Math 631, Lecture 4

Notes by Bryden Cais

Sept. 13, 2002

In this lecture we will define and study maps between varieties.

Definition 1. A regular map or morphism between two affine k varieties X, Y is a collection of maps {f_{K}} for all k algebras K such that for any homomorphism of k algebras φ : K ⟶ K′ the following diagram is commutative:

X(K) ⟶^{f_{K}} Y (K)
φ^{⨁ n}↓  ↓_{φ^{⨁ n}}
X(K′) ⟶^{f_{K′}} Y (K′)

where φ^{⨁ n} : K^{n} ⟶ K^{′n} is the natural map induced by φ and X(K) ⊂ K^{n}, Y (K) ⊂ K^{m}. We will denote the set of morphisms from X to Y by Mor_{Aff/k}(X, Y).

Definition 2. The coordinate ring of a k variety X is the ring

O(X) := k[T]/I(X).

Recall that we have a bijection X(K) ↔ Hom_{k} (O(X), K). Let us now look at maps between varieties in these terms. Given a homomorphism φ : O(Y) → O(X), we have a natural map of sets

Hom_{k} (O(X), K) ⟶ Hom_{k} (O(Y), K)

given by

ψ ↦ ψ ∘ φ.

Therefore, any homomorphism (of k algebras) φO(Y) → O(Y) defines a morphism f : X → Y by setting, for each K and any α ∈ Hom_{k} (O(X), K) = X(K)

f_{K} = α ∘ φ.

Observe that f_{K} : O(Y) → K is in Y (K) (as it ought to be). We thus have a natural map

ξ : Hom_{k} (O(Y), O(X)) ⟶ Mor_{Aff/k}(X, Y)

given by

ξ(ψ)(α) := α ∘ ψ ∈ Hom_{k} (O(Y), K)

for any α ∈ Hom_{k} (O(X), K) (i.e. ξ(ψ) : X(K) → Y (K)).

Theorem 1. The map ξ defined in this way is in fact a bijection.

Proof. Let f : X → Y be a morphism. Then

f_{O(X)} : Hom_{k} (O(X), O(X)) ⟶ Hom_{k} (O(Y, O(X)) .

The image of the identity homomorphism id_{O(X)} ∈ Hom_{k} (O(X), O(X)) under f_{O(X)} is some homomorphism ψ ∈ Hom_{k} (O(Y), O(X)) . If we can show that ξ(ψ) = f then we will have shown that ξ is surjective. To do this, we must show that for any k algebra K and any α ∈ Hom_{k} (O(X), K) we have ξ(ψ)(α) = f_{K}(α). This follows from the definition of a morphism: the diagram

X(O(X)) = Hom_{k} (O(X), O(X)) ⟶^{f_{O(X)}} Hom_{k} (O(Y), O(X)) = Y (O(X))
α∘↓  ↓^{α∘}
X(K) = Hom_{k} (O(X), K)  ⟶^{f_{K}}  Hom_{k} (O(Y), K) = Y (K)

is commutative. We have id_{O(X)} ∈ Hom_{k} (O(X), O(X)) mapping to α ∘ id_{O(X)} = α : O(X) → K. The image of α in Hom_{k} (O(Y), K) = Y (K) under f_{K} is just f_{K}(α). By commutativity of the diagram, this must coincide with the image of ψ ∈ Hom_{k} (O(Y), O(X)) under α∘ in Hom_{k} (O(Y), K), which is just α ∘ ψ = ξ(ψ)(α). We therefore have

ξ(ψ)(α) = f_{K}(α)

for any α ∈ Hom_{k} (O(X), K), and therefore that ξ(ψ) = f and we are done with surjectivity. To prove that ξ is injective, let ψ, ϕ ∈ Hom_{k} (O(Y), O(X)) be distinct and suppose that ξ(ψ) = ξ(ϕ). Then for all k algebras K and any α ∈ Hom_{k} (O(X), K) we have

α ∘ ψ = α ∘ ϕ.

However, if we choose K = O(X) and α = id_{O(X)} ∈ Hom_{k} (O(X), O(X)) then we must have ψ = ϕ, which shows that ξ is injective. Therefore, ξ is a bijection, as claimed.

This bijection allows us to think of morphisms between varieties X, Y as homomorphisms of the corresponding coordinate rings, going in the reverse dirction: O(Y) → O(X) corresponds to a morphism X → Y.

Definition 3. An isomorphism of k varieties X, Y is an invertible morphism.

Using this definition and our interpretation of morphisms X → Y as homomorphisms O(Y) → O(X), we see that

Proposition 1. Two affine algebraic k varieties X, Y are isomorphic if and only if their coordinate k algebras O(X) and O(Y) are isomorphic.

So we have this fine identification of Mor_{Aff/k}(X, Y) with Hom_{k} (O(Y), O(X)) , so what? The point is that this enables us to prove something really profound:

Theorem 2. Every morphism f : X → Y of affine k varieties is given by a collection of polynomials.

Proof. Any morphism f : X → Y corresponds to a homomorphism ψ : O(Y) → O(X) with f_{K}(α) = α ∘ ψ : O(Y) → K for any α ∈ Hom_{k} (O(X), K). Now O(Y), O(X) are quotients of the polynomial rings k[U_{1}, …, U_{r}], k[T_{1}, …, T_{s}] respectively, so ψ is determined by where it sends the generators U_{i}. Moreover, these generators must be cosets of I(X) in k[T_{1}, …, T_{s}], that is

ψ(U_{i}) = P_{i}(T_{1}, …, T_{s}) + I(X)

for P_{i} ∈ k[T_{1}, …, T_{s}]. Now since ψ as a homomorphism k[U_{1}, …, U_{r}] → k[T_{1}, …, T_{s}]/I(X) factors through I(Y), we have, for any F ∈ I(Y),

F (P_{1}(T_{1}, …, T_{s}), …, P_{r}(T_{1}, …, T_{s})) ∈ I(X).

Our morphism f : X → Y is given by f_{K}(α) = α ∘ ψ. Recall, however, that α ∈ Hom_{k} (O(X), K) is given by sending T_{i} ↦ a_{i} ∈ K for 1 ≤ i ≤ s. Therefore, we have

α ∘ ψ(U_{i}) = α ∘ (P_{i}(T_{1}, …, T_{s}) + I(X)) = P_{i}(a_{1}, …, a_{s})

since I(X) is contained in the kernel of the map α : k[T_{1}, …, T_{s}] → K. Therefore, we have

f_{K}(a_{1}, …, a_{s}) = (P_{1}(a_{1}, …, a_{s}), …, P_{r}(a_{1}, …, a_{s}))

for any (a_{1}, …, a_{s}) ∈ X(K). In short, f is given by a collection of polynomials.

Recall that we can interpret points a ∈ Sol(S, K) for any system of algebraic equations S defining a variety X as homomorphisms a : O(X) → K by defining a(p(T)) := P (a) for any P ∈ k[T]. This kind of duality is ubiquitous in algebraic geometry. In this case, we move from considering elements of k[T] as functions on K^{n} to considering elements of K^{n} as functions on k[T]. Another example is the following: any morphism

X ⟶ 𝔸_{k}^{1}

may be thought of as a homomorphism

k[T_{1}] ⟶ O(X).

Any such homomorphism is determined by its value at T_{1}. We can therefore think of such homomorphisms as elements of O(X). In summary, Proposition 2. There is a bijection between elements of O(X) and morphisms f : X ⟶ 𝔸_{k}^{1}.

This enables us to make sense out of the following: let f ∈ Mor_{Aff/k}(X, Y). Then we have a homomorphism

f^{∗} : O(Y) ⟶ O(X).

Now let φ ∈ Mor_{Aff/k}(Y, 𝔸_{k}^{1}) = O(Y). Then the composition of morphisms φ ∘ f is well defined and is a map X ⟶ 𝔸_{k}^{1}, i.e. an element of O(X). By our correspondence, we therefore have

f^{∗}(φ) = φ ∘ f.