Computing Minimal Injective Resolutions of Sheaves on Finite Posets

TL;DR

This paper presents two new methods for constructing minimal injective resolutions of sheaves on finite posets, providing an algorithm, topological interpretation, and complexity bounds.

Contribution

Introduces two novel methods and an algorithm for minimal injective resolutions of sheaves on finite posets, with topological insights and complexity analysis.

Findings

Existence and uniqueness of minimal injective resolutions established

Algorithm for constructing minimal injective resolutions provided

Asymptotically tight bounds on computational complexity derived

Abstract

In this paper we introduce two new methods for constructing injective resolutions of sheaves of finite-dimensional vector spaces on finite posets. Our main result is the existence and uniqueness of a minimal injective resolution of a given sheaf and an algorithm for its construction. For the constant sheaf on a simplicial complex, we give a topological interpretation of the multiplicities of indecomposable injective sheaves in the minimal injective resolution, and give asymptotically tight bounds on the complexity of computing the minimal injective resolution with our algorithm.