Depth

Remarks

The definition makes sense because all maximal $M$-regular sequences in $I$ have the same length, a theorem of Rees. That length is also $\min \set{i : \operatorname{Ext}^i_R(R/I, M) \ne 0}$. If $IM = M$, one sets $\operatorname{depth}_I M = \infty$. Example: for $R = K[x_1, \ldots, x_n]$ localized at $(x_1, \ldots, x_n)$, one has $\operatorname{depth} R = n$, with the maximal regular sequence $x_1, \ldots, x_n$. Example: for $R = K[x,y]/(x^2, xy)$ localized at $\mathfrak{m} = (x, y)$, one has $\operatorname{depth} R = 0$, since $x \ne 0$ and $x \mathfrak{m} = 0$, so no element of $\mathfrak{m}$ is a non-zero-divisor. For a nonzero finitely generated module over a Noetherian local ring, $\operatorname{depth} M \le \dim M$.