Lemma 115.4.15. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $s$ and $d$ be integers. Assume

$A$ has a dualizing complex,

$\text{cd}(A, I) \leq d$,

if $\mathfrak p \not\in V(I)$ then $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) > s$ or $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim (A/\mathfrak p) > d + s$.

Then the assumptions of Algebraic and Formal Geometry, Lemma 52.10.4 hold for $A, I, \mathfrak m, M$ and $H^ i_\mathfrak m(M) \to \mathop{\mathrm{lim}}\nolimits H^ i_\mathfrak m(M/I^ nM)$ is an isomorphism for $i \leq s$ and these modules are annihilated by a power of $I$.

## Comments (0)