History of Algebraic Geometry: Motivation behind definition of schemes

Solution 1:

Well, any field $K$ has a unique maximal ideal, the zero ideal. Hence, the maximal spectrum (as a topological space!) $\mathrm{Specm}(K)$ doesn't suffice to reconstruct $K$. One needs to introduce the sheaf of regular functions on a variety. A space together with a sheaf of rings on it is called a ringed space. We may reconstruct any finitely generated integral commutative $k$-algebra $A$ from the ringed space $\mathrm{Specm}(A)$ (and more generally, but this is already scheme-land, any commutative ring $A$ from the ringed space $\mathrm{Spec}(A)$.) What is suggested in the text is to keep track of the stalks of the sheaf of regular functions. For $\mathrm{Specm}(A)$ these are the localization of $A$ at maximal ideals of $A$. This also suffices to reconstruct a field $K$, but for general commutative rings we need more than just the stalks.

Solution 2:

I will clarify the part "A way to correct to this suggestion is again suggested by Riemann....." which is the only part left, as you point out in the comments above. $A$ is the ring of functions (coordinate ring) on $V$, and $A_z$ the localization of $A$ at $z\in V$ (if you don't know about localization: Atiyah-Macdonald); thus one is led to consider local rings (as he also points in the parts in the article below your quote). The point is that for a field $k$ you have only one maximal ideal and the localization at this maximal ideals is the field itself, so as he says for different fields we obtain different pairs. By the way these things are discussed in another way in Grothendieck's introduction to EGA (GRUNDLEHREN DER MATH. WISS. edition)