Download Algebraic Geometry I - Problem Set 4

Algebraic Geometry I - Problem Set 4
Please pick eight of the following problems. Please write up solutions as legibly and clearly
as you can, preferably in LaTeX.
All ringed spaces in this problem set are assumed to have the structure sheaf a sheaf of
k-valued functions.
Problems on morphisms of ringed spaces/prevarieties:
1. Let (X, OX ) and (Y, OY S
) be ringed spaces.
S Let f : X → Y be a map of sets. Assume
there are open covers X = i Ui and Y = i Vi such that f (Ui ) ⊆ Vi for all i. Prove that
f is a morphism if and only if f|Ui : Ui → Vi is a morphism for all i. Prove that f is an
isomorphism if and only if f|f −1 (Vi ) : f −1 (Vi ) → Vi is an isomorphism for all i.
2. Let (X, OX ) be any ringed space. A map f : Pn → X is a morphism if and only if
f ◦ π : An+1 \ {0} → X is a morphism.
3. Prove that any morphism Pn → A1 is constant (i.e., OPn (Pn ) = k).
4. Let X be the prevariety that is the “line with the origin doubled”. Find OX (X). Prove
that X is not affine by proving that the diagonal ∆X is not closed in X × X.
5. (Projection from a point.) Prove that the map Pn \ {[1, 0, . . . , 0]} → Pn−1 , given by
[z0 , z1 , . . . , zn ] 7→ [z1 , . . . , zn ] is a morphism of prevarieties.
6. Assume k is a field of characteristic 6= 2. A conic in P2 is the vanishing locus of a
homogeneous polynomial of degree 2. Let C be the conic given by x2 + y 2 = z 2 .
(a) Show that P2 \ {[0, 1, 1]} → P1 given by [x, y, z] 7→ [x, y − z] is a morphism. (This is
the projection from [0, 1, 1] - why?).
(b) By (a), the map π 0 : C \ {[0, 1, 1]} → P1 given by [x, y, z] 7→ [x, y − z] is also morphism (composition of the inclusion morphism C \ {[0, 1, 1]} → P2 \ {[0, 1, 1]} and the
projection from (a)). Describe a morphism π : C → P1 which extends π 0 .
(c) Prove that π is an isomorphism by finding a morphism σ : P1 → C and showing that
π ◦ σ and σ ◦ π are identity maps (on P1 and on C respectively).
7. Use the previous exercise to find all solutions to x2 + y 2 = z 2 , where x, y, z lie in a field
k of characteristic 6= 2 (not necessarily algebraically closed, for. ex., Q). No justification
required!
Problems on prevarieties/ringed spaces
8. Let X be a prevariety such that for each pair of points x, y ∈ X, there is an open affine
U ⊆ X containing both x and y. (Hint: Use Problem 12).
(a) Prove that X is separated.
1
2
(b) Prove that Pn has this property.
9. Prove that a prevariety is a Noetherian topological space.
10. (Subspaces of ringed spaces.) Let (X, OX ) be a ringed space. Let Y be an arbitrary
subset of X. Prove that Y inherits a structure of a ringed space when considering Y with
the induced topology and defining OY by the following property: if U is an open set in Y ,
a map f : U → k is in OY (U ) if for every y ∈ U , there is an open set U 0 ⊆ X, y ∈ U 0 , and
a function F : U → k in OX (U 0 ), such that f (x) = F (x) for all x ∈ U ∩ U 0 . Prove also the
following:
(a) The inclusion map i : Y → X is a morphism.
(b) If (Z, OZ ) is another ringed space, and f : Z → Y is a map of sets, then f is a
morphism if and only if i ◦ f : Z → X is a morphism.
(c) The ringed space structure on Y is determined uniquely by (a) and (b).
(d) If Z ⊆ Y ⊆ X, then Z inherits the same structure from X and Y .
Example: If X ⊆ An is an algebraic set, then X inherits its structure from An . Moreover,
if Y ⊆ X ⊆ An are algebraic sets, then Y inherits its structure from X.
Problems on products
11. (Products of ringed spaces.) Let (X, OX ) and (Y, OY ) be ringed spaces. A product
of X and Y is a ringed space X × Y , together with morphisms πX : X × Y → X and
πY : X × Y → Y , such that for any ringed space (Z, OZ ) and morphisms f : Z → X
and g : Z → Y , there is a unique morphism h : Z → X × Y such that f = πX ◦ h and
g = πY ◦ h. Prove if P with morphisms πX : P → X and πY : P → Y , and P 0 with
0
: P 0 → X and πY0 : P 0 → Y are both products of X and Y , then there is a
morphisms πX
0
unique isomorphism h : P → P 0 such that πX = πX
◦ h and πY = πY0 ◦ h.
12. (Products of prevarieties.) Sketch a proof for the following fact: Let X and Y be
prevarieties with affine open covers X = U1 ∪ . . . ∪ Ur and Y = V1 ∪ . . . ∪ Vs . Then the product
of X and Y is the prevariety obtained by glueing the affine opens {Ui × Vj }. (Explain how
you would glue these opens, what should the projections maps be, and why does this satisfy
the universal property of products).
Topology
13. A topological space is Hausdorff if for every two distinct points x and y, there are disjoint
open sets U and V such that x ∈ U and y ∈ V . Prove that the following are euqivalent for a
topological space X:
(a) X is Hausdorff.
(b) The diagonal ∆X = {(x, x)|x ∈ X} is closed in X × X (where X × X has the product
topology, i.e., a base of open sets are sets of the form U × V , for U, V open sets in X).
(c) For any topological space Y and any two continuous maps f : Y → X, g : Y → X
the locus {y ∈ Y |f (y) = g(y)} is closed in Y .