Abstract

We provide a unified approach, via a deformation theory for incidence algebras that we introduce, to several types of representations with finiteness conditions, as well as to the combinatorial algebras which produce them. We show that for finite-dimensional algebras over infinite fields, modules with finitely many orbits, or with finitely many invariant subspaces, or that are distributive, coincide (and further coincide with thin modules in the acyclic case). Incidence algebras produce examples of such modules via their principal projective modules, and we show that algebras which are locally hereditary, and whose indecomposable projectives are distributive, or equivalently, which have finitely many ideals, are precisely the deformations of incidence algebras. New characterizations of incidence algebras are obtained, such as that they are exactly the algebras which have a faithful thin module. As a main consequence, we show that every thin module comes from an incidence algebra, i.e., if $V$ is thin (and, in particular, if $V$ is distributive and $A$ is acyclic), then $A∕\mathrm{ann}<!--FUNCTION\; APPLICATION-->(V)$ is an incidence algebra and $V$ can be presented as its defining representation. As applications, other results in the literature are rederived and a positive answer to the accessibility question of Ringel and Bongartz, in the distributive case, is given.

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