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

In Lemma 42.2.4. the should change to 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

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