Lemma 13.16.4 (015C)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 13: Derived Categories
Section 13.16: Higher derived functors
Lemma 13.16.4 (cite)

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Lemma 13.16.4. Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor between abelian categories and assume $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is everywhere defined. Let $A$ be an object of $\mathcal{A}$ .

$A$ is right acyclic for $F$ if and only if $F(A) \to R^0F(A)$ is an isomorphism and $R^ iF(A) = 0$ for all $i > 0$ ,
if $F$ is left exact, then $A$ is right acyclic for $F$ if and only if $R^ iF(A) = 0$ for all $i > 0$ .

Proof. If $A$ is right acyclic for $F$ , then $RF(A[0]) = F(A)[0]$ and in particular $F(A) \to R^0F(A)$ is an isomorphism and $R^ iF(A) = 0$ for $i \not= 0$ . Conversely, if $F(A) \to R^0F(A)$ is an isomorphism and $R^ iF(A) = 0$ for all $i > 0$ then $F(A[0]) \to RF(A[0])$ is a quasi-isomorphism by Lemma 13.16.3 part (1) and hence $A$ is acyclic. If $F$ is left exact then $F = R^0F$ , see Lemma 13.16.3. $\square$

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