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On the Cohen–Macaulay Property of Modular Invariant Rings

✍ Scribed by Gregor Kemper

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
203 KB
Volume
215
Category
Article
ISSN
0021-8693

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✦ Synopsis

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. ᮊ 1999 Academic Press * The author thanks Ian Hughes, Eddy Campbell, Jim Shank, and David Wehlau for their hospitality during his visit to Queen's University in Kingston, Ontario, where most of this paper was prepared.

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