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

Okoh, F. ; Zorzitto, F.

Canadian journal of mathematics, Tome 60 (2008) no. 4, pp. 923-957

Résumé

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 $K\times K$ .

Détail

DOI : 10.4153/CJM-2008-039-2

Mots-clés : 16S50, 15A27

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