Commutative algebras of rational function matrices as endomorphisms of Kronecker modules I

Given a field K, a Kronecker module is a pair of K-linear spaces (U,V) together with a K-bilinear map K2×U→V. In finite dimensions this is also the notion of pencils of matrices. Every K[X]-module can be construed as a Kronecker module. In particular the K[X]-submodules of K(X) give rise to the Kronecker modules Rh where h is a height function, i.e. a function h:K {∞}→{∞,0,1,2,…}. The K[X]-module K[X] itself gives the Kronecker module that goes with the height function which is ∞ at ∞ and 0 on K. The modules Rh that are infinite-dimensional come up precisely when h attains the value ∞ or when h is stictly positive on an infinite subset of K {∞}. The endomorphism algebra of Rh is called a pole algebra. Those Kronecker modules that are extensions of finite-dimensional submodules of by infinite-dimensional Rh lead to some engaging problems with matrix algebras. This is because the endomorphisms of such constitute a K-subalgebra of n×n matrices over K(X), which is commutative if the extension is indecomposable. Among the algebras that are known to arise in the 2 × 2 case, when the extension is by are the coordinate rings of all elliptic curves. In this paper we replace by an arbitrary infinite-dimensional Rh. The following new algebras are realized: infinite-dimensional pole algebras End Rh where h=0 on an infinite subset of K; maximal subalgberas of A×B for some pole algebras A,B; the quasi local ring K∝S where S is a K-vector space of dimension at most card K. In the process we identify those height functions h that will tolerate an indecomposable extension having non-trivial endomorphisms