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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).