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We explore several variations of the notion of purity for the action ofFrobenius on schemes defined over finite fields. In particular, we study howthese notions are preserved under certain natural operations like quotients forprincipal bundles and also geometric quotients for reductive group actions. Wethen apply these results to study the cohomology of quiver moduli. We provethat a natural stratification of the space of representations of a quiver witha fixed dimension vector is equivariantly perfect and from it deduce that eachof the $l$-adic cohomology groups of the quiver moduli space is strongly pure.

Notions of purity and the cohomology of quiver moduli