Lemma 32.5.1. Let $W$ be a quasi-affine scheme of finite type over $\mathbf{Z}$. Suppose $W \to \mathop{\mathrm{Spec}}(R)$ is an open immersion into an affine scheme. There exists a finite type $\mathbf{Z}$-algebra $A \subset R$ which induces an open immersion $W \to \mathop{\mathrm{Spec}}(A)$. Moreover, $R$ is the directed colimit of such subalgebras.
32.5 Absolute Noetherian Approximation
A nice reference for this section is Appendix C of the article by Thomason and Trobaugh [TT]. See Categories, Section 4.21 for our conventions regarding directed systems. We will use the existence result and properties of the limit from Section 32.2 without further mention.
Proof. Choose an affine open covering $W = \bigcup _{i = 1, \ldots , n} W_ i$ such that each $W_ i$ is a standard affine open in $\mathop{\mathrm{Spec}}(R)$. In other words, if we write $W_ i = \mathop{\mathrm{Spec}}(R_ i)$ then $R_ i = R_{f_ i}$ for some $f_ i \in R$. Choose finitely many $x_{ij} \in R_ i$ which generate $R_ i$ over $\mathbf{Z}$. Pick an $N \gg 0$ such that each $f_ i^ Nx_{ij}$ comes from an element of $R$, say $y_{ij} \in R$. Set $A$ equal to the $\mathbf{Z}$-algebra generated by the $f_ i$ and the $y_{ij}$ and (optionally) finitely many additional elements of $R$. Then $A$ works. Details omitted. $\square$
Lemma 32.5.2. Suppose given a cartesian diagram of rings Let $W' \subset \mathop{\mathrm{Spec}}(R')$ be an open of the form $W' = D(f_1) \cup \ldots \cup D(f_ n)$ such that $t(f_ i) = s(g_ i)$ for some $g_ i \in B$ and $B_{g_ i} \cong R_{s(g_ i)}$. Then $B' \to R'$ induces an open immersion of $W'$ into $\mathop{\mathrm{Spec}}(B')$.
Proof. Set $h_ i = (g_ i, f_ i) \in B'$. More on Algebra, Lemma 15.5.3 shows that $(B')_{h_ i} \cong (R')_{f_ i}$ as desired. $\square$
The following lemma is a precise statement of Noetherian approximation.
Lemma 32.5.3. Let $S$ be a quasi-compact and quasi-separated scheme. Let $V \subset S$ be a quasi-compact open. Let $I$ be a directed set and let $(V_ i, f_{ii'})$ be an inverse system of schemes over $I$ with affine transition maps, with each $V_ i$ of finite type over $\mathbf{Z}$, and with $V = \mathop{\mathrm{lim}}\nolimits V_ i$. Then there exist
a directed set $J$,
an inverse system of schemes $(S_ j, g_{jj'})$ over $J$,
an order preserving map $\alpha : J \to I$,
open subschemes $V'_ j \subset S_ j$, and
isomorphisms $V'_ j \to V_{\alpha (j)}$
such that
the transition morphisms $g_{jj'} : S_ j \to S_{j'}$ are affine,
each $S_ j$ is of finite type over $\mathbf{Z}$,
$g_{jj'}^{-1}(V'_{j'}) = V'_ j$,
$S = \mathop{\mathrm{lim}}\nolimits S_ j$ and $V = \mathop{\mathrm{lim}}\nolimits V'_ j$, and
the diagrams
\[ \vcenter { \xymatrix{ V \ar[d] \ar[rd] \\ V'_ j \ar[r] & V_{\alpha (j)} } } \quad \text{and}\quad \vcenter { \xymatrix{ V'_ j \ar[r] \ar[d] & V_{\alpha (j)} \ar[d] \\ V'_{j'} \ar[r] & V_{\alpha (j')} } } \]are commutative.
Proof. Set $Z = S \setminus V$. Choose affine opens $U_1, \ldots , U_ m \subset S$ such that $Z \subset \bigcup _{l = 1, \ldots , m} U_ l$. Consider the opens
If we can prove the lemma successively for each of the cases
then the lemma will follow for $V \subset S$. In each case we are adding one affine open. Thus we may assume
$S = U \cup V$,
$U$ affine open in $S$,
$V$ quasi-compact open in $S$, and
$V = \mathop{\mathrm{lim}}\nolimits _ i V_ i$ with $(V_ i, f_{ii'})$ an inverse system over a directed set $I$, each $f_{ii'}$ affine and each $V_ i$ of finite type over $\mathbf{Z}$.
Denote $f_ i : V \to V_ i$ the projections. Set $W = U \cap V$. As $S$ is quasi-separated, this is a quasi-compact open of $V$. By Lemma 32.4.11 (and after shrinking $I$) we may assume that there exist opens $W_ i \subset V_ i$ such that $f_{ii'}^{-1}(W_{i'}) = W_ i$ and such that $f_ i^{-1}(W_ i) = W$. Since $W$ is a quasi-compact open of $U$ it is quasi-affine. Hence we may assume (after shrinking $I$ again) that $W_ i$ is quasi-affine for all $i$, see Lemma 32.4.12.
Write $U = \mathop{\mathrm{Spec}}(B)$. Set $R = \Gamma (W, \mathcal{O}_ W)$, and $R_ i = \Gamma (W_ i, \mathcal{O}_{W_ i})$. By Lemma 32.4.7 we have $R = \mathop{\mathrm{colim}}\nolimits _ i R_ i$. Now we have the maps of rings
We set $B_ i = \{ (b, r) \in B \times R_ i \mid s(b) = t_ i(r)\} $ so that we have a cartesian diagram
for each $i$. The transition maps $R_ i \to R_{i'}$ induce maps $B_ i \to B_{i'}$. It is clear that $B = \mathop{\mathrm{colim}}\nolimits _ i B_ i$. In the next paragraph we show that for all sufficiently large $i$ the composition $W_ i \to \mathop{\mathrm{Spec}}(R_ i) \to \mathop{\mathrm{Spec}}(B_ i)$ is an open immersion.
As $W$ is a quasi-compact open of $U = \mathop{\mathrm{Spec}}(B)$ we can find a finitely many elements $g_ l \in B$, $l = 1, \ldots , m$ such that $D(g_ l) \subset W$ and such that $W = \bigcup _{l = 1, \ldots , m} D(g_ l)$. Note that this implies $D(g_ l) = W_{s(g_ l)}$ as open subsets of $U$, where $W_{s(g_ l)}$ denotes the largest open subset of $W$ on which $s(g_ l)$ is invertible. Hence
where the last equality is Properties, Lemma 28.17.1. Since $W_{s(g_ l)}$ is affine this also implies that $D(s(g_ l)) = W_{s(g_ l)}$ as open subsets of $\mathop{\mathrm{Spec}}(R)$. Since $R = \mathop{\mathrm{colim}}\nolimits _ i R_ i$ we can (after shrinking $I$) assume there exist $g_{l, i} \in R_ i$ for all $i \in I$ such that $s(g_ l) = t_ i(g_{l, i})$. Of course we choose the $g_{l, i}$ such that $g_{l, i}$ maps to $g_{l, i'}$ under the transition maps $R_ i \to R_{i'}$. Then, by Lemma 32.4.11 we can (after shrinking $I$ again) assume the corresponding opens $D(g_{l, i}) \subset \mathop{\mathrm{Spec}}(R_ i)$ are contained in $W_ i$ for $l = 1, \ldots , m$ and cover $W_ i$. We conclude that the morphism $W_ i \to \mathop{\mathrm{Spec}}(R_ i) \to \mathop{\mathrm{Spec}}(B_ i)$ is an open immersion, see Lemma 32.5.2.
By Lemma 32.5.1 we can write $B_ i$ as a directed colimit of subalgebras $A_{i, p} \subset B_ i$, $p \in P_ i$ each of finite type over $\mathbf{Z}$ and such that $W_ i$ is identified with an open subscheme of $\mathop{\mathrm{Spec}}(A_{i, p})$. Let $S_{i, p}$ be the scheme obtained by glueing $V_ i$ and $\mathop{\mathrm{Spec}}(A_{i, p})$ along the open $W_ i$, see Schemes, Section 26.14. Here is the resulting commutative diagram of schemes:
The morphism $S \to S_{i, p}$ arises because the upper right square is a pushout in the category of schemes. Note that $S_{i, p}$ is of finite type over $\mathbf{Z}$ since it has a finite affine open covering whose members are spectra of finite type $\mathbf{Z}$-algebras. We define a preorder on $J = \coprod _{i \in I} P_ i$ by the rule $(i', p') \geq (i, p)$ if and only if $i' \geq i$ and the map $B_ i \to B_{i'}$ maps $A_{i, p}$ into $A_{i', p'}$. This is exactly the condition needed to define a morphism $S_{i', p'} \to S_{i, p}$: namely make a commutative diagram as above using the transition morphisms $V_{i'} \to V_ i$ and $W_{i'} \to W_ i$ and the morphism $\mathop{\mathrm{Spec}}(A_{i', p'}) \to \mathop{\mathrm{Spec}}(A_{i, p})$ induced by the ring map $A_{i, p} \to A_{i', p'}$. The relevant commutativities have been built into the constructions. We claim that $S$ is the directed limit of the schemes $S_{i, p}$. Since by construction the schemes $V_ i$ have limit $V$ this boils down to the fact that $B$ is the limit of the rings $A_{i, p}$ which is true by construction. The map $\alpha : J \to I$ is given by the rule $j = (i, p) \mapsto i$. The open subscheme $V'_ j$ is just the image of $V_ i \to S_{i, p}$ above. The commutativity of the diagrams in (5) is clear from the construction. This finishes the proof of the lemma. $\square$
Proposition 32.5.4. Let $S$ be a quasi-compact and quasi-separated scheme. There exist a directed set $I$ and an inverse system of schemes $(S_ i, f_{ii'})$ over $I$ such that
the transition morphisms $f_{ii'}$ are affine
each $S_ i$ is of finite type over $\mathbf{Z}$, and
$S = \mathop{\mathrm{lim}}\nolimits _ i S_ i$.
Proof. This is a special case of Lemma 32.5.3 with $V = \emptyset $. $\square$
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