On form ideals and Artin-Rees conditions

We look for conditions which make two ideals and in a noetherian ring A have the same form ideal in an associated graded ring GA(a^r). More precisely, when and and fi–f'i lies in a^r m m>t0, forall i , we give a necessary and sufficient condition to have, involving the first syzygies modules both of (f1,...,fn) and (f'1,...,f'n); our proof is based on the Artin-Rees lemma. Finally we show that, when the sequence f1,...,fn is regular and for an integer q, then f1–f'i in a^q+1 forall i implies T*=T'*