On Chow Rings of Fine Moduli Spaces of Modules

Let M be a complete nonsingular fine moduli space of modules over an algebra S. A set of conditions is given for the Chow ring of M to be generated by the Chern classes of certain universal bundles occurring in a projective resolution of the universal S-module on M . This result is then applied to the varieties GT parametrizing homogeneous ideals of k[x, y] of Hilbert function T , to moduli spaces of representations of quivers, and finally to moduli spaces of sheaves on P, reinterpreting a result of Ellingsrud and Strømme. In a recent paper [ES] Ellingsrud and Strømme identified a set of generators of the Chow ring of the moduli space of stable sheaves of given rank and Chern classes on P2 (in the case where the moduli space is smooth and projective). In this paper we formulate a part of their argument as a general theorem about fine moduli spaces of modules over an associative algebra. This provides a more widely applicable method for showing that the Chow ring of a fine moduli space is generated by the Chern classes of appropriate universal sheaves. In particular we apply the method to verify a conjecture of Iarrobino and Yaméogo concerning the Chow rings of the varieties GT parametrizing homogeneous ideals in k[x, y] with a given Hilbert function. We also verify a conjecture of the first author concerning Chow rings of moduli spaces of representations of quivers. Let S be an associative algebra over an algebraically closed field k. By convention we will consider only left S-modules in this paper. A flat family of S-modules over a k-scheme X is a sheaf F of S ⊗ OX -modules on X, quasi-coherent and flat over OX . At a (closed) point x ∈ X the fiber F(x) is an S-module. If C is a class of S-modules, then a fine moduli space for C is a scheme M equipped with a flat family U all of whose fibers are in C and with the usual universal property. Our general theorem is the following: Theorem 1. Let C be a class of S-modules, and M a fine moduli space for C which is a complete nonsingular variety. Suppose further that (i) If E ∈ C, then HomS(E,E) ∼= k, Ext 1 S(E,E) ∼= T[E]M , and Ext p S(E,E) = 0 for p ≥ 2; Supported by a research grant from the SERC. Supported in part by NSA research grant MDA904-92-H-3009.