Stabilization of the Spread-Global Dimension

Abstract

Motivated by constructions from applied topology, there has been recent interest in the homological algebra of linear representations of posets, particularly in the context of homological algebra relative to non-standard exact structures. A prominent example is the spread exact structure on the category of representations of a fixed poset, in which the indecomposable projectives are the spread representations (that is, the indicator representations of convex and connected subsets). The spread-global dimension is known to be finite for finite posets and not uniformly bounded on the collection of all Cartesian products between two arbitrary finite total orders. It was conjectured in [AENY23] that the spread-global dimension is uniformly bounded on the collection of all Cartesian products between a fixed finite total order and an arbitrary finite total order. We provide a positive answer to this conjecture and, more generally, prove that the spread-global dimension is uniformly bounded on the collection of all Cartesian products between a fixed finite poset and an arbitrary finite total order. In doing so, we also establish the existence of finite spread-resolutions for finitely presented representations of arbitrary grid posets.