Limit Behavior of the Rational Powers of Monomial Ideals

TL;DR

This paper explores the properties of rational powers of monomial ideals, establishing their connection with symbolic powers and demonstrating convergence results for depths, regularities, and associated primes.

Contribution

It provides a characterization of rational powers for monomial ideals, linking them to symbolic powers and proving convergence of various algebraic invariants.

Findings

Symbolic powers of squarefree monomial ideals are rational powers.

Depths and regularities converge for rational powers.

Finiteness of asymptotic associated primes and convergence of local cohomology lengths.

Abstract

We investigate the rational powers of ideals. We find that in the case of monomial ideals, the canonical indexing leads to a characterization of the rational powers yielding that symbolic powers of squarefree monomial ideals are indeed rational powers themselves. Using the connection with symbolic powers techniques, we use splittings to show the convergence of depths and normalized Castelnuovo-Mumford regularities. We show the convergence of Stanley depths for rational powers, and as a consequence of this we show the before-now unknown convergence of Stanley depths of integral closure powers. In addition, we show the finiteness of asymptotic associated primes, and we find that the normalized lengths of local cohomology modules converge for rational powers, and hence for symbolic powers of squarefree monomial ideals.