The Stacks project

Lemma 52.7.1. Let $X$ be a locally Noetherian scheme. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $K$ be a pseudo-coherent object of $D(\mathcal{O}_ X)$ with derived completion $K^\wedge $. Then

\[ H^ p(U, K^\wedge ) = \mathop{\mathrm{lim}}\nolimits H^ p(U, K)/I^ nH^ p(U, K) = H^ p(U, K)^\wedge \]

for any affine open $U \subset X$ where $I = \mathcal{I}(U)$ and where on the right we have the derived completion with respect to $I$.

Proof. Write $U = \mathop{\mathrm{Spec}}(A)$. The ring $A$ is Noetherian and hence $I \subset A$ is finitely generated. Then we have

\[ R\Gamma (U, K^\wedge ) = R\Gamma (U, K)^\wedge \]

by Remark 52.6.21. Now $R\Gamma (U, K)$ is a pseudo-coherent complex of $A$-modules (Derived Categories of Schemes, Lemma 36.10.2). By More on Algebra, Lemma 15.93.4 we conclude that the $p$th cohomology module of $R\Gamma (U, K^\wedge )$ is equal to the $I$-adic completion of $H^ p(U, K)$. This proves the first equality. The second (less important) equality follows immediately from a second application of the lemma just used. $\square$


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