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