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

$e_ R(M, N, \varphi , \psi ) = \text{length}_ R(H^0(M, N, \varphi , \psi )) - \text{length}_ R(H^1(M, N, \varphi , \psi ))$

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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