Consider $\mathcal{X}$ a projective and flat scheme over $\text{Spec}(\mathcal{O}_K)$, with $\mathcal{O}_K$ the ring of integers of a number field $K$.
Let $F/K$ vary over all finite Galois number field extensions and define $\mathbf{A} := \mathbf{A}_{\overline{K}}$ as the direct limit of the topological rings $\mathbf{A}_F$ of the adèles of each $F$.
Question 1 Is there a good and intrinsic definition of $\mathcal{X}(\mathbf{A})$ and does it agree with the direct limit topological space $\varinjlim_{F/K}\mathcal{X}(\mathbf{A}_F)$? (or is this latter the definition usually given?)
Question 2 Is $\mathcal{X}(\mathbf{A})$ compact?
The ring $\mathbf{A}$ must probably be replaced with a restricted product of countably many copies of $\mathbf{C}$, $\mathbf{R}$, $(\mathbf{C}_p,\mathcal{O}_{\mathbf{C}_p})$ for infinitely many $p$. Even so, $\mathbf{C}_p$ is not locally compact, so I expect the answer to question 2 is "no", no matter how we put it.
Best Answer
Since $\mathcal X$ is projective, a section is given by finitely many coordinates. If the $i$'th coordinate lies in $\mathbf A_{F_i}$ for some extension $F_i$ of $K$, then all the coordinates lie in $\mathbf A_{F}$ for $F$ the composition of the $F_i$.
So $\mathcal X(\mathbf A_{\overline {K}})$ agrees with the direct limit of $\mathcal X(\mathbf A_{\overline {F}})$ - at least as sets: I didn't check the topology but I imagine it's fine.
I'm not sure what you mean by an intrinsic definition. Both these definitions seem fine to me.
$\mathcal X(\mathbf A_{\overline {K}})$ is not compact. Choose a $p$-adic valuation on $\overline{K}$, We have maps $$\mathcal X ( \mathbf A_{\overline{K}}) \to \mathcal X (\overline{K}_p) = \mathcal X( \overline{\mathbb Q}_p) = \mathcal X( \overline{\mathbb Z}_p) \to \mathcal X ( \overline{\mathbb F}_p)$$ and the composition is continuous with the discrete topology on $\mathcal X ( \overline{\mathbb F}_p)$, but $\mathcal X ( \overline{\mathbb F}_p)$ is infinite.
Restricted to each individual $\mathcal X( \mathbf A_{F})$, we're restricting the valuation to $F$, mapping to the appropriate completion, clearing denominators (because we are working in a projective variety), and reducing modulo the uniformizer.