On the Cohen–Macaulay Property of Modular Invariant Rings

Author: Littelmann G.P.

Source: Journal of Algebra, Volume 215, Number 1, May 1999 , pp. 330-351(22)

Publisher: Academic Press

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Abstract:

If V is a faithful module for a finite group G over a field of characteristic p, then the ring of invariants need not be Cohen–Macaulay if p divides the order of G. In this article the cohomology of G is used to study the question of Cohen–Macaulayness of the invariant ring. One of the results is a classification of all groups for which the invariant ring with respect to the regular representation is Cohen–Macaulay. Moreover, it is proved that if p divides the order of G, then the ring of vector invariants of sufficiently many copies of V is not Cohen–Macaulay. A further result is that if G is a p-group and the invariant ring is Cohen–Macaulay, then G is a bireflection group, i.e., it is generated by elements which fix a subspace of V of codimension at most 2. Copyright 1999 Academic Press.