The Stacks project

Lemma 115.4.15. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $s$ and $d$ be integers. Assume

  1. $A$ has a dualizing complex,

  2. $\text{cd}(A, I) \leq d$,

  3. if $\mathfrak p \not\in V(I)$ then $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) > s$ or $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim (A/\mathfrak p) > d + s$.

Then the assumptions of Algebraic and Formal Geometry, Lemma 52.10.4 hold for $A, I, \mathfrak m, M$ and $H^ i_\mathfrak m(M) \to \mathop{\mathrm{lim}}\nolimits H^ i_\mathfrak m(M/I^ nM)$ is an isomorphism for $i \leq s$ and these modules are annihilated by a power of $I$.

Proof. The assumptions of Algebraic and Formal Geometry, Lemma 52.10.4 by the more general Algebraic and Formal Geometry, Lemma 52.10.5. Then the conclusion of Algebraic and Formal Geometry, Lemma 52.10.4 gives the second statement. $\square$

Comments (0)

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Your email address will not be published.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 0EFS. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 115.4.15 as pdf