The Stacks project

Email from Kovacs of 23/02/2018.

Lemma 15.100.2. 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 15.100.1. The final statement follows then from Lemma 15.97.6. $\square$

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Your email address will not be published.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