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Use earlier lemma in local-cohomology

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aisejohan committed Jan 25, 2019
1 parent 6ca9041 commit b247a3f18d4da6d3afbf282662bd937b6fc80720
Showing with 2 additions and 18 deletions.
  1. +2 −18 local-cohomology.tex
@@ -3104,24 +3104,8 @@ \section{Improving coherent modules}
\end{lemma}

\begin{proof}
Let us show that $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}'_x) \geq 2$
for $x \in X$, $x \not \in U$. Namely, let $V$ be the punctured spectrum of
$\mathcal{O}_{X, x}$. Then $V$ contains the inverse image of
$U$ along $\Spec(\mathcal{O}_{X, x}) \to X$.
Since $\mathcal{F}' = j_*(\mathcal{F}|_U) =
j_*(\mathcal{F}'|_U)$, the same is true after
base change by the flat morphism $\Spec(\mathcal{O}_{X, z}) \to X$
(Cohomology of Schemes, Lemma
\ref{coherent-lemma-flat-base-change-cohomology}).
A fortiori, the canonical map
$\mathcal{F}'_x \to H^0(V, \mathcal{F}'|_V)$
is an isomorphism. This means that $H^i_{\mathfrak m_x}(\mathcal{F}'_x)$
is zero for $i = 0, 1$, see
Lemma \ref{lemma-finiteness-pushforwards-and-H1-local}.
Thus the depth is at least $2$ by
Dualizing Complexes, Lemma \ref{dualizing-lemma-depth}.

\medskip\noindent
We have $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}'_x) \geq 2$
by Divisors, Lemma \ref{divisors-lemma-depth-pushforward} part (3).
The uniqueness of $\mathcal{F} \to \mathcal{F}'$ follows from
Divisors, Lemma \ref{divisors-lemma-depth-2-hartog}.
The compatibility with flat pullbacks follows from

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