Lemma 15.74.13 (068T)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.74: Perfect complexes
Lemma 15.74.13 (cite)

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Lemma 15.74.13. Let $R$ be a ring. Let $K^\bullet$ be a complex of $R$ -modules. Let $R \to R'$ be a faithfully flat ring map. If the complex $K^\bullet \otimes _ R R'$ is perfect, then $K^\bullet$ is perfect.

Proof. Using Lemma 15.74.2 this translates into the corresponding results for pseudo-coherent modules and modules of finite tor dimension. See Lemma 15.66.17 and Lemma 15.64.15 for those results. $\square$

Comments (4)

Comment #1689 by David Rydh on November 12, 2015 at 10:17

This result should be about the descent of perfectness, not tor-amplitude (which is 068S). The proof is correct, but does not match the statement...

Comment #1737 by Johan on December 15, 2015 at 18:47

Thanks, fixed here.

Comment #7918 by Peng Du on December 10, 2022 at 15:20

What is a, b relevant here?

Comment #8170 by Aise Johan de Jong on February 28, 2023 at 10:50

Thanks and fixed here.

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