Lemma 52.11.3. Let $(A, \mathfrak m)$ be a Noetherian local ring which has a dualizing complex and is complete with respect to an ideal $I$. Set $X = \mathop{\mathrm{Spec}}(A)$, $Y = V(I)$, and $U = X \setminus \{ \mathfrak m\} $. Let $\mathcal{F}$ be a coherent sheaf on $U$. Assume for any associated point $x \in U$ of $\mathcal{F}$ we have $\dim (\overline{\{ x\} }) > \text{cd}(A, I) + 1$ where $\overline{\{ x\} }$ is the closure in $X$. Then the map

is an isomorphism of finite $A$-modules where the colimit is over opens $V \subset U$ containing $U \cap Y$.

## Comments (3)

Comment #3438 by Kestutis Cesnavicius on

Comment #3439 by Kestutis Cesnavicius on

Comment #3494 by Johan on