PROBLEMS, MATH 214B

1. Let F be a presheaf on X. Denote by T ot(F) the disjoint union of the stalks F_{x} for all x ∈ X and by p : T ot(F) → X the natural projection. Any section a ∈ F(U) over an open subset U ⊂ X and every point x ∈ X define an element a_{x} ∈ F_{x} which then can be viewed as an element of T ot(F) with p(a_{x}) = x. Define topology on T ot(F) with the basis given by the subsets {a_{x}, x ∈ U} ⊂ T ot(F) for all open U ⊂ X and all sections a ∈ F(U). A section of p over an open set U ⊂ X is a continuous map s : U → T ot(F) such that p(s(x)) = x for every x ∈ U. Denote by F̃ the sheaf of sections of p. The sheaf F̃ is called the sheaf associated to the presheaf F.

a) Prove that there is a natural morphism of presheaves i : F → F̃.

b) Prove that F is a sheaf iff i is an isomorphism.

c) Prove that for every morphism j : F → G to a sheaf G there exists a unique morphism j̃ : F̃ → G such that j̃ ∘ i = j.

2. Let A be a set, X a topological space. Let F be the constant presheaf associated with A. Consider the presheaf G with G(U) being the set of continuous maps U → A (we view A as a discrete topological space).

a) Prove that G is a sheaf.

b) Prove that the natural morphism i : F → G is isomorphic to i : F → F̃ considered in 1a).

3. Describe Spec ℤ[X].

4. Describe (Spec ℂ) ×_{Spec ℝ} (Spec ℂ).

5. Find an example of a commutative ring A that is not Noetherian with Noetherian topological space Spec A.

6. Prove that Spec A is not connected if and only if A has a nontrivial idempotent (an element a ≠ 0, 1 such that a^{2} = a).

7. Let A be a commutative ring, f ∈ A. Prove that the set {P ∈ Spec A such that f(P) = 0} is closed in Spec A.

8. Give an example of a commutative ring A and an open set U ⊂ Spec A that is not principal.

9. Prove that the functor Rings^{op} → T opSpaces, A ↦ Spec A is neither full nor faithful.

10. Let X be a scheme, f ∈ O_{X}(X).

a) Prove that the set X_{f} = {x ∈ X such that f(x) ≠ 0} is open in U.

b) Prove that the restriction of f ∈ O_{X}(X) on the set X_{f} is invertible.

c) Let x ∈ X, P_{x} = {f ∈ O_{X}(X) such that f(x) = 0}. Show that P_{x} is a prime ideal in A = O_{X}(X) and the map θ : X → Spec A, x ↦ P_{x} is continuous. d) Prove that θ^{−1}(D(f)) = X_{f}.

e) Show that the map θ gives rise to a morphism of schemes X → Spec A.

11. Let X be a quasi-projective variety such that the scheme X̃ is affine. Prove that the variety X is affine.

12. Let X be a discrete topological space, k a field. For every subset U ⊂ X let F(U) be the ring of all maps U → k. Prove that F is a sheaf of rings with respect to obvious restriction maps and the ringed space (X, F) is a scheme. For which X is this scheme affine?

13. Let K be the rational function field over a field k in infinitely many variables X_{1}, X_{2}, …, X_{n}, . . .. For every i ≥ 1, let v_{i} be the X_{i}-adic discrete valuation on K, that is v_{i}(f) = n for a nonzero function f ∈ K if f can be written in the form f = (X_{i})^{n} · \frac{g}{h} with polynomials g and h not divisible by X_{i}. For every nonzero function f ∈ K we assign a sequence of integers a(f) = (a_{1}, a_{2}, …) as follows. Let a_{1} = v_{1}(f), f_{2} = (f/X_{1}^{a_{1}})_{X_{1}=0}. Set a_{2} = v_{2}(f_{2}), f_{3} = (f_{2}/X_{2}^{a_{2}})_{X_{2}=0} and a_{3} = v_{3}(f_{3}) and so on.

a) We order the set of sequences a = (a_{1}, a_{2}, …) lexicographically, i.e., a ≥ b if for the smallest index k such that a_{k} ≠ b_{k} we have a_{k} ≥ b_{k}. Prove that the set

A = {0} ∪ {f ∈ K \ {0} such that a(f) ≥ 0}

is a local ring with the maximal ideal Q generated by all the X_{i}. b)

For every i ≥ 1, let P_{i} be the union of {0} and the set of all nonzero f ∈ A such that a(f) > (0, …, 0, b_{i+1}, b_{i+2}, …) for all b_{i+1}, b_{i+2}, . . .. Prove that P_{i} is a prime ideal in A and

P_{1} ⊂ P_{2} ⊂ ⋯ ⊂ P_{n} ⊂ ⋯ ⊂ Q.

c) Prove that

Spec A = {0, Q, P_{1}, P_{2}, …, P_{n}, … }

and describe the Zariski topology on Spec A.

d) Prove that the scheme Spec A \ {Q} has no closed points.

14. Let f : X → Y be a morphism of schemes, U ⊂ Y an open subset. Assume that f(X) ⊂ U. Prove that f factors as the composition of a morphism of schemes X → U and the inclusion U ↪ Y.

15. Let X be a scheme. Consider the presheaf F of rings on the topological space X defined by

F(U) = O_{X}(U)/Nil(O_{X}(U))

for every open U ⊂ X. Let O_{X}^{red} be the sheaf of rings associated to the presheaf F (see Ex. 1).

a) Prove that X^{red} := (X, O_{X}^{red}) is a reduced scheme.

b) Construct a canonical morphism of schemes i : X^{red} → X.

c) Prove that every morphism of schemes Y → X with Y reduced factors into the composition of a morphism Y → X^{red} with i.

PROBLEMS, MATH 214B 3

16. Let X be a scheme and let f : X → Spec ℤ be the canonical morphism. Prove that for every x ∈ X, f(x) = pℤ, where p is the characteristic of the residue field k(x).

17. Prove that any two quasi-projective varieties X and Y over F there is a natural isomorphism of schemes Z̃ ≃ X̃ ×_{Spec F} Ỹ, where Z = X × Y.

18. Let X be a separated scheme over an affine scheme. Let U and V be open affine subschemes of X. Prove that U ∩ V is also affine.

19. Let S = ∐_{i≥0} S_{i} be a graded commutative ring. Show that there is a natural morphism of schemes Proj S → Spec S_{0}.

20. Let A = F [t_{0}, t_{1}, … t_{n}] be the polynomial algebra. Let I ⊂ A be a homogeneous ideal. The saturation of I is defined as Ī = {a ∈ A | for every i = 0, … n there exists an m with at_{i}^{m} ∈ I}. Prove that Ī is a homogeneous ideal of A.

21. Let A = F [t_{0}, t_{1}, … t_{n}] be the polynomial algebra and let I and J be two homogeneous ideal of A. Prove that I and J define the same closed subscheme of Proj A if and only if they have the same saturations.