On the length of generalized fractions

✍ Scribed by Nguyen Tu Cuong; Marcel Morales; Le Thanh Nhan

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 149 KB
• Volume: 265
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00224-2

✦ Synopsis

Let M be a finitely generated module over a Noetherian local ring (R, m) with dim M = d. Let (x 1 , . . . , x d ) be a system of parameters of M and (n 1 , . . . , n d ) a set of positive integers. Consider the length of generalized fraction 1/(x n 1 1 , . . . , x n d d , 1) as a function in n 1 , . . . , n d . Sharp and Hamieh [J. Pure Appl. Algebra 38 (1985) 323-336] asked whether this function is a polynomial for n 1 , . . . , n d large enough. In this paper, we will give counterexamples to this question. We also study conditions on the system of parameters x, in order to show that the length of the generalized fraction 1/(x n 1 1 , . . . , x n d d , 1) is not a polynomial for n 1 , . . . , n d large enough.