The rate of graded modules over some graded algebras

Let k be a field, R a standard graded k -algebra and M be a finitely generated graded R -module.##In this paper, we find upper bounds for the rate of M , rate_R ( M ). More precisely, let ( A , n ) be a regular local ring and I ⊆ n^t be an ideal of A , where t ≥ 2. We prove that if ( B = A/ I, m = n/ I ) is a Cohen-Macaulay local ring with multiplicity e ( B ) = ( h+t-1 h ) , where h = embdim ( B ) − dim B , then rate_{gr_m ( B )} ( gr_m ( N )) ≤ t − 1 for every B -module N which is annihilated by a minimal reduction of m