n-Abelian and n-exact categories

Jasso, Gustavo

doi:10.1007/s00209-016-1619-8

Cited by 136 publications

(193 citation statements)

References 41 publications

Supporting

Mentioning

189

Contrasting

Order By: Relevance

“…The following proposition provides motivation for introducing the class of

d

‐cluster‐tilting subcategories of

mod Λ

, cf. [, Theorem 2.2.3] (see [, Proposition 3.17] for a proof). Proposition Let

scriptM

be a

d

‐rigid subcategory of

mod Λ

.…”

Section: D‐cluster‐tilting Subcategoriesmentioning

confidence: 99%

“…The following result shows that

d

‐cluster‐tilting subcategories of

mod Λ

satisfy an analogous property, cf. [, Theorem 3.16, Proposition 3.18]. Proposition Let

scriptM

be a

d

‐cluster‐tilting subcategory of

mod Λ

and

δ : 0 \to L \to M^{1} \to \dots \to M^{d} \overset{g}{\to} N

a left

d

‐exact sequence in

scriptM

such that

g

is an epimorphism.…”

Section: D‐cluster‐tilting Subcategoriesmentioning

confidence: 99%

“…Proposition Let

scriptM

be a

d

‐cluster‐tilting subcategory of

mod Λ

. Then,

scriptM

is a

d

‐abelian category in the sense of [, Definition 3.1].…”

Section: D‐cluster‐tilting Subcategoriesmentioning

confidence: 99%

“…We recall that a (right)

normalΛ

‐module

M

d

‐ cluster‐tilting if

\begin{matrix} true \\ right prefixadd M & left = ({} X \in mod Λ ∣ \forall i \in \{1, \dots, d - 1\} : {prefixExt}_{normalΛ}^{i} false(X, M false) = 0) \\ left = ({} Y \in mod Λ ∣ \forall i \in \{1, \dots, d - 1\} : {prefixExt}_{normalΛ}^{i} false(M, Y false) = 0) . \end{matrix}

Note that a 1‐cluster‐tilting

normalΛ

‐module is precisely a representation generator of

mod Λ

. The intrinsic properties of

d

‐cluster‐tilting subcategories have been investigated in . Theorem There is a one‐to‐one correspondence between the equivalence classes of

d

‐cluster‐tilting modules for Artin

R

‐algebras

normalΛ

and Morita‐equivalence classes of

d

‐Auslander

R

‐algebras, that is Artin

R

‐algebras

normalΓ

satisfying

gl.dim Γ ⩽ d + 1 ⩽ dom.dim Γ .

The correspondence is given by

M \mapsto {prefixEnd}_{normalΛ} false(M false)

, where

M

is a

d

‐cluster‐tilting

normalΛ

‐module.…”

Section: Introductionmentioning

confidence: 99%

“…Note that a 1-cluster-tilting Λ-module is precisely a representation generator of mod Λ. The intrinsic properties of d-cluster-tilting subcategories have been investigated in [14,25,26].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

An introduction to higher Auslander-Reiten theory

Jasso

Kvamme

2018

Bull. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

This article consists of an introduction to Iyama's higher Auslander–Reiten theory for Artin algebras from the viewpoint of higher homological algebra. We provide alternative proofs of the basic results in higher Auslander–Reiten theory, including the existence of d‐almost‐split sequences in d‐cluster‐tilting subcategories, following the approach to classical Auslander–Reiten theory due to Auslander, Reiten, and Smalø. We show that Krause's proof of Auslander's defect formula can be adapted to give a new proof of the defect formula for d‐exact sequences. We use the defect formula to establish the existence of morphisms determined by objects in d‐cluster‐tilting subcategories.

show abstract

“…The following proposition provides motivation for introducing the class of

d

‐cluster‐tilting subcategories of

mod Λ

, cf. [, Theorem 2.2.3] (see [, Proposition 3.17] for a proof). Proposition Let

scriptM

be a

d

‐rigid subcategory of

mod Λ

.…”

Section: D‐cluster‐tilting Subcategoriesmentioning

confidence: 99%

“…The following result shows that

d

‐cluster‐tilting subcategories of

mod Λ

satisfy an analogous property, cf. [, Theorem 3.16, Proposition 3.18]. Proposition Let

scriptM

be a

d

‐cluster‐tilting subcategory of

mod Λ

and

δ : 0 \to L \to M^{1} \to \dots \to M^{d} \overset{g}{\to} N

a left

d

‐exact sequence in

scriptM

such that

g

is an epimorphism.…”

Section: D‐cluster‐tilting Subcategoriesmentioning

confidence: 99%

“…Proposition Let

scriptM

be a

d

‐cluster‐tilting subcategory of

mod Λ

. Then,

scriptM

is a

d

‐abelian category in the sense of [, Definition 3.1].…”

Section: D‐cluster‐tilting Subcategoriesmentioning

confidence: 99%

“…We recall that a (right)

normalΛ

‐module

M

d

‐ cluster‐tilting if

\begin{matrix} true \\ right prefixadd M & left = ({} X \in mod Λ ∣ \forall i \in \{1, \dots, d - 1\} : {prefixExt}_{normalΛ}^{i} false(X, M false) = 0) \\ left = ({} Y \in mod Λ ∣ \forall i \in \{1, \dots, d - 1\} : {prefixExt}_{normalΛ}^{i} false(M, Y false) = 0) . \end{matrix}

Note that a 1‐cluster‐tilting

normalΛ

‐module is precisely a representation generator of

mod Λ

. The intrinsic properties of

d

‐cluster‐tilting subcategories have been investigated in . Theorem There is a one‐to‐one correspondence between the equivalence classes of

d

‐cluster‐tilting modules for Artin

R

‐algebras

normalΛ

and Morita‐equivalence classes of

d

‐Auslander

R

‐algebras, that is Artin

R

‐algebras

normalΓ

satisfying

gl.dim Γ ⩽ d + 1 ⩽ dom.dim Γ .

The correspondence is given by

M \mapsto {prefixEnd}_{normalΛ} false(M false)

, where

M

is a

d

‐cluster‐tilting

normalΛ

‐module.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An introduction to higher Auslander-Reiten theory

Jasso

Kvamme

2018

Bull. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

n-extension closed subcategories of $$(n+2)$$-angulated categories

Zhou

2022

Arch. Math.

View full text Add to dashboard Cite

$$d\mathbb {Z}$$-Cluster tilting subcategories of singularity categories

Kvamme

2020

Math. Z.

View full text Add to dashboard Cite

For an exact category $${{\mathcal {E}}}$$ E with enough projectives and with a $$d\mathbb {Z}$$ d Z -cluster tilting subcategory, we show that the singularity category of $${{\mathcal {E}}}$$ E admits a $$d\mathbb {Z}$$ d Z -cluster tilting subcategory. To do this we introduce cluster tilting subcategories of left triangulated categories, and we show that there is a correspondence between cluster tilting subcategories of $${{\mathcal {E}}}$$ E and $${\underline{{{\mathcal {E}}}}}$$ E ̲ . We also deduce that the Gorenstein projectives of $${{\mathcal {E}}}$$ E admit a $$d\mathbb {Z}$$ d Z -cluster tilting subcategory under some assumptions. Finally, we compute the $$d\mathbb {Z}$$ d Z -cluster tilting subcategory of the singularity category for a finite-dimensional algebra which is not Iwanaga–Gorenstein.

show abstract

n-Abelian and n-exact categories

Cited by 136 publications

References 41 publications

An introduction to higher Auslander-Reiten theory

An introduction to higher Auslander-Reiten theory

n-extension closed subcategories of $$(n+2)$$-angulated categories

$$d\mathbb {Z}$$-Cluster tilting subcategories of singularity categories

Contact Info

Product

Resources

About