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Transactions of the American Mathematical Society

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



Multiplicative structure of generalized Koszul complexes

Author: Eugene H. Gover
Journal: Trans. Amer. Math. Soc. 185 (1973), 287-307
MSC: Primary 13D99
MathSciNet review: 0332769
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Abstract: A multiplicative structure is defined for the generalized Koszul complexes $ K({ \wedge ^p}f)$ associated with the exterior powers of a map $ f:{R^m} \to {R^n}$ where R is a commutative ring and $ m \geq n$. With this structure $ K({ \wedge ^n}f)$ becomes a differential graded R-algebra over which each $ K({ \wedge ^p}f),1 \leq p \leq n$, is a DG right $ K({ \wedge ^n}f)$-module. For $ f = 0$ and $ n > 1$, the multiplication and all higher order Massey operations of $ K({ \wedge ^n}f)$ are shown to be trivial. When R is noetherian local, $ K({ \wedge ^n}f)$ is used to define a class of local rings which includes the local complete intersections. The rings obtained for $ n > 1$ are Cohen-Macaulay but not Gorenstein. Their Betti numbers and Poincaré series are computed but these do not characterize the rings.

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Keywords: Generalized Koszul complex, differential graded algebra, trivial multiplicative structure, Massey operations, generalized local complete intersection, Betti numbers, Poincaré series
Article copyright: © Copyright 1973 American Mathematical Society