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.
Best Answer
$\newcommand{cF}{\mathcal{F}}$ $\newcommand{cG}{\mathcal{G}}$ $\newcommand{cI}{\mathcal{I}}$ $\newcommand{cO}{\mathcal{O}}$ $\newcommand{G}{\Gamma}$
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:
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.)