Growth of primary decompositions of Frobenius powers of ideals

TL;DR

This paper investigates the growth patterns of primary decompositions of Frobenius powers of ideals, establishing linear growth in positively graded rings and polynomial growth in non-negatively graded rings, with explicit examples provided.

Contribution

It proves the linear growth property for primary decompositions in positively graded rings and introduces polynomial growth results for non-negatively graded rings, expanding understanding of Frobenius power decompositions.

Findings

Linear growth holds in positively graded rings.

Polynomial growth occurs in non-negatively graded rings.

Explicit primary decompositions with infinitely many associated primes are constructed.

Abstract

It was previously known, by work of Smith-Swanson and of Sharp-Nossem, that the linear growth property of primary decompositions of Frobenius powers of ideals in rings of prime characteristic has strong connections to the localization problem in tight closure theory. The localization problem has recently been settled in the negative by Brenner and Monsky, but the linear growth question is still open. We study growth of primary decompositions of Frobenius powers of dimension one homogeneous ideals in graded rings over fields. If the ring is positively graded we prove that the linear growth property holds. For non-negatively graded rings we are able to show that there is a "polynomial growth". We present explicit primary decompositions of Frobenius powers of an ideal, which were known to have infinitely many associated primes, having this linear growth property. We also discuss some other interesting examples.