## Nagoya Mathematical Journal

### New estimates of Hilbert–Kunz multiplicities for local rings of fixed dimension

#### Abstract

We present results on the Watanabe–Yoshida conjecture for the Hilbert–Kunz multiplicity of a local ring of positive characteristic. By improving on a “volume estimate” giving a lower bound for Hilbert–Kunz multiplicity, we obtain the conjecture when the ring has either Hilbert–Samuel multiplicity less than or equal to 5 or dimension less than or equal to 6. For nonregular rings with fixed dimension, a new lower bound for the Hilbert–Kunz multiplicity is obtained.

#### Article information

Source
Nagoya Math. J., Volume 212 (2013), 59-85.

Dates
First available in Project Euclid: 1 August 2013

https://projecteuclid.org/euclid.nmj/1375362750

Digital Object Identifier
doi:10.1215/00277630-2335204

Mathematical Reviews number (MathSciNet)
MR3290680

Zentralblatt MATH identifier
1282.13010

#### Citation

Aberbach, Ian M.; Enescu, Florian. New estimates of Hilbert–Kunz multiplicities for local rings of fixed dimension. Nagoya Math. J. 212 (2013), 59--85. doi:10.1215/00277630-2335204. https://projecteuclid.org/euclid.nmj/1375362750

#### References

• [1] I. M. Aberbach and F. Enescu, Lower bounds for Hilbert-Kunz multiplicities in local rings of fixed dimension, Michigan Math. J. 57 (2008), 1–16.
• [2] M. Blickle and F. Enescu, On rings with small Hilbert-Kunz multiplicity, Proc. Amer. Math. Soc. 132 (2004), 2505–2509.
• [3] O. Celikbas, H. Dao, C. Huneke, and Y. Zhang, Bounds on the Hilbert-Kunz multiplicity, Nagoya Math. J. 205 (2012), 149–165.
• [4] D. Chakerian and D. Logothetti, Cube slices, pictorial triangles, and probability, Math. Mag. 64 (1991), 219–241.
• [5] F. Enescu and K. Shimomoto, On the upper semi-continuity of the Hilbert-Kunz multiplicity, J. Algebra 285 (2005), 222–237.
• [6] K. Eto and K.-i. Yoshida, Notes on Hilbert-Kunz multiplicity of Rees Algebra, Comm. Algebra 31 (2003), 5943–5976.
• [7] S. Goto and Y. Nakamura, Multiplicity and tight closures of parameters, J. Algebra 244 (2001), 302–311.
• [8] M. Hochster and C. Huneke, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), 31–116.
• [9] C. Huneke and I. Swanson, Integral closure of ideals, rings, and modules, London Math. Soc. Lecture Note Ser. 336, Cambridge University Press, Cambridge, 2006.
• [10] P. Monsky, The Hilbert-Kunz function, Math. Ann. 263 (1983), 43–49.
• [11] P. Monsky and I. Gessel, The limit as $p\to\infty$ of the Hilbert-Kunz multiplicity of $\sum x_{i}^{d_{i}}$, preprint, 2010, arXiv:1007.2004 [math.AC].
• [12] M. Nagata, Local Rings, Pure Appl. Math. (Hoboken) 13, Wiley, New York, 1962.
• [13] D. G. Northcott and D. Rees, Reductions of ideals in local rings, Math. Proc. Cambridge Philos. Soc. 50 (1954), 145–158.
• [14] D. Rees, Valuations associated with ideals, II, J. London Math. Soc. 31 (1956), 221–228.
• [15] J. D. Sally, Numbers of Generators of Ideals in Local Rings, Marcel Dekker, New York, 1978.
• [16] J. Sally, Cohen-Macaulay local rings of maximal embedding dimension, J. Algebra 56 (1979), 168–183.
• [17] J. Sally, Tangent cones at Gorenstein singularities, Compos. Math. 40 (1980), 167–175.
• [18] K.-i. Watanabe and K.-i. Yoshida, Hilbert-Kunz multiplicity and an inequality between multiplicity and colength, J. Algebra 230 (2000), 295–317.
• [19] K.-i. Watanabe, K.-i.Yoshida, Hilbert-Kunz multiplicity of two-dimensional local rings, Nagoya Math. J. 162 (2001), 87–110.
• [20] K.-i. Watanabe, K.-i.Yoshida, Hilbert-Kunz multiplicity of three-dimensional local rings, Nagoya Math. J. 177 (2005), 47–75.
• [21] Wolfram Research, Mathematica, Version 7.0, Champaign, IL, 2008.
• [22] K.-i. Yoshida, “Small Hilbert-Kunz multiplicity and $(A_{1})$-type singularity” in Proceedings of the 4th Japan-Vietnam Joint Seminar on Commutative Algebra by and for Young Mathematicians, Meiji University, Japan, 2009.