Generic bounds for Frobenius closure and tight closure

TL;DR

This paper establishes geometric and cohomological methods to derive generic degree bounds for tight and Frobenius closures of ideals in polynomial rings and standard-graded algebras, revealing more uniform behavior of tight closure.

Contribution

It introduces a new approach to obtain generic bounds for tight closure and Frobenius closure in arbitrary standard-graded algebras based on known bounds in polynomial rings.

Findings

Provides explicit generic bounds for tight closure in dimension up to 2.

Shows tight closure behaves more uniformly than the ideal itself.

Connects bounds in polynomial rings to those in general standard-graded algebras.

Abstract

We use geometric and cohomological methods to show that given a degree bound for membership in ideals of a fixed degree type in the polynomial ring P=k[x_0,..., x_d], one obtains a good generic degree bound for membership in the tight closure of an ideal of that degree type in any standard-graded k-algebra R of dimension d+1. This indicates that the tight closure of an ideal behaves more uniformly than the ideal itself. Moreover, if R is normal, one obtains a generic bound for membership in the Frobenius closure. If d is at most 2, then the bound for ideal membership in P can be computed from the known cases of the Froeberg conjecture and yields explicit generic tight closure bounds.