Prüfer algebraic spaces

Abstract

This is the first in a series of two papers concerned with relative birational geometry of algebraic spaces. In this paper, we study Prüfer spaces and Prüfer pairs of algebraic spaces that generalize spectra of Prüfer rings. As a particular case of Prüfer spaces we introduce valuation algebraic spaces, and use them to establish valuative criteria of separatedness and properness that sharpen the standard criteria. In a sequel paper, we introduce a version of Riemann–Zariski spaces, and prove Nagata’s compactification theorem for algebraic spaces.

Appendix: Approximation for pro-open immersions

The aim of the “Appendix” is to show that if a pro-open subspace U is the intersection of open subspaces $U_{\alpha }$ then the usual approximation theory applies to the limit $U=\lim _{\alpha }U_{\alpha }$. This task is not so simple because the transition morphisms do not have to be affine and it is even unclear whether there exists a cofinal subfamily with affine transition morphisms.

1.1 Preparation

Note that in the following result it is not enough to assume that f is of finite type as can be easily seen using examples from Sect. 3.3.3.

Proposition 5.2.22

Assume that U, Y, X are qcqs algebraic spaces, $j{:}\,U\rightarrow Y$ is a schematically dominant morphism, $f{:}\,Y\rightarrow X$ is a separated morphism such that the composition $i{:}\,U\rightarrow X$ is a pro-open immersion, and one of the following conditions is satisfied:

• (i) f is of finite presentation,

• (ii) f is schematically dominant and of finite type.

Then, for a sufficiently small quasi-compact open neighborhood $V\subset X$ of i(U), the restriction $f\times _XV$ is an isomorphism.

Proof

In view of Proposition 3.5.3 it suffices to show that the isomorphism locus of f contains i(U). After replacing X by a presentation we may assume that X is a scheme. Choose a Zariski point $u\rightarrow U$ and set $y=j(u)$, $x=i(u)$, $X_x=\mathrm{Spec}({{\mathcal {O}}}_{X,x})$, $Y_x=Y\times _XX_x$ and $U_x=U\times _XX_x$. We should prove that the morphism $f_x{:}\,Y_x\rightarrow X_x$ is an isomorphism. Note that $U_x\rightarrow X_x$ is a surjective pro-open immersion by claims (i) and (iv) of Proposition 3.1.4, hence an isomorphism. Hence $f_x$ is an isomorphism by the following lemma. $\square$

Lemma 5.2.23

Let $\pi {:}\,Y\rightarrow X$ be a separated morphism of qcqs algebraic spaces, and $s{:}\,X\rightarrow Y$ a section of $\pi$. Then s is a closed immersion. In particular, if s is schematically dominant then $\pi$ and s are isomorphisms.

Proof

Note that s is the equalizer of the morphisms $s\circ \pi$ and $\mathrm{id}_Y$. Indeed, any morphism $f{:}\,Z\rightarrow Y$ with $f=s\circ \pi \circ f$ factors as $Z\mathop {\rightarrow }\limits ^{\pi \circ f}X\mathop {\rightarrow }\limits ^{s}Y$, and this gives rise to the identification $X=\mathrm{Eq}(s\circ \pi ,\mathrm{id}_Y)$. On the other hand, the equalizer can be expressed as the base change of the morphism $(s\circ \pi ,\mathrm{id}_Y){:}\,Y\rightarrow Y\times _XY$ with respect to the diagonal $\Delta {:}\,Y\rightarrow Y\times _XY$. Since $\pi$ is separated, $\Delta$ is a closed immersion, and hence s is so. $\square$

1.2 Pro-open approximation

Now, we can extend the approximation theory to pro-open immersions.

Theorem 5.2.24

Assume that $i{:}\,U\rightarrow X$ is a schematically dominant pro-open immersion of qcqs algebraic spaces, and $\{U_{\alpha }\}_{{\alpha }\in A}$ is the family of open qcqs subspaces of X with $|U|\subset |U_{\alpha }|$. Then,

• (i) $U_{\alpha }$ are cofinal among the family $\{Y_\beta \}_{\beta \in B}$ of all strict U-quasi-modifications of X.

• (ii) All results of the classical approximation theory mentioned in Sect. 2.1.8 hold true for the limit $U=\lim _{{\alpha }\in A}U_{\alpha }$.

Proof

Assertion (i) follows from the definition of quasi-modifications, see Sect. 2.3.2. Let us use it to prove assertion (ii). By Lemma 2.1.10, U is isomorphic to the limit of a family $\{Z_\gamma \}_{\gamma \in C}$ of X-separated X-spaces of finite type with affine and schematically dominant transition morphisms. In particular, the projections $U\rightarrow Z_\gamma$ are schematically dominant, so each $Z_\gamma$ is a strict U-quasi-modification of X by Proposition 5.2.22(ii). By Lemma 2.1.11, any morphism $U\rightarrow Y_\beta$ factors through some $Z_\gamma$, so the family $\{Z_\gamma \}$ is cofinal in the family $\{Y_\beta \}$. The transition morphisms in the former family are affine, so the classical approximation applies to it, and by the cofinality of $\{Z_\gamma \}$ and $\{U_{\alpha }\}$ in $\{Y_\beta \}$, all approximation results apply also to the families $\{Y_\beta \}_{\beta \in B}$ and $\{U_{\alpha }\}_{{\alpha }\in A}$. $\square$