Definition 42.2.2. Let $(M, N, \varphi , \psi )$ be a $2$-periodic complex over a ring $R$ whose cohomology modules have finite length. In this case we define the *multiplicity* of $(M, N, \varphi , \psi )$ to be the integer

In the case of a $(2, 1)$-periodic complex $(M, \varphi , \psi )$, we denote this by $e_ R(M, \varphi , \psi )$ and we will sometimes call this the *(additive) Herbrand quotient*.

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