On the multiplicity of graded fiber cones with arbitrary dimensions

Let \((A,\mathfrak{m})\) be a Noetherian local ring with maximal ideal \(\mathfrak{m}\), \(J\subset A\) an -\(\mathfrak{m}\) primary ideal, \(S=\mathop\oplus \limits_{n\ge 0}S_n\) a finitely generated standard graded algebra over A and \(M=\mathop\oplus \limits_{n\ge 0}M_n\) a finitely generated graded -module. Then \(F_J(M)=\mathop\oplus \limits_{n\ge 0}\dfrac{M_n}{JM_n}\) is called the fiber cone of the graded module \(M\) with respect to \(J\) . 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 \(F_J(M)=\mathop\oplus \limits_{n\ge 0}\dfrac{M_n}{JM_n}\) 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.