The Stacks project

Remark 22.36.1. Let $(A, \text{d})$ be a differential graded algebra. Is there a characterization of those differential graded $A$-modules $P$ for which we have

for all differential graded $A$-modules $M$? Let $\mathcal{D} \subset K(A, \text{d})$ be the full subcategory whose objects are the objects $P$ satisfying the above. Then $\mathcal{D}$ is a strictly full saturated triangulated subcategory of $K(A, \text{d})$. If $P$ is projective as a graded $A$-module, then to see where $P$ is an object of $\mathcal{D}$ it is enough to check that $\mathop{\mathrm{Hom}}\nolimits _{K(A, \text{d})}(P, M) = 0$ whenever $M$ is acyclic. However, in general it is not enough to assume that $P$ is projective as a graded $A$-module. Example: take $A = R = k[\epsilon ]$ where $k$ is a field and $k[\epsilon ] = k[x]/(x^2)$ is the ring of dual numbers. Let $P$ be the object with $P^ n = R$ for all $n \in \mathbf{Z}$ and differential given by multiplication by $\epsilon$. Then $\text{id}_ P \in \mathop{\mathrm{Hom}}\nolimits _{K(A, \text{d})}(P, P)$ is a nonzero element but $P$ is acyclic.