The Stacks project

Lemma 22.13.1. Let $(A, \text{d})$ be a differential graded algebra. Let $P$ be a differential graded $A$-module. If $F_\bullet $ is a filtration as in property (P), then we obtain an admissible short exact sequence

\[ 0 \to \bigoplus \nolimits F_ iP \to \bigoplus \nolimits F_ iP \to P \to 0 \]

of differential graded $A$-modules.

Proof. The second map is the direct sum of the inclusion maps. The first map on the summand $F_ iP$ of the source is the sum of the identity $F_ iP \to F_ iP$ and the negative of the inclusion map $F_ iP \to F_{i + 1}P$. Choose homomorphisms $s_ i : F_{i + 1}P \to F_ iP$ of graded $A$-modules which are left inverse to the inclusion maps. Composing gives maps $s_{j, i} : F_ jP \to F_ iP$ for all $j > i$. Then a left inverse of the first arrow maps $x \in F_ jP$ to $(s_{j, 0}(x), s_{j, 1}(x), \ldots , s_{j, j - 1}(x), 0, \ldots )$ in $\bigoplus F_ iP$. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 22.13: P-resolutions

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Your email address will not be published.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 09KL. Beware of the difference between the letter 'O' and the digit '0'.