On a-invariant formulas
Let \(A\) be a local ring of dimension \(d\) having an infinite residue field, and let \(I\subset A\) be an ideal of positive height \(h>0\). Let us denote by \(R_A(I)\) the Rees algebra of \(I\), by \(G_A(I)\) the form ring of \(A\) with respect to \(I\), and by \(r(I)\) the reduction exponent of \(I\). The authors first prove a general a-invariant formula for standard graded algebras. In the case of the form ring it states \(\text{a}(G_A(I))=\max\{r(I)-l(I),\delta(G_A(I))\}\), under the assumption that \(A\) is Cohen-Macaulay and that \(G_A(I)\) fulfills Serre's condition \((S_l)\) only, where \(l=l(I)\) stands for the analytic spread of \(I\). Next it is shown how the a-invariant formula can be used to prove new upper bounds for the reduction exponent of an ideal. For an ideal \(I\) in a regular local ring \(A\) whose form ring \(G_A(I)\) satisfies Serre's condition \((S_l)\) the reduction exponent \(r(I)\) is bounded above by \(l-1\). In the case that \(G_A(I)\) is Gorenstein and \(h>1\) one gets even more: \(r(I)\leq l-2\). Finally, one uses the above formula to describe the a-invariant of the form ring in terms of reduction exponents and analytic spreads of localizations of \(I\) at finitely many primes ideals only (theorem 4.5).
scientific article; zbMATH DE number 1448645
scientific article