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Cycles of modules and finite representation type
Author(s) -
Białkowski Jerzy,
Skowroński Andrzej
Publication year - 2016
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdw030
Subject(s) - indecomposable module , mathematics , type (biology) , representation (politics) , pure mathematics , algebra over a field , infinity , mathematical analysis , ecology , politics , political science , law , biology
We prove that an artin algebra A is of finite representation type if and only if almost all finitely generated indecomposable A ‐modules are cycle‐finite. Moreover, we prove that every cycle‐finite module category of an artin algebra A of infinite representation type contains infinitely many left stable and infinitely many right stable directing modules. In particular, the main results of the paper provide solution of the open problem concerning infinity of directing modules over cycle‐finite artin algebras of infinite representation type.