A new proof of a theorem of Nazarova and Roiter. (Une nouvelle preuve d’un théorème de Nazarova et Roiter.) (French) Zbl 0586.16012

Author’s summary: “Our objective is to give a new proof of the following theorem: Given a representation-infinite finite-dimensional algebra \(A\) over an algebraically closed field \(k\), there are infinitely many dimensions for which \(A\) admits infinitely many isoclasses of indecomposable representations.”
Let us sketch the idea of the proof: By results of R. Bautista, P. Gabriel, A. Roĭter and L. Salmerón [Invent. Math. 81, 217–285 (1985; Zbl 0575.16012)] resp. K. Bongartz [Math. Ann. 269, 1–12 (1984; Zbl 0552.16012)] the situation can be reduced to a finite ray category \(\mathcal P\). Suppose \(\mathcal P\) is a phantom, that is \(\mathcal P\) is of infinite representation type with only finitely many indecomposables in almost all dimensions; the author then shows (1) the quiver of \(\mathcal P\) contains a cycle and (2) the universal cover \(\tilde{\mathcal P}\) of \({\mathcal P}\) contains a phantom as full subcategory.
Proving that the quiver of the universal cover of a mild phantom doesn’t contain a cycle the author gets a contradiction.


16Gxx Representation theory of associative rings and algebras
16P10 Finite rings and finite-dimensional associative algebras
16B50 Category-theoretic methods and results in associative algebras (except as in 16D90)