In old fashioned algebraic geometry a point of an algebraic variety is an n tuple of "numbers" at which all of a given collection of equations vanish. In modern scheme theory, a "point" of an affine scheme is a prime ideal in the ring of regular functions on that scheme. I have tried below to explain how the old concept evolved into the new one. Notice this whole discussion discusses only the concept of a "point" in scheme theory. In general the discussion in EGA is quite unmotivated, and I believe cannot be appreciated without extensive knowledge of classical geometry. Shafarevich is excellent in regard to motivation, as well as being mostly self contained, even more so in its earlier editions.
Algebraic geometry arises from the problem of solving systems of polynomial equations. In modern times, this study has become the theory of “schemes”. You may know the definition of an “affine scheme” as the set of prime ideals of a commutative ring with identity. To see what these have to do with one another, we want to retrace, in the classical case of polynomials over a field, how prime ideals arise in looking for common solutions of systems of equations.
Let k be any field, and {fi} in k[T1,...,Tn] = k[T] a collection of polynomials in n variables over k. A “solution” or “common zero” of the system {fi=0} will be a vector, or point, p = (p1,...,pn) in En, where E is a field extension of k, such that for all i, fi(p) = 0, in E. The key to connecting a solution vector p with a prime ideal of k[T] is to look at the associated evaluation map π:k[T]—>E, taking f(T) to f(p), and especially at its kernel ker(π) = Ip. Then p is a common zero of the polynomials {fi} if and only if {fi} ⊂ Ip. Moreover since E is a field, the ideal Ip is prime in k[T]. Thus a “solution” of the system {fi} in k[T], which might be thought of as a “point” of the algebraic variety defined by this system, determines a prime ideal of k[T] containing {fi}, or equivalently, a prime ideal of the quotient ring k[T]/({fi}), where ({fi}) is the ideal generated by the set {fi}.
Conversely, if {fi} ⊂ I ⊂ k[T] where I is a prime ideal, then there is an associated k algebra map π:k[T]—>k[T]/I ⊂ E where E is the field of fractions of the domain k[T]/I. If we define the point p in En by setting pi = π(Ti) in E, then this map π is evaluation at p, in the sense above.
Thus prime ideals of the quotient ring k[T]/({fi}) correspond to common zeroes of the system {fi}, with components in extensions of k. This correspondence is not however one-one, since e.g. both i and -i in C define the same prime ideal in R[T], since R[T]/(T2+1) has two isomorphisms with C, taking T to either i or -i. Thus considering solutions in a field extension of k involves some Galois theory over k. Indeed if E is an algebraic closure of k, then maximal prime ideals m ⊂ k[T] correspond to orbits of Gal(E/k) in En, parametrizing different embeddings of k[T]/m in E. (See Mumford, II.4., pp. 96 - 99.)
Natural questions:
1) Which prime ideals correspond to solutions in kn, i.e. when do the solutions have coordinates in the original field k itself?
2) Which prime ideals correspond to solution vectors p = (p1,...,pn) with coordinates pi lying in some algebraic extension E of k?
3) What is the geometric meaning of solution vectors whose coordinates lie in transcendental extensions of k?
Question 1). Let p in kn be a solution of the system {fi}. Then the evaluation map at p, taking k[T1,...,Tn]—>k, is surjective since it equals the identity on k, hence its kernel is a maximal ideal m ⊃ {fi}. Thus p in kn corresponds to a maximal ideal m ⊃ {fi} such that the composition k—>k[T]/m—>k is an isomorphism, and since k[T]/m—>k is injective, equivalently such that k—>k[T]/m is an isomorphism.
Conversely let {fi} ⊂ m ⊂ k[T] and assume the map k—>k[T]/m is an isomorphism. Thus for each i, there is a unique pi in k that maps to Ti, mod m. Thus the composition with the inverse isomorphism k[T]—>k[T]/m—>k takes Ti to pi, and this is a k algebra map, and by definition equals evaluation at p = (p1,..., pn). Moreover it takes every element of m to 0. Thus p is a common zero of the ideal m, and hence of the system {fi} having coefficients in k.
Thus common solutions with coefficients in k, of the system {fi}⊂ k[T], correspond one - one with maximal ideals m ⊂ k[T], with {fi} contained in m, and such that k—>k[T]/m is an isomorphism. Equivalently they correspond to maximal ideals m ⊂ k[T]/({fi}) such that k—>k[T]/m is an isomorphism. Of course there might not be any such ideals, as in the case of {T2+1} in R[T], since ({T2+1}) is already maximal and has quotient larger than R. This is because as we know, many systems of equations have no solutions in the coefficient field.
Note that a maximal ideal of k[T] with p = (p1,...,pn) in kn as common zero, contains the functions Ti-pi. But the ideal (T1-p1,...,Tn-pn) is already maximal since the composition k—>k[T]/(T-p) takes pi to Ti, thus surjects, so is isomorphic.
So arguments like those above give one - one correspondences between the following sets:
{common solutions (a1,...,an) in kn of {fi} ⊂ k[X1,...,Xn]}
≈ {k algebra maps π:k[X1,...,Xn]—>k with π(fi) = 0, for all i}
≈ {k-algebra maps π:k[X1,...,Xn]/({fi})—>k}
≈ {maximal ideals M: {fi} ⊂ M⊂k[X1,...,Xn] and such that k—>k[X1,...,Xn]/M is an isomorphism}
≈ {ideals of form {fi} ⊂ (X1-a1,...,Xn-an) ⊂ k[X1,...,Xn]}
These correspondences are as follows: a point a in kn at which all fi = 0 yields the k algebra map π = evaluation at a, which takes all fi to 0; a k algebra map π:k[X1,...,Xn]—>k is always surjective since it already is on k, so π has a maximal ideal kernel M such that the composition
k—>k[X1,...,Xn]—>k[X1,...,Xn]/M —> k is an isomorphism, and since π(fi) = 0 for all i, thus {fi} ⊂ M; a maximal ideal M ⊃ {fi} such that the composition
k—>k[X1,...,Xn]/M is an isomorphism is always of form (X1-a1,...,Xn-an) where ai = the unique element of k such that Xi = ai mod M; finally a maximal ideal of form M = (X1-a1,...,Xn-an) ⊃ {fi} determines the point a by setting aj equal again to the unique element of k congruent mod M to Xj, and since the point a belongs to M, all fi vanish at a.
Question 2). The question is more complicated but the answer is simpler: solutions p of the system {fi}, with coordinates pi in some algebraic extension of k, correspond to all maximal ideals of k[T]/({fi}), i.e. to those maximal ideals of k[T] containing the system {fi}. This time however the correspondence is not one to one, since several solutions can correspond to the same maximal ideal. To prove it will take a little work, but one direction is elementary.
Let p be a common zero of all fi, where each pj is in E, and E is algebraic over k. Consider the evaluation map π:k[T]—>E taking f(T) to f(p). The image is a domain k[p1,...,pn], since it is contained in the field E, and its fraction field
k(p1,...,pn) is a finitely generated algebraic extension field of k, hence a finite dimensional k vector subspace of E. Multiplication by any non zero element u in the domain k[p1,...,pn] is an injective k linear map of the finite dimensional k vector space k[p1,...,pn] to itself, hence also surjective. Thus there is some v in k[p1,...,pn] such that uv = 1. Thus the image k[p1,...,pn] of the evaluation map is actually a field, i.e. k[p1,...,pn] = k(p1,...,pn). Hence the kernel, ker(π) is a maximal ideal of k[T] containing {fi}.
The converse is less elementary, but we claim it is true: i.e. if {fi} ⊂ m⊂ k[T] is any maximal ideal, then the quotient k[T]/m is always a finite dimensional algebraic extension field of k, hence embeds in E = algebraic closure of k, via some embedding π:k[T]/m—>E. If pi = π(Ti), then the map π with kernel m, is evaluation at p in En, which is a common solution of the system {fi} with coordinates in E. Here we see that different embeddings of k[T]/m into E give rise to different solutions p corresponding to the same ideal m.
It seems we get correspondences analogous to those above as follows, where E is an algebraic extension of k:
{common solutions (a1,...,an) in En of {fi} ⊂ k[X1,...,Xn]}
≈ {k algebra maps π:k[X1,...,Xn]—>E with π(fi) = 0, for all i}
≈ {k-algebra maps π:k[X1,...,Xn]/({fi})—>E}
≈ {maximal ideals M: {fi} ⊂ M⊂k[X1,...,Xn] and such that k[X1,...,Xn]/M is k-isomorphic to a subfield of E}.
If E is an algebraic closure of k, this last set is the set of all maximal ideals of k[X1,...,Xn].
The simplest statement arises if k = E is already algebraically closed. Then we can combine the statements in 1) and 2) and see that there is a one one correspondence between solutions of the system {fi} with coordinates in k, and the set of all maximal ideals of k[T] containing {fi}. In particular, if the set {fi} does not generate the unit ideal, then there must be some maximal ideals containing the system, hence there are some common solutions. This is a several variables analog of the fundamental theorem of algebra. In its simplest form it says if k is algebraically closed, there is a one one correspondence between points of kn and maximal ideals in k[T1,...,Tn], where p = (p1,...,pn) in kn corresponds to (T1-p1,...,Tn-pn) in k[T]. This is Hilbert’s famous (“weak”) nullstellensatz, the foundation result of the whole subject of algebraic geometry, the precise dictionary between geometry and polynomial algebra.
If {fi} ⊂ k[T] where k is any field and E its algebraic closure, maximal ideals of k[T]/({fi}) correspond to (finite) Gal(E/k) orbits of the common solution set of {fi} in En. Solutions of the system {fi} lying in kn are the fixed points of the action. [see Mumford’s picture in the redbook chapter II.4, of the scheme associated to the circle X^2+Y^2 = 1, essentially consisting of the closed disc bounded by the usual circle of real points, with the interior of the disc corresponding to algebraic but non real points.]
Question 3), the interpretation of points on the algebraic variety V({fi}) with coordinates in non algebraic (“transcendental”) extensions of k. For simplicity we assume k is algebraically closed, as we will henceforth always assume in this course. We know from the discussion above, such points p correspond to prime ideals Ip in k[T]/({fi}), hence the maximal ideals containing this prime ideal constitute a certain subcollection of the k valued points of the variety V({fi}) ⊂ kn. I.e. a point p with values in a transcendental extension E of k corresponds to a subcollection V(Ip) of k valued points.
Thus if {fi}⊂ Ip ⊂ k[T], then the prime ideal Ip corresponds to a subvariety V(Ip) ⊂ V({fi}) ⊂ kn, and this subvariety V(Ip) is our geometric interpretation, in kn, of the “point” p with values in the transcendental extension E of k. Moreover, if we think of a k - valued point, i.e. a maximal ideal, as having dimension zero, then a “point” p with coordinates pi in E ⊃ k which generate a transcendence degree r extension k(p1,...,pn) of k in E, corresponds to a prime ideal Ip ⊂ k[T]/({fi}) of “coheight r”, so the variety V(Ip) has dimension r.
I.e. the prime ideal Ip can be joined to a maximal ideal by a chain of prime ideals of length r, but not longer, and since k(p1,...,pn) = fraction field of k[p1,...,pn] ≈ k[T]/Ip, thus as commutative algebra students may know, the transcendence degree of k(p1,...,pn) equals the Krull dimension of k[p1,...,pn], equals the coheight of Ip ⊂ k[T]. (We will prove this later.) The subvariety V(I) ⊂ V({fi}) thus has dimension r. Note that “dimension” is a relative term, and here it is taken relative to the base field k. Since not all fields are algebraic over k, not all “points” are zero dimensional over k.
To sum up, if k is algebraically closed, and {fi} ⊂ I ⊂ k[T], where I is a prime ideal of coheight r, then I corresponds equivalently to an (“irreducible”) r dimensional subvariety of the variety V({fi}), and to a point with coordinates in the extension field E = fraction field(k[T]/I) of k of transcendence degree r. (Since a prime ideal is not the proper intersection of two other ideals, here “irreducible” means the variety it defines is not the proper union of two other varieties.)
Grothendieck’s theory of schemes makes this correspondence more intrinsic. I.e. if I is a prime ideal of k[T], then I represents a point of the scheme spec(k[T]) (= set of prime ideals of k[T]), and the fraction field E of k[T]/I is just the residue field of the local ring at the “point” I. The fact that a function in k[T] vanishes at this point iff it lies in I, iff it vanishes at every point of V(I) with coordinates in k, is why this point is called a generic point for V(I). This is reflected in the topology for spec(k[T]) where the generic point of V(I) is dense in V(I). Thus we see more clearly that in scheme theory, even though the residue field at a closed point is the algebraically closed base field k, the residue field at a non closed point is transcendental over k.
So it seems that in a scheme, an abstract “point”, represented by a prime ideal of a ring in the case of an affine scheme, corresponds to an embedded point with coefficients in the residue field of the scheme theoretic point. I.e. if the affine ring is k[X], then the point represented by the prime ideal I, defines the point ([X1],...,[Xn]) in E^n, where E is the residue field of the point I in spec(k[X]).