Lemma 3.9.9. Let $\alpha $ be an ordinal as in Lemma 3.9.2 above. The category $\mathit{Sch}_\alpha $ satisfies the following properties:

If $X, Y, S \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha )$, then for any morphisms $f : X \to S$, $g : Y \to S$ the fibre product $X \times _ S Y$ in $\mathit{Sch}_\alpha $ exists and is a fibre product in the category of schemes.

Given any at most countable collection $S_1, S_2, \ldots $ of elements of $\mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha )$, the coproduct $\coprod _ i S_ i$ exists in $\mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha )$ and is a coproduct in the category of schemes.

For any $S \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha )$ and any open immersion $U \to S$, there exists a $V \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha )$ with $V \cong U$.

For any $S \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha )$ and any closed immersion $T \to S$, there exists an $S' \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha )$ with $S' \cong T$.

For any $S \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha )$ and any finite type morphism $T \to S$, there exists an $S' \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha )$ with $S' \cong T$.

Suppose $S$ is a scheme which has an open covering $S = \bigcup _{i \in I} S_ i$ such that there exists a $T \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha )$ with (a) $\text{size}(S_ i) \leq \text{size}(T)^{\aleph _0}$ for all $i \in I$, and (b) $|I| \leq \text{size}(T)^{\aleph _0}$. Then $S$ is isomorphic to an object of $\mathit{Sch}_\alpha $.

For any $S \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha )$ and any morphism $f : T \to S$ locally of finite type such that $T$ can be covered by at most $\text{size}(S)^{\aleph _0}$ open affines, there exists an $S' \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha )$ with $S' \cong T$. For example this holds if $T$ can be covered by at most $|\mathbf{R}| = 2^{\aleph _0} = \aleph _0^{\aleph _0}$ open affines.

For any $S \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha )$ and any monomorphism $T \to S$ which is either locally of finite presentation or quasi-compact, there exists an $S' \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha )$ with $S' \cong T$.

Suppose that $T \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha )$ is affine. Write $R = \Gamma (T, \mathcal{O}_ T)$. Then any of the following schemes is isomorphic to a scheme in $\mathit{Sch}_\alpha $:

For any ideal $I \subset R$ with completion $R^* = \mathop{\mathrm{lim}}\nolimits _ n R/I^ n$, the scheme $\mathop{\mathrm{Spec}}(R^*)$.

For any finite type $R$-algebra $R'$, the scheme $\mathop{\mathrm{Spec}}(R')$.

For any localization $S^{-1}R$, the scheme $\mathop{\mathrm{Spec}}(S^{-1}R)$.

For any prime $\mathfrak p \subset R$, the scheme $\mathop{\mathrm{Spec}}(\overline{\kappa (\mathfrak p)})$.

For any subring $R' \subset R$, the scheme $\mathop{\mathrm{Spec}}(R')$.

Any scheme of finite type over a ring of cardinality at most $|R|^{\aleph _0}$.

And so on.

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