Adjoint Functors and Representation Dimensions

Xi, Chang Chang

doi:10.1007/s10114-005-0685-3

Cited by 13 publications

(6 citation statements)

References 23 publications

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“…Note that in this case, T is separating and splitting. So, the above theorem also generalizes the result [11,Theorem 6.5]. For APR-tilting modules, the next step is: when the simple module S has injective dimension 2, what can we say about the representation dimensions of A and B.…”

Section: Introductionmentioning

confidence: 65%

“…In [11,Theorem 6.5], the tilting modules considered are separating and splitting. (2) Note that under the assumption of the above theorem, the algebras A and B are not stably equivalent in general.…”

Section: Remarkmentioning

confidence: 99%

“…In [11], Xi proved that the endomorphism algebra of certain APR-tilting module T over an algebra A has the same representation dimension as A. This motivates the following question.…”

Section: Introductionmentioning

confidence: 96%

“…Let B be the endomorphism algebra of the APR-tilting module T . In [11], it is proved that if the injective dimension of S is 1, then A and B have the same representation dimension. Note that in this case, T is separating and splitting.…”

Section: Introductionmentioning

confidence: 99%

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Tilting modules and representation dimensions

Chen

2010

Journal of Algebra

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In this paper, we compare the representation dimensions of two algebras linked by certain tilting modules. Our main results can be stated as follows: Suppose T is a tilting module over A and B = End A (T ). Then: (1) If T is separating and splitting, then rep.dim( A) = rep.dim(B); (2) If T = P ⊕ τ −1 S is an APR-tilting module and the injective dimension of S is at most 2, then rep.dim(B) rep.dim( A) + 1.

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Section: Introductionmentioning

confidence: 65%

Section: Remarkmentioning

confidence: 99%

“…In [11], Xi proved that the endomorphism algebra of certain APR-tilting module T over an algebra A has the same representation dimension as A. This motivates the following question.…”

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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Tilting modules and representation dimensions

Chen

2010

Journal of Algebra

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“…For further information about advances on the finitistic dimension conjecture as well as its relationship with other homological conjectures in the representation theory of Artin algebras, we may refer to [7,29,30,33,34] and the references therein. For representation dimension, we refer to [2,4,10,28,31,32] and the references therein.…”

Section: Lemma 21 For Any Artin Algebra a There Is A Function ψ Whmentioning

confidence: 99%

On the finitistic dimension conjecture, III: Related to the pair eAe⊆A

2008

Journal of Algebra

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Let A be an Artin algebra and e an idempotent element in A. In this paper, we use co-homological conditions on A to control the finitistic dimension of eAe. Such a consideration is of particular interest for understanding the finitistic dimension conjecture. Let us denote the finitistic dimension and global dimension of A by fin.dim(A) and gl.dim(A), respectively. Suppose gl.dim(A) 4. Then fin.dim(eAe) < ∞ if one of the following conditions holds: (1) A/AeA has representation dimension at most 3; (2) Ω −3 A (A) is an A/AeA-module; (3) proj.dim( A S) 3 for all simple A/AeA-modules S. This result can be considered as a first step to the question of whether gl.dim(A) 4 implies fin.dim(eAe) < ∞. Moreover, we show the following: Let A be an arbitrary Artin algebra and e an idempotent element of A such that the * -syzygy dimension or the Gorenstein dimension of the eAe-module Ae is finite. If fin.dim(A) < ∞, then fin.dim(eAe) < ∞.

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Hopf Extensions of CM-finite Artin Algebras

Gao

2011

Algebr Represent Theor

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Adjoint Functors and Representation Dimensions

Cited by 13 publications

References 23 publications

Tilting modules and representation dimensions

Tilting modules and representation dimensions

On the finitistic dimension conjecture, III: Related to the pair eAe⊆A

Hopf Extensions of CM-finite Artin Algebras

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