Notes on Chow rings of G/B and BG

TL;DR

This paper investigates the relationship between Chow rings of certain algebraic varieties associated with a compact Lie group and its maximal torus, revealing near-exactness properties and connections to the generalized Rost motive.

Contribution

It extends classical cohomological results to Chow rings of twisted forms of G/T, analyzing their near-exactness and relation to the generalized Rost motive.

Findings

The composition of Chow ring maps is near to exact but not exact.

The difference in exactness relates to the generalized Rost motive.

Chow rings of twisted forms exhibit near-exactness properties similar to cohomology.

Abstract

Let $G$ be a compact Lie group and $T$ its maximal torus. The composition of maps $ H^*(BG)\to H^*(BT) \to H^*(G/T)$ is zero for positive degree, while it is far from exact. We change $H^*(G/T)$ by Chow ring $CH^*(X)$ for $X$ some twisted form of $G/T$, and change $H^*(BG)$ by $CH^*(BG)$. Then we see that it becomes near to exact but still not exact, in general. We also see that the difference for exactness relates to the generalized Rost motive in $X$.