Some properties of the multiplicity sequence for arbitrary ideals (Q618662)

For a local noetherian ring \((R,\mathfrak{m})\) of dimension \(d\) and an \(\mathfrak{m}\)-primary ideal \(I\), the Hilbert-Samuel multiplicity \(e(I)=\lim_{n \to \infty} (d! \lambda(R/I^n)/n^d)\) is an important invariant in local algebra. A well known result of Rees shows that for a formally equidimensional local ring this invariant characterizes the integral closure of \(I\); more precisely, given \(I \subseteq J\), the ideals \(I\) and \(J\) have the same integral closure if and only if \(e(I)=e(J)\). In the case when \(I\) is not necessarily \(\mathfrak{m}\)-primary, \textit{R.~Achilles} and \textit{M.~Manaresi} [Math. Ann. 309, 573--591 (1997; Zbl 0894.14005)] extended the concept of multiplicity by introducing a multiplicity sequence \(c_0(I), \ldots, c_d(I)\). When \(I\) is \(\mathfrak{m}\)-primary, one has \(c_0(I)=e(I)\) and \(c_i(I)=0\) for \(i > 0\). A result of \textit{H.~Flenner} and \textit{M.~Manaresi} [Math. Z. 238, 205--214 (2001; Zbl 1037.13001)] extends the theorem of Rees as follows: if \(R\) is a formally equidimensional local ring and \(I \subseteq J\), then the ideals \(I\) and \(J\) have the same integral closure if and only if \(c_0(I_{\mathfrak{p}})=c_0(J_{\mathfrak{p}})\) for every prime ideal \(\mathfrak{p}\) of \(R\). \textit{C. Ciupercă} [J. Pure Appl. Algebra 178, 25--48 (2003; Zbl 1088.13501)] also proved that if \(I \subseteq J\) have the same integral closure, then \(c_i(I)=c_i(J)\) for all \(i=0,\ldots, d\). In this paper the authors prove two properties of the multiplicity sequence of \(I\): the additivity with respect to exact sequences and the associativity formula. As a consequence, new proofs are obtained for the results of Flenner--Manaresi and Ciupercă that we mentioned above.