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On the upper semi-continuity of the Hilbert-Kunz multiplicity. (English) Zbl 1080.13011

For a local ring \((R,m,k)\) containing a field of characteristic \(p>0\) the following conjecture was considered by K. I. Watanabe and K. Yoshida [Nagoya Math. J. 177, 47–75 (2005; Zbl 1076.13009)]:
Suppose that \(k\) is the algebraic closure of \({\mathbb F}_p\). Let \(d\geq 2\) and \(p\) a prime , \(p\neq 2.\) Let also \(R_{p,d}:=k[[X_0,\dots,X_d]]/(X_0^2+\cdots+X_d^2).\) If \(R\) is \(d\)-dimensional, unmixed and not regular, then \[ e_{HK}(R)\geq e_{HK}(R_{p,d}). \] The cases \(d=2,3,4\) have been solved in general by M. Blickle and F. Enescu [Proc. Am. Math. Soc. 132, 2505–2509 (2004; Zbl 1099.13009)] and K. I. Watanabe and K. Yoshida [J. Algebra 230, 295–317(2000; Zbl 0964.13008)].
The aim of the paper under review is to prove this conjecture for complete intersections. The proof is given by doing it first for hypersurfaces and reducing the general case to this one.

MSC:

13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13H15 Multiplicity theory and related topics
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
14M10 Complete intersections
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References:

[1] Blickle, M.; Enescu, F., On rings with small Hilbert-Kunz multiplicity, Proc. Amer. Math. Soc., 132, 9, 2505-2509 (2004) · Zbl 1099.13009
[2] Eisenbud, D., Commutative Algebra with a View toward Algebraic Geometry (1996), Springer-Verlag
[3] Flenner, H.; O’Carroll, L.; Vogel, W., Joins and Intersections (1999), Springer-Verlag · Zbl 0939.14003
[4] Gersten, S. M., A short proof of the algebraic Weierstrass Preparation Theorem, Proc. Amer. Math. Soc., 88, 4, 751-752 (1983) · Zbl 0514.13014
[5] Huneke, C.; Yao, Y., Unmixed local rings with minimal Hilbert-Kunz multiplicity are regular, Proc. Amer. Math. Soc., 130, 661-665 (2002) · Zbl 0990.13011
[6] Kunz, E., Characterizations of regular local rings of characteristic \(p > 0\), Amer. J. Math., 41, 772-784 (1969) · Zbl 0188.33702
[7] Kunz, E., On Noetherian rings of characteristic \(p\), Amer. J. Math., 98, 4, 999-1003 (1976) · Zbl 0341.13009
[8] Monsky, P., Hilbert-Kunz functions in a family: point-\(S_4\) quartics, J. Algebra, 208, 343-358 (1998) · Zbl 0932.13010
[9] Watanabe, K.-I.; Yoshida, K., Hilbert-Kunz multiplicity and an inequality between multiplicity and colength, J. Algebra, 230, 295-317 (2000) · Zbl 0964.13008
[10] Watanabe, K.-I.; Yoshida, K., Hilbert-Kunz multiplicity of three-dimensional local rings, preprint · Zbl 1076.13009
[11] Watanabe, K.-I.; Yoshida, K., Minimal relative Hilbert-Kunz multiplicity, Illinois J. Math., 48, 273-294 (2004) · Zbl 1089.13007
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