Abstract
In this chapter we will focus on the study of multiplicities and intersection numbers. We will introduce in Section 6.1 a version of the j-multiplicity for arbitrary local rings which was first studied by Achilles and Manaresi [AMa2]. It assigns a new multiplicity j(a, M) to every finite module M over a local ring A, where a is an arbitrary (not necessarily mA-primary) ideal of A. We will develop the basic properties of this new multiplicity in analogy with the theory of classical multiplicities, in particular the behaviour under exact sequences and the effect of taking hyperplane sections. Moreover we will show that it coincides with our intersection number j for distinguished components of the intersection.
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