The Stacks project

Example 52.12.4. Let $A$ be a Noetherian domain which has a dualizing complex and which is complete with respect to a nonzero $f \in A$. Let $f \in \mathfrak a \subset A$ be an ideal. Assume every irreducible component of $Z = V(\mathfrak a)$ has codimension $> 2$ in $X = \mathop{\mathrm{Spec}}(A)$. Equivalently, assume every irreducible component of $Z$ has codimension $> 1$ in $Y = V(f)$. Then with $U = X \setminus Z$ every element of

\[ \mathop{\mathrm{lim}}\nolimits _ n \Gamma (U, \mathcal{O}_ U/f^ n \mathcal{O}_ U) \]

is the restriction of a section of $\mathcal{O}_ U$ defined on an open neighbourhood of

\[ V(f) \setminus Z = V(f) \cap U = Y \setminus Z = U \cap Y \]

In particular we see that $Y \setminus Z$ is connected. See Lemma 52.14.2 below.


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