Definition 42.2.1. Let $R$ be a ring.

A

*$2$-periodic complex*over $R$ is given by a quadruple $(M, N, \varphi , \psi )$ consisting of $R$-modules $M$, $N$ and $R$-module maps $\varphi : M \to N$, $\psi : N \to M$ such that\[ \xymatrix{ \ldots \ar[r] & M \ar[r]^\varphi & N \ar[r]^\psi & M \ar[r]^\varphi & N \ar[r] & \ldots } \]is a complex. In this setting we define the

*cohomology modules*of the complex to be the $R$-modules\[ H^0(M, N, \varphi , \psi ) = \mathop{\mathrm{Ker}}(\varphi )/\mathop{\mathrm{Im}}(\psi ) \quad \text{and}\quad H^1(M, N, \varphi , \psi ) = \mathop{\mathrm{Ker}}(\psi )/\mathop{\mathrm{Im}}(\varphi ). \]We say the $2$-periodic complex is

*exact*if the cohomology groups are zero.A

*$(2, 1)$-periodic complex*over $R$ is given by a triple $(M, \varphi , \psi )$ consisting of an $R$-module $M$ and $R$-module maps $\varphi : M \to M$, $\psi : M \to M$ such that\[ \xymatrix{ \ldots \ar[r] & M \ar[r]^\varphi & M \ar[r]^\psi & M \ar[r]^\varphi & M \ar[r] & \ldots } \]is a complex. Since this is a special case of a $2$-periodic complex we have its

*cohomology modules*$H^0(M, \varphi , \psi )$, $H^1(M, \varphi , \psi )$ and a notion of exactness.

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