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 $A = A_1 \cap B$. $\square$

Comment #5050 by sidm on

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

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