Situation 32.22.1. Let S = \mathop{\mathrm{lim}}\nolimits _{i \in I} S_ i be the limit of a directed system of Noetherian schemes with affine transition morphisms S_{i'} \to S_ i for i' \geq i.
32.22 Descending finite type schemes
This section continues the theme of Section 32.9 in the spirit of the results discussed in Section 32.10.
Lemma 32.22.2. In Situation 32.22.1. Let X \to S be quasi-separated and of finite type. Then there exists an i \in I and a diagram
32.22.2.1
\begin{equation} \label{limits-equation-good-diagram} \vcenter { \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ S \ar[r] & S_ i } } \end{equation}
such that W \to S_ i is of finite type and such that the induced morphism X \to S \times _{S_ i} W is a closed immersion.
Proof. By Lemma 32.9.3 we can find a closed immersion X \to X' over S where X' is a scheme of finite presentation over S. By Lemma 32.10.1 we can find an i and a morphism of finite presentation X'_ i \to S_ i whose pull back is X'. Set W = X'_ i. \square
Lemma 32.22.3. In Situation 32.22.1. Let X \to S be quasi-separated and of finite type. Given i \in I and a diagram
\vcenter { \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ S \ar[r] & S_ i } }
as in (32.22.2.1) for i' \geq i let X_{i'} be the scheme theoretic image of X \to S_{i'} \times _{S_ i} W. Then X = \mathop{\mathrm{lim}}\nolimits _{i' \geq i} X_{i'}.
Proof. Since X is quasi-compact and quasi-separated formation of the scheme theoretic image of X \to S_{i'} \times _{S_ i} W commutes with restriction to open subschemes (Morphisms, Lemma 29.6.3). Hence we may and do assume W is affine and maps into an affine open U_ i of S_ i. Let U \subset S, U_{i'} \subset S_{i'} be the inverse image of U_ i. Then U, U_{i'}, S_{i'} \times _{S_ i} W = U_{i'} \times _{U_ i} W, and S \times _{S_ i} W = U \times _{U_ i} W are all affine. This implies X is affine because X \to S \times _{S_ i} W is a closed immersion. This also shows the ring map
\mathcal{O}(U) \otimes _{\mathcal{O}(U_ i)} \mathcal{O}(W) \to \mathcal{O}(X)
is surjective. Let I be the kernel. Then we see that X_{i'} is the spectrum of the ring
\mathcal{O}(X_{i'}) = \mathcal{O}(U_{i'}) \otimes _{\mathcal{O}(U_ i)} \mathcal{O}(W)/I_{i'}
where I_{i'} is the inverse image of the ideal I (see Morphisms, Example 29.6.4). Since \mathcal{O}(U) = \mathop{\mathrm{colim}}\nolimits \mathcal{O}(U_{i'}) we see that I = \mathop{\mathrm{colim}}\nolimits I_{i'} and we conclude that \mathop{\mathrm{colim}}\nolimits \mathcal{O}(X_{i'}) = \mathcal{O}(X). \square
Lemma 32.22.4. In Situation 32.22.1. Let f : X \to Y be a morphism of schemes quasi-separated and of finite type over S. Let
\vcenter { \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ S \ar[r] & S_{i_1} } } \quad \text{and}\quad \vcenter { \xymatrix{ Y \ar[r] \ar[d] & V \ar[d] \\ S \ar[r] & S_{i_2} } }
be diagrams as in (32.22.2.1). Let X = \mathop{\mathrm{lim}}\nolimits _{i \geq i_1} X_ i and Y = \mathop{\mathrm{lim}}\nolimits _{i \geq i_2} Y_ i be the corresponding limit descriptions as in Lemma 32.22.3. Then there exists an i_0 \geq \max (i_1, i_2) and a morphism
(f_ i)_{i \geq i_0} : (X_ i)_{i \geq i_0} \to (Y_ i)_{i \geq i_0}
of inverse systems over (S_ i)_{i \geq i_0} such that such that f = \mathop{\mathrm{lim}}\nolimits _{i \geq i_0} f_ i. If (g_ i)_{i \geq i_0} : (X_ i)_{i \geq i_0} \to (Y_ i)_{i \geq i_0} is a second morphism of inverse systems over (S_ i)_{i \geq i_0} such that such that f = \mathop{\mathrm{lim}}\nolimits _{i \geq i_0} g_ i then f_ i = g_ i for all i \gg i_0.
Proof. Since V \to S_{i_2} is of finite presentation and X = \mathop{\mathrm{lim}}\nolimits _{i \geq i_1} X_ i we can appeal to Proposition 32.6.1 to find an i_0 \geq \max (i_1, i_2) and a morphism h : X_{i_0} \to V over S_{i_2} such that X \to X_{i_0} \to V is equal to X \to Y \to V. For i \geq i_0 we get a commutative solid diagram
\xymatrix{ X \ar[d] \ar[r] & X_ i \ar[r] \ar@{..>}[d] \ar@/_2pc/[dd] |!{[d];[ld]}\hole & X_{i_0} \ar[d]^ h \\ Y \ar[r] \ar[d] & Y_ i \ar[r] \ar[d] & V \ar[d] \\ S \ar[r] & S_ i \ar[r] & S_{i_0} }
Since X \to X_ i has scheme theoretically dense image and since Y_ i is the scheme theoretic image of Y \to S_ i \times _{S_{i_2}} V we find that the morphism X_ i \to S_ i \times _{S_{i_2}} V induced by the diagram factors through Y_ i (Morphisms, Lemma 29.6.6). This proves existence.
Uniqueness. Let E_ i \subset X_ i be the equalizer of f_ i and g_ i for i \geq i_0. By Schemes, Lemma 26.21.5 E_ i is a locally closed subscheme of X_ i. Since X_ i is a closed subscheme of S_ i \times _{S_{i_0}} X_{i_0} and similarly for Y_ i we see that
E_ i = X_ i \times _{(S_ i \times _{S_{i_0}} X_{i_0})} (S_ i \times _{S_{i_0}} E_{i_0})
Thus to finish the proof it suffices to show that X_ i \to X_{i_0} factors through E_{i_0} for some i \geq i_0. To do this we will use that X \to X_{i_0} factors through E_{i_0} as both f_{i_0} and g_{i_0} are compatible with f. Since X_ i is Noetherian, we see that the underlying topological space |E_{i_0}| is a constructible subset of |X_{i_0}| (Topology, Lemma 5.16.1). Hence X_ i \to X_{i_0} factors through E_{i_0} set theoretically for large enough i by Lemma 32.4.10. For such an i the scheme theoretic inverse image (X_ i \to X_{i_0})^{-1}(E_{i_0}) is a closed subscheme of X_ i through which X factors and hence equal to X_ i since X \to X_ i has scheme theoretically dense image by construction. This concludes the proof. \square
Remark 32.22.5. In Situation 32.22.1 Lemmas 32.22.2, 32.22.3, and 32.22.4 tell us that the category of schemes quasi-separated and of finite type over S is equivalent to certain types of inverse systems of schemes over (S_ i)_{i \in I}, namely the ones produced by applying Lemma 32.22.3 to a diagram of the form (32.22.2.1). For example, given X \to S finite type and quasi-separated if we choose two different diagrams X \to V_1 \to S_{i_1} and X \to V_2 \to S_{i_2} as in (32.22.2.1), then applying Lemma 32.22.4 to \text{id}_ X (in two directions) we see that the corresponding limit descriptions of X are canonically isomorphic (up to shrinking the directed set I). And so on and so forth.
Lemma 32.22.6. Notation and assumptions as in Lemma 32.22.4. If f is flat and of finite presentation, then there exists an i_3 \geq i_0 such that for i \geq i_3 we have f_ i is flat, X_ i = Y_ i \times _{Y_{i_3}} X_{i_3}, and X = Y \times _{Y_{i_3}} X_{i_3}.
Proof. By Lemma 32.10.1 we can choose an i \geq i_2 and a morphism U \to Y_ i of finite presentation such that X = Y \times _{Y_ i} U (this is where we use that f is of finite presentation). After increasing i we may assume that U \to Y_ i is flat, see Lemma 32.8.7. As discussed in Remark 32.22.5 we may and do replace the initial diagram used to define the system (X_ i)_{i \geq i_1} by the system corresponding to X \to U \to S_ i. Thus X_{i'} for i' \geq i is defined as the scheme theoretic image of X \to S_{i'} \times _{S_ i} U.
Because U \to Y_ i is flat (this is where we use that f is flat), because X = Y \times _{Y_ i} U, and because the scheme theoretic image of Y \to Y_ i is Y_ i, we see that the scheme theoretic image of X \to U is U (Morphisms, Lemma 29.25.16). Observe that Y_{i'} \to S_{i'} \times _{S_ i} Y_ i is a closed immersion for i' \geq i by construction of the system of Y_ j. Then the same argument as above shows that the scheme theoretic image of X \to S_{i'} \times _{S_ i} U is equal to the closed subscheme Y_{i'} \times _{Y_ i} U. Thus we see that X_{i'} = Y_{i'} \times _{Y_ i} U for all i' \geq i and hence the lemma holds with i_3 = i. \square
Lemma 32.22.7. Notation and assumptions as in Lemma 32.22.4. If f is smooth, then there exists an i_3 \geq i_0 such that for i \geq i_3 we have f_ i is smooth.
Lemma 32.22.8. Notation and assumptions as in Lemma 32.22.4. If f is proper, then there exists an i_3 \geq i_0 such that for i \geq i_3 we have f_ i is proper.
Proof. By the discussion in Remark 32.22.5 the choice of i_1 and W fitting into a diagram as in (32.22.2.1) is immaterial for the truth of the lemma. Thus we choose W as follows. First we choose a closed immersion X \to X' with X' \to S proper and of finite presentation, see Lemma 32.13.2. Then we choose an i_3 \geq i_2 and a proper morphism W \to Y_{i_3} such that X' = Y \times _{Y_{i_3}} W. This is possible because Y = \mathop{\mathrm{lim}}\nolimits _{i \geq i_2} Y_ i and Lemmas 32.10.1 and 32.13.1. With this choice of W it is immediate from the construction that for i \geq i_3 the scheme X_ i is a closed subscheme of Y_ i \times _{Y_{i_3}} W \subset S_ i \times _{S_{i_3}} W and hence proper over Y_ i. \square
Lemma 32.22.9. In Situation 32.22.1 suppose that we have a cartesian diagram
\xymatrix{ X^1 \ar[r]_ p \ar[d]_ q & X^3 \ar[d]^ a \\ X^2 \ar[r]^ b & X^4 }
of schemes quasi-separated and of finite type over S. For each j = 1, 2, 3, 4 choose i_ j \in I and a diagram
\xymatrix{ X^ j \ar[r] \ar[d] & W^ j \ar[d] \\ S \ar[r] & S_{i_ j} }
as in (32.22.2.1). Let X^ j = \mathop{\mathrm{lim}}\nolimits _{i \geq i_ j} X^ j_ i be the corresponding limit descriptions as in Lemma 32.22.4. Let (a_ i)_{i \geq i_5}, (b_ i)_{i \geq i_6}, (p_ i)_{i \geq i_7}, and (q_ i)_{i \geq i_8} be the corresponding morphisms of systems constructed in Lemma 32.22.4. Then there exists an i_9 \geq \max (i_5, i_6, i_7, i_8) such that for i \geq i_9 we have a_ i \circ p_ i = b_ i \circ q_ i and such that
(q_ i, p_ i) : X^1_ i \longrightarrow X^2_ i \times _{b_ i, X^4_ i, a_ i} X^3_ i
is a closed immersion. If a and b are flat and of finite presentation, then there exists an i_{10} \geq \max (i_5, i_6, i_7, i_8, i_9) such that for i \geq i_{10} the last displayed morphism is an isomorphism.
Proof. According to the discussion in Remark 32.22.5 the choice of W^1 fitting into a diagram as in (32.22.2.1) is immaterial for the truth of the lemma. Thus we may choose W^1 = W^2 \times _{W^4} W^3. Then it is immediate from the construction of X^1_ i that a_ i \circ p_ i = b_ i \circ q_ i and that
(q_ i, p_ i) : X^1_ i \longrightarrow X^2_ i \times _{b_ i, X^4_ i, a_ i} X^3_ i
is a closed immersion.
If a and b are flat and of finite presentation, then so are p and q as base changes of a and b. Thus we can apply Lemma 32.22.6 to each of a, b, p, q, and a \circ p = b \circ q. It follows that there exists an i_9 \in I such that
(q_ i, p_ i) : X^1_ i \to X^2_ i \times _{X^4_ i} X^3_ i
is the base change of (q_{i_9}, p_{i_9}) by the morphism by the morphism X^4_ i \to X^4_{i_9} for all i \geq i_9. We conclude that (q_ i, p_ i) is an isomorphism for all sufficiently large i by Lemma 32.8.11. \square
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