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.