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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)