On the depth of the fiber cone of filtrations
Let \(A\) be a Noetherian local ring with maximal ideal \({\mathfrak m}\) and \(I\) an ideal. The Rees algebra \(R(I)=\bigoplus_{n\geq 0} I^nt^n\), the associated graded ring \(G(I) =R(I)/IR(I)\) and the fiber cone \(F_{\mathfrak m} (I)=R(I)/{\mathfrak m} R(I)\) represent the blowing-up \(\pi\): \(\text{Proj} R(I) \to\text{Spec} A\), the exceptional fiber \(\pi^{-1} (V(I))= \text{Proj} G(I)\) and the closed fiber \(\pi^{-1} (V({\mathfrak m})) =\text{Proj} F_{\mathfrak m} (I)\), respectively. In this paper the authors are interested in the Cohen-Macaulay property of \(F_{\mathfrak m} (I)\). If \(A/{\mathfrak m}\) is an infinite field and \(s= \dim F_{\mathfrak m} (I)\) then there exist \(a_1, \dots, a_s\in I\) such that \(a_1t, \dots, a_st\in R(I)\) is a system of parameters for \(F_{\mathfrak m} (I)\). In other words, there is a positive integer \(r\) such that \(I^{n+1}= (a_1, \dots, a_s)I^n\) for all \(n\geq r\). We know that \(a_1t, \dots, a_st\) is a \(G(I)\)-sequence if and only if \(a_1, \dots, a_s\) is an \(A\)-sequence and \((a_1, \dots, a_s) \cap I^n= (a_1, \dots, a_s)I^{n-1}\) for all \(n\leq r\). In this case, the authors show that \(a_1t, \dots, a_st\) is an \(F_{\mathfrak m} (I)\)-sequence if and only if \((a_1, \dots, a_s)\cap {\mathfrak m} I^n= (a_1, \dots, a_s) {\mathfrak m} I^{n-1}\) for all \(n\leq r\).
scientific article; zbMATH DE number 1106618