Definition 3.23 Choose coordinates x1, . . . , xn and y1, . . . , ym on An(k) and
Am(k). Let φ : An(k) → Am(k) be a morphism given by the rule
φ(x1, . . . , xn) = (φ1(x1, . . . , xn), . . . , φm(x1, . . . , xn)), φj ∈ k[x1, . . . , xn].
Am(k). Let φ : An(k) → Am(k) be a morphism given by the rule
φ(x1, . . . , xn) = (φ1(x1, . . . , xn), . . . , φm(x1, . . . , xn)), φj ∈ k[x1, . . . , xn].
Conversely, any k-algebra homomorphism
ψ : k[y1, . . . , ym] → k[x1, . . . , xn]
is determined by its values on the generators. Writing ψj (x1, . . . , xn) = ψ(yj ), we
obtain a morphism
An(k) → Am(k)
(x1, . . . , xn) → (ψ1(x1, . . . , xn), . . . , ψm(x1, . . . , xn)).
To summarize:
Proposition 3.24 There is a natural correspondence between morphisms φ :
An(k) → Am(k) and k-algebra homomorphisms
ψ : k[y1, . . . , ym] → k[x1, . . . , xn]
identifying φ∗ and ψ.
ψ : k[y1, . . . , ym] → k[x1, . . . , xn]
is determined by its values on the generators. Writing ψj (x1, . . . , xn) = ψ(yj ), we
obtain a morphism
An(k) → Am(k)
(x1, . . . , xn) → (ψ1(x1, . . . , xn), . . . , ψm(x1, . . . , xn)).
To summarize:
Proposition 3.24 There is a natural correspondence between morphisms φ :
An(k) → Am(k) and k-algebra homomorphisms
ψ : k[y1, . . . , ym] → k[x1, . . . , xn]
identifying φ∗ and ψ.
Let V ⊂ An(k) be affine
with ideal I (V). We restrict polynomial functions on An(k) to V; elements of I (V)
are zero along V, so these functions can be identified with the quotient k[x1, . . . ,
xn]/I (V).
with ideal I (V). We restrict polynomial functions on An(k) to V; elements of I (V)
are zero along V, so these functions can be identified with the quotient k[x1, . . . ,
xn]/I (V).
Example 3.25 Consider the circle V = {(x, y) : x2 + y2 = 1} ⊂ A2(R) with
I (V) = x2 + y2 − 1. The polynomials x2 and 1 − y2 define the same function on
the circle.
I (V) = x2 + y2 − 1. The polynomials x2 and 1 − y2 define the same function on
the circle.
x2 ≡ 1 − y2 mod I (V)
Definition 3.26 Let V ⊂ An(k) be an affine variety. The coordinate ring is defined
as the quotient ring
k[V] = k[x1, . . . , xn]/I (V).
as the quotient ring
k[V] = k[x1, . . . , xn]/I (V).
Definition 3.27 Fix an affine variety V ⊂ An(k). Two morphisms ¯ φ, ˆφ : An(k) →
Am(k) are equivalent on V if the induced pull-back homomorphisms
¯φ
∗ : k[Am] → k[V], ˆφ ∗ : k[Am] → k[V]
are equal. The resulting equivalence classes are called morphisms φ : V → Am(k).
Each ˆφ : An(k) → Am(k) in the equivalence class is called an extension of φ to affine
space.
Am(k) are equivalent on V if the induced pull-back homomorphisms
¯φ
∗ : k[Am] → k[V], ˆφ ∗ : k[Am] → k[V]
are equal. The resulting equivalence classes are called morphisms φ : V → Am(k).
Each ˆφ : An(k) → Am(k) in the equivalence class is called an extension of φ to affine
space.
Proposition 3.29 Let V ⊂ An(k) be an affine variety and
¯ φ, ˆφ : An(k) → Am(k)
two morphisms equivalent on V. Then we have ¯φ(v) = ˆφ (v) for each v ∈ V .
Proof If ¯φ(v) = ˆφ (v) then they can be differentiated by coordinate functions
from k[Am], i.e., we have
yi (¯φ (v)) = yi (ˆφ(v))
for some i . It follows that ¯φ ∗ yi (v) = ˆφ∗ yi (v), which violates the equivalence
assumption.
¯ φ, ˆφ : An(k) → Am(k)
two morphisms equivalent on V. Then we have ¯φ(v) = ˆφ (v) for each v ∈ V .
Proof If ¯φ(v) = ˆφ (v) then they can be differentiated by coordinate functions
from k[Am], i.e., we have
yi (¯φ (v)) = yi (ˆφ(v))
for some i . It follows that ¯φ ∗ yi (v) = ˆφ∗ yi (v), which violates the equivalence
assumption.
Definition 3.30 Fix affine varieties V ⊂ An(k) and W ⊂ Am(k). A morphism
φ : V → W is defined to be morphism φ : V → Am(k) with φ(V) ⊂ W.
φ : V → W is defined to be morphism φ : V → Am(k) with φ(V) ⊂ W.
Proposition 3.31 Let V ⊂ An(k) and W ⊂ Am(k) be affine varieties. Any morphism
φ : V → W induces a k-algebra homomorphism φ∗ : k[W] → k[V]. Conversely,
each k-algebra homomorphism ψ : k[W] → k[V] can be expressed as φ∗
for some morphism φ.
φ : V → W induces a k-algebra homomorphism φ∗ : k[W] → k[V]. Conversely,
each k-algebra homomorphism ψ : k[W] → k[V] can be expressed as φ∗
for some morphism φ.
Corollary 3.32 Let V and W be affine varieties. There is a one-to-one correspondence
between morphisms V → W and k-algebra homomorphisms k[W] → k[V].
Definition 3.33 An isomorphism of affine varieties is a morphism φ : V → W
admitting an inverse morphism φ−1 : W → V. An automorphism of an affine variety
is an isomorphism φ : V → V.
One important consequence of Corollary 3.32 is that automorphisms of V correspond
to k-algebra isomorphisms k[V] → k[V].
between morphisms V → W and k-algebra homomorphisms k[W] → k[V].
Definition 3.33 An isomorphism of affine varieties is a morphism φ : V → W
admitting an inverse morphism φ−1 : W → V. An automorphism of an affine variety
is an isomorphism φ : V → V.
One important consequence of Corollary 3.32 is that automorphisms of V correspond
to k-algebra isomorphisms k[V] → k[V].
Example 3.34 Consider V = A2(k) and the homomorphism
ψ(x1) = x1, ψ(x2) = x2 + g(x1), g ∈ k[x1],
with inverse
ψ−1(x1) = x1, ψ−1(x2) = x2 − g(x1).
Each ψ = φ∗ for some automorphism φ : A2(k) → A2(k). Thus each polynomial
g ∈ k[x1] yields an automorphism of the affine plane.
ψ(x1) = x1, ψ(x2) = x2 + g(x1), g ∈ k[x1],
with inverse
ψ−1(x1) = x1, ψ−1(x2) = x2 − g(x1).
Each ψ = φ∗ for some automorphism φ : A2(k) → A2(k). Thus each polynomial
g ∈ k[x1] yields an automorphism of the affine plane.
Show that every automorphism of the affine line A1(Q) takes the form
x →ax + b, a, b ∈ Q, a = 0.
x →ax + b, a, b ∈ Q, a = 0.
