Notions of Purity and the Cohomology of Quiver Moduli

Abstract

We explore several variations of the notion of purity for the action of Frobenius on schemes defined over finite fields. In particular, we study how these notions are preserved under certain natural operations like quotients for principal bundles and also geometric quotients for reductive group actions. We then apply these results to study the cohomology of quiver moduli. We prove that a natural stratification of the space of representations of a quiver with a fixed dimension vector is equivariantly perfect and from it deduce that each of the l-adic cohomology groups of the quiver moduli space is strongly pure.In [5], the first-named author and Peyre studied in detail the properties of the counting function n → |X(F q n )|, when X is a homogeneous variety under a linear algebraic group (all defined over F q ). For this, they introduced the notion of a weakly pure variety X, by requesting that all eigenvalues of F in H * c (X, Q l ) are of the form ζq j , where ζ is a root of unity, and j a nonnegative integer. This implies that the counting function of X is a periodic polynomial with integer coefficients, i.e. there exist a positive integer N and polynomials P 0 (t), . . . , P N −1 (t) in Z[t] such that |X(F q n )| = P r (q n ) whenever n ≡ r (mod N ). They also showed that homogeneous varieties under linear algebraic groups are weakly pure.The present paper arose out of an attempt to study the notion of weak purity in more detail, and to see how it behaves with respect to torsors and geometric invariant theory quotients. While applying this notion to moduli spaces of quiver representations, it also became clear that they satisfy a stronger notion of purity, which in fact differs from the notion of a strongly pure variety, introduced in [5] as a technical device. Thus, we were led to define weak and strong purity in a more general setting, and to modify the notion of strong purity so that it applies to GIT quotients.Here is an outline of the paper. In Sec. 2, we introduce a notion of weak purity for equivariant local systems (generalizing that in [5] where only the constant local system is considered) and a closely related notion of strong purity. The basic definitions are in Definition 2.6. Then we study how these notions behave with respect to torsors and certain associated fibrations. The main result is the following.Theorem 1.1 (See Theorem 2.10). Let π : X → Y denote a torsor under a linear algebraic group G, all defined over F q . Let C X denote a class of G-equivariant l-adic local systems on X, and C Y a class of G-equivariant l-adic local systems on Y, where Y is provided with the trivial G-action.As explained just after Theorem 2.10, the above theorem applies with the following choice of the classes C X and C Y : Let C X denote the class of G-equivariant l-adic local systems on X obtained as split summands of ρ X * (Q ⊕ n l ) for some n > 0, where ρ X : X → X is a G-equivariant finiteétale map and Q ⊕ n l is the constant local system of rank n on X , and define C Y similarly.In Sec. 3, we first apply so...