6. Finite Maps

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Throughout this section, k is an algebraically closed field.

Recall that an A-algebra B is said to be finite if it is finitely generated as an Amodule.

This is equivalent to B being finitely generated as an A-algebra and integral

over A.

Definition 6.1. A regularmap ϕ : W → V is said to be finite if for all open affine

subsets U of V , ϕ−1(U) is an affine variety, and k[ϕ−1(U)] is a finite k[U]-algebra.

Proposition 6.2. It suffices to check the condition in the definition for all subsets

in one open affine covering (Ui) of V .

Proof. Omitted. (See Mumford 1966, III.1, proposition 5).

Hence a map ϕ :S pecm(B) → Specm(A) of affine varieties is finite if and only if

B is a finite A-algebra.

Proposition 6.3. (a) For any closed subvariety Z of V , the inclusion Z 8→ V

is finite.

(b) The composite of two finite morphisms is finite.

(c) The product of two finite morphisms is finite.

Proof. (a) Let U be an open affine subvariety of V. Then Z ∩ U is a closed

subvariety of U. It is therefore affine, and the map Z ∩ U → U corresponds to a map

A → A/a of rings, which is obviously finite.

(b) If B is a finite A-algebra and C is a finite B-algebra, then C is a finite Aalgebra:

i ndeed, if {bi} is a set of generators for B as an A-module, and {cj} is a

set of generators for C as a B-module, then {bicj} is a set of generators for C as an

A-module.

(c) If B and B_ are respectively finite A and A_-algebras, then B ⊗k B_ is a finite

A⊗k A_-algebra:in deed, if {bi} is a set of generators for B as an A-module, and {b_

j

}

is a set of generators for B_ as an A-module, the {bi ⊗ b_

j

} is a set of generators for

B ⊗A B_ as an A-module.

By way of contrast, an open immersion is rarely finite. For example, the inclusion

A1 − {0} 8→ A1 is not finite because the ring k[T, T−1] is not finitely generated as

a k[T ]-module. (Any finite set of elements in k[T, T−1] has a fixed power of T as a

common denominator.)

The fibres of a regular map ϕ : W → V are the subvarieties ϕ−1(P) of W for

P ∈ V . When the fibres are all finite, ϕ is said to be quasi-finite.

Proposition 6.4. A finite map ϕ : W → V is quasi-finite.

Proof. Let P ∈ V ; we wish to show ϕ−1(P) is finite. After replacing V with an

affine neighbourhood of P, we can suppose that it is affine, and then W will be affine

also. The map ϕ then corresponds to a map α : A → B of affine k-algebras, and a

point Q of W maps to P if and only α−1(mQ) = mP . But this holds if and only if20

mQ ⊃ α(mP ), and so the points of W mapping to P are in one-to-one correspondence

20Clearly then α−1(mQ) ⊃ mP , and we know it is a maximal ideal.

102 Algebraic Geometry: 6. Finite Maps

with the maximal ideals of B/α(m)B. Clearly B/α(m)B is generated as a k-vector

space by the image of any generating set for B as an A-module, and the next lemma

shows that it has only finitely many maximal ideals.

Lemma 6.5. A finite k-algebra A has only finitely many maximal ideals.

Proof. Let m1, . . . ,mn be maximal ideals in A. They are obviously coprime in

pairs, and so the Chinese Remainder Theorem (see below) shows that the map

A → A/m1 ×· · ·×A/mn, a_→ (. . . , ai mod mi, . . . ),

is surjective. It follows that dimk A ≥ _

dimk(A/mi) ≥ n (dimensions as k-vector

spaces).

Lemma 6.6 (Chinese Remainder Theorem). Let a1, . . . , an be ideals in a ring A.

If ai is coprime to aj (i.e., ai + aj = A) whenever i _= j, then the map

A → A/a1 ×· · ·×A/am

is surjective, with kernel

_

ai = ∩ai.

Proof. The proof is elementary (see Atiyah and MacDonald 1969, 1.10).

Theorem 6.7. A finite map ϕ : W → V is closed.

Proof. Again we can assume V and W to be affine. Let Z be a closed subset of

W. The restriction of ϕ to Z is finite (by 6.3a and b), and so we can replace W with

Z; we then we have to show that Im(ϕ) is closed. The map corresponds to a finite

map of rings A → B. This will factor, A → A/a 8→ B, from which we obtain maps

Specm(B) → Specm(A/a) 8→ Specm(A).

The second map identifies Specm(A/a) with the closed subvariety V (a) of Specm(A),

and so it remains to show that the first map is surjective. This is a consequence of

the next lemma.

Lemma 6.8 (Going-Up Theorem). Let A ⊂ B be rings with B integral over A.

(a) For every prime ideal p of A, there is a prime ideal q of B such that q ∩ A = p.

(b) Let p = q ∩ A; then p is maximal if and only if q is maximal.

Proof. (a) If S is a multiplicative subset of a ring A, then the prime ideals of

S−1A are in one-to-one correspondence with the prime ideals of A not meeting S (see

4.15). It therefore suffices to prove (a) after A and B have been replaced by S−1A

and S−1B, where S = A − p. Thus we may assume that A is local, and that p is its

unique maximal ideal. In this case, for all proper ideals b of B, b ∩ A ⊂ p (otherwise

b ⊃ A ) 1). To complete the proof of (a), I shall show that for all maximal ideals n

of B, n ∩ A = p.

Consider B/n ⊃ A/(n ∩ A). Here B/n is a field, which is integral over its subring

A/(n ∩ A), and n ∩ A will be equal to p if and only if A/(n ∩ A) is a field. Thus the

claim follows from the next lemma.

Lemma 6.9. Let A be a subring of a field K. If K is integral over A, then A also

is a field.

Algebraic Geometry: 6. Finite Maps 103

Proof. Let a ∈ A, a _= 0. Then a−1 ∈ K, and it is integral over A:

(a

−1)n + a1(a

−1)n−1 + · · · + an = 0, ai ∈ A.

On multiplying through by an−1, we find that

a−1 + a1 + · · · + anan−1 = 0,

from which it follows that a−1 ∈ A.

Proof. (of 6.8b)The ring B/q contains A/p, and it is integral over A/p. If q is

maximal, then (6.9) shows that p is also. For the converse, note that any integral

domain algebraic over a field is a field — it is a union of integral domains finite over

k, and multiplication by any nonzero element of an integral domain finite over a field

is an isomorphism (it is injective by definition, and an injective endomorphism of a

finite-dimensional vector space is also surjective).

Corollary 6.10. Let ϕ : W → V be finite; if V is complete, then so also is W.

Proof. Consider

W × T → V × T → T, (w, t) _→ (ϕ(w), t) _→ t.

Because W × T → V × T is finite (see 6.3c), it is closed, and because V is complete,

V × T → T is closed. A composite of closed maps is closed, and therefore the

projection W × T → T is closed.

Example 6.11. labelFM11 (a) Project XY = 1 onto the X axis. This map is

quasi-finite but not finite, because k[X,X−1] is not finite over k[X].

(b) The map A2 −{origin} 8→ A2 is quasi-finite but not finite, because the inverse

image of A2 is not affine (2.20).

(c) Let V = V (Xn + T1Xn−1 + · · · + Tn) ⊂ An+1, and consider the projection map

(a1, . . . , an, x) _→ (a1, . . . , an) : V → An.

The fibre over any point (a1, . . . , an) ∈ An is the set of solutions of

Xn + a1Xn−1 + · · · + an = 0,

and so it has exactly n points, counted with multiplicities. The map is certainly

quasi-finite; it is also finite because it corresponds to the finite map of k-algebras,

k[T1, . . . , Tn] → k[T1, . . . , Tn,X]/(Xn + T1Xn−1 + · · · + Tn).

(d) Let V = V (T0Xn+T1Xn−1+· · ·+Tn) ⊂ An+2. The projection ϕ : V → An+1 has

finite fibres except for the fibre above (0, . . . , 0), which is A1. The restriction ϕ|V    

ϕ−1(origin) is quasi-finite, but not finite. Above points of the form (0, . . . , 0, ∗, . . . , ∗)

some of the roots “vanish off to ∞”. (Example (a) is a special case of this.)

(e) Let P(X, Y) = T0Xn + T1Xn−1Y + ... +TnY n, and let V be its zero set in

P1 × (An+1 − {origin}). In this case, the projection map V → An+1 − {origin} is

finite. (Prove this directly, or apply 6.24 below.)

(f) The morphism A1 → A2, t _→ (t2, t3) is finite because the image of k[X, Y ] in

k[T ] is k[T 2, T 3], and {1, T } is a set of generators for k[T ] over this subring.

(g) The morphism A1 → A1, a _→ am is finite (special case of (c)).

104 Algebraic Geometry: 6. Finite Maps

(h) The obvious map

(A1 with the origin doubled ) → A1

is quasi-finite but not finite (the inverse image of A1 is not affine).

Exercise 6.12. Prove that a finite map is an isomorphism if and only if it is

bijective and ´etale. (Cf. Harris 1992, 14.9.)

The Frobenius map t _→ tp : A1 → A1 in characteristic p _= 0 and the map t _→

(t2, t3) : A1 → V (Y 2 − X3) ⊂ A2 from the line to the cuspidal cubic (see 2.17c) are

examples of finite bijective regular maps that are not isomorphisms.

Noether Normalization Theorem. This theorem sometimes allows us to reduce

the proofs of statements about affine varieties to the case of An.

Theorem 6.13. For any irreducible affine algebraic variety V of a variety of dimension

d, there is a finite surjective map ϕ : V → Ad.

Proof. This is a geometric re-statement of the original theorem.

Theorem 6.14 (Noether Normalization Theorem). Let A be a finitely generated

k-algebra, and assume that A is an integral domain. Then there exist elements

y1, . . . , yd ∈ A that are algebraically independent over k and such that A is integral

over k[y1, . . . , yd].

Proof. Let x1, . . . , xn generate A as a k-algebra. We can renumber the xi so that

x1, . . . , xd are algebraically independent and xd+1, . . . , xn are algebraically dependent

on x1, . . . , xd (see 6.12 of my notes on Fields and Galois Theory).

Because xn is algebraically dependent on x1, . . . , xd, there exists a nonzero polynomial

f(X1, . . . ,Xd, T ) such that f(x1, . . . , xd, xn) = 0. Write

f(X1, . . . ,Xd, T) = a0Tm + a1Tm−1 + · · · + am

with ai ∈ k[X1, . . . ,Xd] (≈ k[x1, . . . , xd]). If a0 is a nonzero constant, we can divide

through by it, and then xn will satisfy a monic polynomial with coefficients in

k[x1, . . . , xd], that is, xn will be integral (not merely algebraic) over k[x1, . . . , xd].

The next lemma suggest how we might achieve this happy state by making a linear

change of variables.

Lemma 6.15. If F(X1, . . . ,Xd, T ) is a homogeneous polynomial of degree r, then

F(X1 + λ1T, . . . ,Xd + λdT, T) = F(λ1, . . . , λd, 1)T r + terms of degree < r in T.

Proof. The polynomial F(X1 + λ1T, . . . ,Xd + λdT, T) is still homogeneous of

degree r (in X1, . . . ,Xd, T ), and the coefficient of the monomial T r in it can be

obtained by substituting 0 for each Xi and 1 for T .

Proof. (of the Noether Normalization Theorem, continued). Note that unless

F(X1, . . . ,Xd, T ) is the zero polynomial, it will always be possible to choose

(λ1, . . . , λd) so that F(λ1, . . . , λd, 1) _= 0 —substituting T = 1 merely dehomogenizes

the polynomial (no cancellation of terms occurs), and a nonzero polynomial can’t be

zero on all of kn (this can be proved by induction on the number of variables; it uses

only that k is infinite).

Algebraic Geometry: 6. Finite Maps 105

Let F be the homogeneous part of highest degree of f, and choose (λ1, . . . , λd) so

that F(λ1, . . . , λd, 1) _= 0. The lemma then shows that

f(X1 + λ1T, . . . ,Xd + λdT, T) = cT r + b1T r−1 + · · · + b0,

with c = F(λ1, . . . , λd, 1) ∈ k×, bi ∈ k[X1, . . . ,Xd], deg bi < r. On substituting

xn for T and xi − λixn for Xi we obtain an equation demonstrating that xn

is integral over k[x1 − λ1xn, . . . , xd − λdxn]. Put x_

i = xi − λixn, 1 ≤ i ≤ d.

Then xn is integral over the ring k[x_

1, . . . , x_

d], and it follows that A is integral over

A_ = k[x_

1, . . . , x_

d, xd+1, . . . , xn−1]. Repeat the process for A_, and continue until the

theorem is proved.

Remark 6.16. The above proof uses only that k is infinite, not that it is algebraically

closed (that’s all one needs for a nonzero polynomial not to be zero on all of

kn). There are other proofs that work also for finite fields (see Mumford 1966, p4-6),

but the above proof gives us the additional information that the yi’s can be chosen

to be linear combinations of the xi. This has the following geometric interpretation:

let V be a closed subvariety of An of dimension d; then there exists a linear map

An → Ad whose restriction to V is a finite map V _ Ad.

Zariski’s main theorem. An obvious way to construct a nonfinite quasi-finite map

W → V is to take a finite map W_ → V and remove a closed subset of W_. Zariski’s

Main Theorem show that, when W and V are separated, every quasi-finite map arises

in this way.

Theorem 6.17 (Zariski’s Main Theorem). Any quasi-finite map of varieties ϕ :

W → V factors into W

ι

8→ W_ ϕ_

→ V with ϕ_ finite and ι an open immersion.

Proof. Omitted — see the references below.

Remark 6.18. Assume (for simplicity) that V and W are irreducible and affine.

The proof of the theorem provides the following description of the factorization:i t

corresponds to the maps

k[V ] → k[W_] → k[W]

with k[W_] the integral closure of k[V ] in k[W].

A regular map ϕ : W → V of irreducible varieties is said to be birational if it

induces an isomorphism k(V ) → k(W) on the fields of rational functions (that is, if

it demonstrates that W and V are birationally equivalent).

Remark 6.19. One may ask how a birational regular map ϕ : W → V can fail to

be an isomorphism. Here are three examples.

(a) The inclusion of an open subset into a variety is birational.

(b) The map A1 → C, t _→ (t2, t3), is birational. Here C is the cubic Y 2 = X3, and

the map k[C] → k[A1] = k[T ] identifies k[C] with the subring k[T 2, T 3] of k[T ].

Both rings have k(T ) as their fields of fractions.

(c) For any smooth variety V and point P ∈ V , there is a regular birational map

ϕ : V _ → V such that the restriction of ϕ to V _−ϕ−1(P) is an isomorphism onto

V − P, but ϕ−1(P) is the projective space attached to the vector space TP (V ).

106 Algebraic Geometry: 6. Finite Maps

The next result says that, if we require the target variety to be normal (thereby

excluding example (b)), and we require the map to be quasi-finite (thereby excluding

example (c)), then we are left with (a).

Corollary 6.20. Let ϕ : W → V be a birational regular map of irreducible

varieties. Assume

(a) V is normal, and

(b) ϕ is quasi-finite.

Then ϕ is an isomorphism of W onto an open subset of V .

Proof. Factor ϕ as in the theorem. For each open affine subset U of V , k[ϕ_−1(U)]

is the integral closure of k[U] in k(W). But k(W) = k(V ) (because ϕ is birational),

and k[U] is integrally closed in k(V ) (because V is normal), and so U = ϕ_−1(U) (as

varieties). It follows that W_ = V .

Remark 6.21. Let W and V be irreducible varieties, and let ϕ : W → V be a

dominating map. It induces a map k(V ) 8→ k(W), and if dimW = dimV , then k(W)

is a finite extension of k(V ). We shall see later that, if n is the separable degree of

k(V ) over k(W), then there is an open subset U of W such that ϕ is n : 1 on U, i.e.,

for P ∈ ϕ(U), ϕ−1(P) has exactly n points.

Now suppose that ϕ is a bijective regular map W → V . We shall see later that

this implies that W and V have the same dimension. Assume:

(a) k(W) is separable over k(V );

(b) V is normal.

From (i) and the preceding remark, we find that ϕ is birational, and from (ii) and

the corollary, we find that ϕ is an isomorphism of W onto an open subset of V ; as

it is surjective, it must be an isomorphism of W onto V . We conclude:a bijective

regular map ϕ : W → V satisfying the conditions (i) and (ii) is an isomorphism.

Remark 6.22. The full name of Theorem 6.17 is “the main theorem of Zariski’s

paper Trans. AMS, 53 (1943), 490-532”. Zariski’s original statement is that in (6.20).

Grothendieck proved it in the stronger form (6.17) for all schemes. There is a good

discussion of the theorem in Mumford 1966, III.9. See also:N owak, Krzysztof Jan, A

simple algebraic proof of Zariski’s main theorem on birational transformations, Univ.

Iagel. Acta Math. No. 33 (1996), 115–118; MR 97m:14016.

Fibre products. Consider a variety S and two regular maps ϕ : V → S and ψ :

W → S. Then the set

V ×S W df = {(v,w) ∈ V ×W | ϕ(v) = ψ(w)}

is a closed subvariety of V ×W, called the fibred product of V and W over S. Note

that if S consists of a single point, then V ×S W = V ×W.

Algebraic Geometry: 6. Finite Maps 107

Write ϕ_ for the map (v,w) _→ w : V ×S W → W and ψ_ for the map (v,w) _→ v :

V ×S W → V . We then have a commutative diagram:

V ×S W

ψ_

−−−→ W _ϕ_

 _

ϕ

W

−−ψ−→ S.

The fibred product has the following universal property:c onsider a pair of regular

maps α : T → V , β : T → W; then

(α, β) = t _→ (α(t), β(t)) : T → V ×W

factors through V ×S W (as a map of sets) if and only if ϕα = ψβ, in which case

(α, β) is regular (because it is regular as a map into V ×W).

Suppose V , W, and S are affine, and let A, B, and R be their rings of regular

functions. Then A ⊗R B has the same universal property as V ×S W, except with

the directions of the arrows reversed. Since both objects are uniquely determined by

their universal properties, this shows that k[V ×S W] = A⊗R B/ N, where N is the

nilradical of A ⊗R B (that is, the set of nilpotent elements of A ⊗R B).

The map ϕ_ in the above diagram is called the base change of ϕ with respect to ψ.

For any point P ∈ S, the base change of ϕ : V → S with respect to P 8→ S is the

map ϕ−1(P) → P induced by ϕ.

Proposition 6.23. The base change of a finite map is finite.

Proof. We may assume that all the varieties concerned are affine. Then the

statement becomes:if A is a finite R-algebra, then A ⊗R B/ N is a finite B-algebra,

which is obvious.

Proper maps. A regular map ϕ : V → S of varieties is said to be proper if it is

“universally closed”, that is, if for all maps T → S, the base change ϕ_ : V ×S T → T

of ϕ is closed. Note that a variety V is complete if and only if the map V → {point}

is proper. From its very definition, it is clear that the base change of a proper map is

proper. In particular, if ϕ : V → S is proper, then ϕ−1(P) is a complete variety for

all P ∈ S.

Proposition 6.24. A finite map of varieties is proper.

Proof. The base change of a finite map is finite, and hence closed.

The next result (whose proof requires Zariski’s Main Theorem) gives a purely geometric

criterion for a regular map to be finite.

Proposition 6.25. A proper quasi-finite map ϕ: W → V of varieties is finite.

Proof. Factor ϕ into W

ι

8→ W_ →α W with α finite and ι an open immersion.

Factor ι into

W

w_→→(w,ιw) W ×V W

_ (w,w_)_→w_

→ W

_

.

The image of the first map is Γι, which is closed because W_ is a variety (see 3.25;

W_ is separated because it is finite over a variety — exercise). Because ϕ is proper,

108 Algebraic Geometry: 6. Finite Maps

the second map is closed. Hence ι is an open immersion with closed image. It follows

that its image is a connected component of W_, and that W is isomorphic to that

connected component.

If W and V are curves, then any surjective map W → V is closed. Thus it is easy

to give examples of closed surjective quasi-finite nonfinite maps. For example, the

map

a _→ an : A1     {0} → A1,

which corresponds to the map on rings

k[T ]→ k[T, T

−1], T _→ T n,

is such a map. This doesn’t violate the theorem, because the map is only closed, not

universally closed.

Algebraic Geometry: 7. Dimension Theory 109

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