TY - JOUR

T1 - Factoring the adjoint and maximal Cohen-Macaulay modules over the generic determinant

AU - Buchweitz, Ragnar Olaf

AU - Leuschke, Graham J.

PY - 2007/8

Y1 - 2007/8

N2 - 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.

AB - 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.

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U2 - 10.1353/ajm.2007.0022

DO - 10.1353/ajm.2007.0022

M3 - Article

AN - SCOPUS:34548305029

VL - 129

SP - 943

EP - 981

JO - American Journal of Mathematics

JF - American Journal of Mathematics

SN - 0002-9327

IS - 4

ER -