Let \(X\) be a scheme, let \(f \in \Gamma(X, \OO_X)\), and define \(X_f\) to be the subset of points \(x \in X\) such that the stalk \(f_x\) of \(f\) at \(x\) is not contained in the maximal ideal \(\mfm_x\) of the local ring \(\OO_x\).
If \(U = \Spec B\) is an open affine subscheme of \(X\) and \(\bar{f} \in B = \Gamma\qty{U, \ro{\OO_X}{U}}\) is the restriction of \(f\), show that \(U \intersect X_f = D(\bar{f})\). Conclude that \(X_f\) is an open subset of \(X\).
Assume that \(X\) is quasi-compact. Let \(A = \Gamma(X, \OO_X)\) and let \(a \in A\) be an element whose restriction to \(X_f\) is \(0\). Show that \(f^n a = 0\) for some \(n > 0\). Use an open affine cover of \(X\).
Now assume that \(X\) has a finite cover by open affines \(U_i\) such that each intersection \(U_i \intersect U_j\) is quasi-compact. This hypothesis is satisfied, for example, if \(\operatorname{sp}(X)\) is noetherian. Let \(b \in \Gamma(X_f, \OO_{X_f})\). Show that for some \(n > 0\) the section \(f^n b\) is the restriction of an element of \(A\).
With the hypothesis of (c), conclude that \(\Gamma(X_f, \OO_{X_f}) \cong A_f\).
Solution
1. Let\[ U=\operatorname{Spec}B\subseteq X \]be affine, and let\[ \bar f=f|_U\in B. \]For a point \(x\in U\) corresponding to a prime \(\mathfrak p\subseteq B\),\[ x\in X_f \iff \bar f\notin\mathfrak p. \]
The local ring at \(x\) is \[ \mathcal O_{X,x}=B_{\mathfrak p} \] with maximal ideal \[ \mathfrak m_x=\mathfrak pB_{\mathfrak p}. \] The germ \(f_x\) is the image \[ \bar f/1\in B_{\mathfrak p}. \] Now \[ \bar f/1\in\mathfrak pB_{\mathfrak p} \iff \bar f\in\mathfrak p, \] because contraction of the localized prime is \[ \mathfrak pB_{\mathfrak p}\cap B=\mathfrak p. \] Thus \(f_x\) is a unit, equivalently is not in the maximal ideal, exactly when \(\bar f\notin\mathfrak p\).
2. Therefore\[ \boxed{U\cap X_f=D(\bar f).} \]In particular \(X_f\) is open in \(X\).
1. Distinguished opens are open in \(U\), and the affine opens \(U\) cover \(X\). Hence \(X_f\) has open intersection with every member of an open cover of \(X\), so \(X_f\) is open.
3. Assume \(X\) is quasi-compact. Choose a finite affine open cover\[ X=U_1\cup\cdots\cup U_r, \qquad U_i=\operatorname{Spec}B_i. \]Let\[ f_i=f|_{U_i}, \qquad a_i=a|_{U_i}. \]If \(a|_{X_f}=0\), then for every \(i\) there is an integer \(n_i\ge0\) such that\[ f_i^{n_i}a_i=0 \]in \(B_i\).
By 2, \[ U_i\cap X_f=D(f_i). \] The hypothesis says \[ a_i|_{D(f_i)}=0. \] Since \[ \Gamma(D(f_i),\mathcal O_X) =(B_i)_{f_i}, \] this means \[ \frac{a_i}{1}=0 \] in \((B_i)_{f_i}\). By the definition of localization, this is equivalent to \[ f_i^{n_i}a_i=0 \] for some \(n_i\ge0\).
4. Under the hypotheses of 3, there is one integer \(n>0\) such that\[ \boxed{f^na=0\text{ in }\Gamma(X,\mathcal O_X).} \]
Take \[ n>\max_i n_i. \] Then on every \(U_i\), \[ (f^na)|_{U_i} =f_i^na_i =0. \] The \(U_i\) cover \(X\), so the sheaf uniqueness axiom gives \[ f^na=0 \] globally.
5. Now assume the hypotheses of part (c):\[ X=U_1\cup\cdots\cup U_r \]with each \(U_i\) affine and each intersection\[ U_{ij}=U_i\cap U_j \]quasi-compact. Let\[ b\in\Gamma(X_f,\mathcal O_{X_f}). \]For every \(i\) there is an integer \(n_i\ge0\) and a section\[ c_i\in\Gamma(U_i,\mathcal O_X) \]such that on\[ U_i\cap X_f \]one has\[ c_i=f^{n_i}b. \]
Write \[ U_i=\operatorname{Spec}B_i, \qquad f_i=f|_{U_i}. \] By 2, \[ U_i\cap X_f=D(f_i), \] and hence \[ \Gamma(U_i\cap X_f,\mathcal O_X) =(B_i)_{f_i}. \] The restriction of \(b\) to this distinguished open is some fraction \[ \frac{a_i}{f_i^{n_i}} \] with \(a_i\in B_i\). Let \(c_i\) be the section corresponding to \(a_i\). Then \[ f^{n_i}b=c_i \] on \(U_i\cap X_f\).
6. After increasing the exponents in 5, there is a single integer \(n_0\ge0\) such that for every \(i\) there is\[ c_i\in\Gamma(U_i,\mathcal O_X) \]with\[ \boxed{c_i=f^{n_0}b\text{ on }U_i\cap X_f.} \]
Take \[ n_0=\max_i n_i. \] Replace the section from 5 by \[ c_i':=f_i^{\,n_0-n_i}c_i. \] On \(U_i\cap X_f\), \[ c_i' =f^{n_0-n_i}f^{n_i}b =f^{n_0}b. \] Rename \(c_i'\) as \(c_i\).
7. On every overlap \(U_{ij}\), the difference\[ d_{ij}=c_i|_{U_{ij}}-c_j|_{U_{ij}} \]restricts to zero on\[ (U_{ij})_f=U_{ij}\cap X_f. \]
By 6, both restrictions agree with \[ f^{n_0}b \] on the common open subset \[ U_i\cap U_j\cap X_f. \] Hence their difference vanishes there.
8. For every pair \((i,j)\) there is an integer \(m_{ij}\ge0\) such that\[ f^{m_{ij}}d_{ij}=0 \]on \(U_{ij}\).
The scheme \(U_{ij}\) is quasi-compact by hypothesis. Apply part (b), proved in <1>3–<1>4, to the quasi-compact scheme \(U_{ij}\), the global function \[ f|_{U_{ij}}, \] and the section \(d_{ij}\). Step 7 is exactly the hypothesis that \(d_{ij}\) vanishes on its nonvanishing locus. Therefore some power of \(f\) annihilates \(d_{ij}\).
9. There is a single integer \(m\ge0\) such that the sections\[ f^mc_i\in\Gamma(U_i,\mathcal O_X) \]agree on every pairwise overlap.
There are only finitely many pairs \((i,j)\). Take \[ m=\max_{i,j}m_{ij}. \] Then on \(U_{ij}\), \[ f^m(c_i-c_j) =f^md_{ij} =0. \] Thus \[ f^mc_i=f^mc_j \] on every overlap.
10. The sections in 9 glue to a global section\[ a\in A=\Gamma(X,\mathcal O_X) \]such that\[ \boxed{a|_{X_f}=f^{n_0+m}b.} \]
By 9, the sections \(f^mc_i\) form a compatible family on the finite open cover \(\{U_i\}\). The sheaf gluing axiom gives a unique \[ a\in\Gamma(X,\mathcal O_X) \] whose restriction to \(U_i\) is \(f^mc_i\).
On \(U_i\cap X_f\), 6 gives \[ a =f^mc_i =f^mf^{n_0}b =f^{n_0+m}b. \] The opens \(U_i\cap X_f\) cover \(X_f\), so the equality holds globally on \(X_f\).
11. Thus for every\[ b\in\Gamma(X_f,\mathcal O_{X_f}) \]there is an integer \(N>0\) such that\[ \boxed{f^Nb\text{ extends to a section of }\Gamma(X,\mathcal O_X).} \]
Take \[ N>n_0+m \] and multiply the global extension in 10 by the additional power of \(f\) if necessary. Equivalently, 10 already gives the statement with exponent \(n_0+m\), except in the harmless case that this exponent is zero; increasing it to a positive integer preserves extendability.
12. Under the hypotheses of part (c), \(X\) is quasi-compact.
It has the finite open cover \[ X=U_1\cup\cdots\cup U_r \] by affine schemes. Each affine \(U_i\) is quasi-compact, and a finite union of quasi-compact open subsets is quasi-compact.
13. Restriction induces a natural ring homomorphism\[ \theta:A_f \longrightarrow \Gamma(X_f,\mathcal O_{X_f}), \qquad \frac a{f^r} \longmapsto \frac{a|_{X_f}}{(f|_{X_f})^r}. \]
On \(X_f\), the section \(f\) is a unit in every stalk and hence is a unit of the sheaf of rings locally. In fact its pointwise inverse glues uniquely, so \[ f|_{X_f} \] is an invertible global section. Therefore the restriction map \[ A\to\Gamma(X_f,\mathcal O_{X_f}) \] sends \(f\) to a unit and factors uniquely through the localization \(A_f\).
14. The homomorphism \(\theta\) is injective.
Suppose \[ \theta\left(\frac a{f^r}\right)=0. \] Since \(f|_{X_f}\) is invertible, this implies \[ a|_{X_f}=0. \] By 12, \(X\) is quasi-compact, so part (b), namely 4, gives an integer \(n>0\) with \[ f^na=0 \] in \(A\). This is exactly the condition that \[ \frac a{f^r}=0 \] in the localization \(A_f\).
15. The homomorphism \(\theta\) is surjective.
Let \[ b\in\Gamma(X_f,\mathcal O_{X_f}). \] By 11, there is an integer \(n>0\) and a global section \[ a\in A \] such that \[ a|_{X_f}=f^nb. \] Since \(f\) is invertible on \(X_f\), \[ b =\frac{a|_{X_f}}{f^n} =\theta\left(\frac a{f^n}\right). \] Thus every section lies in the image.
16. Consequently\[ \boxed{ \Gamma(X_f,\mathcal O_{X_f}) \cong A_f. } \]
The natural map \(\theta\) from 13 is injective by 14 and surjective by 15.
17. Q.E.D.
Steps <1>1–<1>2 prove (a), <1>3–<1>4 prove (b), <1>5–<1>11 prove (c), and <1>12–<1>16 prove (d).