Free Lie subalgebras of the cohomology of local rings

Avramov, Luchezar L.

doi:10.1090/S0002-9947-1982-0645332-0

Free Lie subalgebras of the cohomology of local rings
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by Luchezar L. Avramov PDF

Trans. Amer. Math. Soc. 270 (1982), 589-608 Request permission

Abstract:

A criterion is established, in terms of the Massey products structure carried by the homology of partial resolutions, for the Yoneda cohomology algebra ${\operatorname {Ext} _A}(k, k)$ to be a free module over the universal envelope of a free graded Lie subalgebra. It is shown that several conjectures on the (co)homology of local rings, in particular on the asymptotic behaviour of the Betti numbers, follow from such a structure. For all rings with $\operatorname {edim} A - \operatorname {depth} A \leqslant 3$, and for Gorenstein rings with $\operatorname {edim} A - \operatorname {depth} A = 4$, the following dichotomy is proved: Either $A$ is a complete intersection, or ${\operatorname {Ext} _A}(k, k)$ contains a nonabelian free graded Lie subalgebra.

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