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2. Affine Algebraic Varieties

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In this section we define on an algebraic set the structure of a ringed space, and

then we define the notion of affine algebraic variety—roughly speaking, this is an

algebraic set with no preferred embedding into kn. This is in preparation for §3,

where we define an algebraic variety to be a ringed space that is a finite union of

affine algebraic varieties satisfying a natural separation axiom (in the same way that

a topological manifold is a union of subsets homeomorphic to open subsets of Rn

satisfying the Hausdorff axiom).

Ringed spaces. Let V be a topological space and k a field.

Definition 2.1. Suppose that for every open subset U of V we have a set OV (U)

of functions U → k. Then OV is called a sheaf of k-algebras if it satisfies the following

conditions:

(a) OV (U) is an k-subalgebra of the algebra of all functions U → k, i.e., for each

c ∈ k, the constant function c is in OV (U), and if f, g ∈ OV (V ), then so also do

f ± g, and fg.

(b) If U_ is an open subset of U and f ∈ OV (U), then f|U_ ∈ OV (U_).

f : U → k is in OV (U) if f|Uα ∈ OV (Uα) for all α (i.e., the condition for f to be

in OV (U) is local .

Example 2.2. (a) Let V be any topological space, and for each open subset U of

V let OV (U) be the set of all continuous real-valued functions on U. Then OV is a

sheaf of R-algebras.

(b) Recall that a function f : U → R, where U is an open subset of Rn, is said

to be C∞ (or infinitely differentiable) if its partial derivatives of all orders exist and

are continous. Let V be an open subset of Rn, and for each open subset U of V let

OV (U) be the set of all infinitely differentiable functions on U. Then OV is a sheaf

of R-algebras.

to be analytic (or holomorphic) if it is described by a convergent power series in a

neighbourhood of each point of U. Let V be an open subset of Cn, and for each open

subset U of V let OV (U) be the set of all analytic functions on U. Then OV is a sheaf

of C-algebras.

(d) Nonexample:l et V be a topological space, and for each open subset U of V let

OV (U) be the set of all real-valued constant functions on U; then OV is not a sheaf,

unless V is irreducible! If “constant” is replaced with “locally constant”, then OV

becomes a sheaf of R-algebras (in fact, the smallest such sheaf).

A pair (V,OV ) consisting of a topological space V and a sheaf of k-algebras will be

called a ringed space. For historical reasons, we often write Γ(U,OV ) for OV (U) and

call its elements sections of OV over U.

Let (V,OV ) be a ringed space. For any open subset U of V , the restriction OV |U

of OV to U, defined by

Γ(U_ ,OV |U) = Γ(U_,OV ), all open U_ ⊂ U,

Algebraic Geometry: 2. Affine Algebraic Varieties 31

is a sheaf again.

Let (V,OV ) be ringed space, and let P ∈ V . Consider pairs (f,U) consisting of

an open neighbourhood U of P and an f ∈ OV (U). We write (f,U) ∼ (f_, U_)

if f|U__ = f_|U__ for some U__ ⊂ U ∩ U_. This is an equivalence relation, and an

equivalence class of pairs is called a germ of a function at P. The set of equivalence

classes of such pairs forms a k-algebra denoted OV,P or OP . In all the interesting

cases, it is a local ring with maximal ideal the set of germs that are zero at P.

In a fancier terminology,

OP = −li→mOV (U), (direct limit over open neighbourhoods U of P).

Example 2.3. Let V = C, and let OV be the sheaf of holomorphic functions on

C. For c ∈ C, call a power series

n≥0 an(z − c)n, an ∈ C, convergent if it converges

on some neighbourhood of c. The set of such power series is a C-algebra, and I claim

that it is canonically isomorphic to the ring of germs of functions Oc. From basic

complex analysis, we know that if f is a holomorphic function on a neighbourhood U

of c, then f has a power series expansion f =

an(z−c)n in some (possibly smaller)

neighbourhood. Moreover another pair (g,U_) will define the same power series if and

only if g agrees with f on some neighbourhood of c contained in U ∩ U_. Thus we

have injective map from the ring of germs of holomorphic functions at c to the ring

of convergent power series, and it is obvious that it is an isomorphism.

Review of rings of fractions. Before defining the sheaf of regular functions on an

algebraic set, we need to review some of the theory of rings of fractions. When the

initial ring is an integral domain (the most important case), the theory is very easy

because all the rings are subrings of the field of fractions.

A multiplicative subset of a ring A is a subset S with the property:

1 ∈ S, a, b ∈ S ⇒ ab ∈ S.

Define an equivalence relation on A × S by

(a, s) ∼ (b, t) ⇐⇒ u(at − bs) = 0 for some u ∈ S.

Write a

s for the equivalence class containing (a, s), and define addition and multiplication

in the obvious way:

at + bs

We then obtain a ring S−1A = {a

| a ∈ A, s ∈ S}, and a canonical homomorphism

a _→ a

1 : A → S−1A, not necessarily injective. For example, if S contains 0, then S−1A

is the zero ring.

Write i for the homomorphism a _→ a

1 : A → S−1A. Then (S−1A, i) has the following

universal property:ev ery element s ∈ S maps to a unit in S−1A, and any other

homomorphism α: A → B with this property factors uniquely through i:

i✲ S

−1A

❅

❅ α❘

❄

.....

∃!

32 Algebraic Geometry: 2. Affine Algebraic Varieties

The uniqueness is obvious—the map S−1A → B must be a

_→ α(a) · α(s)−1 — and

it is easy to check that this formula does define a homomorphism S−1A → B. For

example, to see that it is well-defined, note that

⇒ s(ad − bc) = 0 some s ∈ S ⇒ α(a)α(d) − α(b)α(c) = 0,

because α(s) is a unit in B, and so

α(a)α(c)−1 = α(b)α(d)−1.

As usual, this universal property determines the pair (S−1A, i) uniquely up to a unique

isomorphism.

In the case that A is an integral domain we can form the field of fractions F = S−1A,

S = A − {0}, and then for any other multiplicative subset S of A not containing 0,

S−1A can be identified with {a

∈ F | a ∈ A, s ∈ S}.

We shall be especially interested in the following examples.

(i) Let h ∈ A. Then Sh

df = {1, h, h2, . . . } is a multiplicative subset of A, and we

write Ah = S

−1

h A. Thus every element of Ah can be written in the form a/hm, a ∈ A,

and

hm =

⇐⇒ hN(ahn − bhm) = 0, some N.

In the case that A is an integral domain, with field of fractions F, Ah is the subring

of F of elements of the form a/hm, a ∈ A, m ∈ N.

(ii) Let p be a prime ideal in A. Then Sp

df = A p is a multiplicative subset of A,

and we write Ap = S−1

p A. Thus each element of Ap can be written in the form a

c ,

c /∈ p, and

⇐⇒ s(ad − bc) = 0, some s /∈ p.

The subset m = {a

| a ∈ p, s /∈ p} is a maximal ideal in Ap, and it is the only

maximal ideal 9 . Therefore Ap is a local ring. Again, when A is an integral domain

with field of fractions F, Ap is the subring of F consisting of elements expressible in

the form a

s , a ∈ A, s /∈ p.

Lemma 2.4. For any ring A, the map

aiXi _→ _ ai

hi defines an isomorphism

A[X]/(1 − hX)

≈→

Ah.

Proof. In the ring A[x] = A[X]/(1 − hX), 1 = hx, and so h is a unit. Consider

a homomorphism of rings α: A → B such that α(h) is a unit in B. Then α extends

to a homomorphism

aiXi _→

α(ai)α(h)

−i : A[X]→ B.

Under this homomorphism 1 − hX _→ 1 − α(h)α(h)−1 = 0, and so the map factors

through A[x]. The resulting homomorphism γ : A[x] → B has the property that

its composite with A → A[x] is α, and (because hx = 1 in A[x]) it is the unique

9First check m is an ideal. Next, if m = Ap, then 1 ∈ m; but 1 = a

s, a ∈ p, s /∈ p means

u(s − a) = 0 some u /∈ p, and so a = us /∈

p. Finally, m is maximal, because any element of Ap not

in m is a unit.

Algebraic Geometry: 2. Affine Algebraic Varieties 33

homomorphism with this property. Therefore A[x] has the same universal property

as Ah, and so the two are (uniquely) isomorphic by an isomorphism that makes h−1

correspond to x.

For more on rings of fractions, see Atiyah and MacDonald 1969, Chapt 3.

The ringed space structure on an algebraic set. We now take k to be an

algebraically closed field. Let V be an algebraic subset of kn. An element h of k[V ]

defines functions

a _→ h(a) : V → k, and a _→ 1/h(a) : D(h) → k.

Thus a pair of elements g, h ∈ k[V ] with h _= 0 defines a function

a _→ g(a)

h(a)

: D(h) → k.

We say that a function f : U → k on an open subset U of V is regular if it is of

this form in a neighbourhood of each of its points, i.e., if for all a ∈ U, there exist

g, h ∈ k[V ] with h(a) _= 0 such that the functions f and g

h agree in a neighbourhood

of a. Write OV (U) for the set of regular functions on U.

For example, if V = kn, then a function f : U → k is regular at a point a ∈ U if there

are polynomials g(X1, . . . ,Xn) and h(X1, . . . ,Xn) with h(a) _= 0 and f(b) = g(b)

h(b) for

all b such that the expression on the right is defined.

Proposition 2.5. The map U _→ OV (U) defines a sheaf of k-algebras on V .

Proof. We have to check the conditions (2.1).

(a) Clearly, a constant function is regular. Suppose f and f_ are regular on U, and

let a ∈ U. By assumption, there exist g, g_, h, h_ ∈ k[V ], with h(a) _= 0 _= h_(a) such

that f and f_ agree with g

h and g_

h_ respectively near a. Then ff_ agrees with gh_+g_h

hh_

near a, and so ff_ is regular on U. Similarly f ± f_ are regular on U. Thus OV (U)

is a k-algebra.

(b) It is clear from the definition that the restriction of a regular function to an

open subset is again regular.

Lemma 2.6. The element g/hm of k[V ]h defines the zero function on D(h) if and

only if gh = 0 (in k[V ]) (and hence g/hm = 0 in k[V ]h).

Proof. If g/hm is zero on D(h), then gh is zero on V because h is zero on the

complement of D(h). Therefore gh is zero in k[V ]. Conversely, if gh = 0, then

g(a)h(a) = 0 for all a ∈ kn, and so g(a) = 0 for all a ∈ D(h).

Proposition 2.7. (a) The canonical map k[V ]h →OV (D(h)) is an isomorphism.

(b) For any a ∈ V , there is a canonical isomorphism Oa → k[V ]ma, where ma is

the maximal ideal (x1 − a1, . . . , xn − an).

Proof. (a) The preceding lemma shows that k[V ]h →OV (D(h)) is injective, and

so it remains to show that every regular function f on D(h) arises from an element

of k[V ]h.

34 Algebraic Geometry: 2. Affine Algebraic Varieties

By definition, we know that there is an open covering D(h) = ∪Vi and elements

gi, hi ∈ k[V] with hi nowhere zero on Vi such that f|Vi = gi

. Since the sets of the

form D(a) form a basis for the topology on V , we can assume that Vi = D(ai), some

ai ∈ k[V ]. By assumption D(ai) ⊂ D(hi), and so aNi

= hig_

i for some h_

∈ k[V ] (see

paragraph after 1.14). On D(ai), f = gi

= gig_

hig_

= gig_

aNi

. Note that D(aNi

) = D(ai).

Therefore, after replacing gi with gig_

i and hi with aNi

, we can suppose that Vi = D(hi).

We now have that D(h) = ∪D(hi) and that f|D(hi) = gi

. Because D(h) is

quasicompact10, we can assume that the covering is finite. As gi

= gj

on D(hi) ∩

D(hj) = D(hihj), we have (by the lemma) that

hihj(gihj − gjhi) = 0. (*)

Because D(h) = ∪D(hi) = ∪D(h2i

), V ((h)) = V ((h21

, . . . , h2

m)), and so there exist

ai ∈ k[V ] such that

hN =

aih2i

. (**)

I claim that f is the function on D(h) defined by

Paigihi

hN .

Let a be a point of D(h). Then a will be in one of the D(hi), say D(hj ). We have

the following equalities in k[V ]:

h2j

i=1

aigihi =

i=1

aigjh2i

hj by (*)

= gjhjhN by (**).

But f|D(hj) = gj

, i.e., fhj and gj agree as functions on D(hj). Therefore we have

the following equality of functions on D(hj ):

h2j

aigihi = fh2j

hN.

Since h2j

is never zero on D(hj ), we can cancel it, to find that, as claimed, the function

fhN on D(hj ) equals that defined by

aigihi.

(b) First a general observation:i n the definition of the germs of a sheaf at a, it

suffices to consider pairs (f,U) with U lying in a fixed basis for the neighbourhoods

of a. Thus each element of Oa is represented by a pair (f,D(h)) where h(a) _= 0 and

f ∈ k[V ]h, and two pairs (f1,D(h1)) and (f2,D(h2)) represent the same element of

Oa if and only if f1 and f2 restrict to the same function on D(h) for some a ∈ D(h) ⊂

D(h1h2).

For each h /∈ p, there is a canonical homomorphism αh : k[V ]h → k[V ]p, and we

map the element of Oa represented by (f,D(h)) to αh(f). It is now an easy exercise

to check that this map is well-defined, injective, and surjective.

The proposition gives us an explicit description of the value of OV on any basic

open set and of the ring of germs at any point a of V. When V is irreducible, this

10Recall (1.13) that V is Noetherian, i.e., has the ascending chain condition on open subsets. This

implies that any open subset of V is also Noetherian, and hence is quasi-compact.

Algebraic Geometry: 2. Affine Algebraic Varieties 35

becomes a little simpler because all the rings are subrings of k(V ). We have:

Γ(D(h),OV) = { g

∈ k(V ) | g ∈ k[V ], N∈ N};

Oa = {g

∈ k(V ) | h(a) _= 0};

Γ(U,OV) = ∩Oa (intersection over all a ∈ U}

= ∩Γ(D(hi),OV ) if U = ∪D(hi).

Note that every element of k(V ) defines a function on some nonempty open subset of

V . Following tradition, we call the elements of k(V ) rational functions on V (even

though they are not functions on V ). The last equality then says that the regular

functions on U are the rational functions on V that are defined at each point of U.

Example 2.8. (a) Let V = kn. Then the ring of regular functions on V , Γ(V,OV ),

is k[X1, . . . ,Xn]. For any nonzero polynomial h(X1, . . . ,Xn), the ring of regular

functions on D(h) is

{ g

∈ k(X1, . . . ,Xn) | g, h ∈ k[X1, . . . ,Xn]}.

For any point a = (a1, . . . , an), the ring of germs of functions at a is

Oa = {g

∈ k(X1, . . . ,Xn) | h(a) _= 0} = k[X1, . . . ,Xn](X1−a1,... ,Xn−an),

and its maximal ideal consists of those g/h with g(a) = 0.

(b) Let U = {(a, b) ∈ k2 | (a, b) _= (0, 0)}. It is an open subset of k2, but it is not

a basic open subset, because its complement {(0, 0)} has dimension 0, and therefore

can’t be of the form V ((f)) (see 1.21). Since U = D(X) ∪ D(Y ), the ring of regular

functions on U is

Γ(D(X),O) ∩ Γ(D(Y ),O) = k[X, Y ]X ∩ k[X, Y ]Y .

Thus (as an element of k(X, Y )), a regular function on U can be written

f =

g(X, Y )

XN =

h(X, Y )

Y M .

Since k[X, Y ] is a unique factorization domain, we can assume that the fractions are

in their lowest terms. On multiplying through by XNY M, we find that

g(X, Y )Y M = h(X, Y )XN.

Because X doesn’t divide the left hand side, it can’t divide the right either, and so

N = 0. Similarly, M = 0, and so f ∈ k[X, Y ]:ev ery regular function on U extends

to a regular function on k2.

Morphisms of ringed spaces. A morphism of ringed spaces (V,OV ) → (W,OW)

is a continuous map ϕ: V → W such that

f ∈ OW(U) ⇒ f ◦ ϕ ∈ OV (ϕ−1U)

for all open subsets U of W. Sometimes we write ϕ∗(f) for f ◦ ϕ. If U is an open

subset of V , then the inclusion (U,OV |V ) 8→ (V,OV ) is a morphism of ringed spaces.

A morphism of ringed spaces is an isomorphism if it is bijective and its inverse is also

a morphism of ringed spaces (in particular, it is a homeomorphism).

36 Algebraic Geometry: 2. Affine Algebraic Varieties

Example 2.9. (a) Let V and V _ be topological spaces endowed with their sheaves

OV and OV _ of continuous real valued functions. Any continuous map ϕ: V → V _ is

a morphism of ringed structures (V,OV ) → (V _,OV _).

(b) Let U and U_ be open subsets of Rn and Rm respectively. Recall from advanced

calculus that a mapping

ϕ = (ϕ1, . . . , ϕm) : U → U

_ ⊂ Rm

is said to be infinitely differentiable (or C∞) if each ϕi is infinitely differentiable, in

which case f ◦ ϕ is infinitely differentiable for every infinitely differentiable function

f : U_ → R. Note that ϕi = xi ◦ϕ, where xi is the coordinate function (a1, . . . , an) _→

ai.

Let V and V _ be open subsets of Rn and Rm respectively, endowed with their

sheaves of infinitely differentiable functions OV and OV _ . The above statements show

that a continuous map ϕ: V → V _ is infinitely differentiable if and only if it is a

morphism of ringed spaces.

Remark 2.10. A morphism of ringed spaces maps germs of functions to germs of

functions. More precisely, a morphism ϕ: (V,OV ) → (V _,OV _) induces a map

OV,P ←OV _,ϕ(P),

namely, [(f,U)] _→ [(f ◦ ϕ, ϕ−1(U))].

Affine algebraic varieties. We have just seen that every algebraic set gives rise to

a ringed space (V,OV ). We define an affine algebraic variety over k to be a ringed

space that is isomorphic to a ringed space of this form. A morphism of affine algebraic

varieties is a morphism of ringed spaces; we often call it a regular map V → W or

a morphism V → W, and we write Mor(V,W) for the set of such morphisms. With

these definitions, the affine algebraic varieties become a category. Since we consider

no nonalgebraic affine varieties, we shall often drop the “algebraic”.

In particular, every algebraic set has a natural structure of an affine variety. We

usually write An for kn regarded as an affine algebraic variety. Note that the affine

varieties we have constructed so far have all been embedded in An. We shall now see

how to construct “unembedded” affine varieties.

A reduced finitely generated k-algebra is called an affine k-algebra. For such an

algebra A, there exist xi ∈ A (not necessarily algebraically independent), such that

A = k[x1, . . . , xn], and the kernel of the homomorphism

Xi _→ xi : k[X1, . . . ,Xn] → A

is a radical ideal. Zariski’s Lemma 1.7 implies that, for any maximal ideal m ∈ A,

the map k → A → A/m is an isomorphism. Thus we can identify A/m with k. For

f ∈ A, we write f(m) for the image of f in A/m = k, i.e., f(m) = f (mod m).

We can associate with any affine k-algebra A a ringed space (V,OV ). First, V is

the set of maximal ideals in A. For h ∈ A, h _= 0, let

D(h) = {m | h(m) _= 0, i.e., h /∈ m},

Algebraic Geometry: 2. Affine Algebraic Varieties 37

and endow V with the topology for which the D(h) form a basis. A pair of elements

g, h ∈ A, h _= 0, defines a function

m _→ g(m)

h(m)

: D(h) → k,

and we call a function f : U → k on an open subset U of V regular if it is of this form

on a neighbourhood of each point of U. Write OV (U) for the set of regular functions

on U.

Proposition 2.11. The pair (V,OV ) is an affine variety with Γ(V,OV) = A.

Proof. Represent A as a quotient k[X1, . . . ,Xn]/a = k[x1, . . . , xn]. Then the

map

(a1, . . . , an) _→ (x1 − a1, . . . , xn − an) (idealinA)

is a bijection ϕ: V (a) → V with inverse

m _→ (x1(m), . . . , xn(m)) : V → V (a) ⊂ kn.

It is easy to check that this is a homeomorphism, and that a function f on an open

subset of V is regular (according to the above definition) if and only if f ◦ ϕ is

regular.

We write specm(A) for the topological space V , and Specm(A) for the ringed

space (V,OV ). If we start with an affine variety V and let A = Γ(V,OV ), then the

Specm(A) ≈ (V,OV ) (canonically). We again write k[V ] for Γ(V,OV ), the ring of

functions regular on the whole of V.

Thus, for each affine k-algebra A, we have an affine variety Specm(A), and conversely,

for each affine variety (V,OV ), we have an affine k-algebra Γ(V,OV ). We now

make this correspondence into an equivalence of categories.

Remark 2.12. I claim that a radical ideal a in k[X1, . . . ,Xn] is equal to the

intersection of the maximal ideals containing it. Indeed, the maximal ideals in

k[X1, . . . ,Xn] are all of the form ma = (X1 − a1, . . . ,Xn − an), and f ∈ ma ⇐⇒

f(a) = 0. Thus ma ⊃ a ⇐⇒ a ∈ V (a), and if f ∈ ma for all a ∈ V (a), then f is

zero on V (a), i.e., f ∈ IV (a) = a.

This remark implies that, for any affine k-algebra A, the intersection of the maximal

ideals of A is zero, because A is isomorphic to a k-algebra k[X1, . . . ,Xn]/a and we

can apply the remark to a. Hence the map that associates with f ∈ A the map

specmA → k, m _→ f(m), is injective: A can be identified with a ring of functions on

specmA.

The category of affine algebraic varieties. Let α: A → B be a homomorphism of

affine k-algebras. For any h ∈ A, α(h) is invertible in Bα(h), and so the homomorphism

A → B → Bα(h) extends to a homomorphism

_→ α(g)

α(h)m : Ah → Bα(h).

For any maximal ideal n of B, m df = α−1(n) is maximal in A, because A/m → B/n = k

is an injective map of k-algebras and this implies A/m = k. Thus α defines a map

ϕ:s pecmB → specm A, ϕ(n) = α−1(n) = m.

38 Algebraic Geometry: 2. Affine Algebraic Varieties

For m = α−1(n) = ϕ(n), we have a commutative diagram:

A −−α−→ B _

_

A/m −−=−→ A/n.

Recall that the image of an element f of A in A/m = k is denoted f(m). Therefore,

the commutativity of the diagram means that, for f ∈ A,

f(ϕ(n)) = α(f)(n), i.e., f ◦ ϕ = α. (*).

Since ϕ−1D(f) = D(f ◦ ϕ) (obviously), it follows from (*) that

−1(D(f)) = D(α(f)),

and so ϕ is continuous.

Let f be a regular function on D(h), and write f = g/hm, g ∈ A. Then, from (*)

we see that f ◦ ϕ is the function on D(α(h)) defined by α(g)/α(h)m. In particular,

it is regular, and so f _→ f ◦ ϕ maps regular functions on D(h) to regular functions

on D(α(h)). It follows that f _→ f ◦ ϕ sends regular functions on any open subset

of specm(A) to regular functions on the inverse image of the open subset. Thus α

defines a morphism Specm(B) → Specm(A).

Conversely, by definition, a morphism of ϕ: (V,OV ) → (W,OW) of affine algebraic

varieties defines a homomorphism of the associated affine k-algebras k[W] → k[V ].

Since these maps are inverse, we have shown:

Proposition 2.13. For any affine algebras A and B,

Homk-alg(A,B)

≈→

Mor(Specm(B), Specm(A));

for any affine varieties V and W,

Mor(V,W)

≈→

Homk-alg(k[W], k[V ]).

A covariant functor F : A → B of categories is said to be an equivalence of categories

(a) for all objects A, A_ of A,

Hom(A,A_) → Hom(F(A), F(A_))

is a bijection (F is fully faithful);

(b) every object of B is isomorphic to an object of the form F(A), A ∈ Ob(A) (F

is essentially surjective).

One can show that such a functor F has a quasi -inverse, i.e., there is a functor

G: B → A, which is also an equivalence, and is such that G(F(A)) ≈ A (functorially)

and F(G(B)) ≈ B (functorially). Hence the relation of equivalence is an equivalence

relation. (In fact one can do better—see, for example, Bucur and Deleanu, Introduction

to the Theory of Categories and Functors, 1968, I.6.)

Similarly one defines the notion of a contravariant functor being an equivalence of

categories. Proposition 2.13 can now be restated in stronger form as:

Algebraic Geometry: 2. Affine Algebraic Varieties 39

Proposition 2.14. The functor A _→ SpecmA is a (contravariant) equivalence

from the category of affine k-algebras to that of affine varieties with quasi-inverse

(V,OV ) _→ Γ(V,OV ).

Explicit description of morphisms of affine varieties.

Proposition 2.15. Let V = V (a) ⊂ km, W = V (b) ⊂ kn. The following conditions

on a continuous map ϕ: V → W are equivalent:

(a) ϕ is regular;

(b) the components ϕ1, . . . , ϕm of ϕ are all regular;

Proof. (a) ⇒ (b). By definition ϕi = yi ◦ ϕ where yi is the coordinate function

(b1, . . . , bn) _→ bi : W → k. Hence this implication follows directly from the definition

of a regular map.

(b) ⇒ (c). The map f _→ f ◦ ϕ is a k-algebra homomorphism from the ring of

all functions W → k to the ring of all functions V → k, and (b) says that the map

sends the coordinate functions yi on W into k[V ]. Since the yi’s generate k[W] as a

k-algebra, this implies that this map sends k[W] into k[V ].

defines a map specmk[V ] → specm k[W], and it remains to show that this coincides

with ϕ when we identify specm k[V ] with V and specm k[W] with W. Let a ∈ V , let

b = ϕ(a), and let ma and mb be the ideals of elements of k[V ] and k[W] that are zero

at a and b respectively. Then, for f ∈ k[W],

α(f) ∈ ma ⇐⇒ f(ϕ(a)) = 0 ⇐⇒ f(b) = 0 ⇐⇒ f ∈ mb.

Therefore α−1(ma) = mb, which is what we needed to show.

Remark 2.16. For all a ∈ V , f _→ f ◦ ϕ maps germs of regular functions at

ϕ(a) to germs of regular functions at a; in fact, it induces a local homomorphism

OV ,ϕ(a) →OV,a.

Now consider equations

Y1 = P1(X1, . . . ,Xm)

. . .

Yn = Pn(X1, . . . ,Xm).

On the one hand, they define a mapping ϕ: km → kn, namely,

(a1, . . . , am) _→ (P1(a1, . . . , am), . . . , Pn(a1, . . . , am)).

On the other, they define a homomorphism of k-algebras α: k[Y1, . . . , Yn] →

k[X1, . . . ,Xn], namely, that sending

Yi _→ Pi(X1, . . . ,Xn).

This map coincides with f _→ f ◦ ϕ, because

α(f)(a) = f(. . . , Pi(a), . . .) = f(ϕ(a)).

40 Algebraic Geometry: 2. Affine Algebraic Varieties

Now consider closed subsets V (a) ⊂ km and V (b) ⊂ kn with a and b radical ideals. I

claim that ϕ maps V (a) into V (b) if and only if α(b) ⊂ a. Indeed, suppose ϕ(V (a)) ⊂

V (b), and let f ∈ b; for b ∈ V (b),

α(f)(b) = f(ϕ(b)) = 0,

and so α(f) ∈ IV (b) = b. Conversely, suppose α(b) ⊂ a, and let a ∈ V (a); for f ∈ a,

f(ϕ(a)) = α(f)(a) = 0,

and so ϕ(a) ∈ V (a). When these conditions hold, ϕ is the morphism of affine

varieties V (a) → V (b) corresponding to the homomorphism k[Y1, . . . , Ym]/b →

k[X1, . . . ,Xn]/a defined by α.

Thus, we see that the morphisms

V (a) → V (b)

are all of the form

a _→ (P1(a), . . . , Pm(a)), Pi ∈ k[X1, . . . ,Xn].

Example 2.17. (a) Consider a k-algebra R. From a k-algebra homomorphism

α: k[X] → R, we obtain an element α(X) ∈ R, and α(X) determines α completely.

Moreover, α(X) can be any element of R. Thus

α _→ α(X):H omk−alg(k[X], R)

≈→

According to (2.13)

Mor(V,A1) = Homk-alg(k[X], k[V ]).

Thus the regular maps V → A1 are simply the regular functions on V (as we would

hope).

(b) Define A0 to be the ringed space (V0,OV0) with V0 consisting of a single point,

and Γ(V0,OV0) = k. Equivalently, A0 = Specm k. Then, for any affine variety V ,

Mor(A0, V )

∼

= Homk-alg(k[V ], k)

∼=

where the last map sends α to the point corresponding to the maximal ideal Ker(α).

V : Y 2 = X3, but it is not an isomorphism onto its image — the inverse map is

not a morphism. Because of (2.14), it suffices to show that t _→ (t2, t3) doesn’t

induce an isomorphism on the rings of regular functions. We have k[A1] = k[T] and

k[V ] = k[X, Y ]/(Y 2 − X3) = k[x, y]. The map on rings is

x _→ T 2, y_→ T 3, k[x, y]→ k[T ],

which is injective, but the image is k[T 2, T 3] _= k[T ]. In fact, k[x, y] is not integrally

closed:( y/x)2−x = 0, and so (y/x) is integral over k[x, y], but y/x /∈

k[x, y] (it maps

to T under the inclusion k(x, y) 8→ k(T )).

(d) Assume that k has characteristic p _= 0, and consider x _→ xp : An → An. This

is a bijection, but it is not an isomorphism because the corresponing map on rings,

Xi _→ Xp

i : k[X1, . . . ,Xn] → k[X1, . . . ,Xn],

is not surjective.

Algebraic Geometry: 2. Affine Algebraic Varieties 41

This map is the famous Frobenius map. Take k to be the algebraic closure of Fp,

the field with p elements, and write F for the map. Then the fixed points of Fm are

precisely the points of An with coordinates in Fpm, the field with pm-elements (recall

from Galois theory that Fpm is the subfield of k consisting of those elements satisfying

the equation Xpm = X). Let P(X1, . . . ,Xn) be a polynomial with coefficients in Fpm,

P =

cαXα, cα ∈ Fpm. If P(a) = 0, a ∈ kn, i.e.,

cαai1

· · · ain

n = 0, then

0 =

cαai1

· · · ain

_pm

cαapmi1

· · · apmin

n ,

and so P(Fma) = 0. Thus Fm maps V (P) into V (P), and its fixed points are the

solutions of

P(X1, . . . ,Xn) = 0

in Fpm.

In one of the most beautiful pieces of mathematics of the last fifty years,

Grothendieck defined a cohomology theory (´etale cohomology) that allowed him to

obtain an expression for the number of solutions of a system of polynomial equations

with coordinates in Fpn in terms of a Lefschetz fixed point formula, and Deligne used

the theory to obtain very precise estimates for the number of solutions. See my course

notes:Le ctures on Etale Cohomology.

Subvarieties. For any ideal a in A, we define

V (a) = {P ∈ specmA | f(P) = 0 all f ∈ a}

= {m maximal ideal in A | a ⊂ m}.

This is a closed subset of specmA, and every closed subset is of this form.

Now assume a is radical, so that A/a is again reduced. Corresponding to the

homomorphism A → A/a, we get a regular map

Specm A/a → SpecmAA

The image is V (a), and specm A/a → V (a) is a homeomorphism. Thus every closed

subset of specmA has a natural ringed structure making it into an affine algebraic

variety. We call V (a) with this structure a closed subvariety of V.

Aside 2.18. If (V,OV ) is a ringed space, and Z is a closed subset of V , we can

define a ringed space structure on Z as follows:let U be an open subset of Z, and let

f be a function U → k; then f ∈ Γ(U,OZ) if for each P ∈ U there is a germ (U_, f_)

of a function at P (regarded as a point of V ) such that f_|Z ∩ U_ = f. One can

check that when this construction is applied to Z = V (a), the ringed space structure

obtained is that described above.

Proposition 2.19. Let (V,OV ) be an affine variety and let h ∈ k[V ], h _= 0.

Then (D(h),OV |D(h)) is an affine variety; in fact if V = specm(A), then D(h) =

specm(Ah). More explicitly, if V = V (a) ⊂ kn, then

(a1, . . . , an) _→ (a1, . . . , an, h(a1, . . . , an)

−1) : D(h) → kn+1,

defines an isomorphism of D(h) onto V (a, 1 − hXn+1).

Proof. The map A → Ah defines a morphism specmAh → specmA. The image

is D(h), and it is routine (using (2.4)) to verify the rest of the statement.

42 Algebraic Geometry: 2. Affine Algebraic Varieties

For example, there is an isomorphism of affine varieties

x _→ (x, 1/x) : A1 − {0} → V ⊂ A2,

where V is the subvariety XY = 1 of A2 — the reader should draw a picture.

Remark 2.20. We have seen that all closed subsets, and all basic open subsets, of

an affine variety V are again affine varieties, but it need not be true that (U,OV |U)

is an affine variety when U open in V. Note that if (U,OV |U) is an affine variety,

then we must have (U,OV )

∼=

Specm(A), A = Γ(U,OV ). In particular, the map

P _→ mP = {f ∈ A | f(P) = 0}

will be a bijection from U onto specm(A).

Consider U ⊂ A2 \ (0, 0) = D(X) ∪ D(Y ). We saw in (2.8b) that Γ(U,OA2) =

k[X, Y ]. Now U → specm k[X, Y ] is not a bijection, because the ideal (X, Y ) is not

in the image.

However, U is clearly a union of affine algebraic varieties — we shall see in the next

section that it is a (nonaffine) algebraic variety.

Properties of specm(α).

Proposition 2.21. Let α: A → B be a homomorphism of affine k-algebras, and

let ϕ:S pecm(B) → Specm(A) be the corresponding morphism of affine varieties (so

that α(f) = ϕ ◦ f).

(a) The image of ϕ is dense for the Zariski topology if and only if α it is injective.

(b) ϕ defines an isomorphism of Specm(B) onto a closed subvariety of Specm(A) if

and only if α is surjective.

Proof. (a) Let f ∈ A. If the image of ϕ is dense, then

f ◦ ϕ = 0 ⇒ f = 0.

Conversely, if the image of ϕ is not dense, there will be a nonzero function f ∈ A

that is zero on its image, i.e., such that f ◦ ϕ = 0.

(b) If α is surjective, then it defines an isomorphism A/a → B where a is the kernel

of α. This induces an isomorphism of Specm(B) with its image in Specm(A).

A regular map ϕ: V → W of affine algebraic varieties is said to be a dominating if

the image is dense in W. The proposition then says that:

ϕ is dominating ⇐⇒ f _→ f ◦ ϕ:Γ( W,OW) → Γ(V,OV ) is injective.

A little history. We have associated with any affine k-algebra A an affine variety

whose underlying topological space is the set of maximal ideals in A. It may seem

strange to be describing a topological space in terms of maximal ideals in a ring, but

the analysts have been doing this for more than 50 years. Gel’fand and Kolmogorov

in 1939 proved that if S and T are completely regular topological spaces, and the

rings of real-valued continuous functions on S and T are isomorphic (just as rings),

then S and T are homeomorphic. The first step in the proof showed that, for such a

space S, the map

P _→ mP = {f : S → R | f(P) = 0}

defines a one-to-one correspondence between the points in the space and maximal

ideals in the ring. (See Shields’s article in Math. Intelligencer, Summer 1989, pp

15-17.) (A space S is completely regular if it is T1 and for every closed subset C

and point P /∈ C, there is a real-valued continuous function f on the space such that

f(P) = 0 and f is identically 1 on C.)

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