Lemma 15.88.2. Let $R$ be a ring. Let $I$ be an ideal of $R$. Let $M$ be an $I$-power torsion module. Then $M$ admits a resolution

with each $K_ i$ a direct sum of copies of $R/I^ n$ for $n$ variable.

Lemma 15.88.2. Let $R$ be a ring. Let $I$ be an ideal of $R$. Let $M$ be an $I$-power torsion module. Then $M$ admits a resolution

\[ \ldots \to K_2 \to K_1 \to K_0 \to M \to 0 \]

with each $K_ i$ a direct sum of copies of $R/I^ n$ for $n$ variable.

**Proof.**
There is a canonical surjection

\[ \oplus _{m \in M} R/I^{n_ m} \to M \to 0 \]

where $n_ m$ is the smallest positive integer such that $I^{n_ m} \cdot m = 0$. The kernel of the preceding surjection is also an $I$-power torsion module. Proceeding inductively, we construct the desired resolution of $M$. $\square$

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