I have a question in the following setting:
Let $k$ be a number field with ring of integers $O_k$. For a finite set of places $S$ of $k$ we can form the ring of $S$-integers $O_{k,S}$. Let $\mathbb{A}_k$ be the adéle ring of $k$.
For a $k$-variety (or more generally any scheme over the spectrum of the adéles), I want to understand the set of adélic points $X(\mathbb{A}_k)$. Everything I need is contained in Exercise 3.4. in Poonen's "Rational points on varieties", so I will present to you my struggles with it.
The first part of the exercise says to find, for a $k$-variety $X$, a separated, finite-type scheme $\mathcal{X}$ over $O_{k,S}$, for some $S$, such that $\mathcal{X} \times_{O_{k,S}} k = X$.
I have constructed the scheme $\mathcal{X}$ by covering $X$ with spectra of finitely generated $k$-algebras $k[x_1,\dots, x_{n_i}]/I_i$ and setting $S_i$ to be the set of places under which some coefficients of the polynomials in $I_i$ have negative valuation. (Note that this is finite because $I_i$ is finitely generated.) The ideal $I_i$ is then an ideal of the same polynomial ring considered with coefficients in the ring $O_{k,S_i}$. We can now set $S= \bigcup S_i$ and glue these to get $\mathcal{X}$ over $O_{k,S}$. One then sees that this becomes isomorphic to $X$ over $k$; by construction it is of finite type over $O_{k,S}$.
My problem now is to show that it is separated. My intuition tells me that this should be almost trivially true, because everything is so well behaved.
I failed using the valuative criterion, but I have the following idea using Stacks 26.21.7:
Choose $p_i,p_j \in \mathcal{X}$ lying over $\mathfrak{p} \in \text{Spec }O_{k,S}$. Let $A_i =\text{Spec }O_{k,S_i}[x_1,\dots, x_{n_i}]/I_i$ and $A_j = \text{Spec }O_{k,S_j}[x_1,\dots, x_{n_j}]/I_j$ be open affines around $p_i$ and $p_j$, respectively. Then we can take the maximum $n_{ij}$ of $i,j$ and take $S_{ij}= S_i \cup S_j$ and $I_i, I_j$ to be ideals in a polynomial ring over $O_{k,S_{ij}}$ in $n_{ij}$ variables.
The intersection $A_i \cap A_j$ should then be $\text{Spec }O_{k,S_{ij}}[x_1,\dots, x_{n_{ij}}]/(I_i + I_j)$, hence affine. The natural map $$ O_{\mathcal{X}}(A_i) \otimes O_{\mathcal{X}}(A_j)\rightarrow O_{\mathcal{X}}(A_i\cap A_j) $$ is an isomorphism, because $$ O_{\mathcal{X}}(A_i) \otimes O_{\mathcal{X}}(A_j) = \Gamma(A_i, O_{A_i}) \otimes \Gamma(A_j, O_{A_j}) = O_{k,S_{i}}[x_1,\dots, x_{n_{i}}]/I_i \otimes O_{k,S_{j}}[x_1,\dots, x_{n_{j}}]/I_j $$ and $$O_{\mathcal{X}}(A_i\cap A_j) = \Gamma(A_i\cap A_j, O_{A_i\cap A_j}) = O_{k,S_{ij}}[x_1,\dots, x_{n_{ij}}]/(I_i + I_j). $$ If this is correct, it would follow that $\mathcal{X}$ is separated. Are there any mistakes here? If there is a better way to argue for this, I would love to hear it.
Thanks already and if I should clarify more, just let me know!