The Stacks project

Email from Kovacs of 23/02/2018.

Lemma 15.99.2. Let $I$ be an ideal of a Noetherian ring $A$. Let $K \in D(A)$ be pseudo-coherent. Set $K_ n = K \otimes _ A^\mathbf {L} A/I^ n$. Then for all $i \in \mathbf{Z}$ the system $H^ i(K_ n)$ satisfies Mittag-Leffler and $\mathop{\mathrm{lim}}\nolimits H^ i(K)/I^ nH^ i(K)$ is equal to $\mathop{\mathrm{lim}}\nolimits H^ i(K_ n)$.

Proof. We may represent $K$ by a bounded above complex $P^\bullet $ of finite free $A$-modules. Then $K_ n$ is represented by $P^\bullet /I^ nP^\bullet $. Hence the Mittag-Leffler property by Lemma 15.99.1. The final statement follows then from Lemma 15.96.6. $\square$


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