Let q denote an m-primary ideal of a d-dimensional local ring (A, m). Let a = a1, . . . , a d ⊂ q be a system of parameters. Then there is the following inequality for the multiplicities c ⋅ e(q; A) ≤ e(a; A) where c denotes the product of the initial degrees of ai in the form ring G A (q). The aim of the paper is a characterization of the equality as well as a description of the difference by various homological methods via Koszul homology. To this end we have to characterize when the sequence of initial elements a ⋆ = a ⋆ 1 , . . . , a ⋆ d is a homogeneous system of parameters of G A (q). In the case of dim A = 2 this leads to results on the local Bezout inequality. In particular, we give several equations for improving the classical Bezout inequality to an equality.

Abstract. This paper provides effective methods for computing the local intersection multiplicity as the length of a well-defined ideal (see Theorem and Proposition 1). There are other ways of obtaining such an ideal (see [2], [9], [12], [18]) but ours is simpler because of our use of reducing systems of parameters. Applying these ideal theoretic methods we will give a new and simple proof of Bezout's Theorem (see §4). Hence this proof again provides the connection between the different viewpoints which are treated in the work of Lasker-Macaulay-Gröbner and Severi-van der Waerden-Weil concerning the multiplicity theory.

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Let (R,m) = k[x 1,..., x n](x 1,...,x n) be a local polynomial ring (k being an algebraically closed field), and Q:= (F 1,..., F r)R be a primary ideal in R with respect to a maximal ideal m ⊂ R. In this short note we give a formula for the multiplicity e 0 (QR/(F 1)R, R/(F 1)R).