The Stacks project

Compare with [Proposition 2.3, Etale-coverings]

Lemma 28.15.3. Let $X$ be a Noetherian scheme. The following are equivalent

  1. $X$ is geometrically unibranch (Definition 28.15.1),

  2. for every point $x \in X$ which is not the generic point of an irreducible component of $X$, the punctured spectrum of the strict henselization $\mathcal{O}_{X, x}^{sh}$ is connected.

Proof. More on Algebra, Lemma 15.106.5 shows that (1) implies that the punctured spectra in (2) are irreducible and in particular connected.

Assume (2). Let $x \in X$. We have to show that $\mathcal{O}_{X, x}$ is geometrically unibranch. By induction on $\dim (\mathcal{O}_{X, x})$ we may assume that the result holds for every nontrivial generalization of $x$. We may replace $X$ by $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$. In other words, we may assume that $X = \mathop{\mathrm{Spec}}(A)$ with $A$ local and that $A_\mathfrak p$ is geometrically unibranch for each nonmaximal prime $\mathfrak p \subset A$.

Let $A^{sh}$ be the strict henselization of $A$. If $\mathfrak q \subset A^{sh}$ is a prime lying over $\mathfrak p \subset A$, then $A_\mathfrak p \to A^{sh}_\mathfrak q$ is a filtered colimit of étale algebras. Hence the strict henselizations of $A_\mathfrak p$ and $A^{sh}_\mathfrak q$ are isomorphic. Thus by More on Algebra, Lemma 15.106.5 we conclude that $A^{sh}_\mathfrak q$ has a unique minimal prime ideal for every nonmaximal prime $\mathfrak q$ of $A^{sh}$.

Let $\mathfrak q_1, \ldots , \mathfrak q_ r$ be the minimal primes of $A^{sh}$. We have to show that $r = 1$. By the above we see that $V(\mathfrak q_1) \cap V(\mathfrak q_ j) = \{ \mathfrak m^{sh}\} $ for $j = 2, \ldots , r$. Hence $V(\mathfrak q_1) \setminus \{ \mathfrak m^{sh}\} $ is an open and closed subset of the punctured spectrum of $A^{sh}$ which is a contradiction with the assumption that this punctured spectrum is connected unless $r = 1$. $\square$


Comments (0)


Post a comment

Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BQ4. Beware of the difference between the letter 'O' and the digit '0'.