Wide subcategories of d-cluster tilting subcategories

TY - JOUR

T1 - Wide subcategories of d-cluster tilting subcategories

AU - Herschend, Martin

AU - Jørgensen, Peter

AU - Vaso, Laertis

PY - 2020

Y1 - 2020

N2 - A subcategory of an abelian category is wide if it is closed under sums, summands, kernels, cokernels, and extensions. Wide subcategories provide a significant interface between representation theory and combinatorics. If Φ is a finite dimensional algebra, then each functorially finite wide subcategory of mod(Φ) is of the form φ∗(mod(Γ)) in an essentially unique way, where Γ is a finite dimensional algebra and Φ –→ φ Γ is an algebra epimorphism satisfying Tor Φ 1 (Γ, Γ) = 0. Let F ⊆ mod(Φ) be a d-cluster tilting subcategory as defined by Iyama. Then F is a d-abelian category as defined by Jasso, and we call a subcategory of F wide if it is closed under sums, summands, d-kernels, d-cokernels, and d-extensions. We generalise the above description of wide subcategories to this setting: Each functorially finite wide subcategory of F is of the form φ∗(G) in an essentially unique way, where Φ –→ φ Γ is an algebra epimorphism satisfying Tor Φ d (Γ, Γ) = 0, and G ⊆ mod(Γ) is a d-cluster tilting subcategory. We illustrate the theory by computing the wide subcategories of some d-cluster tilting subcategories F ⊆ mod(Φ) over algebras of the form Φ = kA m/(rad kA m) l.

AB - A subcategory of an abelian category is wide if it is closed under sums, summands, kernels, cokernels, and extensions. Wide subcategories provide a significant interface between representation theory and combinatorics. If Φ is a finite dimensional algebra, then each functorially finite wide subcategory of mod(Φ) is of the form φ∗(mod(Γ)) in an essentially unique way, where Γ is a finite dimensional algebra and Φ –→ φ Γ is an algebra epimorphism satisfying Tor Φ 1 (Γ, Γ) = 0. Let F ⊆ mod(Φ) be a d-cluster tilting subcategory as defined by Iyama. Then F is a d-abelian category as defined by Jasso, and we call a subcategory of F wide if it is closed under sums, summands, d-kernels, d-cokernels, and d-extensions. We generalise the above description of wide subcategories to this setting: Each functorially finite wide subcategory of F is of the form φ∗(G) in an essentially unique way, where Φ –→ φ Γ is an algebra epimorphism satisfying Tor Φ d (Γ, Γ) = 0, and G ⊆ mod(Γ) is a d-cluster tilting subcategory. We illustrate the theory by computing the wide subcategories of some d-cluster tilting subcategories F ⊆ mod(Φ) over algebras of the form Φ = kA m/(rad kA m) l.

KW - Algebra epimorphism

KW - D-abelian category

KW - D-cluster tilting subcategory

KW - D-homological pair

KW - D-pseudoflat morphism

KW - Functorially finite subcategory

KW - Higher homological algebra

KW - Wide subcategory

UR - http://www.scopus.com/inward/record.url?scp=85080947677&partnerID=8YFLogxK

U2 - 10.1090/tran/8051

DO - 10.1090/tran/8051

M3 - Journal article

SN - 0002-9947

VL - 373

SP - 2281

EP - 2309

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 4

ER -