Factoring the Adjoint and Maximal Cohen--Macaulay Modules over the Generic Determinant

Ragnar-Olaf Buchweitz; Graham J. Leuschke

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Factoring the Adjoint and Maximal Cohen--Macaulay Modules over the Generic Determinant

Ragnar-Olaf Buchweitz·Graham J. Leuschke

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Abstract

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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Ragnar-Olaf Buchweitz, Graham J. Leuschke. 2006-04-17. Factoring the Adjoint and Maximal Cohen--Macaulay Modules over the Generic Determinant. https://arxiv.org/abs/math/0505315

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