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

Published online by Cambridge University Press:  20 November 2018

F. Okoh  and
F. Zorzitto
F. Okoh
Affiliation:
Department of Mathematics, Wayne State University, Detroit, MI, 48202 e-mail:okoh@math.wayne.edu
F. Zorzitto
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1 e-mail:fazorzit@uwaterloo.ca
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Abstract

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The Kronecker modules, $\mathbb{V}\left( m,h,\alpha \right)$, 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 $\mathbb{V}\left( m,h,\alpha \right)$ 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 $\mathbb{V}\left( m,h,\alpha \right)$ that are domains have zero radical. In addition, each semi-local End $\mathbb{V}\left( m,h,\alpha \right)$ must be either a trivial extension $K\ltimes S$ or the product $K\times K$.

Keywords

16S5015A27
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 4 , 01 August 2008 , pp. 923 - 957
Copyright
Copyright © Canadian Mathematical Society 2008

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