Generic free resolutions and root systems

Abstract: In this paper I give an explicit construction of the generic ring R_{gen} for finite free resolutions of length 3. The corresponding problem for resolutions of length 2 was solved in 1970'ies by Hochster and Huneke. The key role is played by the defect Lie algebra introduced in my old work on the subject. The defect Lie algebra turns out to be a parabolic Lie algebra 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} is a Dynkin graph.