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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Structure theory for a class of grade four Gorenstein ideals

Authors: Andrew Kustin and Matthew Miller
Journal: Trans. Amer. Math. Soc. 270 (1982), 287-307
MSC: Primary 13C13; Secondary 13H10, 16A03
MathSciNet review: 642342
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Abstract: An ideal $ I$ in a commutative noetherian ring $ R$ is a Gorenstein ideal of $ \operatorname{grade} g$ if $ {\operatorname{pd} _R}(R / I) = \operatorname{grade} \,I = g$ and the canonical module $ \operatorname{Ext} _R^g(R / I,\,R)$ is cyclic. Serre showed that if $ g = 2$ then $ I$ is a complete intersection, and Buchsbaum and Eisenbud proved a structure theorem for the case $ g = 3$. We present generic resolutions for a class of Gorenstein ideals of $ \operatorname{grade} 4$, and we illustrate the structure of the resolution with various specializations. Among these examples there are Gorenstein ideals of $ \operatorname{grade} \,4$ in $ k[[x,\,y,\,z,\,v]]$ that are $ n$-generated for any odd integer $ n \geqslant 7$. We construct other examples from almost complete intersections of $ \operatorname{grade} \,3$ and their canonical modules. In the generic case the ideals are shown to be normal primes. Finally, we conclude by giving an explicit associative algebra structure for the resolutions. It is this algebra structure that we use to classify the different Gorenstein ideals of $ \operatorname{grade} \,4$, and which may be the key to a complete structure theorem.

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Article copyright: © Copyright 1982 American Mathematical Society