Wide subcategories of 𝑑-cluster tilting subcategories

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 \Phi is a finite dimensional algebra, then each functorially finite wide subcategory of mod ⁡ ( Φ ) \operatorname {mod}( \Phi ) is of the form ϕ ∗ ( mod ⁡ ( Γ ) ) \phi _{ {\textstyle *}}\big ( \operatorname {mod}( \Gamma ) \big ) in an essentially unique way, where Γ \Gamma is a finite dimensional algebra and Φ ⟶ ϕ Γ \Phi \stackrel { \phi }{ \longrightarrow } \Gamma is an algebra epimorphism satisfying Tor 1 Φ ⁡ ( Γ , Γ ) = 0 \operatorname {Tor}^{ \Phi }_1( \Gamma ,\Gamma ) = 0 . Let F ⊆ mod ⁡ ( Φ ) \mathscr {F} \subseteq \operatorname {mod}( \Phi ) be a d d -cluster tilting subcategory as defined by Iyama. Then F \mathscr {F} is a d d -abelian category as defined by Jasso, and we call a subcategory of F \mathscr {F} wide if it is closed under sums, summands, d d -kernels, d d -cokernels, and d d -extensions. We generalise the above description of wide subcategories to this setting: Each functorially finite wide subcategory of F \mathscr {F} is of the form ϕ ∗ ( G ) \phi _{ {\textstyle *}}( \mathscr {G} ) in an essentially unique way, where Φ ⟶ ϕ Γ \Phi \stackrel { \phi }{ \longrightarrow } \Gamma is an algebra epimorphism satisfying Tor d Φ ⁡ ( Γ , Γ ) = 0 \operatorname {Tor}^{ \Phi }_d( \Gamma ,\Gamma ) = 0 , and G ⊆ mod ⁡ ( Γ ) \mathscr {G} \subseteq \operatorname {mod}( \Gamma ) is a d d -cluster tilting subcategory. We illustrate the theory by computing the wide subcategories of some d d -cluster tilting subcategories F ⊆ mod ⁡ ( Φ ) \mathscr {F} \subseteq \operatorname {mod}( \Phi ) over algebras of the form Φ = k A m / ( rad k A m ) ℓ \Phi = kA_m / (\operatorname {rad}\,kA_m )^{ \ell } .