Multiplicities and Volumes of Filtrations - Steven Dale Cutkosky (Mathematics)
In this article, we survey some aspects of the theory of multiplicities of <semantics>mR</semantics>-primary ideals in a local ring <semantics>(R,mR)</semantics> and the extension of this theory to multiplicities of graded families of <semantics>mR</semantics>-primary ideals. We first discuss the existence of multiplicities as a limit. Then, we focus on a theorem of Rees, characterizing when two <semantics>mR</semantics>-primary ideals <semantics>I⊂J</semantics> have the same multiplicity, and discuss extensions of this theorem to filtrations of <semantics>mR</semantics>-primary ideals. In the final sections, we give outlines of the proof of existence of the multiplicity of a graded family of <semantics>mR</semantics>-primary ideals as a limit, with mild conditions on R, and the proof of the extension of Rees’ theorem to divisorial filtrations.