crystalline-proposition-compute-cohomology

With notations as above assume that \begin{enumerate} \item $\mathcal{F}$ is locally quasi-coherent, and \item for any morphism $(U, T, \delta) \to (U', T', \delta')$ of $\text{Cris}(X/S)$ where $f : T \to T'$ is a closed immersion the map $c_f : f^*\mathcal{F}_{T'} \to \mathcal{F}_T$ is surjective. \end{enumerate} Then the complex $$ M(0) \to M(1) \to M(2) \to \ldots $$ computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$.


crystalline-proposition-compute-cohomology-crystal

Assumptions as in Proposition \ref{proposition-compute-cohomology} but now assume that $\mathcal{F}$ is a crystal in quasi-coherent modules. Let $(M, \nabla)$ be the corresponding module with connection over $D$, see Proposition \ref{proposition-crystals-on-affine}. Then the complex $$ M \otimes^\wedge_D \Omega^*_D $$ computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$.


crystalline-proposition-compare-with-de-Rham

In Situation \ref{situation-global}. Let $\mathcal{F}$ be a crystal in quasi-coherent modules on $\text{Cris}(X/S)$. The truncation map of complexes $$ (\mathcal{F} \to \mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^1_{X/S} \to \mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^2_{X/S} \to \ldots) \longrightarrow \mathcal{F}[0], $$ while not a quasi-isomorphism, becomes a quasi-isomorphism after applying $Ru_{X/S, *}$. In fact, for any $i > 0$, we have  $$ Ru_{X/S, *}(\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^i_{X/S}) = 0. $$


spaces-pushouts-lemma-colimit-agrees

Let $S$ be a scheme. Let $\mathcal{I} \to (\Sch/S)_{fppf}$, $i \mapsto X_i$ be a diagram of schemes over $S$ as above. Assume that \begin{enumerate} \item $X = \colim X_i$ exists in the category of schemes, \item $\coprod X_i \to X$ is surjective, \item if $U \to X$ is \'etale and $U_i = X_i \times_X U$, then $U = \colim U_i$ in the category of schemes, and \item every object $(U_i \to X_i)$ of $\lim X_{i, \etale}$ with $U_i \to X_i$ separated is in the essential image the functor $X_\etale \to \lim X_{i, \etale}$. \end{enumerate} Then $X = \colim X_i$ in the category of algebraic spaces over $S$ also.


spaces-pushouts-lemma-pushout-fpqc-local

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $\mathcal{I} \to (\Sch/S)_{fppf}$, $i \mapsto X_i$ be a diagram of algebraic spaces over $B$. Let $(X, X_i \to X)$ be a cocone for the diagram in the category of algebraic spaces over $B$ (Categories, Remark \ref{categories-remark-cones-and-cocones}). If there exists a fpqc covering $\{U_a \to X\}_{a \in A}$ such that \begin{enumerate} \item for all $a \in A$ we have $U_a = \colim X_i \times_X U_a$ in the category of algebraic spaces over $B$, and \item for all $a, b \in A$ we have $U_a \times_X U_b = \colim X_i \times_X U_a \times_X U_b$ in the category of algebraic spaces over $B$, \end{enumerate} then $X = \colim X_i$ in the category of algebraic spaces over $B$.


spaces-pushouts-lemma-colimit-check-etale-locally

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $\mathcal{I} \to (\Sch/S)_{fppf}$, $i \mapsto X_i$ be a diagram of algebraic spaces over $B$. Let $(X, X_i \to X)$ be a cocone for the diagram in the category of algebraic spaces over $B$ (Categories, Remark \ref{categories-remark-cones-and-cocones}). Assume that \begin{enumerate} \item the base change functor $X_{spaces, \'etale} \to \lim X_{i, spaces, \etale}$, sending $U$ to $U_i = X_i \times_X U$ is an equivalence, \item given \begin{enumerate} \item $B'$ affine and \'etale over $B$, \item $Z$ an affine scheme over $B'$, \item $U \to X \times_B B'$ an \'etale morphism of algebraic spaces with $U$ affine, \item $f_i : U_i \to Z$ a cocone over $B'$ of the diagram $i \mapsto U_i = U \times_X X_i$, \end{enumerate} there exists a unique morphism $f : U \to Z$ over $B'$ such that $f_i$ equals the composition $U_i \to U \to Z$. \end{enumerate} Then $X = \colim X_i$ in the category of all algebraic spaces over $B$.


spaces-pushouts-lemma-glue-etale-sheaf-etale

Let $S$ be a scheme. Let $\{f_i : X_i \to X\}$ be an \'etale covering of algebraic spaces. The functor $$ \Sh(X_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{f_i : X_i \to X\} $$ is an equivalence of categories.


spaces-pushouts-lemma-reduce-to-scheme-base

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\{Y_i \to Y\}_{i \in I}$ be an \'etale covering of algebraic spaces. If for each $i \in I$ the functor $$ \Sh(Y_{i, \etale}) \longrightarrow \text{descent data for \'etale sheaves wrt }\{X \times_Y Y_i \to Y_i\} $$ is an equivalence of categories and for each $i, j \in I$ the functor $$ \Sh((Y_i \times_Y Y_j)_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt } \{X \times_Y Y_i \times_Y Y_j \to Y_i \times_Y Y_j\} $$ is an equivalence of categories, then $$ \Sh(Y_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{X \to Y\} $$ is an equivalence of categories.


spaces-pushouts-lemma-representable-case

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is representable (by schemes) and $f$ has one of the following properties: surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation Then  $$ \Sh(Y_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{X \to Y\} $$ is an equivalence of categories.


spaces-pushouts-lemma-reduce-to-scheme-source

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\pi : X' \to X$ be a morphism of algebraic spaces. Assume \begin{enumerate} \item $f \circ \pi$ is representable (by schemes), \item $f \circ \pi$ has one of the following properties: surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation. \end{enumerate} Then  $$ \Sh(Y_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{X \to Y\} $$ is an equivalence of categories.


spaces-pushouts-lemma-glue-etale-sheaf-proper-surjective

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which has one of the following properties: surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation. Then the functor $$ \Sh(Y_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{X \to Y\} $$ is an equivalence of categories.


spaces-pushouts-lemma-glue-etale-sheaf-modification

Let $S$ be a scheme. Let $f : Y' \to Y$ be a proper morphism of algebraic spaces over $S$. Let $i : Z \to Y$ be a closed immersion. Set $E = Z \times_Y Y'$. Picture $$ \xymatrix{ E \ar[d]_g \ar[r]_j & Y' \ar[d]^f \\ Z \ar[r]^i & Y } $$ If $f$ is an isomorphism over $Y \setminus Z$, then the functor $$ \Sh(Y_\etale) \longrightarrow \Sh(Y'_\etale) \times_{\Sh(E_\etale)} \Sh(Z_\etale) $$ is an equivalence of categories.


spaces-pushouts-lemma-descend-etale-proper-surjective

Let $S$ be a scheme. Let $f : X \to Y$ be a proper surjective morphism of algebraic spaces over $S$. Any descent datum $(U/X, \varphi)$ relative to $f$ (Descent on Spaces, Definition \ref{spaces-descent-definition-descent-datum}) with $U$ \'etale over $X$ is effective (Descent on Spaces, Definition \ref{spaces-descent-definition-effective}). More precisely, there exists an \'etale morphism $V \to Y$ of algebraic spaces whose corresponding canonical descent datum is isomorphic to $(U/X, \varphi)$.


spaces-pushouts-lemma-pushout-along-thickening-schemes

Let $S$ be a scheme. Let $X \to X'$ be a thickening of schemes over $S$ and let $X \to Y$ be an affine morphism of schemes over $S$. Let $Y' = Y \amalg_X X'$ be the pushout in the category of schemes (see More on Morphisms, Lemma \ref{more-morphisms-lemma-pushout-along-thickening}). Then $Y'$ is also a pushout in the category of algebraic spaces over $S$.


spaces-pushouts-lemma-pushout-along-thickening

Let $S$ be a scheme. Let $X \to X'$ be a thickening of algebraic spaces over $S$ and let $X \to Y$ be an affine morphism of algebraic spaces over $S$. Then there exists a pushout $$ \xymatrix{ X \ar[r] \ar[d]_f & X' \ar[d]^{f'} \\ Y \ar[r] & Y \amalg_X X' } $$ in the category of algebraic spaces over $S$. Moreover $Y' = Y \amalg_X X'$ is a thickening of $Y$ and $$ \mathcal{O}_{Y'} = \mathcal{O}_Y \times_{f_*\mathcal{O}_X} f'_*\mathcal{O}_{X'} $$ as sheaves on $Y_\etale = (Y')_\etale$.


spaces-pushouts-lemma-categories-spaces-over-pushout

Let $S$ be a base scheme. Let $X \to X'$ be a thickening of algebraic spaces over $S$ and let $X \to Y$ be an affine morphism of algebraic spaces over $S$. Let $Y' = Y \amalg_X X'$ be the pushout (see Lemma \ref{lemma-pushout-along-thickening}). Base change gives a functor $$ F : (\textit{Spaces}/Y') \longrightarrow (\textit{Spaces}/Y) \times_{(\textit{Spaces}/Y')} (\textit{Spaces}/X') $$ given by $V' \longmapsto (V' \times_{Y'} Y, V' \times_{Y'} X', 1)$ which sends $(\Sch/Y')$ into $(\Sch/Y) \times_{(\Sch/Y')} (\Sch/X')$. The functor $F$ has a left adjoint $$ G : (\textit{Spaces}/Y) \times_{(\textit{Spaces}/Y')} (\textit{Spaces}/X') \longrightarrow (\textit{Spaces}/Y') $$ which sends the triple $(V, U', \varphi)$ to the pushout $V \amalg_{(V \times_Y X)} U'$ in the category of algebraic spaces over $S$. The functor $G$ sends $(\Sch/Y) \times_{(\Sch/Y')} (\Sch/X')$ into $(\Sch/Y')$.


spaces-pushouts-lemma-diagram

Let $S$ be a scheme. Let $$ \xymatrix{ A \ar[r] \ar[d] & C \ar[d] \ar[r] & E \ar[d] \\ B \ar[r] & D \ar[r] & F } $$ be a commutative diagram of algebraic spaces over $S$. Assume that $A, B, C, D$ and $A, B, E, F$ form cartesian squares and that $B \to D$ is surjective \'etale. Then $C, D, E, F$ is a cartesian square.


spaces-pushouts-lemma-equivalence-categories-spaces-over-pushout

In the situation of Lemma \ref{lemma-categories-spaces-over-pushout} the functor $F \circ G$ is isomorphic to the identity functor.


spaces-pushouts-lemma-space-over-pushout-flat-modules

Let $S$ be a base scheme. Let $X \to X'$ be a thickening of algebraic spaces over $S$ and let $X \to Y$ be an affine morphism of algebraic spaces over $S$. Let $Y' = Y \amalg_X X'$ be the pushout (see Lemma \ref{lemma-pushout-along-thickening}). Let $V' \to Y'$ be a morphism of algebraic spaces over $S$. Set $V = Y \times_{Y'} V'$, $U' = X' \times_{Y'} V'$, and $U = X \times_{Y'} V'$. There is an equivalence of categories between \begin{enumerate} \item quasi-coherent $\mathcal{O}_{V'}$-modules flat over $Y'$, and \item the category of triples $(\mathcal{G}, \mathcal{F}', \varphi)$ where \begin{enumerate} \item $\mathcal{G}$ is a quasi-coherent $\mathcal{O}_V$-module flat over $Y$, \item $\mathcal{F}'$ is a quasi-coherent $\mathcal{O}_{U'}$-module flat over $X$, and \item $\varphi : (U \to V)^*\mathcal{G} \to (U \to U')^*\mathcal{F}'$ is an isomorphism of $\mathcal{O}_U$-modules. \end{enumerate} \end{enumerate} The equivalence maps $\mathcal{G}'$ to $((V \to V')^*\mathcal{G}', (U' \to V')^*\mathcal{G}', can)$. Suppose $\mathcal{G}'$ corresponds to the triple $(\mathcal{G}, \mathcal{F}', \varphi)$. Then \begin{enumerate} \item[(a)] $\mathcal{G}'$ is a finite type $\mathcal{O}_{V'}$-module if and only if $\mathcal{G}$ and $\mathcal{F}'$ are finite type $\mathcal{O}_Y$ and $\mathcal{O}_{U'}$-modules. \item[(b)] if $V' \to Y'$ is locally of finite presentation, then $\mathcal{G}'$ is an $\mathcal{O}_{V'}$-module of finite presentation if and only if $\mathcal{G}$ and $\mathcal{F}'$ are $\mathcal{O}_Y$ and $\mathcal{O}_{U'}$-modules of finite presentation. \end{enumerate}


spaces-pushouts-lemma-equivalence-categories-spaces-pushout-flat

In the situation of Lemma \ref{lemma-equivalence-categories-spaces-over-pushout}. If $V' = G(V, U', \varphi)$ for some triple $(V, U', \varphi)$, then \begin{enumerate} \item $V' \to Y'$ is locally of finite type if and only if $V \to Y$ and $U' \to X'$ are locally of finite type, \item $V' \to Y'$ is flat if and only if $V \to Y$ and $U' \to X'$ are flat, \item $V' \to Y'$ is flat and locally of finite presentation if and only if $V \to Y$ and $U' \to X'$ are flat and locally of finite presentation, \item $V' \to Y'$ is smooth if and only if $V \to Y$ and $U' \to X'$ are smooth, \item $V' \to Y'$ is \'etale if and only if $V \to Y$ and $U' \to X'$ are \'etale, and \item add more here as needed. \end{enumerate} If $W'$ is flat over $Y'$, then the adjunction mapping $G(F(W')) \to W'$ is an isomorphism. Hence $F$ and $G$ define mutually quasi-inverse functors between the category of spaces flat over $Y'$ and the category of triples $(V, U', \varphi)$ with $V \to Y$ and $U' \to X'$ flat.


spaces-pushouts-lemma-pushout-along-thickening-derived

Let $S$ be a scheme. Consider a pushout $$ \xymatrix{ X \ar[r]_i \ar[d]_f & X' \ar[d]^{f'} \\ Y \ar[r]^j & Y' } $$ in the category of algebraic spaces over $S$ as in Lemma \ref{lemma-pushout-along-thickening}. Assume $i$ is a thickening. Then the essential image of the functor\footnote{All functors given by derived pullback.} $$ D(\mathcal{O}_{Y'}) \longrightarrow D(\mathcal{O}_Y) \times_{D(\mathcal{O}_X)} D(\mathcal{O}_{X'}) $$ contains every triple $(M, K', \alpha)$ where $M \in D(\mathcal{O}_Y)$ and $K' \in D(\mathcal{O}_{X'})$ are pseudo-coherent.


spaces-pushouts-lemma-elementary-distinguished-square-pushout

Let $S$ be a scheme. Let $(U \subset W, f : V \to W)$ be an elementary distinguished square. Then $$ \xymatrix{ U \times_W V \ar[r] \ar[d] & V \ar[d]^f \\ U \ar[r] & W } $$ is a pushout in the category of algebraic spaces over $S$.


spaces-pushouts-lemma-construct-elementary-distinguished-square

Let $S$ be a scheme. Let $V$, $U$ be algebraic spaces over $S$. Let $V' \subset V$ be an open subspace and let $f' : V' \to U$ be a separated \'etale morphism of algebraic spaces over $S$. Then there exists a pushout $$ \xymatrix{ V' \ar[r] \ar[d] & V \ar[d]^f \\ U \ar[r] & W } $$ in the category of algebraic spaces over $S$ and moreover $(U \subset W, f : V \to W)$ is an elementary distinguished square.


spaces-pushouts-lemma-stalk-pushforward-with-support

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Let $Z \subset X$ closed subspace such that $f^{-1}Z \to Z$ is integral and universally injective. Let $\overline{y}$ be a geometric point of $Y$ and $\overline{x} = f(\overline{y})$. We have $$ (Rf_*Q)_{\overline{x}} = Q_{\overline{y}} $$ in $D(\textit{Ab})$ for any object $Q$ of $D(Y_\etale)$ supported on $|f^{-1}Z|$.


spaces-pushouts-lemma-stalk-formal-glueing

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Let $Z \subset X$ closed subspace such that $f^{-1}Z \to Z$ is integral and universally injective. Let $\overline{y}$ be a geometric point of $Y$ and $\overline{x} = f(\overline{y})$. Let $\mathcal{G}$ be an abelian sheaf on $Y$. Then the map of two term complexes $$ \left(f_*\mathcal{G}_{\overline{x}} \to (f \circ j')_*(\mathcal{G}|_V)_{\overline{x}}\right) \longrightarrow \left(\mathcal{G}_{\overline{y}} \to j'_*(\mathcal{G}|_V)_{\overline{y}}\right) $$ induces an isomorphism on kernels and an injection on cokernels. Here $V = Y \setminus f^{-1}Z$ and $j' : V \to Y$ is the inclusion.


spaces-pushouts-lemma-stalk-of-pushforward

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism. Let $\overline{x}$ be a geometric point of $X$ and let $\Spec(\mathcal{O}_{X, \overline{x}}) \to X$ be the canonical morphism. For a quasi-coherent module $\mathcal{G}$ on $Y$ we have $$ f_*\mathcal{G}_{\overline{x}} = \Gamma(Y \times_X \Spec(\mathcal{O}_{X, \overline{x}}), p^*\mathcal{F}) $$ where $p : Y \times_X \Spec(\mathcal{O}_{X, \overline{x}}) \to Y$ is the projection.


spaces-pushouts-lemma-stalk-of-module-with-support

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $i : Z \to X$ be a closed immersion of finite presentation. Let $Q \in D_\QCoh(\mathcal{O}_X)$ be supported on $|Z|$. Let $\overline{x}$ be a geometric point of $X$ and let $I_{\overline{x}} \subset \mathcal{O}_{X, \overline{x}}$ be the stalk of the ideal sheaf of $Z$. Then the cohomology modules $H^n(Q_{\overline{x}})$ are $I_{\overline{x}}$-power torsion (see More on Algebra, Definition \ref{more-algebra-definition-f-power-torsion}).


spaces-pushouts-lemma-formal-glueing-on-closed

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Let $Z \subset X$ be a closed subspace. Assume $f^{-1}Z \to Z$ is an isomorphism and that $f$ is flat in every point of $f^{-1}Z$. For any $Q$ in $D_\QCoh(\mathcal{O}_Y)$ supported on $|f^{-1}Z|$ we have $Lf^*Rf_*Q = Q$.


spaces-pushouts-lemma-adjoint

In Situation \ref{situation-formal-glueing}. The functor (\ref{equation-reverse}) is right adjoint to the functor (\ref{equation-formal-glueing-modules}).


spaces-pushouts-lemma-dominate-by-fpqc-covering

In Situation \ref{situation-formal-glueing} there exists an fpqc covering $\{X_i \to X\}_{i \in I}$ refining the family $\{U \to X, Y \to X\}$.


spaces-pushouts-lemma-equivalence-on-affine

In Situation \ref{situation-formal-glueing} the functor (\ref{equation-formal-glueing-spaces}) restricts to an equivalence \begin{enumerate} \item from the category of algebraic spaces affine over $X$ to the full subcategory of $\textit{Spaces}(Y \to X, Z)$ consisting of $(U' \leftarrow V' \rightarrow Y')$ with $U' \to U$, $V' \to V$, and $Y' \to Y$ affine, \item from the category of closed immersions $X' \to X$ to the full subcategory of $\textit{Spaces}(Y \to X, Z)$ consisting of $(U' \leftarrow V' \rightarrow Y')$ with $U' \to U$, $V' \to V$, and $Y' \to Y$ closed immersions, and \item same statement as in (2) for finite morphisms. \end{enumerate}


spaces-pushouts-lemma-reflects-isomorphisms

In Situation \ref{situation-formal-glueing} the functor (\ref{equation-formal-glueing-spaces}) reflects isomorphisms.


spaces-pushouts-lemma-glueing-affines

Let $(R \to R', f)$ be a glueing pair, see above. The functor (\ref{equation-beauville-laszlo-glueing-spaces}) restricts to an equivalence between the category of affine $Y/X$ which are glueable for $(R \to R', f)$ and the full subcategory of objects $(V, V', Y')$ of $\textit{Spaces}(U \leftarrow U' \to X')$ with $V$, $V'$, $Y'$ affine.


spaces-pushouts-lemma-glueing-affines-etale

Let $P$ be one of the following properties of morphisms: ``finite'', ``closed immersion'', ``flat'', ``finite type'', ``flat and finite presentation'', ``\'etale''. Under the equivalence of Lemma \ref{lemma-glueing-affines} the morphisms having $P$ correspond to morphisms of triples whose components have $P$.


spaces-pushouts-lemma-glueing-f

Let $(R \to R', f)$ be a glueing pair, see above. The functor (\ref{equation-beauville-laszlo-glueing-spaces}) is faithful on the full subcategory of algebraic spaces $Y/X$ glueable for $(R \to R', f)$.


spaces-pushouts-lemma-coequalizer

Let $S$ be a scheme. Let $$ g : Y \longrightarrow X $$ be a morphism of algebraic spaces over $S$. Assume $X$ is locally Noetherian, and $g$ is proper. Let $R = Y \times_X Y$ with projection morphisms $t, s : R \to Y$. There exists a coequalizer $X'$ of $s, t : R \to Y$ in the category of algebraic spaces over $S$. Moreover \begin{enumerate} \item The morphism $X' \to X$ is finite. \item The morphism $Y \to X'$ is proper. \item The morphism $Y \to X'$ is surjective. \item The morphism $X' \to X$ is universally injective. \item If $g$ is surjective, the morphism $X' \to X$ is a universal homeomorphism. \end{enumerate}


spaces-pushouts-lemma-essentially-constant

In Situation \ref{situation-coequalizer-glue} assume $X$ quasi-compact. In (\ref{equation-system-coequalizers}) for all $n$ large enough, there exists an $m$ such that $X_n \to X_{n + m}$ factors through a closed immersion $X \to X_{n + m}$.


spaces-pushouts-lemma-check-separated

Let $S$ be a scheme. Let $X \to Y$ be a morphism of algebraic spaces over $S$. If $(U \subset X, f : V \to X)$ is an elementary distinguished square such that $U \to Y$ and $V \to Y$ are separated and $U \times_X V \to U \times_Y V$ is closed, then $X \to Y$ is separated.


spaces-pushouts-lemma-separate-disjoint-locally-closed-by-blowing-up

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U \subset X$ be a quasi-compact open. \begin{enumerate} \item If $Z_1, Z_2 \subset X$ are closed subspaces of finite presentation such that $Z_1 \cap Z_2 \cap U = \emptyset$, then there exists a $U$-admissible blowing up $X' \to X$ such that the strict transforms of $Z_1$ and $Z_2$ are disjoint. \item If $T_1, T_2 \subset |U|$ are disjoint constructible closed subsets, then there is a $U$-admissible blowing up $X' \to X$ such that the closures of $T_1$ and $T_2$ are disjoint. \end{enumerate}


spaces-pushouts-lemma-blowup-iso-along

Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of quasi-compact and quasi-separated algebraic spaces over $S$. Let $V \subset Y$ be a quasi-compact open and $U = f^{-1}(V)$. Let $T \subset |V|$ be a closed subset such that $f|_U : U \to V$ is an isomorphism over an open neighbourhood of $T$ in $V$. Then there exists a $V$-admissible blowing up $Y' \to Y$ such that the strict transform $f' : X' \to Y'$ of $f$ is an isomorphism over an open neighbourhood of the closure of $T$ in $|Y'|$.


spaces-pushouts-lemma-blowup-etale-along

Let $S$ be a scheme. Consider a diagram $$ \xymatrix{ X \ar[d]_f & U \ar[l] \ar[d]_{f|_U} & A \ar[d] \ar[l] \\ Y & V \ar[l] & B \ar[l] } $$ of quasi-compact and quasi-separated algebraic spaces over $S$. Assume \begin{enumerate} \item $f$ is proper, \item $V$ is a quasi-compact open of $Y$, $U = f^{-1}(V)$, \item $B \subset V$ and $A \subset U$ are closed subspaces, \item $f|_A : A \to B$ is an isomorphism, and $f$ is \'etale at every point of $A$. \end{enumerate} Then there exists a $V$-admissible blowing up $Y' \to Y$ such that the strict transform $f' : X' \to Y'$ satisfies: for every geometric point $\overline{a}$ of the closure of $|A|$ in $|X'|$ there exists a quotient $\mathcal{O}_{X', \overline{a}} \to \mathcal{O}$ such that $\mathcal{O}_{Y', f'(\overline{a})} \to \mathcal{O}$ is finite flat.


spaces-pushouts-lemma-replaced-by-strict-transform

Let $S$ be a scheme. Let $X \to B$ and $Y \to B$ be morphisms of algebraic spaces over $S$. Let $U \subset X$ be an open subspace. Let $V \to X \times_B Y$ be a quasi-compact morphism whose composition with the first projection maps into $U$. Let $Z \subset X \times_B Y$ be the scheme theoretic image of $V \to X \times_B Y$. Let $X' \to X$ be a $U$-admissible blowup. Then the scheme theoretic image of $V \to X' \times_B Y$ is the strict transform of $Z$ with respect to the blowing up.


spaces-pushouts-lemma-compactification-dominates

Let $S$ be a scheme. Let $B$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U$ be an algebraic space of finite type and separated over $B$. Let $V \to U$ be an \'etale morphism. If $V$ has a compactification $V \subset Y$ over $B$, then there exists a $V$-admissible blowing up $Y' \to Y$ and an open $V \subset V' \subset Y'$ such that $V \to U$ extends to a proper morphism $V' \to U$.


spaces-pushouts-lemma-two-compactifications

Let $B$ be an algebraic space of finite type over $\mathbf{Z}$. Let $U$ be an algebraic space of finite type and separated over $B$. Let $(U_2 \subset U, f : U_1 \to U)$ be an elementary distinguished square. Assume $U_1$ and $U_2$ have compactifications over $B$ and $U_1 \times_U U_2 \to U$ has dense image. Then $U$ has a compactification over $B$.


spaces-pushouts-lemma-filter-Noetherian-space

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $U \subset X$ be a proper dense open subspace. Then there exists an affine scheme $V$ and an \'etale morphism $V \to X$ such that \begin{enumerate} \item the open subspace $W = U \cup \Im(V \to X)$ is strictly larger than $U$, \item $(U \subset W, V \to W)$ is a distinguished square, and \item $U \times_W V \to U$ has dense image. \end{enumerate}


spaces-pushouts-proposition-formal-glueing-modules

In Situation \ref{situation-formal-glueing} the functor (\ref{equation-formal-glueing-modules}) is an equivalence with quasi-inverse given by (\ref{equation-reverse}).


varieties-theorem-varieties-rational-maps

Let $k$ be a field. The category of varieties and dominant rational maps is equivalent to the category of finitely generated field extensions $K/k$.


varieties-lemma-product-varieties

\begin{slogan} Products of varieties are varieties over algebraically closed fields. \end{slogan} Let $k$ be an algebraically closed field. Let $X$, $Y$ be varieties over $k$. Then $X \times_{\Spec(k)} Y$ is a variety over $k$.


varieties-lemma-change-fields-flat

Let $K/k$ be an extension of fields. Let $X$ be scheme over $k$ and set $Y = X_K$. If $y \in Y$ with image $x \in X$, then \begin{enumerate} \item $\mathcal{O}_{X, x} \to \mathcal{O}_{Y, y}$ is a faithfully flat local ring homomorphism, \item with $\mathfrak p_0 = \Ker(\kappa(x) \otimes_k K \to \kappa(y))$ we have $\kappa(y) = \kappa(\mathfrak p_0)$, \item $\mathcal{O}_{Y, y} = (\mathcal{O}_{X, x} \otimes_k K)_\mathfrak p$ where $\mathfrak p \subset \mathcal{O}_{X, x} \otimes_k K$ is the inverse image of $\mathfrak p_0$. \item we have $\mathcal{O}_{Y, y}/\mathfrak m_x\mathcal{O}_{Y, y} = (\kappa(x) \otimes_k K)_{\mathfrak p_0}$ \end{enumerate}


varieties-lemma-change-fields-algebraic-dim

Notation as in Lemma \ref{lemma-change-fields-flat}. Assume $X$ is locally of finite type over $k$. Then $$ \dim(\mathcal{O}_{Y, y}/\mathfrak m_x\mathcal{O}_{Y, y}) = \text{trdeg}_k(\kappa(x)) - \text{trdeg}_K(\kappa(y)) = \dim(\mathcal{O}_{Y, y}) - \dim(\mathcal{O}_{X, x}) $$


varieties-lemma-change-fields-algebraic-unramified

Notation as in Lemma \ref{lemma-change-fields-flat}. Assume $X$ is locally of finite type over $k$, that $\dim(\mathcal{O}_{X, x}) = \dim(\mathcal{O}_{Y, y})$ and that $\kappa(x) \otimes_k K$ is reduced (for example if $\kappa(x)/k$ is separable or $K/k$ is separable). Then $\mathfrak m_x \mathcal{O}_{Y, y} = \mathfrak m_y$.


varieties-lemma-geometrically-reduced-at-point

\begin{slogan} Geometric reducedness can be checked on local rings. \end{slogan} Let $k$ be a field. Let $X$ be a scheme over $k$. Let $x \in X$. The following are equivalent \begin{enumerate} \item $X$ is geometrically reduced at $x$, and \item the ring $\mathcal{O}_{X, x}$ is geometrically reduced over $k$ (see Algebra, Definition \ref{algebra-definition-geometrically-reduced}). \end{enumerate}


varieties-lemma-perfect-reduced

Let $X$ be a scheme over a perfect field $k$ (e.g.\ $k$ has characteristic zero). Let $x \in X$. If $\mathcal{O}_{X, x}$ is reduced, then $X$ is geometrically reduced at $x$. If $X$ is reduced, then $X$ is geometrically reduced over $k$.


varieties-lemma-geometrically-reduced

Let $k$ be a field of characteristic $p > 0$. Let $X$ be a scheme over $k$. The following are equivalent \begin{enumerate} \item $X$ is geometrically reduced, \item $X_{k'}$ is reduced for every field extension $k \subset k'$, \item $X_{k'}$ is reduced for every finite purely inseparable field extension $k \subset k'$, \item $X_{k^{1/p}}$ is reduced, \item $X_{k^{perf}}$ is reduced, \item $X_{\bar k}$ is reduced, \item for every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is geometrically reduced (see Algebra, Definition \ref{algebra-definition-geometrically-reduced}). \end{enumerate}


varieties-lemma-check-only-finite-inseparable-extensions

Let $k$ be a field of characteristic $p > 0$. Let $X$ be a scheme over $k$. Let $x \in X$. The following are equivalent \begin{enumerate} \item $X$ is geometrically reduced at $x$, \item $\mathcal{O}_{X_{k'}, x'}$ is reduced for every finite purely inseparable field extension $k'$ of $k$ and $x' \in X_{k'}$ the unique point lying over $x$, \item $\mathcal{O}_{X_{k^{1/p}}, x'}$ is reduced for $x' \in X_{k^{1/p}}$ the unique point lying over $x$, and \item $\mathcal{O}_{X_{k^{perf}}, x'}$ is reduced for $x' \in X_{k^{perf}}$ the unique point lying over $x$. \end{enumerate}


varieties-lemma-geometrically-reduced-upstairs

Let $k$ be a field. Let $X$ be a scheme over $k$. Let $k'/k$ be a field extension. Let $x \in X$ be a point, and let $x' \in X_{k'}$ be a point lying over $x$. The following are equivalent \begin{enumerate} \item $X$ is geometrically reduced at $x$, \item $X_{k'}$ is geometrically reduced at $x'$. \end{enumerate} In particular, $X$ is geometrically reduced over $k$ if and only if $X_{k'}$ is geometrically reduced over $k'$.


varieties-lemma-geometrically-reduced-any-base-change

Let $k$ be a field. Let $X$, $Y$ be schemes over $k$. \begin{enumerate} \item If $X$ is geometrically reduced at $x$, and $Y$ reduced, then $X \times_k Y$ is reduced at every point lying over $x$. \item If $X$ geometrically reduced over $k$ and $Y$ reduced. Then $X \times_k Y$ is reduced. \end{enumerate}


varieties-lemma-generic-points-geometrically-reduced

Let $k$ be a field. Let $X$ be a scheme over $k$. \begin{enumerate} \item If $x' \leadsto x$ is a specialization and $X$ is geometrically reduced at $x$, then $X$ is geometrically reduced at $x'$. \item If $x \in X$ such that (a) $\mathcal{O}_{X, x}$ is reduced, and (b) for each specialization $x' \leadsto x$ where $x'$ is a generic point of an irreducible component of $X$ the scheme $X$ is geometrically reduced at $x'$, then $X$ is geometrically reduced at $x$. \item If $X$ is reduced and geometrically reduced at all generic points of irreducible components of $X$, then $X$ is geometrically reduced. \end{enumerate}


varieties-lemma-finite-extension-geometrically-reduced

Let $k$ be a field. Let $X \to \Spec(k)$ be locally of finite type. Assume $X$ has finitely many irreducible components. Then there exists a finite purely inseparable extension $k \subset k'$ such that $(X_{k'})_{red}$ is geometrically reduced over $k'$.


varieties-lemma-geometrically-connected-check-after-extension

Let $X$ be a scheme over the field $k$. Let $k \subset k'$ be a field extension. Then $X$ is geometrically connected over $k$ if and only if $X_{k'}$ is geometrically connected over $k'$.


varieties-lemma-bijection-connected-components

Let $k$ be a field. Let $X$, $Y$ be schemes over $k$. Assume $X$ is geometrically connected over $k$. Then the projection morphism $$ p : X \times_k Y \longrightarrow Y $$ induces a bijection between connected components.


varieties-lemma-affine-geometrically-connected

Let $k$ be a field. Let $A$ be a $k$-algebra. Then $X = \Spec(A)$ is geometrically connected over $k$ if and only if $A$ is geometrically connected over $k$ (see Algebra, Definition \ref{algebra-definition-geometrically-connected}).


varieties-lemma-separably-closed-field-connected-components

Let $k \subset k'$ be an extension of fields. Let $X$ be a scheme over $k$. Assume $k$ separably algebraically closed. Then the morphism $X_{k'} \to X$ induces a bijection of connected components. In particular, $X$ is geometrically connected over $k$ if and only if $X$ is connected.


varieties-lemma-characterize-geometrically-connected

Let $k$ be a field. Let $X$ be a scheme over $k$. Let $\overline{k}$ be a separable algebraic closure of $k$. Then $X$ is geometrically connected if and only if the base change $X_{\overline{k}}$ is connected.


varieties-lemma-descend-open

Let $k$ be a field. Let $X$ be a scheme over $k$. Let $A$ be a $k$-algebra. Let $V \subset X_A$ be a quasi-compact open. Then there exists a finitely generated $k$-subalgebra $A' \subset A$ and a quasi-compact open $V' \subset X_{A'}$ such that $V = V'_A$.


varieties-lemma-Galois-action-quasi-compact-open

Let $k$ be a field. Let $X$ be a scheme over $k$. Let $\overline{k}$ be a (possibly infinite) Galois extension of $k$. Let $V \subset X_{\overline{k}}$ be a quasi-compact open. Then \begin{enumerate} \item there exists a finite subextension $k \subset k' \subset \overline{k}$ and a quasi-compact open $V' \subset X_{k'}$ such that $V = (V')_{\overline{k}}$, \item there exists an open subgroup $H \subset \text{Gal}(\overline{k}/k)$ such that $\sigma(V) = V$ for all $\sigma \in H$. \end{enumerate}


varieties-lemma-closed-fixed-by-Galois

Let $k$ be a field. Let $k \subset \overline{k}$ be a (possibly infinite) Galois extension. Let $X$ be a scheme over $k$. Let $\overline{T} \subset X_{\overline{k}}$ have the following properties \begin{enumerate} \item $\overline{T}$ is a closed subset of $X_{\overline{k}}$, \item for every $\sigma \in \text{Gal}(\overline{k}/k)$ we have $\sigma(\overline{T}) = \overline{T}$. \end{enumerate} Then there exists a closed subset $T \subset X$ whose inverse image in $X_{\overline{k}}$ is $\overline{T}$.


varieties-lemma-characterize-geometrically-disconnected

Let $k$ be a field. Let $X$ be a scheme over $k$. The following are equivalent \begin{enumerate} \item $X$ is geometrically connected, \item for every finite separable field extension $k \subset k'$ the scheme $X_{k'}$ is connected. \end{enumerate}


varieties-lemma-tricky

Let $k$ be a field. Let $k \subset \overline{k}$ be a (possibly infinite) Galois extension. Let $f : T \to X$ be a morphism of schemes over $k$. Assume $T_{\overline{k}}$ connected and $X_{\overline{k}}$ disconnected. Then $X$ is disconnected.


varieties-lemma-geometrically-connected-criterion

\begin{reference} \cite[IV Corollary 4.5.13.1(i)]{EGA} \end{reference} Let $k$ be a field. Let $T \to X$ be a morphism of schemes over $k$. Assume $T$ is geometrically connected and $X$ connected. Then $X$ is geometrically connected.


varieties-lemma-geometrically-connected-if-connected-and-point

Let $k$ be a field. Let $X$ be a scheme over $k$. Assume $X$ is connected and has a point $x$ such that $k$ is algebraically closed in $\kappa(x)$. Then $X$ is geometrically connected. In particular, if $X$ has a $k$-rational point and $X$ is connected, then $X$ is geometrically connected.


varieties-lemma-inverse-image-connected-component

Let $k \subset K$ be an extension of fields. Let $X$ be a scheme over $k$. For every connected component $T$ of $X$ the inverse image $T_K \subset X_K$ is a union of connected components of $X_K$.


varieties-lemma-image-connected-component-finite-extension

Let $k \subset K$ be a finite extension of fields and let $X$ be a scheme over  $k$. Denote by $p : X_K \to X$ the projection morphism. For every connected  component $T$ of $X_K$ the image $p(T)$ is a connected component of  $X$.


varieties-lemma-geometrically-irreducible-check-after-extension

Let $X$ be a scheme over the field $k$. Let $k \subset k'$ be a field extension. Then $X$ is geometrically irreducible over $k$ if and only if $X_{k'}$ is geometrically irreducible over $k'$.


varieties-lemma-separably-closed-irreducible

Let $X$ be a scheme over a separably closed field $k$. If $X$ is irreducible, then $X_K$ is irreducible for any field extension $k \subset K$. I.e., $X$ is geometrically irreducible over $k$.


varieties-lemma-bijection-irreducible-components

Let $k$ be a field. Let $X$, $Y$ be schemes over $k$. Assume $X$ is geometrically irreducible over $k$. Then the projection morphism $$ p : X \times_k Y \longrightarrow Y $$ induces a bijection between irreducible components.


varieties-lemma-geometrically-irreducible-local

\begin{slogan} Geometric irreductibility is Zariski local modulo connectedness. \end{slogan} Let $k$ be a field. Let $X$ be a scheme over $k$. The following are equivalent \begin{enumerate} \item $X$ is geometrically irreducible over $k$, \item for every nonempty affine open $U$ the $k$-algebra $\mathcal{O}_X(U)$ is geometrically irreducible over $k$ (see Algebra, Definition \ref{algebra-definition-geometrically-irreducible}), \item $X$ is irreducible and there exists an affine open covering $X = \bigcup U_i$ such that each $k$-algebra $\mathcal{O}_X(U_i)$ is geometrically irreducible, and \item there exists an open covering $X = \bigcup_{i \in I} X_i$ with $I \not = \emptyset$ such that $X_i$ is geometrically irreducible for each $i$ and such that $X_i \cap X_j \not = \emptyset$ for all $i, j \in I$. \end{enumerate} Moreover, if $X$ is geometrically irreducible so is every nonempty open subscheme of $X$.


varieties-lemma-geometrically-irreducible-function-field

Let $X$ be an irreducible scheme over the field $k$. Let $\xi \in X$ be its generic point. The following are equivalent \begin{enumerate} \item $X$ is geometrically irreducible over $k$, and \item $\kappa(\xi)$ is geometrically irreducible over $k$. \end{enumerate}


varieties-lemma-separably-closed-field-irreducible-components

Let $k \subset k'$ be an extension of fields. Let $X$ be a scheme over $k$. Set $X' = X_{k'}$. Assume $k$ separably algebraically closed. Then the morphism $X' \to X$ induces a bijection of irreducible components.


varieties-lemma-characterize-geometrically-irreducible

\begin{slogan} Geometric irreducibility can be tested over a separable algebraic closure of the base field. \end{slogan} Let $k$ be a field. Let $X$ be a scheme over $k$. The following are equivalent: \begin{enumerate} \item $X$ is geometrically irreducible over $k$, \item for every finite separable field extension $k \subset k'$ the scheme $X_{k'}$ is irreducible, and \item $X_{\overline{k}}$ is irreducible, where $k \subset \overline{k}$ is a separable algebraic closure of $k$. \end{enumerate}


varieties-lemma-inverse-image-irreducible

Let $k \subset K$ be an extension of fields. Let $X$ be a scheme over $k$. For every irreducible component $T$ of $X$ the inverse image $T_K \subset X_K$ is a union of irreducible components of $X_K$.


varieties-lemma-image-irreducible

Let $k \subset K$ be an extension of fields. Let $X$ be a scheme over $k$. For every irreducible component $\overline{T} \subset X_K$ the image of $\overline{T}$ in $X$ is an irreducible component in $X$. This defines a canonical map $$ \text{IrredComp}(X_K) \longrightarrow \text{IrredComp}(X) $$ which is surjective.


varieties-lemma-galois-action-irreducible-components

Let $k$ be a field, with separable algebraic closure $\overline{k}$. Let $X$ be a scheme over $k$. There is an action $$ \text{Gal}(\overline{k}/k)^{opp} \times \text{IrredComp}(X_{\overline{k}}) \longrightarrow \text{IrredComp}(X_{\overline{k}}) $$ with the following properties: \begin{enumerate} \item An element $\overline{T} \in \text{IrredComp}(X_{\overline{k}})$ is fixed by the action if and only if there exists an irreducible component $T \subset X$, which is geometrically irreducible over $k$, such that $T_{\overline{k}} = \overline{T}$. \item For any field extension $k \subset k'$ with separable algebraic closure $\overline{k}'$ the diagram $$ \xymatrix{ \text{Gal}(\overline{k}'/k') \times \text{IrredComp}(X_{\overline{k}'}) \ar[r] \ar[d] & \text{IrredComp}(X_{\overline{k}'}) \ar[d] \\ \text{Gal}(\overline{k}/k) \times \text{IrredComp}(X_{\overline{k}}) \ar[r] & \text{IrredComp}(X_{\overline{k}}) } $$ is commutative (where the right vertical arrow is a bijection according to Lemma \ref{lemma-separably-closed-field-irreducible-components}). \end{enumerate}


varieties-lemma-orbit-irreducible-components

Let $k$ be a field, with separable algebraic closure $\overline{k}$. Let $X$ be a scheme over $k$. The fibres of the map $$ \text{IrredComp}(X_{\overline{k}}) \longrightarrow \text{IrredComp}(X) $$ of Lemma \ref{lemma-image-irreducible} are exactly the orbits of $\text{Gal}(\overline{k}/k)$ under the action of Lemma \ref{lemma-galois-action-irreducible-components}.


varieties-lemma-galois-action-irreducible-components-locally-finite-type

Let $k$ be a field. Assume $X \to \Spec(k)$ locally of finite type. In this case \begin{enumerate} \item the action $$ \text{Gal}(\overline{k}/k)^{opp} \times \text{IrredComp}(X_{\overline{k}}) \longrightarrow \text{IrredComp}(X_{\overline{k}}) $$ is continuous if we give $\text{IrredComp}(X_{\overline{k}})$ the discrete topology, \item every irreducible component of $X_{\overline{k}}$ can be defined over a finite extension of $k$, and \item given any irreducible component $T \subset X$ the scheme $T_{\overline{k}}$ is a finite union of irreducible components of $X_{\overline{k}}$ which are all in the same $\text{Gal}(\overline{k}/k)$-orbit. \end{enumerate}


varieties-lemma-finite-extension-geometrically-irreducible-components

Let $k$ be a field. Let $X \to \Spec(k)$ be locally of finite type. Assume $X$ has finitely many irreducible components. Then there exists a finite separable extension $k \subset k'$ such that every irreducible component of $X_{k'}$ is geometrically irreducible over $k'$.


varieties-lemma-irreducible-components-geometrically-irreducible

Let $X$ be a scheme over the field $k$. Assume $X$ has finitely many irreducible components which are all geometrically irreducible. Then $X$ has finitely many connected components each of which is geometrically connected.


varieties-lemma-geometrically-integral

Let $k$ be a field. Let $X$ be a scheme over $k$. Then $X$ is geometrically integral over $k$ if and only if $X$ is both geometrically reduced and geometrically irreducible over $k$.


varieties-lemma-proper-geometrically-reduced-global-sections

Let $k$ be a field. Let $X$ be a proper scheme over $k$. \begin{enumerate} \item $A = H^0(X, \mathcal{O}_X)$ is a finite dimensional $k$-algebra, \item $A = \prod_{i = 1, \ldots, n} A_i$ is a product of Artinian local $k$-algebras, one factor for each connected component of $X$, \item if $X$ is reduced, then $A = \prod_{i = 1, \ldots, n} k_i$ is a product of fields, each a finite extension of $k$, \item if $X$ is geometrically reduced, then $k_i$ is finite separable over $k$, \item if $X$ is geometrically connected, then $A$ is geometrically irreducible over $k$, \item if $X$ is geometrically irreducible, then $A$ is geometrically irreducible over $k$, \item if $X$ is geometrically reduced and connected, then $A = k$, and \item if $X$ is geometrically integral, then $A = k$. \end{enumerate}


varieties-lemma-baby-stein

Let $X$ be a proper scheme over a field $k$. Set $A = H^0(X, \mathcal{O}_X)$. The fibres of the canonical morphism $X \to \Spec(A)$ are geometrically connected.


varieties-lemma-geometrically-reduced-stein

Let $k$ be a field. Let $X$ be a proper geometrically reduced scheme over $k$. The following are equivalent \begin{enumerate} \item $H^0(X, \mathcal{O}_X) = k$, and \item $X$ is geometrically connected. \end{enumerate}


varieties-lemma-geometrically-normal-at-point

Let $k$ be a field. Let $X$ be a scheme over $k$. Let $x \in X$. The following are equivalent \begin{enumerate} \item $X$ is geometrically normal at $x$, \item for every finite purely inseparable field extension $k'$ of $k$ and $x' \in X_{k'}$ lying over $x$ the local ring $\mathcal{O}_{X_{k'}, x'}$ is normal, and \item the ring $\mathcal{O}_{X, x}$ is geometrically normal over $k$ (see Algebra, Definition \ref{algebra-definition-geometrically-normal}). \end{enumerate}


varieties-lemma-geometrically-normal

Let $k$ be a field. Let $X$ be a scheme over $k$. The following are equivalent \begin{enumerate} \item $X$ is geometrically normal, \item $X_{k'}$ is a normal scheme for every field extension $k'/k$, \item $X_{k'}$ is a normal scheme for every finitely generated field extension $k'/k$, \item $X_{k'}$ is a normal scheme for every finite purely inseparable field extension $k'/k$, \item for every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is geometrically normal (see Algebra, Definition \ref{algebra-definition-geometrically-normal}), and \item $X_{k^{perf}}$ is a normal scheme. \end{enumerate}


varieties-lemma-geometrically-normal-upstairs

Let $k$ be a field. Let $X$ be a scheme over $k$. Let $k'/k$ be a field extension. Let $x \in X$ be a point, and let $x' \in X_{k'}$ be a point lying over $x$. The following are equivalent \begin{enumerate} \item $X$ is geometrically normal at $x$, \item $X_{k'}$ is geometrically normal at $x'$. \end{enumerate} In particular, $X$ is geometrically normal over $k$ if and only if $X_{k'}$ is geometrically normal over $k'$.


varieties-lemma-fibre-product-normal

Let $k$ be a field. Let $X$ be a geometrically normal scheme over $k$ and let $Y$ be a normal scheme over $k$. Then $X \times_k Y$ is a normal scheme.


varieties-lemma-base-change-normal-by-separable

Let $k$ be a field. Let $X$ be a normal scheme over $k$. Let $K/k$ be a separable field extension. Then $X_K$ is a normal scheme.


varieties-lemma-geometrically-normal-stein

Let $k$ be a field. Let $X$ be a proper geometrically normal scheme over $k$. The following are equivalent \begin{enumerate} \item $H^0(X, \mathcal{O}_X) = k$, \item $X$ is geometrically connected, \item $X$ is geometrically irreducible, and \item $X$ is geometrically integral. \end{enumerate}


varieties-lemma-locally-Noetherian-base-change

Let $k$ be a field. Let $X$ be a scheme over $k$. Let $k \subset k'$ be a finitely generated field extension. Then $X$ is locally Noetherian if and only if $X_{k'}$ is locally Noetherian.


varieties-lemma-geometrically-regular-at-point

Let $k$ be a field. Let $X$ be a locally Noetherian scheme over $k$. Let $x \in X$. The following are equivalent \begin{enumerate} \item $X$ is geometrically regular at $x$, \item for every finite purely inseparable field extension $k'$ of $k$ and $x' \in X_{k'}$ lying over $x$ the local ring $\mathcal{O}_{X_{k'}, x'}$ is regular, and \item the ring $\mathcal{O}_{X, x}$ is geometrically regular over $k$ (see Algebra, Definition \ref{algebra-definition-geometrically-regular}). \end{enumerate}


varieties-lemma-geometrically-regular

Let $k$ be a field. Let $X$ be a locally Noetherian scheme over $k$. The following are equivalent \begin{enumerate} \item $X$ is geometrically regular, \item $X_{k'}$ is a regular scheme for every finitely generated field extension $k \subset k'$, \item $X_{k'}$ is a regular scheme for every finite purely inseparable field extension $k \subset k'$, \item for every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is geometrically regular (see Algebra, Definition \ref{algebra-definition-geometrically-regular}), and \item there exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ is geometrically regular over $k$. \end{enumerate}


varieties-lemma-geometrically-regular-upstairs

Let $k$ be a field. Let $X$ be a scheme over $k$. Let $k'/k$ be a finitely generated field extension. Let $x \in X$ be a point, and let $x' \in X_{k'}$ be a point lying over $x$. The following are equivalent \begin{enumerate} \item $X$ is geometrically regular at $x$, \item $X_{k'}$ is geometrically regular at $x'$. \end{enumerate} In particular, $X$ is geometrically regular over $k$ if and only if $X_{k'}$ is geometrically regular over $k'$.


varieties-lemma-flat-under-geometrically-regular

Let $k$ be a field. Let $f : X \to Y$ be a morphism of locally Noetherian schemes over $k$. Let $x \in X$ be a point and set $y = f(x)$. If $X$ is geometrically regular at $x$ and $f$ is flat at $x$ then $Y$ is geometrically regular at $y$. In particular, if $X$ is geometrically regular over $k$ and $f$ is flat and surjective, then $Y$ is geometrically regular over $k$.


varieties-lemma-geometrically-regular-smooth

Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$. Let $x \in X$. Then $X$ is geometrically regular at $x$ if and only if $X \to \Spec(k)$ is smooth at $x$ (Morphisms, Definition \ref{morphisms-definition-smooth}).


varieties-lemma-CM-base-change

Let $X$ be a locally Noetherian scheme over the field $k$. Let $k \subset k'$ be a finitely generated field extension. Let $x \in X$ be a point, and let $x' \in X_{k'}$ be a point lying over $x$. Then we have $$ \mathcal{O}_{X, x}\text{ is Cohen-Macaulay} \Leftrightarrow \mathcal{O}_{X_{k'}, x'}\text{ is Cohen-Macaulay} $$ If $X$ is locally of finite type over $k$, the same holds for any field extension $k \subset k'$.


varieties-lemma-locally-finite-type-Jacobson

Let $X$ be a scheme which is locally of finite type over $k$. Then \begin{enumerate} \item for any closed point $x \in X$ the extension $k \subset \kappa(x)$ is algebraic, and \item $X$ is a Jacobson scheme (Properties, Definition \ref{properties-definition-jacobson}). \end{enumerate}


varieties-lemma-make-Jacobson

Let $X$ be a scheme over a field $k$. For any field extension $k \subset K$ whose cardinality is large enough we have \begin{enumerate} \item for any closed point $x \in X_K$ the extension $K \subset \kappa(x)$ is algebraic, and \item $X_K$ is a Jacobson scheme (Properties, Definition \ref{properties-definition-jacobson}). \end{enumerate}


varieties-lemma-ample-after-field-extension

Let $k$ be a field. Let $X$ be a scheme over $k$. If there exists an ample invertible sheaf on $X_K$ for some field extension $k \subset K$, then $X$ has an ample invertible sheaf.


varieties-lemma-quasi-projective-after-field-extension

Let $k$ be a field. Let $X$ be a scheme over $k$. If $X_K$ is quasi-projective over $K$ for some field extension $k \subset K$, then $X$ is quasi-projective over $k$.


varieties-lemma-proper-after-field-extension

Let $k$ be a field. Let $X$ be a scheme over $k$. If $X_K$ is proper over $K$ for some field extension $k \subset K$, then $X$ is proper over $k$.


varieties-lemma-tangent-space-cotangent-space

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. There is a canonical isomorphism $$ T_{X/S, x} = \Hom_{\mathcal{O}_{X, x}}(\Omega_{X/S, x}, \kappa(x)) $$ of vector spaces over $\kappa(x)$.


varieties-lemma-tangent-space-rational-point

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point and let $s = f(x) \in S$. Assume that $\kappa(x) = \kappa(s)$. Then there are canonical isomorphisms $$ \mathfrak m_x/(\mathfrak m_x^2 + \mathfrak m_s\mathcal{O}_{X, x}) = \Omega_{X/S, x} \otimes_{\mathcal{O}_{X, x}} \kappa(x) $$ and $$ T_{X/S, x} = \Hom_{\kappa(x)}( \mathfrak m_x/(\mathfrak m_x^2 + \mathfrak m_s\mathcal{O}_{X, x}), \kappa(x)) $$ This works more generally if $\kappa(x)/\kappa(s)$ is a separable algebraic extension.


varieties-lemma-map-tangent-spaces

Let $f : X \to Y$ be a morphism of schemes over a base scheme $S$. Let $x \in X$ be a point. Set $y = f(x)$. If $\kappa(y) = \kappa(x)$, then $f$ induces a natural linear map $$ \text{d}f : T_{X/S, x} \longrightarrow T_{Y/S, y} $$ which is dual to the linear map $\Omega_{Y/S, y} \otimes \kappa(y) \to \Omega_{X/S, x}$ via the identifications of Lemma \ref{lemma-tangent-space-cotangent-space}.


varieties-lemma-tangent-space-product

Let $X$, $Y$ be schemes over a base $S$. Let $x \in X$ and $y \in Y$ with the same image point $s \in S$ such that $\kappa(s) = \kappa(x)$ and $\kappa(s) = \kappa(y)$. There is a canonical isomorphism $$ T_{X \times_S Y/S, (x, y)} = T_{X/S, x} \oplus T_{Y/S, y} $$ The map from left to right is induced by the maps on tangent spaces coming from the projections $X \times_S Y \to X$ and $X \times_S Y \to Y$. The map from right to left is induced by the maps $1 \times y : X_s \to X_s \times_s Y_s$ and $x \times 1 : Y_s \to X_s \times_s Y_s$ via the identification (\ref{equation-tangent-space-fibre}) of tangent spaces with tangent spaces of fibres.


varieties-lemma-injective-tangent-spaces-unramified

Let $f : X \to Y$ be a morphism of schemes locally of finite type over a base scheme $S$. Let $x \in X$ be a point. Set $y = f(x)$ and assume that $\kappa(y) = \kappa(x)$. Then the following are equivalent \begin{enumerate} \item $\text{d}f : T_{X/S, x} \longrightarrow T_{Y/S, y}$ is injective, and \item $f$ is unramified at $x$. \end{enumerate}


varieties-lemma-quasi-finite-in-codim-1

Let $f : X \to Y$ be locally of finite type. Let $y \in Y$ be a point such that $\mathcal{O}_{Y, y}$ is Noetherian of dimension $\leq 1$. Assume in addition one of the following conditions is satisfied \begin{enumerate} \item for every generic point $\eta$ of an irreducible component of $X$ the field extension $\kappa(\eta) \supset \kappa(f(\eta))$ is finite (or algebraic), \item for every generic point $\eta$ of an irreducible component of $X$ such that $f(\eta) \leadsto y$ the field extension $\kappa(\eta) \supset \kappa(f(\eta))$ is finite (or algebraic), \item $f$ is quasi-finite at every generic point of an irreducible component of $X$, \item $Y$ is locally Noetherian and $f$ is quasi-finite at a dense set of points of $X$, \item add more here. \end{enumerate} Then $f$ is quasi-finite at every point of $X$ lying over $y$.


varieties-lemma-finite-in-codim-1

Let $f : X \to Y$ be a proper morphism. Let $y \in Y$ be a point such that $\mathcal{O}_{Y, y}$ is Noetherian of dimension $\leq 1$. Assume in addition one of the following conditions is satisfied \begin{enumerate} \item for every generic point $\eta$ of an irreducible component of $X$ the field extension $\kappa(\eta) \supset \kappa(f(\eta))$ is finite (or algebraic), \item for every generic point $\eta$ of an irreducible component of $X$ such that $f(\eta) \leadsto y$ the field extension $\kappa(\eta) \supset \kappa(f(\eta))$ is finite (or algebraic), \item $f$ is quasi-finite at every generic point of $X$, \item $Y$ is locally Noetherian and $f$ is quasi-finite at a dense set of points of $X$, \item add more here. \end{enumerate} Then there exists an open neighbourhood $V \subset Y$ of $y$ such that $f^{-1}(V) \to V$ is finite.


varieties-lemma-modification-normal-iso-over-codimension-1

Let $X$ be a Noetherian scheme. Let $f : Y \to X$ be a birational proper morphism of schemes with $Y$ reduced. Let $U \subset X$ be the maximal open over which $f$ is an isomorphism. Then $U$ contains \begin{enumerate} \item every point of codimension $0$ in $X$, \item every $x \in X$ of codimension $1$ on $X$ such that $\mathcal{O}_{X, x}$ is a discrete valuation ring, \item every $x \in X$ such that the fibre of $Y \to X$ over $x$ is finite and such that $\mathcal{O}_{X, x}$ is normal, and \item every $x \in X$ such that $f$ is quasi-finite at some $y \in Y$ lying over $x$ and $\mathcal{O}_{X, x}$ is normal. \end{enumerate}


varieties-lemma-noether-normalization-affine

Let $f : X \to S$ be a finite type morphism of affine schemes. Let $s \in S$. If $\dim(X_s) = d$, then there exists a factorization $$ X \xrightarrow{\pi} \mathbf{A}^d_S \to S $$ of $f$ such that the morphism $\pi_s : X_s \to \mathbf{A}^d_{\kappa(s)}$ of fibres over $s$ is finite.


varieties-lemma-geometric-structure-unramified

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. Let $V = \Spec(A)$ be an affine open neighbourhood of $f(x)$ in $S$. If $f$ is unramified at $x$, then there exist exists an affine open $U \subset X$ with $x \in U$ and $f(U) \subset V$ such that we have a commutative diagram $$ \xymatrix{ X \ar[d] & U \ar[l] \ar[rd] \ar[r]^-j & \Spec(A[t]_{g'}/(g)) \ar[d] \ar[r] & \Spec(A[t]) = \mathbf{A}^1_V \ar[ld] \\ Y & & V \ar[ll] } $$ where $j$ is an immersion, $g \in A[t]$ is a monic polynomial, and $g'$ is the derivative of $g$ with respect to $t$. If $f$ is \'etale at $x$, then we may choose the diagram such that $j$ is an open immersion.


varieties-lemma-unramfied-over-affine

Let $f : X \to S$ be a finite type morphism of affine schemes. Let $x \in X$ with image $s \in S$. Let $$ r = \dim_{\kappa(x)} \Omega_{X/S, x} \otimes_{\mathcal{O}_{X, x}} \kappa(x) = \dim_{\kappa(x)} \Omega_{X_s/s, x} \otimes_{\mathcal{O}_{X_s, x}} \kappa(x) = \dim_{\kappa(x)} T_{X/S, x} $$ Then there exists a factorization $$ X \xrightarrow{\pi} \mathbf{A}^r_S \to S $$ of $f$ such that $\pi$ is unramified at $x$.


varieties-lemma-dimension-fibre-in-codim-1

Let $f : X \to Y$ be locally of finite type. Let $x \in X$ be a point with image $y \in Y$ such that $\mathcal{O}_{Y, y}$ is Noetherian of dimension $\leq 1$. Let $d \geq 0$ be an integer such that for every generic point $\eta$ of an irreducible component of $X$ which contains $x$, we have $\dim_\eta(X_{f(\eta)}) = d$. Then $\dim_x(X_y) = d$.


varieties-lemma-dimension-fibre-in-higher-codimension

Let $f : X \to Y$ be locally of finite type. Let $x \in X$ be a point with image $y \in Y$ such that $\mathcal{O}_{Y, y}$ is Noetherian. Let $d \geq 0$ be an integer such that for every generic point $\eta$ of an irreducible component of $X$ which contains $x$, we have $f(\eta) \not = y$ and $\dim_\eta(X_{f(\eta)}) = d$. Then $\dim_x(X_y) \leq d + \dim(\mathcal{O}_{Y, y}) - 1$.


varieties-lemma-algebraic-scheme-dim-0

Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme of dimension $0$. Then $X$ is a disjoint union of spectra of local Artinian $k$-algebras $A$ with $\dim_k(A) < \infty$. If $X$ is an algebraic $k$-scheme of dimension $0$, then in addition $X$ is affine and the morphism $X \to \Spec(k)$ is finite.


varieties-lemma-dimension-locally-algebraic

Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. \begin{enumerate} \item \label{item-catenary} The topological space of $X$ is catenary (Topology, Definition \ref{topology-definition-catenary}). \item \label{item-dimension-at-closed-point} For $x \in X$ closed, we have $\dim_x(X) = \dim(\mathcal{O}_{X, x})$. \item \label{item-dimension-irreducible} For $X$ irreducible we have $\dim(X) = \dim(U)$ for any nonempty open $U \subset X$ and $\dim(X) = \dim_x(X)$ for any $x \in X$. \item \label{item-irreducible-maximal-chains} For $X$ irreducible any chain of irreducible closed subsets can be extended to a maximal chain and all maximal chains of irreducible closed subsets have length equal to $\dim(X)$. \item \label{item-dimension-irreducibles-passing-through} For $x \in X$ we have $\dim_x(X) = \max \dim(Z) = \min \dim(\mathcal{O}_{X, x'})$ where the maximum is over irreducible components $Z \subset X$ containing $x$ and the minimum is over specializations $x \leadsto x'$ with $x'$ closed in $X$. \item \label{item-dimension-irreducible-trdeg} If $X$ is irreducible with generic point $x$, then $\dim(X) = \text{trdeg}_k(\kappa(x))$. \item \label{item-immediate-specialization} If $x \leadsto x'$ is an immediate specialization of points of $X$, then we have $\text{trdeg}_k(\kappa(x)) = \text{trdeg}_k(\kappa(x')) + 1$. \item \label{item-dimension-sup-trdeg} The dimension of $X$ is the supremum of the numbers $\text{trdeg}_k(\kappa(x))$ where $x$ runs over the generic points of the irreducible components of $X$. \item \label{item-specialization} If $x \leadsto x'$ is a nontrivial specialization of points of $X$, then \begin{enumerate} \item $\dim_x(X) \leq \dim_{x'}(X)$, \item $\dim(\mathcal{O}_{X, x}) < \dim(\mathcal{O}_{X, x'})$, \item $\text{trdeg}_k(\kappa(x)) > \text{trdeg}_k(\kappa(x'))$, and \item any maximal chain of nontrivial specializations $x = x_0 \leadsto x_1 \leadsto \ldots \leadsto x_n = x$ has length $n = \text{trdeg}_k(\kappa(x)) - \text{trdeg}_k(\kappa(x'))$. \end{enumerate} \item \label{item-dimension-formula} For $x \in X$ we have $\dim_x(X) = \text{trdeg}_k(\kappa(x)) + \dim(\mathcal{O}_{X, x})$. \item \label{item-immediate-specialization-local-ring} If $x \leadsto x'$ is an immediate specialization of points of $X$ and $X$ is irreducible or equidimensional, then $\dim(\mathcal{O}_{X, x'}) = \dim(\mathcal{O}_{X, x}) + 1$. \end{enumerate}


varieties-lemma-dimension-fibres-locally-algebraic

Let $k$ be a field. Let $f : X \to Y$ be a morphism of locally algebraic $k$-schemes. \begin{enumerate} \item For $y \in Y$, the fibre $X_y$ is a locally algebraic scheme over $\kappa(y)$ hence all the results of Lemma \ref{lemma-dimension-locally-algebraic} apply. \item Assume $X$ is irreducible. Set $Z = \overline{f(X)}$ and $d = \dim(X) - \dim(Z)$. Then \begin{enumerate} \item $\dim_x(X_{f(x)}) \geq d$ for all $x \in X$, \item the set of $x \in X$ with $\dim_x(X_{f(x)}) = d$ is dense open, \item if $\dim(\mathcal{O}_{Z, f(x)}) \geq 1$, then $\dim_x(X_{f(x)}) \leq d + \dim(\mathcal{O}_{Z, f(x)}) - 1$, \item if $\dim(\mathcal{O}_{Z, f(x)}) = 1$, then $\dim_x(X_{f(x)}) = d$, \end{enumerate} \item For $x \in X$ with $y = f(x)$ we have $\dim_x(X_y) \geq \dim_x(X) - \dim_y(Y)$. \end{enumerate}


varieties-lemma-dimension-product-locally-algebraic

\begin{slogan} The dimension of the product is the sum of the dimensions. \end{slogan} Let $k$ be a field. Let $X$, $Y$ be locally algebraic $k$-schemes. \begin{enumerate} \item For $z \in X \times Y$ lying over $(x, y)$ we have $\dim_z(X \times Y) = \dim_x(X) + \dim_y(Y)$. \item We have $\dim(X \times Y) = \dim(X) + \dim(Y)$. \end{enumerate}


varieties-lemma-complete-local-ring-structure-as-algebra

Let $k$ be a field. Let $X$ be a locally Noetherian scheme over $k$. Let $x \in X$ be a point with residue field $\kappa$. There is an isomorphism \begin{equation} \label{equation-complete-local-ring} \kappa[[x_1, \ldots, x_n]]/I \longrightarrow \mathcal{O}_{X, x}^\wedge \end{equation} inducing the identity on residue fields. In general we cannot choose (\ref{equation-complete-local-ring}) to be a $k$-algebra isomorphism. However, if the extension $\kappa/k$ is separable, then we can choose (\ref{equation-complete-local-ring}) to be an isomorphism of $k$-algebras.


varieties-lemma-base-change-complete-local-ring

Let $K/k$ be an extension of fields. Let $X$ be a locally algebraic $k$-scheme. Set $Y = X_K$. Let $y \in Y$ be a point with image $x \in X$. Assume that $\dim(\mathcal{O}_{X, x}) = \dim(\mathcal{O}_{Y, y})$ and that $\kappa(x)/k$ is separable. Choose an isomorphism $$ \kappa(x)[[x_1, \ldots, x_n]]/(g_1, \ldots, g_m) \longrightarrow \mathcal{O}_{X, x}^\wedge $$ of $k$-algebras as in (\ref{equation-complete-local-ring}). Then we have an isomorphism $$ \kappa(y)[[x_1, \ldots, x_n]]/(g_1, \ldots, g_m) \longrightarrow \mathcal{O}_{Y, y}^\wedge $$ of $K$-algebras as in (\ref{equation-complete-local-ring}). Here we use $\kappa(x) \to \kappa(y)$ to view $g_j$ as a power series over $\kappa(y)$.


varieties-lemma-globally-generated-base-change

Let $X \to \Spec(A)$ be a morphism of schemes. Let $A \subset A'$ be a faithfully flat ring map. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $\mathcal{F}$ is globally generated if and only if the base change $\mathcal{F}_{A'}$ is globally generated.


varieties-lemma-very-ample-vanish-at-point

Let $k$ be an infinite field. Let $X$ be a scheme of finite type over $k$. Let $\mathcal{L}$ be a very ample invertible sheaf on $X$. Let $n \geq 0$ and $x, x_1, \ldots, x_n \in X$ be points with $x$ a $k$-rational point, i.e., $\kappa(x) = k$, and $x \not = x_i$ for $i = 1, \ldots, n$. Then there exists an $s \in H^0(X, \mathcal{L})$ which vanishes at $x$ but not at $x_i$.


varieties-lemma-separate-points-tangent-vectors

Let $k$ be an algebraically closed field. Let $X$ be a proper $k$-scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $V \subset H^0(X, \mathcal{L})$ be a $k$-subvector space. If \begin{enumerate} \item for every pair of distinct closed points $x, y \in X$ there is a section $s \in V$ which vanishes at $x$ but not at $y$, and \item for every closed point $x \in X$ and nonzero tangent vector $\theta \in T_{X/k, x}$ there exist a section $s \in V$ which vanishes at $x$ but whose pullback by $\theta$ is nonzero, \end{enumerate} then $\mathcal{L}$ is very ample and the canonical morphism $\varphi_{\mathcal{L}, V} : X \to \mathbf{P}(V)$ is a closed immersion.


varieties-lemma-variant-separate-points-tangent-vectors

Let $k$ be an algebraically closed field. Let $X$ be a proper $k$-scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Suppose that for every closed subscheme $Z \subset X$ of dimension $0$ and degree $2$ over $k$ the map $$ H^0(X, \mathcal{L}) \longrightarrow H^0(Z, \mathcal{L}|_Z) $$ is surjective. Then $\mathcal{L}$ is very ample on $X$ over $k$.


varieties-lemma-closure-of-product

Let $k$ be a field. Let $X$, $Y$ be schemes over $k$, and let $A \subset X$, $B \subset Y$ be subsets. Set $$ AB = \{z \in X \times_k Y \mid \text{pr}_X(z) \in A, \ \text{pr}_Y(z) \in B\} \subset X \times_k Y $$ Then set theoretically we have $$ \overline{A} \times_k \overline{B} = \overline{AB} $$


varieties-lemma-closure-image-product-map

Let $k$ be a field. Let $f : A \to X$, $g : B \to Y$ be morphisms of schemes over $k$. Then set theoretically we have $$ \overline{f(A)} \times_k \overline{g(B)} = \overline{(f \times g)(A \times_k B)} $$


varieties-lemma-scheme-theoretic-image-product-map

Let $k$ be a field. Let $f : A \to X$, $g : B \to Y$ be quasi-compact morphisms of schemes over $k$. Let $Z \subset X$ be the scheme theoretic image of $f$, see Morphisms, Definition \ref{morphisms-definition-scheme-theoretic-image}. Similarly, let $Z' \subset Y$ be the scheme theoretic image of $g$. Then $Z \times_k Z'$ is the scheme theoretic image of $f \times g$.


varieties-lemma-char-zero-differentials-free-smooth

Let $k$ be a field. Let $X$ be a scheme over $k$. Assume \begin{enumerate} \item $X$ is locally of finite type over $k$, \item $\Omega_{X/k}$ is locally free, and \item $k$ has characteristic zero. \end{enumerate} Then the structure morphism $X \to \Spec(k)$ is smooth.


varieties-lemma-char-p-differentials-free-smooth

Let $k$ be a field. Let $X$ be a scheme over $k$. Assume \begin{enumerate} \item $X$ is locally of finite type over $k$, \item $\Omega_{X/k}$ is locally free, \item $X$ is reduced, and \item $k$ is perfect. \end{enumerate} Then the structure morphism $X \to \Spec(k)$ is smooth.


varieties-lemma-smooth-regular

\begin{slogan} Smooth over a field implies regular \end{slogan} Let $X \to \Spec(k)$ be a smooth morphism where $k$ is a field. Then $X$ is a regular scheme.


varieties-lemma-smooth-geometrically-normal

Let $X \to \Spec(k)$ be a smooth morphism where $k$ is a field. Then $X$ is geometrically regular, geometrically normal, and geometrically reduced over $k$.


varieties-lemma-affine-space-over-field

Let $k$ be a field. Let $d \geq 0$. Let $W \subset \mathbf{A}^d_k$ be nonempty open. Then there exists a closed point $w \in W$ such that $k \subset \kappa(w)$ is finite separable.


varieties-lemma-smooth-separable-closed-points-dense

Let $k$ be a field. If $X$ is smooth over $\Spec(k)$ then the set $$ \{x \in X\text{ closed such that }k \subset \kappa(x) \text{ is finite separable}\} $$ is dense in $X$.


varieties-lemma-geometrically-reduced-dense-smooth-open

Let $X$ be a scheme over a field $k$. If $X$ is locally of finite type and geometrically reduced over $k$ then $X$ contains a dense open which is smooth over $k$.


varieties-lemma-dense-smooth-open-variety-over-perfect-field

Let $k$ be a perfect field. Let $X$ be a locally algebraic reduced $k$-scheme, for example a variety over $k$. Then we have $$ \{x \in X \mid X \to \Spec(k)\text{ is smooth at }x\} = \{x \in X \mid \mathcal{O}_{X, x}\text{ is regular}\} $$ and this is a dense open subscheme of $X$.


varieties-lemma-flat-under-smooth

Let $k$ be a field. Let $f : X \to Y$ be a morphism of schemes locally of finite type over $k$. Let $x \in X$ be a point and set $y = f(x)$. If $X \to \Spec(k)$ is smooth at $x$ and $f$ is flat at $x$ then $Y \to \Spec(k)$ is smooth at $y$. In particular, if $X$ is smooth over $k$ and $f$ is flat and surjective, then $Y$ is smooth over $k$.


varieties-lemma-variety-with-smooth-rational-point

Let $k$ be a field. Let $X$ be a variety over $k$ which has a $k$-rational point $x$ such that $X$ is smooth at $x$. Then $X$ is geometrically integral over $k$.


varieties-lemma-regular-functions-proper-variety

Let $X$ be a proper variety over $k$. Then \begin{enumerate} \item $K = H^0(X, \mathcal{O}_X)$ is a field which is a finite extension of the field $k$, \item if $X$ is geometrically reduced, then $K/k$ is separable, \item if $X$ is geometrically irreducible, then $K/k$ is purely inseparable, \item if $X$ is geometrically integral, then $K = k$. \end{enumerate}


varieties-lemma-normalization-locally-algebraic

Let $k$ be a field. Let $X$ be a locally algebraic scheme over $k$. Let $\nu : X^\nu \to X$ be the normalization morphism, see Morphisms, Definition \ref{morphisms-definition-normalization}. Then \begin{enumerate} \item $\nu$ is finite, dominant, and $X^\nu$ is a disjoint union of normal irreducible locally algebraic schemes over $k$, \item $\nu$ factors as $X^\nu \to X_{red} \to X$ and the first morphism is the normalization morphism of $X_{red}$, \item if $X$ is a reduced algebraic scheme, then $\nu$ is birational, \item if $X$ is a variety, then $X^\nu$ is a variety and $\nu$ is a finite birational morphism of varieties. \end{enumerate}


varieties-lemma-relative-normalization-finite

Let $k$ be a field. Let $f : Y \to X$ be a quasi-compact morphism of locally algebraic schemes over $k$. Let $X'$ be the normalization of $X$ in $Y$. If $Y$ is reduced, then $X' \to X$ is finite.


varieties-lemma-finite-extension-geometrically-normal

Let $k$ be a field. Let $X$ be an algebraic $k$-scheme. Then there exists a finite purely inseparable extension $k \subset k'$ such that the normalization $Y$ of $X_{k'}$ is geometrically normal over $k'$.


varieties-lemma-normalization-and-change-of-fields

Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. Let $K/k$ be an extension of fields. Let $\nu : X^\nu \to X$ be the normalization of $X$ and let $Y^\nu \to X_K$ be the normalization of the base change. Then the canonical morphism $$ Y^\nu \longrightarrow X^\nu \times_{\Spec(k)} \Spec(K) $$ is an isomorphism if $K/k$ is separable and a universal homeomorphism in general.


varieties-lemma-geometrically-normal-in-codim-1

Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. Let $\nu : X^\nu \to X$ be the normalization of $X$. Let $x \in X$ be a point such that (a) $\mathcal{O}_{X, x}$ is reduced, (b) $\dim(\mathcal{O}_{X, x}) = 1$, and (c) for every $x' \in X^\nu$ with $\nu(x') = x$ the extension $\kappa(x')/k$ is separable. Then $X$ is geometrically reduced at $x$ and $X^\nu$ is geometrically regular at $x'$ with $\nu(x') = x$.


varieties-lemma-open-in-normal-proper

Let $k$ be an algebraically closed field. Let $\overline{X}$ be a proper variety over $k$. Let $X \subset \overline{X}$ be an open subscheme. Assume $X$ is normal. Then $\mathcal{O}^*(X)/k^*$ is a finitely generated abelian group.


varieties-lemma-units-integral-finite-type-algebraically-closed

Let $k$ be an algebraically closed field. Let $X$ be an integral scheme locally of finite type over $k$. Then $\mathcal{O}^*(X)/k^*$ is a finitely generated abelian group.


varieties-lemma-units-general-algebraically-closed

Let $k$ be an algebraically closed field. Let $X$ be a connected reduced scheme which is locally of finite type over $k$ with finitely many irreducible components. Then $\mathcal{O}^*(X)/k^*$ is a finitely generated abelian group.


varieties-lemma-integral-closure-ground-field

Let $k$ be a field. Let $X$ be a scheme over $k$ which is connected and reduced. Then the integral closure of $k$ in $\Gamma(X, \mathcal{O}_X)$ is a field.


varieties-lemma-kunneth

Let $k$ be a field. Let $X$ and $Y$ be schemes over $k$ and let $\mathcal{F}$, resp.\ $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_X$-module, resp.\ $\mathcal{O}_Y$-module. Then we have a canonical isomorphism $$ H^n(X \times_{\Spec(k)} Y, \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_{\Spec(k)} Y}} \text{pr}_2^*\mathcal{G}) = \bigoplus\nolimits_{p + q = n} H^p(X, \mathcal{F}) \otimes_k H^q(Y, \mathcal{G}) $$ provided $X$ and $Y$ are quasi-compact and have affine diagonal\footnote{The case where $X$ and $Y$ are quasi-separated will be discussed in Lemma \ref{lemma-kunneth-general} below.} (for example if $X$ and $Y$ are separated).


varieties-lemma-kunneth-general

Let $k$ be a field. Let $X$ and $Y$ be schemes over $k$ and let $\mathcal{F}$, resp.\ $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_X$-module, resp.\ $\mathcal{O}_Y$-module. Then we have a canonical isomorphism $$ H^n(X \times_{\Spec(k)} Y, \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_{\Spec(k)} Y}} \text{pr}_2^*\mathcal{G}) = \bigoplus\nolimits_{p + q = n} H^p(X, \mathcal{F}) \otimes_k H^q(Y, \mathcal{G}) $$ provided $X$ and $Y$ are quasi-compact and quasi-separated.


varieties-lemma-change-rings-pic-pre

Let $A \to B$ be a faithfully flat ring map. Let $X$ be a quasi-compact and quasi-separated scheme over $A$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module whose pullback to $X_B$ is trivial. Then $H^0(X, \mathcal{L})$ and $H^0(X, \mathcal{L}^{\otimes -1})$ are invertible $H^0(X, \mathcal{O}_X)$-modules and the multiplication map induces an isomorphism $$ H^0(X, \mathcal{L}) \otimes_{H^0(X, \mathcal{O}_X)} H^0(X, \mathcal{L}^{\otimes -1}) \longrightarrow H^0(X, \mathcal{O}_X) $$


varieties-lemma-change-rings-pic

Let $A \to B$ be a faithfully flat ring map. Let $X$ be a scheme over $A$ such that \begin{enumerate} \item $X$ is quasi-compact and quasi-separated, and \item $R = H^0(X, \mathcal{O}_X)$ is a semi-local ring. \end{enumerate} Then the pullback map $\Pic(X) \to \Pic(X_B)$ is injective.


varieties-lemma-change-fields-pic

Let $k'/k$ be a field extension. Let $X$ be a scheme over $k$ such that \begin{enumerate} \item $X$ is quasi-compact and quasi-separated, and \item $R = H^0(X, \mathcal{O}_X)$ is semi-local, e.g., if $\dim_k R < \infty$. \end{enumerate} Then the pullback map $\Pic(X) \to \Pic(X_{k'})$ is injective.


varieties-lemma-rational-equivalence-for-Pic

Let $k$ be a field. Let $X$ be a normal variety over $k$. Let $U \subset \mathbf{A}^n_k$ be an open subscheme with $k$-rational points $p, q \in U(k)$. For every invertible module $\mathcal{L}$ on $X \times_{\Spec(k)} U$ the restrictions $\mathcal{L}|_{X \times p}$ and $\mathcal{L}|_{X \times q}$ are isomorphic.


varieties-lemma-euler-characteristic-additive

Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ be a short exact sequence of coherent modules on $X$. Then $$ \chi(X, \mathcal{F}_2) = \chi(X, \mathcal{F}_1) + \chi(X, \mathcal{F}_3) $$


varieties-lemma-chi-tensor-finite

Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{F}$ be a coherent sheaf with $\dim(\text{Supp}(\mathcal{F})) \leq 0$. Then \begin{enumerate} \item $\mathcal{F}$ is generated by global sections, \item $H^i(X, \mathcal{F}) = 0$ for $i > 0$, \item $\chi(X, \mathcal{F}) = \dim_k\Gamma(X, \mathcal{F})$, and \item $\chi(X, \mathcal{F} \otimes \mathcal{E}) = n\chi(X, \mathcal{F})$ for every locally free module $\mathcal{E}$ of rank $n$. \end{enumerate}


varieties-lemma-euler-characteristic-extend-base-field

Let $k \subset k'$ be an extension of fields. Let $X$ be a proper scheme over $k$. Let $\mathcal{F}$ be a coherent sheaf on $X$. Let $\mathcal{F}'$ be the pullback of $\mathcal{F}$ to $X_{k'}$. Then $\chi(X, \mathcal{F}) = \chi(X', \mathcal{F}')$.


varieties-lemma-euler-characteristic-morphism

Let $k$ be a field. Let $f : Y \to X$ be a morphism of proper schemes over $k$. Let $\mathcal{G}$ be a coherent $\mathcal{O}_Y$-module. Then $$ \chi(Y, \mathcal{G}) = \sum (-1)^i \chi(X, R^if_*\mathcal{G}) $$


varieties-lemma-projective-space-smooth

\begin{slogan} Projective space is smooth. \end{slogan} Let $k$ be a field and $n \geq 0$. Then $\mathbf{P}^n_k$ is a smooth projective variety of dimension $n$ over $k$.


varieties-lemma-intersection-in-affine-space

Let $k$ be a field and $n \geq 0$. Let $X, Y \subset \mathbf{A}^n_k$ be closed subsets. Assume that $X$ and $Y$ are equidimensional, $\dim(X) = r$ and $\dim(Y) = s$. Then every irreducible component of $X \cap Y$ has dimension $\geq r + s - n$.


varieties-lemma-intersection-in-projective-space

Let $k$ be a field and $n \geq 0$. Let $X, Y \subset \mathbf{P}^n_k$ be nonempty closed subsets. If $\dim(X) = r$ and $\dim(Y) = s$ and $r + s \geq n$, then $X \cap Y$ is nonempty and $\dim(X \cap Y) \geq r + s - n$.


varieties-lemma-equation-codim-1-in-projective-space

Let $k$ be a field. Let $Z \subset \mathbf{P}^n_k$ be a closed subscheme which has no embedded points such that every irreducible component of $Z$ has dimension $n - 1$. Then the ideal $I(Z) \subset k[T_0, \ldots, T_n]$ corresponding to $Z$ is principal.


varieties-lemma-hyperplane

Let $k$ be a field. Let $n \geq 1$. Let $i : H \to \mathbf{P}^n_k$ be a hyperplane. Then there exists an isomorphism $$ \varphi : \mathbf{P}^{n - 1}_k \longrightarrow H $$ such that $i^*\mathcal{O}(1)$ pulls back to $\mathcal{O}(1)$.


varieties-lemma-exact-sequence-induction

Let $k$ be an infinite field. Let $n \geq 1$. Let $\mathcal{F}$ be a coherent module on $\mathbf{P}^n_k$. Then there exist a nonzero section $s \in \Gamma(\mathbf{P}^n_k, \mathcal{O}(1))$ and a short exact sequence $$ 0 \to \mathcal{F}(-1) \to \mathcal{F} \to i_*\mathcal{G} \to 0 $$ where $i : H \to \mathbf{P}^n_k$ is the hyperplane $H$ associated to $s$ and $\mathcal{G} = i^*\mathcal{F}$.


varieties-lemma-m-regular-extend-base-field

Let $k \subset k'$ be an extension of fields. Let $n \geq 0$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbf{P}^n_k$. Let $\mathcal{F}'$ be the pullback of $\mathcal{F}$ to $\mathbf{P}^n_{k'}$. Then $\mathcal{F}$ is $m$-regular if and only if $\mathcal{F}'$ is $m$-regular.


varieties-lemma-m-regular

In the situation of Lemma \ref{lemma-exact-sequence-induction}, if $\mathcal{F}$ is $m$-regular, then $\mathcal{G}$ is $m$-regular on $H \cong \mathbf{P}^{n - 1}_k$.


varieties-lemma-m-regular-up

Let $k$ be a field. Let $n \geq 0$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbf{P}^n_k$. If $\mathcal{F}$ is $m$-regular, then $\mathcal{F}$ is $(m + 1)$-regular.


varieties-lemma-m-regular-multiply

Let $k$ be a field. Let $n \geq 0$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbf{P}^n_k$. If $\mathcal{F}$ is $m$-regular, then the multiplication map $$ H^0(\mathbf{P}^n_k, \mathcal{F}(m)) \otimes_k H^0(\mathbf{P}^n_k, \mathcal{O}(1)) \longrightarrow H^0(\mathbf{P}^n_k, \mathcal{F}(m + 1)) $$ is surjective.


varieties-lemma-m-regular-globally-generated

Let $k$ be a field. Let $n \geq 0$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbf{P}^n_k$. If $\mathcal{F}$ is $m$-regular, then $\mathcal{F}(m)$ is globally generated.


varieties-lemma-hilbert-polynomial

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


varieties-lemma-bound-quotients-free

Let $k$ be a field. Let $n \geq 0$. Let $r \geq 1$. Let $P \in \mathbf{Q}[t]$. There exists an integer $m$ depending on $n$, $r$, and $P$ with the following property: if $$ 0 \to \mathcal{K} \to \mathcal{O}^{\oplus r} \to \mathcal{F} \to 0 $$ is a short exact sequence of coherent sheaves on $\mathbf{P}^n_k$ and $\mathcal{F}$ has Hilbert polynomial $P$, then $\mathcal{K}$ is $m$-regular.


varieties-lemma-frobenius-endomorphism-identity

Let $p > 0$ be a prime number. Let $f : X \to Y$ be a morphism of schemes in characteristic $p$. Then the diagram $$ \xymatrix{ X \ar[d]_f \ar[r]_{F_X} & X \ar[d]^f \\ Y \ar[r]^{F_Y} & Y } $$ commutes.


varieties-lemma-frobenius

Let $p > 0$ be a prime number. Let $X$ be a scheme in characteristic $p$. Then the absolute frobenius $F_X : X \to X$ is a universal homeomorphism, is integral, and induces purely inseparable residue field extensions.


varieties-lemma-relative-frobenius-endomorphism-identity

Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$. Let $f : X \to Y$ be a morphism of schemes over $S$ . Then the diagram $$ \xymatrix{ X \ar[d]_f \ar[r]_{F_{X/S}} & X^{(p)} \ar[d]^{f^{(p)}} \\ Y \ar[r]^{F_{Y/S}} & Y^{(p)} } $$ commutes.


varieties-lemma-relative-frobenius

Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$. Let $X$ be a scheme over $S$. Then the relative frobenius $F_{X/S} : X \to X^{(p)}$ is a universal homeomorphism, is integral, and induces purely inseparable residue field extensions.


varieties-lemma-relative-frobenius-omega

Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$. Let $X$ be a scheme over $S$. Then $\Omega_{X/S} = \Omega_{X/X^{(p)}}$.


varieties-lemma-relative-frobenius-finite

Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$. Let $X$ be a scheme over $S$. If $X \to S$ is locally of finite type, then $F_{X/S}$ is finite.


varieties-lemma-geometrically-reduced-p

Let $k$ be a field of characteristic $p > 0$. Let $X$ be a scheme over $k$. Then $X$ is geometrically reduced if and only if $X^{(p)}$ is reduced.


varieties-lemma-inseparable-deg-p-smooth

Let $k$ be a field of characteristic $p > 0$. Let $X$ be a variety over $k$. The following are equivalent \begin{enumerate} \item $X^{(p)}$ is reduced, \item $X$ is geometrically reduced, \item there is a nonempty open $U \subset X$ smooth over $k$. \end{enumerate} In this case $X^{(p)}$ is a variety over $k$ and $F_{X/k} : X \to X^{(p)}$ is a finite dominant morphism of degree $p^{\dim(X)}$.


varieties-lemma-glue-separated

In Situation \ref{situation-glue} assume $A$ is a Noetherian ring of dimension $1$. The following are equivalent \begin{enumerate} \item $A \otimes B \to K$ is not surjective, \item there exists a discrete valuation ring $\mathcal{O} \subset K$ containing both $A$ and $B$. \end{enumerate}


varieties-lemma-semi-local

In Situation \ref{situation-glue} assume \begin{enumerate} \item $A$ is a Noetherian semi-local domain of dimension $1$, \item $B$ is a discrete valuation ring, \end{enumerate} Then we have the following two possibilities \begin{enumerate} \item[(a)] If $A^*$ is not contained in $R$, then $\Spec(A) \to \Spec(R)$ and $\Spec(B) \to \Spec(R)$ are open immersions covering $\Spec(R)$ and $K = A \otimes_R B$. \item[(b)] If $A^*$ is contained in $R$, then $B$ dominates one of the local rings of $A$ at a maximal ideal and $A \otimes B \to K$ is not surjective. \end{enumerate}


varieties-lemma-semi-local-dimension-one-conductor

Let $B$ be a semi-local Noetherian domain of dimension $1$. Let $B'$ be the integral closure of $B$ in its fraction field. Then $B'$ is a semi-local Dedekind domain. Let $x$ be a nonzero element of the Jacobson radical of $B'$. Then for every $y \in B'$ there exists an $n$ such that $x^n y \in B$.


varieties-lemma-semi-local-both-side

In Situation \ref{situation-glue} assume \begin{enumerate} \item $A$ is a Noetherian semi-local domain of dimension $1$, \item $B$ is a Noetherian semi-local domain of dimension $1$, \item $A \otimes B \to K$ is surjective. \end{enumerate} Then $\Spec(A) \to \Spec(R)$ and $\Spec(B) \to \Spec(R)$ are open immersions covering $\Spec(R)$ and $K = A \otimes_R B$.


varieties-lemma-glue-a-bunch-of-local-rings

Let $K$ be a field. Let $A_1, \ldots, A_r \subset K$ be Noetherian semi-local rings of dimension $1$ with fraction field $K$. If $A_i \otimes A_j \to K$ is surjective for all $i \not = j$, then there exists a Noetherian semi-local domain $A \subset K$ of dimension $1$ contained in $A_1, \ldots, A_r$ such that \begin{enumerate} \item $A \to A_i$ induces an open immersion $j_i : \Spec(A_i) \to \Spec(A)$, \item $\Spec(A)$ is the union of the opens $j_i(\Spec(A_i))$, \item each closed point of $\Spec(A)$ lies in exactly one of these opens. \end{enumerate}


varieties-lemma-create-globally-generated

Let $A$ be a domain with fraction field $K$. Let $B_1, \ldots, B_r \subset K$ be Noetherian $1$-dimensional semi-local rings whose fraction fields are $K$. If $A \otimes B_i \to K$ are surjective for $i = 1, \ldots, r$, then there exists an $x \in A$ such that $x^{-1}$ is in the Jacobson radical of $B_i$ for $i = 1, \ldots, r$.


varieties-lemma-power-equal

Let $A$ be a Noetherian local ring of dimension $1$. Let $L = \prod A_\mathfrak p$ where the product is over the minimal primes of $A$. Let $a_1, a_2 \in \mathfrak m_A$ map to the same element of $L$. Then $a_1^n = a_2^n$ for some $n > 0$.


varieties-lemma-power-works

Let $A$ be a Noetherian local ring of dimension $1$. Let $L = \prod A_\mathfrak p$ and $I = \bigcap \mathfrak p$ where the product and intersection are over the minimal primes of $A$. Let $f \in L$ be an element of the form $f = i + a$ where $a \in \mathfrak m_A$ and $i \in IL$. Then some power of $f$ is in the image of $A \to L$.


varieties-lemma-affine

Let $X$ be a scheme all of whose local rings are Noetherian of dimension $\leq 1$. Let $U \subset X$ be a retrocompact open. Denote $j : U \to X$ the inclusion morphism. Then $R^pj_*\mathcal{F} = 0$, $p > 0$ for every quasi-coherent $\mathcal{O}_U$-module $\mathcal{F}$.


varieties-lemma-find-globally-generated

Let $X$ be an integral separated scheme. Let $U \subset X$ be a nonempty affine open such that $X \setminus U$ is a finite set of points $x_1, \ldots, x_r$ with $\mathcal{O}_{X, x_i}$ Noetherian of dimension $1$. Then there exists a globally generated invertible $\mathcal{O}_X$-module $\mathcal{L}$ and a section $s$ such that $U = X_s$.


varieties-lemma-enough-globally-generated-ample

Let $X$ be a quasi-compact scheme. If for every $x \in X$ there exists a pair $(\mathcal{L}, s)$ consisting of a globally generated invertible sheaf $\mathcal{L}$ and a global section $s$ such that $x \in X_s$ and $X_s$ is affine, then $X$ has an ample invertible sheaf.


varieties-lemma-dim-1-noetherian-integral-separated-has-ample

Let $X$ be a Noetherian integral separated scheme of dimension $1$. Then $X$ has an ample invertible sheaf.


varieties-lemma-surjective-pic-birational-finite

Let $f : X \to Y$ be a finite morphism of schemes. Assume there exists an open $V \subset Y$ such that $f^{-1}(V) \to V$ is an isomorphism and $Y \setminus V$ is a discrete space. Then every invertible $\mathcal{O}_X$-module is the pullback of an invertible $\mathcal{O}_Y$-module.


varieties-lemma-glue-invertible-sheaves

Let $X$ be a scheme. Let $Z_1, \ldots, Z_n \subset X$ be closed subschemes. Let $\mathcal{L}_i$ be an invertible sheaf on $Z_i$. Assume that \begin{enumerate} \item $X$ is reduced, \item $X = \bigcup Z_i$ set theoretically, and \item $Z_i \cap Z_j$ is a discrete topological space for $i \not = j$. \end{enumerate} Then there exists an invertible sheaf $\mathcal{L}$ on $X$ whose restriction to $Z_i$ is $\mathcal{L}_i$. Moreover, if we are given sections $s_i \in \Gamma(Z_i, \mathcal{L}_i)$ which are nonvanishing at the points of $Z_i \cap Z_j$, then we can choose $\mathcal{L}$ such that there exists a $s \in \Gamma(X, \mathcal{L})$ with $s|_{Z_i} = s_i$ for all $i$.


varieties-lemma-dim-1-noetherian-reduced-separated-has-ample

Let $X$ be a Noetherian reduced separated scheme of dimension $1$. Then $X$ has an ample invertible sheaf.


varieties-lemma-lift-line-bundle-from-reduction-dimension-1

Let $i : Z \to X$ be a closed immersion of schemes. If the underlying topological space of $X$ is Noetherian and $\dim(X) \leq 1$, then $\Pic(X) \to \Pic(Z)$ is surjective.


varieties-lemma-pre-pre-delta-invariant

Let $(A, \mathfrak m)$ be a Noetherian $1$-dimensional local ring. Let $f \in \mathfrak m$. The following are equivalent \begin{enumerate} \item $\mathfrak m = \sqrt{(f)}$, \item $f$ is not contained in any minimal prime of $A$, and \item $A_f = \prod_{\mathfrak p\text{ minimal}} A_\mathfrak p$ as $A$-algebras. \end{enumerate} Such an $f \in \mathfrak m$ exists. If $\text{depth}(A) = 1$ (for example $A$ is reduced), then (1) -- (3) are also equivalent to \begin{enumerate} \item[(4)] $f$ is a nonzerodivisor, \item[(5)] $A_f$ is the total ring of fractions of $A$. \end{enumerate} If $A$ is reduced, then (1) -- (5) are also equivalent to \begin{enumerate} \item[(6)] $A_f$ is the product of the residue fields at the minimal primes of $A$. \end{enumerate}


varieties-lemma-pre-delta-invariant

Let $(A, \mathfrak m)$ be a reduced Nagata $1$-dimensional local ring. Let $A'$ be the integral closure of $A$ in the total ring of fractions of $A$. Then $A'$ is a normal Nagata ring, $A \to A'$ is finite, and $A'/A$ has finite length as an $A$-module.


varieties-lemma-delta-invariant-is-zero

Let $A$ be a reduced Nagata local ring of dimension $1$. The $\delta$-invariant of $A$ is $0$ if and only if $A$ is a discrete valuation ring.


varieties-lemma-normalization-same-after-completion

Let $A$ be a reduced Nagata local ring of dimension $1$. Let $A \to A'$ be as in Lemma \ref{lemma-pre-delta-invariant}. Let $A^h$, $A^{sh}$, resp.\ $A^\wedge$ be the henselization, strict henselization, reps.\ completion of $A$. Then $A^h$, $A^{sh}$, resp. $A^\wedge$ is a reduced Nagata local ring of dimension $1$ and $A' \otimes_A A^h$, $A' \otimes_A A^{sh}$, resp. $A' \otimes_A A^\wedge$ is the integral closure of $A^h$, $A^{sh}$, resp.\ $A^\wedge$ in its total ring of fractions.


varieties-lemma-delta-same-after-completion

Let $A$ be a reduced Nagata local ring of dimension $1$. The $\delta$-invariant of $A$ is the same as the $\delta$-invariant of the henselization, strict henselization, or the completion of $A$.


varieties-lemma-delta-invariant-and-change-of-fields

Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. Let $K/k$ be a field extension and set $Y = X_K$. Let $y \in Y$ with image $x \in X$. Assume $X$ is geometrically reduced at $x$ and $\dim(\mathcal{O}_{X, x}) = \dim(\mathcal{O}_{Y, y}) = 1$. Then $$ \delta\text{-invariant of }X\text{ at }x \leq \delta\text{-invariant of }Y\text{ at }y $$


varieties-lemma-delta-invariant-and-change-of-fields-better

Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. Let $K/k$ be a field extension and set $Y = X_K$. Let $y \in Y$ with image $x \in X$. Assume assumptions (a), (b), (c) of Lemma \ref{lemma-geometrically-normal-in-codim-1} hold for $x \in X$ and that $\dim(\mathcal{O}_{Y, y}) = 1$. Then the $\delta$-invariant of $X$ at $x$ is $\delta$-invariant of $Y$ at $y$.


varieties-lemma-number-of-branches

Let $X$ be a scheme. Assume every quasi-compact open of $X$ has finitely many irreducible components. Let $\nu : X^\nu \to X$ be the normalization of $X$. Let $x \in X$. \begin{enumerate} \item The number of branches of $X$ at $x$ is the number of inverse images of $x$ in $X^\nu$. \item The number of geometric branches of $X$ at $x$ is $\sum_{\nu(x^\nu) = x} [\kappa(x^\nu) : \kappa(x)]_s$. \end{enumerate}


varieties-lemma-geometric-branches-and-change-of-fields

Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. Let $K/k$ be an extension of fields. Let $y \in X_K$ be a point with image $x$ in $X$. Then the number of geometric branches of $X$ at $x$ is the number of geometric branches of $X_K$ at $y$.


varieties-lemma-geometrically-unibranch-and-change-of-fields

Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. Let $K/k$ be an extension of fields. Let $y \in X_K$ be a point with image $x$ in $X$. Then $X$ is geometrically unibranch at $x$ if and only if $X_K$ is geometrically unibranch at $y$.


varieties-lemma-delta-number-branches-inequality-sh

Let $(A, \mathfrak m)$ be a strictly henselian $1$-dimensional reduced Nagata local ring. Then $$ \delta\text{-invariant of }A \geq \text{number of geometric branches of }A - 1 $$ If equality holds, then $A$ is a wedge of $n \geq 1$ strictly henselian discrete valuation rings.


varieties-lemma-delta-number-branches-inequality

Let $(A, \mathfrak m)$ be a $1$-dimensional reduced Nagata local ring. Then $$ \delta\text{-invariant of }A \geq \text{number of geometric branches of }A - 1 $$


varieties-lemma-normalize-noetherian-dim-1

Let $X$ be a locally Noetherian scheme of dimension $1$. Let $\nu : X^\nu \to X$ be the normalization. Then \begin{enumerate} \item $\nu$ is integral, surjective, and induces a bijection on irreducible components, \item there is a factorization $X^\nu \to X_{red} \to X$ and the morphism $X^\nu \to X_{red}$ is the normalization of $X_{red}$, \item $X^\nu \to X_{red}$ is birational, \item for every closed point $x \in X$ the stalk $(\nu_*\mathcal{O}_{X^\nu})_x$ is the integral closure of $\mathcal{O}_{X, x}$ in the total ring of fractions of $(\mathcal{O}_{X, x})_{red} = \mathcal{O}_{X_{red}, x}$, \item the fibres of $\nu$ are finite and the residue field extensions are finite, \item $X^\nu$ is a disjoint union of integral normal Noetherian schemes and each affine open is the spectrum of a finite product of Dedekind domains. \end{enumerate}


varieties-lemma-prepare-delta-invariant

Let $X$ be a reduced Nagata scheme of dimension $1$. Let $\nu : X^\nu \to X$ be the normalization. Let $x \in X$ denote a closed point. Then \begin{enumerate} \item $\nu : X^\nu \to X$ is finite, surjective, and birational, \item $\mathcal{O}_X \subset \nu_*\mathcal{O}_{X^\nu}$ and $\nu_*\mathcal{O}_{X^\nu}/\mathcal{O}_X$ is a direct sum of skyscraper sheaves $\mathcal{Q}_x$ in the singular points $x$ of $X$, \item $A' = (\nu_*\mathcal{O}_{X^\nu})_x$ is the integral closure of $A = \mathcal{O}_{X, x}$ in its total ring of fractions, \item $\mathcal{Q}_x = A'/A$ has finite length equal to the $\delta$-invariant of $X$ at $x$, \item $A'$ is a semi-local ring which is a finite product of Dedekind domains, \item $A^\wedge$ is a reduced Noetherian complete local ring of dimension $1$, \item $(A')^\wedge$ is the integral closure of $A^\wedge$ in its total ring of fractions, \item $(A')^\wedge$ is a finite product of complete discrete valuation rings, and \item $A'/A \cong (A')^\wedge/A^\wedge$. \end{enumerate}


varieties-lemma-characterize-open-immersion

Let $f : X \to Y$ be a morphism of schemes. Let $X^0$ denote the set of generic points of irreducible components of $X$. If \begin{enumerate} \item $f$ is separated, \item there is an open covering $X = \bigcup U_i$ such that $f|_{U_i} : U_i \to Y$ is an open immersion, and \item if $\xi, \xi' \in X^0$, $\xi \not = \xi'$, then $f(\xi) \not = f(\xi')$, \end{enumerate} then $f$ is an open immersion.


varieties-lemma-local-isomorphism

Let $X \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s \in S$. If \begin{enumerate} \item $\mathcal{O}_{X, x} = \mathcal{O}_{S, s}$, \item $X$ is reduced, \item $X \to S$ is of finite type, and \item $S$ has finitely many irreducible components, \end{enumerate} then there exists an open neighbourhood $U$ of $x$ such that $f|_U$ is an open immersion.


varieties-lemma-points-in-affine

Let $f : T \to X$ be a morphism of schemes. Let $X^0$, resp.\ $T^0$ denote the sets of generic points of irreducible components. Let $t_1, \ldots, t_m \in T$ be a finite set of points with images $x_j = f(t_j)$. If \begin{enumerate} \item $T$ is affine, \item $X$ is quasi-separated, \item $X^0$ is finite \item $f(T^0) \subset X^0$ and $f : T^0 \to X^0$ is injective, and \item $\mathcal{O}_{X, x_j} = \mathcal{O}_{T, t_j}$, \end{enumerate} then there exists an affine open of $X$ containing $x_1, \ldots, x_r$.


varieties-lemma-finite-set-codim-1-points-in-affine

Let $X$ be an integral separated scheme. Let $x_1, \ldots, x_r \in X$ be a finite set of points such that $\mathcal{O}_{X, x_i}$ is Noetherian of dimension $\leq 1$. Then there exists an affine open subscheme of $X$ containing all of $x_1, \ldots, x_r$.


varieties-lemma-extra-silly

Let $A$ be a ring, $I \subset A$ an ideal, $\mathfrak p_1, \ldots, \mathfrak p_r$ primes of $A$, and $\overline{f} \in A/I$ an element. If $I \not \subset \mathfrak p_i$ for all $i$, then there exists an $f \in A$, $f \not \in \mathfrak p_i$ which maps to $\overline{f}$ in $A/I$.


varieties-lemma-finite-set-codim-1-points-in-affine-per-component

Let $X$ be a scheme. Let $T \subset X$ be finite set of points. Assume \begin{enumerate} \item $X$ has finitely many irreducible components $Z_1, \ldots, Z_t$, and \item $Z_i \cap T$ is contained in an affine open of the reduced induced subscheme corresponding to $Z_i$. \end{enumerate} Then there exists an affine open subscheme of $X$ containing $T$.


varieties-lemma-proper-minus-point

Let $X$ be an irreducible scheme of dimension $> 0$ over a field $k$. Let $x \in X$ be a closed point. The open subscheme $X \setminus \{x\}$ is not proper over $k$.


varieties-lemma-dim-1-quasi-projective

Let $X$ be a separated finite type scheme over a field $k$. If $\dim(X) \leq 1$ then $X$ is H-quasi-projective over $k$.


varieties-lemma-dim-1-proper-projective

Let $X$ be a proper scheme over a field $k$. If $\dim(X) \leq 1$ then $X$ is H-projective over $k$.


varieties-lemma-dim-1-projective-completion

Let $X$ be a separated scheme of finite type over $k$. If $\dim(X) \leq 1$, then there exists an open immersion $j : X \to \overline{X}$ with the following properties \begin{enumerate} \item $\overline{X}$ is H-projective over $k$, i.e., $\overline{X}$ is a closed subscheme of $\mathbf{P}^d_k$ for some $d$, \item $j(X) \subset \overline{X}$ is dense and scheme theoretically dense, \item $\overline{X} \setminus X = \{x_1, \ldots, x_n\}$ for some closed points $x_i \in \overline{X}$. \end{enumerate}


varieties-lemma-reduced-dim-1-projective-completion

Let $X$ be a separated scheme of finite type over $k$. If $X$ is reduced and $\dim(X) \leq 1$, then there exists an open immersion $j : X \to \overline{X}$ such that \begin{enumerate} \item $\overline{X}$ is H-projective over $k$, i.e., $\overline{X}$ is a closed subscheme of $\mathbf{P}^d_k$ for some $d$, \item $j(X) \subset \overline{X}$ is dense and scheme theoretically dense, \item $\overline{X} \setminus X = \{x_1, \ldots, x_n\}$ for some closed points $x_i \in \overline{X}$, \item the local rings $\mathcal{O}_{\overline{X}, x_i}$ are discrete valuation rings for $i = 1, \ldots, n$. \end{enumerate}


varieties-lemma-curve-affine-projective

Let $X$ be a curve over $k$. Then either $X$ is an affine scheme or $X$ is H-projective over $k$.


varieties-lemma-degree-base-change

Let $k \subset k'$ be an extension of fields. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_X$-module of constant rank $n$. Then the degree of $\mathcal{E}/X/k$ is equal to the degree of $\mathcal{E}_{k'}/X_{k'}/k'$.


varieties-lemma-degree-additive

Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Let $0 \to \mathcal{E}_1 \to \mathcal{E}_2 \to \mathcal{E}_3 \to 0$ be a short exact sequence of locally free $\mathcal{O}_X$-modules each of finite constant rank. Then $$ \deg(\mathcal{E}_2) = \deg(\mathcal{E}_1) + \deg(\mathcal{E}_3) $$


varieties-lemma-degree-birational-pullback

Let $k$ be a field. Let $f : X' \to X$ be a birational morphism of proper schemes of dimension $\leq 1$ over $k$. Then $$ \deg(f^*\mathcal{E}) = \deg(\mathcal{E}) $$ for every finite locally free sheaf of constant rank. More generally it suffices if $f$ induces a bijection between irreducible components of dimension $1$ and isomorphisms of local rings at the corresponding generic points.


varieties-lemma-degree-on-proper-curve

Let $k$ be a field. Let $X$ be a proper curve over $k$ with generic point $\xi$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_X$-module of rank $n$ and let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then $$ \chi(X, \mathcal{E} \otimes \mathcal{F}) = r \deg(\mathcal{E}) + n \chi(X, \mathcal{F}) $$ where $r = \dim_{\kappa(\xi)} \mathcal{F}_\xi$ is the rank of $\mathcal{F}$.


varieties-lemma-degree-in-terms-of-components

Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_X$-module of rank $n$. Then $$ \deg(\mathcal{E}) = \sum m_i \deg(\mathcal{E}|_{C_i}) $$ where $C_i \subset X$, $i = 1, \ldots, t$ are the irreducible components of dimension $1$ with reduced induced scheme structure and $m_i$ is the multiplicity of $C_i$ in $X$.