Jan-Erik Roos

A computer-aided study of the graded Lie algebra of a local commutative noetherian ring

We initiate a systematic study of the homotopy Lie algebra gR of a local commutative noetherian ring R. Particular emphasis is put on the sub Lie algebra ηitR, generated by elements of degree 1. The computer is used in an essential way to detect new phenomena and new structure results, which are then proved by hand calculations. In Appendix B (by Clas Lofwall) it is proved that in many cases (including many examples encountered in this paper) gitR is a nice semi-direct product of ηitR by means of a free graded Lie algebra.

A toric ring with irrational Poincaré-Betti series

We show that there exists a toric curve in P8 whose homogeneous coordinate ring has a presentation with 12 quadratic relations and whose Poincare-Betti series is irrational. The example was found by a computer search, aiming at a homological classification of those toric curves that have a quadratic presentation in Pn-1 for n ≤ 9. Some other consequences of this search are also presented.

On computer-assisted research in homological algebra

We give a survey of how computer algebra can be used to help the mathematician to guess results and to prove theorems in homological algebra. Our main point is that the Poincare-Betti series of a commutative graded algebra contains much deeper information and is harder to calculate than the Hilbert series of the same algebra. However, in many cases (going far beyond the so-called Koszul algebras), the two series are closely related, and this gives an interesting theory. This theory could hardly have been revealed without an intensive use of the programs MACAULAY by Dave Bayer and Michael Stillman and BERGMAN by Jorgen Backelin. We also present new results and conjectures inspired by these studies and indicate how our results are related to problems in algebraic geometry and algebraic topology.

The Homotopy Lie Algebra of a Complex Hyperplane Arrangement Is Not Necessarily Finitely Presented

We present a theory that produces several examples in which the homotopy Lie algebra of a complex hyperplane arrangement is not finitely presented. We also present examples of hyperplane arrangements in which the enveloping algebra of this Lie algebra has an irrational Hilbert series. This answers two questions of Denham and Suciu.

Some Non-Koszul Algebras

The aim of this note is to show that the quadratic algebras enstudied by S. Fomin and A. N. Kirillov in [FK] arenotKoszul algebras for anyn≥ 3. The algebrae n (of type A) has generators T ij for 1 ≤i≤j≤nsubject to the following relations:

Three-dimensional manifolds, skew-Gorenstein rings and their cohomology

\tau _{{ij}}^{2} = 0{\text{ for }}i < j;

Explicit formulae for the global homological dimensions of trivial extensions of rings

(i)