On the homotopy Lie algebra of a local ring

Abstract

For a commutative, local, noetherian ring (R, m, k), the Ext-algebra, ExtR(k, k), is the enveloping algebra of a positively graded Lie algebra, X(R), called the homotopy Lie algebra of R, see [l] [7]. Very few results about this Lie structure are known. We will consider some special cases of local rings, where it is possible to describe the Lie structure in a rather explicit way. In all cases n(R) is a semi-direct product of a Lie algebra with a free Lie algebra. This situation occurs, for example, when there is a surjective Golod homomorphism S + R. In this case n(R) is a semidirect product of rc(S) with L(V), the free Lie algebra on I’, where Y”+’ = Extg(R, k) for n L 1. Levin [3] proved that the induced action of n(S) on I/ is the natural action of Exts(k, k) on Exts(R, k). He has also asked, when this action induces the action of n(S) on L( I’); i.e., when does n(S) map V to I’ inside L(V)? This does not hold in general, as is shown by an example of Sjodin [9, Remark 1 to Theorem 41. In this note we give sufficient conditions for this to be true. In the first part of the paper we give a general theorem, which we in the second part apply to the following four cases: Equi-characteristic, local rings with m3 = 0. Here we describe n(R) in terms of the Lie algebra generated by n’(R) in z(R). Trivial extensions. These rings give rise to a special type of Golod homomorphisms (not surjective, however), where we give a positive answer to Levin’s question. A ‘Frtiberg’ local ring (S, n, k) module an ideal Q such that n2rp 3 c R C nr. The map S+S/s2 is Golod, and, again, Levin’s question has a positive answer. The case S= k[[x,, . . . , x,,]], D = (x1, . . . , x,)‘. Here we give an explicit presentation of n(S/Q). The third application generalizes Sjodins result [9, Theorem 41, and his proof contains the main idea to get our result. Sjodin [9, Section 71 also gives a different presentation of 7c(S/Q) in the last case.