2023/08/29 by Toshitaka Aoki, Aoki, Toshitaka, Emerson G. Escolar +3 · 2 citations
Computer Science · Mathematics · #16E05 #16G20 #18G25 #55N31 #Algebraic Topology (math.AT) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Representation Theory (math.RT) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2308.14979
openalex publication_date 2023/08/29 · openalex created_date 2023/08/31 · openalex updated_date 2026/07/28
Recently, there is growing interest in the use of relative homology algebra to develop invariants using interval covers and interval resolutions (i.e., right minimal approximations and resolutions relative to interval-decomposable modules) for multi-parameter persistence modules. In this paper, the set of all interval modules over a given poset plays a central role. Firstly, we show that the restriction of interval covers of modules to each indecomposable direct summand is injective. This result suggests a way to simplify the computation of interval covers. Secondly, we show the monotonicity of the interval resolution global dimension, i.e., if Q is a full subposet of P, then the interval resolution global dimension of Q is not larger than that of P. Finally, we provide a complete classification of posets whose interval resolution global dimension is zero.