Relations Between the Strong Global Dimension, Complexes of Fized Size and Derived Category

2020/11/10 by Yohny Calderón-Henao, Calderón-Henao, Y., F. Gallego-Olaya +3
Mathematics · #18Gxx (Primary) #Advanced Topics in Algebra #Algebraic structures and combinatorial models #F.2.2 #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2011.05220

openalex publication_date 2020/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let ℤ be the integer numbers, \mathbbK an algebraically closed field, Λ a finite dimensional \mathbbK-algebra, modΛ the category of finitely generated right modules, projΛ the full subcategory of modΛ consisting of all projective Λ-modules, and C_n(projΛ) the bounded complexes of projective Λ-modules of fixed size for any integer n≥2. We find an algorithm to calculate the strong global dimension of Λ, when Λ is a finite strong global dimension and derived discrete, using the Auslander-Reiten quivers of the categories C_n(projΛ). Also, we show the relation between the Auslander-Reiten quiver of the bounded derived category D^b(Λ) and the Auslander-Reiten quiver of C_η+1(projΛ), where η=s.gl.dim(Λ) (strong global dimension of Λ).

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