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