The Stacks project

Email from Kovacs of 23/02/2018.

Lemma 15.97.6 (Kollár-Kovács). Let $I$ be an ideal of a Noetherian ring $A$. Let $K \in D(A)$. Set $K_ n = K \otimes _ A^\mathbf {L} A/I^ n$. Assume for all $i \in \mathbf{Z}$ we have

  1. $H^ i(K)$ is a finite $A$-module, and

  2. the system $H^ i(K_ n)$ satisfies Mittag-Leffler.

Then $\mathop{\mathrm{lim}}\nolimits H^ i(K)/I^ nH^ i(K)$ is equal to $\mathop{\mathrm{lim}}\nolimits H^ i(K_ n)$ for all $i \in \mathbf{Z}$.

Proof. Recall that $K^\wedge = R\mathop{\mathrm{lim}}\nolimits K_ n$ is the derived completion of $K$, see Proposition 15.94.2. By Lemma 15.94.4 we have $H^ i(K^\wedge ) = \mathop{\mathrm{lim}}\nolimits H^ i(K)/I^ nH^ i(K)$. By Lemma 15.87.4 we get short exact sequences

\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{i - 1}(K_ n) \to H^ i(K^\wedge ) \to \mathop{\mathrm{lim}}\nolimits H^ i(K_ n) \to 0 \]

The Mittag-Leffler condition guarantees that the left terms are zero (Lemma 15.87.1) and we conclude the lemma is true. $\square$

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