Invariants of Hopf Algebras

In the winter of 1999 I gave a series of lectures at Queen’s university about some recent results concerning the Cohen-Macaulay property of invariants of Hopf algebras. Tony Geramita asked me to write up my notes for the Queen’s Papers, and I happily took up his suggestion. Although this article focuses on the proof of one main theorem (Theorem 2.11 on page 12), it has some of the character of a survey article, since the results already appeared in my (German) habilitation thesis [10], and I am trying to explain concepts in more detail and give more background than in an original article. This work originated in the study of the Cohen-Macaulay property of invariant rings of finite groups [11], which led to results about invariants of algebraic groups as well [12]. Actions of Hopf algebras are a natural generalization of group actions. Apart from group rings, Lie algebras (or, more precisely, their universal enveloping algebras) are an interesting incarnation of Hopf algebras, so everything that we say in this article will apply to invariants if Lie algebras as well. By a celebrated theorem of Hochster and Roberts [8], invariant rings of linearly reductive groups are always Cohen-Macaulay, i.e., they are free modules of finite rank over a subalgebra which is isomorphic to a polynomial ring. With the proper definition, this carries over to invariants of linearly reductive Hopf algebras (see Theorem 3.4 in this paper). For finite groups, Hochster and Roberts’s result means that the invariant ring is Cohen-Macaulay if the characteristic of the ground field does not divide the group order (see Hochster and Eagon [7]). On the other hand, it has been generally observed that invariant rings of finite groups tend not to be Cohen-Macaulay if the characteristic divides the group order. However, it is still an open question to characterize exactly those groups and representations for which the invariant ring is Cohen-Macaulay. In this article we prove that if a Hopf algebra is geometrically reductive but not linearly reductive, then there exists a representation whose invariant ring is not Cohen-Macaulay. This constitutes a partial converse to the (generalized) theorem of Hochster and Roberts. This means, for example, that for each of the classical algebraic groups in positive characteristic there exists a representation such that the invariant ring is not Cohen-Macaulay. Moreover, every finite group of order divisible by the characteristic of the ground field has a representation with non-Cohen-Macaulay invariant ring. This gives a partial answer to the question raised above. For (universal enveloping algebras of) Lie algebras, the main