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Add reference to Nakayama

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aisejohan committed Dec 10, 2019
1 parent b38bb61 commit 2cfc834a5f9d8a0129914d5dea28db80476d8f1a
Showing with 2 additions and 1 deletion.
  1. +2 −1 algebra.tex
As $M_{n + 1} \to M_n$ is surjective, the map $M \to M_1$ is surjective.
Pick $x_t \in M$, $t \in T$ mapping to generators of $M_1$. This gives a map
$\bigoplus_{t \in T} A \to M$. Note that the images of $x_t$ in $M_n$
generate $M_n$ for all $n$ too. Consider the exact sequences
generate $M_n$ for all $n$ too by Nakayama's lemma (Lemma \ref{lemma-NAK}).
Consider the exact sequences
$$
0 \to K_n \to \bigoplus\nolimits_{t \in T} A/I^n \to M_n \to 0
$$

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