Mathematics – Commutative Algebra
Scientific paper
2005-05-15
Mathematics
Commutative Algebra
44 pages, final version of the work announced in math.RA/0408425, to appear in the American Journal of Mathematics
Scientific paper
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.
Buchweitz Ragnar-Olaf
Leuschke Graham J.
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