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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.

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