Wide subcategories of d-cluster tilting subcategories

Martin Herschend, Peter Jørgensen, Laertis Vaso

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Abstract

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.

Original languageEnglish
JournalTransactions of the American Mathematical Society
Volume373
Issue4
Pages (from-to)2281-2309
Number of pages29
ISSN0002-9947
DOIs
Publication statusPublished - 2020
Externally publishedYes

Keywords

  • Algebra epimorphism
  • D-abelian category
  • D-cluster tilting subcategory
  • D-homological pair
  • D-pseudoflat morphism
  • Functorially finite subcategory
  • Higher homological algebra
  • Wide subcategory

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