2005/05/15 by Ragnar-Olaf Buchweitz, Buchweitz, Ragnar-Olaf, Graham J. Leuschke +1
Mathematics · #13C14 #14C40 #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA) #math.AC #math.RA #msc:13C14 #msc:14C40
paper · pdf · doi:10.48550/arxiv.math/0505315
44 pages, final version of the work announced in math.RA/0408425, to appear in the American Journal of Mathematics
arxiv created 2006/04/17 · arxiv updated 2009/12/01
A question of Bergman asks whether the adjoint of the generic square matrix over a field can be factored nontrivially as a product of square matrices. We show that such factorizations indeed exist over any coefficient ring when the matrix has even size. Establishing a correspondence between such factorizations and extensions of maximal Cohen--Macaulay modules over the generic determinant, we exhibit all factorizations where one of the factors has determinant equal to the generic determinant. The classification shows not only that the Cohen--Macaulay representation theory of the generic determinant is wild in the tame-wild dichotomy, but that it is quite wild: even in rank two, the isomorphism classes cannot be parametrized by a finite-dimensional variety over the coefficients. We further relate the factorization problem to the multiplicative structure of the \Ext--algebra of the two nontrivial rank-one maximal Cohen--Macaulay modules and determine it completely.