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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Free Lie subalgebras of the cohomology of local rings


Author: Luchezar L. Avramov
Journal: Trans. Amer. Math. Soc. 270 (1982), 589-608
MSC: Primary 13D03; Secondary 13H99, 55N99, 55S20
MathSciNet review: 645332
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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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0645332-0
PII: S 0002-9947(1982)0645332-0
Keywords: Yoneda products, Massey products, homology of local rings, deviations, local complete intersections
Article copyright: © Copyright 1982 American Mathematical Society