Free summands of conormal modules and central elements in homotopy Lie algebras of local rings

Abstract

If (Q, n) -* (R, m) is a surjective local homomorphism with kernel I, such that I C n2 and the conormal module I/12 has a free summand of rank n, then the degree 2 central subspace of the homotopy Lie algebra of R has dimension greater than or equal to n. This is a corollary of the Main Theorem of this note. The techniques involved provide new proofs of some well known results concerning the conormal module. Let R be a noetherian local ring with maximal ideal m and residue field k. Ring theoretic properties of R are reflected on the algebra structures carried by TorR (k, k) and EXtR (k, k) . Recall that the former has the rh-product of Cartan and Eilenberg, and the latter the Yoneda multiplication. The k-algebra structure of TorR (k, k) is rather simple: It is free in the appropriate category, and so determined by the dimension of the k-vector space TornR (k, k), for all integers i > 0, that is to say, the Betti numbers of k over R. The multiplicative structure of EXtR (k, k) is quite another matter. The simplicity of the product on TorR (k, k) arises from the fact that it is endowed with additional structures: It is a commutative algebra, in the graded sense, with a family of divided powers, and has a diagonal map that is compatible with the divided powers algebra structure on it. In other words, the k-algebra TorR (k, k) is a commutative Hopf algebra with divided powers, and so is free as a divided powers algebra. In contrast, the multiplication on EXtR(k, k) is decidedly non-commutative, unless R happens to be a complete intersection of a special kind. The graded kdual of the product on TorR (k, k) turns EXtR (k, k) = TorR (k, k) * into a Hopf algebra. Attention has focussed on a certain subspace of primitives of this Hopf algebra, denoted ir(R), which is a Lie algebra with the bracket operation defined by the commutator in EXtR (k, k) . This object is called the homotopy Lie algebra of R. The importance of this Lie algebra is attested to by its defining property: Its universal enveloping algebra is EXtR (k, k) . In this note, we are concerned with the centre of the homotopy Lie algebra of R, denoted ((R). This subspace, besides being a measure of the non-commutativity of ir(R), and hence of ExtR (k, k), plays an important part in the change of rings Received by the editors April 7, 1999 and, in revised form, May 12, 1999. 1991 Mathematics Subject Classification. Primary 13C15, 13D03, 13D07, 18G15.