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.
Categories: math.RT math.AT