ABSTRACT: In a recent preprint, math.AC/0306126 "Can one factor the classical adjoint of a generic matrix?", George M. Bergman asks essentially what are the possible Maximal Cohen-Macaulay modules (MCMs) with a small number of generators over the generic determinant, equivalently, what are the matrix factorizations of that determinant of small size. More specifically, he inquires about possible extensions of MCMs that have the cokernel of the adjoint matrix as their middle term. He shows, using a recent result by De Concini and Reichstein about maps between Grassmannians, that there are no such extensions in characteristic zero for generic matrices of odd size and that for even size only extensions where one of the ends is of rank one could be possible. We show that the latter case indeed occurs over any ring and classify all such extensions. It is known by work of Bruns-Vetter that there are just three MCMs of rank one over the generic determinant, and one may thus ask, what about MCMs of small rank in general? As a first step, we classify all extensions of rank one MCMs, producing "lots"(= not parametrizable by a finite dimensional algebraic variety) of orientable rank two MCMs with any, necessarily even, number of generators between n and 2n over the determinant of the generic (n x n)-matrix as soon as n is at least 3. This implies essentially that the situation for MCMs of rank two is already (very) "wild". This is joint work with Graham Leuschke.

Date: March 26, 2004

Time: 11:00 - 12:00
(Notice unusual time)
511 Lake Hall