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

Ragnar Olaf Buchweitz, Graham J. Leuschke

Research output: Contribution to journalArticle

5 Scopus citations

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 languageEnglish (US)
Pages (from-to)943-981
Number of pages39
JournalAmerican Journal of Mathematics
Volume129
Issue number4
DOIs
StatePublished - Aug 2007

ASJC Scopus subject areas

  • Mathematics(all)

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