Let $k$ be a field and $X, Y$ k-schemes locally of finite type.
Let $x\in X, y\in Y$, and $g: \mathscr O_{Y,y} \to \mathscr O_{X,x}$ be an isomorphism of k-algebras.
Prove $\exists U, V$ open neigbourhoods of $x,y$ and $\exists h:U\to V$ an isomorphism of k-schemes that induces $g$.
My attempt:
$X,Y$ are locally of finite type over $k$, so locally noetherian. Hence there are neighbourhoods $x\in U'\subset X, y\in V'\subset Y$ s.t. $U'=\operatorname{Spec} A, V'=\operatorname{Spec} B$ for some noetherian rings $A,B$.
Since $\mathscr O_{Y,y} = B_b=\mathscr O_Y(D(b))$ and $\mathscr O_{X,x} = A_a=\mathscr O_X(D(a))$ for some $a\in A, b\in B$ we have an induced isomorphism $h:\ \ D(a)\to D(b)$ that induces $g$. (here notation $A_a$ means localization at $a\in A$)
Setting $U:=D(a)\subset U',V:=D(b)\subset V'$ we are done.
I suspect lots of mistakes in this proof as I am new to scheme theory.
Thank you for all your help.