## A note on the computation of multiplicities.(English)Zbl 1107.13026

Let $$(A, \mathfrak m, k)$$ be a noetherian local ring of dimension $$d$$. Let us suppose that the residue field $$k$$ is contained in $$A$$. Let us consider a system of parameters $$\underline x=x_1,\dots, x_d$$ of $$A$$. Therefore the polynomial ring $$P=k[x_1,\dots, x_d]$$ is a subring of $$A$$. Let us denote by $$R$$ the localization $$P_{\underline xP}$$. Let us consider a sequence of polynomials $$\underline f=f_1,\dots, f_d\in P$$ such that the ideal $$\underline f R$$ is an $$\underline x R$$-primary ideal of $$R$$. Then the author consider the problem of relating $$e_0(\underline fR, R)$$ with $$e_0(\underline fA, A)$$, where $$e_0$$ denotes Hilbert-Samuel multiplicity. In particular, the authors prove that $e_0(\underline fA, A)=e_0(\underline fR, R)e_0(\underline xA, A).$ As a corollary, under the previous hypothesis, the authors also prove that $e_0(\underline fA, A)\leq e_0(\underline fR, R)L_A(A/\underline xA)$ and equality holds if and only if $$A$$ is a Cohen-Macaulay ring. Furthermore the authors also obtain a direct proof of Serre’s formula for the multiplicity of a system of parameters.

### MSC:

 13H15 Multiplicity theory and related topics 13B02 Extension theory of commutative rings