Lemma 42.2.4 (0EA8)—The Stacks project

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. Observe that on the category of $2$ -periodic complexes with $M$ , $N$ of finite length the quantity “ $\text{length}_ R(M) - \text{length}_ R(N)$ ” is additive in short exact sequences (precise statement left to the reader). Consider the short exact sequence

$0 \to (M, \mathop{\mathrm{Im}}(\varphi ), \varphi , 0) \to (M, N, \varphi , \psi ) \to (0, N/\mathop{\mathrm{Im}}(\varphi ), 0, 0) \to 0$

The initial remark combined with the additivity of Lemma 42.2.3 reduces us to the cases (a) $M = 0$ and (b) $\varphi$ is surjective. We leave those cases to the reader. $\square$

Comments (2)

Comment #8258 by Runlei Xiao on March 14, 2023 at 05:21

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.

Comment #8900 by Stacks project on April 09, 2024 at 17:05

Thanks! The statement is correct but the proof was not. Now fixed here.

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