Depth of modular invariant rings
This paper investigates the depth of modular rings of invariants for certain finite groups. Although the ring of invariants of a non-modular representation of a finite group is Cohen-Macaulay, and hence has depth equal to the dimension of the representation, the ring of invariants of a modular representation may fail to be Cohen-Macaulay. In this case computing the depth of the ring of invariants can be a difficult problem. In this paper the authors introduce the concept of a shallow representation and prove that the depth of the ring of invariants of a shallow representation, \(V\), is given by \(\min(\dim (V^P)+2, \dim(V))\) where \(P\) is any \(p\)-Sylow subgroup. For an abelian group with a cyclic \(p\)-Sylow subgroup, every representation is shallow. Thus the paper provides a relatively elementary proof of the well known result of \textit{G. Ellingsrud} and \textit{T. Skjelbred} [Compos. Math. 41, 233-244 (1980; Zbl 0438.13007)]. Additional examples of shallow representations are given in the paper and the observation is made that the direct sum of \(m\) copies of a shallow representation is still shallow. The final section of the paper gives an alternate proof of the fact, originally due to \textit{H. Nakajima} [Tsukuba J. Math. 3, 109-122 (1979; Zbl 0418.20041)], that the ring of invariants of a \(p\)-group is Cohen-Macaulay if and only if it is Buchsbaum. The authors then conjecture that this should hold for any finite linear group acting on a polynomial algebra. The conjecture has since been proven by one of the authors [see \textit{G. Kemper}, ``Loci in quotients by finite groups, pointwise stabilizers and the Buchsbaum property'' (preprint, Univ. Heidelberg 2000)].
scientific article; zbMATH DE number 1435994
scientific article