2.5. Dimension
77
5.8. Let OK be as in Exercise 5.3, and let K = Frac(OK ), k = OK /tOK be
the residue field of OK . Let us set A = K × k and let ϕ : OK → A
be the homomorphism induced by OK → K and OK → k. Show that
fϕ : Spec A → Spec OK is surjective and that dim OK > dim A. Also show
that A is a finitely generated OK -algebra (i.e., quotient of a polynomial
ring over OK ).
5.9. Let X be a scheme over a field k. Show that the points of X(k) are closed
in X. If X is an algebraic variety over k, then x ∈ X is closed if and only
if k(x) is algebraic over k.
5.10. Let k be an non-countable infinite field. Let X be an algebraic variety over
k with dim X ≥ 1, (Yn )n a sequence of closed subsets of X with dim Yn <
dim X. We want to show that ∪n Yn0 6= X 0 (where the superscript 0 stands
for closed points).
(a) Show the result for X = A1k and then for Am
k . there exists an a ∈ k
such that V (T1 − a) is not contained in any Yn
(b) By using Noether’s normalization lemma, show the result for an arbitrary affine variety. Deduce the general case from this.
(c) Let Z be a Noetherian scheme and let {Zi }i be any family of closed
subsets with dim Zi < dim Z. Show that ∪i Zi 6= Z (consider the
generic points of Z).
5.11. (Schemes of dimension 0)
(a) Let X be a scheme which is a (finite Q
or not) disjoint union of open
subschemes Xi . Show that OX (X) ' i OX (Xi ).
(b) Show that any scheme of finite cardinal and dimension 0 is affine.
(c) Let X = Spec A be a scheme of finite cardinal and dimension
0. Show that every point x ∈ X is open. Deduce from this that
A ' ⊕p∈Spec A Ap .
(d) Show that statement (c) is false if we do not suppose Spec A of dimension 0.
(e) (An example of infinite cardinality) Let k be a field, let I be an infinite
set, A = k[Ti ]i∈I /(Ti2 − Ti )i and X = Spec A.
(1) Denote by ti the image of Ti in A. Let λ = (ε)i∈I ∈ {0, 1}I . Show
that mλ := (ti − εi )i is a maximal ideal of A.
(2) Let R a k-algebra such that Spec R is connected and let ϕ : A →
R be a k-algebra homomorphism. Show that ϕ factorizes through
A → A/mλ → R for some λ ∈ {0, 1}I . Deduce from this that the
map θ : {0, 1}I → X, λ 7→ mλ is a bijection.
(3) Show that OX,x = k for all x ∈ X; dim X = 0, and that each
point of X is its own connected component.
(4) We endow {0, 1}I with the product topology (the coarsest one
which makes continuous the projection to any component). Let
78
2. General properties of schemes
I0 be a finite subset of I and let F be a subset of {0, 1}I0 . Show
that there exists an idempotent element f ∈ A such that D(f ) =
V (f − 1) = θ(F × {0, 1}I\I0 ). So θ is a homeomorphism, D(f ) is
open and closed, and no connected component of X is open.
5.12. Let k be a field. We will determine the affine open subsets of Ank and of
Pnk . See also Exercise 4.1.15.
(a) Show that the principal open subsets of Ank and of Pnk (not equal to
Pnk ) are affine.
(b) Let X = ∪i D(fi ) be a finite union of principal open subsets of Ank .
Show that OAnk (X) = k[T1 , . . . , Tn ]f , where f = gcd{fi }i . Show that
every affine open subset of Ank is principal.
(c) Show that every irreducible closed subset of Pnk of dimension n − 1 is
principal.
(d) Let X be an affine open subset of Pnk . Show that the irreducible
components of Pnk \X are of dimension n−1, and that X is a principal
open subset of Pnk .
5.13. Let k be a field. A function field in n variables over k is a field K that
is finitely generated over k (i.e., there exist f1 , . . . , fr ∈ K such that
K = k(f1 , . . . , fr )), of transcendence degree trdegk K = n.
(a) Show that an extension K of k is a function field in n variables over
k if and only if K = K(X), where X is an integral algebraic variety
over k, of dimension n. If this is the case, we can take X projective.
(b) Let L be a subextension of K. Show that L is a function field in
≤ n variables. In particular, the algebraic closure of k in K is a finite
extension of k.
5.14. Let L/K be a finite field extension. Let x ∈ L. Then the multiplication
by x is an endomorphism of L as K-vector space. We let NormL/K (x)
denote the determinant of this endomorphism. We also call it the norm
of x over K.
(a) Show that NormL/K is a multiplicative map from L to K.
(b) Let A ⊆ B be rings such that K = Frac(A), L = Frac(B), and that
B is integral over A. Show that for any b ∈ B, NormL/K (b) is integral
over A.
(c) Let us moreover suppose that A is a polynomial ring over a field k.
Show that NormL/K (B) ⊆ A.
5.15. (Intersection in projective varieties) Let X = Proj B be a projective variety over a field k.
(a) Let f ∈ B+ be a homogeneous element. Let p be a prime ideal of
B, minimal among those containing f . Show that p is homogeneous,
contained in B+ , and that ht(p) ≤ 1. Show that if V+ (f ) = ∅, then
B+ = p and dim X ≤ 0.
2.5. Dimension
79
(b) Let Y be a closed subvariety of X, of dimension r. Show that for
any sequence f1 , . . . , fr of homogeneous elements of B+ , we have
V+ (f1 , . . . , fr ) ∩ Y 6= ∅. In particular, if a projective space Pnk , n ≥ 1,
a hypersurface intersects any closed subvariety of dimension ≥ 1.
5.16. We construct an algebraic variety X and a regular function f ∈ OX (X)
such that (f OX )(X) 6= f OX (X) (cf. Exercise 3.4). Let k be a field of
characteristic different from 2. Let Z be the subvariety V+ (x((x+y)y−z 2 ))
of P2k , let X be the complementary in Z of the point (0, 0, 1).
(a) Show that X is reduced with two irreducible components Γ1 ' A1k ,
Γ2 ' P1k and Γ12 := Γ1 ∩ Γ2 consists in two distinct points.
(b) For any open subset U of X, show that we have a canonical exact
sequence
θ
0 → OX (U ) → OΓ1 (Γ1 ∩ U ) ⊕ OΓ2 (Γ2 ∩ U ) −
→ ⊕p∈Γ12 ∩U k(p) → 0
where θ(a, b) = (a(p) − b(p))p∈Γ12 ∩U . Let t = y/z be a parameter
of Γ1 identified to the affine line. Then Γ12 = {t = ±1}. Show that
OX (X) = k + (t2 − 1)k[t].
(c) Let f = t2 −1 ∈ OX (X). Show that (f OX )(X) = (t2 −1)k[t] contains
strictly f OX (X).