On the cohomology and the Chow ring of the classifying space of PGL_p

TL;DR

This paper studies the algebraic structures of the classifying space of PGL_p, determining its additive cohomology and Chow rings, and linking the multiplicative structure to invariant theory for odd primes.

Contribution

It fully determines the additive structure of the cohomology and Chow rings of B PGL_p and connects the multiplicative structure to invariant theory, advancing understanding of these algebraic invariants.

Findings

Complete determination of the additive structure of the cohomology ring.

Complete determination of the additive structure of the Chow ring.

Reduction of the multiplicative structure problem to invariant theory.

Abstract

We investigate the integral cohomology ring and the Chow ring of the classifying space of the complex projective linear group PGL_p, when p is an odd prime. In particular, we determine its additive structure completely, and we reduce the problem of determing its multiplicative structure to a problem in invariant theory.