On the ring of constants of a diagonalizable higher derivation
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.
scientific article; zbMATH DE number 64210