Dimensions of triangulated categories with respect to subcategories

Aihara, Takuma; Araya, Tokuji; Iyama, Osamu; Takahashi, Ryo; Yoshiwaki, Michio

doi:10.1016/j.jalgebra.2013.09.035

Cited by 14 publications

(11 citation statements)

References 33 publications

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“…Then our main result is the following, which completes a main result Theorem 5.3 in [1]. Theorem 1.1.…”

Section: Introductionsupporting

confidence: 68%

Relative derived dimensions for cotilting modules

Yoshiwaki

2017

Journal of Algebra

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Abstract. For a Noetherian ring R and a cotilting R-module T of injective dimension at least 1, we prove that the derived dimension of R with respect to the category X T is precisely the injective dimension of T by applying Auslander-Buchweitz theory and Ghost Lemma. In particular, when R is a commutative Noetherian local ring with a canonical module ω R and dim R ≥ 1, the derived dimension of R with respect to the category of maximal Cohen-Macaulay modules is precisely dim R.

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“…Then our main result is the following, which completes a main result Theorem 5.3 in [1]. Theorem 1.1.…”

Section: Introductionsupporting

confidence: 68%

Relative derived dimensions for cotilting modules

Yoshiwaki

2017

Journal of Algebra

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“…Setup 3. 1 We mostly work over the category Mod-S. In this case, our standing assumption is that S is a small category.…”

Section: Recollements Of 'Rings With Several Objects'mentioning

confidence: 99%

“…dim A, while the dimension of the stable category of finitely generated Gorenstein projective modules, Gp-A is called the stable dimension of A [1], and we will denote it by st. dim A.…”

Section: Remark 322mentioning

confidence: 99%

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On the Recollements of Functor Categories

Asadollahi

Hafezi

Vahed

2015

Appl Categor Struct

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This paper is devoted to the study of recollements of functor categories in different levels. In the first part of the paper, we start with a small category S and a maximal object s of S and construct a recollement of Mod-S in terms of Mod-End S (s) and Mod-(S \ {s}) in four different levels. In case S is a finite directed category, by iterating this argument, we get chains of recollements having some interesting applications. In the second part, we start with a recollement of rings and construct a recollement of their path rings, with respect to a finite quiver. Third part of the paper presents some applications, including recollements of triangular matrix rings, an example of a recollement in Gorenstein derived level and recollements of derived categories of N -complexes.

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“…Applying Theorem 3.2(1) to A = mod R, X = add T and n = d − 1, we have M / ∈ ⊥ T d . Combine this with[AAITY, Theorem 5.3].…”

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confidence: 95%

Remarks on dimensions of triangulated categories

Aihara

Takahashi

2019

Journal of Algebra

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In this paper, we explore when a locally finite triangulated category has dimension zero or finite representation type. We also study generation of derived categories by orthogonal subcategories.Applying the first assertion of this theorem, we deduce that for an isolated hypersurface singularity R, the stable category CM(R) is locally finite if and only if it has dimension zero, if and only if R has finite CM representation type (Corollary 2.14).The second subject of this paper, which is discussed in Section 3, is to find a generator G of a derived category D such that the dimension of D with respect to G in the sense of [AAITY] is as small as possible. This brings us a lower bound of the dimension of the derived category. The main result on this subject is the following.Theorem 1.2 (Theorem 3.2). Let A be an abelian category and X a full subcategory. Let M be an object of A admitting an exact sequence with X 0 , . . . , X n ∈ Xsuch that the corresponding element in Ext n+1 A (M, X n ) (resp. Ext n+1 A (X n , M)) is nonzero. Then M is outside ⊥ X n+1 (resp. X ⊥ n+1 ) in the derived category D(A) of A.

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Dimensions of triangulated categories with respect to subcategories

Cited by 14 publications

References 33 publications

Relative derived dimensions for cotilting modules

Relative derived dimensions for cotilting modules

On the Recollements of Functor Categories

Remarks on dimensions of triangulated categories

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