Multiplicity for ideals of maximal analytic spread and intersection theory
Let \((A, {\mathfrak m})\) be a local ring and \(I\) an ideal in \(A\) having maximal analytic spread. The authors define a multiplicity \(\mu (I,A)\) for \(I\) which coincides with the usual multiplicity in case \(I\) is \({\mathfrak m}\)-primary. If \(K\) is an algebraically closed field, the Stückrad-Vogel intersection number for a \(K\)-rational component of an intersection in a projective space over \(K\) can be interpreted as such a multiplicity. Denote by \(G\) the associated graded ring \(\text{gr}_ I (A)\) of \(A\) with respect to \(I\). Then \(\mu (I,A) : = \sum e (G/{\mathfrak p})\) length\((G_{\mathfrak p})\), where \({\mathfrak p}\) runs over all associated prime ideals of \(G/{\mathfrak m} G\) in \(G\) having dimension \(\dim (G/{\mathfrak m} G)\), and \(e (G/{\mathfrak p})\) is the (usual) multiplicity of the graded ring \(G/{\mathfrak p}\). \(\mu (I,A)\) can be computed by means of `super-reductions' for \(I\) which generalize superficial systems of parameters for \(I\) in case \(I\) is \({\mathfrak m}\)-primary.
scientific article; zbMATH DE number 584580