Cohomology of the moduli stack of algebraic vector bundles

TL;DR

This paper proves that for a broad class of cohomology theories, the cohomology of the moduli stack of vector bundles is generated by Chern classes, extending known results to a more general setting.

Contribution

It establishes a general freeness result for cohomology of the moduli stack of vector bundles using sheaves of ring spectra with specific properties.

Findings

Cohomology of the moduli stack is freely generated by Chern classes.

Applicable to all multiplicative localizing invariants.

Generalizes previous results to a broader class of cohomology theories.

Abstract

Let $\mathscr{V}\mathrm{ect}_n$ 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 $E^*(\mathscr{V}\mathrm{ect}_{n,S})$ is freely generated by Chern classes $c_1,\dotsc,c_n$ over $E^*(S)$ for any scheme $S$. Examples include all multiplicative localizing invariants.