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

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 $G_I(A)={⨁}_{n\ge 0}{I}^n/I^{n+1}$ be the associated graded ring of $I$ and let its depth be $t$. We prove $\text{depth}A/I^n\ge t-\ell (I)$ for all $n\ge 1$.