LOCALLY DYNKIN QUIVERS AND HEREDITARY COALGEBRAS WHOSE LEFT COMODULES ARE DIRECT SUMS OF FINITE DIMENSIONAL COMODULES
2002; Taylor & Francis; Volume: 30; Issue: 1 Linguagem: Inglês
10.1081/agb-120006503
ISSN1532-4125
AutoresSebastian Nowak, Daniel Simson,
Tópico(s)Advanced Topics in Algebra
ResumoABSTRACT Let C be an indecomposable hereditary K-coalgebra, where K is an algebraically closed field. We prove that every left C-comodule is a direct sum of finite dimensional C-comodules if and only if C is comodule Morita equivalent (see [19] Castaño Iglesias, F. and Gómez Torrecillas, J. 1998. Wide Morita Context and Equivalences of Comodule Categories. J. Pure Appl. Algebra, 131: 213–225. [Crossref], [Web of Science ®] , [Google Scholar]) with a path K-coalgebra , where Q is a pure semisimple locally Dynkin quiver, that is, Q is either a finite quiver whose underlying graph is any of the Dynkin diagrams , , , , , , , or Q is any of the infinite quivers , , , with , shown in Sec. 2. In particular, we get in Corollaries 2.5 and 2.6 a K-coalgebra analogue of Gabriel's theorem [11] Gabriel, P. and Unzerlegbare Darstellungen, I. 1972. Manuscripta Math., 6: 71–103. [Crossref], [Web of Science ®] , [Google Scholar] characterising representation-finite hereditary K-algebras (see also [[6] Doi, Y. 1981. Homological Coalgebra. J. Math. Soc. Japan, 33: 31–50. [Crossref], [Web of Science ®] , [Google Scholar], Sec. VIII.5]). It is shown in Sec. 3 that if , then the Auslander-Reiten quiver of the category of finite dimensional left comodules has at most four connected components, and is connected if and only if Q has no sink vertices and .
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