ABSTRACT
S. C A E N E P E E L a n d T . G U E D E N O N Faculty of Applied Sciences, Vrije Universiteit Brussel, VUB, B-1050 Brussels, Belgium e-mail address: scaenepe@vub.ac.be, guedenon@caramail.com
Let i f be a Hopf algebra with bijective antipode, and A a left H -module algebra. We can then form the smash product A # H . In [8 ], the second author investigated necessary and sufficient conditions for the projectivity of an A #H -m odu \e M over the subring of invariants A H. Our starting point is the following: if H is finitely generated and projective, then H* is also a Hopf algebra, A is a right i/*-comodule algebra, and A H = A coH The category of left j4#/f-m odules is isomorphic to a M h \ the category of relative (A ,H *)-Hopf modules, i.e. fc-modules together with a left A-action and a right H*-coaction, satisfying an appropriate compatibility condition. Thus [8 ] brings us necessary and sufficient conditions for a relative Hopf module to be projective as a module over the ring of coinvariants. In this paper, we will generalize these results to relative (A , i/)-H opf modules, where H is an arbitrary Hopf algebra with bijective antipode over a commutative ring k , and A is a (A;-flat) right H -comodule algebra. Our main result is Theorem 2 .1 , where we give necessary and sufficient conditions for projectivity of a relative Hopf module over the subring of coinvariants B — A coH. The main tool - based on the methods developed in [7] and [8 ] - is the basic fact th a t the canonical structure of A as left-right (A, i/)-H opf module is such th a t ,4 HomH(A, A), the fc-module consisting of ^4-linear and if-colinear maps, is isomorphic to B (see Section 1 ). The result can be improved if we assume tha t there exists a total integral <j>: H —> A, see Proposition 2 .2 . In Proposition 2.5, we look at relative modules th a t are coinvariantly generated, and we will present some conditions th a t are sufficient, but in general not necessary for projectivity. These conditions have the advantage th a t they are easier to verify than the ones from
Theorem 2.1; they tu rn out to be necessary if the coinvariants functor is exact. In Sections 3 and 4, we will work over a field k. Our methods will be applied to discuss properties of injective and projective dimension in the category of relative Hopf modules (Section 3), and semisimplicity of the the category of relative Hopf modules. Our main result is Corollary 4.5, where we give a sufficient condition for the category of relative Hopf modules to be semisimple. Let us finally mention th a t our results may be applied in the following particular cases:
- A = H with comodule structure map A. In this case, B is isomorphic to k as fc-algebra and trivial //-comodule.
