On the mixed multiplicity and the multiplicity of blow-up rings of equimultiple ideals.
Let \(A\) be a standard graded algebra over a field and \(M\) the maximal graded ideal of \(A\). The main results of this paper (theorems 1.1 and 1.2) give formulae for the multiplicity of the Rees algebra \(A[It]\) and for mixed multiplicities of the pair \((M, I)\), where \(I\) is an arbitrary homogeneous equimultiple ideal in \(A\). However, the formulae of Theorem 1.1, Theorem 1.3 as well as Theorem 2.11, Theorem 3.1, Theorem 3.7 and their consequences are not valid in the sense that their proofs were based on a false statement, namely on the existence of weak-(FC)-sequences of homogeneous elements in \(I\) (introduced by the author). The following observation shows that such a sequence does not always exist. By the definition and lemma 2.8 of the paper, an arbitrary homogeneous weak-(FC)-sequence \(x_1,\dots,x_h\) (with respect to \((M,I)\)) satisfies the following conditions: (1) \(0:x_1 \subseteq \bigcup_{n \geq 1}0:(M I)^n\), (2) \(\deg x_1\) is the initial degree of \(I\). \noindent One can easily construct a homogeneous parameter ideal \(I\) which does not contain any homogeneous element \(x_1\) satisfying both (1) and (2), e.g. \(A = k[x,y,z] = k[X,Y,Z]/(X^2,XY)\) and \(I = (y,z^2)\). The paper also uses another false result on the maximal length of weak-(FC)-sequences in \(I\) [proved in an earlier paper; see \textit{Duong Quôc Viêt}, Proc. Am. Math. Soc. 131, 45--53 (2003; Zbl 1027.13001)]. This length is not ht(\(I)\) (as stated) but the minimal number of generators of \(I\). Since the main results of this paper hold in the case \(I\) is generated by a filter-regular sequence of non-decreasing degrees [see \textit{N. V. Trung}, J. Math. Kyoto Univ. 33, 665--683 (1993; Zbl 0816.13020) and \textit{N. D. Hoang}, Beitr. Algebra Geom. 42, 463--473 (2001; Zbl 1037.13009)], it remains to be seen whether they can be proved in general. Warning: The author said in the introduction that these results cover a similar result of Huneke-Sally on the multiplicity of Rees algebras of equimultiple ideals in a local ring and another result of Herzog-Trung-Ulrich in the case \(I\) is generated by a \(d\)-sequence. But that is wrong because the local case can not be derived from the graded case and ideals generated by \(d\)-sequences do not belong to the class of equimultiple ideals.
scientific article; zbMATH DE number 2005431
scientific article