On the structure of free resolutions of length 3

Abstract

The sources of structure theory of finite free resolutions go back to Hilbert. In his famous paper on invariant theory he proved that every ideal of projective dimension 1 over a polynomial ring is determinantal. This theorem was then extended by Burch to arbitrary commutative rings. There are various motivations for extending this results. One of them is a variety of applications of the Hilbert-Burch theorem in deformation theory and algebraic geometry (cf., for example, [S]). Another reason comes from Serre’s conjecture. One would like to understand in terms of resolutions why the lifting property of Grothendieck does not hold. In 1974 Buchsbaum and Eisenbud in [BEl] introduced so-called structure theorems-various factorisations which occur for acyclic complexes of arbitrary length. Then Hochster in [H] generalized the Hilbert-Burch theorem to arbitrary free complexes of length 2. His results were then improved in [Br, PW]. Still, the structure of complexes of length bigger than 2 is not understood. The first steps in this direction are the results of [B] and the construction of a generic ring for complexes of length 3 of type (1,rz 1, 1) given in [PW]. In this paper I construct a ring which, I believe, is the generic ring for complexes of length 3. The construction involves new structure theorems and a new idea-the notion of defects of structure theorems. The paper is organized as follows. Section 1 is introductory. I give there the background from representation theory and the cohomology groups of a Lie algebra. Section 2 contains the proof of the new structure theorems we need, the construction of the generic ring R,, and of the defect Lie algebra L. In Section 3 I show that genericity of R,, follows from vanising of the homology of a certain family of complexes over U(L)-the enveloping algebra of L. In Section 4 I apply the results of Section 2 to study multi-