The Stacks project

Proposition 13.28.2. Let $F : \mathcal{A} \to \mathcal{B}$ be a right exact functor of abelian categories. Let $\mathcal{P} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ be a subset. Assume

  1. every object of $\mathcal{A}$ is a quotient of an element of $\mathcal{P}$,

  2. for any bounded above acyclic complex $P^\bullet $ of $\mathcal{A}$ with $P^ n \in \mathcal{P}$ for all $n$ the complex $F(P^\bullet )$ is exact,

  3. $\mathcal{A}$ and $\mathcal{B}$ have colimits of systems over $\mathbf{N}$,

  4. colimits over $\mathbf{N}$ are exact in both $\mathcal{A}$ and $\mathcal{B}$, and

  5. $F$ commutes with colimits over $\mathbf{N}$.

Then $LF$ is defined on all of $D(\mathcal{A})$.

Proof. By (1) and Lemma 13.16.5 for any bounded above complex $K^\bullet $ there exists a quasi-isomorphism $P^\bullet \to K^\bullet $ with $P^\bullet $ bounded above and $P^ n \in \mathcal{P}$ for all $n$. Suppose that $s : P^\bullet \to (P')^\bullet $ is a quasi-isomorphism of bounded above complexes consisting of objects of $\mathcal{P}$. Then $F(P^\bullet ) \to F((P')^\bullet )$ is a quasi-isomorphism because $F(C(s)^\bullet )$ is acyclic by assumption (2). This already shows that $LF$ is defined on $D^{-}(\mathcal{A})$ and that a bounded above complex consisting of objects of $\mathcal{P}$ computes $LF$, see Lemma 13.15.15.

Next, let $K^\bullet $ be an arbitrary complex of $\mathcal{A}$. Choose a diagram

\[ \xymatrix{ P_1^\bullet \ar[d] \ar[r] & P_2^\bullet \ar[d] \ar[r] & \ldots \\ \tau _{\leq 1}K^\bullet \ar[r] & \tau _{\leq 2}K^\bullet \ar[r] & \ldots } \]

as in Lemma 13.28.1. Note that the map $\mathop{\mathrm{colim}}\nolimits P_ n^\bullet \to K^\bullet $ is a quasi-isomorphism because colimits over $\mathbf{N}$ in $\mathcal{A}$ are exact and $H^ i(P_ n^\bullet ) = H^ i(K^\bullet )$ for $n > i$. We claim that

\[ F(\mathop{\mathrm{colim}}\nolimits P_ n^\bullet ) = \mathop{\mathrm{colim}}\nolimits F(P_ n^\bullet ) \]

(termwise colimits) is $LF(K^\bullet )$, i.e., that $\mathop{\mathrm{colim}}\nolimits P_ n^\bullet $ computes $LF$. To see this, by Lemma 13.15.15, it suffices to prove the following claim. Suppose that

\[ \mathop{\mathrm{colim}}\nolimits Q_ n^\bullet = Q^\bullet \xrightarrow {\ \alpha \ } P^\bullet = \mathop{\mathrm{colim}}\nolimits P_ n^\bullet \]

is a quasi-isomorphism of complexes, such that each $P_ n^\bullet $, $Q_ n^\bullet $ is a bounded above complex whose terms are in $\mathcal{P}$ and the maps $P_ n^\bullet \to \tau _{\leq n}P^\bullet $ and $Q_ n^\bullet \to \tau _{\leq n}Q^\bullet $ are quasi-isomorphisms. Claim: $F(\alpha )$ is a quasi-isomorphism.

The problem is that we do not assume that $\alpha $ is given as a colimit of maps between the complexes $P_ n^\bullet $ and $Q_ n^\bullet $. However, for each $n$ we know that the solid arrows in the diagram

\[ \xymatrix{ & R^\bullet \ar@{..>}[d] \\ P_ n^\bullet \ar[d] & L^\bullet \ar@{..>}[l] \ar@{..>}[r] & Q_ n^\bullet \ar[d] \\ \tau _{\leq n}P^\bullet \ar[rr]^{\tau _{\leq n}\alpha } & & \tau _{\leq n}Q^\bullet } \]

are quasi-isomorphisms. Because quasi-isomorphisms form a multiplicative system in $K(\mathcal{A})$ (see Lemma 13.11.2) we can find a quasi-isomorphism $L^\bullet \to P_ n^\bullet $ and map of complexes $L^\bullet \to Q_ n^\bullet $ such that the diagram above commutes up to homotopy. Then $\tau _{\leq n}L^\bullet \to L^\bullet $ is a quasi-isomorphism. Hence (by the first part of the proof) we can find a bounded above complex $R^\bullet $ whose terms are in $\mathcal{P}$ and a quasi-isomorphism $R^\bullet \to L^\bullet $ (as indicated in the diagram). Using the result of the first paragraph of the proof we see that $F(R^\bullet ) \to F(P_ n^\bullet )$ and $F(R^\bullet ) \to F(Q_ n^\bullet )$ are quasi-isomorphisms. Thus we obtain a isomorphisms $H^ i(F(P_ n^\bullet )) \to H^ i(F(Q_ n^\bullet ))$ fitting into the commutative diagram

\[ \xymatrix{ H^ i(F(P_ n^\bullet )) \ar[r] \ar[d] & H^ i(F(Q_ n^\bullet )) \ar[d] \\ H^ i(F(P^\bullet )) \ar[r] & H^ i(F(Q^\bullet )) } \]

The exact same argument shows that these maps are also compatible as $n$ varies. Since by (4) and (5) we have

\[ H^ i(F(P^\bullet )) = H^ i(F(\mathop{\mathrm{colim}}\nolimits P_ n^\bullet )) = H^ i(\mathop{\mathrm{colim}}\nolimits F(P_ n^\bullet )) = \mathop{\mathrm{colim}}\nolimits H^ i(F(P_ n^\bullet )) \]

and similarly for $Q^\bullet $ we conclude that $H^ i(\alpha ) : H^ i(F(P^\bullet ) \to H^ i(F(Q^\bullet )$ is an isomorphism and the claim follows. $\square$


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