Endomorphisms of Kronecker Modules Regulated by Quadratic Algebra Extensions of a Function Field

Abstract

Abstract. The Kronecker modules V(m, h, α), where m is a positive integer, h is a height function, and α is a K-linear functional on the space K(X) of rational functions in one variable X over an algebraically closed field K, are models for the family of all torsion-free rank-2 modules that are extensions of finitedimensional rank-1 modules. Every such module comes with a regulating polynomial f in K(X) [Y ]. When the endomorphism algebra of V(m, h, α) is commutative and non-trivial, the regulator f must be quadratic in Y . If f has one repeated root in K(X), the endomorphism algebra is the trivial extension K⋉S for some vector space S. If f has distinct roots in K(X), then the endomorphisms form a structure that we call a bridge. These include the coordinate rings of some curves. Regardless of the number of roots in the regulator, those End V(m, h, α) that are domains have zero radical. In addition, each semi-local End V(m, h, α) must be either a trivial extension K ⋉ S or the product K × K.