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Much better depth >= 2 of pushforward thing

Finally! Not sure why this was missed earlier...
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aisejohan committed Jan 25, 2019
1 parent 9a03196 commit 6ca9041058161da6b0da1bdeed17dc97e6f063cf
Showing with 31 additions and 4 deletions.
  1. +31 −4 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 $\text{depth}(f_*\mathcal{O}_{S, s}) \geq 1$.
is Noetherian, then $\text{depth}(f_*\mathcal{O}_{S, s}) \geq 2$.
\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 $a, b \in \mathfrak m_s$
@@ -940,9 +940,36 @@ \section{Morphisms and weakly associated points}
\begin{proof}
Part (1) is Lemma \ref{lemma-weakass-pushforward}.
Part (2) follows from (1) and Lemma \ref{lemma-ass-weakly-ass}.
Part (3) follows from (2) and
Algebra, Lemma \ref{algebra-lemma-ass-zero-divisors} or
Algebra, Lemma \ref{algebra-lemma-ideal-nonzerodivisor}.

\medskip\noindent
Proof of part (3). To show the depth is $\geq w$ 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}.
Using the exact sequence
$0 \to \mathfrak m_s \to \mathcal{O}_{S, s} \to \kappa(s) \to 0$
it suffices to prove that the map
$$
\Hom_{\mathcal{O}_{S, s}}(\mathcal{O}_{S, s}, (f_*\mathcal{F})_s)
\to
\Hom_{\mathcal{O}_{S, s}}(\mathfrak m_s, (f_*\mathcal{F})_s)
$$
is an isomorphism. By flat base change (Cohomology of Schemes, Lemma
\ref{coherent-lemma-flat-base-change-cohomology})
we may replace $S$ by
$\Spec(\mathcal{O}_{S, s})$ and $X$ by $\Spec(\mathcal{O}_{S, s}) \times_S X$.
Denote $\mathfrak m \subset \mathcal{O}_S$ the ideal sheaf of $s$.
Then we see that
$$
\Hom_{\mathcal{O}_{S, s}}(\mathfrak m_s, (f_*\mathcal{F})_s) =
\Hom_{\mathcal{O}_S}(\mathfrak m, f_*\mathcal{F}) =
\Hom_{\mathcal{O}_X}(f^*\mathfrak m, \mathcal{F})
$$
the first equality because $S$ is local with closed point $s$
and the second equality
by adjunction for $f^*, f_*$ on quasi-coherent modules. However, since
$s \not \in f(X)$ we see that $f^*\mathfrak m = \mathcal{O}_X$.
Working backwards through the arguments we get the desired equality.

\medskip\noindent
For the proof of (4), (5), and (6) we use flat base change

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