On form ideals and Artin-Rees condition

We look for conditions which make two ideals and in a noetherian ring A have the same form ideal in an associated graded ring G_A(α). More precisely, when and and f_i?f'_iεα ^m m?0, ? i , we give a necessary and sufficient condition to have , involving the first syzygies modules both of (f₁,...,f_n) and (f'₁,...,f'_n); our proof is based on the Artin-Rees lemma. Finally we show that, when the sequence f₁,...,f_n is regular and for an integer q, then f₁?f'_i ? α^q+1 ? i implies .