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 15.94.6 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 15.87.1. Part (2) then follows from More on Algebra, Lemma 15.87.4. $\square$

Comments (0)

There are also:

  • 5 comment(s) on Section 47.12: Torsion versus complete modules

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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0EEW. Beware of the difference between the letter 'O' and the digit '0'.