**Proof.**
Assume every object of $\mathcal{A}$ injects into an object acyclic for $RF$. Let $\mathcal{I}$ be the set of objects acyclic for $RF$. Let $K^\bullet $ be a bounded below complex in $\mathcal{A}$. By Lemma 13.15.5 there exists a quasi-isomorphism $\alpha : K^\bullet \to I^\bullet $ with $I^\bullet $ bounded below and $I^ n \in \mathcal{I}$. Hence in order to prove (1) it suffices to show that $F(I^\bullet ) \to F((I')^\bullet )$ is a quasi-isomorphism when $s : I^\bullet \to (I')^\bullet $ is a quasi-isomorphism of bounded below complexes of objects from $\mathcal{I}$, see Lemma 13.14.15. Note that the cone $C(s)^\bullet $ is an acyclic bounded below complex all of whose terms are in $\mathcal{I}$. Hence it suffices to show: given an acyclic bounded below complex $I^\bullet $ all of whose terms are in $\mathcal{I}$ the complex $F(I^\bullet )$ is acyclic.

Say $I^ n = 0$ for $n < n_0$. Setting $J^ n = \mathop{\mathrm{Im}}(d^ n)$ we break $I^\bullet $ into short exact sequences $0 \to J^ n \to I^{n + 1} \to J^{n + 1} \to 0$ for $n \geq n_0$. These sequences induce distinguished triangles $(J^ n, I^{n + 1}, J^{n + 1})$ in $D^+(\mathcal{A})$ by Lemma 13.12.1. For each $k \in \mathbf{Z}$ denote $H_ k$ the assertion: For all $n \leq k$ the object $J^ n$ is in $\mathcal{I}$. Then $H_ k$ holds trivially for $k < n_0$. If $H_ n$ holds, then Lemma 13.14.12 shows that $J^{n + 1}$ is in $\mathcal{I}$ and we have $H_{n + 1}$. By Proposition 13.14.8 we have a distinguished triangle $(RF(J^ n), RF(I^{n + 1}), RF(J^{n + 1}))$. Since $J^ n, I^{n + 1}, J^{n + 1}$ are in $\mathcal{I}$ the long exact cohomology sequence (13.11.1.1) associated to this distinguished triangle collapses to an exact sequence

\[ 0 \to F(J^ n) \to F(I^{n + 1}) \to F(J^{n + 1}) \to 0 \]

This in turn proves that $F(I^\bullet )$ is exact.

The proof in the case of $LF$ is dual.
$\square$

## Comments (9)

Comment #1274 by JuanPablo on

Comment #1275 by JuanPablo on

Comment #1276 by JuanPablo on

Comment #1301 by Johan on

Comment #2601 by Rogier Brussee on

Comment #2626 by Johan on

Comment #8404 by Elías Guisado on

Comment #8405 by Elías Guisado on

Comment #9018 by Stacks project on