Multiplicity bounds in graded rings

TL;DR

This paper investigates bounds on the $F$-threshold in graded rings, proving a conjecture relating it to multiplicities for certain ideals, and extends results to jumping numbers of parameter test submodules.

Contribution

It proves a conjecture bounding the $F$-threshold using multiplicities in graded rings and extends the bounds to jumping numbers of parameter test submodules.

Findings

Proved the conjecture for homogeneous systems in graded rings.

Established bounds relating $F$-thresholds to multiplicities.

Extended results to generalized parameter test submodules.

Abstract

The $F$-threshold $c^J(\a)$ of an ideal $\a$ with respect to an ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\a$ with the Frobenius powers of $J$. We study a conjecture formulated in an earlier paper \cite{HMTW} by the same authors together with M. Musta\c{t}\u{a}, which bounds $c^J(\a)$ in terms of the multiplicities $e(\a)$ and $e(J)$, when $\a$ and $J$ are zero-dimensional ideals and $J$ is generated by a system of parameters. We prove the conjecture when $\a$ and $J$ are generated by homogeneous systems of parameters in a Noetherian graded $k$-algebra. We also prove a similar inequality involving, instead of the $F$-threshold, the jumping number for the generalized parameter test submodules introduced in \cite{ST}.