Contramodules for algebraic groups: Induction and projective covers

In this paper we investigate contramodules for algebraic groups. Namely, we give contramodule analogs to two ${20}^{\text{th}}$ century results about comodules. Firstly, we show that induction of contramodules over coordinate rings of algebraic groups is exact if and only if the associated quotient variety is affine. Secondly, we give an inverse limit theorem for constructing projective covers of simple G-modules using G-structures of projective covers of simple modules of the first Frobenius kernel, $G_1$. We conclude by showing that the inverse limit theorem is a special case of a more general phenomenon between injective covers in $k\left[G\right]$-Comod and projective covers in $k\left[G\right]$-Contra.