Lemma 32.2.1. Let I be a directed set. Let (S_ i, f_{ii'}) be an inverse system of schemes over I. If all the schemes S_ i are affine, then the limit S = \mathop{\mathrm{lim}}\nolimits _ i S_ i exists in the category of schemes. In fact S is affine and S = \mathop{\mathrm{Spec}}(\mathop{\mathrm{colim}}\nolimits _ i R_ i) with R_ i = \Gamma (S_ i, \mathcal{O}).
32.2 Directed limits of schemes with affine transition maps
In this section we construct the limit.
Proof. Just define S = \mathop{\mathrm{Spec}}(\mathop{\mathrm{colim}}\nolimits _ i R_ i). It follows from Schemes, Lemma 26.6.4 that S is the limit even in the category of locally ringed spaces. \square
Lemma 32.2.2. Let I be a directed set. Let (S_ i, f_{ii'}) be an inverse system of schemes over I. If all the morphisms f_{ii'} : S_ i \to S_{i'} are affine, then the limit S = \mathop{\mathrm{lim}}\nolimits _ i S_ i exists in the category of schemes. Moreover,
each of the morphisms f_ i : S \to S_ i is affine,
for an element 0 \in I and any open subscheme U_0 \subset S_0 we have
f_0^{-1}(U_0) = \mathop{\mathrm{lim}}\nolimits _{i \geq 0} f_{i0}^{-1}(U_0)in the category of schemes.
Proof. Choose an element 0 \in I. Note that I is nonempty as the limit is directed. For every i \geq 0 consider the quasi-coherent sheaf of \mathcal{O}_{S_0}-algebras \mathcal{A}_ i = f_{i0, *}\mathcal{O}_{S_ i}. Recall that S_ i = \underline{\mathop{\mathrm{Spec}}}_{S_0}(\mathcal{A}_ i), see Morphisms, Lemma 29.11.3. Set \mathcal{A} = \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathcal{A}_ i. This is a quasi-coherent sheaf of \mathcal{O}_{S_0}-algebras, see Schemes, Section 26.24. Set S = \underline{\mathop{\mathrm{Spec}}}_{S_0}(\mathcal{A}). By Morphisms, Lemma 29.11.5 we get for i \geq 0 morphisms f_ i : S \to S_ i compatible with the transition morphisms. Note that the morphisms f_ i are affine by Morphisms, Lemma 29.11.11 for example. By Lemma 32.2.1 above we see that for any affine open U_0 \subset S_0 the inverse image U = f_0^{-1}(U_0) \subset S is the limit of the system of opens U_ i = f_{i0}^{-1}(U_0), i \geq 0 in the category of schemes.
Let T be a scheme. Let g_ i : T \to S_ i be a compatible system of morphisms. To show that S = \mathop{\mathrm{lim}}\nolimits _ i S_ i we have to prove there is a unique morphism g : T \to S with g_ i = f_ i \circ g for all i \in I. For every t \in T there exists an affine open U_0 \subset S_0 containing g_0(t). Let V \subset g_0^{-1}(U_0) be an affine open neighbourhood containing t. By the remarks above we obtain a unique morphism g_ V : V \to U = f_0^{-1}(U_0) such that f_ i \circ g_ V = g_ i|_{U_ i} for all i. The open sets V \subset T so constructed form a basis for the topology of T. The morphisms g_ V glue to a morphism g : T \to S because of the uniqueness property. This gives the desired morphism g : T \to S.
The final statement is clear from the construction of the limit above. \square
Lemma 32.2.3. Let I be a directed set. Let (S_ i, f_{ii'}) be an inverse system of schemes over I. Assume all the morphisms f_{ii'} : S_ i \to S_{i'} are affine, Let S = \mathop{\mathrm{lim}}\nolimits _ i S_ i. Let 0 \in I. Suppose that T is a scheme over S_0. Then
T \times _{S_0} S = \mathop{\mathrm{lim}}\nolimits _{i \geq 0} T \times _{S_0} S_ i
Proof. The right hand side is a scheme by Lemma 32.2.2. The equality is formal, see Categories, Lemma 4.14.10. \square
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