Mayer–Vietoris sequence for sheaf cohomology

Fix $q\ge 0$ and consider the sequence of abelian groups $0\to \mathrm{\Gamma }(X,I^q)\stackrel{\alpha }{\to }\mathrm{\Gamma }(U,I^q)\oplus \mathrm{\Gamma }(V,I^q)\stackrel{\beta }{\to }\mathrm{\Gamma }(U\cap V,I^q)\to 0$ with $\alpha (s):=(s{\mathrm{\mid }}_U,s{\mathrm{\mid }}_V)$ and $\beta (s_U,s_V):=s_V{\mathrm{\mid }}_{U\cap V}-s_U{\mathrm{\mid }}_{U\cap V}$. It is exact: $\alpha$ is injective because the restrictions to $U$ and to $V$ determine a section of the sheaf $I^q$ by [F2] and $U\cup V=X$; the kernel of $\beta$ consists of the pairs $(s_U,s_V)$ with equal restrictions, which glue to a section of $I^q$ over $X=U\cup V$ by [F1], so it equals the image of $\alpha$; and $\beta$ is surjective because $I^q$ is injective, hence flasque by [F3], so the restriction $\mathrm{\Gamma }(U,I^q)\to \mathrm{\Gamma }(U\cap V,I^q)$ is onto and a preimage $s_U$ of a given $t$ gives $\beta (-s_U,0)=0-(-t)=t$.

The maps $\alpha$ and $\beta$ commute with the differentials $d^q$ of $I^{\bullet }$, which are morphisms of sheaves, so degreewise they form a short exact sequence $0\to \mathrm{\Gamma }(X,I^{\bullet })\to \mathrm{\Gamma }(U,I^{\bullet })\oplus \mathrm{\Gamma }(V,I^{\bullet })\to \mathrm{\Gamma }(U\cap V,I^{\bullet })\to 0$ of cochain complexes; moreover the whole construction is natural in the sheaf, since the supplied resolutions are functorial. Applying [F5] gives the natural long exact sequence $\cdots \to H^q(\mathrm{\Gamma }(X,I^{\bullet }))\to H^q(\mathrm{\Gamma }(U,I^{\bullet }))\oplus H^q(\mathrm{\Gamma }(V,I^{\bullet }))\to H^q(\mathrm{\Gamma }(U\cap V,I^{\bullet }))\stackrel{\mathrm{\partial }}{\to }{H}^{q+1}(\mathrm{\Gamma }(X,I^{\bullet }))\to \cdots$ of the cohomology of these three complexes.

The middle and right complexes compute sheaf cohomology on the open subspaces: the restriction $I^{\bullet }{\mathrm{\mid }}_W$ of the injective resolution to an open $W\subseteq X$ is a resolution of $\mathcal{F}{\mathrm{\mid }}_W$ (restriction is exact) by objects which are injective by [F11], hence flasque by [F3] and acyclic on every open by [F4], and it is even acyclic for the functor $\mathrm{\Gamma }(W,-)$ there. The supplied datum on $\mathrm{A}\mathrm{b}(W)$ covers every abelian sheaf, including $\mathcal{F}{\mathrm{\mid }}_W$ and all cycles of this restricted resolution. Hence by the acyclic resolution theorem [F6], whose choice hypothesis is available through [F8], $H^q(\mathrm{\Gamma }(W,I^{\bullet }))\cong H^q(W,\mathcal{F}{\mathrm{\mid }}_W)$ for $W\in \{U,V,U\cap V\}$ and every $q\ge 0$; on the left $H^q(\mathrm{\Gamma }(X,I^{\bullet }))=H^q(X,\mathcal{F})$ by definition [F7]. Substituting these identifications into the sequence of [step 2.1] yields the exact sequence of the statement, whose degree-zero middle map is by construction the difference of restrictions displayed in [F9]; the identification in degree zero with global sections is [F10].

Collecting: [step 1.1] provides the degreewise short exact sequence, [step 2.1] the long exact cohomology sequence and its naturality, and [step 3.1] the identification of its terms with sheaf cohomology on $X$, $U$, $V$ and $U\cap V$; the left end $0\to H^0(X,\mathcal{F})$ is the injectivity statement inside [step 1.1] read in degree zero. This proves the theorem. ∎