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Baby stein

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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

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