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?