The Stacks project

Lemma 33.37.7. Let $K$ be a field. Let $A_1, \ldots , A_ r \subset K$ be Noetherian semi-local rings of dimension $1$ with fraction field $K$. If $A_ i \otimes A_ j \to K$ is surjective for all $i \not= j$, then there exists a Noetherian semi-local domain $A \subset K$ of dimension $1$ contained in $A_1, \ldots , A_ r$ such that

  1. $A \to A_ i$ induces an open immersion $j_ i : \mathop{\mathrm{Spec}}(A_ i) \to \mathop{\mathrm{Spec}}(A)$,

  2. $\mathop{\mathrm{Spec}}(A)$ is the union of the opens $j_ i(\mathop{\mathrm{Spec}}(A_ i))$,

  3. each closed point of $\mathop{\mathrm{Spec}}(A)$ lies in exactly one of these opens.

Proof. Namely, we can take $A = A_1 \cap \ldots \cap A_ r$. First we note that (3), once (1) and (2) have been proven, follows from the assumption that $A_ i \otimes A_ j \to K$ is surjective since if $\mathfrak m \in j_ i(\mathop{\mathrm{Spec}}(A_ i)) \cap j_ j(\mathop{\mathrm{Spec}}(A_ j))$, then $A_ i \otimes A_ j \to K$ ends up in $A_\mathfrak m$. To prove (1) and (2) we argue by induction on $r$. If $r > 1$ by induction we have the results (1) and (2) for $B = A_2 \cap \ldots \cap A_ r$. Then we apply Lemma 33.37.6 to see they hold for $A = A_1 \cap B$. $\square$

Comments (2)

Comment #5050 by sidm on

"containing" in the statement should be "contained in".

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Your email address will not be published.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 09N4. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 33.37.7 as pdf