On algebras of strongly unbounded representation type.

*(English)*Zbl 0584.16017The 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)].

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) |