History of Algebraic Geometry: Motivation behind definition of schemes

Question

I am trying to read an article by Jean Dieudonne which talks about development of Algebraic Geometry. The article was being published in the journal "Advances in Mathematics" Volume 3, Issue 3, Pages 233-413 (July 1969). One can find the article from the link http://www.sciencedirect.com/science/journal/00018708/3/3

I quote from the article:

Thus we are led to associate with every variety $V$ a ring of finite type $A = K[X_1, \dots, X_n]/\mathfrak a$, where, for the sake of simplicity, we assume that the field $K$ is algebraically closed. Our problem is to establish a one-to-one correspondence between algebraic and geometric objects. Given a point $z\in V$, we associate with it the set of all functions on $V$ which vanish at $z$; this set is a maximal ideal in $A$. A subvariety $W$ of $V$ is defined by an ideal $\mathfrak b$ which contains $\mathfrak a$: the set of all functions on $V$ which vanish on $W$ forms an ideal in $A$, and we associate this ideal with the subvariety $W$.

In the opposite direction, given an algebraically closed field $K$ and a ring of finite type $A = K[X_1, \dots, X_n]/\mathfrak a$, we want to associate with it a variety $V$. By what precedes, it is natural to consider as the points of $V$ the elements of the maximal spectrum $\operatorname{Specm}(A)$ of $A$, i.e., the maximal ideals of $A$. This, however, will certainly not give a one-to-one correspondence between algebraic and geometric objects, since to any field $K$ there would correspond the variety consisting of one point. A way to correct this situation is again suggested by Riemann’s approach who considered the rings $A_z$ formed by the functions on $V$ which have no poles at $z$. Thus we should take as elements of the geometric object we want to associate with the ring $A$, the pairs $(\mathfrak m, A_{\mathfrak m})$, where $\mathfrak m$ is a maximal ideal of $A$ and $A_{\mathfrak m}$ is the local ring at $\mathfrak m$. In this way, for different fields we obtain different pairs $(z, K)$.

I don't understand “This, however, will certainly not give a one-to-once correspondence between algebraic and geometric objects,since to any field $K$ there would correspond the variety consisting of one point.” and whatever he says after that about considering the local ring at a maximal ideal.

Can someone please clarify.

Answer 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.

Answer 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)