A computer-aided study of the graded Lie algebra of a local commutative noetherian ring. -- Appendix A: Some technical details about how the computer was used. -- Appendix B (by Clas Löfwall): The Lie algebra structure of a ring satisfying \({\mathcal M}_ 3\) and variants (Q1313788)

Let \((R,m)\) be a local commutative noetherian ring with maximal ideal \(m\) and residue field \(k=R/m\). The graded algebra \(\text{Ext}^*_ R (k,k)\) is then a Hopf algebra and in fact the enveloping algebra of a graded Lie algebra \(g_ R\). The author initiates a systematic study of \(g_ R\) with a particular emphasis on the sub Lie algebra \(\eta_ R\) generated by elements of degree 1. The computer algebra program MACAULAY is described in the paper and used to detect new phenomena and new structure theorems. The paper contains in particular a classification of the rings \(R = k[x,y,z,t]/(f_ 1, f_ 2, f_ 3)\) where the \(f_ i\)'s are quadratic forms. In each case the author describes the Koszul homology, the Hilbert series and the resolution of \(k\) up to a certain degree. In appendix B, \textit{C. Löfwall} proves that in many cases, \(g_ R\) is a nice semidirect product of \(\eta_ R\) by means of a free graded Lie algebra.