Algebra Seminar

A theorem of Hochster and Eagon says that invariant rings of a finite group G over a field K such that char(K) does not divide |G| are Cohen-Macaulay, i.e., they are free modules of finite rank over a polynomial subalgebra. This is a very important structural property with high relevance for computations.

This talk gives an introduction into the Cohen-Macaulay property and then goes on to explain the following converse to the theorem of Hochster and Eagon: if for every representation of G over K the invariant ring is Cohen-Macaulay, then char(K) does not divide |G|. More results come out of this work, such as a classification of all finite groups G and fields K such that the invariant ring of the regular representation over K is Cohen-Macaulay. Furthermore, if G is a linear p-group, p = char(K), and the invariant ring of G is Cohen-Macaulay, then G is generated by bireflections.