### 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.

Original language | English (US) |
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Pages (from-to) | 943-981 |

Number of pages | 39 |

Journal | American Journal of Mathematics |

Volume | 129 |

Issue number | 4 |

DOIs | |

State | Published - Aug 2007 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*American Journal of Mathematics*,

*129*(4), 943-981. https://doi.org/10.1353/ajm.2007.0022