A study of the length function of generalized fractions of modules

Abstract

Abstract: Let $(R,\ frak m) $ be a Noetherian local ring and $ M $ a finitely generated $ R $- module of dimension $ d $. Let $\ underline {x}= x_1,..., x_d $ be a system of parameters of $ M $ and $\ underline {n}=(n_1,..., n_d) $ a $ d $-tuple of positive integers. In this paper we study the length of generalized fractions $ M (1/(x_1,..., x_d, 1)) $ which was introduced by Sharp and Hamieh in\ cite {ShH85}. First, we study the growth of the function $ J_ {\ underline {x}, M}(\ underline {n})=\ ell (M (1/(x_1^{n_1},..., x_d^{n_d}, 1)))-n_1... n_d e (\ ...