Currently I am learning some alg. geometry and I would like to show the following claim:
Let $\mathfrak{p}$ be some prime ideal of $A$. Then $$ \varprojlim_{f\notin \mathfrak{p}} \operatorname{Spec}A_f= \operatorname{Spec}A_\mathfrak{p}.$$
It is rather straightforward to construct a map from right to left and showing that it is injective is rather easy as well, however I can't show that it is surjective.
Any hint is appreciated.