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

For $X$ a smooth scheme acted on by a linear algebraic group $G$ and $p$ a prime, the equivariant Chow ring $C{H}_G^{\ast }(X)\otimes {\mathbb{F}}_p$ is an unstable algebra over the Steenrod algebra. We compute Lannes’s $T$-functor applied to $C{H}_G^{\ast }(X)\otimes {\mathbb{F}}_p$. As an application, we compute the localization of $C{H}_G^{\ast }(X)\otimes {\mathbb{F}}_p$ 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.