Lannes's T –functor and equivariant Chow rings

Abstract

For $X$a smooth scheme acted on by a linear algebraic group$G$ and$p$ a prime, theequivariant Chow ring ${CH}_G^{\ast }\left(X\right)\otimes {\mathbb{𝔽}}_p$is an unstable algebra over the Steenrod algebra. We compute Lannes’s$T$ –functorapplied to ${CH}_G^{\ast }\left(X\right)\otimes {\mathbb{𝔽}}_p$.As an application, we compute the localization of${CH}_G^{\ast }\left(X\right)\otimes {\mathbb{𝔽}}_p$ away from$n$–nilpotentmodules over the Steenrod algebra, affirming a conjecture of Totaro as a special case. Thecase when $X$ isa point and $n=1$generalizes and recovers an algebrogeometric version of Quillen’s stratificationtheorem proved by Yagita and Totaro.