In this section, we want to sort out the various meanings of the word "point," as used by algebraic geometers. This word is used in at least three distinct ways. As a result, beginners in the subject often find themselves confused.
Let k be a field, and let S be an algebraic variety over k. (An algebraic variety is a scheme of finite type over a field. This usage differs from that of other authors, who reserve the word variety for an irreducible scheme of finite type. We want to work over arbitrary fields, and the property of being irreducible is not stable under the operation of extending the base field.) The most primitive notion of a point is also the rarest usage among algebraic geometers. We call any element of the underlying topological space of S a point. When using the term this way, we often make special mention of the closed points. This term emphasizes the fact that the underlying topological space is not Hausdorff, and many of its points are, in fact, not closed in S.
Let's assume for the moment that the variety in question can be embedded in a projective space Pnk. Then we can give an elementary definition of a k-rational point of S. A closed point s∈ S⊂Pn is called a k-rational point if it has the form s = (s0:…:sn) with all si∈ k. We write S(k) for the set of all k-rational points of the scheme S.
If the field k is not algebraically closed, then the k-rational points usually form a very small subset of the collection of all closed points of S. We can reinterpret the definition of a k-rational point in a coordinate-free manner that will allow us to generalize the above definition in several ways. Notice that to give a k-rational point of S (in the sense described above) is equivalent to specifying a k-morphism Spec(k)→ S. The first generalization, then, is to drop the assumption that S can be embedded in projective space. Next, suppose that s∈ S is a closed point. Then the residue field k(s) = OS,s/ms is an extension field of k of finite degree. The inclusion of this closed point in S corresponds (loosely) to a k-morphism Spec(k(s)) → S. The looseness of the correspondence comes from the fact that we can compose this morphism with any k-automorphism of k(s) without detecting any difference in the closed point s∈ S. To elaborate, let L/k be any field extension of finite degree. Let us define an L-rational point of the k-scheme S to be a k-morphism Spec(L)→ S, and denote the set of all L-rational points of S by S(L). In general, the set S(L) is strictly larger than the set of closed points of S whose coefficients lie in L.
We are now ready to define yet another kind of point on a k-scheme S. There is no reason to restrict the previous definition to finite extension fields of k. If we let k- denote the algebraic closure of k, then we can consider the set
S(k-) = {α : Spec(k-) → S} = Homk(Spec(k-), S).
An element of this set is often called a geometric point of S. Quite often, an algebraic geometer will use the word "point" to refer to any geometric point of a variety.S(T) = Homk(T, S).
This new notion does not look like much of a leap beyond our earlier definitions, but it has some fascinating consequences. We have gotten here by considering more and more general notions of the idea of a point. We have ended up considering points with values in an arbitrary k-scheme. Moreover, suppose that we have a morphism f:T→ U of k-schemes. Composition of morphisms defines a function° f : S(U) = Homk(U, S) → Homk(T, S) = S(T).
This gives us a completely new way to look at a scheme: A scheme S gives rise to a functor, which assigns to any k-scheme T a certain set. We call this new thing the functor of points associated to the scheme S.S(T) = Mork(T,S) = Homk(k, Γ(T, OT))
contains exactly one element for each k-scheme T, the scheme S corresponds to the constant functor T |→ {*}.S(T) = Mork(T,S) = Homk(k[x], Γ(T, OT)) = Γ(T, OT).
Thus, the affine line over k is equivalent to the functor T |→ Γ(T, OT). In this case, the functor takes values not just in the category of sets, but in the category of k-algebras; this observation suggests that there is more structure to the affine line than we have so far used.S(T) = Mork(T,S) = Homk(k[x, x-1], Γ(T, OT)) = Γ(T, OT)*.
Thus, S is equivalent to the functor that maps a scheme T to the multiplicative group Γ(T, OT)*. For this reason, this scheme is often denoted Gm.Let us denote the functor of points of S by the notation hS, so that hS(T) = S(T) = Homk(T, S). In this way, we can distinguish the scheme S from the functor hS. Now we realize that there is another functor staring us in the face. The assignment S |→ hS defines a functor
h : SCHk → FUN(SCHkop, SET).
For the functor h to be faithful, we must show that if two morphisms of schemes f, g : S → T induce the same natural transformation hS → hT, then they must have started out as the same morphism. But this is also easy, since the natural transformation is defined by composing with the morphism, and we can test it on an identity morphism.
h0 : SCHk → FUN(ALGk, SET),
defined by S |→ h0S, where h0S(R) = hS(Spec(R)).Comments on this web site should be addressed to the author:
Kevin R. Coombes