2025/09/22 by Peter Jørgensen, Jorgensen, Peter, Emre Sen +1
Mathematics · #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2509.17922
In this work, we provide a simple way to construct d-abelian categories via bounded derived categories for certain values of d. Namely, let \mathcal C be an abelian category, and let \mathcal C[0,m] denote the full subcategory of the bounded derived category of \mathcal C whose objects X satisfy that H_*(X) is concentrated in degrees j where 0 ≤ j ≤ m. We prove that if \mathcal C is hereditary, then \mathcal C[0,m] is a d-abelian category where d = 3m + 1. Beyond offering a uniform method for constructing d-abelian categories, this construction allows us to create d-abelian categories that exhibit some unexpected properties depending on the choice of the category \mathcal C. For instance, if \mathcal C is the category of abelian groups, then \mathcal C[0,m] is a d-abelian category which is not \mathbbK-linear over a field \mathbbK but has set indexed products and coproducts. Similarly, if \mathcal C is the category of coherent sheaves over certain algebraic curves, then \mathcal C[0,m] is a d-abelian category without enough injectives. We extend our results to (n+2)-angulated categories. Namely, let M be an n-cluster tilting object over an n-representation finite algebra and let \mathcal T be the corresponding (n+2)-angulated category with n-suspension functor Σn. We prove that the full subcategory \mathcal T[0,m] = add \bigoplusmj=0Σjn M is a d-abelian category where d = (n+2)(m+1)-2. Furthermore, we show that there is a bijection between the functorially finite wide subcategories of add M and the functorially finite repetitive wide subcategories of \mathcal T[0,m].