[Paper Review] Primary decomposition over partially ordered groups

Ezra Miller|arXiv (Cornell University)|Jul 31, 2020

Homotopy and Cohomology in Algebraic Topology9 references4 citations

TL;DR

This paper establishes finite primary decomposition for modules over partially ordered abelian groups with closed, polyhedral positive cones having finitely many faces. By introducing notions like faces, coprimary modules, and localization functors within this generality, it proves that downset-finite modules admit canonical, finite primary decompositions, generalizing classical monomial ideal theory to continuous and non-discrete settings.

ABSTRACT

Over any partially ordered abelian group whose positive cone is closed in an appropriate sense and has finitely many faces, modules that satisfy a weak finiteness condition admit finite primary decompositions. This conclusion rests on the introduction of basic notions in the relevant generality, such as closedness of partially ordered abelian groups, faces and their coprimary modules, and finiteness conditions as well local and global support functors for modules over partially ordered groups.

Motivation & Objective

• To extend primary decomposition theory beyond discrete, finitely generated settings to continuous, non-discrete partially ordered abelian groups.
• To define and characterize coprimary modules and faces in the context of partially ordered groups with closed positive cones.
• To establish a finiteness condition (downset-finiteness) sufficient for finite primary decomposition in this generalized setting.
• To develop localization and support functors that preserve the ambient module structure, enabling functorial isolation of coprimary elements.
• To resolve the lack of minimality in primary decompositions by identifying the geometric obstructions in non-discrete settings.

Proposed method

• Introduces the concept of a closed partially ordered abelian group, where the positive cone is topologically closed and has finitely many faces.
• Defines faces as submonoids of the positive cone that are also downsets, enabling localized analysis without altering the ambient group.
• Constructs localization functors for modules along faces, preserving the module structure and allowing functorial identification of coprimary elements.
• Introduces the downset-finiteness condition as a finiteness hypothesis for modules, ensuring finite hulls by direct sums of downset modules.
• Applies the syzygy theorem for poset modules to relate finiteness conditions to structural properties of modules.
• Uses the canonical decomposition of downsets into coprimary components (via Theorem 3.10 and Corollary 3.11) as a foundation for module-level primary decomposition.

Experimental results

Research questions

• RQ1 Can primary decomposition be generalized to modules over non-discrete, partially ordered abelian groups?
• RQ2 What conditions on the positive cone of a partially ordered group ensure finite primary decomposition of modules?
• RQ3 How can localization and support functors be defined in a way that preserves the module structure over arbitrary partially ordered groups?
• RQ4 Why does minimality fail in primary decompositions over non-finitely generated or non-discrete partially ordered groups?
• RQ5 What is the role of the geometry of the positive cone’s faces in determining the structure of primary decompositions?

Key findings

• Every downset-finite module over a polyhedral partially ordered group with finitely many faces admits a finite primary decomposition into coprimary submodules.
• The primary decomposition is canonical and functorial, relying on localization and support functors that identify coprimary elements without altering the ambient module.
• The failure of minimality in primary decompositions arises from geometric obstructions—such as the absence of translates of faces in the boundary of a downset—when the group is not finitely generated or discrete.
• The module $\Bbbk[\partial Q_{+}]$ over the right circular cone in $\mathbb{R}^3$ provides a counterexample: it does not admit a finite primary decomposition due to infinitely many faces.
• The downset-finiteness condition is strictly weaker than tameness and strictly stronger than the noetherian condition in the context of monoid algebras.
• The theory generalizes classical primary decomposition for monomial ideals in polynomial rings to the continuous setting of $\mathbb{R}^n$-graded modules over real-exponent polynomial rings.