Chow-Witt Rings of Classifying Spaces of Products of Multiplicative and Cyclic Groups

TL;DR

This paper computes the Chow-Witt rings and I^j-cohomology of classifying spaces of roots of unity, multiplicative groups, and their products, extending previous strategies to new cases involving products of these algebraic groups.

Contribution

It extends the computation of Chow-Witt rings to products of classifying spaces of roots of unity and multiplicative groups, generalizing prior work for even n.

Findings

Computed Chow-Witt rings of Bμ_n for all n ≥ 1.

Determined I^j-cohomology and Chow-Witt rings of P^q x P^r and BG_m x BG_m.

Extended existing methods to new classes of algebraic groups and their products.

Abstract

We compute the total Chow-Witt rings of the classifying space B\mu_n of the roots of unity, as well as the products BG_m x B\mu_n and B\mu_m x B\mu_n for all m,n greater than or equal to 1 based on the strategy by di Lorenzo and Mantovani (2023) for B\mu_n with n even. Moreover we compute the total I^j-cohomology and Chow-Witt rings of P^q x P^r for all q,r greater than or equal to 1 and of BG_m x BG_m.