The Stacks project

Lemma 47.12.4. Let $I$ and $J$ be ideals in a Noetherian ring $A$. Let $M$ be a finite $A$-module. Set $Z =V(J)$. Consider the derived $I$-adic completion $R\Gamma _ Z(M)^\wedge $ of local cohomology. Then

  1. we have $R\Gamma _ Z(M)^\wedge = R\mathop{\mathrm{lim}}\nolimits R\Gamma _ Z(M/I^ nM)$, and

  2. there are short exact sequences

    \[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{i - 1}_ Z(M/I^ nM) \to H^ i(R\Gamma _ Z(M)^\wedge ) \to \mathop{\mathrm{lim}}\nolimits H^ i_ Z(M/I^ nM) \to 0 \]

In particular $R\Gamma _ Z(M)^\wedge $ has vanishing cohomology in negative degrees.

Proof. Suppose that $J = (g_1, \ldots , g_ m)$. Then $R\Gamma _ Z(M)$ is computed by the complex

\[ M \to \prod M_{g_{j_0}} \to \prod M_{g_{j_0}g_{j_1}} \to \ldots \to M_{g_1g_2\ldots g_ m} \]

by Lemma 47.9.1. By More on Algebra, Lemma 15.87.6 the derived $I$-adic completion of this complex is given by the complex

\[ \mathop{\mathrm{lim}}\nolimits M/I^ nM \to \prod \mathop{\mathrm{lim}}\nolimits (M/I^ nM)_{g_{j_0}} \to \ldots \to \mathop{\mathrm{lim}}\nolimits (M/I^ nM)_{g_1g_2\ldots g_ m} \]

of usual completions. Since $R\Gamma _ Z(M/I^ nM)$ is computed by the complex $ M/I^ nM \to \prod (M/I^ nM)_{g_{j_0}} \to \ldots \to (M/I^ nM)_{g_1g_2\ldots g_ m}$ and since the transition maps between these complexes are surjective, we conclude that (1) holds by More on Algebra, Lemma 15.81.1. Part (2) then follows from More on Algebra, Lemma 15.81.4. $\square$


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