The Stacks project

Lemma 15.101.5. In Situation 15.91.15 assume $A$ is Noetherian. With notation as above, the inverse system $(I^ n)$ is pro-isomorphic in $D(A)$ to the inverse system $(I_ n^\bullet )$.

Proof. It is elementary to show that the inverse system $I^ n$ is pro-isomorphic to the inverse system $(f_1^ n, \ldots , f_ r^ n)$ in the category of $A$-modules. Consider the inverse system of distinguished triangles

\[ I_ n^\bullet \to (f_1^ n, \ldots , f_ r^ n) \to C_ n^\bullet \to I_ n^\bullet [1] \]

where $C_ n^\bullet $ is the cone of the first arrow. By Derived Categories, Lemma 13.42.4 it suffices to show that the inverse system $C_ n^\bullet $ is pro-zero. The complex $I_ n^\bullet $ has nonzero terms only in degrees $i$ with $-r + 1 \leq i \leq 0$ hence $C_ n^\bullet $ is bounded similarly. Thus by Derived Categories, Lemma 13.42.3 it suffices to show that $H^ p(C_ n^\bullet )$ is pro-zero. By the discussion above we have $H^ p(C_ n^\bullet ) = H^ p(K_ n^\bullet )$ for $p \leq -1$ and $H^ p(C_ n^\bullet ) = 0$ for $p \geq 0$. The fact that the inverse systems $H^ p(K_ n^\bullet )$ are pro-zero was shown in the proof of Lemma 15.94.1 (and this is where the assumption that $A$ is Noetherian is used). $\square$

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Your email address will not be published.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 0G3M. Beware of the difference between the letter 'O' and the digit '0'.