Consider two schemes $X,Y$ over a locally noetherian scheme $S$. Let $p \in X$ and assume that $X$ is irreducible and *not* affine spectrum of a semilocal ring.

We assume moreover we have a morphism $f: \operatorname{Spec}O_{X,p} \to Y$. One may ask what are the *neccessary* conditions to extend $f$ to an open subscheme $U \subset X$ which contains $p$, that is to extend to a morphism $h: U \to Y$ which restricts under composition with $ \operatorname{Spec}O_{X,p} \to U$ to $f$.

Certainly, a well known *sufficient* condition is if we assume that $Y$ is locally of finite type over $S$. Then we can choose an affine subscheme $\operatorname{Spec} \ R= S_0 \subset S$ and open subscheme $\operatorname{Spec} \ T= Y_0 \subset Y$. Since $Y$ locally of finite type $T= R[x_1,x_2,..., x_n]/I$. Since $R$ is noetherian, $R[x_1,x_2,..., x_n]$ is also noetherian and the ideal $I$ is finitely generated. Let $\operatorname{Spec} \ A\subset X$ be an affine open neighbourhood of $p \in X$.

Then we consider a morphism $\phi:R[x_1,\dots,x_n]/I\rightarrow \mathscr{O}_ {X,p}$. Now, write $\phi(x_i)=a_i/r$ for some $r \in A$ not vanishing at $p$ and let $s \in A$ not vanishing at $p$ be such that $s \cdot g(\phi(x_1),\dots,\phi(x_n))=0$ for all $g \in I$. Here it is crucial that $I$ is finitely generated ideal, otherwise such $s$ might not exist. then the open set $D(sf)$ makes the job, where $l:V→Y$ corresponds to the morphism $\psi:k[x_1,\dots,x_n]/I\rightarrow A[(sf)^{-1}]$ mapping $x_i$ to $a_i/r$.

Now I ask if to require $Y$ is locally of finite type over $S$ around $y=f(x)$ is also a *neccessary* condition to obtain the extension above. I guess so but I haven't found a conterexample.

every$S$-scheme $X$ with a morphism $\operatorname{Spec} \mathcal O_{X,x} \to Y$ taking $x$ to $y$ there exists an extension on an open"? $\endgroup$ – R. van Dobben de Bruyn Apr 29 at 18:47neccessaryto require that $Y$ is locally of finite type over $S$ near $y$? I will improve my question... $\endgroup$ – user7391733 Apr 29 at 19:01equivalentto $Y$ is locally of finite type over $S$ near $y$. One implicaition I solved. For the other I conjecture that it also true but I need a conterexample in order to argue by contraposition $\endgroup$ – user7391733 Apr 29 at 19:09