# When is Hilbert-Samuel multiplicity of a local ring non-increasing along localization at prime ideals?

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 $$R$$?

• I think you also need something like $R/\mathfrak p$ is analytically irreducible ...
• The book reference does say $R/\mathfrak p$ has to be analytically unramified ... so I believe only Cohen-Macaulay is not enough ...