On the multiplicity of multigraded modules over Artinian local rings (Q632480)

Let \(S\) be a finitely generated standard multigraded algebra over an Artinian local ring \(A\) and let \(M\) be a finitely generated multigraded \(S\)-module. The paper under review first investigates the relationship between the multiplicity and mixed multiplicities of \(M\). More precisely, the main result of Section 2 of the paper expresses the multiplicity of \(M\) as a sum of its mixed multiplicities. In that way the authors achieve a ``perfect version'' of Theorem 4.3 from [\textit{M. Herrmann, E. Hyry, J. Ribbe} and \textit{Z. Tang}, J.~Algebra 197, No. 2, 311--341 (1997; Zbl 0931.13002)]. Then in Section 3 the authors give some applications of Section 2 to multigraded fiber cones.