On Krull-Gabriel dimension and Galois coverings

Pastuszak, Grzegorz

doi:10.1016/j.aim.2019.04.035

Cited by 4 publications

(8 citation statements)

References 34 publications

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“…where i g and π g are the canonical injection and projection, respectively, and if g = h, then Φ(ι) X,g,h = 0. As it is shown in [P,Theorem 5.5] the functor Φ is a G-precovering.…”

Section: Galois Precovering For Functor Categoriesmentioning

confidence: 90%

“…The above natural isomorphisms follow from [P,Proposition 5.3] (or Proposition 4.2) and the exactness of the functor F λ .…”

Section: S(mod-a)mentioning

confidence: 99%

“…According to the diagram given in Proposition 4.2, we observe that the functor Φ is dense. See [P,Remark 5.6] for a discussion about density of the functor Φ. We also add that Proposition 4.4 implies that Φ is dense if and only if so is HF λ .…”

Section: Gabriel's Theorem For (Mono)morphism Categorymentioning

confidence: 99%

“…Observe that the group G acts on F(mod-A). Indeed, given a functor T ∈ F(mod-A) and g ∈ G, we define g T ∈ F(mod-A) (see [P,Section 5] for a argument to show that it belongs to F(mod-A)) such that g T (X) = T ( g −1 X) and g T (f ) = T ( g −1 f ), for any object X ∈ mod-A and A-homomorphism f . Note that g −1 X and g −1 f are defined by the action G on mod-A, which introduced earlier.…”

Section: Galois Precovering For Functor Categoriesmentioning

confidence: 99%

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Covering theory, (mono)morphism categories and stable Auslander algebras

Hafezi,

Mahdavi

2020

Preprint

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Let A be a locally bounded k-category and G a torsion-free group of k-linear automorphisms of A acting freely on the objects of A, and F : A → B is a Galois functor. We extend naturally the push-down functor F λ to the functor HF λ : H(mod-A) → H(mod-B), resp. SF λ : S(mod-A) → S(mod-B), between the corresponding morphism categories, resp. monomorphism categories, of mod-A and mod-B. Under some additional conditions, we show that H(mod-A), resp. S(mod-A), is locally bounded if and only if H(mod-B), resp. S(mod-B), is of finite representation type. As an application, we show that the stable Auslander algebra of a representation-finite selfinjective algebra Λ is again representationfinite if and only if Λ is of Dynkin type An with n 4.

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“…where i g and π g are the canonical injection and projection, respectively, and if g = h, then Φ(ι) X,g,h = 0. As it is shown in [P,Theorem 5.5] the functor Φ is a G-precovering.…”

Section: Galois Precovering For Functor Categoriesmentioning

confidence: 90%

“…The above natural isomorphisms follow from [P,Proposition 5.3] (or Proposition 4.2) and the exactness of the functor F λ .…”

Section: S(mod-a)mentioning

confidence: 99%

Section: Gabriel's Theorem For (Mono)morphism Categorymentioning

confidence: 99%

Section: Galois Precovering For Functor Categoriesmentioning

confidence: 99%

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Covering theory, (mono)morphism categories and stable Auslander algebras

Hafezi,

Mahdavi

2020

Preprint

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“…The case when R is a finite-dimensional algebra over a field k is studied in many papers, see [10] for an up-to-date list of results in this direction and [14,Theorem 8.3] for the most recent one concerning self-injective algebras. Of course all the known results support the conjecture of Prest.…”

Section: Introductionmentioning

confidence: 99%

On Wild Algebras and Super-Decomposable Pure-Injective Modules

Pastuszak

2022

Algebr Represent Theor

Self Cite

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Assume that k is an algebraically closed field and A is a finite-dimensional wild k-algebra. Recently, L. Gregory and M. Prest proved that in this case the width of the lattice of all pointed A-modules is undefined. Hence the result of M. Ziegler implies that there exists a super-decomposable pure-injective A-module, if the base field k is countable. Here we give a different and straightforward proof of this fact. Namely, we show that there exists a special family of pointed A-modules, called an independent pair of dense chains of pointed A-modules. This also yields the existence of a super-decomposable pure-injective A-module.

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A functorial approach to categorical resolutions

Hafezi

Keshavarz

2019

Sci. China Math.

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Using a relative version of Auslander's formula, we give a functorial approach to show that the bounded derived category of every Artin algebra admits a categorical resolution. This, in particular, implies that the bounded derived categories of Artin algebras of finite global dimension determine bounded derived categories of all Artin algebras. Hence, this paper can be considered as a typical application of functor categories, introduced in representation theory by Auslander, to categorical resolutions.

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On Krull-Gabriel dimension and Galois coverings

Cited by 4 publications

References 34 publications

Covering theory, (mono)morphism categories and stable Auslander algebras

Covering theory, (mono)morphism categories and stable Auslander algebras

On Wild Algebras and Super-Decomposable Pure-Injective Modules

A functorial approach to categorical resolutions

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