## 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 language | English |
---|---|

Journal | Transactions of the American Mathematical Society |

Volume | 373 |

Issue | 4 |

Pages (from-to) | 2281-2309 |

Number of pages | 29 |

ISSN | 0002-9947 |

DOIs | |

Publication status | Published - 2020 |

Externally published | Yes |

## Keywords

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