The Stacks project

Lemma 33.42.3. Let $f : T \to X$ be a morphism of schemes. Let $X^0$, resp. $T^0$ denote the sets of generic points of irreducible components. Let $t_1, \ldots , t_ m \in T$ be a finite set of points with images $x_ j = f(t_ j)$. If

  1. $T$ is affine,

  2. $X$ is quasi-separated,

  3. $X^0$ is finite

  4. $f(T^0) \subset X^0$ and $f : T^0 \to X^0$ is injective, and

  5. $\mathcal{O}_{X, x_ j} = \mathcal{O}_{T, t_ j}$,

then there exists an affine open of $X$ containing $x_1, \ldots , x_ r$.

Proof. Using Limits, Proposition 32.11.2 there is an immediate reduction to the case where $X$ and $T$ are reduced. Details omitted.

Assume $X$ and $T$ are reduced. We may write $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ as a directed limit of schemes of finite presentation over $X$ with affine transition morphisms, see Limits, Lemma 32.7.2. Pick $i \in I$ such that $T_ i$ is affine, see Limits, Lemma 32.4.13. Say $T_ i = \mathop{\mathrm{Spec}}(R_ i)$ and $T = \mathop{\mathrm{Spec}}(R)$. Let $R' \subset R$ be the image of $R_ i \to R$. Then $T' = \mathop{\mathrm{Spec}}(R')$ is affine, reduced, of finite type over $X$, and $T \to T'$ dominant. For $j = 1, \ldots , r$ let $t'_ j \in T'$ be the image of $t_ j$. Consider the local ring maps

\[ \mathcal{O}_{X, x_ j} \to \mathcal{O}_{T', t'_ j} \to \mathcal{O}_{T, t_ j} \]

Denote $(T')^0$ the set of generic points of irreducible components of $T'$. Let $\xi \leadsto t'_ j$ be a specialization with $\xi \in (T')^0$. As $T \to T'$ is dominant we can choose $\eta \in T^0$ mapping to $\xi $ (warning: a priori we do not know that $\eta $ specializes to $t_ j$). Assumption (3) applied to $\eta $ tells us that the image $\theta $ of $\xi $ in $X$ corresponds to a minimal prime of $\mathcal{O}_{X, x_ j}$. Lifting $\xi $ via the isomorphism of (5) we obtain a specialization $\eta ' \leadsto t_ j$ with $\eta ' \in T^0$ mapping to $\theta \leadsto x_ j$. The injectivity of (4) shows that $\eta = \eta '$. Thus every minimal prime of $\mathcal{O}_{T', t'_ j}$ lies below a minimal prime of $\mathcal{O}_{T, t_ j}$. We conclude that $\mathcal{O}_{T', t'_ j} \to \mathcal{O}_{T, t_ j}$ is injective, hence both maps above are isomorphisms.

By Lemma 33.42.2 there exists an open $U \subset T'$ containing all the points $t'_ j$ such that $U \to X$ is a local isomorphism as in Lemma 33.42.1. By that lemma we see that $U \to X$ is an open immersion. Finally, by Properties, Lemma 28.29.5 we can find an open $W \subset U \subset T'$ containing all the $t'_ j$. The image of $W$ in $X$ is the desired affine open. $\square$

Comments (2)

Comment #4208 by 羽山籍真 on

The second last paragraph: "we obtain a specialization \eta' to t_j with \eta' \in X^0". Here \eta' is in T^0 (not X^0), right?

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