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typos in local-cohomology

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aisejohan committed Jan 28, 2019
1 parent b247a3f commit 80b642e484208a6e4f52ee9fe89c9a15fff8e1b1
Showing with 4 additions and 4 deletions.
  1. +4 −4 local-cohomology.tex
@@ -1439,8 +1439,8 @@ \section{Finiteness of pushforwards, I}
Note that $\text{Ass}(\mathcal{F}') \subset \text{Ass}(\mathcal{F})$, see
Divisors, Lemma \ref{divisors-lemma-ses-ass}.
Since
$\text{length}_{\mathcal{O}_{X, x}}(\mathcal{F}') =
\text{length}_{\mathcal{O}_{X, x}}(\mathcal{F}) - 1$
$\text{length}_{\mathcal{O}_{X, x}}(\mathcal{F}'_x) =
\text{length}_{\mathcal{O}_{X, x}}(\mathcal{F}_x) - 1$
we may apply the
induction hypothesis to conclude $j_*\mathcal{F}'$ is coherent.
Since $\mathcal{G} = j_*(\mathcal{G}|_U) = j_*i_{x, *}\mathcal{O}_{W_x}$
@@ -1468,7 +1468,7 @@ \section{Finiteness of pushforwards, I}
$\varphi : \mathcal{G}|_U \to \mathcal{F}$.
Then $\varphi$ is injective (for example by
Divisors, Lemma \ref{divisors-lemma-check-injective-on-ass})
and we find and injective map
and we find an injective map
$\mathcal{G} = j_*(\mathcal{G}|_V) \to j_*\mathcal{F}$.
Thus (1) holds.
\end{proof}
@@ -1661,7 +1661,7 @@ \section{Finiteness of pushforwards, I}
$$
We have $j_*i_{x, *}\mathcal{O}_{W_x} = i_*j'_*\mathcal{O}_{W_x}$.
As the left vertical arrow is a closed immersion we see that
$j_*i_{x, *}\mathcal{O}_{W_x}$ is coherent if and only of
$j_*i_{x, *}\mathcal{O}_{W_x}$ is coherent if and only if
$j'_*\mathcal{O}_{W_x}$ is coherent.
\end{remark}

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