The Stacks project

27.29 Finding suitable affine opens

In this section we collect some results on the existence of affine opens in more and less general situations.

Lemma 27.29.1. Let $X$ be a quasi-separated scheme. Let $Z_1, \ldots , Z_ n$ be pairwise distinct irreducible components of $X$, see Topology, Section 5.8. Let $\eta _ i \in Z_ i$ be their generic points, see Schemes, Lemma 25.11.1. There exist affine open neighbourhoods $\eta _ i \in U_ i$ such that $U_ i \cap U_ j = \emptyset $ for all $i \not= j$. In particular, $U = U_1 \cup \ldots \cup U_ n$ is an affine open containing all of the points $\eta _1, \ldots , \eta _ n$.

Proof. Let $V_ i$ be any affine open containing $\eta _ i$ and disjoint from the closed set $Z_1 \cup \ldots \hat Z_ i \ldots \cup Z_ n$. Since $X$ is quasi-separated for each $i$ the union $W_ i = \bigcup _{j, j \not= i} V_ i \cap V_ j$ is a quasi-compact open of $V_ i$ not containing $\eta _ i$. We can find open neighbourhoods $U_ i \subset V_ i$ containing $\eta _ i$ and disjoint from $W_ i$ by Algebra, Lemma 10.25.4. Finally, $U$ is affine since it is the spectrum of the ring $R_1 \times \ldots \times R_ n$ where $R_ i = \mathcal{O}_ X(U_ i)$, see Schemes, Lemma 25.6.8. $\square$

Remark 27.29.2. Lemma 27.29.1 above is false if $X$ is not quasi-separated. Here is an example. Take $R = \mathbf{Q}[x, y_1, y_2, \ldots ]/((x-i)y_ i)$. Consider the minimal prime ideal $\mathfrak p = (y_1, y_2, \ldots )$ of $R$. Glue two copies of $\mathop{\mathrm{Spec}}(R)$ along the (not quasi-compact) open $\mathop{\mathrm{Spec}}(R) \setminus V(\mathfrak p)$ to get a scheme $X$ (glueing as in Schemes, Example 25.14.3). Then the two maximal points of $X$ corresponding to $\mathfrak p$ are not contained in a common affine open. The reason is that any open of $\mathop{\mathrm{Spec}}(R)$ containing $\mathfrak p$ contains infinitely many of the “lines” $x = i$, $y_ j = 0$, $j \not= i$ with parameter $y_ i$. Details omitted.

Notwithstanding the example above, for “most” finite sets of irreducible closed subsets one can apply Lemma 27.29.1 above, at least if $X$ is quasi-compact. This is true because $X$ contains a dense open which is separated.

Lemma 27.29.3. Let $X$ be a quasi-compact scheme. There exists a dense open $V \subset X$ which is separated.

Proof. Say $X = \bigcup _{i = 1, \ldots , n} U_ i$ is a union of $n$ affine open subschemes. We will prove the lemma by induction on $n$. It is trivial for $n = 1$. Let $V' \subset \bigcup _{i = 1, \ldots , n - 1} U_ i$ be a separated dense open subscheme, which exists by induction hypothesis. Consider

\[ V = V' \amalg (U_ n \setminus \overline{V'}). \]

It is clear that $V$ is separated and a dense open subscheme of $X$. $\square$

It turns out that, even if $X$ is quasi-separated as well as quasi-compact, there does not exist a separated, quasi-compact dense open, see Examples, Lemma 104.21.2. Here is a slight refinement of Lemma 27.29.1 above.

Lemma 27.29.4. Let $X$ be a quasi-separated scheme. Let $Z_1, \ldots , Z_ n$ be pairwise distinct irreducible components of $X$. Let $\eta _ i \in Z_ i$ be their generic points. Let $x \in X$ be arbitrary. There exists an affine open $U \subset X$ containing $x$ and all the $\eta _ i$.

Proof. Suppose that $x \in Z_1 \cap \ldots \cap Z_ r$ and $x \not\in Z_{r + 1}, \ldots , Z_ n$. Then we may choose an affine open $W \subset X$ such that $x \in W$ and $W \cap Z_ i = \emptyset $ for $i = r + 1, \ldots , n$. Note that clearly $\eta _ i \in W$ for $i = 1, \ldots , r$. By Lemma 27.29.1 we may choose affine opens $U_ i \subset X$ which are pairwise disjoint such that $\eta _ i \in U_ i$ for $i = r + 1, \ldots , n$. Since $X$ is quasi-separated the opens $W \cap U_ i$ are quasi-compact and do not contain $\eta _ i$ for $i = r + 1, \ldots , n$. Hence by Algebra, Lemma 10.25.4 we may shrink $U_ i$ such that $W \cap U_ i = \emptyset $ for $i = r + 1, \ldots , n$. Then the union $U = W \cup \bigcup _{i = r + 1, \ldots , n} U_ i$ is disjoint and hence (by Schemes, Lemma 25.6.8) a suitable affine open. $\square$

Lemma 27.29.5. Let $X$ be a scheme. Assume either

  1. The scheme $X$ is quasi-affine.

  2. The scheme $X$ is isomorphic to a locally closed subscheme of an affine scheme.

  3. There exists an ample invertible sheaf on $X$.

  4. The scheme $X$ is isomorphic to a locally closed subscheme of $\text{Proj}(S)$ for some graded ring $S$.

Then for any finite subset $E \subset X$ there exists an affine open $U \subset X$ with $E \subset U$.

Proof. By Properties, Definition 27.18.1 a quasi-affine scheme is a quasi-compact open subscheme of an affine scheme. Any affine scheme $\mathop{\mathrm{Spec}}(R)$ is isomorphic to $\text{Proj}(R[X])$ where $R[X]$ is graded by setting $\deg (X) = 1$. By Proposition 27.26.13 if $X$ has an ample invertible sheaf then $X$ is isomorphic to an open subscheme of $\text{Proj}(S)$ for some graded ring $S$. Hence, it suffices to prove the lemma in case (4). (We urge the reader to prove case (2) directly for themselves.)

Thus assume $X \subset \text{Proj}(S)$ is a locally closed subscheme where $S$ is some graded ring. Let $T = \overline{X} \setminus X$. Recall that the standard opens $D_{+}(f)$ form a basis of the topology on $\text{Proj}(S)$. Since $E$ is finite we may choose finitely many homogeneous elements $f_ i \in S_{+}$ such that

\[ E \subset D_{+}(f_1) \cup \ldots \cup D_{+}(f_ n) \subset \text{Proj}(S) \setminus T \]

Suppose that $E = \{ \mathfrak p_1, \ldots , \mathfrak p_ m\} $ as a subset of $\text{Proj}(S)$. Consider the ideal $I = (f_1, \ldots , f_ n) \subset S$. Since $I \not\subset \mathfrak p_ j$ for all $j = 1, \ldots , m$ we see from Algebra, Lemma 10.56.6 that there exists a homogeneous element $f \in I$, $f \not\in \mathfrak p_ j$ for all $j = 1, \ldots , m$. Then $E \subset D_{+}(f) \subset D_{+}(f_1) \cup \ldots \cup D_{+}(f_ n)$. Since $D_{+}(f)$ does not meet $T$ we see that $X \cap D_{+}(f)$ is a closed subscheme of the affine scheme $D_{+}(f)$, hence is an affine open of $X$ as desired. $\square$

Lemma 27.29.6. Let $X$ be a scheme. Let $\mathcal{L}$ be an ample invertible sheaf on $X$. Let

\[ E \subset W \subset X \]

with $E$ finite and $W$ open in $X$. Then there exists an $n > 0$ and a section $s \in \Gamma (X, \mathcal{L}^{\otimes n})$ such that $X_ s$ is affine and $E \subset X_ s \subset W$.

Proof. The reader can modify the proof of Lemma 27.29.5 to prove this lemma; we will instead deduce the lemma from it. By Lemma 27.29.5 we can choose an affine open $U \subset W$ such that $E \subset U$. Consider the graded ring $S = \Gamma _*(X, \mathcal{L}) = \bigoplus _{n \geq 0} \Gamma (X, \mathcal{L}^{\otimes n})$. For each $x \in E$ let $\mathfrak p_ x \subset S$ be the graded ideal of sections vanishing at $x$. It is clear that $\mathfrak p_ x$ is a prime ideal and since some power of $\mathcal{L}$ is globally generated, it is clear that $S_{+} \not\subset \mathfrak p_ x$. Let $I \subset S$ be the graded ideal of sections vanishing on all points of $X \setminus U$. Since the sets $X_ s$ form a basis for the topology we see that $I \not\subset \mathfrak p_ x$ for all $x \in E$. By (graded) prime avoidance (Algebra, Lemma 10.56.6) we can find $s \in I$ homogeneous with $s \not\in \mathfrak p_ x$ for all $x \in E$. Then $E \subset X_ s \subset U$ and $X_ s$ is affine by Lemma 27.26.4. $\square$

Lemma 27.29.7. Let $X$ be a quasi-affine scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $E \subset W \subset X$ with $E$ finite and $W$ open. Then there exists an $s \in \Gamma (X, \mathcal{L})$ such that $X_ s$ is affine and $E \subset X_ s \subset W$.

Proof. The proof of this lemma has a lot in common with the proof of Algebra, Lemma 10.14.2. Say $E = \{ x_1, \ldots , x_ n\} $. If $E = W = \emptyset $, then $s = 0$ works. If $W \not= \emptyset $, then we may assume $E \not= \emptyset $ by adding a point if necessary. Thus we may assume $n \geq 1$. We will prove the lemma by induction on $n$.

Base case: $n = 1$. After replacing $W$ by an affine open neighbourhood of $x_1$ in $W$, we may assume $W$ is affine. Combining Lemmas 27.27.1 and Proposition 27.26.13 we see that every quasi-coherent $\mathcal{O}_ X$-module is globally generated. Hence there exists a global section $s$ of $\mathcal{L}$ which does not vanish at $x_1$. On the other hand, let $Z \subset X$ be the reduced induced closed subscheme on $X \setminus W$. Applying global generation to the quasi-coherent ideal sheaf $\mathcal{I}$ of $Z$ we find a global section $f$ of $\mathcal{I}$ which does not vanish at $x_1$. Then $s' = fs$ is a global section of $\mathcal{L}$ which does not vanish at $x_1$ such that $X_{s'} \subset W$. Then $X_{s'}$ is affine by Lemma 27.26.4.

Induction step for $n > 1$. If there is a specialization $x_ i \leadsto x_ j$ for $i \not= j$, then it suffices to prove the lemma for $\{ x_1, \ldots , x_ n\} \setminus \{ x_ i\} $ and we are done by induction. Thus we may assume there are no specializations among the $x_ i$. By either Lemma 27.29.5 or Lemma 27.29.6 we may assume $W$ is affine. By induction we can find a global section $s$ of $\mathcal{L}$ such that $X_ s \subset W$ is affine and contains $x_1, \ldots , x_{n - 1}$. If $x_ n \in X_ s$ then we are done. Assume $s$ is zero at $x_ n$. By the case $n = 1$ we can find a global section $s'$ of $\mathcal{L}$ with $\{ x_ n\} \subset X_{s'} \subset W \setminus \overline{\{ x_1, \ldots , x_{n - 1}\} }$. Here we use that $x_ n$ is not a specialization of $x_1, \ldots , x_{n - 1}$. Then $s + s'$ is a global section of $\mathcal{L}$ which is nonvanishing at $x_1, \ldots , x_ n$ with $X_{s + s'} \subset W$ and we conclude as before. $\square$

Lemma 27.29.8. Let $X$ be a scheme and $x \in X$ a point. There exists an affine open neighbourhood $U \subset X$ of $x$ such that the canonical map $\mathcal{O}_ X(U) \to \mathcal{O}_{X, x}$ is injective in each of the following cases:

  1. $X$ is integral,

  2. $X$ is locally Noetherian,

  3. $X$ is reduced and has a finite number of irreducible components.

Proof. After translation into algebra, this follows from Algebra, Lemma 10.30.9. $\square$


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