The modules in any component of theAR-quiver of a wild hereditary Artin algebra are uniquely determined by their composition factors

Let A be an Artin algebra. Then there are finitely many non-isomorphic simple A-modules. Suppose S1, S2, …, Sn form a complete list of all non-isomorphic simple A-modules and we fix this ordering of simple modules. Let Pi and Qi be the projective cover and the injective envelope of Si respectively. With this order of simple modules we define for each i the standard module Δ(i) to be the maximal quotient of Pi with composition factors Sj with j [les ] i. Let Δ be the set of all these standard modules Δ(i). We denote by [Fscr ](Δ) the subcategory of A-mod whose objects are the modules M which have a Δ-filtration, namely there is a finite chain0 = M0 ⊂ M1 ⊂ M2 ⊂ … ⊂ Mt = Mof submodules of M such that Mi/Mi−1 is isomorphic to a module in Δ for all i. The modules in [Fscr ](Δ) are called Δ-good modules. Dually, we define the costandard module ∇(i) to be the maximal submodule of Qi with composition factors Sj with j [les ] i and denote by ∇ the collection of all costandard modules. In this way, we have also the subcategory [Fscr ](∇) of A-mod whose objects are these modules which have a ∇-filtration. Of course, we have the notion of ∇-good modules. Note that Δ(n) is always projective and ∇(n) is always injective.

Download Full-text

On Algebras Stably Equivalent to an Hereditary Artin Algebra

Canadian Journal of Mathematics ◽

10.4153/cjm-1978-070-8 ◽

1978 ◽

Vol 30 (4) ◽

pp. 817-829 ◽

Cited By ~ 7

Author(s):

María Inés Platzeck

Keyword(s):

Finitely Generated ◽

Artin Algebra ◽

Finitely Generated Module ◽

Artin Ring ◽

Finitely Generated Modules ◽

Hereditary Artin Algebra

Let Λ be an artin algebra, that is, an artin ring that is a finitely generated module over its center C which is also an artin ring. We denote by mod Λ the category of finitely generated left Λ-modules. We recall that the category of finitely generated modules modulo projectives is the category given by the following data: the objects are the finitely generated Λ-modules.

Download Full-text

The Global Dimension of the Endomorphism Algebra of a Generator-cogenerator for a Subcategory

Algebra Colloquium ◽

10.1142/s1005386713000424 ◽

2013 ◽

Vol 20 (03) ◽

pp. 443-456

Author(s):

Jingjing Guo

Keyword(s):

Global Dimension ◽

Endomorphism Algebra ◽

Artin Algebra ◽

Hereditary Artin Algebra

Let A be a hereditary Artin algebra and T a tilting A-module. The possibilities for the global dimension of the endomorphism algebra of a generator-cogenerator for the subcategory T⊥ in A-mod are determined in terms of relative Auslander-Reiten orbits of indecomposable A-modules in T⊥.

Download Full-text