TITLE: "Points" in algebraic geometry: Why shift from m-Spec to Spec?
QUESTION [37 upvotes]: Why were algebraic geometers in the 19th Century thinking of
m-Spec as the set of points of an affine variety associated to the
ring whereas, sometime in the middle of the 20 Century, people started to think Spec was more appropriate as the "set of points".
What are advantages of the Spec approach? Specific theorems?

REPLY [6 votes]: The reason why $\operatorname{Spec} A$ is an important notion is because it solves the following problem for a commutative ring $A$:
Find a local ring $\mathcal O$ together with a localisation morphism $A \to \mathcal O$ such that every other localisation morphism $A \to B$ to a local ring $B$ factors as a local morphism over $A \to \mathcal O$, i.e. one looks for a kind of universal localisation of $A$.
Stated as above, this problem has no solution at least as long one is not willing to leave the world of rings in the category of sets. However it has a solution in the following more general setting:
There is a topos $X$ endowed with a local ring object $\mathcal O$ and a localisation morphism $A \to \mathcal O$ such that for every other topos $Y$ together with a local ring object $B$ and a localisation morphism $A \to B$ there is a pair of a geometric morphism $f\colon Y \to X$ and a morphism $f^* \mathcal O \to B$ of local rings, which is unique up to a unique natural isomorphism, such that $A \to B$ is given by the composition of $f^* \mathcal O \to B$ and $A \to f^* \mathcal O$.
In fact, the solution to this problem is the topos $X$ of sheaves on $\operatorname{Spec} A$ together with the structure sheaf $\mathcal O_X$ as a local ring object. Now if you replace $\operatorname{Spec} A$ by the max-spectrum, the locally ringed sheaf topos you get will not solve the universal localisation problem in general.
This means that the usual definition of $\operatorname{Spec} A$ with prime ideals is a correct one (as long as one is working in classical logic with the axiom of choice) but it  does not mean that it is the only correct definition: You can, for example, replace $\operatorname{Spec}$ by any other topological space or, more generally, by any other site such that the sheaf topos over it is still equivalent to $X$.