Stability of the depth function of good filtrations

Let A be a Noetherian local ring‎, ‎and let I \subset J be two ideals of A‎. ‎Let M be a finitely generated A-module‎. ‎Brodmann proved that the function n \mapsto \mathrm{depth}_{J}(\frac{M}{I^{n}M}) is constant for large n‎.‎In this paper‎, ‎we consider a filtration \phi = (M_n)_{n \in \mathbb{N}} of M and a filtration f = (I_n)_{n \in \mathbb{N}} of A‎. ‎Generalizing Brodmann's result‎, ‎we first show that the function n \mapsto \mathrm{depth}_{J}(\frac{M}{M_{n}}) is constant for large n of value \mathrm{depth}_{J}(f‎, ‎M)‎, ‎provided that \phi is f-good and f is strongly Noetherian‎. ‎Secondly‎, ‎we establish the inequality‎ ‎\gamma_{J}(f‎, ‎M) \leq \dim_A(M)‎ - ‎\mathrm{depth}_{J}(f‎, ‎M)‎, ‎where \gamma_{J}(f‎, ‎M) denotes the analytic spread of f at J with respect to M‎.

کلیدواژه ها:

Filtration ، dimension ، analytic spread ، Depth