A note on local reduction numbers and \(a^*\)-invariants of graded rings (Q1890462)

The local reduction numbers of a standard graded algebra \(S\) over a Noetherian local ring have been defined by \textit{I. M. Aberbach, C. Huneke} and \textit{Ngo Viet Trung} [Compos. Math. 97, No. 3, 403--434 (1995; Zbl 0854.13003)]. They applied them in the study of the Rees algebra of an ideal in a Cohen-Macaulay ring finding necessary and sufficient conditions for such algebras to be Cohen-Macaulay or Gorenstein. The paper under review shows how one can change the notion of local reduction numbers of a standard graded algebra \(S\) in order to apply some of the ideas in the paper mentioned above to graded algebras \(S\) satisfying the Serre condition \((S_l)\) where \(l\) is the analytic spread of \(S_+\), the ideal generated by all homogeneous elements of positive degree in \(S\). The author succeeds in obtaining results where the Cohen-Macaulay hypothesis is replaced by a single Serre condition. For example, one of the main theorems of the paper under review states the following: Theorem 4.2. Let \((A, m)\) be a Noetherian ring of dimension \(d \geq 2\), \(I\) an equidimensional ideal of \(A\) with \(\text{grade}(I) \geq 2\). Assume that \(A\) satisfies the Serre condition \(S_{\text{ht}(I)}\). Then the Rees algebra of \(I\), \(R(I)\), is Gorenstein if and only if the associated graded ring of \(I\), \(G(I)\), is Gorenstein and the reduction number of \(I\), \(r(I)\), satisfies \(r(I) =\text{ht}(I)-2\). In this case \(A\) is Gorenstein.