Graded 1-parameter subgroups and detection properties

TL;DR

This paper develops a theory of graded 1-parameter subgroups for graded group schemes over fields of positive characteristic, linking cohomology to coordinate rings and suggesting a potential Quillen-type detection result.

Contribution

It introduces graded 1-parameter subgroups and constructs a homomorphism connecting cohomology to coordinate rings, advancing understanding of graded group schemes.

Findings

The homomorphism   o oldsymbol{ m k}[V_r^* (G)] is an F-monomorphism for certain schemes.

Provides evidence for a Quillen-type detection property in the graded setting.

Establishes a framework for analyzing cohomology via graded 1-parameter subgroups.

Abstract

We use tools of representation theory to get a better understanding of the cohomology of graded group schemes. For that, we focus our attention on the case in which the base field is of characteristic $p > 0$. Using as inspiration the work of Friedlander, et al, we build the theory of graded $1$-parameter subgroups denoted by $V_r^*(G)$. We give a natural homomorphism of bigraded $\boldsymbol{\rm k}$-algebras $\psi: H^{*, *}(G, \boldsymbol{\rm k}) \to \boldsymbol{\rm k}[V_r^* (G)],$ where $\boldsymbol{\rm k}[V_r^* (G)]$ is the bigraded coordinate ring for $V_r^*(G)$. We show that $\psi$ is an $F$-monomorphism for a class of graded group schemes. This provides evidence that with the appropriate detection property, a Quillen-type result could exist for graded group schemes.