Lemma 47.6.4. Assumptions and notation as in Lemma 47.6.3. Let $I \subset R$ be an ideal and $M$ a finite $R$-module. Then

\[ D(M[I]) = D(M)/ID(M) \quad \text{and}\quad D(M/IM) = D(M)[I] \]

**Proof.**
Say $I = (f_1, \ldots , f_ t)$. Consider the map

\[ M^{\oplus t} \xrightarrow {f_1, \ldots , f_ t} M \]

with cokernel $M/IM$. Applying the exact functor $D$ we conclude that $D(M/IM)$ is $D(M)[I]$. The other case is proved in the same way. $\square$

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