Existence of a multiplicative basis for a finitely spaced module over an aggregate

Abstract

It is proved that a finitely spaced module over a k-category admits a multiplicative basis (such a module gives rise to a matrix problem in which the allowed column transformations are determined by a module structure, the row transformations are arbitrary, and the number of canonical matrices is finite).It was proved in [1] that a finite-dimensional algebra having finitely many isoclasses of indecomposable representations admits a multiplicative basis. In [2, Secs. 4.10-4.12], an analogous hypothesis was formulated for finitely spaced modules over an aggregate and an approach to its proof was proposed. The purpose of the present paper is to prove this hypothesis. Below, k always denotes an algebraically closed field.Let us recall some definitions given in [2] (see also [3]).By definition, an aggregate A over k is a category that satisfies the following conditions: As a consequence, each X ~ A is a finite direct sum of indecomposables and the algebra of endomorphisms of each indecomposable is local.We denote by flA a spectroid of N, i.e., a full subcategory formed by chosen representatives of the isoclasses of indecomposables; let RA be the radical of A. Suppose that flA has finitely many objects. For each a, b ~ fiN,

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