Representation theory of the diagram An over the ring k[[x]]

Abstract

Fix R = k[[x]]. Let Qn be the category whose objects are ((M₁,...,Mn),(f₁,...,fn-1)) where each Mi is a free R-module and fi:Mi⟶Mi+1 for each i=1,...,n-1, and in which the morphisms are the obvious ones. Let βn be the full subcategory of Ωn in which each map fi is a monomorphism whose cokernel is a torsion module. It is shown that there is a full dense functor Ωn⟶βn. If X is an object of βn, we say that X diagonalizes if X is isomorphic to a direct sum of objects ((M₁,...,Mn),(f₁,...,fn-1)) in which each Mi is of rank one. We establish an algorithm which diagonalizes any diagonalizable object X of βn, and which fails only in case X is not diagonalizable. Let Λ be an artin algebra of finite type. We prove that for a fixed C in mod(Λ) there are only finitely many modules A in mod(Λ) (up to isomorphism) for which a short exact sequence of the form 0⟶A⟶B⟶C⟶0 is indecomposable.