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For Noetherian local ring $(R,\mathfrak m)$, let $e(R)$ denote the Hilbert-Samuel multiplicity of $R$ with respect to $\mathfrak m$ (https://en.m.wikipedia.org/wiki/Hilbert%E2%80%93Samuel_function#Multiplicity. https://stacks.math.columbia.edu/tag/0AZU).

My question is: If $R$ is a Gorenstein local ring, then is it true that $e(R_{\mathfrak p})\le e(R)$ for every prime ideal $\mathfrak p$ of $R$?

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This is a theorem of Nagata and is true for even Cohen Macaulay rings.

(Reference: Nagata 'Local Rings' Theorem 40.1, p.153)

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    $\begingroup$ I think you also need something like $R/\mathfrak p$ is analytically irreducible ... $\endgroup$
    – sdey
    Oct 4, 2022 at 2:26
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    $\begingroup$ The book reference does say $R/\mathfrak p$ has to be analytically unramified ... so I believe only Cohen-Macaulay is not enough ... $\endgroup$
    – Alex
    Oct 9, 2022 at 23:50

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