On Artin algebras whose indecomposable modules are determined by composition factors
Victor Blasco
TL;DR
The paper investigates when Artin algebras are determined by the composition factors of their indecomposable modules, formalizing this as property $\\mathfrak{X}$. It develops a suite of reductions and equivalences—full exact embeddings, algebraic bimodules, $K$-species, and tensor algebras—to transfer the problem to combinatorial settings, proving that $\\mathfrak{X}$ holds for commutative artinian rings and characterizes it for hereditary algebras: they satisfy $\\mathfrak{X}$ exactly in finite representation type. For Artin algebras with radical square zero, $\\mathfrak{X}$ implies finite representation type, but the converse fails, with explicit counterexamples provided. Overall, $\\mathfrak{X}$ is Morita-invariant and its validity aligns with finite representation type in key classes, linking module-length data to representation type via categorical and combinatorial methods.
Abstract
It was conjectured at the end of the book "Representation theory of Artin algebras" by M. Auslander, I. Reiten and S. Smalo that an Artin algebra with the property that its finitely generated indecomposable modules are up to isomorphism completely determined by theirs composition factors is of finite representation type. Examples of rings with this property are the semisimple artinian rings and the rings of the form $\mathbb{Z}_n$. An affirmative answer is obtained for some special cases, namely, the commutative, the hereditary and the radical square zero case.