The Cohomology Of Doi-Hopf Modues

Let k be a field, and H a Hopf algebra with bijective antipode s . Let M ,N be right H -comodules. The right H -comodule HOM ( M , N ) was defined by Ulbrich in [3]. In [7], Caenepeel and Guédénon studied the right derived functors of the Coinvaria -nt functor coH and the functor HOM . They also studied the cohomology of left- right relative ( A , H )-Hopf modules. They mainly investigated the injective modules, minimal injective resolutions and cohomology groups, where A is a right H -comodule algebra. In this paper, we first generalize the definition of HOM ( M , N ) (to the case that M ,N are right C -comodules), and give some properties of the functors coC and HOM . Then, we extend some results of left-right relative (A,H)-Hopf modules in [7] to left-right ( H , A , C )-Doi-Hopf modules, where C is a left H -module coalgebra.In section 1, we introduce some basic notions and properties about comodule algeb -ras, module coalgebras, Doi-Hopf modules and functors.In section 2, given right C -comodules M and N , we give the definition of right C -comodule HOM ( M , N ) and a new comodule structure on M ? N. Then, we discuss the properties of the functors coC and HOM . Under some suitable assumptions, the functor Hom C can be viewed as the composition of the functors HOM and coC . So we can consider the right derived functors of the functors coC and HOM . In order to compute the cohomology groups, we describe one methold to give an injective resolution of any right C -comodule M .In section 3, for two left-right ( H , A ,C )-Doi-Hopf modules M and N , we co -nsider the space _AHom~C( M , N ) of A -linear and C -colinear maps from M to N , and the right C -comodule _AHOM ( M , N ), consisting of rational A -linear maps betw -een M and N . It is shown that, _AHom~C is the composition of the functors _AHOM and coC . For the functors _AHom~C, _AHOM and coC , we study their right derived functors and discuss the relations between them.