A formula for the associated Buchsbaum-Rim multiplicities of a direct sum of cyclic modules (Q1635337)

Let \((R, \mathfrak{m})\) be a local noetherian ring of dimension \(d\) and let \(C\) be an \(R\)-module of finite length. If \(R^n \overset{\varphi}{\rightarrow} {R^r} {C} 0\) is a minimal free presentation of \(C\), let \(S=\text{Sym}_R(R^r)\) and \(M=\text{Im} \varphi\) with homogeneous components of degee \(p\) denoted \(S_p\) and \(M^p\), respectively. Generalizing the construction of the Buchsbaum-Rim multiplicity [\textit{D. A. Buchsbaum} and \textit{D. S. Rim}, Trans. Am. Math. Soc. 111, 197--224 (1964; Zbl 0131.27802)], \textit{S. Kleiman} and \textit{A. Thorup} [J. Algebra 167, No. 1, 168--231 (1994; Zbl 0815.13012); Am. J. Math. 118, No. 3, 529--569 (1996; Zbl 0858.14002)] and \textit{D. Kirby} and \textit{D. Rees} [Contemp. Math. 159, 209--267 (1994; Zbl 0805.13001), Math. Proc. Camb. Philos. Soc. 119, No. 3, 425--445 (1996; Zbl 0877.13023)] considered the length function \(\Lambda(p,q)=\ell_R(S_{p+q}/M^pS_q)\) and showed that \(\Lambda(p,q)\) is eventually a polynomial function of two variables of total degree \(d+r-1\). For \(j=0,\ldots, d+r-1\), the \(j\)-th Buchsbaum-Rim multiplicity \(e^j(C)\) is defined as the normalized coefficient of bi-degree \((d+r-1-j,j)\) of this polynomial, i.e., \((d-r+1-j)!j!\) times the coefficient of \(p^{d+r-1-j}q^{j}\). They proved that these coefficients are non-negative integers and do not depend on the choice of the presentation \(\varphi\). Moreover, \[ e^0(C) \geq e^1(C) \geq \ldots \geq e^{r-1}(C) > e^r(C)= \ldots = e^{d+r-1}(C)=0, \] and \(e^0(C)\) coincides with the classical Buchsbaum-Rim multiplicity of \(C\). If \(I_1\subseteq I_2 \subseteq \ldots \subseteq I_r\) are \(\mathfrak{m}\)-primary ideals of \(R\), \textit{D. Kirby} and \textit{D. Rees} [Math. Proc. Camb. Philos. Soc. 119, No. 3, 425--445 (1996; Zbl 0877.13023)] proved that \[ e^j(R/I_1\oplus \cdots \oplus R/I_r)=e^0(R/I_{j+1}\oplus \cdots \oplus R/I_r). \] In particular, for the last non-zero Buchsbaum-Rim multiplicity, \[ e^{r-1}(R/I_1\oplus \cdots \oplus R/I_r)=e^0(R/I_r). \] The main result of this paper generalizes this equality for an arbitrary family of \(\mathfrak{m}\)-primary ideals \(I_1,\ldots, I_r\) by showing that \[ e^{r-1}(R/I_1\oplus \cdots \oplus R/I_r)=e^0(R/(I_1+\cdots +I_r)). \]