# stacks/stacks-project

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>
 @@ -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.