2017/05/05 by Herschend, Martin, Jorgensen, Peter, Vaso, Laertis · 2 citations
#FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1705.02246
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\stackrel ϕ \longrightarrow Γ is an algebra epimorphism satisfying Tor Φ1( Γ,Γ) = 0. Let \mathcal F ⊆ mod( Φ) be a d-cluster tilting subcategory as defined by Iyama. Then \mathcal F is a d-abelian category as defined by Jasso, and we call a subcategory of \mathcal 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 \mathcal F is of the form ϕ * ( \mathcal G ) in an essentially unique way, where Φ\stackrel ϕ \longrightarrow Γ is an algebra epimorphism satisfying Tor Φd( Γ,Γ) = 0, and \mathcal G ⊆ mod( Γ) is a d-cluster tilting subcategory. We illustrate the theory by computing the wide subcategories of some d-cluster tilting subcategories \mathcal F ⊆ mod( Φ) over algebras of the form Φ= kAm / (rad kAm ) ℓ .