Local Bezout estimates and multiplicities of parameter and primary ideals (Q2401614)

Let \((A,\mathfrak{m})\) be a local Noetherian ring of dimension \(d\) and \(\mathfrak{q}\) an \(\mathfrak{m}\)-primary ideal in \(A\). Let \(\underline{a} = a_1, \ldots, a_d \subset \mathfrak{q}\) be a system of parameters of \(A\) such that the initial degree of \(a_i\) is \(c_i\) for \(i = 1,\ldots, d\). The main aim of the paper under review concerns a comparison between two Hilbert-Samuel multiplicities \(e(\underline{a}; A)\) and \(e(\mathfrak{q}; A)\). To this end, first it is proved that \(e(\underline{a};A) \geq c \cdot e(\mathfrak{q}; A)\), where \(c = c_1 \cdots c_d\), and then the authors improves this inequality and present a characterization of the equality infact: \[ e(\underline{a}; A) = c \cdot e(\mathfrak{q}; A) + \chi(\underline{a}, \mathfrak{q}), \] where \(\chi(\underline{a}, \mathfrak{q})\) is the Euler characteristic of a certain variation of Koszul homology. Also, the authors gives two counter examples to the claim of \textit{F. L. Pritchard} which says that \(e(\underline{a}; A) = c \cdot e(\mathfrak{m}; A)\) if and only if the sequence of initial forms \(a_1^\star, \ldots, a_d^\star\) form a \(G_A(\mathfrak{m})\)-regular sequence (see [Manuscripta Math. 39, 267--292 (1984; Zbl 0581.12011)]). As an application of these investigation, the authors improve the classical local Bézout inequality in the affine plane \(\mathbb{A}^2_k\), where \(k\) is an algebraically closed field and mentions a complete characterization of the equality via blowing up algebras.