Filter-regular sequences and multiplicity of blow-up rings of ideals of the principal class
Let \(R\) be a graded algebra generated by finitely many forms of degree one over a field \(k\) and \(I\) a homogeneous ideal of \(R\) of principal class, that is, \(I\) is generated by ht\((I)\) homogeneous elements. This paper computes the multiplicity of the associated graded ring gr\(_ I(R)\), the Rees algebra \(R[It]\), and the extended Rees algebra \(R[It,t^{-1}]\). Suppose that \(a_ 1, \dots, a_ n\) are the degrees of the elements of a homogeneous minimal basis (of length ht\((I))\) of \(I\). Then the multiplicity of gr\(_ I(R)\) is given by \(e(\text{gr}_ I(R)) = a_ 1 \cdots a_ n e(R)\). (Here \(e(\;)\) denotes the multiplicity with respect to the maximal homogeneous ideal.) To compute the multiplicities of the other two algebras, it is further assumed that \(I\) is generated by a homogeneous filter-regular sequence \(x_ 1, \dots, x_ n\) with respect to \(I\) where \(a_ 1 \leq \cdots \leq a_ n\) \((a_ i = \deg x_ i)\). (A sequence \(x_ 1, \dots, x_ n\) is a filter-regular sequence with respect to \(I\) if \(x_ i \notin p\) for all associated primes \(p \nsupseteq I\) of \((x_ 1, \dots, x_{i-1})\), \(i = 1, \dots, n\).) Then \(e(R[It]) = (1 + \sum^{n-1}_{i=1} a_ 1 \cdots a_ i) e(R)\) and \(e(R[It,t^{-1}]) = (1 + \sum^{n-1}_{i=l} a_ 1 \cdots a_ i) e(R)\) where \(l\) is the largest integer for which \(a_ l = 1\) \((l = 0\) and \(a_ 1 \cdots a_ l = 1\) if \(a_ i > 1\) for all \(i = 1, \dots, n)\).
scientific article; zbMATH DE number 567069