Global dimension of real-exponent polynomial rings

TL;DR

This paper determines the global and flat dimensions of rings of real-exponent polynomials and related monoid algebras, providing explicit resolutions and extending results to broader algebraic structures.

Contribution

It establishes the global and flat dimensions of real-exponent polynomial rings and their monoid algebra counterparts, including explicit projective and flat resolutions.

Findings

Global dimension of R is n+1

Flat dimension of R is n

Resolutions of all R-modules constructed from the residue field

Abstract

The ring R of real-exponent polynomials in n variables over any field has global dimension n+1 and flat dimension n. In particular, the residue field k = R/m of R modulo its maximal graded ideal m has flat dimension n via a Koszul-like resolution. Projective and flat resolutions of all R-modules are constructed from this resolution of k. The same results hold when R is replaced by the monoid algebra for the positive cone of any subgroup of $\mathbb{R}^n$ satisfying a mild density condition.