Lemma 15.91.8 (0A05)—The Stacks project

Processing math: 100%

Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.91: Derived Completion
Lemma 15.91.8 (cite)

numberstags

Lemma 15.91.8. Let $A$ be a ring and $I \subset A$ an ideal. If $A$ is derived complete (eg. $I$ -adically complete) then any pseudo-coherent object of $D(A)$ is derived complete.

Proof. (Lemma 15.91.3 explains the parenthetical statement of the lemma.) Let $K$ be a pseudo-coherent object of $D(A)$ . By definition this means $K$ is represented by a bounded above complex $K^\bullet$ of finite free $A$ -modules. Since $A$ is derived complete it follows that $H^ n(K)$ is derived complete for all $n$ , by part (1) of Lemma 15.91.6. This in turn implies that $K$ is derived complete by part (2) of the same lemma. $\square$

Comments (3)

Comment #6207 by Zhouhang MAO on May 04, 2021 at 06:39

I find it weird to phrase the assumption as $A$ being $I$ -adically complete. Here what we really need is $A$ being derived $I$ -complete.

Comment #6208 by Johan on May 04, 2021 at 11:34

This is a good point! I will strengthen the lemma in the way you mention the next time I go through all the comments. Thanks.

Comment #6348 by Johan on July 13, 2021 at 13:05

THanks and fixed here.

There are also:

14 comment(s) on Section 15.91: Derived Completion

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).

Comments (3)

Post a comment