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