Partial orders related to the Hom-order and degenerations

Abstract

Given a finite length module M over a K-algebra Λ, an n× n-matrix A over Λ induces a K-homomorphism AM : M →M. We then define the relation ≤n by M ≤n N ⇔ `(cokerAM ) ≤ `(cokerAN ). We will show that ≤n is a partial order on the set of modules of length d (modulo isomorphisms) if n ≥ d. The results presented in this paper are from the author’s Master thesis. The author thanks S.O. Smalo for the help with both the thesis and this paper. Throughout the paper let Λ be an artin algebra with center K, and let mod Λ denote the the category of finitely generated left Λ-modules. For a Λ-module X, `(X) denotes the length of X as a K-module. For a homomorphism φ, imφ denotes its image and cokerφ denotes its cokernel. For a natural number d, let repd Λ = {X ∈ mod Λ | `(X) = d}. One can define several partial orders on repd Λ modulo isomorphisms (see [3]). Here we will look at the Hom-order and the quasiorders ≤n, and investigate for which n ≤n is a partial order on repd Λ. Definition. The relation ≤Hom on repd Λ is defined by M ≤Hom N if `(HomΛ(X,M)) ≤ `(HomΛ(X,N)) for all X ∈ mod Λ. This relation is called the Hom-order. 2000 Mathematics Subject Classification: 16E30, 16G20, 16G60.