The Stacks project

Proposition 13.29.2. 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. $\mathcal{P}$ contains $0$, is closed under (finite) direct sums, and 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 13.15.4 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 13.14.15.

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 13.29.1. 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 13.14.15, 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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0794. Beware of the difference between the letter 'O' and the digit '0'.