Essential graded algebra over polynomial rings with real exponents

TL;DR

This paper develops a comprehensive algebraic and geometric framework for monomial ideals and multigraded modules over polynomial rings with real exponents, extending classical theories to a continuous setting.

Contribution

It generalizes key algebraic concepts like Nakayama's lemma, primary decomposition, and Matlis duality to real-exponent polynomial rings, providing canonical decompositions and geometric insights.

Findings

Generalized Nakayama's lemma for real exponents

Established primary and secondary decompositions in the real-exponent setting

Provided geometric analysis of staircases and module structures

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.