Multiplicity bounds in graded rings

Abstract

The $F$-threshold $c^J\left(\mathfrak{a}\right)$ of an ideal $\mathfrak{a}$ with respect to an ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\mathfrak{a}$ with the Frobenius powers of $J$. We study a conjecture formulated in an earlier article that we authored with M. Mustaţă, which bounds $c^J\left(\mathfrak{a}\right)$ in terms of the multiplicities $e\left(\mathfrak{a}\right)$ and $e\left(J\right)$ when $\mathfrak{a}$ and $J$ are zero-dimensional ideals and $J$ is generated by a system of parameters. We prove the conjecture when $\mathfrak{a}$ 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.