Lemma 33.37.7 (09N4)—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

$A \to A_ i$ induces an open immersion $j_ i : \mathop{\mathrm{Spec}}(A_ i) \to \mathop{\mathrm{Spec}}(A)$ ,
$\mathop{\mathrm{Spec}}(A)$ is the union of the opens $j_ i(\mathop{\mathrm{Spec}}(A_ i))$ ,
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 $A = A_1 \cap B$ . $\square$

The Stacks project

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