Conjuncted: Affine Schemes: How Would Functors Carry the Same Information?


If we go to the generality of schemes, the extra structure overshadows the topological points and leaves out crucial details so that we have little information, without the full knowledge of the sheaf. For example the evaluation of odd functions on topological points is always zero. This implies that the structure sheaf of a supermanifold cannot be reconstructed from its underlying topological space.

The functor of points is a categorical device to bring back our attention to the points of a scheme; however the notion of point needs to be suitably generalized to go beyond the points of the topological space underlying the scheme.

Grothendieck’s idea behind the definition of the functor of points associated to a scheme is the following. If X is a scheme, for each commutative ring A, we can define the set of the A-points of X in analogy to the way the classical geometers used to define the rational or integral points on a variety. The crucial difference is that we do not focus on just one commutative ring A, but we consider the A-points for all commutative rings A. In fact, the scheme we start from is completely recaptured only by the collection of the A-points for every commutative ring A, together with the admissible morphisms.

Let (rings) denote the category of commutative rings and (schemes) the category of schemes.

Let (|X|, OX) be a scheme and let T ∈ (schemes). We call the T-points of X, the set of all scheme morphisms {T → X}, that we denote by Hom(T, X). We then define the functor of points hX of the scheme X as the representable functor defined on the objects as

hX: (schemes)op → (sets), haX(A) = Hom(Spec A, X) = A-points of X

Notice that when X is affine, X ≅ Spec O(X) and we have

haX(A) = Hom(Spec A, O(X)) = Hom(O(X), A)

In this case the functor haX is again representable.

Consider the affine schemes X = Spec O(X) and Y = Spec O(Y). There is a one-to-one correspondence between the scheme morphisms X → Y and the ring morphisms O(X) → O(Y). Both hX and haare defined on morphisms in the natural way. If φ: T → S is a morphism and ƒ ∈ Hom(S, X), we define hX(φ)(ƒ) = ƒ ○ φ. Similarly, if ψ: A → Bis a ring morphism and g ∈ Hom(O(X), A), we define haX(ψ)(g) = ψ ○ g. The functors hX and haare for a given scheme X not really different but carry the same information. The functor of points hof a scheme X is completely determined by its restriction to the category of affine schemes, or equivalently by the functor

haX: (rings) → (sets), haX(A) = Hom(Spec A, X)

Let M = (|M|, OM) be a locally ringed space and let (rspaces) denote the category of locally ringed spaces. We define the functor of points of locally ringed spaces M as the representable functor

hM: (rspaces)op → (sets), hM(T) = Hom(T, M)

hM is defined on the manifold as

hM(φ)(g) = g ○ φ

If the locally ringed space M is a differentiable manifold, then

Hom(M, N) ≅ Hom(C(N), C(M))

This takes us to the theory of Yoneda’s Lemma.

Let C be a category, and let X, Y be objects in C and let hX: Cop → (sets) be the representable functors defined on the objects as hX(T) = Hom(T, X), and on the arrows as hX(φ)(ƒ) = ƒ . φ, for φ: T → S, ƒ ∈ Hom(T, X)

If F: Cop → (sets), then we have a one-to-one correspondence between sets:

{hX → F} ⇔ F(X)

The functor

h: C → Fun(Cop, (sets)), X ↦ hX,

is an equivalence of C with a full subcategory of functors. In particular, hX ≅ hY iff X ≅ Y and the natural transformations hX → hY are in one-to-one correspondence with the morphisms X → Y.

Two schemes (manifolds) are isomorphic iff their functors of points are isomorphic.

The advantages of using the functorial language are many. Morphisms of schemes are just maps between the sets of their A-points, respecting functorial properties. This often simplifies matters, allowing allowing for leaving the sheaves machinery in the background. The problem with such an approach, however, is that not all the functors from (schemes) to (sets) are the functors of points of a scheme, i.e., they are representable.

A functor F: (rings) → (sets) is of the form F(A) = Hom(Spec A, X) for a scheme X iff:

F is local or is a sheaf in Zariski Topology. This means that for each ring R and for every collection αi ∈ F(Rƒi), with (ƒi, i ∈ I) = R, so that αi and αj map to the same element in F(Rƒiƒj) ∀ i and j ∃ a unique element α ∈ F(R) mapping to each αi, and

F admits a cover by open affine subfunctors, which means that ∃ a family Ui of subfunctors of F, i.e. Ui(R) ⊂ F(R) ∀ R ∈ (rings), Ui = hSpec Ui, with the property that ∀ natural transformations ƒ: hSpec A  → F, the functors ƒ-1(Ui), defined as ƒ-1(Ui)(R) = ƒ-1(Ui(R)), are all representable, i.e. ƒ-1(Ui) = hVi, and the Vi form an open covering for Spec A.

This states the conditions we expect for F to be the functor of points of a scheme. Namely, locally, F must look like the functor of points of a scheme, moreover F must be a sheaf, i.e. F must have a gluing property that allows us to patch together the open affine cover.


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