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Filtration‐Closed Auslander–Reiten Components for Wild Hereditary Algebras
Author(s) -
Kerner Otto
Publication year - 2000
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/s0024611500012478
Subject(s) - filtration (mathematics) , mathematics , indecomposable module , quiver , closure (psychology) , composition (language) , component (thermodynamics) , pure mathematics , combinatorics , physics , thermodynamics , linguistics , philosophy , economics , market economy
Let H = k Q be a finite‐dimensional connected wild hereditary path algebra, over some field k . Denote by H ‐reg the category of finite‐dimensional regular H ‐modules, that is, the category of modules M with τ H − m ( τ H m M ) ≅ M for all integers m , where τ H denotes the Auslander–Reiten translation. Call a filtration( ∗ ) M = M 0 ⊃ M 1 ⊃ … ⊃ M r ⊃ M r + 1 = 0of a regular H ‐module M a regular filtration if all subquotients M i / M i +1 are regular. Call a regular filtration (*) a regular composition series if it is strictly decreasing and has no proper refinement. A regular component C in the Auslander–Reiten quiver Γ ( H ) of H ‐mod is called filtration closed if, for each M ∈ add C, the additive closure of C, and each regular filtration (*) of M , all the subquotients M i / M i +1 are also in add C. We show that most wild hereditary algebras have filtration‐closed Auslander–Reiten components. Moreover, we deduce from this that there are also almost serial components, that is regular components C, such that any indecomposable X ∈C has a unique regular composition series. This composition series coincides with the Auslander–Reiten filtration of X , given by the maximal chain of irreducible monos ending at X . 1991 Mathematics Subject Classification : 16G70, 16G20, 16G60, 16E30.