Lemma 20.8.1.slogan Let $X$ be a ringed space. Let $U' \subset U \subset X$ be open subspaces. For any injective $\mathcal{O}_ X$-module $\mathcal{I}$ the restriction mapping $\mathcal{I}(U) \to \mathcal{I}(U')$ is surjective.
20.8 Mayer-Vietoris
Below will construct the Čech-to-cohomology spectral sequence, see Lemma 20.11.5. A special case of that spectral sequence is the Mayer-Vietoris long exact sequence. Since it is such a basic, useful and easy to understand variant of the spectral sequence we treat it here separately.
Proof. Let $j : U \to X$ and $j' : U' \to X$ be the open immersions. Recall that $j_!\mathcal{O}_ U$ is the extension by zero of $\mathcal{O}_ U = \mathcal{O}_ X|_ U$, see Sheaves, Section 6.31. Since $j_!$ is a left adjoint to restriction we see that for any sheaf $\mathcal{F}$ of $\mathcal{O}_ X$-modules
\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(j_!\mathcal{O}_ U, \mathcal{F}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{O}_ U, \mathcal{F}|_ U) = \mathcal{F}(U) \]
see Sheaves, Lemma 6.31.8. Similarly, the sheaf $j'_!\mathcal{O}_{U'}$ represents the functor $\mathcal{F} \mapsto \mathcal{F}(U')$. Moreover there is an obvious canonical map of $\mathcal{O}_ X$-modules
\[ j'_!\mathcal{O}_{U'} \longrightarrow j_!\mathcal{O}_ U \]
which corresponds to the restriction mapping $\mathcal{F}(U) \to \mathcal{F}(U')$ via Yoneda's lemma (Categories, Lemma 4.3.5). By the description of the stalks of the sheaves $j'_!\mathcal{O}_{U'}$, $j_!\mathcal{O}_ U$ we see that the displayed map above is injective (see lemma cited above). Hence if $\mathcal{I}$ is an injective $\mathcal{O}_ X$-module, then the map
\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(j_!\mathcal{O}_ U, \mathcal{I}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(j'_!\mathcal{O}_{U'}, \mathcal{I}) \]
is surjective, see Homology, Lemma 12.27.2. Putting everything together we obtain the lemma. $\square$
Lemma 20.8.2 (Mayer-Vietoris). Let $X$ be a ringed space. Suppose that $X = U \cup V$ is a union of two open subsets. For every $\mathcal{O}_ X$-module $\mathcal{F}$ there exists a long exact cohomology sequence
\[ 0 \to H^0(X, \mathcal{F}) \to H^0(U, \mathcal{F}) \oplus H^0(V, \mathcal{F}) \to H^0(U \cap V, \mathcal{F}) \to H^1(X, \mathcal{F}) \to \ldots \]
This long exact sequence is functorial in $\mathcal{F}$.
Proof. The sheaf condition says that the kernel of $(1, -1) : \mathcal{F}(U) \oplus \mathcal{F}(V) \to \mathcal{F}(U \cap V)$ is equal to the image of $\mathcal{F}(X)$ by the first map for any abelian sheaf $\mathcal{F}$. Lemma 20.8.1 above implies that the map $(1, -1) : \mathcal{I}(U) \oplus \mathcal{I}(V) \to \mathcal{I}(U \cap V)$ is surjective whenever $\mathcal{I}$ is an injective $\mathcal{O}_ X$-module. Hence if $\mathcal{F} \to \mathcal{I}^\bullet $ is an injective resolution of $\mathcal{F}$, then we get a short exact sequence of complexes
\[ 0 \to \mathcal{I}^\bullet (X) \to \mathcal{I}^\bullet (U) \oplus \mathcal{I}^\bullet (V) \to \mathcal{I}^\bullet (U \cap V) \to 0. \]
Taking cohomology gives the result (use Homology, Lemma 12.13.12). We omit the proof of the functoriality of the sequence. $\square$
Lemma 20.8.3 (Relative Mayer-Vietoris). Let $f : X \to Y$ be a morphism of ringed spaces. Suppose that $X = U \cup V$ is a union of two open subsets. Denote $a = f|_ U : U \to Y$, $b = f|_ V : V \to Y$, and $c = f|_{U \cap V} : U \cap V \to Y$. For every $\mathcal{O}_ X$-module $\mathcal{F}$ there exists a long exact sequence
\[ 0 \to f_*\mathcal{F} \to a_*(\mathcal{F}|_ U) \oplus b_*(\mathcal{F}|_ V) \to c_*(\mathcal{F}|_{U \cap V}) \to R^1f_*\mathcal{F} \to \ldots \]
This long exact sequence is functorial in $\mathcal{F}$.
Proof. Let $\mathcal{F} \to \mathcal{I}^\bullet $ be an injective resolution of $\mathcal{F}$. We claim that we get a short exact sequence of complexes
\[ 0 \to f_*\mathcal{I}^\bullet \to a_*\mathcal{I}^\bullet |_ U \oplus b_*\mathcal{I}^\bullet |_ V \to c_*\mathcal{I}^\bullet |_{U \cap V} \to 0. \]
Namely, for any open $W \subset Y$, and for any $n \geq 0$ the corresponding sequence of groups of sections over $W$
\[ 0 \to \mathcal{I}^ n(f^{-1}(W)) \to \mathcal{I}^ n(U \cap f^{-1}(W)) \oplus \mathcal{I}^ n(V \cap f^{-1}(W)) \to \mathcal{I}^ n(U \cap V \cap f^{-1}(W)) \to 0 \]
was shown to be short exact in the proof of Lemma 20.8.2. The lemma follows by taking cohomology sheaves and using the fact that $\mathcal{I}^\bullet |_ U$ is an injective resolution of $\mathcal{F}|_ U$ and similarly for $\mathcal{I}^\bullet |_ V$, $\mathcal{I}^\bullet |_{U \cap V}$ see Lemma 20.7.1. $\square$
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