On reduction exponents of ideals with Gorenstein formring.

*(English)*Zbl 0842.13001The authors prove two main results concerning ideals of analytic deviation 1 and 2, respectively. To explain, let \(A\) be a noetherian local ring, let \(I \subset A\) be a proper ideal and let \(G(I)\) denote the form ring associated to \(I\). The symbols \(\ell (I)\), \(\text{ad} (I)\), \(r(I)\) and \(a (G(I))\) denote the analytic spread, the analytic deviation of \(I\), the reduction exponent of \(I\) and the \(a\)-invariant of \(G(I)\), respectively. The first theorem says that if \(A\) is a Gorenstein ring and \(I\) is generically a complete intersection with \(\text{ad} (I) = 1\) then \(G(I)\) Gorenstein implies that \(r(I) \leq 1\); conversely, \(r(I) \leq 1\) (plus \(A/I\) Cohen-Macaulay) implies \(G(I)\) Gorenstein and already triggers \(r(I) = 0\).

The second theorem assumes that \(A\) is Gorenstein, \(A/I\) is Cohen-Macaulay, \(\text{ad} (I) = 2\) and \(\ell (I) \geq 4\). Then, provided \(G(I)\) is Cohen-Macaulay and \(I\) is locally a complete intersection in (relative) codimension one, the authors prove that \(a(G(I)) = \max \{r(I) - \ell (I)\), \(- \ell (I)\), \(-\ell (I) + 2\}\) and if, moreover, \(G(I)\) is Gorenstein, that \(r(I) \leq 1\).

The proofs by and large involve a careful analysis of the socle of \(G(I)\) and draw fully on techniques of (graded) local cohomology. The general philosophy for this sort of result is that, provided \(G(I)\) is Cohen-Macaulay, there is an expected (or predicted) value for the reduction exponent of \(I\) and, correspondingly, for the \(a\)-invariant of \(G(I)\). Moreover, if \(G(I)\) happens to be Gorenstein, then these predictions can roughly be improved by one.

As a matter of update, the present results have been simultaneously obtained or subsequently extended by various authors, including the present authors in other joint collaborations. A great deal of these results have been reported and discussed in the Workshop on commutative algebra and its relation to combinatorics and computer algebra, Internat. Centre Theor. Phys. (Trieste 16-27 May, 1994).

The second theorem assumes that \(A\) is Gorenstein, \(A/I\) is Cohen-Macaulay, \(\text{ad} (I) = 2\) and \(\ell (I) \geq 4\). Then, provided \(G(I)\) is Cohen-Macaulay and \(I\) is locally a complete intersection in (relative) codimension one, the authors prove that \(a(G(I)) = \max \{r(I) - \ell (I)\), \(- \ell (I)\), \(-\ell (I) + 2\}\) and if, moreover, \(G(I)\) is Gorenstein, that \(r(I) \leq 1\).

The proofs by and large involve a careful analysis of the socle of \(G(I)\) and draw fully on techniques of (graded) local cohomology. The general philosophy for this sort of result is that, provided \(G(I)\) is Cohen-Macaulay, there is an expected (or predicted) value for the reduction exponent of \(I\) and, correspondingly, for the \(a\)-invariant of \(G(I)\). Moreover, if \(G(I)\) happens to be Gorenstein, then these predictions can roughly be improved by one.

As a matter of update, the present results have been simultaneously obtained or subsequently extended by various authors, including the present authors in other joint collaborations. A great deal of these results have been reported and discussed in the Workshop on commutative algebra and its relation to combinatorics and computer algebra, Internat. Centre Theor. Phys. (Trieste 16-27 May, 1994).

Reviewer: A.Simis (Salvador)

##### MSC:

13A30 | Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics |

13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |

13D03 | (Co)homology of commutative rings and algebras (e.g., Hochschild, AndrĂ©-Quillen, cyclic, dihedral, etc.) |

13D45 | Local cohomology and commutative rings |

##### Keywords:

\(a\)-invariant; analytic deviation; form ring; analytic spread; reduction exponent; Gorenstein; Cohen-Macaulay
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\textit{M. Herrmann} et al., Proc. Edinb. Math. Soc., II. Ser. 38, No. 3, 449--463 (1995; Zbl 0842.13001)

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