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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
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