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.

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