Lemma 15.28.10. Let $R$ be a ring. Let $\varphi : E \to R$ be an $R$-module map. Let $f, g \in R$. Set $E' = E \oplus R$ and define $\varphi '_ f, \varphi '_ g, \varphi '_{fg} : E' \to R$ by $\varphi$ on $E$ and multiplication by $f, g, fg$ on $R$. The complex $K_\bullet (\varphi '_{fg})$ is homotopy equivalent to the cone of a map of complexes

$K_\bullet (\varphi '_ f)[1] \longrightarrow K_\bullet (\varphi '_ g).$

Proof. By Lemma 15.28.7 the complex $K_\bullet (\varphi '_ f)$ is isomorphic to the cone of multiplication by $f$ on $K_\bullet (\varphi )$ and similarly for the other two cases. Hence the lemma follows from Lemma 15.28.9. $\square$

Comment #7391 by Elie Studnia on

Doesn't lemma 062A imply that $K_{\bullet}(\varphi_{fg}')$ should be homotopy equivalent (rather than isomorphic) to the cone of some map of complexes $K_{\bullet}(\varphi_f')[1] \rightarrow K_{\bullet}(\varphi_g')$?

There are also:

• 2 comment(s) on Section 15.28: The Koszul complex

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).