Relative homological algebra of filter modules

Abstract

Over an arbitrary ordered set we consider the class of filter modules, i.e. indicator modules of filters in the ordered set. We show that the induced exact structure of filter modules, the so-called filter exact structure, which has filter modules as relative projective modules, admits an explicit projectivization by the distributive lattice of filters. This allows the reduction of the projective-homological properties of the induced exact structure to the standard homological algebra of $k{\operatorname{Fil}(Q)}$, the incidence algebra of $k$-representations on the lattice of filters. In the case of finite join-semilattices, we leverage the combinatorial structure of the distributive lattice of filters to determine the standard global dimension of $k{\operatorname{Fil}(Q)}$. By virtue of projectivization we provide lower and upper bounds for the global dimension of the filter exact structure. Moreover, we call an ordered set filter-tiny if its relative injective modules one-to-one-correspond to the standard injective modules. We exploit relative derived functors to show that filter-tiny ordered sets do not admit relative injective prehulls if the relative global dimension is larger than 1. It is open to which classes of ordered sets $Q$ the filter-tininess condition applies, and what the relative injective modules are in the case of non-filter-tiny ordered sets.