Generic free resolutions and a family of generically perfect ideals

Abstract

Beginning with Hilbert’s construction of what is now called the Koszul complex [18], the study of finite free resolutions of modules over commutative rings has always proceeded by a study of certain particular generic resolutions. This has led to information about the structure of all finite free resolutions, as in [5] and [II], and to theorems on the structure and deformation of certain classes of “generically perfect” ideals and other ideals whose resolutions are of a known type [5, 6,9, 15, 22, 24, 261. In this paper we will describe some new classes of finite free resolutions and generically perfect ideals. Under “generic” circumstances we will construct the minimal free resolution of the cokernel of a map of the form Ak C# or Sk+, where 4: F --t G is a map between free modules F and G over a noetherian commutative ring, with rank F >, rank G, and where A” and Sk denote the kth exterior and symmetric powers, respectively. We will also describe a family of finite complexes associated with the (n - 1)st order minors of an n x n matrix (a minor is the determinant of a submatrix). Finally, we will consider a class of ideals that is related to inclusions of one ideal generated by an R-sequence in another. This class includes, for example, the ideal defining the singular locus of a projective algebraic variety that is a complete intersection in P. Our main innovation is the construction and use of a (doubly indexed) family of “multilinear” functors LpQ defined on finitely generated free modules, which includes both the symmetric and exterior powers. For q > 1, L,‘J g Aq, while for p > 0, LP1 s S, . These arise naturally in the resolutions of cokernels of the maps A”$ and S,$; and it turns out more generally that the free modules occuring in the generic minimal free resolutions of the cokernel of a map of the form L,‘+ can all be expressed in terms of tensor products of the form L,qF @I (LrSG)*.