On the ring of constants of a diagonalizable higher derivation
type: Item
instance of: scholarly article
title: On the ring of constants of a diagonalizable higher derivation
zbMATH Open document ID: 0763.13001
DOI: 10.1016/0022-4049(92)90133-Z
author: Yasunori Ishibashi
author: Nobuharu Onoda
published in: Journal of Pure and Applied Algebra
publication date: 1992-09-27T00:00:00Z
review text: Let \(k\) be a field of positive characteristic \(p>0\), let \(q=p^ e\), and let \(\mu_ q\) denote the group scheme of \(q\)-th roots of unity over \(k\). Suppose the \(\mu_ q\) acts linearly on the affine \(n\)-space \(\mathbb{A}^ n\) over \(k\). The authors compute the ring of invariants of such an action, show that it is always Cohen-Macaulay, compute its canonical module, and give a numerical criterion for it to be Gorenstein. Their work begins by observing that any such action is given by a diagonalizable higher derivation on the polynomial ring \(R=k[X_ 1,\dots,X_ n]\). A higher derivation is a sequence of \(k\)-linear endomorphisms \(D=\{d_ 0=1,\dots,d_{m-1}\}\) of \(R\) such that \(d_ j(ab)=\sum d_ i(a)d_{j-i}(b)\); \(D\) is diagonalizable if \(d_ i(X_ j)=\lambda_{ji}X_ j\) for all \(i\) and \(j\), with \(\lambda\) in \(k\). The ring of invariants above corresponds to \(S=\{a\in R| d_ i(a)=0\), \(i>0\}\). The authors give an explicit description of \(S\) in terms of the \(\lambda_{ij}\) and show that \(S\) is a direct summand of \(R\) as a graded \(S\)-module (and hence Cohen-Macaulay). They then give an explicit description of the canonical module for \(S\), also in terms of the \(\lambda\)'s, from which they derive an explicit condition for \(S\) to be Gorenstein.
reviewed by: Andy R. Magid
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 13B10
zbMATH DE Number: 64210
zbMATH Keywords: ring of invariants
zbMATH Keywords: Cohen-Macaulay
zbMATH Keywords: canonical module
zbMATH Keywords: Gorenstein
zbMATH Keywords: diagonalizable higher derivation
MaRDI profile type: Publication
cites work: Reductive derivations of local rings of characteristic p
cites work: On p-radical descent of higher exponent
cites work: On graded rings. I
cites work: Q3903085
full work available at URL: 0022-4049(92)90133-z
OpenAlex ID: W2066622079
description: scientific article; zbMATH DE number 64210