Lemma 21.53.5. Let \mathcal{C} be a site. Let \Lambda be a Noetherian ring. Let I \subset \Lambda be an ideal. Let K \in D^-(\mathcal{C}, \Lambda ). If the cohomology sheaves of K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I} are locally constant sheaves of \Lambda /I-modules of finite type, then the cohomology sheaves of K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n} are locally constant sheaves of \Lambda /I^ n-modules of finite type for all n \geq 1.
Proof. Recall that the locally constant sheaves of \Lambda -modules of finite type form a weak Serre subcategory of all \underline{\Lambda }-modules, see Modules on Sites, Lemma 18.43.5. Thus the subcategory of D(\mathcal{C}, \Lambda ) consisting of complexes whose cohomology sheaves are locally constant sheaves of \Lambda -modules of finite type forms a strictly full, saturated triangulated subcategory of D(\mathcal{C}, \Lambda ), see Derived Categories, Lemma 13.17.1. Next, consider the distinguished triangles
K \otimes _\Lambda ^\mathbf {L} \underline{I^ n/I^{n + 1}} \to K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^{n + 1}} \to K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n} \to K \otimes _\Lambda ^\mathbf {L} \underline{I^ n/I^{n + 1}}[1]
and the isomorphisms
K \otimes _\Lambda ^\mathbf {L} \underline{I^ n/I^{n + 1}} = \left(K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I}\right) \otimes _{\Lambda /I}^\mathbf {L} \underline{I^ n/I^{n + 1}}
Combined with Lemma 21.53.4 we obtain the result. \square
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