Document Type
Article
Date
4-17-2006
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 - All Scholarship. 33.
https://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