The Stacks project

Remark 22.36.2. Let $(A, \text{d})$ be a differential graded algebra. Let us say a differential graded $A$-module $M$ is finite if $M$ is generated, as a right $A$-module, by finitely many elements. If $P$ is a differential graded $A$-module which is finite graded projective, then we can ask: Does $P$ give a compact object of $D(A, \text{d})$? Presumably, this is not true in general, but we do not know a counter example. However, if $P$ is also an object of the category $\mathcal{D}$ of Remark 22.36.1, then this is the case (this follows from the fact that direct sums in $D(A, \text{d})$ are given by direct sums of modules; details omitted).

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