#### Volume 21, issue 4 (2021)

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

### David Hemminger

Algebraic & Geometric Topology 21 (2021) 1881–1910
##### Abstract

For $X$ a smooth scheme acted on by a linear algebraic group $G$ and $p$ a prime, the equivariant Chow ring ${CH}_{G}^{\ast }\left(X\right)\otimes {\mathbb{𝔽}}_{p}$ is an unstable algebra over the Steenrod algebra. We compute Lannes’s $T$–functor applied 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 $n$–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 algebrogeometric version of Quillen’s stratification theorem proved by Yagita and Totaro.

##### Keywords
Chow ring, equivariant Chow ring, group cohomology, Steenrod algebra, unstable modules, unstable algebras, $T$–functor
##### Mathematical Subject Classification 2010
Primary: 14C15, 55S10
Secondary: 14L30, 55N91