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Fix two typos in divisors

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aisejohan committed Feb 1, 2019
1 parent a03896c commit 799bb8a66fda1e7f4bb2cbde4b5c9d3df11c0e69
Showing with 2 additions and 2 deletions.
  1. +2 −2 divisors.tex
@@ -924,7 +924,7 @@ \section{Morphisms and weakly associated points}
then $s$ is not associated to $f_*\mathcal{F}$.
\item If $s \not \in f(X)$, $(f_*\mathcal{F})_s$ is a finite
$\mathcal{O}_{S, s}$-module, and $\mathcal{O}_{S, s}$
is Noetherian, then $\text{depth}(f_*\mathcal{O}_{S, s}) \geq 2$.
is Noetherian, then $\text{depth}((f_*\mathcal{F})_s) \geq 2$.
\item If $\mathcal{F}$ is flat over $S$ and $a \in \mathfrak m_s$
is a nonzerodivisor, then $a$ is a nonzerodivisor on $(f_*\mathcal{F})_s$.
\item If $\mathcal{F}$ is flat over $S$ and $a, b \in \mathfrak m_s$
@@ -942,7 +942,7 @@ \section{Morphisms and weakly associated points}
Part (2) follows from (1) and Lemma \ref{lemma-ass-weakly-ass}.

\medskip\noindent
Proof of part (3). To show the depth is $\geq w$ it suffices to show that
Proof of part (3). To show the depth is $\geq 2$ it suffices to show that
$\Hom_{\mathcal{O}_{S, s}}(\kappa(s), (f_*\mathcal{F})_s) = 0$ and
$\Ext^1_{\mathcal{O}_{S, s}}(\kappa(s), (f_*\mathcal{F})_s) = 0$, see
Algebra, Lemma \ref{algebra-lemma-depth-ext}.

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