Local Bezout estimates and multiplicities of parameter and primary ideals

Let q denote an m-primary ideal of a d-dimensional local ring (A,m). Let a_=a1,…,ad⊂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 GA(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⋆_=a1⋆,…,ad⋆ is a homogeneous system of parameters of GA(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.