Lemma 42.2.4. Let $R$ be a ring. If $(M, N, \varphi , \psi )$ is a $2$-periodic complex such that $M$, $N$ have finite length, then $e_ R(M, N, \varphi , \psi ) = \text{length}_ R(M) - \text{length}_ R(N)$. In particular, if $(M, \varphi , \psi )$ is a $(2, 1)$-periodic complex such that $M$ has finite length, then $e_ R(M, \varphi , \psi ) = 0$.

Proof. This follows from the additity of Lemma 42.2.3 and the short exact sequence $0 \to (M, 0, 0, 0) \to (M, N, \varphi , \psi ) \to (0, N, 0, 0) \to 0$. $\square$

Comment #8258 by Runlei Xiao on

In Lemma 42.2.4. the $(M, N, \varphi , \psi )$ should change to $(M,N,0,0)$ if not the following exact sequence in your proof will be not well-defined.

There are also:

• 2 comment(s) on Section 42.2: Periodic complexes and Herbrand quotients

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.

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 0EA8. Beware of the difference between the letter 'O' and the digit '0'.