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.

## Comments (2)

Comment #5050 by sidm on

Comment #5270 by Johan on