Algebraic geometry and Ring theory

Yoshifumi Tsuchimoto

Note that a sequence

$$0 \to \mathcal A \to \mathcal B \to \mathcal C \to 0$$

of sheaves of abelian groups is exact if and only if it is exact stalkwise.

PROPOSITION 09.1   Let

$$0 \to \mathcal A \to \mathcal B \to \mathcal C \to 0$$

be an exact sequence of sheaves of abelian groups on a topological space $X$ . Then:

1. For any open subset $U$ of $X$ , the corresponding sequence
   $$0 \to \mathcal A(U) \to \mathcal B(U) \to \mathcal C(U)$$
   of sections is exact.

2. $$B(U) \to \mathcal C(U)$$
   may not be surjective in general.

In a language of category theory, the global section function

$$\Gamma(U,\bullet):$$   (Sheaf of abelian groups on $X$)$$\to$$   (Ab)

is a left exact functior. (But not exact.) To treat it, we employ derived functors.

LEMMA 09.2  

1. For any ring $A$ , if an $A$ -module $I$ is injective, then the associated sheaf $\tilde I$ on $\operatorname{Spec}A$ is injective.
2. The category ($A$ -modules) has enough injectives.
3. For any scheme $X$ with an affine open covering $X=\sum_j U_j$ , for any $\mathcal O_X$ -quasi coherent sheaf $\mathcal F$ on $X$ , we have:

   $\mathcal F$ : injective $\iff$ $\mathcal F_{U_j}$ injective for any $j$ .

DEFINITION 09.3  

$$H^i (X, \mathcal F) = R^i\Gamma(X,\mathcal F)$$

THEOREM 09.4 (Serre)   For any affine scheme $X=\operatorname{Spec}(A)$ and for any quasi coherent $\mathcal O_X$ -module $\mathcal F$ on $X$ , we have

$$H^i(X,\mathcal F)=0 \qquad ($$for $$\forall i>0)$$

PROPOSITION 09.5   Let $X$ be a scheme with an affine open covering $\mathfrak{U}=\{ U_j\}$ . Then for any quasi coherent $\mathcal O_X$ -module $\mathcal F$ on $X$ , The cohomology $H^i(X,\mathcal F)$ may be computed by usin the Cech cohomology with respect to $\mathfrak{U}$ .