On the Cohen–Macaulay Property of Modular Invariant Rings

IfVis a faithful module for a finite groupGover a field of characteristicp, then the ring of invariants need not be Cohen–Macaulay ifpdivides the order ofG. In this article the cohomology ofGis 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 ifpdivides the order ofG, then the ring of vector invariants of sufficiently many copies ofVis not Cohen–Macaulay. A further result is that ifGis ap-group and the invariant ring is Cohen–Macaulay, thenGis a bireflection group, i.e., it is generated by elements which fix a subspace ofVof codimension at most 2.