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Typos

Dear stack project maintainers.

Here are some typos. I can split the patch in two (or send a pr via github)
if you prefer.

Best regards,
Damien Robert

---------- 8< ---------------------

From c58fb79c4f576678ab273c951b8cda8015906aac Mon Sep 17 00:00:00 2001
From: Damien Robert <damien.olivier.robert+git@gmail.com>
Date: Tue, 12 Nov 2019 21:24:52 +0100
Subject: [PATCH 1/1] Fix some typos

In descent.tex: fix a simple typo U <-> V (lemma-etale-on-fiber)

In more-morphisms.tex: in sections ZMT and Stein factorization, sometimes
the base is denoted Y while it is usually denoted S.

In tag 0BUI for instance, the lemma switches from S to Y.

Fix this by using S everywhere.

Signed-off-by: Damien Robert <damien.olivier.robert+git@gmail.com>
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Damien Robert authored and aisejohan committed Nov 12, 2019
1 parent 063a1aa commit 1f6661839e5d6c934a07e8e4daed7f5521dc3397
Showing with 28 additions and 28 deletions.
  1. +1 −1 descent.tex
  2. +27 −27 more-morphisms.tex
@@ -7206,7 +7206,7 @@ \section{Properties of morphisms of germs local on source-and-target}
U \ar[r] & V
}
$$
with \'etale vertical arrows and a point $v' \in U'$ mapping to $v \in U$.
with \'etale vertical arrows and a point $v' \in V'$ mapping to $v \in V$.
Then the morphism of fibres $U'_{v'} \to U_v$ is \'etale.
\end{lemma}

@@ -10822,7 +10822,7 @@ \section{Zariski's Main Theorem}
Lemma \ref{lemma-quasi-finite-separated-pass-through-finite}.
Assume moreover that $f$ is locally of finite presentation. Then we can
choose the factorization such that $T$ is finite and of
finite presentation over $Y$.
finite presentation over $S$.
\end{lemma}

\begin{proof}
\begin{lemma}
\label{lemma-stein-universally-closed-residue-fields}
In Lemma \ref{lemma-stein-universally-closed} assume in addition that
$f$ is locally of finite type. Then for $y \in Y$ the fibre
$\pi^{-1}(\{y\}) = \{y_1, \ldots, y_n\}$ is finite and the field extensions
$\kappa(y_i)/\kappa(y)$ are finite.
$f$ is locally of finite type. Then for $s \in S$ the fibre
$\pi^{-1}(\{s\}) = \{s_1, \ldots, s_n\}$ is finite and the field extensions
$\kappa(s_i)/\kappa(s)$ are finite.
\end{lemma}

\begin{proof}
Recall that there are no specializations among the points of $\pi^{-1}(\{y\})$,
Recall that there are no specializations among the points of $\pi^{-1}(\{s\})$,
see Algebra, Lemma \ref{algebra-lemma-integral-no-inclusion}.
As $f'$ is surjective, we find that $|X_y| \to \pi^{-1}(\{y\})$ is surjective.
Observe that $X_y$ is a quasi-separated scheme of finite type
As $f'$ is surjective, we find that $|X_s| \to \pi^{-1}(\{s\})$ is surjective.
Observe that $X_s$ is a quasi-separated scheme of finite type
over a field (quasi-compactness was shown in the proof of the
referenced lemma). Thus $X_y$ is Noetherian
referenced lemma). Thus $X_s$ is Noetherian
(Morphisms, Lemma \ref{morphisms-lemma-finite-type-noetherian}).
A topological argument (omitted) now shows that $\pi^{-1}(\{y\})$ is finite.
For each $i$ we can pick a finite type point $x_i \in X_y$ mapping to $y_i$
A topological argument (omitted) now shows that $\pi^{-1}(\{s\})$ is finite.
For each $i$ we can pick a finite type point $x_i \in X_s$ mapping to $s_i$
(Morphisms, Lemma \ref{morphisms-lemma-enough-finite-type-points}).
We conclude that $\kappa(y_i)/\kappa(y)$ is finite:
$x_i$ can be represented by a morphism $\Spec(k_i) \to X_y$
We conclude that $\kappa(s_i)/\kappa(s)$ is finite:
$x_i$ can be represented by a morphism $\Spec(k_i) \to X_s$
of finite type (by our definition of finite type points)
and hence $\Spec(k_i) \to y = \Spec(\kappa(y))$ is of finite type
and hence $\Spec(k_i) \to s = \Spec(\kappa(s))$ is of finite type
(as a composition of finite type morphisms),
hence $k_i/\kappa(y)$ is finite (Morphisms, Lemma
hence $k_i/\kappa(s)$ is finite (Morphisms, Lemma
\ref{morphisms-lemma-point-finite-type}).
\end{proof}


\begin{lemma}
\label{lemma-geometrically-connected-fibres-towards-normal}
Let $f : X \to Y$ be a morphism of schemes. Assume
Let $f : X \to S$ be a morphism of schemes. Assume
\begin{enumerate}
\item $f$ is proper,
\item $Y$ is integral with generic point $\xi$,
\item $Y$ is normal,
\item $S$ is integral with generic point $\xi$,
\item $S$ is normal,
\item $X$ is reduced,
\item every generic point of an irreducible component of $X$ maps to $\xi$,
\item we have $H^0(X_\xi, \mathcal{O}) = \kappa(\xi)$.
\end{enumerate}
Then $f_*\mathcal{O}_X = \mathcal{O}_Y$ and $f$
Then $f_*\mathcal{O}_X = \mathcal{O}_S$ and $f$
has geometrically connected fibres.
\end{lemma}

\begin{proof}
Apply Theorem \ref{theorem-stein-factorization-general} to get a
factorization $X \to Y' \to Y$. It is enough to show that $Y' = Y$.
factorization $X \to S' \to S$. It is enough to show that $S' = S$.
This will follow from Morphisms, Lemma
\ref{morphisms-lemma-finite-birational-over-normal}.
Namely, $Y'$ is reduced because $X$ is reduced
Namely, $S'$ is reduced because $X$ is reduced
(Morphisms, Lemma \ref{morphisms-lemma-normalization-in-reduced}).
The morphism $Y' \to Y$ is integral by the theorem cited above.
Every generic point of $Y'$ lies over $\xi$ by
The morphism $S' \to S$ is integral by the theorem cited above.
Every generic point of $S'$ lies over $\xi$ by
Morphisms, Lemma \ref{morphisms-lemma-normalization-generic}
and assumption (5). On the other hand, since $Y'$ is the relative
and assumption (5). On the other hand, since $S'$ is the relative
spectrum of $f_*\mathcal{O}_X$ we see that the scheme theoretic fibre
$Y'_\xi$ is the spectrum of $H^0(X_\xi, \mathcal{O})$ which is
equal to $\kappa(\xi)$ by assumption. Hence $Y'$ is an integral
scheme with function field equal to the function field of $Y$.
$S'_\xi$ is the spectrum of $H^0(X_\xi, \mathcal{O})$ which is
equal to $\kappa(\xi)$ by assumption. Hence $S'$ is an integral
scheme with function field equal to the function field of $S$.
This finishes the proof.
\end{proof}

\begin{lemma}
\label{lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous}
Let $X \to S$ be a flat proper morphism of finite presentation. Let
$n_{X/S}$ be the function on $Y$ counting the numbers of geometric
$n_{X/S}$ be the function on $S$ counting the numbers of geometric
connected components of fibres of $f$ introduced in
Lemma \ref{lemma-base-change-fibres-nr-geometrically-connected-components}.
Then $n_{X/S}$ is lower semi-continuous.

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