On the multiplicity of graded fiber cones with arbitrary dimensions

Tóm tắt

Let  be a Noetherian local ring with maximal ideal ,  an -primary ideal,  a finitely generated standard graded algebra over A and   a finitely generated graded -module.  Then  is called the fiber cone of the graded module  with respect to  As we know, the concept of multiplicities in commutative algebra is an object that plays an important role in determining the properties and classifying the structure of rings and modules including the Cohen-Macaulay property. In this paper, we establish the multiplicity formula of the fiber cone  with arbitrary dimensions. The concept we used to prove the results is filter-regular sequences of graded modules. Our approach is based on the formulas of multiplicities of fiber cones of graded modules in the case that  the dimensions of  those fiber cones equal 1 and by induction on dimensions of fiber cones of graded modules.