Induction Functors Research Articles

Hopf extensions of algebras and Maschke type theorems

LetB/C be anA-extension for a Hopf algebraA. We consider two Maschketype questions: first, for an exact sequence of (A, B)-Hopf modules which splitsB-linearly, when does it split (A, B)-linearly? Second, for an exact sequence ofB-modules which splitsC-linearly, when does it splitB-linearly? Finally, we consider when the pair of the restriction functor from mod-B to mod-C and the induction functor ( ) ⊗ C B is an adjoint pair of functors.

Separable functors applied to graded rings

In this paper we introduce the notion of separability for functors. As special cases we mention the restriction (of scalars) functor, ‘p*, and the induction functor, cp *, associated to a ring homomorphism ‘p: R + S; separability of ‘p* relates to separability of S over R which is defined in terms of splitting of the canonical map II/: S 0 R S -+ S as an S-bimodule map. Separable functors and their properties have various applications, but here we restrict to applications in the theory of graded rings where the morphism cp is usually taken to be the map R, + R, where e is the neutral element of the group G. However, there are a few very natural functors around that allow interesting applications. After the general properties of separability studied in Section 1 we turn to separability of the restriction functor for graded rings in Section 2. In case R is strongly graded by G then R is separable over R, if and only if G is finite and the trace function is surjective. The forgetful functor U: R-gr + R-mod (U forgets the gradiation) is always a separable functor. The body of this paper deals with the properties of the right adjoint F of U in case G is finite of order n. Let Cal,(M) be the set of n x l-columns over it4 and H: R-mod + R-gr the functor obtained by putting H(M) = Cal.(M) viewing it as a graded R-module in some way, then the functors F and H are equivalent. In Theorem 3.6 we provide a separability criterion for Fusing smash-product constructions appearing in the duality for coac-

A cohomological characterization of parabolic subgroups of reductive algebraic groups

Let G be an algebraic group defined over an algebraically closed field k, and RAT(G) the category of rational G-modules. For any closed subgroup H of G and any rational H-module V, let V Ig denote the rational G-module induced from I’. The induction functor ( ) Ig: RAT(H) + RAT(G) is left exact and we denote its derived functors by L;,J ) for n=o, 1, 2, . . . . Of course these derived functors are 0 for n 2 1 whenever ( ) 1 g is exact, and by a result of Cline, Parshall, and Scott [3] this happens precisely when G/H is an afftne variety. So the case of an afftne quotient is characterized by the vanishing of the higher derived functors. In a like vein, in this paper we will attempt to find a property of these functors which characterizes the case of a projective quotient. Since by definition G/H is projective iff H = P is a parabolic subgroup of G, we are in effect looking for a cohomological characterization of parabolic subgroups of G. The property of the functors L;,J ) we consider is a certain finiteness condition explained below. The author was originally led to consider this question in [14], where induction from certain nonparabolic subgroups is studied and is shown to have this finiteness property. It is well known that for every rational H-module V there is an induced bundle L, on G/H with rank equal to dim I’, such that L;,J V) identifies naturally with the sheaf cohomology group H”((G/H, L.). See [4] for details. Thus we may apply standard results from algebraic geometry to obtain some basic facts. For example, Lg,G( ) = 0 for n > dim(G/H), and Serre’s theorem gives the easy half of the result of Cline, Parshall, and Scott men-