2023/07/13 by Hideto Asashiba, Asashiba, Hideto · 2 citations
Computer Science · Mathematics · #16E05 #16G20 #16G70 #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Representation Theory (math.RT) #Topological and Geometric Data Analysis
paper · doi:10.48550/arxiv.2307.06559
openalex publication_date 2023/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Let G be a finitely generated right A-module for a finite-dimensional algebra A over a filed \Bbbk, and I the additive closure of G. We will define a I-relative Koszul coresolution K\bullet(V) of an indecomposable direct summand V of G, and show that for a finitely generated A-module M, the I-relative i-th Betti number for M at V is given as the \Bbbk-dimension of the i-th homology of the I-relative Koszul complex KV(M)\bullet:=HomA(K\bullet(V),M) of M at V for all i ≥ 0. This is applied to investigate the minimal interval resolution/coresolution of a persistence module M, e.g., to check the interval decomposability of M, and to compute the interval approximation of M.