Lemma 22.7.2 (09JU)—The Stacks project

Processing math: 100%

Table of contents
Part 1: Preliminaries
Chapter 22: Differential Graded Algebra
Section 22.7: Admissible short exact sequences
Lemma 22.7.2 (cite)

numberstags

Lemma 22.7.2. Let $(A, \text{d})$ be a differential graded algebra. Let $0 \to K \to L \to M \to 0$ be an admissible short exact sequence of differential graded $A$ -modules. Let $s : M \to L$ and $\pi : L \to K$ be splittings such that $\mathop{\mathrm{Ker}}(\pi ) = \mathop{\mathrm{Im}}(s)$ . Then we obtain a morphism

$\delta = \pi \circ \text{d}_ L \circ s : M \to K[1]$

of $\text{Mod}_{(A, \text{d})}$ which induces the boundary maps in the long exact sequence of cohomology (22.4.2.1).

Proof. The map $\pi \circ \text{d}_ L \circ s$ is compatible with the $A$ -module structure and the gradings by construction. It is compatible with differentials by Homology, Lemmas 12.14.10. Let $R$ be the ring that $A$ is a differential graded algebra over. The equality of maps is a statement about $R$ -modules. Hence this follows from Homology, Lemmas 12.14.10 and 12.14.11. $\square$

Comments (0)

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).

Comments (0)

Post a comment