Topics in Geometry - Homework 5 | MATH 6490, Assignments of Geometry

Download Topics in Geometry - Homework 5 | MATH 6490 and more Assignments Geometry in PDF only on Docsity! TOPICS IN GEOMETRY: SHEAF THEORY MATH 6490, SPRING 2009 HOMEWORK 5 For all the exercises below we work in the category of (pre)sheaves of abelian groups over a topological space X. Exercise 1. Show that a sequence · · · → F i−1 → F i → F i+1 → · · · of sheaves and morphisms is exact if and only if the corresponding sequence of stalks is exact (as a sequence of abelian groups) for every point of X. Exercise 2. Show that a morphism of sheaves is an isomorphism if and only if it is both injective and surjective. Exercise 3. Let φ : F → G be a morphism of sheaves. (1) Show that im(φ) ' F/ker(φ). (2) Show that coker(φ) ' G/im(φ). Exercise 4. Let F and G be sheaves on X. Show that the presheaf U 7→ F(U)⊕G(U) is a sheaf. (This is called the direct sum of F and G, and is denoted F ⊕ G.) Show that F ⊕ G plays the role of direct sum and of direct product of F and G in the category of sheaves of abelian groups on X. Exercise 5. (Extending a Sheaf by Zero.) Let X be a topological space, Z a closed subset of X, and U = X − Z the complementary open subset. Let i : Z → X and j : U → X be the inclusion maps. (1) Let G be a sheaf on Z, and let i∗G be its direct image sheaf on X. For x ∈ X, show that the stalk (i∗G)x is Gx if x ∈ Z and (i∗G)x = 0 if x /∈ Z. (This justifies the terminology that i∗G is obtained by extending G by 0 outside of Z.) (2) Let H be a sheaf on U . Let j!H be the sheaf on X associated to the presheaf: V 7→ H(V ) if V ⊂ U , and V 7→ 0 if V 6⊂ U . For x ∈ X, show that the stalk (j!H)x is Hx if x ∈ U and (j!H)x = 0 if x /∈ U . (This justifies the terminology that j!H is obtained by extending H by 0 outside of Z.) Explain why j!H is not the same as the direct image sheaf j∗H. (3) Let F be a sheaf on X. Show that one has an exact sequence of sheaves on X: 0 → j!(F|U ) → F → i∗(F|Z) → 0. ∗ ∗ ∗ ∗ ∗ ∗ ∗