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Essential graded algebra over polynomial rings with real exponents

Ezra Miller

TL;DR

The paper develops a comprehensive algebraic theory for modules over real-exponent polynomial rings $\Bbbk[\mathbb{R}^n_+]$, where non-discrete exponents destroy Noetherianity and standard prime-related tools fail. It introduces a topological framework built from dense generator/cogenerator functors, upper closures $\delta M$, and socles along faces, enabling canonical primary and irreducible decompositions, Matlis duality, and staircases geometry in both real and discrete polyhedral groups. Core achievements include Nakayama-type surjection/injection criteria via tops/socc, density-enhanced minimal primary and interval decompositions (canonical up to density), and preservation of tameness/semialgebraic/PL structure under socle/cogenerator constructions. The results have direct implications for multipersistence and quantum noncommutative toric geometry, providing robust foundations for real-parameter multigraded algebra and practical computability through structured, finite presentations of otherwise unwieldy non-noetherian objects.

Abstract

The geometric and algebraic theory of monomial ideals and multigraded modules is initiated over real-exponent polynomial rings and, more generally, monoid algebras for real polyhedral cones. The main results include the generalization of Nakayama's lemma; complete theories of minimal and dense primary, secondary, and irreducible decomposition, including associated and attached faces; socles and tops; minimality and density for downset hulls, upset covers, and fringe presentations; Matlis duality; and geometric analysis of staircases. Modules that are semialgebraic or piecewise-linear (PL) have the relevant property preserved by functorial constructions as well as by minimal primary and secondary decompositions. And when the modules in question are subquotients of the group itself, such as monomial ideals and quotients modulo them, minimal primary and secondary decompositions are canonical, as are irreducible decompositions up to the new real-exponent notion of density.

Essential graded algebra over polynomial rings with real exponents

TL;DR

The paper develops a comprehensive algebraic theory for modules over real-exponent polynomial rings , where non-discrete exponents destroy Noetherianity and standard prime-related tools fail. It introduces a topological framework built from dense generator/cogenerator functors, upper closures , and socles along faces, enabling canonical primary and irreducible decompositions, Matlis duality, and staircases geometry in both real and discrete polyhedral groups. Core achievements include Nakayama-type surjection/injection criteria via tops/socc, density-enhanced minimal primary and interval decompositions (canonical up to density), and preservation of tameness/semialgebraic/PL structure under socle/cogenerator constructions. The results have direct implications for multipersistence and quantum noncommutative toric geometry, providing robust foundations for real-parameter multigraded algebra and practical computability through structured, finite presentations of otherwise unwieldy non-noetherian objects.

Abstract

The geometric and algebraic theory of monomial ideals and multigraded modules is initiated over real-exponent polynomial rings and, more generally, monoid algebras for real polyhedral cones. The main results include the generalization of Nakayama's lemma; complete theories of minimal and dense primary, secondary, and irreducible decomposition, including associated and attached faces; socles and tops; minimality and density for downset hulls, upset covers, and fringe presentations; Matlis duality; and geometric analysis of staircases. Modules that are semialgebraic or piecewise-linear (PL) have the relevant property preserved by functorial constructions as well as by minimal primary and secondary decompositions. And when the modules in question are subquotients of the group itself, such as monomial ideals and quotients modulo them, minimal primary and secondary decompositions are canonical, as are irreducible decompositions up to the new real-exponent notion of density.

Paper Structure

This paper contains 38 sections, 89 theorems, 81 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Real exponent issues
  3. Nakayama's lemma and primary decomposition
  4. Socles, cogenerators, and staircases
  5. Algebra over partially ordered abelian groups
  6. Real and discrete polyhedral groups
  7. Localization and restriction
  8. Support and primary decomposition
  9. Tame, semialgebraic, and PL modules and morphisms
  10. Matlis duality
  11. Geometry of real staircases
  12. Tangent cones of downsets
  13. Upper closure functors
  14. Socles and cogenerators
  15. Closed socles and closed cogenerator functors
  16. ...and 23 more sections

Key Result

Lemma 2.15

The subgroup ${\mathbb Z}\tau \subseteq Q$ of a partially ordered abelian group $Q$ acts freely on the localization $M_\tau$ of any $Q$-module $M$ along a face $\tau$. Consequently, if $I_\tau \subseteq \Bbbk[Q_+]$ is the augmentation ideal $\langle m - 1 \mid m \in \Bbbk[\tau]$ is a monomial$\rangl

Figures (1)

  • Figure :

Theorems & Definitions (378)

  • Example 1.1
  • Example 1.2
  • Example 1.5
  • Example 1.6
  • Example 1.7
  • Example 1.8
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • ...and 368 more