Niveau: Supérieur, Doctorat, Bac+8
NOTIONS OF PURITY AND THE COHOMOLOGY OF QUIVER MODULI M. BRION AND R. JOSHUA 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. 0. Introduction Consider a scheme X of finite type over a finite field Fq. Then the number of points of X that are rational over a finite extension Fqn is expressed by the trace formula (0.0.1) |X(Fqn)| = ∑ i≥0 (?1)i Tr ( Fn,H ic(X¯,Ql) ) where X¯ = X ? Spec Fq Spec F¯q and F denotes the Frobenius morphism of X¯. The results of Deligne (see [De74a, De77, De80]) show that every eigenvalue of F on H ic(X¯,Ql) has absolute value qw/2 for some non-negative integer w ≤ i.
- pure variety
- map ?
- then rmf?
- geometric invariant
- also connected
- etale map
- equivariant local
- theory quotients
- connected
- sequences