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bigcup -> bigoplus
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aisejohan committed Jan 16, 2023
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6 changes: 3 additions & 3 deletions dualizing.tex
Expand Up @@ -296,13 +296,13 @@ \section{Injective modules}

\begin{proof}
Let $E_i$ be a family of injective modules parametrized by a set $I$.
Set $E = \bigcup E_i$. To show that $E$ is injective we use
Set $E = \bigoplus E_i$. To show that $E$ is injective we use
Injectives, Lemma \ref{injectives-lemma-criterion-baer}.
Thus let $\varphi : I \to E$ be a module map from an ideal of $R$
into $E$. As $I$ is a finite $R$-module (because $R$ is Noetherian)
we can find finitely many elements $i_1, \ldots, i_r \in I$
such that $\varphi$ maps into $\bigcup_{j = 1, \ldots, r} E_{i_j}$.
Then we can extend $\varphi$ into $\bigcup_{j = 1, \ldots, r} E_{i_j}$
such that $\varphi$ maps into $\bigoplus_{j = 1, \ldots, r} E_{i_j}$.
Then we can extend $\varphi$ into $\bigoplus_{j = 1, \ldots, r} E_{i_j}$
using the injectivity of the modules $E_{i_j}$.
\end{proof}

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