Download Algebraic Geometry I - Problem Set 3

Algebraic Geometry I - Problem Set 3
You may pick six of the following problems and make sure to include three of the first four!
Please write up solutions as legibly and clearly as you can, preferably in LaTeX.
1. Let U ⊆ X be an open in an affine variety X over a field k. Prove directly (using only
definitions) that a map f : U → Am
k is a morphism of ringed spaces if and only if each
coordinate function fi : U → k (i = 1, . . . , m) is a regular function. Prove that if Y ⊆ Am is
an algebraic set and f : U → Am is a morphism whose image is in Y , then the induced map
f : U → Y is a morphism.
2. Prove that the map f : A1 → X = V (Y 3 − X 2 ) ⊆ A2k given by f (t) = (t3 , t2 ) is not
an isomorphism by proving directly that the induced map on coordinate rings is not an
isomorphism of k-algebras. More generally, prove that there is no isomorphism between A1
and X.
3. Prove that A2 \ {(0, 0)} is not an affine variety (i.e., not isomorphic to any algebraic set
in any An ).
4. Recall that if X and Y are topological spaces, the product topology on the set X × Y is
the smallest topology for which the projections
πX : X × Y → X,
πY : X × Y → Y
are continuous. The sets U × V , with U open in X and V open in Y , form a basis for the
product topology.
Prove that the Zariski toplogy on A2 is different than the product topology on A1 × A1 .
5. Let F be a presheaf of rings on a topological space X. Recall that the stalk Fp at a
point p ∈ X is the set of equivalence classes of pairs (U, f ), with U open in X and p ∈ U ,
f ∈ F(U ), subject to the equivalence relation: (U, f ) ∼ (V, g) if there is an open W ⊆ U ∩ V
such that f|W = g|W in F(W ).
a. Prove that the stalk Fp is also a ring.
b. Prove that the stalk Fp is the direct limit of the direct system {F(U )} given by the
open sets U in X such that p ∈ U . (See Exercises 14-21 on page 33 in AtiyahMacDonald for definitions and properties of direct limits.)
6. Let (X, OX ) be an affine variety and let p be a point in X. Prove that the stalk of OX at
p is isomorphic (as a ring) to the localization of the ring A(X) at the maximal ideal mp of
regular functions on X that vanish at p.
7. Let R be any reduced finitely generated k-algebra. Let spm(R) denote the set of maximal
ideals in R. Give spm(R) a topology by taking the closed sets to be those of the form
V (I) = { m ∈ spm(R) | I ⊆ m },
1
2
for all ideals I of R. If f ∈ R, then define a function f : spm(R) → k by taking f (m) to
be the image of f under R → R/m ∼
= k. If U ⊆ spm(R) is an open set, then a function
f : U → k is called regular if it is locally of the form f (m) = p(m)
for some p, q ∈ R. This
q(m)
gives the structure of a ringed space on spm(R).
Prove that the ringed space spm(R) is an affine variety: if R ∼
= k[X1 , . . . , Xn ]/I for some
radical ideal I, then spm(R) is isomorphic (as a ringed space) to X = V (I) ⊆ An .
8. Let f : X → Y be a morphism of affine varieties over k and let f ∗ : A(Y ) → A(X) be
the corresponding map of k-algebras. Which of the following statements are true? Here mP
denotes the maximal ideal of regular functions that vanish at P .
a. If P ∈ X, Q ∈ Y then f (P ) = Q if and only if (f ∗ )−1 (mP ) = mQ .
b. f ∗ is injective if and only if f is surjective.
c. f ∗ is surjective if and only if f is injective.
d. f ∗ is an isomorphism if and only if f is an isomorphism.
9. Let k be an algebraically closed field of characteristic 6= 2. An algebraic set X ⊆ A2k
defined by a polynomial of degree 2 is called a conic. Show that any irreducible conic is
isomorphic either to V (Y − X 2 ) or V (XY − 1).
10. Let X be an irreducible algebraic set. Prove that for any open set U ⊆ X, we have:
\
OX (U ) =
OX,p ,
p∈U
where the intersection is taken in the function field K(X) (the fraction field of A(X)) and
OX (U ) is identified with a subring of K(X) as we did in class.