Let $M$ be a smooth manifold, $\mathscr O_M$ the sheaf of vector fields. Then, by an argument involving partition of unity, we can show that the map $$\mathscr O_M(M)\to\mathscr O_{M,x}$$ is surjective for every $x\in M.$
This is a rather strange in my opinion: even in the case of an affine scheme, if $x$ does not correspond to the unique maximal ideal of a local ring, i.e. if there are non-invertible elements outside the ideal corresponding to $x$, then this map is not surjective: $A\to A_x.$
Let $(X, \mathscr O_X)$ be a scheme. I would like to know when is the map $\mathscr O_X\to\mathscr O_{X,x}$ surjective for every $x\in X,$ and if there is a name for such schemes.
Any suggestions, corrections, opinions, or references are sincerely welcomed, and thanks in advance.