The Stacks project

Remark 15.58.13. In fact, we can do better than Lemma 15.58.12. Namely, we can find a quasi-isomorphism $P^\bullet \to M^\bullet $ where $P^\bullet $ is a complex of $R$-modules endowed with a filtration

\[ 0 = F_{-1}P^\bullet \subset F_0P^\bullet \subset F_1P^\bullet \subset \ldots \subset P^\bullet \]

by subcomplexes such that

  1. $P^\bullet = \bigcup F_ pP^\bullet $,

  2. the inclusions $F_ iP^\bullet \to F_{i + 1}P^\bullet $ are termwise split injections,

  3. the quotients $F_{i + 1}P^\bullet /F_ iP^\bullet $ are isomorphic to direct sums of shifts $R[k]$ (as complexes, so differentials are zero).

This will be shown in Differential Graded Algebra, Lemma 22.20.4. Moreover, given such a complex we obtain a distinguished triangle

\[ \bigoplus F_ iP^\bullet \to \bigoplus F_ iP^\bullet \to M^\bullet \to \bigoplus F_ iP^\bullet [1] \]

in $D(R)$. Using this we can sometimes reduce statements about general complexes to statements about $R[k]$ (this of course only works if the statement is preserved under taking direct sums). More precisely, let $T$ be a property of objects of $D(R)$. Suppose that

  1. if $K_ i \in D(R)$, $i \in I$ is a family of objects with $T(K_ i)$ for all $i \in I$, then $T(\bigoplus K_ i)$,

  2. if $K \to L \to M \to K[1]$ is a distinguished triangle and $T$ holds for two, then $T$ holds for the third object,

  3. $T(R[k])$ holds for all $k$.

Then $T$ holds for all objects of $D(R)$.

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