On $\Delta $–good module categories without short cycles

Deng, Bangming; Zhu, Bin

doi:10.1090/s0002-9939-00-05518-0

Cited by 5 publications

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References 9 publications

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“…Recently, it has been given attention to which sufficient finiteness conditions for modA could also be proved true for (∆). For instance, in [4] it was proved for (∆) a similar finiteness condition to the one proved in [5], that is, if (∆) does not have short cycles then (∆) is of finite type. In [9], it is proved that a finite-dimensional quasi-hereditary algebra over an infinite perfect field satisfies Brauer-Thrall II and, as consequence, that if each module in (∆) is either postprojective or preinjective then (∆) is finite.…”

Section: Introductionmentioning

confidence: 66%

Artin Algebras of Finite Type and Finite Categories of Δ-Good Modules

Silva

2016

Communications in Algebra

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We give an alternative proof to the fact that if the square of the infinite radical of the module category of an Artin algebra is equal to zero then the algebra is of finite type by making use of the theory of postprojective and preinjective partitions. Further, we use this new approach in order to get a characterization of finite subcategories of ∆-good modules of a quasihereditary algebra in terms of depth of morphisms similar to a recently obtained characterization of Artin algebras of finite type.

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Section: Introductionmentioning

confidence: 66%

Artin Algebras of Finite Type and Finite Categories of Δ-Good Modules

Silva

2016

Communications in Algebra

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Quasi-Hereditary Algebras and Quadratic Forms

Deng

2001

Journal of Algebra

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The fundamental theorem on representation{ nite quivers in 6] indicates a close connection between the representation type of a quiver and the de niteness of a certain quadratic form. Later on a similar connection has been discovered in other classi cation problems of representation theory. It turns out that there is a strong interaction of quadratic forms and the reperesentation theory of nite{dimensional algebras. Given a quasi{hereditary algebra A, instead of the complete module category, one studies the {good module category F() consisting of A{modules which have a ltration by standard modules. Analogously to a complete module category, one can associate quadratic forms (may also called Euler form and Tits form) with the {good module category F(). The aim of this paper is to study {good module categories in term of these forms. First, we study {directing and {omnipresent modules in the {good module category F() over a quasi{hereditary algebra A and show that the existence of a {directing and {omnipresent module in F() implies that all standard modules have projective dimension at most 2. By using the process of standardization introduced in 5], the study of {directing modules in F() can be reduced to the study of those over certain quasi{hereditary algebras which admit a {directing and {omnipresent module. For these quasi{hereditary algebras the quadratic forms associated with their {good module categories are well behaved. By applying this reduction, we show that F() is nite (i.e. there are only nitely many isomorphism classes of indecomposables in F()) if all indecomposables in F() are {directing. Note that this statement has been proved in 4] in a more general situation in connection with certain vector space category, but the proof presented here is more straightforward. Further, if A is connected and F() has a preprojective component, then the weak positivity of the Tits form of F() implies the niteness of F(). Secondly, we study {good module categories over hereditary algebras. Note that hereditary algebras as a particular class of quasi{hereditary algebras admit the following description: An algebra is hereditary if and only if it is quasi-hereditary Supported by Alexander von Humboldt Foundation

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