Definition 3.35 A rational map ρ : An(k) Am(k) is given by a rule
ρ(x1, . . . , xn) = (ρ1(x1, . . . , xn), . . . , ρm(x1, . . . , xn)), ρj ∈ k(x1, . . . , xn).
ρ(x1, . . . , xn) = (ρ1(x1, . . . , xn), . . . , ρm(x1, . . . , xn)), ρj ∈ k(x1, . . . , xn).
the closed set
V({g1, . . . , gm}) ⊂ An(k) is called the indeterminacy locus of ρ
V({g1, . . . , gm}) ⊂ An(k) is called the indeterminacy locus of ρ
Proposition 3.36 Each rational map ρ : An(k) Am(k) defined over k induces
a k-algebra homomorphism
ρ∗ : k[y1, . . . , ym] → k(x1, . . . , xn),
yj → ρj (x1, . . . , xn).
Conversely, each k-algebra homomorphism
k[y1, . . . , ym] → k(x1, . . . , xn)
arises from a rational map.
Definition3.37 LetW ⊂ Am(k) be an affine variety.Arationalmapρ : An(k)
W is a rational map ρ : An(k) Am(k) with ρ∗ I (W) = 0.
a k-algebra homomorphism
ρ∗ : k[y1, . . . , ym] → k(x1, . . . , xn),
yj → ρj (x1, . . . , xn).
Conversely, each k-algebra homomorphism
k[y1, . . . , ym] → k(x1, . . . , xn)
arises from a rational map.
Definition3.37 LetW ⊂ Am(k) be an affine variety.Arationalmapρ : An(k)
W is a rational map ρ : An(k) Am(k) with ρ∗ I (W) = 0.
Definition 3.40 Let ρ : An(k) Am(k) be a rational map with components ρj =
f j /gj with f j , gj ∈ k[x1, . . . , xn] having no common irreducible factors. Let V ⊂
An(k) be an affine variety with ideal I (V) and coordinate ring k[V]. Assume that
the image of each gj in k[V] does not divide zero. Then we say that ρ is admissible
on V.
f j /gj with f j , gj ∈ k[x1, . . . , xn] having no common irreducible factors. Let V ⊂
An(k) be an affine variety with ideal I (V) and coordinate ring k[V]. Assume that
the image of each gj in k[V] does not divide zero. Then we say that ρ is admissible
on V.
Definition 3.41 The ring of fractions of a ring R is defined
K = {r/s : r, s ∈ R, s not a zero divisor},
where r1/s1 = r2/s2 whenever r1s2 = r2s1.
K = {r/s : r, s ∈ R, s not a zero divisor},
where r1/s1 = r2/s2 whenever r1s2 = r2s1.
Definition 3.42 For an affine variety V, let k(V) denote the ring of fractions of
the coordinate ring k[V].
Proposition 3.43 Let ρ : An(k) Am(k) be a rational map admissible on an
affine variety V ⊂ An(k). Then ρ induces a k-algebra homomorphism
ρ∗ : k[Am] → k(V).
Conversely, each such homomorphism arises from a suitable rational map.
the coordinate ring k[V].
Proposition 3.43 Let ρ : An(k) Am(k) be a rational map admissible on an
affine variety V ⊂ An(k). Then ρ induces a k-algebra homomorphism
ρ∗ : k[Am] → k(V).
Conversely, each such homomorphism arises from a suitable rational map.
Definition 3.44 Let V ⊂ An(k) be an affine variety and
ρ¯, ρˆ : An(k) → Am(k)
rational maps admissible on V. These are equivalent along V if the induced homomorphisms
ρ¯∗, ρˆ∗ : k[Am] → k(V)
are equal.
Definition 3.45 Let V andW be affine varieties realized as closed subsets ofAn(k)
and Am(k) respectively. A rational map ρ : V W is defined as an equivalence
class of rational maps ρ : An(k) W admissible on V. Each such ρ is called an
extension of ρ to affine space.
ρ¯, ρˆ : An(k) → Am(k)
rational maps admissible on V. These are equivalent along V if the induced homomorphisms
ρ¯∗, ρˆ∗ : k[Am] → k(V)
are equal.
Definition 3.45 Let V andW be affine varieties realized as closed subsets ofAn(k)
and Am(k) respectively. A rational map ρ : V W is defined as an equivalence
class of rational maps ρ : An(k) W admissible on V. Each such ρ is called an
extension of ρ to affine space.
Corollary 3.46 Let V and W be affine varieties. There is a one-to-one correspondence
between rational maps V W over k and k-algebra homomorphisms
k[W] → k(V).
between rational maps V W over k and k-algebra homomorphisms
k[W] → k(V).
