Summand-injectivity of interval covers and monotonicity of interval resolution global dimensions

TL;DR

This paper investigates the properties of interval covers and resolutions in multi-parameter persistence modules, establishing injectivity, monotonicity, and classifying posets with zero global dimension.

Contribution

It proves summand-injectivity of interval covers, demonstrates the monotonicity of interval resolution global dimensions, and classifies posets with zero global dimension.

Findings

Interval covers are injective on indecomposable summands.

Interval resolution global dimension is monotonic with respect to subposets.

Posets with zero interval resolution global dimension are fully classified.

Abstract

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.