Mayer-Vietoris Sequence for Cohomology with Supports I am working on problem III.2.4 in Hartshorne, and I have been quite stuck in showing the existence of a Mayer-Vietoris sequence for cohomology with supports. To be more precise, I have $Y_1,Y_2\subseteq X$ closed subsets and I want to show a long exact sequence 
$$ \cdots \to H^i_{Y_1\cap Y_2}(X,\mathscr{F})\to H^i_{Y_1}(X,\mathscr{F})\oplus H^i_{Y_2}(X,\mathscr{F})\to  H^i_{Y_1\cup Y_2}(X,\mathscr{F})\to\cdots.$$
I intend to do this by showing that there is an exact sequence 
$$ 0\to \Gamma_{Y_1\cap Y_2}(X,\mathscr{F})\to \Gamma_{Y_1}(X,\mathscr{F})\oplus\Gamma_{Y_2}(X,\mathscr{F})\to \Gamma_{Y_1\cup Y_2}(X,\mathscr{F})\to 0$$
and then using it to extract a long exact sequence on relative cohomology. Showing exactness at the first and second positions is not very hard. I am stuck trying to show surjectivity of $\Gamma_{Y_1}(X,\mathscr{F})\oplus\Gamma_{Y_2}(X,\mathscr{F})\to \Gamma_{Y_1\cup Y_2}(X,\mathscr{F})$. I have tried a lot of acrobatics to construct for a given $s\in \Gamma_{Y_1\cup Y_2}(X,\mathscr{F})$ a pair $(s_1,s_2)\in \Gamma_{Y_1}(X,\mathscr{F})\oplus\Gamma_{Y_2}(X,\mathscr{F})$ so that $s_1-s_2=s$, but to no avail. 
It occurred to me that I might want to solve the flasque case first, but even the flasque assumption hasn't helped. I'd really appreciate a nudge in the right direction.
 A: $\newcommand{cF}{\mathcal{F}}$
$\newcommand{cG}{\mathcal{G}}$
$\newcommand{cI}{\mathcal{I}}$
$\newcommand{cO}{\mathcal{O}}$
$\newcommand{G}{\Gamma}$
Here is one method. First, we recall a particular method of devising an injective resolution for $\cF \in \mathfrak{Ab}(X)$, as in proposition II.2.2:

For each point $x\in X$, the stalk $\cF_x$ is an $\cO_{X,x}$-module, therefore there is an injection $\cF_x\to I_x$ where $I_x$ is an injective $\cO_{X,x}$-module (2.1A). For each point $x$, let $j$ denote the inclusion of the one-point space $\{x\}$ into $X$, and consider the sheaf $\cI=\prod_{x\in X} j_*(I_x)$. Here we consider $I_x$ as a sheaf on the one point space $\{x\}$ and $j_*$ is the direct image functor (II, section 1).
Now for any sheaf $\cG$ of $\cO_X$-modules, we have $\operatorname{Hom}_{\cO_X}(\cG,\cI)=\prod \operatorname{Hom}_{\cO_X}(\cG,j_*(I_x))$ by the definition of the direct product. On the other hand, for each point $x\in X$, we have $\operatorname{Hom}_{\cO_X}(\cG,j_*(I_x))=\operatorname{Hom}_{\cO_{X,x}}(\cG_x,I_x)$. Thus we conclude that there is a natural morphism $\cF\to \cI$ obtained from the local maps $\cF_x\to I_x$. (con't...)

From here, the proof finishes by observing everything we would require of the map $\cF\to I$ is verified from the above description of the Hom-set and the situation for modules over a local ring. The upshot for us is that we can always find a resolution $0\to\cF\to I_0\to I_1\to \cdots$ where each $I$ is a direct product of sheaves with support on a single point.
Now we claim that $$0\to \G_{Y_1\cap Y_2}(X,I_i) \to \G_{Y_1}(X,I_i) \oplus \G_{Y_2}(X,I_i) \to \G_{Y_1\cup Y_2}(X,I_i) \to 0$$ is an exact sequence, where the first map is the inclusion in to each factor and the second map is the difference. This follows immediately from the description of $I_i$ as supported on a single point. Finally, by applying the snake lemma to the appropriately-stacked diagrams, we get a long exact sequence in cohomology.
(This strategy lets us skip what you're having trouble with because when we want to calculate a derived functor of an object $\cF$, we can instead just calculate homology of that derived functor applied to any appropriate resolution of $\cF$. So we pick a nice resolution, and then we don't have to work as hard.)
