ADDITIONAL NOTES IN ALGEBRAIC GEOMETRY, II
ANDREAS LEOPOLD KNUTSEN
Abstract. These notes supplement [Hart, Chapter II] and form part of the syllabus in the course MAT321-Introduction to sheaves and schemes, taught at the
University of Bergen, Spring 2016.
1. Basic properties of coverings and morphisms of schemes
This section supplements [Hart, II, §2].
The following summarizes basic properties of an affine scheme that in particular
are used in the proof of [Hart, II, Prop. 3.2]:
Lemma 1.1. Let X = Spec A be an affine scheme. Then the following holds:
(a) The open sets of the form D(α), for α ∈ A, form a basis for the topology and
(D(α), OX |D(α) ) ' Spec Aα .
(b) For any affine subscheme Spec B ⊆ X, we have
D(α) ∩ Spec B = D(α),
(1)
where α = ϕ(α), with ϕ : A → B corresponding to the inclusion morphism
Spec B → Spec A as in the proof of [Hart, II,Prop. 2.3] (see also below).
In particular, if D(α) ⊆ Spec B, then Aα ' Bα .
(c) We have
(2)
Spec A = ∪i∈I D(αi ) ⇐⇒ ({αi | i ∈ I}) = A.
(d) Given (2), we have
Spec A = D(α1 ) ∪ · · · ∪ D(αr ),
for finitely many αi . In particular, sp(X) is quasi-compact ([Hart, II, Exc.
2.13(b)]).
(e) sp(X) is not necessarly noetherian, but is noetherian if A is noetherian ([Hart,
II, Exc. 2.13(b,c)]).
Proof. (a) The first part is proved in [Hart, p. 70-71] (bottom and start of pages) and
the second is [Hart, II, Exc. 2.1] and is proved in the same way as [Hart, II, Prop.
2.5(b)] (and is partially proved in [Hart, II, Prop. 2.2(b,c)]).
(b) We have
D(α) ∩ Spec B = {p ∈ Spec A | α 6∈ p} ∩ Spec B
= {q ∈ Spec B | α 6∈ ϕ−1 q}
= {q ∈ Spec B | ϕ(α) 6∈ q} = D(α),
Date: April 18th, 2016.
1
2
ANDREAS LEOPOLD KNUTSEN
as desired. (This will be generalized in Remark 2.4 below). In particular, if D(α) ⊆
Spec B, then D(α) = D(α), so that, by (a), we have Spec Aα ' Spec Bα , so that
Aα ' Bα by (the proof of) [Hart, II, Prop. 2.3].
(c) We have
∪i∈I D(αi ) = X − V (({αi | i ∈ I}))
by [Hart, II, Lemma 2.1(b)] and V (({αi | i ∈ I})) = ∅ if and only if ({αi | i ∈ I}) = A.
(d) ({αi | i ∈ I}) = A means that we can write
X
1A =
ti αi , for some ti ∈ A.
i∈I
But this must hold for finitely many i’s, so that (α1 , . . . , αr ) = A, which means that
Spec A = D(α1 ) ∪ · · · ∪ D(αr ). By (a), this implies that sp(X) is quasi-compact.
(e) One easily sees that Spec k[x1 , x2 , . . .] is not noetherian. However, if A is
noetherian and
V (a1 ) % V (a2 ) % · · ·
is a descending chain of closed subsets, then the corresponding chain
√
√
a1 ⊆ a2 · · ·
√
must eventually stabilize, as A is noetherian. Now the rest follows as V (ai ) = V ( ai ).
An almost immediate consequence of Lemma 1.1 is that any open subset U ⊂ X is
itself a scheme, with structure sheaf OU = OX |U , cf. [Hart, II, Exc. 2.2]. (Indeed, X
can be covered by open affine subschemes Spec A, each of which has a basis for the
topology consisting of open sets that are spectra of rings, whence also U is a union
of such spectra of rings.)
We next study morphisms between affine schemes.
Let X = Spec A and Y = Spec B be affine schemes and
Y
(f,f ] )
p
/
X
/
ϕ−1 p
be a morphism, corresponding to a ring homomorphism
ϕ
A
/
B,
as given in the proof of [Hart, II,Prop. 2.3] (so that ϕ is given by
A ' Γ(X, OX )
f ] (X)
/
Γ(Y, OY ) ' B,
(using [Hart, II, Prop. 2.2(c)]). In particular, we have that for any y ∈ Y , the
induced morphism on the stalks
(3)
OX,f (y)
fy]
/
OY,y
ADDITIONAL NOTES IN ALGEBRAIC GEOMETRY, II
3
is just the natural localization of ϕ:
Aϕ−1 py = Af (py )
(4)
ϕpy
/
Bpy ,
where py ∈ Spec B is the prime ideal corresponding to y.
It was noted in the proof of [Hart, II, Prop. 2.3] that
f −1 (V (a)) = V (ϕ(a)) for any ideal a ⊆ A.
(5)
Conversely, it is easy to check that
f (V (b)) = V (ϕ−1 (b)) for any ideal b ⊆ B.
(6)
The following properties of morphisms is for instance used in [Hart, II, Example
3.2.3]:
Lemma 1.2. ([Hart, II, Exc. 2.18(b,c)]) Let X = Spec A and Y = Spec B be affine
schemes and f , f ] and ϕ as in the beginning of the section. Then
(i) ϕ is injective if and only if f ] : OX → f∗ OY is injective. Moreover, f is
dominant in this case.
(ii) If ϕ is surjective, then f is a closed immersion 1.
Proof. (i) If ϕ is injective, then also (4) is injective for every py ∈ Spec B, whence
also (3), and therefore also f ] is injective.
Conversely, if f ] is injective, then also ϕ = f ] (X) is.
The fact that f is dominant in this case follows from (6):
(∗)
f (Y ) = f (Spec B) = f (V ((0))) = V (ϕ−1 ((0))) = V ((0)) = Spec A
where the injectivity of ϕ is used in (∗).
(ii) If ϕ is surjective, then B ∼
= A/ ker ϕ, and, as is well-known from commutative
algebra, there is a one-to-one correspondence (given by ϕ−1 ) between prime ideals in
B and prime ideals in A containing ker ϕ. This turns (6) into
(7)
f (V (b)) = V (ϕ−1 (b)) for any ideal b ⊆ B
and yields that
f (Y ) = f (Spec B) = f (V ((0))) = V (ker ϕ),
and f induces a bijection between Y = Spec B and the closed subset V (ker ϕ) ⊆ A.
It is continuous as f is, and it is bicontinuous since f maps closed sets into closed
sets by (7).
The surjectivity of f ] can be checked on stalks, where it follows since (4) and (3)
are the same map.
There are similar results concerning the schemes Proj S associated to graded rings,
as we will now see. We use the same notation as in [Hart, p. 76-77].
Proposition 1.3. ([Hart, II, Exc. 2.14(b-c) and Exc. 3.12(a)]) Let ϕ : S → T be a
graded homomorphism of graded rings. Let U := {p ∈ Proj T | ϕ(S+ ) 6⊆ p}.
1By
definition (see [Hart, Def. p. 85]) this means that f is a homeomorphism onto a closed subset
of X (necessarily equal to V (ker ϕ), as seen in the proof) and f ] is surjective.
4
ANDREAS LEOPOLD KNUTSEN
(a) U is an open subset of Proj T and ϕ determines a natural morphism f : U →
Proj S (given on the topological spaces by f (p) = ϕ−1 p).
(b) If ϕ is surjective, then U = Proj T and f : Proj T → Proj S is a closed
immersion.
(c) If ϕd : Sd → Td is an isomorphism for all d ≥ d0 , for some d0 ∈ Z, then
U = Proj T and f : Proj T → Proj S is an isomorphism.
Proof. (a) The ideal ϕ(S+ )T in T is homogeneous, as it is generated by the homogeneous elements ϕ(α) for homogeneous α ∈ S+ . Hence, as
(8)
Proj T − U = V (ϕ(S+ )T ),
the set U is an open subset of Proj T .
We note that if p ∈ U, then S+ 6⊆ ϕ−1 p, whence ϕ−1 p ∈ Proj S and there is a map
f : U → Proj S on the topological spaces by f (p) = ϕ−1 p.
If V (a) ⊆ Proj S is a closed set, for a homogeneous ideal a ⊂ S, then
f −1 V (a) = {p ∈ U | ϕ−1 p ⊇ a} = V (ϕ(a)T ),
whence f is continuous.
We have left to define the morphism of sheaves f ] : OProj S → f∗ OU . We proceed as
in the proof of [Hart, II, Prop. 2.3]. For each p ∈ U, we obtain a local homomorphism
of local rings ϕp : S(ϕ−1 p) → T(p) . For any open set V ⊆ Proj S we thus obtain a
homomorphism of rings
OProj S (V )
s
f ] (V )
/
/
OU (f −1 V )
t[p∈f −1 V ] ϕp ◦ s ◦ f,
such that ff] (p) = ϕp on the stalks. This defines f ] and proves that (f, f ] ) is a
morphism.
(b) Surjectivity of ϕ implies that ϕ(S+ ) = T+ , proving that U = Proj T . As T '
S/ ker ϕ, we have that f (Proj T ) = V (ker ϕ), which is closed in Proj S. Finally, the
surjectivity of f ] can be proved on the stalks, where we get the local homomorphisms
ϕϕ−1 p : S(ϕ−1 p) → T(p) , which are surjective by hypothesis. Hence f is a closed
immersion.
(c) Assume that ϕd : Sd → Td is an isomorphism for all d ≥ d0 . If p ∈ Proj T − U,
then ϕ(S+ ) ⊆ p, whence ⊕d≥d0 Td ⊂ p. Therefore (T+ )d0 ⊆ p. As p is prime, we must
have T+ ⊂ p, a contradiction. Thus U = Proj T .
We now prove that f is injective. If f (p) = f (q), then ϕ−1 p = ϕ−1 q, whence
p ∩ Td = q ∩ Td for all d ≥ d0 . Thus, if a ∈ p is a homogeneous element, then
an ∈ p ∩ Td for some n and d ≥ d0 . Hence an ∈ q, which implies a ∈ q, as q is
prime. This proves that p ⊆ q. By symmetry, we have p = q, which proves that f is
injective.
Now Proj S is covered by the basic open sets D+ (s) for s ∈ S+ . The morphism f
satisfies f −1 D+ (s) = D+ (ϕ(s)). By the construction in (a) and [Hart, II, Prop. 2.5],
ADDITIONAL NOTES IN ALGEBRAIC GEOMETRY, II
5
the morphism f|D+ (ϕ(s)) is simply the morphism
f|D+ (ϕ(s)) : Spec T(ϕ(s)) → Spec S(s)
induced by the localization homomorphisms
ϕs : S(s) → T(ϕ(s)) .
But the latter are isomorphisms by the hypotheses, whence so are also the morphisms
fD+ (s) . Thus f is an isomorphism on all members of an open cover. As f is injective,
it is an isomorphism.
Example 1.4. Part (c) in the previous proposition shows that the morphism f can
be an isomorphism even if ϕ is not, as opposed to the case of affine schemes, where
clearly ϕ is an isomomorphism if f is.
For a concrete such example, let T be the graded ring k[x, y] and S = T0 + T2 +
T3 + · · · . Let ϕ : S → T be the inclusion. Then ϕd is an isomorhism for all d ≥ 2,
but ϕ is not.
Example 1.5. Any homogeneous ideal I ⊂ S gives rise to a closed subscheme of
S. Indeed, let T = S/I. Then Proposition 1.3(b) yields a natural closed immersion
Proj T → Proj S. However, as opposed to the case of affine schemes, cf. [Hart, II,
Cor. 5.10], different ideals can give rise to the same subschemes. For instance, for
any integer d0 , the ideal I 0 := ⊕Id≥d0 determines the same closed subscheme as I, by
Proposition 1.3(c).
2. The residue field, and the distinguished open subsets and zeros of
a section
This section supplements [Hart, II, §2] and the beginning of [Hart, II, §3]. In
particular, the result proved in Lemma 2.3 below is used in the proof of [Hart, II,
Prop. 3.1].
Let X be a scheme and x ∈ X a point (closed or not). Let mx ⊂ Ox denote the
maximal ideal.
Definition 2.1. ([Hart, II, Exc. 2.7]) The residue field of x (on X) is the field
k(x) := Ox /mx .
Note that this only depends on the local ring Ox , so the residue field is a local
property of the point x, that is, it does not change whether we consider the point
x ∈ X, or whether we consider x ∈ U , a neighborhood. By the definition of scheme,
we may find an affine neighbourhood U = Spec A of x. Denoting by p ∈ Spec A the
point corresponding to x, we see from [Hart, II, Prop. 2.2] that
k(x) ' Ap /pAp .
(9)
Let s ∈ Γ(X, OX ) be a global section. Look at the natural morphisms
(10)
Γ(X, OX )
s
/
Ox
/ sx /
k(x) = Ox /mx
/
s(x).
6
ANDREAS LEOPOLD KNUTSEN
We call the images of s in Ox and k(x) the stalk sx of s at x and the evaluation s(x)
of s at x, respectively.
The fields k(x) may vary from point to point, as will be discussed in Section 5
below, but the following set is well defined:
Z(s) = {x ∈ X | s(x) = 0},
and its complement is the set
Xs := {x ∈ X | sx 6∈ mx ⊂ Ox }.
Lemma 2.3 below shows that Xs is open, equivalently that Z(s) is closed, a fact
that is used in the proof of [Hart, II, Prop. 3.1].
Definition 2.2. We often call the open sets Xs , for s ∈ Γ(X, OX ), distinguished
open sets. The closed set Z(s), the complement of Xs , is called the set of zeros of s.
Lemma 2.3. If U = Spec A ⊂ X is an affine subscheme and s|U = ρXU (s) ∈
Γ(U, OX |U ) = A is the restriction of s, then
Xs ∩ U = D(s|U ), whence Z(s) ∩ U = V ((s|U )).
Proof. For any x ∈ Spec A ⊂ X, denote by px the corresponding prime ideal. By
[Hart, II, Prop. 2.2] we have that Ox ' Apx and (10) factorizes as
Γ(X, OX )
s
/
/
Γ(U, OX |U ) = A
Ox ' Apx
/
k(x) = Ox /mx ' Apx /px Apx
/ sx / s|U /
s(x).
Therefore,
D(s|U ) = {p ∈ U | s|U 6∈ px } = {x ∈ U | s|U 6∈ mx } = Xs ∩ U.
Remark 2.4. In particular, if X = Spec A is affine and f ∈ Γ(X, OX ) = A (by
[Hart, II, Prop. 2.2]), then
Xf = D(f ) ⊆ Spec A,
and the lemma generalizes Lemma 1.1(b).
Example 2.5. Let X = A2k = Spec k[x1 , x2 ] be the affine plane over an algebraically
closed field k. Then Γ(X, OX ) = k[x1 , x2 ] by [Hart, II, Prop. 2.2]. Let x be a closed
point in A2k . Then x = (a, b) for some a, b ∈ k, equivalently, x corresponds to the
maximal ideal mx = (x1 − a, x2 − b). Then (10) looks like
(11)
Γ(X, OX ) = k[x1 , x2 ]
s = s(x1 , x2 ) /
/
Ox ' k[x1 , x2 ](x1 −a,x2 −b)
/ sx /
Ox /mx ' k
s(x) = s(a, b),
the evaluation of the polynomial s(x1 , x2 ) at the point (a, b) in the usual sense.
ADDITIONAL NOTES IN ALGEBRAIC GEOMETRY, II
7
The notions above behave nicely when it comes to morphisms, as the two next
lemmas show. (The first one plays an important role in the proof of Corollary 5.2
below, and the second is useful in [Hart, II, Exc. 3.2].)
Lemma 2.6. Let (f, f ] ) : X → Y be a morphism of schemes and x ∈ X a point.
Then k(x) is a field extension of k(f (x)).
In particular, if X is a scheme over k, then the residue field at any point of X is
a field extension of k.
Proof. By definition, the induced homomorphism on the local rings (or stalks)
fx] : OY,f (x) −→ OX,x
−1
is local, that is, fx] mx = mf (x) . (In particular, it is nonzero.) Hence, we can mod
out by the maximal ideals on both sides and obtain an injective homomorphism
k(f (x)) → k(x), proving that k(x) is a field extension of k(f (x)).
The last assertion follows setting Y = Spec k and using that the residue field of
the only point of Spec k is k itself.
Lemma 2.7. Let (f, f ] ) : X → Y be any morphism of schemes and σ ∈ Γ(Y, OY ).
Then
f −1 Yσ = Xf ] (Y )(σ)
(where f ] (Y ) : Γ(Y, OY ) → Γ(X, OX ) denotes the map induced on global sections).
Proof. By definition of a morphism of schemes, we have, for any x ∈ X, a commutative diagram
Γ(Y, OY )
OY,f (x)
f ] (Y )
fx]
/
Γ(X, OX )
/
OX,x ,
−1
such that fx] mx = mf (x) . Hence
f (x) ∈ Yσ ⇔ σf (x) 6∈ mf (x) ⇔ fx] (σf (x) ) 6∈ mx ⇔ (f ] (Y )(σ))x 6∈ mx ⇔ x ∈ Xf ] (Y )(σ)
and the result follows (we have used that fx] (σf (x) ) = (f ] (Y )(σ))x by the commutativity of the diagram).
3. Reduced and integral schemes
This section supplements [Hart, II, 3.0.1 and Prop. 3.1].
Proposition 3.1. ([Hart, II, Exc. 2.3]) Let X be a scheme. The following conditions
are equivalent:
(i) OX (U ) has no nilpotent elements for all open sets U ⊆ X;
(ii) OX,P has no nilpotent elements for all points P ∈ X.
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ANDREAS LEOPOLD KNUTSEN
Proof. Assume that there is a nonzero f ∈ OX,P such that f m = 0 for some m > 0.
Then f has a representative hU, gi, where U ⊆ X is open and g ∈ OX (U ), and g 6= 0,
g m = 0 in OX (U ), proving that there is a U such that OX (U ) has nilpotent elements.
This proves that (i) implies (ii).
Conversely, assume that there is an open set U ⊂ X and a nonzero g ∈ OX (U ) such
that g m = 0 for some m > 0. Localizing at any point P ∈ U , we obtain gP ∈ OX,P .
Now gP cannot be zero for all P , as otherwise, by the sheaf properties, g would be
zero also on U . Hence, there is a P ∈ U such that 0 6= gP ∈ OX,P and gPm = 0,
whence OX,P has nilpotent elements. This proves that (ii) implies (i).
If the equivalent conditions above are satisfied, then we say that X is reduced, cf.
[Hart, Def. p. 82]. It is easy to see that it is enough to check conditions (i) on any
base for the topology.
We also recall the definition of X being integral, cf. [Hart, Def. p. 82], as well as
[Hart, II, Prop. 3.1], which states that a scheme X is integral if and only if it is both
reduced and irreducible.
Example 3.2. ([Hart, II, Example 3.0.1]) If X = Spec A, then
(a) X is irreducible if and only if the nilradical nil A is prime;
(b) X is reduced if and only if nil A = 0;
(c) X integral if and only if A is an integral domain.
Proof. (a) X is reducible if and only if there are two nonempty open subsets of X
with empty intersection. These can be taken as basic open subsets, that is, we have
a, b ∈ A such that D(a), D(b) 6= ∅, D(a) ∩ D(b) = D(ab) = ∅
Recalling that
D(f ) = ∅ ⇔ f ∈ p for all p ∈ Spec A ⇔ f ∈ nil A,
we get
X is reducible ⇔ there exist a, b ∈ A such that a, b 6∈ nil A, ab ∈ nil A
⇔ nil A is not prime.
(b) By Proposition 3.1, X is reduced if and only if Ap has no nilpotents for all
p ∈ X = Spec A. The latter is equivalent to A having no nilpotents, by standard
algebra, which means precisely that nil A = 0.
(c) By [Hart, II, Prop. 3.1], X is integral if and only if it is both reduced and
irreducible, which by (a) and (b) occurs if and only if (0) is a prime ideal in A. The
latter is equivalent to A being an integral domain.
Example 3.3. (i) Let A = k[x, y]/(x2 ). Let x denote the residue class of x in
A. Then nil A = (x), which is prime but nonzero. Hence, Spec A is irreducible,
nonreduced, and nonintegral.
(ii) Let A = k[x, y]/(xy). Let x and y denote the residue class of x and y, respectively, in A. Then nil A = (0) is not prime, as x, y 6= 0 but x · y = 0. Hence, Spec A
is reduced, reducible, and nonintegral.
ADDITIONAL NOTES IN ALGEBRAIC GEOMETRY, II
9
4. Generic point and function field
This section supplements [Hart, II, Example 2.3.4 and beginning of §3].
Definition 4.1. ([Hart, II, Exc. 2.9]) Let X be a topological space and Z ⊂ X an
irreducible closed subset.
A point ξ ∈ Z is a generic point for Z if {ξ} = Z.
Note that a generic point has the property that it lies in every nonempty open
subset of Z. Therefore, for any sheaf F on X, and any point z ∈ Z we have natural
maps on stalks
(12)
Fz −→ Fξ .
Lemma 4.2. ([Hart, II, Exc. 2.9]) If X is a scheme, then every (nonempty) irreducible closed subset has a unique generic point.
Proof. Let Z ⊂ X be a (nonempty) irreducible closed subset. Choose any open,
affine V = Spec A ⊂ X such that V ∩ Z 6= ∅. Then V ∩ Z is open, irreducible and
dense in Z, by [Hart, I, 1.1.3]. At the same time, V ∩ Z is closed in V = Spec A,
so that V ∩ Z = V (a) for some ideal a in A, that is, V ∩ Z = Spec(A/a). This is
irreducible if and only if the nilradical of A/a is prime, by [Hart, II, Example 3.0.1]
or by Example 3.2 above. The nilradical is the
A/a, that
√ intersection of all primes in√
is, of all primes in A containing a, which is a, the radical of a. Hence a = p, a
prime, so that
√
V (a) = V ( a) = V (p)
the closure of {p} in Spec A. Letting x ∈ X be the point corresponding to p, we have
Z = V ∩ Z = {x},
proving the existence of a generic point.
If also Z = {x0 }, then x0 ∈ V by the property of a generic point, so that x = p0 ∈
Spec A. But then p0 ∈ V (p), whence p0 ⊇ p. Likewise, p ⊇ p0 , so that p = p0 , proving
the uniqueness.
In particular, if X is an integral scheme, then it is reduced and irreducible by
[Hart, II, Prop. 3.1], so it has a unique generic point ξ. For any x ∈ X, pick an open
affine neighborhood V = Spec A of x, and let p be the prime ideal corresponding to x.
Then Ox ' Ap by [Hart, II, Prop. 2.2] and since A = OX (V ) is an integral domain by
definition, the point ξ ∈ V ⊆ X must be the zero ideal. Hence Oξ ' A((0)) = K(A),
the quotient field of A, again by [Hart, II, Prop. 2.2] so that Oξ is a field. Also note
that we have a natural injective homomorphism
Ox ' Ap −→ A((0)) ' Oξ ,
which is nothing but (12).
We summarize this in the following result, which corresponds to the same facts for
varieties mentioned in [Hart, p. 16]
Lemma 4.3. ([Hart, II, Exc. 3.6]) If X is an integral scheme and ξ its generic point,
then for any point x ∈ X and any neighborhood U of x, the natural homomorphisms
OX (U ) −→ Ox −→ Oξ
10
ANDREAS LEOPOLD KNUTSEN
are both injective. Moreover, Oξ is a field, and for any open, affine U = Spec A ⊆ X,
we have that Oξ is isomorphic to the quotient field of A.
Proof. We only have left to prove the injectivity of the left map. For this, it suffices
to show the injectivity of the composed map to Oξ .
Assume therefore that s ∈ OX (U ) is mapped to zero in Oξ . Let V = Spec A be any
affine open subset of U . (Recall that U , being open in X, has a natural structure of
scheme, see the lines following Lemma 1.1.) Then ξ ∈ V , as ξ is a generic point. Since
there is a base for the topology on V = Spec A consisting of open subsets of the form
D(f ) for f ∈ A (see [Hart, p. 70-71] or Lemma 1.1(a)), we can find a neighborhood
of the form D(f ) ⊆ U ∩ V of ξ such that s|D(f ) = 0 in OX (D(f )) = Af (by [Hart, II,
Prop. 2.2(b)]: here we also use that OX |V = OV by definition of a scheme, so that
OX (D(f )) = OV (D(f )). Now A injects into Af , since A is an integral domain as X
is integral. This means that s|V = 0 in OX (V ) = A (again by [Hart, II, Prop. 2.2]).
Hence s = 0 by the sheaf properties.
Definition 4.4. ([Hart, II, Exc. 3.6]) Let X be an integral scheme and ξ its generic
point. The field Oξ (in Lemma 4.3) is called the function field of X and is denoted
by K(X). Its elements are called rational functions.
We say that a rational function f ∈ K(X) is regular at a point x ∈ X if f ∈ Ox ⊆
K(X) (by the right inclusion in Lemma 4.3).
Example 4.5. We have K(Ank ) ∼
= k(x1 , . . . , xn ), the field of rational functions in n
n
variables. As Pk can be covered by open affine subschemes isomorphic to Ank , we also
have K(Pnk ) ∼
= k(x1 , . . . , xn ) by Lemma 4.3.
5. The residue field of points in a scheme of finite type over a field
This section supplements [Hart, II, §3] by collecting some important result on the
residue field of points in a scheme of finite type over a field k (see [Hart, Def. p. 84]).
As above, let X be a scheme, x ∈ X a point (closed or not) and mx ⊂ Ox denote
the maximal ideal. We recall the definition of the residue field k(x) := Ox /mx from
Definition 2.1.
Proposition 5.1. If X is a scheme of finite type over k, then k(x) is a finitely
generated field extension of k. Moreover x ∈ X is a closed point if and only if k(x)
is a finite algebraic extension of k.
In particular, if k is algebraically closed, then x is a closed point if and only if
k(x) ' k.
Proof. If X is of finite type over a field k, then for any open affine U = Spec A ⊆ X,
we have that A is a finitely generated k-algebra, by [Hart, II, Exc. 3.3(c)]). We
therefore have that A ' k[x1 , . . . , xn ]/I for some ideal I ⊂ A. If x ∈ U , and
p ∈ Spec A is the corresponding prime ideal, then k(x) ' Ap /pAp (cf. (9)), so that
k(x) is a finitely generated field extension of k.
Now x is closed if and only if x is closed in any affine neighborhood U = Spec A.
This happens if and only if the corresponding point p ∈ Spec A is a maximal ideal.
If p is maximal, then k(x) ' A/p by (9), which is a finite algebraic extension of k,
as A is a finitely generated k-algebra, by Hilbert’s Nullstellensatz [A-M, Cor. 5.24].
ADDITIONAL NOTES IN ALGEBRAIC GEOMETRY, II
11
Conversely, if k(x) ⊇ k is finite and algebraic, then, as
k ⊆ A/p ⊆ (A/p)p ' Ap /pAp ' k(x)
(using (9) again), we see in particular that A/p is integral over k, whence a field by
[A-M, Prop. 5.7], so that p is maximal.
We collect a few corollaries of this result:
Corollary 5.2. Let f : X → Y be a morphism of schemes of finite type over k.
Then f maps closed points to closed points.
Proof. By definition of a morphism of schemes over k, we have a commutative diagram
f
/Y
X GG
w
w
GG
w
GG
ww
GG
ww
G#
w
{w
Spec k.
For any x ∈ X, this induces, by Lemma 2.6, a commutative diagram of field extensions
? _ k(f (x))
k(x) o
aBB
;
xx
BB
xx
BB
x
BB xxx
0P -x
k
If x ∈ X is closed, then k(x) is a finite algebraic extension of k, by Proposition 5.1.
The latter diagram implies that also k(f (x)) is, whence also f (x) ∈ Y is closed by
Proposition 5.1.
A consequence of this is a proof of the last part of [Hart, II, Prop. 2.6] left as an
exercise ([Hart, II, Exc. 2.15]):
Corollary 5.3. With notation as in [Hart, II, Prop. 2.6], the natural map
HomVar(k) (V, W ) → HomSch(k) (t(V ), t(W ))
is bijective.
Proof. Injectivity is immediate. Let us prove that any morphism of schemes f :
t(V ) → t(W ) over k is induced from a unique morphism of varieties V → W .
We recall from the proof of [Hart, II, Prop. 2.6] that
t(V ) = {(nonempty) closed irreducible subsets of V }
and that the natural map
αV
V
P
/
/
t(V )
{P }
induces a homeomorphism between V and the subspace t(V )cl of closed points of
t(V ). By [Hart, II, Example 3.2.1], the schemes t(V ) and t(W ) are schemes of finite
type over k. Hence, by Corollary 5.2, the morphism f induces a natural continuous
12
ANDREAS LEOPOLD KNUTSEN
map f cl : t(V )cl → t(W )cl , whence also a continuous map between the homeomorphic
−1
spaces αW
◦ f cl ◦ αV : V → W . To be precise, we have a commutative diagram:
(13)
t(V )
f
/
O
iV
t(W )
O
iW
?
t(V )
O
cl
f cl
∼
=
αV
V
/
?
t(W )cl
O
αW
/
∼
=
W
The unicity of V → W is clear, as it has to make the latter diagram commutative.
The obtained map is continuous. The fact that it takes regular functions on W
to regular functions on V follows since it is proved in [Hart, II, Prop. 2.6] that the
sheaves of regular functions on V and W are obtained by restricting the structure
sheaves on t(V ) (resp. t(W )) via the inclusion iV (resp. iW ) and the homeomorphism
αV (resp. αW ). Hence, it is a morphism, as desired.
Finally, the following consequence of Proposition 5.1 is used in the proof of [Hart,
II, Prop. 4.10]:
Corollary 5.4. ([Hart, II, Exc. 3.14]) Let X be a scheme of finite type over a field
k. Then the set of closed points in X is dense.
Proof. Assume, to get a contradiction, that
Y := {x ∈ X | x is closed} $ X.
Then we could find an open, affine subset Spec A ⊆ X − Y . Since X is of finite
type, A must be a finitely generated k-algebra [Hart, II, Exc. 3.3(c)]. Pick a closed
point x ∈ Spec A. Then k(x) (which depends only on a neighborhood of x) is a finite,
algebraic extension of k by Proposition 5.1. But then, again by the same proposition,
x is also closed in X, a contradiction, as x ∈ X − Y .
Remark 5.5. ([Hart, II, Exc. 3.14]) The same is not true for general schemes.
For instance, Spec Z(2) contains precisely one closed point, namely the ideal (2). In
general, look at Spec of any DVR (see for instance [Hart, II, Example 2.3.2]).
Example 5.6. Consider the affine plane A2k = Spec k[x, y] over an algebraically
closed field k. If p ∈ A2k is a closed point, then p = (a, b) for some a, b ∈ k, and the
corresponding prime ideal in Spec k[x, y] is the maximal ideal mp = (x − a, y − b).
Then
k(p) = k[x, y](x−a,y−b) /(x − a, y − b)k[x, y](x−a,y−b) ' k,
as anticipated by Proposition 5.1.
The prime ideals that are neither maximal nor the zero ideal, are those of height
one, whence principal (see e.g. [Hart, I, Prop. 1.12A]), that is, of the form p = (f ),
for an irreducible polynomial f ∈ k[x, y]. In this case, one easily sees that k(p) is
isomorphic to the function field over k in one variable, where p ∈ A2k denotes the
corresponding point.
ADDITIONAL NOTES IN ALGEBRAIC GEOMETRY, II
13
6. Fibers of a morphism
This section supplements [Hart, II, §3] by supplying some results concerning fibers
of a morphism (cf. [Hart, Def. p. 89]) that are left as exercises.
We start with the following result:
Lemma 6.1. ([Hart, II, Exc. 2.7]) Let K be any field and X a scheme. Then, giving
a morphism Spec K → X is equivalent to giving a point x ∈ X and an extension
k(x) ⊆ K.
In particular, for any x ∈ X, we have a natural morphism
Spec k(x) −→ X
(14)
Proof. Given a morphism f : Spec K → X, we obtain a point x = f ((0)) ∈ X. By
Lemma 2.6, the residue field k((0)) = K is a field extension of k(f ((0)) = k(x).
Conversely, if we are given a point x ∈ X and k(x) ⊆ K, we get an induced local
homomorphism, by composing:
Ox −→ k(x) := Ox /mx −→ K.
This induces a morphism of sheaves
f ] : OX −→ Sx ,
where Sx is the skyscraper sheaf located at x, which can be described as f∗ OSpec K ,
where f : {x} = Spec K −→ X is the inclusion (cf. [Hart, II, Exc. 1.17]). Therefore
f and f ] define a morphism Spec K → X.
The natural morphism (14) is used to define the fiber over a point of a morphism,
cf. [Hart, Def. p. 89]: If f : X −→ Y is a morphism of schemes and y ∈ Y is any
point, then we define the fiber of f over y to be Xy := X ×Y Spec k(y) given by the
morphisms f : X −→ Y and k(y) −→ Y from (14). That is, we have a commutative
diagram
X ×Y Spec k(y)
(15)
X
q
qqq
q
q
qqq
xqqq
NNN
NNN f
NNN
NNN
NN&
QQQ
QQQ
QQQ
QQQ
(
Spec k(y)
Y
m
mmm
mmm
m
m
mmm
mv mm
Therefore, Xy is a scheme over k(y). Furthermore it is the scheme-theoretic fiber
over the point y, that is, it gives the set
f −1 y := {x ∈ X | f (x) = y}
a structure of scheme, by the following result:
Lemma 6.2. ([Hart, II, Exc. 3.10]) The underlying topological space of Xy is homeomorphic to f −1 y.
14
ANDREAS LEOPOLD KNUTSEN
Proof. By taking affine coverings of X and Y we reduce to the affine case.
So let X = Spec B and Y = Spec A and ϕ : A −→ B be the ring homomorphism
inducing f , as in [Hart, II, Prop. 2.3]. This means that
Spec B
(16)
q
f
/
Spec A
/
ϕ−1 q
Let p ∈ Spec A be the prime ideal corresponding to y.
We have
k(y) = Ap /pAp
by (9) and
Xy = X ×Y Spec k(y) = Spec(B ⊗A Ap /pAp )
by Step 1 in the proof of [Hart, II, Thm. 3.3] (the affine case of the theorem).
Moreover, denoting by S is the multiplicative set ϕ(A − p), one can check that there
is a natural isomorphism
B ⊗A Ap /pAp ' S −1 (B/ϕ(p)B),
where we use the notation ϕ(p)B for the ideal generated by ϕ(p) in B (sometimes
called the extension of p by ϕ and denoted by pe , cf. [A-M]). We therefore have that
Xy ' Spec S −1 (B/ϕ(p)B).
Finally, recalling (16), we have
f −1 (y) = f −1 (p) =
=
=
=
{q ∈ Spec B | f (q) = p}
{q ∈ Spec B | ϕ−1 (q) = p}
{q ∈ Spec B | ϕ(p)B ⊆ q and q ∩ S = ∅}
Spec S −1 (B/ϕ(p)B),
where the last equality is a homeomorphism of topological spaces. This proves the
lemma.
Acknowledgements. I would like to thank Mirjam Solberg, Mauricio Godoy, Martin Stolz and Tommy Lundemo for useful comments on and discovering mistakes and
inaccuracies in previous versions of the notes.
References
[A-M] M. F. Atiyah, I. G. Macdonald, Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont.
[Hart] R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics, 52. Springer-Verlag,
New York-Heidelberg, 1977.