Partial Orders in Representation Theory of Algebras

Abstract

In this paper we investigate some partial orders used in representation theory of algebras. Let $K$ be a commutative ring, $Lambda$ a finitely generated $K$-algebra and $d$ a natural number. We then study partial orders on the set of isomorphism classes of $Lambda$-modules of length $d$. The orders degeneration, virtual degeneration and hom-order are discussed. The main purpose of the paper is to study the relation $leq_n$ constructed by considering the ranks of $ntimes n$-matrices over $Lambda$ as $K$-endomorphisms on $M^n$ for a $Lambda$-module $M$. We write $Mleq_n N$ when for any $ntimes n$-matrix the rank with respect to $M$ is greater than or equal to the rank with respect to $N$. We study these relations for various algebras and determine when $leq_n$ is a partial order.