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Using the notion of flat covers and proper flat resolutions, we study modules with periodic flat resolutions. It follows that equivalently, we may study modules with periodic homology. We specialize our results to the category of modules over integral group ring $\mathbb{Z}\Gamma$, where $\Gamma$ is an arbitrary group. Among other results, we show that if a group $\Gamma$ is in a certain class of groups then $\Gamma$ has periodic homology of period $q$ after some steps with the periodicity isomorphisms of homology groups induced by the cap product with an element in $H^{q}(\Gamma,C)$, where $C$ is the cotorsion envelope of the trivial $\Gamma$-module $\mathbb{Z}$, if and only if it has periodic cohomology of period $q$ after some steps with the periodicity isomorphisms of cohomology groups induced by the cup product with an element in $H^{q}(\Gamma,\mathbb{Z})$.
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