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.

MSC:

 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)

Citations:

Zbl 0575.16012; Zbl 0552.16012