Essential graded algebra over polynomial rings with real exponents
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.
Cite
@article{arxiv.2008.03819,
title = {Essential graded algebra over polynomial rings with real exponents},
author = {Ezra Miller},
journal= {arXiv preprint arXiv:2008.03819},
year = {2025}
}
Comments
v2: 85 pages, 18 figures, many more and more detailed examples, additional exposition at start of each section, typos corrected; v1: 73 pages, 13 figures. This supersedes Sections 6-14 of arXiv:1709.08155. (Earlier sections in that preprint are now expanded into separate manuscripts: arXiv:2008.00063 and arXiv:2008.00093; they involve different background and running hypotheses.)