The Stacks project

Lemma 22.13.2. Let $(A, \text{d})$ be a differential graded algebra. Let $P$ be a differential graded $A$-module with property (P). Then

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

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 $P$ as in property (P). The short exact sequence of Lemma 22.13.1 produces a distinguished triangle. Hence by Derived Categories, Lemma 13.4.2 it suffices to show that

\[ \mathop{\mathrm{Hom}}\nolimits _{K(\text{Mod}_{(A, \text{d})})}(F_ iP, N) = 0 \]

for all acyclic differential graded $A$-modules $N$ and all $i$. Each of the differential graded modules $F_ iP$ has a finite filtration by admissible monomorphisms, whose graded pieces are direct sums of shifts $A[k]$. Thus it suffices to prove that

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

for all acyclic differential graded $A$-modules $N$ and all $k$. This follows from Lemma 22.11.2. $\square$


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