By Akhil Mathew

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Suppose b ⊂ B. It is less easy to describe explicitly the image fa (V (b) ⊂ SpecA. When B is the quotient A/a for some ideal, then SpecB is in bijection with V (a). In the general case, we have to use the closure. Remark. Let S ⊂ SpecA be a subset. Its closure S is then V (a) for a the intersection of the prime ideals in S. This is immediate from the definition of the closure as the smallest closed set containing a given set. 10. We have fa (V (b) = V (f −1 (b)) ⊂ SpecA. Proof. First, we show the inclusion ⊂.

39 40 4. 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 → .

So let x ∈ X. We will define a map h : C n (A, F)x → C n−1 (A, F)x . in F(V ∩Ui1 ∩· · ·∩Uin ). Suppose i1 . . in−1 ∈ I. in−1 F(V ∩ Ui1 ∩ · · · ∩ Uin−1 ). To do this, pick some i such that x ∈ Ui and let V = Ui ∩ V . in−1 ∈ F(V ∩ Ui1 ∩ . . Uin−1 ). Let us define the map h to be the map sending α → β as above. Then I claim that h is the claimed homotopy. in−1 ∈ Fx . = k=0 (This is a slight abuse of notation; (dhα)x is the family of the above as i1 , . . in = k=0 When these are added, the germ of α is the result; it follows that dh + hd is the identity.