The Stacks project

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.

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.

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

\[ \xymatrix{ B \ar[r]_ s & R \\ B'\ar[u] \ar[r] & R' \ar[u]_ t } \]

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

  1. a directed set $J$,

  2. an inverse system of schemes $(S_ j, g_{jj'})$ over $J$,

  3. an order preserving map $\alpha : J \to I$,

  4. open subschemes $V'_ j \subset S_ j$, and

  5. isomorphisms $V'_ j \to V_{\alpha (j)}$

such that

  1. the transition morphisms $g_{jj'} : S_ j \to S_{j'}$ are affine,

  2. each $S_ j$ is of finite type over $\mathbf{Z}$,

  3. $g_{jj'}^{-1}(V'_{j'}) = V'_ j$,

  4. $S = \mathop{\mathrm{lim}}\nolimits S_ j$ and $V = \mathop{\mathrm{lim}}\nolimits V'_ j$, and

  5. 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

\[ V \subset V \cup U_1 \subset V \cup U_1 \cup U_2 \subset \ldots \subset V \cup \bigcup \nolimits _{l = 1, \ldots , m} U_ l = S \]

If we can prove the lemma successively for each of the cases

\[ V \cup U_1 \cup \ldots \cup U_ l \subset V \cup U_1 \cup \ldots \cup U_{l + 1} \]

then the lemma will follow for $V \subset S$. In each case we are adding one affine open. Thus we may assume

  1. $S = U \cup V$,

  2. $U$ affine open in $S$,

  3. $V$ quasi-compact open in $S$, and

  4. $V = \mathop{\mathrm{lim}}\nolimits _ i V_ i$ 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

\[ \xymatrix{ B \ar[r]_ s & R \\ & R_ i \ar[u]_{t_ i} } \]

We set $B_ i = \{ (b, r) \in B \times R_ i \mid s(b) = t_ i(t)\} $ so that we have a cartesian diagram

\[ \xymatrix{ B \ar[r]_ s & R \\ B_ i \ar[u] \ar[r] & R_ i \ar[u]_{t_ i} } \]

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

\[ B_{g_ l} = \Gamma (D(g_ l), \mathcal{O}_ U) = \Gamma (W_{s(g_ l)}, \mathcal{O}_ W) = R_{s(g_ l)}, \]

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:

\[ \xymatrix{ & & V \ar[lld] \ar[d] & W \ar[l] \ar[lld] \ar[d] \\ V_ i \ar[d] & W_ i \ar[l] \ar[d] & S \ar[lld] & U \ar[lld] \ar[l] \\ S_{i, p} & \mathop{\mathrm{Spec}}(A_{i, p}) \ar[l] } \]

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

  1. the transition morphisms $f_{ii'}$ are affine

  2. each $S_ i$ is of finite type over $\mathbf{Z}$, and

  3. $S = \mathop{\mathrm{lim}}\nolimits _ i S_ i$.

Proof. This is a special case of Lemma 32.5.3 with $V = \emptyset $. $\square$


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