Homework 7 - Topics in Geometry - Spring 2009 | MATH 6490, Assignments of Geometry

Download Homework 7 - Topics in Geometry - Spring 2009 | MATH 6490 and more Assignments Geometry in PDF only on Docsity! TOPICS IN GEOMETRY: SHEAF THEORY MATH 6490, SPRING 2009 HOMEWORK 7 Exercise 1. Let 0 → F → G → H→ 0 be a short exact sequence of sheaves. Assume that F is a flabby sheaf. For any open set U of X show that the sequence 0 → F(U) → G(U) → H(U) → 0 of sections over U is exact. Exercise 2. Let 0 → F → G → H → 0 be a short exact sequence of sheaves. If F and G are flabby sheaves then show that H is also a flabby sheaf. Exercise 3. Let F be a sheaf on a topological space X. Let s ∈ F(X). For x ∈ X, let s(x) denote the germ at x determined by s. Define the support of s as Supp(s) = {x ∈ X : s(x) 6= 0}. Show that Supp(s) is a closed subset of X. Give an example to show that Supp(s) is not in general open. Define the support of the sheaf F as Supp(F) = {x ∈ X : Fx 6= 0}. Show by an example that Supp(F) need not be closed. Exercise 4. Let R be a sheaf of commutative rings (with identities) on a topological space X. Assume that R has following property: For any open cover {Ui} of X there exists a partition of unity subordinate to that cover, meaning that there exists global sections hi ∈ R(X) such that: (i) Supp(hi) ⊂ Ui; (ii) for any x ∈ X there are only finitely many indices i for which hi(x) 6= 0; and (iii) 1 = ∑ i hi in R(X). Now let F be a sheaf of R-modules over X. This means that for each open set U , F(U) is an R(U)-module, and for V ⊂ U the restriction map F(U) → F(V ) is an R(V )-module homomorphism. Show that F is an acyclic sheaf. (Hint: See Exercise 16 in Harder’s book.) ∗ ∗ ∗ ∗ ∗ ∗ ∗