Homotopy quotients and comodules of supercommutative Hopf algebras

TL;DR

This paper explores the homotopy theory of comodules over supercommutative Hopf algebras, establishing model structures and analyzing the resulting homotopy categories, with applications to supergroup representations.

Contribution

It introduces a framework for inducing monoidal model structures on comodules of supercommutative Hopf algebras and studies their homotopy quotients, extending understanding of supergroup representation categories.

Findings

Vanishing and finiteness theorems for morphisms in homotopy categories

Construction of monoidal model structures on comodules

Application to the representation category of GL(m|n)

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 $Ho \mathcal{T}$.