Homotopy Quotients and Comodules of Supercommutative Hopf Algebras

Abstract

We study model structures on the category of comodules of a supercommutative Hopf algebra A over fields of characteristic 0. Given a graded Hopf algebra quotient A→B satisfying some finiteness conditions, the Frobenius tensor category D of graded B-comodules with its stable model structure induces a monoidal model structure on C. We consider the corresponding homotopy quotient γ:C→HoC and the induced quotient T→HoT for the tensor category T of finite dimensional A-comodules. Under some mild conditions we prove vanishing and finiteness theorems for morphisms in HoT. We apply these results in the Rep(GL(m|n))-case and study its homotopy category HoT associated to the parabolic subgroup of upper triangular block matrices. We construct cofibrant replacements and show that the quotient of HoT by the negligible morphisms is again the representation category of a supergroup scheme.