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Is there an example of a Cohen-Macaulay local domain $R$ of characteristic $p>0$ for which the Hilbert-Kunz multiplicity $e_{HK}(R)$ is not equal to its Hilbert-Samuel multiplicity $e(R)$? If no example, a result that states they may not in general be equal would also be helpful, of course.

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up vote 4 down vote accepted

In general $e_{HK}(R) \leq e(R)$. Most of the time the inequality is strict.

The case $e(R)=2$ and $\dim R=2$ is studied carefully in the paper by Yoshida-Watanabe "Hilbert-Kunz multiplicity of two-dimensional local rings" , available here.

For example, in this case $e_{HK}<2$ if $R$ is a complete rational double point. In this case one can write $R = k[[x,y,z]]^G$ (assume $k$ alg. closed with big enough characteristic), and it turns out that $e_{HK}(R) = 2-1/|G|<2=e(R)$ (see Example 4.1).

Of course, there are many more results on both kinds of multiplicities, but $e_{HK}$ is much more difficult to compute. Paul Monsky is really good at computing them (among many other things), so you can check out his papers for many exotic examples.

Added: Of course, this being MO, Paul himself has added via the comment below some very interesting and relevant results. One thing that may help newcomers: in his comment we look at homogenous hypersurfaces of degree $d$. Those rings automatically have $e(R)=d$. So for example the work by Brenner-Trivedi (not joint) says that $e_{HK}<e$ when there are $3$ variables, $d=4$ and the polynomial is absolutely irreducible.

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As Dao says, very little is known about the H.K multiplicity, mu, of R, even when R is the quotient of a power series ring in s variables by a homogeneous polynomial g of degree d. When s=3, there's a beautiful theory due to Brenner and to Trivedi--for example when d=4 and g is absolutely irreducible, mu can only be 3, 3+1/q^2, or 3+1/4q^2 where q is a power of the characteristic. But when s>3 one has theory only for special cases. (See Teixeira). And there's convincing (to me) evidence that mu can be irrational (for s>4) and even transcendental. – paul Monsky Jul 16 '11 at 11:50
Thank you for great response and comment. – Mahdi Majidi-Zolbanin Jul 16 '11 at 19:58

Can we find any example for local domain rings?

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If you have a new question use the button in the upper right corner. – Julian Kuelshammer Jun 25 '13 at 7:33

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