Lannes's $T$-functor and equivariant Chow rings

TL;DR

This paper computes Lannes's T-functor on equivariant Chow rings with mod p coefficients, applying it to localize these rings and generalize Quillen's stratification theorem in algebraic geometry.

Contribution

It introduces the computation of Lannes's T-functor on equivariant Chow rings and applies it to confirm a conjecture of Totaro, extending Quillen's stratification theorem.

Findings

Computed T-functor on equivariant Chow rings over finite fields.

Localized Chow rings away from n-nilpotent modules, affirming Totaro's conjecture.

Generalized Quillen's stratification theorem to algebraic geometry setting.

Abstract

For $X$ a smooth scheme acted on by a linear algebraic group $G$ and $p$ a prime, the equivariant Chow ring $CH^*_G(X)\otimes \mathbb{F}_p$ is an unstable algebra over the Steenrod algebra. We compute Lannes's $T$-functor applied to $CH^*_G(X)\otimes \mathbb{F}_p$. As an application, we compute the localization of $CH^*_G(X)\otimes \mathbb{F}_p$ away from $n$-nilpotent modules over the Steenrod algebra, affirming a conjecture of Totaro as a special case. The case when $X$ is a point and $n = 1$ generalizes and recovers an algebro-geometric version of Quillen's stratification theorem proved by Yagita and Totaro.