Hilbert-Kunz function and Hilbert-Kunz multiplicity of ideals and Rees algebras

Abstract: Hilbert-Kunz functions were introduced by E. Kunz in 1969 in his work characterizing regular local rings in the prime characteristic setting. The existence of Hilbert-Kunz multiplicity was proved later by P. Monsky in 1983. Since then, Hilbert-Kunz functions and Hilbert-Kunz multiplicities have been extensively studied, partly because of their connections with the theory of tight closure and their unpredictable behaviour. Unlike the Hilbert-Samuel function, the Hilbert-Kunz function need not be a polynomial function.

In this talk, we consider the Hilbert-Kunz function of Rees algebra of ideals and show that, in certain cases, it behaves as a quasi-polynomial, a piece-wise polynomial, or even a polynomial. We also consider Hilbert-Kunz multiplicity of powers of an ideal, in an attempt to write it as a function of the power of the ideal. This involves a surprising connection with the Hilbert-Samuel coefficients of Frobenius powers of an ideal.

commutative algebra

Audience: researchers in the topic

Fellowship of the Ring

Series comments: Description: National Commutative Algebra Seminar

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