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On reduction exponents of ideals with Gorenstein formring. (English) Zbl 0842.13001
The 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).
Reviewer: A.Simis (Salvador)

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, André-Quillen, cyclic, dihedral, etc.)
13D45 Local cohomology and commutative rings
Full Text: DOI
[1] DOI: 10.2307/2375005 · Zbl 0803.13002 · doi:10.2307/2375005
[2] Evans, Syzygies (1985) · doi:10.1017/CBO9781107325661
[3] DOI: 10.1007/BF01459507 · Zbl 0788.13001 · doi:10.1007/BF01459507
[4] Noh, Results in Math. 23 pp 149– (1993) · Zbl 0777.13006 · doi:10.1007/BF03323133
[5] Mcadam, Ring Theory 7 pp 163– (1974)
[6] Herrmann, Equimultiplicity and Blowing Up (1988) · doi:10.1007/978-3-642-61349-4
[7] DOI: 10.2307/2374083 · Zbl 0505.13003 · doi:10.2307/2374083
[8] DOI: 10.2307/2154225 · Zbl 0813.13009 · doi:10.2307/2154225
[9] Hermmann, Math. Z. 213 pp 301– (1993)
[10] DOI: 10.2307/2374708 · Zbl 0758.13001 · doi:10.2307/2374708
[11] Herzog, Koszul homology and blowing up rings pp 79– (1983) · Zbl 0499.13002
[12] Lipman, Math. Res. Letter. 1 pp 1– (1994) · Zbl 0873.32032 · doi:10.4310/MRL.1994.v1.n1.a1
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