Cohomology of the moduli stack of algebraic vector bundles

Abstract

Let $\mathcal{V}{\mathrm{ect}}_n$ be the moduli stack of vector bundles of rank $n$ on schemes. We prove that, if $E$ is a Zariski sheaf of ring spectra which is equipped with finite quasi-smooth transfers and satisfies the projective bundle formula, then $E^{\ast }(\mathcal{V}{\mathrm{ect}}_{n,S})$ is freely generated by Chern classes $c_1,\dots ,c_n$ over $E^{\ast }(S)$ for any scheme $S$. Examples include all multiplicative localizing invariants.

17 pages. Final version + footnotes