Sheaf Theory in Geometry: Exercises from Math 6490, Spring 2009, Homework 10, Assignments of Geometry

Download Sheaf Theory in Geometry: Exercises from Math 6490, Spring 2009, Homework 10 and more Assignments Geometry in PDF only on Docsity! TOPICS IN GEOMETRY: SHEAF THEORY MATH 6490, SPRING 2009 HOMEWORK 10 In the following exercises X is a topological space, U = {Ui} is an open cover of X, and F is a sheaf of abelian groups on X. Exercise 1. Consider the map d : Cq(X,U ,F) → Cq+1(X,U ,F) defined as in the definition of the Čech complex. Show that d2 = 0. (So we are indeed talking about a complex.) Exercise 2. Let Cqalt(X,U ,F) be the alternating cochains in degree q. Show that (1) d(Cqalt(X,U ,F)) ⊂ C q+1 alt (X,U ,F). (That is the alternating Čech complex is a sub- complex of the Čech complex.) (2) The inclusion (C•alt(X,U ,F), d) ↪→ (C•(X,U ,F), d) induces an isomorphism in co- homology. Exercise 3. Let ∆n be the usual n-simplex: ∆n = {(x0, . . . , xn) ∈ Rn : xi ≥ 0, and ∑ i xi = 1}. Consider the open cover U = {U0, . . . , Un} of ∆n where Ui consists of all those points for which xi > 0. Show that Ȟq(∆n,U , Z) = { Z if q = 0, 0 if q > 0. Exercise 4. Note that Sn ≈ ∂∆n+1 ⊂ Rn+2. Let Vi = Ui ∩ Sn. Let V = {Vi}. Show that Ȟq(Sn,V, Z) = { Z if q = 0 or q = n, 0 if q /∈ {0, n}. Exercise 5. Let {Fi} be a collection of sheaves of abelian groups on X. Define a presheaf∏ iFi by U 7→ ∏ iFi(U). Show that this presheaf is in fact a sheaf, and is the categorical product of the sheaves {Fi}. Assuming that for each x ∈ X one has Fi,x = 0 for all but finitely many i, show that ( ∏ iFi)x = ∏ iFi,x. ∗ ∗ ∗ ∗ ∗ ∗ ∗