The Stacks project

Lemma 13.32.3. Let $F : \mathcal{A} \to \mathcal{B}$ be a right exact functor of abelian categories. If

1. every object of $\mathcal{A}$ is a quotient of an object which is left acyclic for $F$,

2. there exists an integer $n \geq 0$ such that $L^ nF = 0$,

Then

1. $LF : D(\mathcal{A}) \to D(\mathcal{B})$ exists,

2. any complex consisting of left acyclic objects for $F$ computes $LF$,

3. any complex is the target of a quasi-isomorphism from a complex consisting of left acyclic objects for $F$,

4. for $E \in D(\mathcal{A})$

   1. $H^ i(LF(\tau _{\leq a + n - 1}E) \to H^ i(LF(E))$ is an isomorphism for $i \leq a$,

   2. $H^ i(LF(E)) \to H^ i(LF(\tau _{\geq b}E))$ is an isomorphism for $i \geq b$,

   3. if $H^ i(E) = 0$ for $i \not\in [a, b]$ for some $-\infty \leq a \leq b \leq \infty $, then $H^ i(LF(E)) = 0$ for $i \not\in [a - n + 1, b]$.