Stabilization of the Spread-Global Dimension

TL;DR

This paper proves that the spread-global dimension remains uniformly bounded when considering Cartesian products of a fixed finite poset with any finite total order, advancing understanding in homological algebra of poset representations.

Contribution

It confirms the conjecture that spread-global dimension is uniformly bounded in this context and establishes finite spread-resolutions for representations of grid posets.

Findings

Spread-global dimension is uniformly bounded for Cartesian products with a fixed finite poset.

Finite spread-resolutions exist for finitely presented representations of grid posets.

The result generalizes previous finiteness results to broader classes of poset products.

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.