Homotopy quotients and comodules of supercommutative Hopf algebras

Abstract

We study induced model structures on Frobenius categories. In particular we consider the case where $\mathcal{C}$ is the category of comodules of a supercommutative Hopf algebra $A$ over a field $k$. Given a graded Hopf algebra quotient $A\to B$ satisfying some finiteness conditions, the Frobenius tensor category $\mathcal{D}$ of graded $B$-comodules with its stable model structure induces a monoidal model structure on $\mathcal{C}$. We consider the corresponding homotopy quotient $\gamma :\mathcal{C}\to Ho\mathcal{C}$ and the induced quotient $\mathcal{T}\to Ho\mathcal{T}$ for the tensor category $\mathcal{T}$ of finite dimensional $A$-comodules. Under some mild conditions we prove vanishing and finiteness theorems for morphisms in $Ho\mathcal{T}$. We apply these results in the $Rep(GL(m|n))$-case and study its homotopy category $Ho\mathcal{T}$.

v3: Minor changes, added lemma 6.2, assumed char 0 even in part 1. v2: Removed a few typos. v1: 77 pages