The Stacks project

Remark 22.22.5. Let $R$ be a ring. Let $(A, \text{d})$ be a differential graded $R$-algebra. Using P-resolutions we can sometimes reduce statements about general objects of $D(A, \text{d})$ to statements about $A[k]$. Namely, let $T$ be a property of objects of $D(A, \text{d})$ and assume that

  1. if $K_ i$, $i \in I$ is a family of objects of $D(A, \text{d})$ and $T(K_ i)$ holds for all $i \in I$, then $T(\bigoplus K_ i)$,

  2. if $K \to L \to M \to K[1]$ is a distinguished triangle of $D(A, \text{d})$ and $T$ holds for two, then $T$ holds for the third object, and

  3. $T(A[k])$ holds for all $k \in \mathbf{Z}$.

Then $T$ holds for all objects of $D(A, \text{d})$. This is clear from Lemmas 22.20.1 and 22.20.4.

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