A computation of Buchsbaum-Rim functions of two variables in a special case (Q2374285)

Let \((R,m)\) be a Noetherian local ring of dimension \(d>0,\) \(\varphi: R^n \rightarrow R^r\) an \(R\)-linear map, and let \(C=\mathrm{Coker} \varphi\) be of finite length. Buchsbaum-Rim function is the following numerical function \[ \lambda : {p} \rightarrow \ell_R ([\mathrm{Coker} \mathrm{Sym}(\varphi)]_{{p}} ) \] where \([~~]_{p}\) is the homogeneous component of degree \(p\). In 1967 for the first time \textit{D. A. Buchsbaum} and \textit{D. S. Rim} [Trans. Am. Math. Soc. 111, 197--224 (1964; Zbl 0131.27802)] introduced this function as the generalized Hilbert-Samuel function. This is eventually a polynomial of degree \(d+r-1.\) The positive integer \(e\) occurs in the polynomial \[ \lambda(p)= \frac{p^{d+r-1}}{(d+r-1)!} e+\text{lower terms for }p\gg0 \] is the Buchsbaum-Rim multiplicity. This multiplicity agrees with Hilbert-Samuel multiplicity when \(C\) is a cyclic module. In the same paper they have generalized the system of parameter notion by introducing a parameter matrix \(\varphi\) when \(\mathrm{Coker}(\varphi)\) has finite length, \(d=n-r+1\) \(\mathrm{Im}\varphi\subset mR^r.\) \textit{S. Kleiman} and \textit{A. Thorup} [Am. J. Math. 118, No. 3, 529--569 (1996; Zbl 0858.14002)] generalized Buchsbaum-Rim multiplicity by geometric approach, involving blow ups and intersection numbers. They considered the function of two variables, \[ \lambda(p,q):=\ell(S_{p+q}/M^p S_q) \] where \(M=\mathrm{Im}\varphi\) and \(S=\mathrm{Sym}(R^r)\). Set \(\mathcal{M}\) as the associated sheaf of \(M\), as \(\lambda(p,q)\) is eventually a polynomial of total degree at most \(d+r-1=\alpha\), its term of total degree \(\alpha\) is the form \[ \Lambda(p,q)=\sum_{i+k=\alpha}e^{i,k}([\mathcal{M}]_{\alpha})p^iq^k/i!k! \] in particular \(e^{d+r-1,0}\) is equal to the Buchsbaum-Rim multiplicity and \(e^{j}:=e^{j,0}\) is called the jth Buchsbaum-Rim multiplicity. In the paper under review, the author computes explicitly \(\lambda(p,q)\) and determines \(e^{j}(C)\) in the case that \(C= \bigoplus_{i=1}^r(R/Q_i)\), where \(Q_i\)s are ascending chain of parameter ideals and \(R\) is a one dimensional Cohen-Macaulay ring. Where \(r=2\) and there is no necessarily inclusion relation between \(Q_i\)s the author characterizes the situations when \(\lambda(p,q)\) coincides with its polynomial function. As a consequence it contrasts the Theorem 3.4 in the paper [\textit{J. Brennan} et al., J. Algebra 241, No. 1, 379--392 (2001; Zbl 1071.13504)].