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arXiv · 1706.07445

Interpolating Between Hilbert-Samuel and Hilbert-Kunz Multiplicity

William D. Taylor
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Abstract

We define a function, called s-multiplicity, that interpolates between Hilbert-Samuel multiplicity and Hilbert-Kunz multiplicity by comparing powers of ideals to the Frobenius powers of ideals. The function is continuous in s, and its value is equal to Hilbert-Samuel multiplicity for small values of s and is equal to Hilbert-Kunz multiplicity for large values of s. We prove that it has an Associativity Formula generalizing the Associativity Formulas for Hilbert-Samuel and Hilbert-Kunz multiplicity. We also define a family of closures such that if two ideals have the same s-closure then they have the same s-multiplicity, and the converse holds under mild conditions. We describe the s-multiplicity of monomial ideals in toric rings as a certain volume in real space

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William D. Taylor. 2017-06-22. Interpolating Between Hilbert-Samuel and Hilbert-Kunz Multiplicity. https://arxiv.org/abs/1706.07445

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