Hilbert-Kunz Multiplicity of Three-Dimensional Local Rings

In this paper, we investigate the lower bound s HK(p, d) of Hilbert-Kunz multiplicities for non-regular unmixed local rings of Krull dimension d containing a field of characteristic p > 0. Especially, we focus on three-dimensional local rings. In fact, as a main result, we will prove that s HK (p...

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Bibliographic Details
Published inNagoya mathematical journal Vol. 177; pp. 47 - 75
Main Authors Watanabe, Kei-ichi, Yoshida, Ken-ichi
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 2005
Duke University Press
Subjects
13A35
13D40
13H05
13H10
13H15
13H15
13H05
13A35
13D40
13H10
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Summary:In this paper, we investigate the lower bound s HK(p, d) of Hilbert-Kunz multiplicities for non-regular unmixed local rings of Krull dimension d containing a field of characteristic p > 0. Especially, we focus on three-dimensional local rings. In fact, as a main result, we will prove that s HK (p, 3) = 4/3 and that a three-dimensional complete local ring of Hilbert-Kunz multiplicity 4/3 is isomorphic to the non-degenerate quadric hypersurface k[[X, Y, Z,W]]/(X 2 + Y 2 + Z 2 + W 2) under mild conditions. Furthermore, we pose a generalization of the main theorem to the case of dim A ≥ 4 as a conjecture, and show that it is also true in case dim A = 4 using the similar method as in the proof of the main theorem.
ISSN:0027-7630
2152-6842
DOI:10.1017/S0027763000009053