Multiplicative structure of generalized Koszul complexes

Gover, Eugene H.

doi:10.1090/S0002-9947-1973-0332769-7

Multiplicative structure of generalized Koszul complexes
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by Eugene H. Gover PDF

Trans. Amer. Math. Soc. 185 (1973), 287-307 Request permission

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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