The Stacks project

Lemma 22.15.1. Let $(A, \text{d})$ be a differential graded algebra. The full subcategory $\text{Ac}$ of $K(\text{Mod}_{(A, \text{d})})$ consisting of acyclic modules is a strictly full saturated triangulated subcategory of $K(\text{Mod}_{(A, \text{d})})$. The corresponding saturated multiplicative system (see Derived Categories, Lemma 13.6.10) of $K(\text{Mod}_{(A, \text{d})})$ is the class $\text{Qis}$ of quasi-isomorphisms. In particular, the kernel of the localization functor

\[ Q : K(\text{Mod}_{(A, \text{d})}) \to \text{Qis}^{-1}K(\text{Mod}_{(A, \text{d})}) \]

is $\text{Ac}$. Moreover, the functor $H^0$ factors through $Q$.

Proof. We know that $H^0$ is a homological functor by the long exact sequence of homology ( The kernel of $H^0$ is the subcategory of acyclic objects and the arrows with induce isomorphisms on all $H^ i$ are the quasi-isomorphisms. Thus this lemma is a special case of Derived Categories, Lemma 13.6.11.

Set theoretical remark. The construction of the localization in Derived Categories, Proposition 13.5.5 assumes the given triangulated category is “small”, i.e., that the underlying collection of objects forms a set. Let $V_\alpha $ be a partial universe (as in Sets, Section 3.5) containing $(A, \text{d})$ and where the cofinality of $\alpha $ is bigger than $\aleph _0$ (see Sets, Proposition 3.7.2). Then we can consider the category $\text{Mod}_{(A, \text{d}), \alpha }$ of differential graded $A$-modules contained in $V_\alpha $. A straightforward check shows that all the constructions used in the proof of Proposition 22.10.3 work inside of $\text{Mod}_{(A, \text{d}), \alpha }$ (because at worst we take finite direct sums of differential graded modules). Thus we obtain a triangulated category $\text{Qis}_\alpha ^{-1}K(\text{Mod}_{(A, \text{d}), \alpha })$. We will see below that if $\beta > \alpha $, then the transition functors

\[ \text{Qis}_\alpha ^{-1}K(\text{Mod}_{(A, \text{d}), \alpha }) \longrightarrow \text{Qis}_\beta ^{-1}K(\text{Mod}_{(A, \text{d}), \beta }) \]

are fully faithful as the morphism sets in the quotient categories are computed by maps in the homotopy categories from P-resolutions (the construction of a P-resolution in the proof of Lemma 22.13.4 takes countable direct sums as well as direct sums indexed over subsets of the given module). The reader should therefore think of the category of the lemma as the union of these subcategories. $\square$

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