Rational symbolic powers of ideals

TL;DR

This paper introduces and analyzes rational symbolic powers of ideals in Noetherian rings, providing criteria, geometric descriptions, and stability results, especially for monomial ideals, advancing understanding of their algebraic and geometric properties.

Contribution

It defines rational symbolic powers, explores their properties, and offers a convex-geometric framework for monomial ideals, including stability and asymptotic behavior analysis.

Findings

Membership criteria for rational symbolic powers

Convex-geometric description of monomial ideals' powers

Asymptotic stability of the filtration of rational symbolic powers

Abstract

We introduce and study rational symbolic powers of ideals in Noetherian rings. We give membership criteria for rational symbolic powers and discuss settings where they agree with integer symbolic powers. We investigate the binomial expansion formula for rational symbolic powers of mixed sums of ideals. Finally, we study rational symbolic powers of monomial ideals. In this case, we give a convex-geometric description of the rational symbolic powers. We also show that the filtration of rational symbolic powers of a monomial ideal is asymptotically stable and, as a consequence, deduce that the asymptotic regularity and asymptotic depth for this filtration exist.