Lemma 22.21.2 (09KS)—The Stacks project

Lemma 22.21.2. Let $(A, \text{d})$ be a differential graded algebra. Let $I$ be a differential graded $A$ -module with property (I). Then

$\mathop{\mathrm{Hom}}\nolimits _{K(\text{Mod}_{(A, \text{d})})}(N, I) = 0$

for all acyclic differential graded $A$ -modules $N$ .

Proof. We will use that $K(\text{Mod}_{(A, \text{d})})$ is a triangulated category (Proposition 22.10.3). Let $F_\bullet$ be a filtration on $I$ as in property (I). The short exact sequence of Lemma 22.21.1 produces a distinguished triangle. Hence by Derived Categories, Lemma 13.4.2 it suffices to show that

$\mathop{\mathrm{Hom}}\nolimits _{K(\text{Mod}_{(A, \text{d})})}(N, I/F_ iI) = 0$

for all acyclic differential graded $A$ -modules $N$ and all $i$ . Each of the differential graded modules $I/F_ iI$ has a finite filtration by admissible monomorphisms, whose graded pieces are products of $A^\vee [k]$ . Thus it suffices to prove that

$\mathop{\mathrm{Hom}}\nolimits _{K(\text{Mod}_{(A, \text{d})})}(N, A^\vee [k]) = 0$

for all acyclic differential graded $A$ -modules $N$ and all $k$ . This follows from Lemma 22.19.3 and the fact that $(-)^\vee$ is an exact functor. $\square$

The Stacks project

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