The intersection theory of the moduli stack of vector bundles on P1

We determine the integral Chow and cohomology rings of the moduli stack B-r,B-d of rank r, degree d vector bundles on P-1-bundles. We work over a field k of arbitrary characteristic. We first show that the rational Chow ring A(Q)*(B-r,B-d) is a free Q-algebra on 2r + 1 generators. The isomorphism class of this ring happens to be independent of d. Then, we prove that the integral Chow ring A*(B-r,B-d) is torsion-free and provide multiplicative generators for A*(B-r,B-d) as a subring of A(Q)*(B-r,B-d). From this description, we see that A*(B-r,B-d) is not finitely generated as a Z-algebra. Finally, when k = C, the cohomology ring of B-r,B-d is isomorphic to its Chow ring.