A lower bound on $\text{depth} \ \mathbf{A/I^n}$

Abstract

Let $(A,\mathfrak{m})$ be a Noetherian local ring and let $I\subseteq \mathfrak{m}$ be an ideal in $A$. Let $\ell(I)$ be the analytic spread of $I$. Let $G_I(A) = \bigoplus_{n \geq 0}I^n/I^{n+1}$ be the associated graded ring of $I$ and let its depth be $t$. We prove $\text{depth} \ A/I^n \geq t - \ell(I)$ for all $n\geq 1$.