Extensions of homologically finite subcategories

Sikko, Svein Arne; Smalø, Sverre O.

doi:10.1007/bf01236075

Cited by 10 publications

(7 citation statements)

References 5 publications

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“…The following result is due to Gentle and Todorov [6], which extends the corresponding results in artin algebras and coherent rings, obtained by Sikko and Smalø (see [11,Theorem 2.6] and [12]). Proof.…”

Section: Main Theoremssupporting

confidence: 69%

“…The following result is due to Gentle and Todorov [6], which extends the corresponding results for Artin algebras and coherent rings, obtained by Sikko and Smalø (see [12,Theorem 2.6] and [13] Example. We will show that Gentle-Todorov's theorem may fail for an arbitrary abelian category.…”

Section: Triangulated Version Of Gentle-todorov's Theoremsupporting

confidence: 68%

“…The author would like to thank Prof. Henning Krause very much, who half a year ago asked him to give a direct proof to their surprising result [8, Corollary 0.3]. Thanks also go to Dr. Yu Ye for pointing out the references [11,12] and special thanks go to Prof. Apostolos Beligiannis who kindly pointed out to the author that Theorem 1.1 and the proof were originally due to Gentle and Todorov.…”

mentioning

confidence: 98%

“…The author is indebted to Prof. Henning Krause, who asked him to give a direct proof to their surprising result [8, Corollary 0.3]. Thanks also go to Prof. Yu Ye for pointing out the references [12,13] and to Prof. Apostolos Beligiannis who kindly pointed out to the author that Theorem 1.1 was due to Gentle and Todorov [6]. The final version of this paper was completed during the author's stay at the University of Paderborn with a support by Alexander von Humboldt Stiftung.…”

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confidence: 99%

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Extensions of covariantly finite subcategories

Chen

2009

Arch. Math.

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Gentle and Todorov proved that in an abelian category with enough projective objects, the extension subcategory of two covariantly finite subcategories is covariantly finite. We give an example to show that Gentle-Todorov's theorem may fail in an arbitrary abelian category; however we prove a triangulated version of Gentle-Todorov's theorem which holds for arbitrary triangulated categories; we apply Gentle-Todorov's theorem to obtain short proofs of a classical result by Ringel and a recent result by Krause and Solberg. (2000). Primary 18E30; Secondary 16G10. Mathematics Subject Classification Triangulated version of Gentle-Todorov's theorem.Let C be an additive category. By a subcategory X of C we always mean a full additive subcategory. Let X be a subcategory of C and let M ∈ C. A morphism x M : M −→ X M is called a left X -approximation of M if X M ∈ X and every morphism from M to an object in X factors through x M . The subcategory X is said to be covariantly finite in C, if every object in C has a left X -approximation. The notions of left X -approximation and covariantly finite subcategories are also known as X -preenvelope and preenveloping subcategories, respectively. For details, see [3][4][5].Let C be an abelian category and let X and Y be two subcategories. Let X * Y be the subcategory consisting of objects Z such that there is a short exact sequence 0 −→ X −→ Z −→ Y −→ 0 with X ∈ X and Y ∈ Y; it is called the extension subcategory of Y by X . Note that the operation " * " on subcategories is associative. Recall that an abelian category C has enough

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Section: Main Theoremssupporting

confidence: 69%

Section: Triangulated Version Of Gentle-todorov's Theoremsupporting

confidence: 68%

mentioning

confidence: 98%

mentioning

confidence: 99%

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Extensions of covariantly finite subcategories

Chen

2009

Arch. Math.

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“…The importance of the operation "⋆" for our purposes lies in the following fact from [33]; see also [16].…”

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confidence: 99%

Finitistic Dimension Conjectures via Gorenstein Projective Dimension

Moradifar¹,

Šaroch²

2020

Preprint

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It is a well-known result of Auslander and Reiten that contravariant finiteness of the class P fin ∞ (of finitely generated modules of finite projective dimension) over an Artin algebra is a sufficient condition for validity of finitistic dimension conjectures. Motivated by the fact that finitistic dimensions of an algebra can alternatively be computed by Gorenstein projective dimension, in this work we examine the Gorenstein counterpart of Auslander-Reiten condition, namely contravariant finiteness of the class GP fin ∞ (of finitely generated modules of finite Gorenstein projective dimension), and its relation to validity of finitistic dimension conjectures. It is proved that contravariant finiteness of the class GP fin ∞ implies validity of the second finitistic dimension conjecture over left artinian rings. In the more special setting of Artin algebras, however, it is proved that the Auslander-Reiten sufficient condition and its Gorenstein counterpart are virtually equivalent in the sense that contravariant finiteness of the class GP fin ∞ implies contravariant finiteness of the class P fin ∞ over any Artin algebra, and the converse holds for Artin algebras over which the class GP fin 0 (of finitely generated Gorenstein projective modules) is contravariantly finite.

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Relations for relative Grothendieck groups of rings of finite representation type

Generalov¹

1998

J Math Sci

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For a ring R of finite representation type the set of free generators is described for the group of relations of the relative Grothendieck group Ko(R, F) corresponding to an additive subfunctor F of the functor Ext,, which yields an extension to the absolute case of the corresponding results of Butler and Auslander. Bibliography: 8 titles.Let R be a ring of finite representation type, i.e., R is a left and a right Artinian ring over which there are only finitely many nonisomorphic finitely generated indecomposable (left and right) modules. The category of finitely generated right R-modules is denoted by mod R, ind R denotes the full set of nonisomorphic indecomposable modules in mod R and p.ind R is the subset of ind R that consists of projective modules. In the sequel, we consider only finitely generated right R-modules.

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Extensions of homologically finite subcategories

Cited by 10 publications

References 5 publications

Extensions of covariantly finite subcategories

Extensions of covariantly finite subcategories

Finitistic Dimension Conjectures via Gorenstein Projective Dimension

Relations for relative Grothendieck groups of rings of finite representation type

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