I am trying to understand a statement in a proof.
The setup is $(R,m)$ is a $3$-dimensional regular local ring with infinite residue field, $\mathfrak{p}$ is a height-$2$ prime ideal and $x$ is not in $\mathfrak{p}$. Also, $\mathfrak{p}^{(m)}$ represents the $m$th symbolic power of $\mathfrak{p}.$ My difficulty is with the following equation:
$$\chi(R/\mathfrak{p}^{(m)},R/(x))=\chi(R/(\mathfrak{p},x))\ell(R_{\mathfrak{p}}/\mathfrak{p}_{\mathfrak{p}}^m)$$
Is the $\chi$ on the right just the Hilbert-Samuel multiplicity of the quotient? And I am also not clear on how, $\ell$, the analytic spread comes in?
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user656543
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Can you define what you mean by $\chi (-,-)$ and $\chi (-)$? – Youngsu Mar 22 '19 at 17:56
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$$\chi(M,N)=\sum_{i=0}^{\infty}(-1)^i\lambda(\mbox{Tor}_i^R(M,N))$$ is Serre’s intersection multiplicity ($\lambda(-)$ represents the length of the module. I am not sure about $\chi(-)$ myself. Maybe the Hilbert-Samuel multiplicity? – user656543 Mar 22 '19 at 18:06