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Let $K_1\subset K_2\subset \ldots$ be a tower of number fields, with $K_n$ of degree $2^n$.

Let $$X := \sqcup_{n=1}^\infty Spec \ K_n$$

This is a $\mathbb{Q}$-scheme. The structure morphism is locally of finite type. Is $X$ naturally an ind-scheme over $\mathbb{Q}$?

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    $\begingroup$ Isn't $X$ the limit of the $X_n:=\sqcup_{i=1}^{n}\operatorname{Spec}(K_i) $? The fact that the $K_i$ form a tower seems irrelevant. $\endgroup$
    – abx
    Commented Feb 21, 2018 at 5:35
  • $\begingroup$ Yes, I understand now. I had somehow confused myself. $\endgroup$
    – Xiboto
    Commented Feb 21, 2018 at 10:35

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