Loading [MathJax]/extensions/tex2jax.js

The Stacks project

Lemma 52.14.2. Let $I \subset \mathfrak a$ be ideals of a Noetherian ring $A$. Assume

  1. $A$ is $I$-adically complete and has a dualizing complex,

  2. if $\mathfrak p \subset A$ is a minimal prime not contained in $V(I)$ and $\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)$, then $\dim ((A/\mathfrak p)_\mathfrak q) > \text{cd}(A, I) + 1$,

  3. any nonempty open $V \subset \mathop{\mathrm{Spec}}(A)$ which contains $V(I) \setminus V(\mathfrak a)$ is connected1.

Then $V(I) \setminus V(\mathfrak a)$ is either empty or connected.

Proof. We may replace $A$ by its reduction. Then we have the inequality in (2) for all associated primes of $A$. By Proposition 52.12.3 we see that

\[ \mathop{\mathrm{colim}}\nolimits H^0(V, \mathcal{O}_ V) = \mathop{\mathrm{lim}}\nolimits H^0(T_ n, \mathcal{O}_{T_ n}) \]

where the colimit is over the opens $V$ as in (3) and $T_ n$ is the $n$th infinitesimal neighbourhood of $T = V(I) \setminus V(\mathfrak a)$ in $U = \mathop{\mathrm{Spec}}(A) \setminus V(\mathfrak a)$. Thus $T$ is either empty or connected, since if not, then the right hand side would have a nontrivial idempotent and we've assumed the left hand side does not. Some details omitted. $\square$

[1] For example if $A$ is a domain.

Comments (0)


Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.