# stacks/stacks-project

Baby stein

aisejohan committed Aug 20, 2019
1 parent 95eefb7 commit 6a681c3f25c699d0d64e0b788008f58aeeed65eb
Showing with 120 additions and 33 deletions.
1. +6 −26 moduli-curves.tex
2. +108 −3 varieties.tex
3. +6 −4 weil.tex
 @@ -1620,33 +1620,13 @@ \section{Smooth curves} \medskip\noindent A smooth scheme over a field is geometrically normal (Varieties, Lemma \ref{varieties-lemma-smooth-geometrically-normal}). A locally Noetherian normal scheme is a disjoint union of its irreducible components (Properties, Lemma \ref{properties-lemma-normal-Noetherian}). Thus (2)(c) implies (2)(d). \medskip\noindent Being smooth is preserved under base change (Morphisms, Lemma \ref{morphisms-lemma-base-change-smooth}). Thus (2)(d) implies (2)(e). \medskip\noindent Being smooth is fpqc local on the target (Varieties, Lemma \ref{varieties-lemma-smooth-geometrically-normal}), smoothness is preserved under base change (Morphisms, Lemma \ref{morphisms-lemma-base-change-smooth}), and being smooth is fpqc local on the target (Descent, Lemma \ref{descent-lemma-descending-property-smooth}). Hence (2)(e) implies (2)(c). \medskip\noindent Lemma \ref{lemma-geomredcon-in-h0-1} shows that the equivalent conditions (2)(c), (d), (e) imply (2)(b). Conversely, if (2)(b) holds, then $\overline{k} = H^0(X_{\overline{k}}, \mathcal{O}_{X_{\overline{k}}})$ by flat base change (Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}) and hence $X_{\overline{k}}$ is connected (Varieties, Lemma \ref{varieties-lemma-proper-geometrically-reduced-global-sections}). Thus (2)(c) holds. Keeping this in mind, the equivalence of (2)(b), (2)(c), 2(d), and (2)(e) follows from Varieties, Lemma \ref{varieties-lemma-geometrically-normal-stein}. \end{proof} \begin{definition}
 @@ -1865,11 +1865,11 @@ \section{Geometrically integral schemes} This proves (3). \medskip\noindent If $X$ is geometrically reduced, then same thing is true for If $X$ is geometrically reduced, then $A \otimes_k \overline{k} = H^0(X_{\overline{k}}, \mathcal{O}_{X_{\overline{k}}})$ (see Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology} for equality). (equality by Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}) is reduced. This implies that $k_i \otimes_k \overline{k}$ is a product of fields and hence $k_i/k$ is separable for example by Algebra, @@ -1893,6 +1893,88 @@ \section{Geometrically integral schemes} that $k_1 = k$. This proves (7). Of course (7) implies (8). \end{proof} \noindent Here is a baby version of Stein factorization; actual Stein factorization will be discussed in More on Morphisms, Section \ref{more-morphisms-section-stein-factorization}. \begin{lemma} \label{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. \end{lemma} \begin{proof} Set $S = \Spec(A)$. The canonical morphism $X \to S$ is the morphism corresponding to $\Gamma(S, \mathcal{O}_S) = A = \Gamma(X, \mathcal{O}_X)$ via Schemes, Lemma \ref{schemes-lemma-morphism-into-affine}. The $k$-algebra $A$ is a finite product $A = \prod A_i$ of local Artinian $k$-algebras finite over $k$, see Lemma \ref{lemma-proper-geometrically-reduced-global-sections}. Denote $s_i \in S$ the point corresponding to the maximal ideal of $A_i$. Choose an algebraic closure $\overline{k}$ of $k$ and set $\overline{A} = A \otimes_k \overline{k}$. Choose an embedding $\kappa(s_i) \to \overline{k}$ over $k$; this determines a $\overline{k}$-algebra map $$\sigma_i : \overline{A} = A \otimes_k \overline{k} \to \kappa(s_i) \otimes_k \overline{k} \to \overline{k}$$ Consider the base change $$\xymatrix{ \overline{X} \ar[r] \ar[d] & X \ar[d] \\ \overline{S} \ar[r] & S }$$ of $X$ to $\overline{S} = \Spec(\overline{A})$. By Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology} we have $\Gamma(\overline{X}, \mathcal{O}_{\overline{X}}) = \overline{A}$. If $\overline{s}_i \in \Spec(\overline{A})$ denotes the $\overline{k}$-rational point corresponding to $\sigma_i$, then we see that $\overline{s}_i$ maps to $s_i \in S$ and $\overline{X}_{\overline{s}_i}$ is the base change of $X_{s_i}$ by $\Spec(\sigma_i)$. Thus we see that it suffices to prove the lemma in case $k$ is algebraically closed. \medskip\noindent Assume $k$ is algebraically closed. In this case $\kappa(s_i)$ is algebraically closed and we have to show that $X_{s_i}$ is connected. The product decomposition $A = \prod A_i$ corresponds to a disjoint union decomposition $\Spec(A) = \coprod \Spec(A_i)$, see Algebra, Lemma \ref{algebra-lemma-spec-product}. Denote $X_i$ the inverse image of $\Spec(A_i)$. It follows from Lemma \ref{lemma-proper-geometrically-reduced-global-sections} part (2) that $A_i = \Gamma(X_i, \mathcal{O}_{X_i})$. Observe that $X_{s_i} \to X_i$ is a closed immersion inducing an isomorphism on underlying topological spaces (because $\Spec(A_i)$ is a singleton). Hence if $X_{s_i}$ isn't connected, then neither is $X_i$. So either $X_i$ is empty and $A_i = 0$ or $X_i$ can be written as $U \amalg V$ with $U$ and $V$ open and nonempty which would imply that $A_i$ has a nontrivial idempotent. Since $A_i$ is local this is a contradiction and the proof is complete. \end{proof} \begin{lemma} \label{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} \end{lemma} \begin{proof} By Lemma \ref{lemma-baby-stein} we have (1) $\Rightarrow$ (2). By Lemma \ref{lemma-proper-geometrically-reduced-global-sections} we have (2) $\Rightarrow$ (1). \end{proof} @@ -2073,6 +2155,29 @@ \section{Geometrically normal schemes} \ref{algebra-lemma-separable-field-extension-geometrically-normal}. \end{proof} \begin{lemma} \label{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} \end{lemma} \begin{proof} By Lemma \ref{lemma-geometrically-reduced-stein} we have the equivalence of (1) and (2). A locally Noetherian normal scheme (such as $X_{\overline{k}}$) is a disjoint union of its irreducible components (Properties, Lemma \ref{properties-lemma-normal-Noetherian}). Thus we see that (2) and (3) are equivalent. Since $X_{\overline{k}}$ is assumed reduced, we see that (3) and (4) are equivalent too. \end{proof}
 @@ -2318,10 +2318,12 @@ \section{Cycles over non-closed fields} \medskip\noindent We may assume $X$ is irreducible of dimension $d$. Observe that $k' = H^0(X, \mathcal{O}_X)$ is a finite separable Then $k' = H^0(X, \mathcal{O}_X)$ is a finite separable field extension of $k$ and that $X$ is geometrically integral over $k'$. We may and do replace $k$ by $k'$ and assume that $X$ is geometrically integral. See Varieties, Lemmas \ref{varieties-lemma-smooth-geometrically-normal}, \ref{varieties-lemma-proper-geometrically-reduced-global-sections}, and \ref{varieties-lemma-baby-stein}. We may and do replace $k$ by $k'$ and assume that $X$ is geometrically integral. \medskip\noindent Let $x \in X$ be a closed point. Choose a sufficiently ample invertible @@ -3516,7 +3518,7 @@ \section{Further properties} Let $A \to A'$ be the maximal separable $F$-algebra quotient. The maps $\sigma, \sigma', \sigma_i$ each factor through $A \to A'$. After replacing $A$ by $A'$ we may assume $A$ is a finnite separable $F$-algebra. $A$ is a finite separable $F$-algebra. \medskip\noindent Choose an algebraic closure $\overline{F}$. Set