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


Published:20080801
Printed: Aug 2008
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Abstract
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
torsionfree rank2 modules that are extensions of finitedimensional
rank1 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 nontrivial, 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 semilocal $\End\mathbb{V}(m,h,\alpha)$ must be either a trivial extension
$K\ltimes S$ or the product $K\times K$.