By Akhil Mathew

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Now it is easy to see that to give a morphism Z → G for any sheaf G is the same as giving a global section of G. Similarly, to give a morphism ZU → G is the same as giving a section of G over U . So suppose G is injective. We will show that G(X) → G(U ) is surjective for any U , which evidently implies flabbiness. But 2If r had not been extended to a global section, this would have only been a section over U ∩ Ui . 44 4. COHOMOLOGY OF SHEAVES ZU → Z is injective, so by assumption any ZU → G can be extended to a Z → G.

COHOMOLOGY OF SHEAVES Recall that the functor F → F(X) = Γ(X, F) is a covariant additive functor from Sh(X) to Ab, the category of abelian groups. It is also leftexact by Proposition ??. Consequently, in view of the existence of enough injectives, we can define its right derived functors. 4. The derived functors of Γ(X, ·) are written H i (X, ·); for a sheaf F, the groups H i (X, F) are called the sheaf cohomology groups. They are defined for i ≥ 0. We shall briefly the definition. To compute H i (X, F), we consider an injective resolution 0 → F → I0 → I1 → .

14. Let C, C be two abelian categories and F : C → C a covariant left-exact functor. 3 Suppose that M ⊂ C is a collection of objects satisfying the following: (1) Any object A ∈ C admits an embedding into an object of M (2) A direct factor of an element of M belongs to M (3) If 0 → A → B → C → 0 is an exact sequence in C and A, B ∈ M, then C ∈ M too and 0 → F (A) → F (B) → F (C) → 0 is exact. Then the objects of M are F -acyclic. In other words, the derived functors Ri F (M ), i ≥ 1 vanish for M ∈ M.

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