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 \(A \rightarrow B\) 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 \(\gamma : {\mathcal {C}} \rightarrow Ho {\mathcal {C}}\) and the induced quotient \({\mathcal {T}} \rightarrow Ho {\mathcal {T}}\) 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 \(Ho {\mathcal {T}}\) associated to the parabolic subgroup of upper triangular block matrices. We construct cofibrant replacements and show that the quotient of \(Ho{\mathcal {T}}\) by the negligible morphisms is again the representation category of a supergroup scheme.