Homological codimension of modular rings of invariants and the Koszul complex
The rings of invariants \(R^G\) of a finite group \(G\) acting on a ring of polynomials \(R = F[V],\) where \(F\) denotes a field, is a Cohen-Macaulay ring provided the characteristic of \(F\) does not divide the order of \(G,\) see e.g. the author's book: \textit{L. Smith}, ``Polynomial invariants of finite groups'' (1997; Zbl 0864.13002). This is not true in the modular case, i.e. when the characteristic is a divisor of the order of \(G,\) as it was first noticed by \textit{M.-J. Bertin} [see C. R. Acad. Sci., Paris, Sér. A 264, 653-656 (1967; Zbl 0147.29503)]. The paper under review is concerned with the following problem: Let \(h_1, \ldots, h_k \in R^G\) be invariant polynomials that form an \(R\)-regular sequence. Do they form a regular sequence in \(R^G\), too? As follows from examples discussed by the author in his book cited above, respectively: \textit{L. Smith}, Bull. Am. Math. Soc., New Ser. 34, No.~3, 211-250 (1997; Zbl 0904.13004) that this is not true in general. Here the author studies Koszul complexes, which determine the property of being a regular sequence by the vanishing of their homology. Then he uses this in order to describe the depth of rings of invariants of permutation representations of the cyclic group of order \(p \not= 0,\) and to answer the above problem in this case. The main technical tool is a spectral sequence which is in some sense the dual to the one used by \textit{G. Ellingsrud} and \textit{T. Skjelbred} [see Compos. Math. 41, 233-244 (1980; Zbl 0438.13007)].
scientific article; zbMATH DE number 1319571