Projectivity of Hopf algebras over subalgebras with semilocal central localizations

Abstract

The purpose of this paper is to extend the class of pairs A, H where H is a Hopf algebra over a field and A a right coideal subalgebra for which H is proved to be either projective or flat as an A-module. The projectivity is obtained under the assumption that H is residually finite dimensional, A has semilocal localizations with respect to a central subring, and there exists a Hopf subalgebra B of H such that the antipode of B is bijective and B is a finitely generated A-module. The flatness of H over A is shown to hold when H is a directed union of residually finite dimensional Hopf subalgebras, and there exists a Hopf subalgebra of H whose center contains A. More general projectivity and flatness results are established for (co)equivariant modules over an H-(co)module algebra under similar assumptions.

Flatness of Noetherian Hopf algebras over coideal subalgebras

“…Many classes of Hopf algebras satisfy this condition. Any residually finite dimensional Hopf algebra is flat over central right coideal subalgebras, and there are considerably better results in the case of Hopf subalgebras (see [18]). This shows once again that dealing with coideal subalgebras incurs extra complications.…”

“…The restriction to central subalgebras is clearly a serious limitation when it comes to noncommutative Hopf algebras. Unfortunately, the technique of central localizations used in [18] is not applicable in other situations. The main result of the present paper is Theorem 4.5.…”

Finiteness of the number of coideal subalgebras

“…Over coideal subalgebras the flatness property turns out to be more elusive. There are several results in which faithful flatness over some coideal subalgebras has been proved, but this requires usually more severe restrictions than in the case of Hopf subalgebras (see [14], [20], [31]).…”

“…Faithful flatness and projectivity over all Hopf subalgebras have been known in the two cases just mentioned [36], [21], and in two other cases rather different by the arguments employed: the Hopf algebras with cocommutative coradicals [14] (including pointed Hopf algebras [23]) and the cosemisimple ones [4]. Partial results on faithful flatness over PI Hopf subalgebras of some type can be found in [28], [31], [37].…”