508 Spring 2012 – HOMEWORKS

Problem 1. Let Y, Z be non-singular varieties over k, ϕ : Y → Z a morphism. Prove that if char k = 0 and ϕ is surjective, then there exists a non-empty open set U ⊆ Y such that T_{P} ϕ is also surjective for all P ∈ U. (difficulty 4) Problem 2. Let S_{1} be the vector space of linear polynomials in the variables x_{0}, …, x_{n} and P ∈ Y ⊆ ℙ^{n} a point on the projective variety Y. Let ϕ_{P} : S_{1} → 𝒪^{P,Y} /.𝔪_{P,Y}^{2} . be the natural map (defined in class). Let B_{P} = ker ϕ_{P} and B = {(P, H)|P ∈ Y, H = Z(f) such that f ∈ B_{P} }. Prove that B is closed and irreducible. (difficulty 3)

Problem 3. Let 0 → ℱ ^{}→ 𝒢 ^{}→ ℋ ^{}→ 0 be a short exact sequence of complexes of objects of an abelian category. Show that for all i ∈ ℤ there exist a natural morphism δ^{i} : h^{i}(ℋ ^{}) → h^{i+1}(ℱ ^{}) such that the induced sequence ⋯ → h^{i}(ℱ ^{}) → h^{i}(𝒢 ^{}) → h^{i}(ℋ ^{}) → h^{i+1}(ℱ ^{}) → … is exact. (difficulty 2)

Problem 4. Cohomology may be computed using acyclic resolutions. I.e., if 0 → 𝒢 → ℱ^{0} → ℱ^{1} → … is exact, and ℱ^{i} is acyclic for a left-exact functor F for every i ∈ ℕ, then R^{i}F(𝒢) ≃ h^{i}(F(ℱ ^{}). (difficulty 3)

Problem 5. Let X be a topological space, ℱ a sheaf of abelian groups on X, j : U ↪ X and open set and i : Z := X \ U ↪ X its complement. Let ℱ_{U} := j_{!} (ℱ |_{U}) and ℱ_{Z} := i_{∗} (ℱ |_{Z}) . Prove that there exist natural morphisms such that the following is a short exact sequence of sheaves: 0 → ℱ_{U} → ℱ → ℱ_{Z} → 0. (difficulty 3)

Problem 6. Let X be a variety and 0 → ℱ → 𝒢 → ℋ → 0 a short exact sequence of 𝒪_{X}-modules. Prove that if ℱ and 𝒢 are flasque, then so is ℋ . (difficulty 2)

Problem 7. Let X be a variety and 0 → ℱ → 𝒢 → ℋ → 0 a short exact sequence of 𝒪_{X}-modules. Prove that if ℱ is flasque, then Γ(X, 𝒢) → Γ(X, ℋ ) is surjective. (difficulty 2)

Problem 8. Let Y be an affine variety and let A = A(Y) be its affine coordinate ring. Let ϕ : X → Y be a morphism from an arbitrary variety. Prove that there exists a coherent 𝒪_{X}-module, Ω_{X/Y}, such that for any open affine subvariety U ⊆ X, for which B = A(U) is the affine coordinate ring of U, Γ(U, Ω_{X/Y}) ≃ Ω_{B/A}. [Hint: use the condition you need to prove to define the sheaf locally and prove that the local definitions agree on the overlaps.] Problem 9. Let Y be an arbitrary variety and ϕ : X → Y a morphism from another variety. Prove that there exists a coherent 𝒪_{X}-module, Ω_{X/Y}, such that for any open affine subvariety V ⊆ Y, we have Ω_{X/Y} |_{ϕ^{−1}V} ≃ Ω_{ϕ^{−1}V/V}, where the latter sheaf is the one defined in the previous exercise. [Hint: use the condition you need to prove to define the sheaf locally and prove that the local definitions agree on the overlaps.] Problem 10. Let ϕ : X → Y be a morphism of varieties. Prove that then there exists an exact sequence, ϕ^{∗}Ω_{Y} → Ω_{X} → Ω_{X/Y} → 0. Problem 11. Let ϕ : X → Y be a morphism of varieties and y ∈ Y a point. Consider the ideal sheaf ℐ_{y} := 𝔪_{y} · 𝒪_{X} ⊆ 𝒪_{X}. Assume that ℐ_{y} is a radical ideal (i.e., for any open subset U ⊆ X, ℐ (U) ⊂ 𝒪_{X}(U) is a radical ideal). Let X_{y} ⊆ X be the vanishing set of the ideal ℐ_{y}, in other words let X_{y} := supp 𝒪_{X}/ℐ_{y}. Prove that then Ω_{X/Y} ⊗ 𝒪_{X_{y}} ≃ Ω_{X_{y}}.