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

  • Ragnar-Olaf Buchweitz, Graham J. Leuschke
  • Published 2005

Abstract

G.M. Bergman recently asked whether the adjoint of the generic square matrix can be factored as a product of square matrices [3]. He showed that in most cases there are no nontrivial factorizations. We recast the existence of such factorizations in terms of extensions of maximal Cohen–Macaulay modules over the hypersurface ring defined by the generic determinant. Specifically, a nontrivial factorization wherein one of the factors has determinant equal to the generic determinant gives an extension of the rank-one maximal Cohen–Macaulay (MCM) modules; we construct and classify all of these, as well as the corresponding factorizations of the adjoint. The classification shows that even in rank two, the MCM-representation theory of the generic determinant is quite wild. We also describe completely the Ext–algebra of the rank-one maximal Cohen–Macaulay modules.

Cite this paper

@inproceedings{Buchweitz2005FactoringTA, title={Factoring the Adjoint and Maximal Cohen-Macaulay Modules over the Generic Determinant}, author={Ragnar-Olaf Buchweitz and Graham J. Leuschke}, year={2005} }