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 $$A \rightarrow B$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>→</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:math> satisfying some finiteness conditions, the Frobenius tensor category $${\mathcal {D}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> of graded B -comodules with its stable model structure induces a monoidal model structure on $${\mathcal {C}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> . We consider the corresponding homotopy quotient $$\gamma : {\mathcal {C}} \rightarrow Ho {\mathcal {C}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>:</mml:mo> <mml:mi>C</mml:mi> <mml:mo>→</mml:mo> <mml:mi>H</mml:mi> <mml:mi>o</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> and the induced quotient $${\mathcal {T}} \rightarrow Ho {\mathcal {T}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>→</mml:mo> <mml:mi>H</mml:mi> <mml:mi>o</mml:mi> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> for the tensor category $${\mathcal {T}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> of finite dimensional A -comodules. Under some mild conditions we prove vanishing and finiteness theorems for morphisms in $$Ho {\mathcal {T}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mi>o</mml:mi> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> . We apply these results in the Rep ( GL ( m | n ))-case and study its homotopy category $$Ho {\mathcal {T}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mi>o</mml:mi> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> associated to the parabolic subgroup of upper triangular block matrices. We construct cofibrant replacements and show that the quotient of $$Ho{\mathcal {T}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mi>o</mml:mi> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> by the negligible morphisms is again the representation category of a supergroup scheme.