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 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.
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}
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 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 $X = \mathop{\mathrm{lim}}\nolimits _{i' \geq i} X_{i'}$.
\[ \vcenter { \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ S \ar[r] & S_ i } } \]
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 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 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 $i \gg i_0$.
\[ \vcenter { \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ S \ar[r] & S_{i_1} } } \quad \text{and}\quad \vcenter { \xymatrix{ Y \ar[r] \ar[d] & V \ar[d] \\ S \ar[r] & S_{i_2} } } \]
\[ (f_ i)_{i \geq i_0} : (X_ i)_{i \geq i_0} \to (Y_ i)_{i \geq i_0} \]
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 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 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 contructed 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 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.
\[ \xymatrix{ X^1 \ar[r]_ p \ar[d]_ q & X^3 \ar[d]^ a \\ X^2 \ar[r]^ b & X^4 } \]
\[ \xymatrix{ X^ j \ar[r] \ar[d] & W^ j \ar[d] \\ S \ar[r] & S_{i_ j} } \]
\[ (q_ i, p_ i) : X^1_ i \longrightarrow X^2_ i \times _{b_ i, X^4_ i, a_ i} X^3_ i \]
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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