Lengths of certain generalized fractions
Let A be a (commutative Noetherian) local ring of dimension \(d\geq 1\). Set \(U(A)_{d+1}=\{(x_ 1,...,x_ d,1)\in A^{d+1}:\) there exists j with \(0\leq j\leq d\) such that \(x_ 1,...,x_ j\) form a subset of a system of parameters (s.o.p.) for A \(and\quad x_{j+1}=...=x_ d=1\}.\) Then \(U(A)_{d+1}\) is a triangular subset of \(A^{d+1}\) and the module of generalized fractions \(U(A)_{d+1}^{-d-1}A\) may be formed: see \textit{R. Y. Sharp} and \textit{H. Zakeri} [Mathematika 29, 32-41 (1982; Zbl 0497.13006)]. Let \(x_ 1,...,x_ d\) form a s.o.p. for A. For positive integers \(n_ 1,...,n_ d\), the generalized fraction \(1/(x_ 1^{n_ 1},...,x_ d^{n_ d},1)\) of \(U(A)_{d+1}^{-d-1}A\) is annihilated by \(\sum^{d}_{i=1}Ax_ i,\quad and\) so the cyclic submodule it generates has finite length \(\ell (A(1/(x_ 1^{n_ 1},...,x_ d^{n_ d},1)))\). This paper is concerned with the following question: given \(x_ 1,...,x_ d\) which form a s.o.p. for A, does there exist a rational polynomial \(h\in {\mathbb{Q}}[X_ 1,...,X_ d]\) such that, provided \(n_ 1,...,n_ d\) are all sufficiently large, \(\ell (A(1/(x_ 1^{n_ 1},...,x_ d^{n_ d},1)))=h(n_ 1,...,n_ d)?\) The paper does not settle this question, but shows that it does have an affirmative answer in the special case when A is a (Cohen-Macaulay or) generalized Cohen-Macaulay local ring, and in the special case when d=(1 or) 2.
length of module of generalized fraction
scientific article; zbMATH DE number 3924946