#### Document Type

Article

#### Date

4-17-2006

#### Embargo Period

11-15-2011

#### Disciplines

Mathematics

#### Description/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.

#### Recommended Citation

Buchweitz, Ragnar-Olaf and Leuschke, Graham J., "Factoring the Adjoint and Maximal Cohen-Macaulay Modules over the Generic Determinant" (2006). *Mathematics Faculty Scholarship.* Paper 33.

http://surface.syr.edu/mat/33

#### Source

Harvested from arXiv.org

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 3.0 License.

## Additional Information

This manuscript is from arXiv.org, for more information see http://arxiv.org/abs/math/0505315