9. Sheaf Cohomology Definition 9.1. Let X be a topological space

... are functors H i from the category of sheaves of abelian groups on X to the category of abelian groups such that (1) H 0 (X, F) = Γ(X, F). (2) Given a short exact sequence, 0 −→ F −→ G −→ H −→ 0, there are coboundary maps H i (X, H) −→ H i+1 (X, F). which can be strung together to get a long exact s ...

... are functors H i from the category of sheaves of abelian groups on X to the category of abelian groups such that (1) H 0 (X, F) = Γ(X, F). (2) Given a short exact sequence, 0 −→ F −→ G −→ H −→ 0, there are coboundary maps H i (X, H) −→ H i+1 (X, F). which can be strung together to get a long exact s ...

Exercise Sheet 4

... (a) Prove that the sheaf of normal vector fields on S n−1 ⊂ Rn is isomorphic to the sheaf of functions C ∞ (−, R). (b) Give an example of a differentiable submanifold of codimension 1 where this does not hold. *2. Let X be a topological space and j : U ,→ X the embedding of an open subset. (a) Prove ...

... (a) Prove that the sheaf of normal vector fields on S n−1 ⊂ Rn is isomorphic to the sheaf of functions C ∞ (−, R). (b) Give an example of a differentiable submanifold of codimension 1 where this does not hold. *2. Let X be a topological space and j : U ,→ X the embedding of an open subset. (a) Prove ...

Topology Qual Winter 2000

... define functors G and H by G(X,A)=Hp(X,A), H(X,A)=Hp-1(X,A). Show that the map * is a natural transformation of G to H. Define and give an example (with proof) of a contravariant functor. 2. State and prove the Kunneth theorem for topological spaces. 3. a) Let F be the closed orientable surface of ...

... define functors G and H by G(X,A)=Hp(X,A), H(X,A)=Hp-1(X,A). Show that the map * is a natural transformation of G to H. Define and give an example (with proof) of a contravariant functor. 2. State and prove the Kunneth theorem for topological spaces. 3. a) Let F be the closed orientable surface of ...

What Is...a Topos?, Volume 51, Number 9

... functor is the stalk functor E → Ex . But new phenomena occur. Deligne has constructed examples of toposes without points (he has also given criteria for the existence of “enough points”) [SGA 4 IV 7, VI 9]. Moreover, if x and y are points of a topos T , there may exist nontrivial morphisms (of fun ...

... functor is the stalk functor E → Ex . But new phenomena occur. Deligne has constructed examples of toposes without points (he has also given criteria for the existence of “enough points”) [SGA 4 IV 7, VI 9]. Moreover, if x and y are points of a topos T , there may exist nontrivial morphisms (of fun ...

Algebraic topology exam

... Answer eight questions, four from part I and four from part II. Give as much detail in your answers as you can. Part I 1. Prove the Zig-Zag lemma: let 0 C D E 0 be a short exact sequence of chain complexes with the above maps being f: C D, g : D E. Show that there is a long exact sequenc ...

... Answer eight questions, four from part I and four from part II. Give as much detail in your answers as you can. Part I 1. Prove the Zig-Zag lemma: let 0 C D E 0 be a short exact sequence of chain complexes with the above maps being f: C D, g : D E. Show that there is a long exact sequenc ...