Lemma 19.12.4 (079M)—The Stacks project

Lemma 19.12.4. Let $\mathcal{A}$ be a Grothendieck abelian category. Let $(K_ i^\bullet )_{i \in I}$ be a set of acyclic complexes. There exists a functor $M^\bullet \mapsto \mathbf{M}^\bullet (M^\bullet )$ and a natural transformation $j_{M^\bullet } : M^\bullet \to \mathbf{M}^\bullet (M^\bullet )$ such

$j_{M^\bullet }$ is a (termwise) injective quasi-isomorphism, and
for every $i \in I$ and $w : K_ i^\bullet \to M^\bullet $ the morphism $j_{M^\bullet } \circ w$ is homotopic to zero.

Proof. For every $i \in I$ choose a (termwise) injective map of complexes $K_ i^\bullet \to L_ i^\bullet $ which is homotopic to zero with $L_ i^\bullet $ quasi-isomorphic to zero. For example, take $L_ i^\bullet $ to be the cone on the identity of $K_ i^\bullet $. We define $\mathbf{M}^\bullet (M^\bullet )$ by the following pushout diagram

\[ \xymatrix{ \bigoplus _{i \in I} \bigoplus _{w : K_ i^\bullet \to M^\bullet } K_ i^\bullet \ar[r] \ar[d] & M^\bullet \ar[d] \\ \bigoplus _{i \in I} \bigoplus _{w : K_ i^\bullet \to M^\bullet } L_ i^\bullet \ar[r] & \mathbf{M}^\bullet (M^\bullet ). } \]

Then $M^\bullet \to \mathbf{M}^\bullet (M^\bullet )$ is a functor. The right vertical arrow defines the functorial injective map $j_{M^\bullet }$. The cokernel of $j_{M^\bullet }$ is isomorphic to the direct sum of the cokernels of the maps $K_ i^\bullet \to L_ i^\bullet $ hence acyclic. Thus $j_{M^\bullet }$ is a quasi-isomorphism. Part (2) holds by construction. $\square$

Comments (1)

Comment #9514 by Elías Guisado on July 19, 2024 at 04:55

In the proof, in “then $M^\bullet\to M^\bullet(M^\bullet)$ is a functor” one could write ' $\mapsto$ ' instead to make it agree with the style displayed in the statement.

When the proof says “the cokernel of $j_{M^\bullet}$ is isomorphic to the direct sum of the cokernels of the maps $K^{\bullet}_i\to L^\bullet_i$ hence acyclic” we are using that $\mathcal{A}$ is AB5. Specifically, we are using firstly that AB5 $\Rightarrow$ AB4 (as it was pointed out in #9401) and secondly that in an AB4 category a direct sum of acyclic complexes is acyclic (actually, this is equivalent to the AB4 condition, see Proposition 1 here).

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