The Stacks project

Remark 15.59.11. In fact, we can do better than Lemma 15.59.10. 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 $D(R)$.

Comments (0)

There are also:

  • 4 comment(s) on Section 15.59: Derived tensor product

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09PB. Beware of the difference between the letter 'O' and the digit '0'.