Let $X = X_1 \sqcup ... \sqcup X_n$ be a disjoint union of schemes, and let $F \in \text{Ab}(X_{et})$ be an abelian sheaf on an etale site of $X$. I need to show that $H^q(X_{et},F) \cong \prod_{i=1}^n H^q(X_{i_{et}}, F|_{X_i}) $.
I am completely stuck on this one. Any help would be very appreciated. Thanks.
There are plenty of way to do this.
The quickest is to use the Cech to derived spectral sequence. It reads $$ E_2^{pq}=\check{H}^p(\mathcal{U},\underline{H}^q(F))\Rightarrow H^{p+q}(X,F) $$ If $\mathcal{U}=\{X_i\}$ is a covering of $X$ by disjoint open subset, then $\check{H}^p(\mathcal{U},\cdot)=0$ for $p>0$ since there is no intersections. Thus the spectral sequence collapse and $$H^q(X,F)=\check{H}^0(\mathcal{U},\underline{H}^q(F))=\prod \underline{H}^q(F)(X_i)=\prod H^q(X_i,F_{|X_i})$$
You can also directly show if $j_i:X_i\rightarrow X$ denotes the inclusion, then the functors $j_i^*,{j_i}_*$ are adjoint in both direction, that is $j_i^*$ is a left (as usual) but also a right adjoint of ${j_i}_*$. In particular, they are both exact, preserving injective (and projective), so you can quickly see that the derived functor of $\Gamma(X,F)=\prod\Gamma(X_i,j_i^*F)$ is $\prod H^p(X_i,j_i^*F)$.
In fact, you even have an equivalence of categories $Sh(X)\simeq \prod Sh(X_i)$.