Reynolds Operator on functors

Abstract

Let $G=\mathrm{Spec}A$ be an affine $R$-monoid scheme. We prove that the category of dual functors (over the category of commutative $R$-algebras) of $G$-modules is equivalent to the category of dual functors of ${\mathcal{A}}^{\ast }$-modules. We prove that $G$ is invariant exact if and only if $A^{\ast }=R\times B^{\ast }$ as $R$-algebras and the first projection $A^{\ast }\to R$ is the unit of $A$. If $\mathbb{M}$ is a dual functor of $G$-modules and $w_G:=(1,0)\in R\times B^{\ast }=A^{\ast }$, we prove that ${\mathbb{M}}^G=w_G\cdot \mathbb{M}$ and $\mathbb{F}=w_G\cdot \mathbb{M}\oplus (1-w_G)\cdot \mathbb{M}$; hence, the Reynolds operator can defined on $\mathcal{M}$.

notations have been improved

Cited by in corpus (2)

• A Characterization of Linearly Semisimple Groups
• Invariant integral on classical groups and algebraic harmonic analysis