The following lemma produces our distinguished triangles.
Lemma 22.8.1. 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. The triangle with $\delta $ as in Lemma 22.7.2 is, up to canonical isomorphism in $K(\text{Mod}_{(A, \text{d})})$, independent of the choices made in Lemma 22.7.2.
22.8.1.1
\begin{equation} \label{dga-equation-triangle-associated-to-admissible-ses} K \to L \to M \xrightarrow {\delta } K[1] \end{equation}
Proof. Namely, let $(s', \pi ')$ be a second choice of splittings as in Lemma 22.7.2. Then we claim that $\delta $ and $\delta '$ are homotopic. Namely, write $s' = s + \alpha \circ h$ and $\pi ' = \pi + g \circ \beta $ for some unique homomorphisms of $A$-modules $h : M \to K$ and $g : M \to K$ of degree $-1$. Then $g = -h$ and $g$ is a homotopy between $\delta $ and $\delta '$. The computations are done in the proof of Homology, Lemma 12.14.12. $\square$
Definition 22.8.2. Let $(A, \text{d})$ be a differential graded algebra.
If $0 \to K \to L \to M \to 0$ is an admissible short exact sequence of differential graded $A$-modules, then the triangle associated to $0 \to K \to L \to M \to 0$ is the triangle (22.8.1.1) of $K(\text{Mod}_{(A, \text{d})})$.
A triangle of $K(\text{Mod}_{(A, \text{d})})$ is called a distinguished triangle if it is isomorphic to a triangle associated to an admissible short exact sequence of differential graded $A$-modules.
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)