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Clarify choice in local-cohomology

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aisejohan committed Jan 29, 2019
1 parent 80b642e commit a03896ca4581ee5d29100953925de28b79686016
Showing with 5 additions and 4 deletions.
  1. +5 −4 local-cohomology.tex
@@ -3771,10 +3771,11 @@ \section{Structure of certain modules}
\end{lemma}

\begin{proof}
Choose a set $J$ and an $A$-module homorphism
$\varphi : M \to \bigoplus_{j \in J} E$ which maps
$M[\mathfrak m]$ isomorphically onto
$(\bigoplus_{j \in J} E)[\mathfrak m] = \bigoplus_{j \in J} k$.
Choose a set $J$ and an isomorphism $M[\mathfrak m] \to \bigoplus_{j \in J} k$.
Since $\bigoplus_{j \in J} E$ is injective
(Dualizing Complexes, Lemma \ref{dualizing-lemma-sum-injective-modules})
we can extend this isomorphism to an $A$-module homomorphism
$\varphi : M \to \bigoplus_{j \in J} E$.
We claim that $\varphi$ is an isomorphism, i.e., bijective.

\medskip\noindent

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