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

http://dx.doi.org/10.4153/CJM-2008-039-2
Canad. J. Math. 60(2008), 923-957

  Published:2008-08-01
 Printed: Aug 2008

The Kronecker modules $\mathbb{V}(m,h,\alpha)$, where $m$ is a positive integer, $h$ is a height function, and $\alpha$ 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 finite-dimensional rank-1 modules. Every such module comes with a regulating polynomial $f$ in $K(X)[Y]$. When the endomorphism algebra of $\mathbb{V}(m,h,\alpha)$ 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 $K\ltimes S$ 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 $\End\mathbb{V}(m,h,\alpha)$ that are domains have zero radical. In addition, each semi-local $\End\mathbb{V}(m,h,\alpha)$ must be either a trivial extension $K\ltimes S$ or the product $K\times K$.