The Stacks project

Proof. Assume $K^\bullet $ satisfies the assumptions of (1). Let $s : K^\bullet \to L^\bullet $ be any quasi-isomorphism. Then it is also true that $K^\bullet \to \tau _{\geq a}L^\bullet $ is a quasi-isomorphism by our assumption on $K^\bullet $. Hence in the category $K^\bullet /\text{Qis}^{+}(\mathcal{A})$ the quasi-isomorphisms $s : K^\bullet \to L^\bullet $ with $L^ n = 0$ for $n < a$ are cofinal. From Categories, Lemma 4.22.11 we deduce that $RF$ is the value of the essentially constant ind-object $F(L^\bullet )$ for these $s$. This means that $\text{id} : RF(K^\bullet ) \to RF(K^\bullet )$ factors through $F(L^\bullet )$ for some complex $L^\bullet $ with $L^ n = 0$ for $n < a$. It follows that $H^ i(RF(K^\bullet )) = 0$ for $i < a$.

To prove (2) we use the distinguished triangle

of Remark 13.12.4 to conclude via Lemma 13.14.6 that $RF$ is defined at $\tau _{\geq a + 1}K^\bullet $ as well and that we have a distinguished triangle

in $D(\mathcal{B})$. By part (1) we see that $RF(\tau _{\geq a + 1}K^\bullet )$ has vanishing cohomology in degrees $< a + 1$. The long exact cohomology sequence of this distinguished triangle then shows what we want. $\square$

Proof. Let $A$ be an object of $\mathcal{A}$. By Lemma 13.16.1 we have $H^ i(RF(A[0]) = 0$ for $i < 0$. This proves (1).

Let $0 \to A \to B \to C \to 0$ be a short exact sequence of $\mathcal{A}$. By Lemma 13.12.1 we obtain a distinguished triangle $(A[0], B[0], C[0], a, b, c)$ in $D^{+}(\mathcal{A})$. From the long exact cohomology sequence (and the vanishing for $i < 0$ proved above) we deduce that $0 \to R^0F(A) \to R^0F(B) \to R^0F(C)$ is exact. Hence $R^0F$ is left exact. Of course this also proves that if $F \to R^0F$ is an isomorphism, then $F$ is left exact.

Assume $F$ is left exact. Recall that $RF(A[0])$ is the value of the essentially constant system $F(K^\bullet )$ for $s : A[0] \to K^\bullet $ quasi-isomorphisms. It follows that $R^0F(A)$ is the value of the essentially constant system $H^0(F(K^\bullet ))$ for $s : A[0] \to K^\bullet $ quasi-isomorphisms, see Categories, Lemma 4.22.8. But if $s : A[0] \to K^\bullet $ is a quasi-isomorphism, then $A[0] \to \tau _{\geq 0}K^\bullet $ is a quasi-isomorphism. Hence in the category $A[0]/\text{Qis}^{+}(\mathcal{A})$ the quasi-isomorphisms $s : A[0] \to K^\bullet $ with $K^ n = 0$ for $n < 0$ are cofinal. It follows from Categories, Lemma 4.22.11 that we may restrict to such $s$. Moreover, for such an $s$ the sequence

is exact. Since $F$ is left exact we see that $0 \to F(A) \to F(K^0) \to F(K^1)$ is exact as well. It follows that $F(A) \to H^0(F(K^\bullet ))$ is an isomorphism and the system is actually constant with value $F(A)$. We conclude $R^0F = F$ as desired. $\square$

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. $\square$

Proof. By Lemma 13.12.1 we obtain a distinguished triangle $(A[0], B[0], C[0], a, b, c)$ in $D^{+}(\mathcal{A})$. As $RF$ is an exact functor and since $R^ iF = 0$ for $i < 0$ and $R^0F = F$ (Lemma 13.16.3) we obtain an exact cohomology sequence

in the abelian category $\mathcal{B}$. Thus the lemma follows from the characterization of acyclic objects in Lemma 13.16.4. $\square$

Proof. The functor $\mathcal{A} \to \text{Comp}^{+}(\mathcal{A})$, $A \mapsto A[0]$ is exact. The functor $\text{Comp}^{+}(\mathcal{A}) \to D^{+}(\mathcal{A})$ is a $\delta $-functor, see Lemma 13.12.1. The functor $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is exact. Finally, the functor $H^0 : D^{+}(\mathcal{B}) \to \mathcal{B}$ is a homological functor, see Definition 13.11.3. Hence we get the structure of a $\delta $-functor from Lemma 13.4.22 and Lemma 13.4.21. Part (2) follows from Homology, Lemma 12.12.4 and the description of acyclics in Lemma 13.16.4. $\square$

Proof. Let $A^\bullet $ be a bounded complex of right $F$-acyclic objects. We claim that $RF$ is defined at $A^\bullet $ and that $F(A^\bullet ) \to RF(A^\bullet )$ is an isomorphism in $D^+(\mathcal{B})$. Namely, it holds for complexes with at most one nonzero right $F$-acyclic object by Definition 13.15.3. Next, suppose that $A^ n = 0$ for $n \not\in [a, b]$. Using the “stupid” truncations we obtain a termwise split short exact sequence of complexes

see Homology, Section 12.15. Thus a distinguished triangle $(\sigma _{\geq a + 1} A^\bullet , A^\bullet , \sigma _{\leq a} A^\bullet )$. By induction hypothesis $RF$ is defined for the two outer complexes and these complexes compute $RF$. Then the same is true for the middle one by Lemma 13.14.12.

Suppose that $A^\bullet $ is a bounded below complex of acyclic objects such that $RF$ is defined at $A^\bullet $. To show that $F(A^\bullet ) \to RF(A^\bullet )$ is an isomorphism in $D^{+}(\mathcal{B})$ it suffices to show that $H^ i(F(A^\bullet )) \to H^ i(RF(A^\bullet ))$ is an isomorphism for all $i$. Pick $i$. Consider the termwise split short exact sequence of complexes

Note that this induces a termwise split short exact sequence

Hence we get distinguished triangles

Since $RF$ is defined at $A^\bullet $ (by assumption) and at $\sigma _{\leq i + 1}A^\bullet $ (by the first paragraph) we see that $RF$ is defined at $\sigma _{\geq i + 1}A^\bullet $ and we get a distinguished triangle

See Lemma 13.14.6. Using these distinguished triangles we obtain a map of exact sequences

By the results of the first paragraph the map $\beta $ is an isomorphism. By inspection the objects on the upper left and the upper right are zero. Hence to finish the proof it suffices to show that $H^ i(RF(\sigma _{\geq i + 2} A^\bullet )) = 0$ and $H^{i + 1}(RF(\sigma _{\geq i + 2} A^\bullet )) = 0$. This follows immediately from Lemma 13.16.1. $\square$

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

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

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

Proof. This is true because $F$ transforms acyclic complexes into acyclic complexes and quasi-isomorphisms into quasi-isomorphisms. Details omitted. $\square$