\psi^2(\alpha)) = 2^i ch^A_i(\alpha)$ for all $\alpha \in K_0(\textit{Vect}(X))$.


weil-lemma-lambda-operations

Let $X$ be a scheme. There are maps $$ \lambda^r : K_0(\textit{Vect}(X)) \longrightarrow K_0(\textit{Vect}(X)) $$ which sends $[\mathcal{E}]$ to $[\wedge^r(\mathcal{E})]$ when $\mathcal{E}$ is a finite locally free $\mathcal{O}_X$-module and which are compatible with pullbacks.


weil-lemma-chern-classes

Assume given (D0), (D1), and (D2') satisfying axioms (A1), (A2), (A3), and (A4). There is a unique rule which assigns to every smooth projective $X$ over $k$ a ring homomorphism $$ ch^H : K_0(\textit{Vec}(X)) \longrightarrow \prod\nolimits_{i \geq 0} H^{2i}(X)(i) $$ compatible with pullbacks such that $ch^H(\mathcal{L}) = \exp(c_1^H(\mathcal{L}))$ for any invertible $\mathcal{O}_X$-module $\mathcal{L}$.


weil-lemma-cycle-classes

Assume given (D0), (D1), and (D2') satisfying axioms (A1), (A2), (A3), and (A4). There is a unique rule which assigns to every smooth projective $X$ over $k$ a graded ring homomorphism $$ \gamma : \CH^*(X) \longrightarrow \bigoplus\nolimits_{i \geq 0} H^{2i}(X)(i) $$ compatible with pullbacks such that $ch^H(\alpha) = \gamma(ch(\alpha))$ for $\alpha$ in $K_0(\textit{Vect}(X))$.


weil-lemma-divide-pullback-good-blowing-up

Let $b : X' \to X$ be the blowing up of a smooth projective scheme over $k$ in a smooth closed subscheme $Z \subset X$. Picture $$ \xymatrix{ E \ar[r]_j \ar[d]_\pi & X' \ar[d]^b \\ Z \ar[r]^i & X } $$ Assume there exists an element of $K_0(X)$ whose restriction to $Z$ is equal to the class of $\mathcal{C}_{Z/X}$ in $K_0(Z)$. Assume every irreducible component of $Z$ has codimension $r$ in $X$. Then there exists a cycle $\theta \in \CH^{r - 1}(X')$ such that $b^![Z] = [E] \cdot \theta$ in $\CH^r(X')$ and $\pi_*j^!(\theta) = [Z]$ in $\CH^r(Z)$.


weil-lemma-A5-A6-imply

Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A4) and (A7). Let $X$ be a smooth projective scheme over $k$. Let $Z \subset X$ be a smooth closed subscheme such that every irreducible component of $Z$ has codimension $r$ in $X$. Assume the class of $\mathcal{C}_{Z/X}$ in $K_0(Z)$ is the restriction of an element of $K_0(X)$. If $a \in H^*(X)$ and $a|_Z = 0$ in $H^*(Z)$, then $\gamma([Z]) \cup a = 0$.


weil-lemma-poincare-duality

Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). Then axiom (A) of Section \ref{section-axioms} holds with $\int_X = \lambda$ as in axiom (A6).


weil-lemma-trace-product

Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). Then axiom (B) of Section \ref{section-axioms} holds.


weil-lemma-trace-base

Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). Then axiom (C)(d) of Section \ref{section-axioms} holds.


weil-lemma-ok-for-projective-bundle

Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). Let $p : P \to X$ be as in axiom (A3) with $X$ nonempty equidimensional. Then $\gamma$ commutes with pushforward along $p$.


weil-lemma-integrate-1

Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). If $k'/k$ is a Galois extension, then we have $\int_{\Spec(k')} 1 = [k' : k]$.


weil-lemma-enough

Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). In order to show that $\gamma$ commutes with pushforward it suffices to show that $i_*(1) = \gamma([Z])$ if $i : Z \to X$ is a closed immersion of nonempty smooth projective equidimensional schemes over $k$.


weil-lemma-grassmanian

Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). Given integers $0 < l < n$ and a nonempty equidimensional smooth projective scheme $X$ over $k$ consider the projection morphism $p : X \times \mathbf{G}(l, n) \to X$. Then $\gamma$ commutes with pushforward along $p$.


weil-lemma-enough-better

Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). In order to show that $\gamma$ commutes with pushforward it suffices to show that $i_*(1) = \gamma([Z])$ if $i : Z \to X$ is a closed immersion of nonempty smooth projective equidimensional schemes over $k$ such that the class of $\mathcal{C}_{Z/X}$ in $K_0(Z)$ is the pullback of a class in $K_0(X)$.


weil-lemma-injective-H0

Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). If $k''/k'/k$ are finite separable field extensions, then $H^0(\Spec(k')) \to H^0(\Spec(k''))$ is injective.


weil-lemma-pushforward-blowup

Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A8). Let $b : X' \to X$ be a blowing up of a smooth projective scheme $X$ over $k$ which is nonempty equidimensional of dimension $d$ in a nonwhere dense smooth center $Z$. Then $b_*(1) = 1$.


weil-lemma-done

Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A8). Then the cycle class map $\gamma$ commutes with pushforward.


weil-lemma-check-over-extension

Let $k'/k$ be an extension of fields. Let $F'/F$ be an extension of fields of characteristic $0$. Assume given \begin{enumerate} \item data (D0), (D1), (D2') for $k$ and $F$ denoted $F(1), H^*, c_1^H$, \item data (D0), (D1), (D2') for $k'$ and $F'$ denoted $F'(1), (H')^*, c_1^{H'}$, and \item an isomorphism $F(1) \otimes_F F' \to F'(1)$, functorial isomorphisms $H^*(X) \otimes_F F' \to (H')^*(X_{k'})$ on the category of smooth projective schemes $X$ over $k$ such that the diagrams $$ \xymatrix{ \Pic(X) \ar[r]_{c_1^H} \ar[d] & H^2(X)(1) \ar[d] \\ \Pic(X_{k'}) \ar[r]^{c_1^{H'}} & (H')^2(X_{k'})(1) } $$ commute. \end{enumerate} In this case, if $F'(1), (H')^*, c_1^{H'}$ satisfy axioms (A1) -- (A9), then the same is true for $F(1), H^*, c_1^H$.


weil-proposition-chern-class

In the situation above there is a unique rule which assigns to every $X \in \Ob(\mathcal{C})$ a ``total Chern class'' $$ c^A : K_0(\textit{Vect}(X)) \longrightarrow  \prod\nolimits_{i \geq 0} A^i(X) $$ with the following properties \begin{enumerate} \item For $X \in \Ob(\mathcal{C})$ we have $c^A(\alpha + \beta) = c^A(\alpha) c^A(\beta)$ and $c^A(0) = 1$. \item If $f : X' \to X$ is a morphism of $\mathcal{C}$, then $f^* \circ c^A =  c^A \circ f^*$. \item Given $X \in \Ob(\mathcal{C})$ and $\mathcal{L} \in \Pic(X)$ we have $c^A([\mathcal{L}]) = 1 + c_1^A(\mathcal{L})$. \end{enumerate}


weil-proposition-chern-character

In the situation above assume $A(X)$ is a $\mathbf{Q}$-algebra for all $X \in \Ob(\mathcal{C})$. Then there is a unique rule which assigns to every $X \in \Ob(\mathcal{C})$ a ``chern character'' $$ ch^A : K_0(\textit{Vect}(X)) \longrightarrow \prod\nolimits_{i \geq 0} A^i(X) $$ with the following properties \begin{enumerate} \item $ch^A$ is a ring map for all $X \in \Ob(\mathcal{C})$. \item If $f : X' \to X$ is a morphism of $\mathcal{C}$, then $f^* \circ ch^A =  ch^A \circ f^*$. \item Given $X \in \Ob(\mathcal{C})$ and $\mathcal{L} \in \Pic(X)$ we have $ch^A([\mathcal{L}]) = \exp(c_1^A(\mathcal{L}))$. \end{enumerate}


weil-proposition-get-weil

Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A8). Then we have a Weil cohomology theory.


morphisms-theorem-chevalley

Let $f : X \to Y$ be a morphism of schemes. Assume $f$ is quasi-compact and locally of finite presentation. Then the image of every locally constructible subset is locally constructible.


morphisms-theorem-main-theorem

Let $f : Y \to X$ be an affine morphism of schemes. Assume $f$ is of finite type. Let $X'$ be the normalization of $X$ in $Y$. Picture: $$ \xymatrix{ Y \ar[rd]_f \ar[rr]_{f'} & & X' \ar[ld]^\nu \\ & X & } $$ Then there exists an open subscheme $U' \subset X'$ such that \begin{enumerate} \item $(f')^{-1}(U') \to U'$ is an isomorphism, and \item $(f')^{-1}(U') \subset Y$ is the set of points at which $f$ is quasi-finite. \end{enumerate}


morphisms-lemma-closed-immersion

Let $i : Z \to X$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item The morphism $i$ is a closed immersion. \item For every affine open $\Spec(R) = U \subset X$, there exists an ideal $I \subset R$ such that $i^{-1}(U) = \Spec(R/I)$ as schemes over $U = \Spec(R)$. \item There exists an affine open covering $X = \bigcup_{j \in J} U_j$, $U_j = \Spec(R_j)$ and for every $j \in J$ there exists an ideal $I_j \subset R_j$ such that $i^{-1}(U_j) = \Spec(R_j/I_j)$ as schemes over $U_j = \Spec(R_j)$. \item The morphism $i$ induces a homeomorphism of $Z$ with a closed subset of $X$ and $i^\sharp : \mathcal{O}_X \to i_*\mathcal{O}_Z$ is surjective. \item The morphism $i$ induces a homeomorphism of $Z$ with a closed subset of $X$, the map $i^\sharp : \mathcal{O}_X \to i_*\mathcal{O}_Z$ is surjective, and the kernel $\Ker(i^\sharp)\subset \mathcal{O}_X$ is a quasi-coherent sheaf of ideals. \item The morphism $i$ induces a homeomorphism of $Z$ with a closed subset of $X$, the map $i^\sharp : \mathcal{O}_X \to i_*\mathcal{O}_Z$ is surjective, and the kernel $\Ker(i^\sharp)\subset \mathcal{O}_X$ is a sheaf of ideals which is locally generated by sections. \end{enumerate}


morphisms-lemma-closed-immersion-ideals

Let $X$ be a scheme. Let $i : Z \to X$ and $i' : Z' \to X$ be closed immersions and consider the ideal sheaves $\mathcal{I} = \Ker(i^\sharp)$ and $\mathcal{I}' = \Ker((i')^\sharp)$ of $\mathcal{O}_X$. \begin{enumerate} \item The morphism $i : Z \to X$ factors as $Z \to Z' \to X$ for some $a : Z \to Z'$ if and only if $\mathcal{I}' \subset \mathcal{I}$. If this happens, then $a$ is a closed immersion. \item We have $Z \cong Z'$ over $X$ if and only if $\mathcal{I} = \mathcal{I}'$. \end{enumerate}


morphisms-lemma-closed-immersion-bijection-ideals

Let $X$ be a scheme. Let $\mathcal{I} \subset \mathcal{O}_X$ be a sheaf of ideals. The following are equivalent: \begin{enumerate} \item $\mathcal{I}$ is locally generated by sections as a sheaf of $\mathcal{O}_X$-modules, \item $\mathcal{I}$ is quasi-coherent as a sheaf of $\mathcal{O}_X$-modules, and \item there exists a closed immersion $i : Z \to X$ of schemes whose corresponding sheaf of ideals $\Ker(i^\sharp)$ is equal to $\mathcal{I}$. \end{enumerate}


morphisms-lemma-base-change-closed-immersion

The base change of a closed immersion is a closed immersion.


morphisms-lemma-closed-immersion-quasi-compact

A closed immersion is quasi-compact.


morphisms-lemma-closed-immersion-separated

A closed immersion is separated.


morphisms-lemma-immersion-permanence

Let $Z \to Y \to X$ be morphisms of schemes. \begin{enumerate} \item If $Z \to X$ is an immersion, then $Z \to Y$ is an immersion. \item If $Z \to X$ is a quasi-compact immersion and $Y \to X$ is quasi-separated, then $Z \to Y$ is a quasi-compact immersion. \item If $Z \to X$ is a closed immersion and $Y \to X$ is separated, then $Z \to Y$ is a closed immersion. \end{enumerate}


morphisms-lemma-factor-quasi-compact-immersion

Let $h : Z \to X$ be an immersion. If $h$ is quasi-compact, then we can factor $h = i \circ j$ with $j : Z \to \overline{Z}$ an open immersion and $i : \overline{Z} \to X$ a closed immersion.


morphisms-lemma-factor-reduced-immersion

Let $h : Z \to X$ be an immersion. If $Z$ is reduced, then we can factor $h = i \circ j$ with $j : Z \to \overline{Z}$ an open immersion and $i : \overline{Z} \to X$ a closed immersion.


morphisms-lemma-check-immersion

Let $f : Y \to X$ be a morphism of schemes. If for all $y \in Y$ there is an open subscheme $f(y) \in U \subset X$ such that $f|_{f^{-1}(U)} : f^{-1}(U) \to U$ is an immersion, then $f$ is an immersion.


morphisms-lemma-i-star-equivalence

Let $i : Z \to X$ be a closed immersion of schemes. Let $\mathcal{I} \subset \mathcal{O}_X$ be the quasi-coherent sheaf of ideals cutting out $Z$. The functor $$ i_* : \QCoh(\mathcal{O}_Z) \longrightarrow \QCoh(\mathcal{O}_X) $$ is exact, fully faithful, with essential image those quasi-coherent $\mathcal{O}_X$-modules $\mathcal{G}$ such that $\mathcal{I}\mathcal{G} = 0$.


morphisms-lemma-largest-quasi-coherent-subsheaf

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{G} \subset \mathcal{F}$ be a $\mathcal{O}_X$-submodule. There exists a unique quasi-coherent $\mathcal{O}_X$-submodule $\mathcal{G}' \subset \mathcal{G}$ with the following property: For every quasi-coherent $\mathcal{O}_X$-module $\mathcal{H}$ the map $$ \Hom_{\mathcal{O}_X}(\mathcal{H}, \mathcal{G}') \longrightarrow \Hom_{\mathcal{O}_X}(\mathcal{H}, \mathcal{G}) $$ is bijective. In particular $\mathcal{G}'$ is the largest quasi-coherent $\mathcal{O}_X$-submodule of $\mathcal{F}$ contained in $\mathcal{G}$.


morphisms-lemma-scheme-theoretic-intersection

Let $X$ be a scheme. Let $Z, Y \subset X$ be closed subschemes. Let $Z \cap Y$ be the scheme theoretic intersection of $Z$ and $Y$. Then $Z \cap Y \to Z$ and $Z \cap Y \to Y$ are closed immersions and $$ \xymatrix{ Z \cap Y \ar[r] \ar[d] & Z \ar[d] \\ Y \ar[r] & X } $$ is a cartesian diagram of schemes, i.e., $Z \cap Y = Z \times_X Y$.


morphisms-lemma-scheme-theoretic-union

Let $S$ be a scheme. Let $X, Y \subset S$ be closed subschemes. Let $X \cup Y$ be the scheme theoretic union of $X$ and $Y$. Let $X \cap Y$ be the scheme theoretic intersection of $X$ and $Y$. Then $X \to X \cup Y$ and $Y \to X \cup Y$ are closed immersions, there is a short exact sequence $$ 0 \to \mathcal{O}_{X \cup Y} \to \mathcal{O}_X \times \mathcal{O}_Y \to \mathcal{O}_{X \cap Y} \to 0 $$ of $\mathcal{O}_S$-modules, and the diagram $$ \xymatrix{ X \cap Y \ar[r] \ar[d] & X \ar[d] \\ Y \ar[r] & X \cup Y } $$ is cocartesian in the category of schemes, i.e., $X \cup Y = X \amalg_{X \cap Y} Y$.


morphisms-lemma-support-affine-open

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\Spec(A) = U \subset X$ be an affine open, and set $M = \Gamma(U, \mathcal{F})$. Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime. The following are equivalent \begin{enumerate} \item $\mathfrak p$ is in the support of $M$, and \item $x$ is in the support of $\mathcal{F}$. \end{enumerate}


morphisms-lemma-support-closed-specialization

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. The support of $\mathcal{F}$ is closed under specialization.


morphisms-lemma-support-finite-type

Let $\mathcal{F}$ be a finite type quasi-coherent module on a scheme $X$. Then \begin{enumerate} \item The support of $\mathcal{F}$ is closed. \item For $x \in X$ we have $$ x \in \text{Supp}(\mathcal{F}) \Leftrightarrow \mathcal{F}_x \not = 0 \Leftrightarrow \mathcal{F}_x \otimes_{\mathcal{O}_{X, x}} \kappa(x) \not = 0. $$ \item For any morphism of schemes $f : Y \to X$ the pullback $f^*\mathcal{F}$ is of finite type as well and we have $\text{Supp}(f^*\mathcal{F}) = f^{-1}(\text{Supp}(\mathcal{F}))$. \end{enumerate}


morphisms-lemma-scheme-theoretic-support

Let $\mathcal{F}$ be a finite type quasi-coherent module on a scheme $X$. There exists a smallest closed subscheme $i : Z \to X$ such that there exists a quasi-coherent $\mathcal{O}_Z$-module $\mathcal{G}$ with $i_*\mathcal{G} \cong \mathcal{F}$. Moreover: \begin{enumerate} \item If $\Spec(A) \subset X$ is any affine open, and $\mathcal{F}|_{\Spec(A)} = \widetilde{M}$ then $Z \cap \Spec(A) = \Spec(A/I)$ where $I = \text{Ann}_A(M)$. \item The quasi-coherent sheaf $\mathcal{G}$ is unique up to unique isomorphism. \item The quasi-coherent sheaf $\mathcal{G}$ is of finite type. \item The support of $\mathcal{G}$ and of $\mathcal{F}$ is $Z$. \end{enumerate}


morphisms-lemma-scheme-theoretic-image

Let $f : X \to Y$ be a morphism of schemes. There exists a closed subscheme $Z \subset Y$ such that $f$ factors through $Z$ and such that for any other closed subscheme $Z' \subset Y$ such that $f$ factors through $Z'$ we have $Z \subset Z'$.


morphisms-lemma-quasi-compact-scheme-theoretic-image

Let $f : X \to Y$ be a morphism of schemes. Let $Z \subset Y$ be the scheme theoretic image of $f$. If $f$ is quasi-compact then \begin{enumerate} \item the sheaf of ideals $\mathcal{I} = \Ker(\mathcal{O}_Y \to f_*\mathcal{O}_X)$ is quasi-coherent, \item the scheme theoretic image $Z$ is the closed subscheme determined by $\mathcal{I}$, \item for any open $U \subset Y$ the scheme theoretic image of $f|_{f^{-1}(U)} : f^{-1}(U) \to U$ is equal to $Z \cap U$, and \item the image $f(X) \subset Z$ is a dense subset of $Z$, in other words the morphism $X \to Z$ is dominant (see Definition \ref{definition-dominant}). \end{enumerate}


morphisms-lemma-reach-points-scheme-theoretic-image

Let $f : X \to Y$ be a quasi-compact morphism. Let $Z$ be the scheme theoretic image of $f$. Let $z \in Z$\footnote{By Lemma \ref{lemma-quasi-compact-scheme-theoretic-image} set-theoretically $Z$ agrees with the closure of $f(X)$ in $Y$.}. There exists a valuation ring $A$ with fraction field $K$ and a commutative diagram $$ \xymatrix{ \Spec(K) \ar[rr] \ar[d] & & X \ar[d] \ar[ld] \\ \Spec(A) \ar[r] & Z \ar[r] & Y } $$ such that the closed point of $\Spec(A)$ maps to $z$. In particular any point of $Z$ is the specialization of a point of $f(X)$.


morphisms-lemma-factor-factor

Let $$ \xymatrix{ X_1 \ar[d] \ar[r]_{f_1} & Y_1 \ar[d] \\ X_2 \ar[r]^{f_2} & Y_2 } $$ be a commutative diagram of schemes. Let $Z_i \subset Y_i$, $i = 1, 2$ be the scheme theoretic image of $f_i$. Then the morphism $Y_1 \to Y_2$ induces a morphism $Z_1 \to Z_2$ and a commutative diagram $$ \xymatrix{ X_1 \ar[r] \ar[d] & Z_1 \ar[d] \ar[r] & Y_1 \ar[d] \\ X_2 \ar[r] & Z_2 \ar[r] & Y_2 } $$


morphisms-lemma-scheme-theoretic-image-reduced

Let $f : X \to Y$ be a morphism of schemes. If $X$ is reduced, then the scheme theoretic image of $f$ is the reduced induced scheme structure on $\overline{f(X)}$.


morphisms-lemma-characterize-scheme-theoretically-dense

Let $j : U \to X$ be an open immersion of schemes. Then $U$ is scheme theoretically dense in $X$ if and only if $\mathcal{O}_X \to j_*\mathcal{O}_U$ is injective.


morphisms-lemma-intersection-scheme-theoretically-dense

Let $X$ be a scheme. If $U$, $V$ are scheme theoretically dense open subschemes of $X$, then so is $U \cap V$.


morphisms-lemma-quasi-compact-immersion

Let $h : Z \to X$ be an immersion. Assume either $h$ is quasi-compact or $Z$ is reduced. Let $\overline{Z} \subset X$ be the scheme theoretic image of $h$. Then the morphism $Z \to \overline{Z}$ is an open immersion which identifies $Z$ with a scheme theoretically dense open subscheme of $\overline{Z}$. Moreover, $Z$ is topologically dense in $\overline{Z}$.


morphisms-lemma-reduced-scheme-theoretically-dense

Let $X$ be a reduced scheme and let $U \subset X$ be an open subscheme. Then the following are equivalent \begin{enumerate} \item $U$ is topologically dense in $X$, \item the scheme theoretic closure of $U$ in $X$ is $X$, and \item $U$ is scheme theoretically dense in $X$. \end{enumerate}


morphisms-lemma-reduced-subscheme-closure

Let $X$ be a scheme and let $U \subset X$ be a reduced open subscheme. Then the following are equivalent \begin{enumerate} \item the scheme theoretic closure of $U$ in $X$ is $X$, and \item $U$ is scheme theoretically dense in $X$. \end{enumerate} If this holds then $X$ is a reduced scheme.


morphisms-lemma-equality-of-morphisms

Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$. Let $f, g : X \to Y$ be morphisms of schemes over $S$. Let $U \subset X$ be an open subscheme such that $f|_U = g|_U$. If the scheme theoretic closure of $U$ in $X$ is $X$ and $Y \to S$ is separated, then $f = g$.


morphisms-lemma-generic-points-in-image-dominant

Let $f : X \to S$ be a morphism of schemes. If every generic point of every irreducible component of $S$ is in the image of $f$, then $f$ is dominant.


morphisms-lemma-quasi-compact-dominant

\begin{slogan} Morphisms whose image contains the generic points are dominant \end{slogan} Let $f : X \to S$ be a quasi-compact morphism of schemes. Then $f$ is dominant (if and) only if for every irreducible component $Z \subset S$ the generic point of $Z$ is in the image of $f$.


morphisms-lemma-quasi-compact-generic-point-not-in-image

Let $f : X \to S$ be a quasi-compact morphism of schemes. Let $\eta \in S$ be a generic point of an irreducible component of $S$. If $\eta \not \in f(X)$ then there exists an open neighbourhood $V \subset S$ of $\eta$ such that $f^{-1}(V) = \emptyset$.


morphisms-lemma-dominant-finite-number-irreducible-components

Let $f : X \to S$ be a morphism of schemes. Suppose that $X$ has finitely many irreducible components. Then $f$ is dominant (if and) only if for every irreducible component $Z \subset S$ the generic point of $Z$ is in the image of $f$. If so, then $S$ has finitely many irreducible components as well.


morphisms-lemma-dominant-between-integral

Let $f : X \to Y$ be a morphism of integral schemes. The following are equivalent \begin{enumerate} \item $f$ is dominant, \item $f$ maps the generic point of $X$ to the generic point of $Y$, \item for some nonempty affine opens $U \subset X$ and $V \subset Y$ with $f(U) \subset V$ the ring map $\mathcal{O}_Y(V) \to \mathcal{O}_X(U)$ is injective, \item for all nonempty affine opens $U \subset X$ and $V \subset Y$ with $f(U) \subset V$ the ring map $\mathcal{O}_Y(V) \to \mathcal{O}_X(U)$ is injective, \item for some $x \in X$ with image $y = f(x) \in Y$ the local ring map $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ is injective, and \item for all $x \in X$ with image $y = f(x) \in Y$ the local ring map $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ is injective. \end{enumerate}


morphisms-lemma-composition-surjective

The composition of surjective morphisms is surjective.


morphisms-lemma-when-point-maps-to-pair

Let $X$ and $Y$ be schemes over a base scheme $S$. Given points $x \in X$ and $y \in Y$, there is a point of $X \times_S Y$ mapping to $x$ and $y$ under the projections if and only if $x$ and $y$ lie above the same point of $S$.


morphisms-lemma-base-change-surjective

The base change of a surjective morphism is surjective.


morphisms-lemma-universally-injective

Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item For every field $K$ the induced map $\Mor(\Spec(K), X) \to \Mor(\Spec(K), S)$ is injective. \item The morphism $f$ is universally injective. \item The morphism $f$ is radicial. \item The diagonal morphism $\Delta_{X/S} : X \longrightarrow X \times_S X$ is surjective. \end{enumerate}


morphisms-lemma-universally-injective-separated

A universally injective morphism is separated.


morphisms-lemma-base-change-universally-injective

A base change of a universally injective morphism is universally injective.


morphisms-lemma-composition-universally-injective

A composition of radicial morphisms is radicial, and so the same holds for the equivalent condition of being universally injective.


morphisms-lemma-characterize-affine

\begin{reference} \cite[II, Corollary 1.3.2]{EGA} \end{reference} Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item The morphism $f$ is affine. \item There exists an affine open covering $S = \bigcup W_j$ such that each $f^{-1}(W_j)$ is affine. \item There exists a quasi-coherent sheaf of $\mathcal{O}_S$-algebras $\mathcal{A}$ and an isomorphism $X \cong \underline{\Spec}_S(\mathcal{A})$ of schemes over $S$. See Constructions, Section \ref{constructions-section-spec} for notation. \end{enumerate} Moreover, in this case $X = \underline{\Spec}_S(f_*\mathcal{O}_X)$.


morphisms-lemma-affine-equivalence-algebras

Let $S$ be a scheme. There is an anti-equivalence of categories $$ \begin{matrix} \text{Schemes affine} \\ \text{over }S \end{matrix} \longleftrightarrow \begin{matrix} \text{quasi-coherent sheaves} \\ \text{of }\mathcal{O}_S\text{-algebras} \end{matrix} $$ which associates to $f : X \to S$ the sheaf $f_*\mathcal{O}_X$. Moreover, this equivalence is compatible with arbitrary base change.


morphisms-lemma-affine-equivalence-modules

Let $f : X \to S$ be an affine morphism of schemes. Let $\mathcal{A} = f_*\mathcal{O}_X$. The functor $\mathcal{F} \mapsto f_*\mathcal{F}$ induces an equivalence of categories $$ \left\{ \begin{matrix} \text{category of quasi-coherent}\\ \mathcal{O}_X\text{-modules} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{category of quasi-coherent}\\ \mathcal{A}\text{-modules} \end{matrix} \right\} $$ Moreover, an $\mathcal{A}$-module is quasi-coherent as an $\mathcal{O}_S$-module if and only if it is quasi-coherent as an $\mathcal{A}$-module.


morphisms-lemma-composition-affine

The composition of affine morphisms is affine.


morphisms-lemma-base-change-affine

The base change of an affine morphism is affine.


morphisms-lemma-closed-immersion-affine

A closed immersion is affine.


morphisms-lemma-affine-s-open

Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{L})$. The inclusion morphism $j : X_s \to X$ is affine.


morphisms-lemma-affine-permanence

Suppose $g : X \to Y$ is a morphism of schemes over $S$. \begin{enumerate} \item If $X$ is affine over $S$ and $\Delta : Y \to Y \times_S Y$ is affine, then $g$ is affine. \item If $X$ is affine over $S$ and $Y$ is separated over $S$, then $g$ is affine. \item A morphism from an affine scheme to a scheme with affine diagonal is affine. \item A morphism from an affine scheme to a separated scheme is affine. \end{enumerate}


morphisms-lemma-morphism-affines-affine

A morphism between affine schemes is affine.


morphisms-lemma-Artinian-affine

Let $S$ be a scheme. Let $A$ be an Artinian ring. Any morphism $\Spec(A) \to S$ is affine.


morphisms-lemma-get-affine

Let $j : Y \to X$ be an immersion of schemes. Assume there exists an open $U \subset X$ with complement $Z = X \setminus U$ such that \begin{enumerate} \item $U \to X$ is affine, \item $j^{-1}(U) \to U$ is affine, and \item $j(Y) \cap Z$ is closed. \end{enumerate} Then $j$ is affine. In particular, if $X$ is affine, so is $Y$.


morphisms-lemma-quasi-affine-separated

A quasi-affine morphism is separated and quasi-compact.


morphisms-lemma-characterize-quasi-affine

Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item The morphism $f$ is quasi-affine. \item There exists an affine open covering $S = \bigcup W_j$ such that each $f^{-1}(W_j)$ is quasi-affine. \item There exists a quasi-coherent sheaf of $\mathcal{O}_S$-algebras $\mathcal{A}$ and a quasi-compact open immersion $$ \xymatrix{ X \ar[rr] \ar[rd] & & \underline{\Spec}_S(\mathcal{A}) \ar[dl] \\ & S & } $$ over $S$. \item Same as in (3) but with $\mathcal{A} = f_*\mathcal{O}_X$ and the horizontal arrow the canonical morphism of Constructions, Lemma \ref{constructions-lemma-canonical-morphism}. \end{enumerate}


morphisms-lemma-composition-quasi-affine

The composition of quasi-affine morphisms is quasi-affine.


morphisms-lemma-base-change-quasi-affine

The base change of a quasi-affine morphism is quasi-affine.


morphisms-lemma-quasi-compact-immersion-quasi-affine

A quasi-compact immersion is quasi-affine.


morphisms-lemma-quasi-affine-permanence

Suppose $g : X \to Y$ is a morphism of schemes over $S$. If $X$ is quasi-affine over $S$ and $Y$ is quasi-separated over $S$, then $g$ is quasi-affine. In particular, any morphism from a quasi-affine scheme to a quasi-separated scheme is quasi-affine.


morphisms-lemma-locally-P

Let $f : X \to S$ be a morphism of schemes. Let $P$ be a property of ring maps. Let $U$ be an affine open of $X$, and $V$ an affine open of $S$ such that $f(U) \subset V$. If $f$ is locally of type $P$ and $P$ is local, then $P(\mathcal{O}_S(V) \to \mathcal{O}_X(U))$ holds.


morphisms-lemma-locally-P-characterize

Let $P$ be a local property of ring maps. Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item The morphism $f$ is locally of type $P$. \item For every affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ we have $P(\mathcal{O}_S(V) \to \mathcal{O}_X(U))$. \item There exists an open covering $S = \bigcup_{j \in J} V_j$ and open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that each of the morphisms $U_i \to V_j$, $j\in J, i\in I_j$ is locally of type $P$. \item There exists an affine open covering $S = \bigcup_{j \in J} V_j$ and affine open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that $P(\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_i))$ holds, for all $j\in J, i\in I_j$. \end{enumerate} Moreover, if $f$ is locally of type $P$ then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f|_U : U \to V$ is locally of type $P$.


morphisms-lemma-composition-type-P

Let $P$ be a property of ring maps. Assume $P$ is local and stable under composition. The composition of morphisms locally of type $P$ is locally of type $P$.


morphisms-lemma-base-change-type-P

Let $P$ be a property of ring maps. Assume $P$ is local and stable under base change. The base change of a morphism locally of type $P$ is locally of type $P$.


morphisms-lemma-properties-local

The following properties of a ring map $R \to A$ are local. \begin{enumerate} \item (Isomorphism on local rings.) For every prime $\mathfrak q$ of $A$ lying over $\mathfrak p \subset R$ the ring map $R \to A$ induces an isomorphism $R_{\mathfrak p} \to A_{\mathfrak q}$. \item (Open immersion.) For every prime $\mathfrak q$ of $A$ there exists an $f \in R$, $\varphi(f) \not \in \mathfrak q$ such that the ring map $\varphi : R \to A$ induces an isomorphism $R_f \to A_f$. \item (Reduced fibres.) For every prime $\mathfrak p$ of $R$ the fibre ring $A \otimes_R \kappa(\mathfrak p)$ is reduced. \item (Fibres of dimension at most $n$.) For every prime $\mathfrak p$ of $R$ the fibre ring $A \otimes_R \kappa(\mathfrak p)$ has Krull dimension at most $n$. \item (Locally Noetherian on the target.) The ring map $R \to A$ has the property that $A$ is Noetherian. \item Add more here as needed\footnote{But only those properties that are not already dealt with separately elsewhere.}. \end{enumerate}


morphisms-lemma-properties-base-change

The following properties of ring maps are stable under base change. \begin{enumerate} \item (Isomorphism on local rings.) For every prime $\mathfrak q$ of $A$ lying over $\mathfrak p \subset R$ the ring map $R \to A$ induces an isomorphism $R_{\mathfrak p} \to A_{\mathfrak q}$. \item (Open immersion.) For every prime $\mathfrak q$ of $A$ there exists an $f \in R$, $\varphi(f) \not \in \mathfrak q$ such that the ring map $\varphi : R \to A$ induces an isomorphism $R_f \to A_f$. \item Add more here as needed\footnote{But only those properties that are not already dealt with separately elsewhere.}. \end{enumerate}


morphisms-lemma-properties-composition

The following properties of ring maps are stable under composition. \begin{enumerate} \item (Isomorphism on local rings.) For every prime $\mathfrak q$ of $A$ lying over $\mathfrak p \subset R$ the ring map $R \to A$ induces an isomorphism $R_{\mathfrak p} \to A_{\mathfrak q}$. \item (Open immersion.) For every prime $\mathfrak q$ of $A$ there exists an $f \in R$, $\varphi(f) \not \in \mathfrak q$ such that the ring map $\varphi : R \to A$ induces an isomorphism $R_f \to A_f$. \item (Locally Noetherian on the target.) The ring map $R \to A$ has the property that $A$ is Noetherian. \item Add more here as needed\footnote{But only those properties that are not already dealt with separately elsewhere.}. \end{enumerate}


morphisms-lemma-locally-finite-type-characterize

Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item The morphism $f$ is locally of finite type. \item For all affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ the ring map $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is of finite type. \item There exists an open covering $S = \bigcup_{j \in J} V_j$ and open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that each of the morphisms $U_i \to V_j$, $j\in J, i\in I_j$ is locally of finite type. \item There exists an affine open covering $S = \bigcup_{j \in J} V_j$ and affine open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that the ring map $\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_i)$ is of finite type, for all $j\in J, i\in I_j$. \end{enumerate} Moreover, if $f$ is locally of finite type then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f|_U : U \to V$ is locally of finite type.


morphisms-lemma-composition-finite-type

The composition of two morphisms which are locally of finite type is locally of finite type. The same is true for morphisms of finite type.


morphisms-lemma-base-change-finite-type

The base change of a morphism which is locally of finite type is locally of finite type. The same is true for morphisms of finite type.


morphisms-lemma-immersion-locally-finite-type

A closed immersion is of finite type. An immersion is locally of finite type.


morphisms-lemma-finite-type-noetherian

Let $f : X \to S$ be a morphism. If $S$ is (locally) Noetherian and $f$ (locally) of finite type then $X$ is (locally) Noetherian.


morphisms-lemma-finite-type-Noetherian-quasi-separated

Let $f : X \to S$ be locally of finite type with $S$ locally Noetherian. Then $f$ is quasi-separated.


morphisms-lemma-permanence-finite-type

Let $X \to Y$ be a morphism of schemes over a base scheme $S$. If $X$ is locally of finite type over $S$, then $X \to Y$ is locally of finite type.


morphisms-lemma-point-finite-type

Let $S$ be a scheme. Let $k$ be a field. Let $f : \Spec(k) \to S$ be a morphism. The following are equivalent: \begin{enumerate} \item The morphism $f$ is of finite type. \item The morphism $f$ is locally of finite type. \item There exists an affine open $U = \Spec(R)$ of $S$ such that $f$ corresponds to a finite ring map $R \to k$. \item There exists an affine open $U = \Spec(R)$ of $S$ such that the image of $f$ consists of a closed point $u$ in $U$ and the field extension $\kappa(u) \subset k$ is finite. \end{enumerate}


morphisms-lemma-artinian-finite-type

Let $S$ be a scheme. Let $A$ be an Artinian local ring with residue field $\kappa$. Let $f : \Spec(A) \to S$ be a morphism of schemes. Then $f$ is of finite type if and only if the composition $\Spec(\kappa) \to \Spec(A) \to S$ is of finite type.


morphisms-lemma-identify-finite-type-points

Let $S$ be a scheme. We have $$ S_{\text{ft-pts}} = \bigcup\nolimits_{U \subset S\text{ open}} U_0 $$ where $U_0$ is the set of closed points of $U$. Here we may let $U$ range over all opens or over all affine opens of $S$.


morphisms-lemma-finite-type-points-morphism

Let $f : T \to S$ be a morphism of schemes. If $f$ is locally of finite type, then $f(T_{\text{ft-pts}}) \subset S_{\text{ft-pts}}$.


morphisms-lemma-finite-type-points-surjective-morphism

Let $f : T \to S$ be a morphism of schemes. If $f$ is locally of finite type and surjective, then $f(T_{\text{ft-pts}}) = S_{\text{ft-pts}}$.


morphisms-lemma-enough-finite-type-points

Let $S$ be a scheme. For any locally closed subset $T \subset S$ we have $$ T \not = \emptyset \Rightarrow T \cap S_{\text{ft-pts}} \not = \emptyset. $$ In particular, for any closed subset $T \subset S$ we see that $T \cap S_{\text{ft-pts}}$ is dense in $T$.


morphisms-lemma-jacobson-finite-type-points

Let $S$ be a scheme. The following are equivalent: \begin{enumerate} \item the scheme $S$ is Jacobson, \item $S_{\text{ft-pts}}$ is the set of closed points of $S$, \item for all $T \to S$ locally of finite type closed points map to closed points, and \item for all $T \to S$ locally of finite type closed points $t \in T$ map to closed points $s \in S$ with $\kappa(s) \subset \kappa(t)$ finite. \end{enumerate}


morphisms-lemma-Jacobson-universally-Jacobson

Let $S$ be a Jacobson scheme. Any scheme locally of finite type over $S$ is Jacobson.


morphisms-lemma-ubiquity-Jacobson-schemes

The following types of schemes are Jacobson. \begin{enumerate} \item Any scheme locally of finite type over a field. \item Any scheme locally of finite type over $\mathbf{Z}$. \item Any scheme locally of finite type over a $1$-dimensional Noetherian domain with infinitely many primes. \item A scheme of the form $\Spec(R) \setminus \{\mathfrak m\}$ where $(R, \mathfrak m)$ is a Noetherian local ring. Also any scheme locally of finite type over it. \end{enumerate}


morphisms-lemma-universally-catenary-local

Let $S$ be a locally Noetherian scheme. The following are equivalent \begin{enumerate} \item $S$ is universally catenary, \item there exists an open covering of $S$ all of whose members are universally catenary schemes, \item for every affine open $\Spec(R) = U \subset S$ the ring $R$ is universally catenary, and \item there exists an affine open covering $S = \bigcup U_i$ such that each $U_i$ is the spectrum of a universally catenary ring. \end{enumerate} Moreover, in this case any scheme locally of finite type over $S$ is universally catenary as well.


morphisms-lemma-ubiquity-uc

The following types of schemes are universally catenary. \begin{enumerate} \item Any scheme locally of finite type over a field. \item Any scheme locally of finite type over a Cohen-Macaulay scheme. \item Any scheme locally of finite type over $\mathbf{Z}$. \item Any scheme locally of finite type over a $1$-dimensional Noetherian domain. \item And so on. \end{enumerate}


morphisms-lemma-finite-type-nagata

Let $f : X \to S$ be a morphism. If $S$ is Nagata and $f$ locally of finite type then $X$ is Nagata. If $S$ is universally Japanese and $f$ locally of finite type then $X$ is universally Japanese.


morphisms-lemma-ubiquity-nagata

The following types of schemes are Nagata. \begin{enumerate} \item Any scheme locally of finite type over a field. \item Any scheme locally of finite type over a Noetherian complete local ring. \item Any scheme locally of finite type over $\mathbf{Z}$. \item Any scheme locally of finite type over a Dedekind ring of characteristic zero. \item And so on. \end{enumerate}


morphisms-lemma-J

Let $X$ be a locally Noetherian scheme. The following are equivalent \begin{enumerate} \item $X$ is J-2, \item there exists an open covering of $X$ all of whose members are J-2 schemes, \item for every affine open $\Spec(R) = U \subset X$ the ring $R$ is J-2, and \item there exists an affine open covering $S = \bigcup U_i$ such that each $\mathcal{O}(U_i)$ is J-2 for all $i$. \end{enumerate} Moreover, in this case any scheme locally of finite type over $X$ is J-2 as well.


morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point. Set $s = f(x)$. If $\kappa(x)/\kappa(s)$ is an algebraic field extension, then \begin{enumerate} \item $x$ is a closed point of its fibre, and \item if in addition $s$ is a closed point of $S$, then $x$ is a closed point of $X$. \end{enumerate}


morphisms-lemma-closed-point-fibre-locally-finite-type

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point. Set $s = f(x)$. Assume $f$ is locally of finite type. Then $x$ is a closed point of its fibre if and only if $\kappa(s) \subset \kappa(x)$ is a finite field extension.


morphisms-lemma-base-change-closed-point-fibre-locally-finite-type

Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $g : S' \to S$ be any morphism. Denote $f' : X' \to S'$ the base change. If $x' \in X'$ maps to a point $x \in X$ which is closed in $X_{f(x)}$ then $x'$ is closed in $X'_{f'(x')}$.


morphisms-lemma-residue-field-quasi-finite

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point. Set $s = f(x)$. If $f$ is quasi-finite at $x$, then the residue field extension $\kappa(s) \subset \kappa(x)$ is finite.


morphisms-lemma-quasi-finite-at-point-characterize

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point. Set $s = f(x)$. Let $X_s$ be the fibre of $f$ at $s$. Assume $f$ is locally of finite type. The following are equivalent: \begin{enumerate} \item The morphism $f$ is quasi-finite at $x$. \item The point $x$ is isolated in $X_s$. \item The point $x$ is closed in $X_s$ and there is no point $x' \in X_s$, $x' \not = x$ which specializes to $x$. \item For any pair of affine opens $\Spec(A) = U \subset X$, $\Spec(R) = V \subset S$ with $f(U) \subset V$ and $x \in U$ corresponding to $\mathfrak q \subset A$ the ring map $R \to A$ is quasi-finite at $\mathfrak q$. \end{enumerate}


morphisms-lemma-finite-fibre

Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Assume that \begin{enumerate} \item $f$ is locally of finite type, and \item $f^{-1}(\{s\})$ is a finite set. \end{enumerate} Then $X_s$ is a finite discrete topological space, and $f$ is quasi-finite at each point of $X$ lying over $s$.


morphisms-lemma-locally-quasi-finite-fibres

\begin{slogan} Finite type morphisms with discrete fibers are quasi-finite. \end{slogan} Let $f : X \to S$ be a morphism of schemes. Assume $f$ is locally of finite type. Then the following are equivalent \begin{enumerate} \item $f$ is locally quasi-finite, \item for every $s \in S$ the fibre $X_s$ is a discrete topological space, and \item for every morphism $\Spec(k) \to S$ where $k$ is a field the base change $X_k$ has an underlying discrete topological space. \end{enumerate}


morphisms-lemma-quasi-finite-locally-quasi-compact

Let $f : X \to S$ be a morphism of schemes. Then $f$ is quasi-finite if and only if $f$ is locally quasi-finite and quasi-compact.


morphisms-lemma-quasi-finite

Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item $f$ is quasi-finite, and \item $f$ is locally of finite type, quasi-compact, and has finite fibres. \end{enumerate}


morphisms-lemma-locally-quasi-finite-characterize

Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item The morphism $f$ is locally quasi-finite. \item For every pair of affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ the ring map $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is quasi-finite. \item There exists an open covering $S = \bigcup_{j \in J} V_j$ and open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that each of the morphisms $U_i \to V_j$, $j\in J, i\in I_j$ is locally quasi-finite. \item There exists an affine open covering $S = \bigcup_{j \in J} V_j$ and affine open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that the ring map $\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_i)$ is quasi-finite, for all $j\in J, i\in I_j$. \end{enumerate} Moreover, if $f$ is locally quasi-finite then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f|_U : U \to V$ is locally quasi-finite.


morphisms-lemma-composition-quasi-finite

The composition of two morphisms which are locally quasi-finite is locally quasi-finite. The same is true for quasi-finite morphisms.


morphisms-lemma-base-change-quasi-finite

\begin{slogan} (Locally) quasi-finite morphisms are stable under base change. \end{slogan} Let $f : X \to S$ be a morphism of schemes. Let $g : S' \to S$ be a morphism of schemes. Denote $f' : X' \to S'$ the base change of $f$ by $g$ and denote $g' : X' \to X$ the projection. Assume $X$ is locally of finite type over $S$. \begin{enumerate} \item Let $U \subset X$ (resp.\ $U' \subset X'$) be the set of points where $f$ (resp.\ $f'$) is quasi-finite. Then $U' = U \times_S S' = (g')^{-1}(U)$. \item The base change of a locally quasi-finite morphism is locally quasi-finite. \item The base change of a quasi-finite morphism is quasi-finite. \end{enumerate}


morphisms-lemma-quasi-finite-at-a-finite-number-of-points

Let $f : X \to S$ be a morphism of schemes of finite type. Let $s \in S$. There are at most finitely many points of $X$ lying over $s$ at which $f$ is quasi-finite.


morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite

Let $f : X \to Y$ be a morphism of schemes. If $f$ is locally of finite type and a monomorphism, then $f$ is separated and locally quasi-finite.


morphisms-lemma-immersion-locally-quasi-finite

Any immersion is locally quasi-finite.


morphisms-lemma-permanence-quasi-finite

Let $X \to Y$ be a morphism of schemes over a base scheme $S$. Let $x \in X$. If $X \to S$ is quasi-finite at $x$, then $X \to Y$ is quasi-finite at $x$. If $X$ is locally quasi-finite over $S$, then $X \to Y$ is locally quasi-finite.


morphisms-lemma-locally-finite-presentation-characterize

Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item The morphism $f$ is locally of finite presentation. \item For every affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ the ring map $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is of finite presentation. \item There exists an open covering $S = \bigcup_{j \in J} V_j$ and open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that each of the morphisms $U_i \to V_j$, $j\in J, i\in I_j$ is locally of finite presentation. \item There exists an affine open covering $S = \bigcup_{j \in J} V_j$ and affine open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that the ring map $\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_i)$ is of finite presentation, for all $j\in J, i\in I_j$. \end{enumerate} Moreover, if $f$ is locally of finite presentation then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f|_U : U \to V$ is locally of finite presentation.


morphisms-lemma-composition-finite-presentation

The composition of two morphisms which are locally of finite presentation is locally of finite presentation. The same is true for morphisms of finite presentation.


morphisms-lemma-base-change-finite-presentation

The base change of a morphism which is locally of finite presentation is locally of finite presentation. The same is true for morphisms of finite presentation.


morphisms-lemma-open-immersion-locally-finite-presentation

Any open immersion is locally of finite presentation.


morphisms-lemma-quasi-compact-open-immersion-finite-presentation

Any open immersion is of finite presentation if and only if it is quasi-compact.


morphisms-lemma-closed-immersion-finite-presentation

\begin{slogan} Closed immersions of finite presentation correspond to quasi-coherent sheaves of ideals of finite type. \end{slogan} A closed immersion $i : Z \to X$ is of finite presentation if and only if the associated quasi-coherent sheaf of ideals $\mathcal{I} = \Ker(\mathcal{O}_X \to i_*\mathcal{O}_Z)$ is of finite type (as an $\mathcal{O}_X$-module).


morphisms-lemma-finite-presentation-finite-type

A morphism which is locally of finite presentation is locally of finite type. A morphism of finite presentation is of finite type.


morphisms-lemma-noetherian-finite-type-finite-presentation

\begin{slogan} Over a locally Noetherian base, finite type is finite presentation. \end{slogan} Let $f : X \to S$ be a morphism. \begin{enumerate} \item If $S$ is locally Noetherian and $f$ locally of finite type then $f$ is locally of finite presentation. \item If $S$ is locally Noetherian and $f$ of finite type then $f$ is of finite presentation. \end{enumerate}


morphisms-lemma-finite-presentation-quasi-compact-quasi-separated

Let $S$ be a scheme which is quasi-compact and quasi-separated. If $X$ is of finite presentation over $S$, then $X$ is quasi-compact and quasi-separated.


morphisms-lemma-finite-presentation-permanence

Let $f : X \to Y$ be a morphism of schemes over $S$. \begin{enumerate} \item If $X$ is locally of finite presentation over $S$ and $Y$ is locally of finite type over $S$, then $f$ is locally of finite presentation. \item If $X$ is of finite presentation over $S$ and $Y$ is quasi-separated and locally of finite type over $S$, then $f$ is of finite presentation. \end{enumerate}


morphisms-lemma-diagonal-morphism-finite-type

Let $f : X \to Y$ be a morphism of schemes with diagonal $\Delta : X \to X \times_Y X$. If $f$ is locally of finite type then $\Delta$ is locally of finite presentation. If $f$ is quasi-separated and locally of finite type, then $\Delta$ is of finite presentation.


morphisms-lemma-inverse-image-constructible

Let $f : X \to Y$ be a morphism of schemes. Let $E \subset Y$ be a subset. If $E$ is (locally) constructible in $Y$, then $f^{-1}(E)$ is (locally) constructible in $X$.


morphisms-lemma-chevalley

Let $f : X \to Y$ be a morphism of schemes. Assume \begin{enumerate} \item $f$ is quasi-compact and locally of finite presentation, and \item $Y$ is quasi-compact and quasi-separated. \end{enumerate} Then the image of every constructible subset of $X$ is constructible in $Y$.


morphisms-lemma-constructible-containing-open

Let $X$ be a scheme. Let $x \in X$. Let $E \subset X$ be a locally constructible subset. If $\{x' \mid x' \leadsto x\} \subset E$, then $E$ contains an open neighbourhood of $x$.


morphisms-lemma-locally-finite-presentation-universally-open

Let $f : X \to S$ be a morphism. \begin{enumerate} \item If $f$ is locally of finite presentation and generalizations lift along $f$, then $f$ is open. \item If $f$ is locally of finite presentation and generalizations lift along every base change of $f$, then $f$ is universally open. \end{enumerate}


morphisms-lemma-composition-open

A composition of (universally) open morphisms is (universally) open.


morphisms-lemma-scheme-over-field-universally-open

Let $k$ be a field. Let $X$ be a scheme over $k$. The structure morphism $X \to \Spec(k)$ is universally open.


morphisms-lemma-open-generizing

\begin{reference} Follows from the implication (a) $\Rightarrow$ (b) in \cite[IV, Corollary 1.10.4]{EGA} \end{reference} Let $\varphi : X \to Y$ be a morphism of schemes. If $\varphi$ is open, then $\varphi$ is generizing (i.e., generalizations lift along $\varphi$). If $\varphi$ is universally open, then $\varphi$ is universally generizing.


morphisms-lemma-base-change-universally-submersive

The base change of a universally submersive morphism of schemes by any morphism of schemes is universally submersive.


morphisms-lemma-composition-universally-submersive

The composition of a pair of (universally) submersive morphisms of schemes is (universally) submersive.


morphisms-lemma-flat-module-characterize

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-modules. The following are equivalent \begin{enumerate} \item The sheaf $\mathcal{F}$ is flat over $S$. \item For every affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ the $\mathcal{O}_S(V)$-module $\mathcal{F}(U)$ is flat. \item There exists an open covering $S = \bigcup_{j \in J} V_j$ and open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that each of the modules $\mathcal{F}|_{U_i}$ is flat over $V_j$, for all $j\in J, i\in I_j$. \item There exists an affine open covering $S = \bigcup_{j \in J} V_j$ and affine open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that $\mathcal{F}(U_i)$ is a flat $\mathcal{O}_S(V_j)$-module, for all $j\in J, i\in I_j$. \end{enumerate} Moreover, if $\mathcal{F}$ is flat over $S$ then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $\mathcal{F}|_U$ is flat over $V$.


morphisms-lemma-flat-characterize

Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item The morphism $f$ is flat. \item For every affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ the ring map $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is flat. \item There exists an open covering $S = \bigcup_{j \in J} V_j$ and open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that each of the morphisms $U_i \to V_j$, $j\in J, i\in I_j$ is flat. \item There exists an affine open covering $S = \bigcup_{j \in J} V_j$ and affine open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that $\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_i)$ is flat, for all $j\in J, i\in I_j$. \end{enumerate} Moreover, if $f$ is flat then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f|_U : U \to V$ is flat.


morphisms-lemma-pushforward-flat-affine

Let $f : X \to Y$ be an affine morphism of schemes over a base scheme $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $\mathcal{F}$ is flat over $S$ if and only if $f_*\mathcal{F}$ is flat over $S$.


morphisms-lemma-composition-module-flat

Let $X \to Y \to Z$ be morphisms of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $x \in X$ with image $y$ in $Y$. If $\mathcal{F}$ is flat over $Y$ at $x$, and $Y$ is flat over $Z$ at $y$, then $\mathcal{F}$ is flat over $Z$ at $x$.


morphisms-lemma-composition-flat

The composition of flat morphisms is flat.


morphisms-lemma-base-change-module-flat

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-modules. Let $g : S' \to S$ be a morphism of schemes. Denote $g' : X' = X_{S'} \to X$ the projection. Let $x' \in X'$ be a point with image $x = g'(x') \in X$. If $\mathcal{F}$ is flat over $S$ at $x$, then $(g')^*\mathcal{F}$ is flat over $S'$ at $x'$. In particular, if $\mathcal{F}$ is flat over $S$, then $(g')^*\mathcal{F}$ is flat over $S'$.


morphisms-lemma-base-change-flat

The base change of a flat morphism is flat.


morphisms-lemma-generalizations-lift-flat

Let $f : X \to S$ be a flat morphism of schemes. Then generalizations lift along $f$, see Topology, Definition \ref{topology-definition-lift-specializations}.


morphisms-lemma-fppf-open

A flat morphism locally of finite presentation is universally open.


morphisms-lemma-pf-flat-module-open

Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume $f$ locally finite presentation, $\mathcal{F}$ of finite type, $X = \text{Supp}(\mathcal{F})$, and $\mathcal{F}$ flat over $Y$. Then $f$ is universally open.


morphisms-lemma-fpqc-quotient-topology

\begin{reference} \cite[Expose VIII, Corollaire 4.3]{SGA1} and \cite[IV, Corollaire 2.3.12]{EGA} \end{reference} Let $f : X \to Y$ be a quasi-compact, surjective, flat morphism. A subset $T \subset Y$ is open (resp.\ closed) if and only $f^{-1}(T)$ is open (resp.\ closed). In other words, $f$ is a submersive morphism.


morphisms-lemma-flat-permanence

Let $h : X \to Y$ be a morphism of schemes over $S$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $Y$. Let $x \in X$ with $y = h(x) \in Y$. If $h$ is flat at $x$, then $$ \mathcal{G}\text{ flat over }S\text{ at }y \Leftrightarrow h^*\mathcal{G}\text{ flat over }S\text{ at }x. $$ In particular: If $h$ is surjective and flat, then $\mathcal{G}$ is flat over $S$, if and only if $h^*\mathcal{G}$ is flat over $S$. If $h$ is surjective and flat, and $X$ is flat over $S$, then $Y$ is flat over $S$.


morphisms-lemma-flat-pullback-support

Let $f : Y \to X$ be a morphism of schemes. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module with scheme theoretic support $Z \subset X$. If $f$ is flat, then $f^{-1}(Z)$ is the scheme theoretic support of $f^*\mathcal{F}$.


morphisms-lemma-flat-morphism-scheme-theoretically-dense-open

Let $f : X \to Y$ be a flat morphism of schemes. Let $V \subset Y$ be a retrocompact open which is scheme theoretically dense. Then $f^{-1}V$ is scheme theoretically dense in $X$.


morphisms-lemma-flat-base-change-scheme-theoretic-image

\begin{slogan} Taking scheme theoretic images commutes with flat base change in the quasi-compact case \end{slogan} Let $f : X \to Y$ be a flat morphism of schemes. Let $g : V \to Y$ be a quasi-compact morphism of schemes. Let $Z \subset Y$ be the scheme theoretic image of $g$ and let $Z' \subset X$ be the scheme theoretic image of the base change $V \times_Y X \to X$. Then $Z' = f^{-1}Z$.


morphisms-lemma-characterize-flat-closed-immersions

Let $X$ be a scheme. The rule which associates to a closed subscheme of $X$ its underlying closed subset defines a bijection $$ \left\{ \begin{matrix} \text{closed subschemes }Z \subset X \\ \text{such that }Z \to X\text{ is flat} \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{closed subsets }Z \subset X \\ \text{closed under generalizations} \end{matrix} \right\} $$ If $Z \subset X$ is such a closed subscheme, every morphism of schemes $g : Y \to X$ with $g(Y) \subset Z$ set theoretically factors (scheme theoretically) through $Z$.


morphisms-lemma-flat-closed-immersions-finite-presentation

A flat closed immersion of finite presentation is the open immersion of an open and closed subscheme.


morphisms-lemma-dimension-fibre-at-a-point

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ and set $s = f(x)$. Assume $f$ is locally of finite type. Then $$ \dim_x(X_s) = \dim(\mathcal{O}_{X_s, x}) + \text{trdeg}_{\kappa(s)}(\kappa(x)). $$


morphisms-lemma-dimension-fibre-at-a-point-additive

Let $f : X \to Y$ and $g : Y \to S$ be morphisms of schemes. Let $x \in X$ and set $y = f(x)$, $s = g(y)$. Assume $f$ and $g$ locally of finite type. Then $$ \dim_x(X_s) \leq \dim_x(X_y) + \dim_y(Y_s). $$ Moreover, equality holds if $\mathcal{O}_{X_s, x}$ is flat over $\mathcal{O}_{Y_s, y}$, which holds for example if $\mathcal{O}_{X, x}$ is flat over $\mathcal{O}_{Y, y}$.


morphisms-lemma-dimension-fibre-after-base-change

Let $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S } $$ be a fibre product diagram of schemes. Assume $f$ locally of finite type. Suppose that $x' \in X'$, $x = g'(x')$, $s' = f'(x')$ and $s = g(s') = f(x)$. Then \begin{enumerate} \item $\dim_x(X_s) = \dim_{x'}(X'_{s'})$, \item if $F$ is the fibre of the morphism $X'_{s'} \to X_s$ over $x$, then $$ \dim(\mathcal{O}_{F, x'}) = \dim(\mathcal{O}_{X'_{s'}, x'}) - \dim(\mathcal{O}_{X_s, x}) = \text{trdeg}_{\kappa(s)}(\kappa(x)) - \text{trdeg}_{\kappa(s')}(\kappa(x')) $$ In particular $\dim(\mathcal{O}_{X'_{s'}, x'}) \geq \dim(\mathcal{O}_{X_s, x})$ and $\text{trdeg}_{\kappa(s)}(\kappa(x)) \geq \text{trdeg}_{\kappa(s')}(\kappa(x'))$. \item given $s', s, x$ there exists a choice of $x'$ such that $\dim(\mathcal{O}_{X'_{s'}, x'}) = \dim(\mathcal{O}_{X_s, x})$ and $\text{trdeg}_{\kappa(s)}(\kappa(x)) = \text{trdeg}_{\kappa(s')}(\kappa(x'))$. \end{enumerate}


morphisms-lemma-openness-bounded-dimension-fibres

\begin{reference} \cite[IV Theorem 13.1.3]{EGA} \end{reference} Let $f : X \to S$ be a morphism of schemes. Let $n \geq 0$. Assume $f$ is locally of finite type. The set $$ U_n = \{x \in X \mid \dim_x X_{f(x)} \leq n\} $$ is open in $X$.


morphisms-lemma-morphism-finite-type-bounded-dimension

Let $f : X \to Y$ be a morphism of finite type with $Y$ quasi-compact. Then the dimension of the fibres of $f$ is bounded.


morphisms-lemma-openness-bounded-dimension-fibres-finite-presentation

Let $f : X \to S$ be a morphism of schemes. Let $n \geq 0$. Assume $f$ is locally of finite presentation. The open $$ U_n = \{x \in X \mid \dim_x X_{f(x)} \leq n\} $$ of Lemma \ref{lemma-openness-bounded-dimension-fibres} is retrocompact in $X$. (See Topology, Definition \ref{topology-definition-quasi-compact}.)