On algebras of strongly unbounded representation type. (English) Zbl 0584.16017

Bautista, Raymundo

Comment. Math. Helv. 60, 392-399 (1985).

The second Brauer-Thrall conjecture asserts that every representation- infinite finite dimensional algebra over an algebraically closed field k is of strongly unbounded type, that is, there is an infinite sequence of positive numbers \(d_ 1<d_ 2<...<d_ s<..\). such that for each \(d_ i\) there are infinitely many isomorphism classes of indecomposable modules with k-dimension \(d_ i\). The first (affirmative) proof of this conjecture, using vector space category methods, due to L. A. Nazarova and A. V. Rojter [Mitt. Math. Sem. Gießen 115, 1-153 (1975; Zbl 0315.16021)] contains ”delicate points”.
A new interesting geometric proof for fields k of characteristic different from 2 is presented here. Together with a recent result of K. Bongartz [Comment. Math. Helv. 60, 400-410 (1985)] this gives a proof for arbitrary characteristic. The proof presented here depends heavily on the work of R. Bautista, P. Gabriel, A. V. Rojter and L. Salmerón on multiplicative bases [Invent. Math. 81, 217-285 (1985; Zbl 0575.16012)] and uses covering techniques. Independent proofs of the second Brauer-Thrall conjecture have been also obtained by U. Fischbacher [C. R. Acad. Sci., Paris, Sér. I 300, 259-262 (1985)] and O. Bretscher and G. Todorov [Lect. Notes Math. 1177, 50-54 (1986)].

Reviewer: A.Skowroński

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)
16Exx	Homological methods in associative algebras
16D70	Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)

Keywords:

ray-category; essential contour; ghost; abelian covering; second Brauer- Thrall conjecture; representation-infinite finite dimensional algebra; strongly unbounded type

Citations:

Zbl 0315.16021; Zbl 0575.16012