I have two questions on scheme morphisms.
Is the property of a scheme morphism to be a closed immersion a local property (as it is for open immersions)?
Let $X=Spec (R)$ be a noetherian scheme and $p$ a prime ideal of $R$.
In Georges Elencwajg's very helpful answer to some Questions on scheme morphisms, he explains that one may think of the embedded scheme $g:Spec (R_p)\to X$ as of the intersection of all the neighbourhoods of $p$ in $X$.
On the other hand, there is the closed immersion $Spec (R/p)\to X$ and as topological spaces $Spec (R/p)=V(p)$ is the closure of the ''generic point'' $p$ of the subscheme.
As $g$ is not always a closed immersion, one can consider the topological space $V(Im(g))$, the closure of the image of $g$. Is my intuition correct that this always has to be the whole space $Spec (R)$?