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 1Is 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 2Is $\mathcal{X}(\mathbf{A})$ 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.