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arXiv · 0912.3853

Multiplicity bounds in graded rings

Craig Huneke·Shunsuke Takagi·Kei-ichi Watanabe
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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}.

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Craig Huneke, Shunsuke Takagi, Kei-ichi Watanabe. 2009-12-21. Multiplicity bounds in graded rings. https://doi.org/10.1215/0023608x-2010-022

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