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Tilting Sets on Cylinders
Author(s) -
Happel Dieter
Publication year - 1988
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-56.2.260-s
Subject(s) - mathematics , endomorphism , iterated function , combinatorics , type (biology) , enumeration , set (abstract data type) , discrete mathematics , structured program theorem , algebra over a field , pure mathematics , mathematical analysis , ecology , computer science , biology , programming language
We are indebted to P. Gabriel who communicated to us the following counter‐example to Theorem 1. Let Δ be the Dynkin diagram D 6 with enumeration of the verticesΔ → = 1 → 2 → 3 → 4 ↘ 6 ↗ 5Consider the subset of ZΔ → given by T ′ = {(0, 5), (2, 1), (2, 5), (4, 1), (6, 1), (8, 1)}. Then T = π( T ′) (compare §1.1) is a cycle‐free tilting set whose endomorphism algebra is not an iterated tilted algebra. However, the following theorem is true and is proved in [1] using methods from §4.2. THEOREM. Let T be a tilting set in M (Δ) such that A = End T is simply‐connected . Then A is an iterated tilted algebra of type Δ.