Lemma 64.7.2 (03TB)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 64: The Trace Formula
Section 64.7: Filtered derived category
Lemma 64.7.2 (cite)

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Lemma 64.7.2. If $\mathcal{A}$ has enough injectives, then $DF^+(\mathcal{A}) \cong K^+(\mathcal{I})$ , where $\mathcal{I}$ is the full additive subcategory of $\text{Fil}^ f(\mathcal{A})$ consisting of filtered injective objects. Similarly, if $\mathcal{A}$ has enough projectives, then $DF^-(\mathcal{A}) \cong K^-(\mathcal{P})$ , where $\mathcal P$ is the full additive subcategory of $\text{Fil}^ f(\mathcal{A})$ consisting of filtered projective objects.

Proof. Omitted. $\square$

Comments (2)

Comment #8298 by Xiaolong Liu on April 07, 2023 at 10:15

We may replace $DF^-(\mathcal{A}) \cong K^+(\mathcal{P})$ to $DF^-(\mathcal{A}) \cong K^-(\mathcal{P})$ .

Comment #8924 by Stacks project on April 10, 2024 at 17:08

Thanks and fixed here.

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