The Stacks project

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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