Graded 1-parameter subgroups and detection properties

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 $V_r^{\ast }(G)$. We give a natural homomorphism of bigraded $\mathrm{k}$-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 $V_r^{\ast }(G)$. 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.