Abstract

In this paper I give an explicit construction of the generic rings R-gen for free resolutions of length 3 over Noetherian commutative C-algebras. The key role is played by the defect Lie algebra introduced earlier on. The defect algebra turns out to be a parabolic subalgebra in a Kac–Moody Lie algebra associated to the graph T-p,q,r corresponding to the format of the resolution. The ring R-gen is Noetherian if and only if the graph T-p,q,r corresponding to a given format is a Dynkin diagram. In such case R-gen has rational singularities so it is Cohen–Macaulay. The ring R-gen is a deformation of a commutative ring R-spec which has a structure of a multiplicity free module over a product of Kac–Moody Lie algebras corresponding to the graph T-p,q,r and a product of two general linear Lie algebras.