Let $k$ be a field, and $K$ a function field over $k$ (i.e. it is finitely generated). Let $A\subseteq K$ be a local ring with maximal $\mathfrak{m}$, and $\{A_s\}_s$ be a set of local rings $A\subseteq A_s\subseteq K$ with maximal $\mathfrak{m}_s$ such that $\mathfrak{m}_s\cap A=\mathfrak{m}$ and $\cap_s A_s=A$.

Now suppose we have a scheme $X$ over $k$ and a morphism $\text{Spec} K\to X$ with extensions $\text{Spec}A_s\to X$ such that all the closed points $\mathfrak{m}_s\in\text{Spec}A_s$ map to the same point $x\in X$. Then the image of $\mathcal{O}_{X,x}\to K$ is contained in $\cap_s A_s=A$, and hence we have an extension $\text{Spec}A\to X$.

Question: is the above still true if $X$ is an algebraic space? If it is useful, we may suppose that $A$ and $A_s$ are normal for every $s$.


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