Hilbert-Kunz density function and Hilbert-Kunz multiplicity
| Autor: | Trivedi, V. |
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| Rok vydání: | 2015 |
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| Druh dokumentu: | Working Paper |
| Popis: | For a pair $(M, I)$, where $M$ is finitely generated graded module over a standard graded ring $R$ of dimension $d$, and $I$ is a graded ideal with $\ell(R/I) < \infty$, we introduce a new invariant $HKd(M, I)$ called the {\em Hilbert-Kunz density function}. In Theorem 1.1, we relate this to the Hilbert-Kunz multiplicity $e_{HK}(M,I)$ by an integral formula. We prove that the Hilbert-Kunz density function is additive. Moreover it satisfies a multiplicative formula for a Segre product of rings. This gives a formula for $e_{HK}$ of the Segre product of rings in terms of the HKd of the rings involved. As a corollary, $e_{HK}$ of the Segre product of any finite number of Projective curves is a rational number. As an another application we see that $e_{HK}(R, {\bf m}^k) - e(R, {\bf m}^k)/d!$ grows at least as a fixed positive multiple of $k^{d-1}$ as $k\to \infty$. Comment: This paper is one part of the earlier submission, which has been split in two parts. This part will introduce and develop the theory of Hilbert-Kunz Density. This will appear in Transactions of AMS. The second part (which will be another paper) will involve the study of asymptotic behaviour of Hilbert-Kunz multiplicities of powers of an ideal |
| Databáze: | arXiv |
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