Lemma 52.11.3 (0DXR)—The Stacks project

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

\[ \mathop{\mathrm{colim}}\nolimits H^0(V, \mathcal{F}) \longrightarrow \mathop{\mathrm{lim}}\nolimits H^0(U, \mathcal{F}/I^ n\mathcal{F}) \]

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

Proof. Apply Theorem 52.11.2 with $s = 1$ (we get finiteness too). $\square$

Comments (3)

Comment #3438 by Kestutis Cesnavicius on July 27, 2018 at 04:19

What is $d$ ? Can't we simply consolidate the two conditions into "for any associated point $x \in U$ of $\mathcal{F}$ , we have $\mathrm{dim}(\overline{\{x\}}) > \mathrm{cd}(A, I) + 1$ "? (Likewise for some preceding results.)

Comment #3439 by Kestutis Cesnavicius on July 27, 2018 at 04:23

Also, it would be nice to mention as in the preceding result that the closure of $x$ is taken in $X$ (not in $U$ ).

Comment #3494 by Johan on August 05, 2018 at 11:45

OK, I have made this change in this case. However, when writing these sections I tried to have some consistency in the numbering of the conditions listed in the lemmas. So I want to be careful in changing this in other lemmas, etc. So unless somebody is going to review the chapter as a whole and look more carefully to see if things can be simplified and/or strengthened (as may very well be the case), I would prefer to leave things as is for now.

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