On the multiplicity of blow-up rings of ideals generated by \(d\)-sequences
Let \(R=R_ 0+R_ 1+R_ 2+\dots\) be a noetherian graded ring generated by \(R_ 1\) over \(R_ 0=k\), a field, and let \(I\) be an ideal generated by a homogeneous \(d\)-sequence \(x_ 1,\dots,x_ n\). Let \(S=R[It]\) be the Rees ring, \(S[t^{-1}]\) be the extended Rees ring and gr\(_ I(R)=S/IS\) be the associated graded ring. The aim of the paper is to calculate the multiplicities (with respect to their respective unique maximal graded ideal) of these ``blow-up rings''. Set \(a_ i=\deg x_ i\) and \(I_ i=(x_ 1,\dots,x_{i-1}):x_ i\), \(1\leq i\leq n\). Assume that \(a_ 1\leq a_ 2\leq\cdots\leq a_ n\). Then the multiplicities \(e(S)\) etc. are explicitly calculated in terms of \(e(R/I_ i)\)'s and the degrees \(a_ i\). In particular, if \(\dim R/I=\dim R-n\), then one gets very simple results like \(e(S)=(1+a_ 1+a_ 1a_ 2+\cdots+a_ 1a_ 2\dots a_{n-1})\cdot e(R)\), \(e(\text{gr}_ I(R))=a_ 1\dots a_{n- 1}e(R)\). The proofs are as interesting as the results. The authors use certain filtration \({\mathcal F}\) on \(S\), finer than the adic filtration with respect to the maximal graded ideal, and get the nice presentation gr\(_{\mathcal F}(S)\simeq R[T_ 1,\dots,T_ n]/(I_ 1T_ 1,\dots,I_ nT_ n)\). The technique is basically similar to that of Gröbner basis.
scientific article; zbMATH DE number 89585
scientific article