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