# Bertini theorems admitting base changes

Hiromu Tanaka  Graduate School of Mathematical Sciences, The University of
Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, JAPAN<EMAIL_ADDRESS>
###### Abstract.

Given a base point free linear system on an algebraic variety, many classes of
singularities are stable under taking suitable members after enlarging the
base field. We establish analogous results when the base ring is an excellent
ring.

###### Key words and phrases:

Bertini theorem, base point free.

###### 2010 Mathematics Subject Classification:

14G17, 14D06.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Notation
    2. 2.2 Singularities
      1. 2.2.1 Singularities for schemes
      2. 2.2.2 Singularities of minimal model program
      3. 2.2.3 $F$-Singularities
    3. 2.3 Localisation of rings
    4. 2.4 $\mathbb{A}^{n}$-localising morphisms
  3. 3 Purely transcendental extensions for rings
    1. 3.1 Purely transcendental extensions for rings
    2. 3.2 Base changes
      1. 3.2.1 Integrality
      2. 3.2.2 Fibres
      3. 3.2.3 Dimension
    3. 3.3 Examples
    4. 3.4 A generalisation
  4. 4 Bertini theorems for generic members
    1. 4.1 Universal members and generic members
    2. 4.2 Bertini theorems for schemes
    3. 4.3 Avoidance property for generic members
    4. 4.4 Restriction to generic members
    5. 4.5 Bertini theorems for pairs
    6. 4.6 Examples
      1. 4.6.1 Positive characteristic
      2. 4.6.2 Mixed characteristic
  5. 5 Bertini theorems over fields
    1. 5.1 Density of good members
    2. 5.2 Generic members for linear systems
    3. 5.3 Generic members vs General members
      1. 5.3.1 Characteristic zero
      2. 5.3.2 Positive characteristic
    4. 5.4 Base points
    5. 5.5 Picard groups for generic members

##  1\. Introduction

Given a smooth projective variety $X$ over an infinite field, the classical
Bertini theorem asserts that general hyperplane sections of $X$ are smooth
[Har77, Ch. II, Theorem 8.18]. In other words, if $L$ is a very ample divisor
on $X$, then general members of its complete linear system are smooth. In
characteristic zero, the same conclusion holds for arbitrary base point free
linear systems [Har77, Ch. III, Corollary 10.9], whilst such a generalisation
is known to fail in positive characteristic (cf. Example 4.23 and Example
4.24). This pathological phenomenon is one of typical obstructions to extend
results in characteristic zero to ones in positive characteristic.

The purpose of this paper is to establish results of Bertini type for base
point free linear systems in positive and mixed characteristic. For a base
point free linear system, all the members can have bad singularities. On the
other hand, we shall prove that the linear system will have good members after
taking a suitable base change. To this end, we first introduce purely
transcendental extensions $R(t_{1},...,t_{n})$ for arbitrary rings $R$:

$R(t_{1},...,t_{n}):=U_{R[t_{1},...,t_{n}]/R}^{-1}(R[t_{1},...,t_{n}]),$

where $R[t_{1},...,t_{n}]$ denotes the polynomial ring over $R$ and

$U_{R[t_{1},...,t_{n}]/R}:=\bigcap_{\mathfrak{p}\in{\operatorname{Spec}}\,R}(R[t_{1},...,t_{n}]\setminus\mathfrak{p}R[t_{1},...,t_{n}]).$

When $R$ is a field, $R(t_{1},...,t_{n})$ coincides with the purely
transcendental extension in the usual sense. After taking the base change
$X\mapsto X\times_{R}R(t_{1},...,t_{n})$, base point free linear systems have
good members as follows.

######  Theorem 1.1 (Theorem 3.22, Theorem 4.7).

Fix $n\in\mathbb{Z}_{>0}$. Let $R$ be a noetherian ring and set
$R(n):=R(t_{1},...,t_{n})$. Then there exists a hyperplane
$(\mathbb{P}^{n}_{R})^{{\operatorname{gen}}}_{{\rm id}}$ of
$\mathbb{P}^{n}_{R(n)}$ such that the following holds.

Let $\varphi:X\to\mathbb{P}^{n}_{R}$ be an $R$-morphism from a scheme $X$ of
finite type over $R$. Set $X_{R(n)}:=X\times_{R}R(n)$. Let (P) be a property
for noetherian schemes which satisfies the following properties (I)–(III)
(e.g. (P) can be regular, reduced, klt, or strongly $F$-regular).

  1. (I)

Let $Y$ be a noetherian scheme and let $Y=\bigcup_{i\in I}Y_{i}$ be an open
cover. Then $Y$ is (P) if and only if $Y_{i}$ is (P) for every $i\in I$.

  2. (II)

For a noetherian ring $A$, if ${\operatorname{Spec}}\,A$ is (P), then also
${\operatorname{Spec}}\,A[t]$ is (P), where $A[t]$ denotes the polynomial ring
over $A$ with one variable.

  3. (III)

For a noetherian ring $A$ and a multiplicatively closed subset $S$ of $A$, if
${\operatorname{Spec}}\,A$ is (P), then also ${\operatorname{Spec}}\,S^{-1}A$
is (P).

If $X$ is (P), then also $X^{{\operatorname{gen}}}_{\varphi}$ is (P), where
$X^{{\operatorname{gen}}}_{\varphi}$ is the scheme-theoretic inverse image of
$(\mathbb{P}^{n}_{R})^{{\operatorname{gen}}}_{{\rm id}}$, i.e. the following
diagram is cartesian:

$\begin{CD}X^{{\operatorname{gen}}}_{\varphi}@>{}>{}>X_{R(n)}\\\
@V{}V{}V@V{}V{\varphi\times_{R}R(n)}V\\\
(\mathbb{P}^{n}_{R})^{{\operatorname{gen}}}_{{\rm
id}}@>{}>{}>\mathbb{P}^{n}_{R(n)}.\end{CD}$

######  Remark 1.2.

We shall establish analogous results for pairs $(X,\Delta)$ (Theorem 4.15,
Theorem 4.17).

######  Remark 1.3.

Theorem 1.1 is folklore for the case when $R$ is a field. Indeed, we can apply
a similar argument to the one of [CGM86, Theorem 1] for this case.

Theorem 1.1 assures the existence of good members
$X^{{\operatorname{gen}}}_{\varphi}$. If $k$ is an infinite field, then the
classical Bertini theorem states that general hyperplane sections are good. We
shall establish the following analogous result.

######  Theorem 1.4 (Theorem 5.3, Remark 5.5).

Let $k$ be a field. Fix $n\in\mathbb{Z}_{\geq 0}$ and let $k^{\prime}$ be a
purely transcendental field extension with $n+1\leq{\rm
tr.deg}_{k}k^{\prime}<\infty$. Then there exists a dense subset
$\Lambda\subset(\mathbb{P}^{n}_{k^{\prime}})^{*}(k^{\prime})$ with respect to
the Zariski topology such that the following holds.

Let (P) be a property for noetherian schemes which satisfies the properties
(I)–(III) listed in Theorem 1.1. Let $\varphi:X\to\mathbb{P}^{n}_{k}$ be a
$k$-morphism from a $k$-scheme $X$ of finite type. If $X$ is (P) and
$\lambda\in\Lambda$, then also $X^{\lambda}_{\varphi}$ is (P), where
$X^{\lambda}_{\varphi}$ denotes the scheme-theoretic inverse image of the
hyperplane $H^{\lambda}\subset\mathbb{P}^{n}_{k^{\prime}}$ corresponding to
$\lambda$, i.e. the following diagram is cartesian:

$\begin{CD}X^{\lambda}_{\varphi}@>{}>{}>X\times_{k}k^{\prime}\\\
@V{}V{}V@V{}V{\varphi\times_{k}k^{\prime}}V\\\
<EMAIL_ADDRESS>
Overview of contents and sketches of proofs: In Section 2, we summarise
notation and recall some known results. Furthermore, we introduce the notion
of $\mathbb{A}^{n}$-localising morphisms in Subsection 2.4, which frequently
appears in this paper.

In Section 3, we introduce purely transcendental extensions for rings:
$R\subset R(t_{1},...,t_{n})$ (Definition 3.1). We then establish some
foundational properties. For example,

  1. (i)

the pullback map
${\operatorname{Spm}}\,R(t_{1},...,t_{n})\to{\operatorname{Spm}}\,R$ is
bijective (Theorem 3.7), and

  2. (ii)

$R(t_{1},...,t_{n})\simeq R(t_{1},...,t_{n-1})(t_{n})$ (Proposition 3.8).

Theorem 1.1 claims that results of Bertini type hold after replacing $X$ by
its base change $X\times_{R}R(t_{1},...,t_{n})$. Then it is natural to compare
$X$ and $X\times_{R}R(t_{1},...,t_{n})$. In this direction, we shall establish
the following properties under the assumption that $R$ is an excellent ring.

  1. (iii)

$R\to R(t_{1},...,t_{n})$ is essentially smooth and faithfully flat
(Proposition 3.6). In particular, $X$ is regular (resp. Cohen-Macaulay, resp.
reduced) if and only if so is $X\times_{R}R(t_{1},...,t_{n})$.

  2. (iv)

$X$ is integral (resp. connected) if and only if so is
$X\times_{R}R(t_{1},...,t_{n})$ (Theorem 3.13).

  3. (v)

$\dim X=\dim X\times_{R}R(t_{1},...,t_{n})$ (Theorem 3.22).

We here give a sketch of the proof only for (v). By (ii), we may assume that
$n=1$. Set $t:=t_{1}$, The key part is to determine the fibres of
$X\times_{R}R(t)\to X$ over the closed points of $X$. More specifically, we
show that $\kappa\otimes_{R}R(t)\simeq\kappa(t)$ if $\kappa$ is a field which
is a finitely generated $R$-algebra (Theorem 3.18). We now describe the
outline of how to prove this. By the formula
$(R/I)\otimes_{R}R(t)\simeq(R/I)(t)$ (Proposition 3.5), we may assume that $R$
is a subring of $\kappa$. Applying Hilbert’s Nullstellensatz and the
Artin–Tate lemma, the problem is reduced to the case when $\kappa$ is the
field of fractions of $R$: $\kappa={\operatorname{Frac}}\,R$. Since
${\operatorname{Frac}}\,R$ is a finitely generated $R$-algebra, it follows
from Ratliff’s existence theorem that $R$ is a semi-local ring with $\dim
R\leq 1$. Taking the normalisation $R^{N}$ of $R$, the problem is finally
reduced to the case when $R$ is a principal ideal domain (Lemma 3.16). In this
case, we can directly check the required equation
$\kappa\otimes_{R}R(t)\simeq\kappa(t)$ (Lemma 3.17). For more details, see
Subsection 3.2.

In Section 4, we prove a main theorem: Theorem 1.1. We now explain the main
idea of the proof of Theorem 1.1, which is similar to the one of [CGM86,
Theorem 1]. For simplicity, we assume that $k:=R$ is a field and (P) is
regular. We have the following diagram in which each square is cartesian:

$\begin{CD}X^{{\operatorname{gen}}}_{\varphi}@>{}>{}>X_{k(n)}@>{}>{}>X\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\ (\mathbb{P}^{n}_{k})^{{\operatorname{gen}}}_{{\rm
id}}<EMAIL_ADDRESS>\end{CD}$

In particular, it suffices to find a hyperplane
$(\mathbb{P}^{n}_{k})^{{\operatorname{gen}}}_{{\rm
id}}(\simeq\mathbb{P}^{n-1}_{k(n)})$ of $\mathbb{P}^{n}_{k(n)}$ such that the
induced morphism $(\mathbb{P}^{n}_{k})^{{\operatorname{gen}}}_{{\rm
id}}\to\mathbb{P}^{n}_{k}$ is essentially smooth. For the coordinates
$\mathbb{P}^{n}_{k}={\operatorname{Proj}}\,k[x_{0},...,x_{n}]$ and
$(\mathbb{P}^{n}_{k})^{*}\simeq\mathbb{P}^{n}_{k}={\operatorname{Proj}}\,k[s_{0},...,s_{n}]$,
we set

$(\mathbb{P}^{n}_{k})^{{\operatorname{univ}}}_{{\rm
id}}:=\\{s_{0}x_{0}+\cdots+s_{n}x_{n}=0\\}\subset\mathbb{P}^{n}_{k}\times_{k}(\mathbb{P}^{n}_{k})^{*}.$

Identifying $k(n)=k(t_{1},...,t_{n})$ with the function field of
$(\mathbb{P}^{n}_{k})^{*}$, we define
$(\mathbb{P}^{n}_{k})^{{\operatorname{gen}}}_{{\rm id}}$ as the generic fibre,
i.e. the following cartesian square

$\begin{CD}(\mathbb{P}^{n}_{k})^{{\operatorname{gen}}}_{{\rm
id}}@>{}>{}>(\mathbb{P}^{n}_{k})^{{\operatorname{univ}}}_{{\rm id}}\\\
@V{}V{}V@V{}V{}V\\\
{\operatorname{Spec}}\,k(n)@>{}>{}>(\mathbb{P}^{n}_{k})^{*}.\end{CD}$

Then it is almost straightforward to prove that
$(\mathbb{P}^{n}_{k})^{{\operatorname{gen}}}_{{\rm id}}\to\mathbb{P}^{n}_{k}$
is essentially smooth. For more details, see Proposition 4.6.

In Section 5, we prove Theorem 1.4. We now overview the proof of Theorem 1.4.
Assume that $X$ is (P). For the universal family
$X^{{\operatorname{univ}}}_{\varphi}\subset
X\times_{k}(\mathbb{P}^{n}_{k})^{*}$, we consider the generic fibre
$X^{{\operatorname{gen}}}_{\varphi}$ of
$X^{{\operatorname{univ}}}_{\varphi}\to(\mathbb{P}^{n}_{k})^{*}$. Set

$\Lambda:=\\{\lambda\in(\mathbb{P}^{n}_{k^{\prime}})^{*}(k^{\prime})\,|\,{\rm(I),(II)}\\}.$

  1. (I)

The image of $\lambda$ to $(\mathbb{P}^{n}_{k})^{*}$ is equal to the generic
point of $(\mathbb{P}^{n}_{k})^{*}$.

  2. (II)

The field extension $K((\mathbb{P}^{n}_{k})^{*})\subset\kappa(\lambda)$
induced by (I) is a purely transcendental extension.

Then it suffices to prove the following.

  1. (1)

$\Lambda$ is a dense subset of
$(\mathbb{P}^{n}_{k^{\prime}})^{*}(k^{\prime})$.

  2. (2)

If $\lambda\in\Lambda$, then $X^{\lambda}_{\varphi}$ is (P).

(1) Fix a complement of a hyperplane:
$\mathbb{A}^{n}_{k}={\operatorname{Spec}}\,k[z_{1},...,z_{n}]\subset(\mathbb{P}^{n}_{k})^{*}$.
For a transcendental basis $\\{t_{i}\\}_{i\in I}$ of the purely transcendental
extension $k\subset k^{\prime}$, it follows from ${\rm
tr.deg}_{k}k^{\prime}\geq n+1$ that $|I|\geq n+1$. Then there exist distinct
elements $i_{0},i_{1},...,i_{n}\in I$. We define $\Lambda^{\prime}$ as the set
of the maximal ideals $\lambda$ of $k^{\prime}[z_{1},...,z_{n}]$ of the form

$\lambda=(z_{1}-t_{i_{1}}-t_{i_{0}}^{d_{1}},...,z_{n}-t_{i_{n}}-t_{i_{0}}^{d_{n}})$

for $(d_{1},...,d_{n})\in(\mathbb{Z}_{>0})^{n}$. Then we can check that
$\Lambda^{\prime}\subset\Lambda$, which implies that $\Lambda$ is dense in
$(\mathbb{P}^{n}_{k^{\prime}})^{*}(k^{\prime})$.

(2) Fix $\lambda\in\Lambda$. By the definition of $\Lambda$,
$X^{\lambda}_{\varphi}$ is obtained as the base change of
$X^{{\operatorname{gen}}}_{\varphi}$ by the purely transcendental field
extension $K(\mathbb{P}(V))\subset\kappa(\lambda)$, i.e.
$X^{\lambda}_{\varphi}=X^{{\operatorname{gen}}}_{\varphi}\times_{K(\mathbb{P}(V))}\kappa(\lambda)$.
Since (P) is stable under taking base changes by purely transcendental field
extensions of finite degree, it is enough to show that
$X^{{\operatorname{gen}}}_{\varphi}$ is (P). This is guaranteed by Theorem
1.1.

Acknowledgements: The author would like to thank Karl Schwede and Shunsuke
Takagi for many constructive suggestions and answering questions. He also
thanks the referee for many constructive suggestions and reading the
manuscript carefully. The author was funded by JSPS KAKENHI Grant numbers
JP18K13386 and JP22H01112.

##  2\. Preliminaries

###  2.1. Notation

In this subsection, we summarise notation and terminologies used in this
paper.

  1. (1)

Throughout this paper, $k$ denotes a field. We will freely use the notation
and terminology in [Har77]. We say that an $\mathbb{F}_{p}$-scheme $X$ is
$F$-finite if the absolute Frobenius morphism $F:X\to X$ is a finite morphism.
Given a ring $A$, we denote by ${\operatorname{Spm}}\,A$ the set of the
maximal ideals of $A$.

  2. (2)

We say that a scheme $X$ is excellent if $X$ is a noetherian scheme such that
$\mathcal{O}_{X,x}$ is an excellent ring for any $x\in X$.

  3. (3)

Given a scheme $X$, $X_{{\operatorname{red}}}$ be its reduced structure, i.e.
$X_{{\operatorname{red}}}$ is the reduced closed subscueme of $X$ such that
the induced closed immersion $X_{{\operatorname{red}}}\to X$ is surjective.
For an integral scheme $X$, its normalisation is denoted by $X^{N}$.

  4. (4)

For an integral scheme $X$, we define the function field $K(X)$ of $X$ as
$\mathcal{O}_{X,\xi}$ for the generic point $\xi$ of $X$. Given an integral
domain $A$, we set ${\operatorname{Frac}}\,A:=K({\operatorname{Spec}}\,A)$,
which is the field of fractions of $A$.

  5. (5)

For a ring homomorphism $R\to R^{\prime}$ and an $R$-scheme $X$, we set
$X\times_{R}R^{\prime}:=X\times_{{\operatorname{Spec}}\,R}{\operatorname{Spec}}\,R^{\prime}$.

  6. (6)

Given a ring $A$, we say that an $A$-algebra $B$ is a localisation of $A$ if
$B$ is isomorphic to $S^{-1}A$ as $A$-algebras for some multiplicatively
closed subset of $A$. In this case, also the induced ring homomorphism $A\to
S^{-1}A,a\mapsto a/1$ is called a localisation.

  7. (7)

A variety $X$ (over $k$) is an integral scheme which is separated and of
finite type over $k$.

  8. (8)

Given a finite-dimensional $k$-vector space $V$, we set $\mathbb{P}(V):={\rm
Proj}\,(S(V^{*}))$, where $S(W):=\bigoplus_{d=0}^{\infty}S^{d}(W)$ denotes the
symmetric algebra for $W$. Note that the definition of $\mathbb{P}(V)$ depends
on literature.

  9. (9)

Given an integral scheme $X$, we define $K(X):=\mathcal{O}_{X,\xi}$ where
$\xi$ is the generic point of $X$.

  10. (10)

Let $X$ be a scheme of finite type over $k$ and $L$ an invertible sheaf. Take
$\varphi_{0},...,\varphi_{n}\in H^{0}(X,L)$ and set
$\mathbb{P}^{n}_{k}:={\operatorname{Proj}}\,k[t_{0},...,t_{n}]$. We define
$\\{t_{0}\varphi_{0}+\cdots+t_{n}\varphi_{n}=0\\}$ as the closed subscheme of
$X\times_{k}\mathbb{P}^{n}_{k}$ as follows. Take an affine open cover
$X=\bigcup_{i\in I}X_{i}$ which trivialises $L$, and hence we have an
isomorphism $\theta_{i}:L|_{X_{i}}\xrightarrow{\simeq}\mathcal{O}_{X_{i}}$ for
each $i\in I$. Then $\\{t_{0}\varphi_{0}+\cdots+t_{n}\varphi_{n}=0\\}$ is
defined as the closed subscheme of $X\times_{k}\mathbb{P}^{n}_{k}$ that
satisfies the following equation for each $i\in I$:

$\\{t_{0}\varphi_{0}+\cdots+t_{n}\varphi_{n}=0\\}|_{X_{i}\times_{k}\mathbb{P}^{n}_{k}}:={\operatorname{Proj}}\,\left(\frac{\mathcal{O}_{X}(X_{i})[t_{0},...,t_{n}]}{(t_{0}\theta_{i}(\varphi_{0}|_{X_{i}})+\cdots+t_{n}\theta_{i}(\varphi_{n}|_{X_{i}})}\right).$

We can check that this definition does not depend on the choice of the open
cover $X=\bigcup_{i\in I}X_{i}$.

  11. (11)

Let $X$ be a scheme of finite type over $k$, $L$ an invertible sheaf, and $V$
a $k$-vector subspace of $H^{0}(X,L)$. For an element $\varphi$ of $V$, we
have the closed subscheme of $X$ defined as the zeros of $\varphi$, which is
called a member of $V$. A general member of $V$ is a member of some fixed
Zariski open subset of $V$. For $\varphi\in H^{0}(X,L)$, we set $Z(\varphi)$
to be the closed subset of $X$ that is the zero locus of $\varphi$. Set

${\operatorname{Bs}}(V):=\bigcap_{\varphi\in V}Z(\varphi),$

which is a closed subset of $X$. Note that we have
${\operatorname{Bs}}(\\{0\\})=X$ by definition. We say that $V$ is base point
free if ${\operatorname{Bs}}(V)=\emptyset$.

  12. (12)

Let $A$ be a ring. For $n\in\mathbb{Z}_{\geq 0}$, $A[x_{1},...,x_{n}]$ denotes
the polynomial ring with $n$ variables $x_{1},...,x_{n}$. For
$n\in\mathbb{Z}_{<0}$, we set $A[x_{1},...,x_{n}]:=\\{0\\}$, i.e. the zero
ring. Similarly, if $n\in\mathbb{Z}_{<0}$, then we set
$\mathbb{A}^{n}_{k}:=\emptyset$ and $\mathbb{P}^{n}_{k}:=\emptyset$.

  13. (13)

Given an $\mathbb{R}$-divisor $D$ on a normal variety $X$, we define
$\mathcal{O}_{X}(D)$ by

$\Gamma(U,\mathcal{O}_{X}(D))=\\{\varphi\in K(X)\,|\,({\rm
div}(\varphi)+D)|_{U}\geq 0\\}$

for any open subset $U$ of $X$. In particular, we have
$\mathcal{O}_{X}(D)=\mathcal{O}_{X}(\llcorner D\lrcorner)$.

  14. (14)

Unless otherwise specified, $R[t_{1},...,t_{n}]$ denotes the polynomial ring
over a ring $R$ with variables $t_{1},...,t_{n}$. Similarly,
$R(t_{1},...,t_{n})$ denotes the purely transcendental extension of $R$ (of
degree $n$) in the sense of Definition 3.1. We frequently use the abbreviated
notation $\underline{t}=(t_{1},...,t_{n})$. For example,
$k[\underline{t}]:=k[t_{1},...,t_{n}]$ denotes the polynomial ring over $k$
with variables $t_{1},...,t_{n}$.

  15. (15)

Let $R$ be a ring. We say that $H$ is a hyperplane of
$\mathbb{P}^{n}_{R}={\operatorname{Proj}}\,R[x_{0},...,x_{n}]$ (over $R$) if
$H$ is flat over $R$ and there exist $a_{0},...,a_{n}\in R$ such that
$(a_{0},...,a_{n})\neq(0,...,0)$ and
$H={\operatorname{Proj}}\,R[x_{0},...,x_{n}]/(a_{0}x_{0}+\cdots+a_{n}x_{n})$.
For example, if $a_{0}\in R^{\times}$, then
${\operatorname{Proj}}\,R[x_{0},...,x_{n}]/(a_{0}x_{0}+\cdots+a_{n}x_{n})$ is
a hyperplane of $\mathbb{P}^{n}_{R}$.

###  2.2. Singularities

####  2.2.1. Singularities for schemes

Let $X$ be a noetherian scheme. Fix $n\in\mathbb{Z}_{\geq 0}$.

  1. (i)

$X$ is $R_{n}$ (resp. $G_{n}$) if $\mathcal{O}_{X,x}$ is regular (resp.
Gorenstein) for any point $x\in X$ such that $\dim\mathcal{O}_{X,x}\leq n$.
$X$ is $S_{n}$ if if $\mathcal{O}_{X,x}$ is Cohen–Macaulay for any point $x\in
X$ such that $\dim\mathcal{O}_{X,x}\leq n$.

  2. (ii)

We say that $X$ is a Mori scheme if $X$ is a reduced noetherian scheme such
that the normalisation $X^{N}\to X$ is a finite morphism. We say that $X$ is
semi-normal (resp. weakly normal) if $X$ is a Mori scheme such that
$\mathcal{O}_{X,x}$ is semi-normal (resp. weakly normal) for any $x\in X$. For
basic properties on semi-normal (resp. weakly normal) schemes, we refer to
[GT80] (resp. [Man80]).

  3. (iii)

We say that $X$ is resolution-rational if

     * •

$X$ is a normal excellent scheme,

     * •

$X$ admits a dualising complex, and

     * •

there exists a proper birational morphism $f:X^{\prime}\to X$ from a regular
scheme $X^{\prime}$ such that
$Rf_{*}\mathcal{O}_{X^{\prime}}\simeq\mathcal{O}_{X}$ and
$Rf_{*}\omega_{X^{\prime}}\simeq\omega_{X}$.

It is known that this condition does not depend on the choice of
$f:X^{\prime}\to X$ when $X$ is of finite type over a perfect field [CR11,
Theorem 1].

  4. (iv)

We say that $X$ is $F$-rational if $X$ is a noetherian $\mathbb{F}_{p}$-scheme
such that $\Gamma(U,\mathcal{O}_{X})$ is $F$-rational in the sense of [HH94,
Definition 4.1] for any affine open subset $U$ of $X$. If $X$ is an
$F$-rational excellent scheme, then $\mathcal{O}_{X,x}$ is $F$-rational
[Vel95, Theorem 3.1].

  5. (v)

We say that $X$ is $F$-injective if $X$ is a noetherian
$\mathbb{F}_{p}$-scheme such that $\mathcal{O}_{X,x}$ is $F$-injective in the
sense of [DM, Definition 2.1] for any $x\in X$ (cf. [DM, Proposition 3.3]).

  6. (vi)

Assume that $X$ is a reduced scheme of finite type over a field $k$ is of
characteristic zero. As for Du Bois singularities, we only use the following
characterisation given by [Sch07, Theorem 4.6]. Assume that $X$ is separated
and a closed subscheme of a smooth scheme $Y$ over $k$. Let
$\pi:\widetilde{Y}\to Y$ is a log resolution of $X$ which is isomorphic over
$Y\setminus X$. Set $E$ to be the reduced closed subscheme of $\widetilde{Y}$
that satisfies the set-theoretic equation $E=\pi^{-1}(X)$. Then $X$ is Du Bois
if and only if $\mathcal{O}_{X}\simeq R\pi_{*}\mathcal{O}_{E}$ holds.

####  2.2.2. Singularities of minimal model program

Let $X$ be an integral normal excellent scheme and let $\Delta$ be an
$\mathbb{R}$-divisor on $X$. Assume that $K_{X}+\Delta$ is
$\mathbb{R}$-Cartier and $X$ admits a dualising complex. For the singularities
in minimal model program (terminal, canonical, klt, plt, dlt, lc), we refer to
[BMP, Definition 2.23]. If $X$ is of finite type over a field and $\Delta$ is
a $\mathbb{Q}$-divisor such that $K_{X}+\Delta$ is $\mathbb{Q}$-Cartier, this
definition coincides with the one in [Kol13, Definition 2.8]. We say that $X$
is terminal, klt,…etc if so is $(X,0)$.

####  2.2.3. $F$-Singularities

######  Definition 2.1.

Let $X$ be an integral normal noetherian $F$-finite $\mathbb{F}_{p}$-scheme
and let $\Delta$ be an effective $\mathbb{R}$-divisor on $X$.

  1. (i)

The pair $(X,\Delta)$ is globally $F$-regular if, for every effective
$\mathbb{Z}$-divisor $D$ on $X$, there exists $e\in\mathbb{Z}_{>0}$ such that

$\mathcal{O}_{X}\xrightarrow{F^{e}}F_{*}^{e}\mathcal{O}_{X}\hookrightarrow
F_{*}^{e}\mathcal{O}_{X}(\ulcorner(p^{e}-1)\Delta\urcorner+D)$

splits as an $\mathcal{O}_{X}$-module homomorphism. The pair $(X,\Delta)$ is
globally purely $F$-regular if, for every effective $\mathbb{Z}$-divisor $D$
on $X$ such that ${\operatorname{Supp}}\,D$ contains no irreducible component
of $\llcorner\Delta\lrcorner$, there exists $e\in\mathbb{Z}_{>0}$ such that

$\mathcal{O}_{X}\xrightarrow{F^{e}}F_{*}^{e}\mathcal{O}_{X}\hookrightarrow
F_{*}^{e}\mathcal{O}_{X}(\ulcorner(p^{e}-1)\Delta\urcorner+D)$

splits as an $\mathcal{O}_{X}$-module homomorphism. The pair $(X,\Delta)$ is
globally sharply $F$-split if there exists $e\in\mathbb{Z}_{>0}$ such that

$\mathcal{O}_{X}\xrightarrow{F^{e}}F_{*}^{e}\mathcal{O}_{X}\hookrightarrow
F_{*}^{e}\mathcal{O}_{X}(\ulcorner(p^{e}-1)\Delta\urcorner)$

splits as an $\mathcal{O}_{X}$-module homomorphism. The pair $(X,\Delta)$ is
globally $F$-split if

$\mathcal{O}_{X}\xrightarrow{F^{e}}F_{*}^{e}\mathcal{O}_{X}\hookrightarrow
F_{*}^{e}\mathcal{O}_{X}(\llcorner(p^{e}-1)\Delta\lrcorner)$

splits as an $\mathcal{O}_{X}$-module homomorphism for any
$e\in\mathbb{Z}_{>0}$.

  2. (ii)

Let $f:X\to Y$ be a morphism to a scheme $Y$. The pair $(X,\Delta)$ is
globally $F$-regular over $Y$ (resp. globally purely $F$-regular over $Y$,
globally sharply $F$-split over $Y$, resp. globally $F$-split over $Y$) if,
for some open cover $Y=\bigcup_{i\in I}Y_{i}$, so is
$(f^{-1}(Y_{i}),\Delta|_{f^{-1}(Y_{i})})$ for any $i\in I$.

  3. (iii)

The pair $(X,\Delta)$ is strongly $F$-regular (resp. purely $F$-regular, resp.
sharply $F$-pure, resp. $F$-pure) if $(X,\Delta)$ is globally $F$-regular over
$X$ (resp. globally purely $F$-regular over $X$, resp. globally sharply
$F$-split over $X$, resp. globally $F$-split over $X$) with respect to the
identity morphism ${\rm id}:X\to X$.

  4. (iv)

We say that $X$ is strongly $F$-regular, globally $F$-regular,…etc if so is
$(X,0)$.

######  Lemma 2.2.

Let $X$ be an integral normal noetherian $F$-finite $\mathbb{F}_{p}$-scheme
and let $f:X\to Y$ be a morphism to a quasi-compact scheme $Y$. Let $E$ be an
effective $\mathbb{Z}$-divisor and let $\Delta$ be an effective
$\mathbb{R}$-divisor on $X$. Then the following hold.

  1. (1)

If $X$ is affine, then the following are equivalent.

    1. (a)

$\mathcal{O}_{X}\to F_{*}^{e}\mathcal{O}_{X}\hookrightarrow
F_{*}^{e}\mathcal{O}_{X}(E)$ splits.

    2. (b)

$\mathcal{O}_{X,x}\to(F_{*}^{e}\mathcal{O}_{X})_{x}\hookrightarrow(F_{*}^{e}\mathcal{O}_{X}(E))_{x}$
splits for any $x\in X$.

  2. (2)

Assume that $(X,\Delta)$ is globally sharply $F$-split over $Y$. Then there
exists a $\mathbb{Q}$-divisor $\Delta^{\prime}$ such that
$\Delta\leq\Delta^{\prime}$,
${\operatorname{Supp}}\,\Delta={\operatorname{Supp}}\,\Delta^{\prime}$, and
$(X,\Delta^{\prime})$ is globally sharply $F$-split over $Y$.

  3. (3)

If $(X,\Delta)$ is globally $F$-regular over $Y$, then there exists an
effective $\mathbb{Q}$-divisor $\Delta^{\prime}$ such that
$\Delta\leq\Delta^{\prime}$ and $(X,\Delta^{\prime})$ is globally $F$-regular
over $Y$.

###### Proof.

The assertion (1) follows from the fact that the splitting of
$F^{e}(E):\mathcal{O}_{X}\xrightarrow{F^{e}}F_{*}^{e}\mathcal{O}_{X}\hookrightarrow
F_{*}^{e}\mathcal{O}_{X}(E)$ is equivalent to the surjectivity of

$(F^{e}(E))^{*}:H^{0}(X,\mathcal{H}om_{\mathcal{O}_{X}}(F_{*}^{e}\mathcal{O}_{X}(E),\mathcal{O}_{X}))\to
H^{0}(X,\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{O}_{X},\mathcal{O}_{X})).$

Let us show (2). We first treat the case when
$Y={\operatorname{Spec}}\,\mathbb{F}_{p}$, i.e. $X$ is globally sharply
$F$-split. In this case, there exists $e\in\mathbb{Z}_{>0}$ such that
$\mathcal{O}_{X}\to F_{*}^{e}\mathcal{O}_{X}\hookrightarrow
F_{*}^{e}\mathcal{O}_{X}(\ulcorner(p^{e}-1)\Delta\urcorner)$ splits. By
definition, also $(X,\Delta^{\prime})$ is globally sharply $F$-split for
$\Delta^{\prime}:=\frac{1}{p^{e}-1}\ulcorner(p^{e}-1)\Delta\urcorner$. This
completes the proof for the case when
$Y={\operatorname{Spec}}\,\mathbb{F}_{p}$.

Let us go back to the general case. By Definition 2.1, there exists an open
cover $Y=\bigcup_{i\in I}Y_{i}$ such that
$(f^{-1}(Y_{i}),\Delta_{i}|_{f^{-1}(Y_{i})})$ is globally sharply $F$-split,
i.e. globally sharply $F$-split over ${\operatorname{Spec}}\,\mathbb{F}_{p}$.
Since $Y$ is quasi-compact, we may assume that $I$ is a finite set. As we have
already proven the case when $Y={\operatorname{Spec}}\,\mathbb{F}_{p}$, if
$i\in I$, then there exists an effective $\mathbb{Q}$-divisor $\Gamma_{i}$
such that $\Delta|_{f^{-1}(Y_{i})}\leq\Gamma_{i}$,
${\operatorname{Supp}}\,\Delta|_{f^{-1}(Y_{i})}={\operatorname{Supp}}\,\Gamma_{i}$,
and $(f^{-1}(Y_{i}),\Gamma_{i})$ is globally sharply $F$-split. Let
$\Delta=\sum_{k}\delta_{k}\Delta_{k}$ and
$\Gamma_{i}=\sum_{k}\gamma_{k,i}\Gamma_{k}|_{f^{-1}(Y_{i})}$ be the
irreducible composition. For any $k$ and $i\in I$, we define
$\delta^{\prime}_{k,i}\in\mathbb{Q}\cup\\{\infty\\}$ as follows:

$\delta^{\prime}_{k,i}:=\begin{cases}\gamma_{k,i}\qquad\text{if
}\Delta_{k}\cap f^{-1}(Y_{i})\neq\emptyset\\\ \infty\qquad\text{if
}\Delta_{k}\cap f^{-1}(Y_{i})=\emptyset.\end{cases}$

We set

$\delta^{\prime}_{k}:=\min_{i\in
I}\delta^{\prime}_{k,i}\in\mathbb{Q}_{>0},\qquad\Delta^{\prime}:=\sum_{k}\delta^{\prime}_{k}\Delta_{k}.$

By $\Delta^{\prime}|_{f^{-1}(Y_{i})}\leq\Gamma_{i}$,
$(f^{-1}(Y_{i}),\Delta^{\prime}|_{f^{-1}(Y_{i})})$ is globally sharply
$F$-split for any $i\in I$, i.e. $(X,\Delta^{\prime})$ is globally sharply
$F$-split over $Y$. Thus, (2) holds.

Let us show (3). By the same argument as in (2), we may assume that
$Y={\operatorname{Spec}}\,\mathbb{F}_{p}$, i.e. $X$ is globally $F$-regular.
Fix an effective $\mathbb{Z}$-divisor $C$ such that
${\operatorname{Supp}}\,\Delta\subset{\operatorname{Supp}}\,C$. Then there
exists $e$ such that

$\mathcal{O}_{X}\to
F_{*}^{e}\mathcal{O}_{X}(\ulcorner(p^{e}-1)\Delta\urcorner+C)$

splits. Enlarging the irrational coefficients of $\Delta$, we can find an
effective $\mathbb{Q}$-divisor $\Delta^{\prime}$ such that
$\Delta\leq\Delta^{\prime}$,
${\operatorname{Supp}}\,\Delta={\operatorname{Supp}}\,\Delta^{\prime}$, and
$\ulcorner(p^{e}-1)\Delta\urcorner=\ulcorner(p^{e}-1)\Delta^{\prime}\urcorner$.
Since $(X\setminus C,\Delta^{\prime}|_{X\setminus C})$ is globally
$F$-regular, it follows from [SS10, Theorem 3.9] that $(X,\Delta^{\prime})$ is
globally $F$-regular. ∎

###  2.3. Localisation of rings

We collect some basic results on localisation of rings. Although the results
in this subsection might be well known, we include the proofs, as the author
could not find appropriate references.

######  Lemma 2.3.

Let $A$ be a ring. Let $S$ and $T$ be multiplicatively closed subsets of $A$
such that an $A$-algebra homomorphism $\varphi:S^{-1}A\to T^{-1}A$ exists:

$\theta^{\prime}:A\xrightarrow{\theta}S^{-1}A\xrightarrow{\varphi}T^{-1}A,$

where $\theta$ and $\theta^{\prime}$ denote the induced ring homomorphisms. If
$\theta^{\prime}:A\to T^{-1}A$ is injective, then also $\varphi:S^{-1}A\to
T^{-1}A$ is injective.

###### Proof.

Take $a/s\in S^{-1}A$ such that $\varphi(a/s)=0$ in $T^{-1}A$, where $a\in A$
and $s\in S$. We also have

$0=\varphi(a/1)=\varphi(\theta(a))=\theta^{\prime}(a).$

Since $\theta^{\prime}$ is injective, we get $a=0$ in $A$. Hence, $a/s=0$
holds in $S^{-1}A$, as required. ∎

######  Lemma 2.4.

Let $A$ be a ring. Let $S$ be a multiplicatively closed subset of $A$ and let
$T$ be a multiplicatively closed subset of $S^{-1}A$. Then there exists a
multiplicatively closed subset $U$ of $A$ such that there is an isomorphim
$T^{-1}(S^{-1}A)\simeq U^{-1}A$ of $A$-algebras.

###### Proof.

Set

  * •

$T^{\prime}:=\\{t^{\prime}\in A\,|\,\frac{t^{\prime}}{s^{\prime}}\in T\text{
for some }s^{\prime}\in S\\}$ and

  * •

$U:=ST^{\prime}:=\\{st^{\prime}\in A\,|\,s\in S,t^{\prime}\in T^{\prime}\\}$.

Then $T^{\prime}$ is a multiplicatively closed subset of $A$. Hence, also $U$
is a multiplicatively closed subset of $A$.

We now prove that the induced ring homomorphism

$\theta:A\to T^{-1}(S^{-1}A)$

factors through the induced one $A\to U^{-1}A$. Pick $st^{\prime}\in U$ with
$s\in S,t^{\prime}\in T^{\prime}$. We have $t^{\prime}/s^{\prime}\in T$ for
some $s^{\prime}\in S$. Then we obtain

$\theta(st^{\prime})=\frac{st^{\prime}/1}{1/1}=\frac{(ss^{\prime})/1}{1/1}\cdot\frac{t^{\prime}/s^{\prime}}{1/1},$

which is contained in $(T^{-1}(S^{-1}A))^{\times}$, as required. Therefore, we
get the induced ring homomorphism:

$\theta^{\prime}:U^{-1}A\to
T^{-1}(S^{-1}A),\qquad\frac{a}{u}\mapsto\theta(u)^{-1}\theta(a).$

for $a\in A$ and $u\in U$.

Let us show that $\theta^{\prime}$ is injective. Fix $a\in A$ and assume that
$\theta^{\prime}(a/1)=0$. Then there exists $t\in T$ such that $t\cdot(a/1)=0$
holds in $S^{-1}A$. We can write $t=t^{\prime}/s^{\prime}$ for some
$t^{\prime}\in T^{\prime}$ and $s^{\prime}\in S^{\prime}$. Therefore, we
obtain $s^{\prime\prime}t^{\prime}a=0$ in $A$ for some $t^{\prime}\in
T^{\prime}$ and $s^{\prime\prime}\in S$. By $s^{\prime\prime}t^{\prime}\in U$,
we get $a/1=0/1$ in $U^{-1}A$. Thus $\theta^{\prime}$ is injective.

Let us show that $\theta^{\prime}$ is surjective. Pick $\zeta\in
T^{-1}(S^{-1}A)$. We can write $\zeta=\frac{a/s}{t}$ for some $a\in A,s\in
S,t\in T$. We further have $t=t^{\prime}/s^{\prime}$ for some $t^{\prime}\in
T^{\prime},s^{\prime}\in S$. It holds that

$\zeta=\frac{a/s}{t}=\frac{a/s}{t^{\prime}/s^{\prime}}.$

We can check that $\zeta=\theta^{\prime}(as^{\prime}/st^{\prime})$ for
$as^{\prime}\in A,st^{\prime}\in U$, as required. ∎

######  Lemma 2.5.

Let $A$ be a ring and let $S$ be a multiplicatively closed subset of $A$. Set

  * •

$M(S):={\rm
Im}(\theta^{*}:{\operatorname{Spm}}\,S^{-1}A\to{\operatorname{Spec}}\,A)$, and

  * •

$\overline{S}:=\bigcap_{\mathfrak{p}\in M(S)}(A\setminus\mathfrak{p})$,

where $\theta^{*}:{\operatorname{Spm}}\,S^{-1}A\to{\operatorname{Spec}}\,A$ is
defined by $\theta^{*}(\mathfrak{n}):=\theta^{-1}(\mathfrak{n})$ for the
natural ring homomorphism $\theta:A\to S^{-1}A,a\mapsto a/1$. Then the
following hold.

  1. (1)

The inclusion $S\subset\overline{S}$ holds. Furthermore, the induce
$A$-algebra homomorphism $S^{-1}A\to\overline{S}^{-1}A$ is an isomorphism.

  2. (2)

Let $S_{1}$ and $S_{2}$ be multiplicatively closed subsets of $A$. If
$M(S_{1})=M(S_{2})$, then an $A$-algebra isomorphism $S_{1}^{-1}A\simeq
S_{2}^{-1}A$ holds.

###### Proof.

Since (1) immediately implies (2), it suffices to show (1). Let us show the
inclusion $S\subset\overline{S}$. Fix $s\in S$ and $\mathfrak{p}\in M(S)$.
Note that $\mathfrak{p}S^{-1}A$ is a maximal ideal of $S^{-1}A$. It follows
from $s/1\in(S^{-1}A)^{\times}$ that $s/1\not\in\mathfrak{p}S^{-1}A$.
Therefore, we obtain $s\not\in\theta^{-1}(\mathfrak{p}S^{-1}A)=\mathfrak{p}$.
Since $\mathfrak{p}$ is chosen to be an arbitrary element of $M(S)$, we get
$s\in\overline{S}$, i.e. $S\subset\overline{S}$.

We then obtain an $A$-algebra homomorphism $S^{-1}A\to\overline{S}^{-1}A$. Fix
$t\in\overline{S}$. It is enough to show that
$\theta(t)\in(S^{-1}A)^{\times}$. Take
$\mathfrak{n}\in{\operatorname{Spm}}\,(S^{-1}A)$. It holds that

$t\in\overline{S}=\bigcap_{\mathfrak{p}\in
M(S)}(A\setminus\mathfrak{p})\subset A\setminus\theta^{-1}(\mathfrak{n}).$

Hence we get $\theta(t)\not\in\mathfrak{n}$. Therefore, we obtain
$t\in(S^{-1}A)^{\times}$, as required. ∎

###  2.4. $\mathbb{A}^{n}$-localising morphisms

In this subsection, we first introduce notions of $\mathbb{A}^{n}$-localising
morphisms and strongly $\mathbb{A}^{n}$-localising morphisms (Definition 2.6).
We then establish some fundamental properties, e.g. we study the behaviour
under base changes and compositions (Proposition 2.7). In a main theorem of
this paper (Theorem 4.7), we shall prove that
$X^{{\operatorname{gen}}}_{\varphi}\to X$ is an $\mathbb{A}^{n}$-localising
morphism under suitable assumptions. In this sense, what we are mainly
interested in is not strongly $\mathbb{A}^{n}$-localising morphisms but
$\mathbb{A}^{n}$-localising morphisms.

######  Definition 2.6.

Fix $n\in\mathbb{Z}$.

  1. (1)

We say that a ring homomorphism $\varphi:A\to B$ is
$\mathbb{A}^{n}$-localising if $B$ is isomorphic to
$S^{-1}(A[x_{1},...,x_{n}])$ as an $A$-algebra for some multiplicatively
closed subset $S$ of $A[x_{1},...,x_{n}]$, where $A[x_{1},...,x_{n}]$ denotes
the polynomial ring over $A$ with variables $x_{1},...,x_{n}$. Note that if
$n<0$, then we have $A[x_{1},...,x_{n}]=\\{0\\}$ and $B=\\{0\\}$ under our
terminology (Subsection 2.1(12)).

  2. (2)

We say that a morphism $f:X\to Y$ of schemes is $\mathbb{A}^{n}$-localising if

    1. (a)

$f$ is affine and

    2. (b)

there exists an affine open cover $Y=\bigcup_{i\in I}Y_{i}$ of $Y$ such that
the induced ring homomorphism
$\mathcal{O}_{Y}(Y_{i})\to\mathcal{O}_{X}(f^{-1}(Y_{i}))$ is
$\mathbb{A}^{n}$-localising for any $i\in I$.

  3. (3)

We say that a morphism $f:X\to Y$ of schemes is strongly
$\mathbb{A}^{n}$-localising if

    1. (a)

$f$ is affine and

    2. (b)

for any affine open subset $Y^{\prime}$ of $Y$, the induced ring homomorphism
$\mathcal{O}_{Y}(Y^{\prime})\to\mathcal{O}_{X}(f^{-1}(Y^{\prime}))$ is
$\mathbb{A}^{n}$-localising.

######  Proposition 2.7.

Fix $n,m\in\mathbb{Z}$.

  1. (1)

Let

$\begin{CD}X^{\prime}@>{f^{\prime}}>{}>Y^{\prime}\\\ @V{}V{}V@V{}V{}V\\\
X@>{f}>{}>Y\end{CD}$

be a cartesian diagram of schemes.

    1. (a)

If $f$ is an $\mathbb{A}^{n}$-localising morphism, then also $f^{\prime}$ is
an $\mathbb{A}^{n}$-localising morphism.

    2. (b)

If $f$ is a strongly $\mathbb{A}^{n}$-localising morphism and $Y$ is an affine
scheme, then also $f^{\prime}$ is a strongly $\mathbb{A}^{n}$-localising
morphism.

  2. (2)

Let $\varphi:A\to B$ be a ring homomorphism. Then $\varphi$ is an
$\mathbb{A}^{n}$-localising ring homomorphism if and only if the induced
morphism ${\operatorname{Spec}}\,B\to{\operatorname{Spec}}\,A$ is strongly
$\mathbb{A}^{n}$-localising.

  3. (3)

Let $A\xrightarrow{\varphi}B\xrightarrow{\psi}C$ be ring homomorphisms. If
$\varphi$ is $\mathbb{A}^{n}$-localising and $\psi$ is
$\mathbb{A}^{m}$-localising, then $\psi\circ\varphi$ is
$\mathbb{A}^{n+m}$-localising.

  4. (4)

Let $X\xrightarrow{f}Y\xrightarrow{g}Z$ be morphisms of schemes. If $f$ is
strongly $\mathbb{A}^{n}$-localising and $g$ is $\mathbb{A}^{m}$-localising,
then $g\circ f$ is $\mathbb{A}^{n+m}$-localising.

  5. (5)

Let $f_{0}:X_{0}\to Y_{0}$ be a morphism of affine schemes such that the
induced ring homomorphism
$\mathcal{O}_{Y_{0}}(Y_{0})\to\mathcal{O}_{X_{0}}(X_{0})$ is
$\mathbb{A}^{n}$-localising. Let

$\begin{CD}X@>{f}>{}>Y\\\ @V{}V{}V@V{}V{}V\\\
X_{0}@>{f_{0}}>{}>Y_{0},\end{CD}$

be a cartesian diagram of schemes. If $g:Y\to Z$ is an
$\mathbb{A}^{m}$-localising morphism of schemes, then the composite morphism
$g\circ f:X\to Z$ is an $\mathbb{A}^{n+m}$-localising morphism.

###### Proof.

Let us show (1). The assertion (a) is clear. We now show (b). Replacing
$Y^{\prime}$ by an affine open subset of $Y^{\prime}$, we may assume that
$Y^{\prime}$ is an affine scheme. Then all the schemes $X,Y,X^{\prime},$ and
$Y^{\prime}$ are affine, and hence the assertion (b) holds, which completes
the proof of (1). The assertion (2) directly follows from (b) of (1).

Let us show (3). We have

$B=S^{-1}(A[x_{1},...,x_{n}])=S^{-1}(A[\underline{x}])\quad\text{and}\quad
C=T^{-1}(B[y_{1},...,y_{m}])=T^{-1}(B[\underline{y}]),$

where $\underline{x}$ and $\underline{y}$ denote the multi-variables. We then
obtain

$C=T^{-1}(B[\underline{y}])=T^{-1}((S^{-1}(A[\underline{x}]))[\underline{y}]).$

We define a multiplicatively closed subset $\widetilde{S}$ of
$A[\underline{x},\underline{y}]$ by

$\widetilde{S}:={\rm Im}(S\hookrightarrow A[\underline{x}]\to
A[\underline{x},\underline{y}]),$

where each of $S\hookrightarrow A[\underline{x}]$ and $A[\underline{x}]\to
A[\underline{x},\underline{y}]$ denotes the induced map. Then it is easy to
see that

$(S^{-1}(A[\underline{x}]))[\underline{y}]\simeq\widetilde{S}^{-1}(A[\underline{x},\underline{y}]).$

Therefore, we obtain

$C=T^{-1}((S^{-1}(A[\underline{x}]))[\underline{y}])\simeq
T^{-1}(\widetilde{S}^{-1}(A[\underline{x},\underline{y}]))\simeq
U^{-1}(A[\underline{x},\underline{y}])$

for some multiplicatively closed subset $U$ of
$A[\underline{x},\underline{y}]$ (Lemma 2.4). Thus (3) holds.

Let us show (4). Let $Z=\bigcup_{i\in I}Z_{i}$ be an affine open cover of $Z$
such that if we set $Y_{i}:=g^{-1}(Z_{i})$, then
$\mathcal{O}_{Z}(Z_{i})\to\mathcal{O}_{Y}(Y_{i})$ is
$\mathbb{A}^{n}$-localising for any $i\in I$. Set $X_{i}:=f^{-1}(Y_{i})$ for
any $i\in I$. As $f$ is strongly $\mathbb{A}^{n}$-localising,
$\mathcal{O}_{Y}(Y_{i})\to\mathcal{O}_{X}(X_{i})$ is
$\mathbb{A}^{n}$-localising. Then (3) implies that
$\mathcal{O}_{Z}(Z_{i})\to\mathcal{O}_{Y}(Y_{i})\to\mathcal{O}_{X}(X_{i})$ is
$\mathbb{A}^{n+m}$-localising. Hence (4) holds.

Let us show (5). By (b) of (1), $f$ is strongly $\mathbb{A}^{n}$-localising.
Then (4) implies that $g\circ f$ is $\mathbb{A}^{n+m}$-localising. ∎

##  3\. Purely transcendental extensions for rings

###  3.1. Purely transcendental extensions for rings

In this subsection, we first introduce purely transcendental extensions for
rings (Definition 3.1):

$R\subset R(t_{1},...,t_{n}),$

which generalise purely transcendental extensions for fields. We then
establish some foundational properties as follows:

  1. (1)

$R(t_{1},...,t_{n})/IR(t_{1},...,t_{n})\simeq(R/I)(t_{1},...,t_{n})$
(Proposition 3.5).

  2. (2)

$R\to R(t_{1},...,t_{n})$ is faithfully flat and essentially smooth
(Proposition 3.6).

  3. (3)

The map ${\operatorname{Spm}}R(t_{1},...,t_{n})\to{\operatorname{Spm}}R$
defined by the pullback is bijective (Theorem 3.7).

  4. (4)

$R(t_{1},...,t_{n})\simeq R(t_{1},...,t_{n-1})(t_{n})$ (Proposition 3.8).

######  Definition 3.1.

Let $R$ be a ring. Set

  1. (1)

$R[\underline{t}]:=R[t_{1},...,t_{n}]$,

  2. (2)

$U_{R[\underline{t}]/R}:=\bigcap_{\mathfrak{p}\in{\operatorname{Spec}}\,R}(R[\underline{t}]\setminus\mathfrak{p}R[\underline{t}])=\bigcap_{\mathfrak{m}\in{\operatorname{Spm}}\,R}(R[\underline{t}]\setminus\mathfrak{m}R[\underline{t}])$,
and

  3. (3)

$R(\underline{t}):=R(t_{1},...,t_{n}):=U_{R[\underline{t}]/R}^{-1}(R[\underline{t}])$.

It is obvious that $U_{R[\underline{t}]/R}$ is a multiplicatively closed
subset of $R[\underline{t}]$. We call $R(t_{1},...,t_{n})$ a purely
transcendental extension of $R$ (of degree $n$).

######  Remark 3.2.

Let $k$ be a field. Let $k(t_{1},...,t_{n})$ be as in Definition 3.1. Then
$k(t_{1},...,t_{n})$ is nothing but a purely transcendental extension of $k$
of degree $n$ in the usual sense. This follows from

$U_{k[\underline{t}]/k}=\bigcap_{\mathfrak{p}\in{\operatorname{Spec}}\,k}k[\underline{t}]\setminus\mathfrak{p}k[\underline{t}]=k[t_{1},...,t_{n}]\setminus\\{0\\}.$

######  Lemma 3.3.

Let $R$ be a ring. Then the induced ring homomorphism

$R[t_{1},...,t_{n}]\to R(t_{1},...,t_{n})$

is injective. In particular, if $R\neq 0$, then $R(t_{1},...,t_{n})\neq 0$.

###### Proof.

In what follows, $\underline{t}$ denotes the multi-variable
$(t_{1},...,t_{n})$. Fix $u\in U_{R[\underline{t}]/R}\subset
R[\underline{t}]$. It suffices to show that $u$ is not a zero-divisor of
$R[\underline{t}]$. Suppose that $u$ is a zero-divisor of $R[\underline{t}]$.
Then there exists $r\in R\setminus\\{0\\}$ such that $ru=0$ (cf. [AM69, iii)
of Exercise 2 in Chapter 1]).

Fix $\mathfrak{p}\in{\operatorname{Spec}}\,R$. By $ru=0$ and

$u\in U_{R[\underline{t}]/R}\subset
R[\underline{t}]\setminus\mathfrak{p}R[\underline{t}],$

we can find a coefficient $u^{\prime}$ of $u$ such that $u^{\prime}\in
R\setminus\mathfrak{p}$ and $ru^{\prime}=0$, which implies that
$r_{\mathfrak{p}}=0$ in $R_{\mathfrak{p}}$, where $r_{\mathfrak{p}}$ denotes
the image: $R\to R_{\mathfrak{p}},r\mapsto r_{\mathfrak{p}}$. Then it follows
from [AM69, Proposition 3.8] that $r=0$. This is a contradiction. ∎

######  Lemma 3.4.

Let $\varphi:R\to R^{\prime}$ be a ring homomorphism and let
$\widetilde{\varphi}:R[t_{1},...,t_{n}]\to R^{\prime}[t_{1},...,t_{n}]$ be the
ring homomorphism such that $\widetilde{\varphi}(t_{i})=t_{i}$ for any $1\leq
i\leq n$ and $\widetilde{\varphi}(r)=\varphi(r)$ for any $r\in R$. Then
$\widetilde{\varphi}$ induces a ring homomorphism

$\psi:R(t_{1},...,t_{n})\to R^{\prime}(t_{1},...,t_{n}),$

i.e., the follwoing diagram is commutative

$\begin{CD}R[t_{1},...,t_{n}]@>{\widetilde{\varphi}}>{}>R^{\prime}[t_{1},...,t_{n}]\\\
@V{}V{}V@V{}V{}V\\\
R(t_{1},...,t_{n})@>{\psi}>{}>R^{\prime}(t_{1},...,t_{n}),\end{CD}$

where the vertical arrows are the induced ring homomorphisms.

###### Proof.

In what follows, $\underline{t}$ denotes the multi-variable
$(t_{1},...,t_{n})$. We have

$\widetilde{\varphi}:R[\underline{t}]\to R^{\prime}[\underline{t}].$

Fix $u(\underline{t})\in U_{R[\underline{t}]/R}$. By
$R(\underline{t})=U_{R[\underline{t}]/R}^{-1}R[\underline{t}]$ and
$R^{\prime}(\underline{t})=U_{R^{\prime}[\underline{t}]/R^{\prime}}^{-1}R^{\prime}[\underline{t}]$,
it suffices to show $\widetilde{\varphi}(u(\underline{t}))\in
U_{R^{\prime}[\underline{t}]/R^{\prime}}$. Take
$\mathfrak{q}\in{\operatorname{Spec}}\,R^{\prime}$. It is enough to prove
$\widetilde{\varphi}(u(\underline{t}))\not\in\mathfrak{q}R^{\prime}[\underline{t}]$.
Set $\mathfrak{p}:=\varphi^{-1}(\mathfrak{q})$. By $u(\underline{t})\in
U_{R[\underline{t}]/R}\subset
R[\underline{t}]\setminus\mathfrak{p}R[\underline{t}]$, we get
$u(\underline{t})\not\in\mathfrak{p}R[\underline{t}]=\varphi^{-1}(\mathfrak{q})R[\underline{t}]$,
which implies that
$\widetilde{\varphi}(u(\underline{t}))\not\in\mathfrak{q}R^{\prime}[\underline{t}]$.
∎

######  Proposition 3.5.

Let $R$ be a noetherian ring and let $I$ be an ideal of $R$. Then the induced
ring homomorphism

$\theta:R(t_{1},...,t_{n})/IR(t_{1},...,t_{n})\to(R/I)(t_{1},...,t_{n})$

is an isomorphim.

###### Proof.

Let $\underline{t}:=(t_{1},...,t_{n})$ be the multi-variable. We first recall
the construction of $\theta$. By the natural surjective ring homomorphisms
$\pi:R\to R/I$ and $\pi^{\prime}:R[\underline{t}]\to(R/I)[\underline{t}]$, we
obtain $\pi^{\prime\prime}:R(\underline{t})\to(R/I)(\underline{t})$ (Lemma
3.4). It follows from $\pi^{\prime\prime}(I)=0$ that

$\theta:R(\underline{t})/IR(\underline{t})\to(R/I)(\underline{t})$

is induced.

Recall that

$U_{R[\underline{t}]/R}=\bigcap_{\mathfrak{p}\in{\operatorname{Spec}}\,R}(R[\underline{t}]\setminus\mathfrak{p}R[\underline{t}])\quad\text{and}\quad
U_{(R/I)[\underline{t}]/(R/I)}=\bigcap_{\mathfrak{q}\in{\operatorname{Spec}}\,(R/I)}((R/I)[\underline{t}]\setminus\mathfrak{q}(R/I)[\underline{t}]).$

Set

$\overline{U}_{R[\underline{t}]/R}:=\pi^{\prime}(U_{R[\underline{t}]/R}).$

Note that we have

$R(\underline{t})/IR(\underline{t})\simeq\overline{U}_{R[\underline{t}]/R}^{-1}((R/I)[\underline{t}])$

and

$(R/I)(\underline{t})=U_{(R/I)[\underline{t}]/(R/I)}^{-1}((R/I)[\underline{t}]).$

In order to prove that $\theta$ is an isomorphism, it suffices to show that
there exists an $(R/I)[\underline{t}]$-algebra homomorphism

$(R/I)(\underline{t})\to R(\underline{t})/IR(\underline{t}).$

Hence it suffices to show $\overline{U}_{R[\underline{t}]/R}\supset
U_{(R/I)[\underline{t}]/(R/I)}$.

Take $\overline{f}(\underline{t})\in
U_{(R/I)[\underline{t}]/(R/I)}\subset(R/I)[\underline{t}]$. Pick an element
$f(\underline{t})\in R[\underline{t}]$ such that
$\pi^{\prime}(f(\underline{t}))=\overline{f}(\underline{t})$. Take a set of
generator of $I$:

$I=(\alpha_{1},...,\alpha_{r}).$

Set $N:=\deg f$. We then take another lift $f^{\prime}(x)$ of
$\overline{f}(x)$ as follows:

$f^{\prime}(\underline{t}):=f(\underline{t})+\alpha_{1}t_{1}^{N+1}+\alpha_{2}t_{1}^{N+2}+\cdots+\alpha_{r}t_{1}^{N+r}\in
R[\underline{t}].$

It suffices to show $f^{\prime}(\underline{t})\in U_{R[\underline{t}]/R}$.

Fix $\mathfrak{p}\in{\operatorname{Spec}}\,R$. It is enough to prove
$f^{\prime}(\underline{t})\not\in\mathfrak{p}R[\underline{t}]$. If
$I\not\subset\mathfrak{p}$, then we have $\alpha_{i}\not\in\mathfrak{p}$ for
some $i$, which implies
$f^{\prime}(\underline{t})\not\in\mathfrak{p}R[\underline{t}]$, as required.
Therefore, we may assume that $I\subset\mathfrak{p}$. In this case,
$\mathfrak{q}:=\pi(\mathfrak{p})$ is a prime ideal of $R/I$. By
$\overline{f}(\underline{t})\in U_{(R/I)[\underline{t}]/(R/I)}$, we have
$\overline{f}(\underline{t})\not\in\mathfrak{q}(R/I)[\underline{t}]$. We then
obtain $f^{\prime}(\underline{t})\not\in\mathfrak{p}R[\underline{t}]$ by
$\mathfrak{p}=\pi^{-1}(\mathfrak{q})$. ∎

######  Proposition 3.6.

Let $R$ be a noetherian ring and let $n\in\mathbb{Z}_{\geq 0}$. Then the
induced morphism

$f:{\operatorname{Spec}}\,R(t_{1},...,t_{n})\to{\operatorname{Spec}}\,R$

is a strongly $\mathbb{A}^{n}$-localising surjective morphism. In particular,
$f$ is essentially smooth and faithfully flat.

###### Proof.

It is clear that $f$ is a strongly $\mathbb{A}^{n}$-localising surjective
morphism (cf. Definition 2.6, Proposition 2.7(2)). In particular, $f$ is flat.
Therefore, it suffices to show that
$R(t_{1},...,t_{n})\otimes_{R}(R/\mathfrak{m})\neq 0$ for any maximal ideal
$\mathfrak{m}$ of $R$. By Proposition 3.5, we obtain

$R(t_{1},...,t_{n})\otimes_{R}(R/\mathfrak{m})\simeq(R/\mathfrak{m})(t_{1},...,t_{n}),$

which is a field (Remark 3.2) and hence not the zero ring. ∎

The following theorem describes the maximal ideals of $R(t_{1},...,t_{n})$.

######  Theorem 3.7.

Let $R$ be a noetherian ring. Then the following hold.

  1. (1)

Let $\mathfrak{q}$ be a prime ideal of $R[t_{1},...,t_{n}]$. Then
$\mathfrak{q}\cap U_{R[t_{1},...,t_{n}]/R}=\emptyset$ if and only if
$\mathfrak{q}\subset\mathfrak{m}R[t_{1},...,t_{n}]$ for some maximal ideal
$\mathfrak{m}$ of $R$.

  2. (2)

For any maximal ideal $\mathfrak{m}$ of $R$, $\mathfrak{m}R(t_{1},...,t_{n})$
is a maximal ideal of $R(t_{1},...,t_{n})$.

  3. (3)

For the induced ring homomorphism $\epsilon:R\to R(t_{1},...,t_{n})$, we have
the bijection

$\displaystyle{\operatorname{Spm}}\,R(t_{1},...,t_{n})$ $\displaystyle\to$
$\displaystyle{\operatorname{Spm}}\,R$ $\displaystyle\mathfrak{n}$
$\displaystyle\mapsto$ $\displaystyle\epsilon^{-1}(\mathfrak{n})$

whose inverse map is given by
${\operatorname{Spm}}\,R\to{\operatorname{Spm}}\,R(t_{1},...,t_{n}),$
$\mathfrak{m}\mapsto\mathfrak{m}R(t_{1},...,t_{n})$.

###### Proof.

In what follows, $\underline{t}$ denotes the multi-variable
$(t_{1},...,t_{n})$.

Let us show (1). Assume $\mathfrak{q}\subset\mathfrak{m}R[\underline{t}]$ for
some maximal ideal $\mathfrak{m}$ of $R$. We obtain
$\mathfrak{q}\cap(R[\underline{t}]\setminus\mathfrak{m}R[\underline{t}])=\emptyset$.
Then we get $\mathfrak{q}\cap U_{R[\underline{t}]/R}=\emptyset$.

Take $\mathfrak{q}\in{\operatorname{Spec}}\,R[\underline{t}]$ such that
$\mathfrak{q}\not\subset\mathfrak{m}R[t]$ for any
$\mathfrak{m}\in{\operatorname{Spm}}\,R$. It suffices to show
$\mathfrak{q}\cap U_{R[\underline{t}]/R}\neq\emptyset$. For any
$\mathfrak{m}\in{\operatorname{Spm}}\,R$, there exists
$f_{\mathfrak{m}}(\underline{t})\in\mathfrak{q}$ such that
$f_{\mathfrak{m}}(\underline{t})\not\in\mathfrak{m}R[\underline{t}]$. For each
$\mathfrak{m}\in{\operatorname{Spm}}\,R$, let $I_{\mathfrak{m}}\subset R$ be
the ideal of $R$ generated by the coefficients of
$f_{\mathfrak{m}}(\underline{t})$. By
$I_{\mathfrak{m}}\not\subset\mathfrak{m}$, we have
$\sum_{\mathfrak{m}\in{\operatorname{Spm}}\,R}I_{\mathfrak{m}}=R$. Since $R$
is a noetherian ring, there exist
$\mathfrak{m}_{1},...,\mathfrak{m}_{r}\in{\operatorname{Spm}}\,R$ such that
$I_{\mathfrak{m}_{1}}+\cdots+I_{\mathfrak{m}_{r}}=R$. For $0\ll N_{2}\ll
N_{3}\ll\cdots\ll N_{r}$, $I_{\mathfrak{m}_{1}}+\cdots+I_{\mathfrak{m}_{r}}$
is nothing but the ideal generated by the coefficients of

$f(\underline{t}):=f_{\mathfrak{m}_{1}}(\underline{t})+t_{1}^{N_{2}}f_{\mathfrak{m}_{2}}(\underline{t})+\cdots+t_{1}^{N_{r}}f_{\mathfrak{m}_{2}}(\underline{t})\in\mathfrak{q}.$

Therefore, we obtain $f(\underline{t})\not\in\mathfrak{m}R[\underline{t}]$ for
any $\mathfrak{m}\in{\operatorname{Spm}}\,R$. Hence, we get
$f(\underline{t})\in\mathfrak{q}\cap U_{R[\underline{t}]/R}$. Thus (1) holds.

Let us show (2) and (3). By (1), we have

${\operatorname{Spm}}\,R(\underline{t})=\\{\mathfrak{m}R(\underline{t})\,|\,\mathfrak{m}\in{\operatorname{Spm}}\,R\\}.$

Thus (2) holds. We obtain
$\epsilon^{-1}(\mathfrak{m}R(\underline{t}))=\mathfrak{m}$. Hence we have the
following two maps

${\operatorname{Spm}}\,R(\underline{t})\to{\operatorname{Spm}}\,R,\qquad\mathfrak{n}\mapsto\epsilon^{-1}(\mathfrak{n})$

and

${\operatorname{Spm}}\,R\to{\operatorname{Spm}}\,R(\underline{t}),\qquad\mathfrak{m}\mapsto\mathfrak{m}R(\underline{t}),$

which are inverse each other. Thus (3) holds. ∎

######  Proposition 3.8.

Let $R$ be a noetherian ring. Then an isomorphism of
$R[t_{1},...,t_{n}]$-algebras

$R(t_{1},...,t_{n})\simeq R(t_{1},...,t_{n-1})(t_{n})$

holds.

###### Proof.

In what follows, $\underline{t}$ denotes the multi-variable
$(t_{1},...,t_{n})$. Both hand sides can be written as
$S^{-1}(R[\underline{t}])$ for some multiplicatively closed subset of
$R[\underline{t}]$. By Lemma 2.5(2), it suffices to show that

${\rm
Im}({\operatorname{Spm}}\,R(\underline{t})\to{\operatorname{Spec}}\,R[\underline{t}])={\rm
Im}({\operatorname{Spm}}\,R(t_{1},...,t_{n-1})(t_{n})\to{\operatorname{Spec}}\,R[\underline{t}]).$

By Theorem 3.7, each of both sides is equal to
$\\{\mathfrak{m}R[\underline{t}]\,|\,\mathfrak{m}\in{\operatorname{Spm}}\,R\\}$.
∎

######  Proposition 3.9.

Let $R$ be a noetherian ring and let $\mathfrak{m}$ be a maximal ideal of $R$.
Then an isomorphism

$R_{\mathfrak{m}}(t_{1},...,t_{n})\simeq
R(t_{1},...,t_{n})_{\mathfrak{m}R(t_{1},...,t_{n})}.$

of $R(t_{1},...,t_{n})$-algebras holds.

###### Proof.

In what follows, $\underline{t}$ denotes the multi-variable
$(t_{1},...,t_{n})$. Note that
$R(\underline{t})_{\mathfrak{m}R(\underline{t})}$ is the local ring of
$R(\underline{t})$ at the maximal ideal $\mathfrak{m}R(\underline{t})$
(Theorem 3.7). We can write $R_{\mathfrak{m}}(\underline{t})\simeq
S^{-1}R(\underline{t})$ for some multiplicatively closed subset $S$ of
$R(\underline{t})$. Since $R_{\mathfrak{m}}(\underline{t})$ is a local ring
whose maximal ideal is $\mathfrak{m}R_{\mathfrak{m}}(\underline{t})$, it is
enough to prove the equation
$\theta^{-1}(\mathfrak{m}R_{\mathfrak{m}}(\underline{t}))=\mathfrak{m}R(\underline{t})$
for the induced ring homomorphism $\theta:R(\underline{t})\to
R_{\mathfrak{m}}(\underline{t})$. The inclusion
$\theta^{-1}(\mathfrak{m}R_{\mathfrak{m}}(\underline{t}))\supset\mathfrak{m}R(\underline{t})$
is clear. Since $\mathfrak{m}R(\underline{t})$ is a maximal ideal, we obtain
the required equation
$\theta^{-1}(\mathfrak{m}R_{\mathfrak{m}}(\underline{t}))=\mathfrak{m}R(\underline{t})$.
∎

######  Remark 3.10.

Let $R$ be a ring and let $S$ be a multiplicatively closed subset of $R$. Then
the induced ring homomorphism
$R(t_{1},,....,t_{n})\to(S^{-1}R)(t_{1},...,t_{n})$ factors through
$R(t_{1},,....,t_{n})\to S^{-1}(R(t_{1},...,t_{n}))$:

$R(t_{1},,....,t_{n})\to
S^{-1}(R(t_{1},...,t_{n}))\xrightarrow{\varphi}(S^{-1}R)(t_{1},...,t_{n}).$

Indeed, the image of $S$ by $R\to(S^{-1}R)(t_{1},...,t_{n})$ is contained in
the multiplicative group $((S^{-1}R)(t_{1},...,t_{n}))^{\times}$. Although it
is tempting to hope that $\varphi$ is an isomorphism, this does not hold in
general (Proposition 3.24).

###  3.2. Base changes

Given $n\in\mathbb{Z}_{>0}$, a noetherian ring $R$, and a morphism
$f:X\to{\operatorname{Spec}}\,R$ of finite type, consider the following
cartesian diagram

$\begin{CD}X\times_{R}R(t_{1},...,t_{n})@>{}>{}>X\\\ @V{}V{}V@V{}V{}V\\\
{\operatorname{Spec}}\,R(t_{1},...,t_{n})@>{}>{}>{\operatorname{Spec}}\,R.\end{CD}$

Since ${\operatorname{Spec}}\,R(t_{1},...,t_{n})\to{\operatorname{Spec}}\,R$
is essentially smooth and faithfully flat (Proposition 3.6), the following
hold.

  1. (i)

$X$ is regular if and only if $X\times_{R}R(t_{1},...,t_{n})$ is regular.

  2. (ii)

$X$ is Cohen–Macaulay if anf only if $X\times_{R}R(t_{1},...,t_{n})$ is
Cohen–Macaulay.

The purpose of this subsection is to establish the following properties.

  1. (iii)

$X$ is integral if and only if $X\times_{R}R(t_{1},...,t_{n})$ is integral
(Theorem 3.13).

  2. (iv)

$\dim X=\dim(X\times_{R}R(t_{1},...,t_{n}))$ if $R$ is excellent (Theorem
3.22).

####  3.2.1. Integrality

######  Lemma 3.11.

Let $R$ and $A$ be rings and let $\kappa$ be a field. Then the following hold.

  1. (1)

If $R\to A$ is a ring homomorphism, then the induced ring homomorphism

$A\otimes_{R}R(t_{1},...,t_{n})\to A(t_{1},...,t_{n})$

is injective.

  2. (2)

If $R\to\kappa$ is a ring homomorphism, then an isomorphism as
$\kappa[t_{1},...,t_{n}]$-algberas

$\kappa\otimes_{R}R(t_{1},...,t_{n})\simeq V^{-1}(\kappa[t_{1},...,t_{n}])$

holds for some multiplicatively closed subset $V$ of $\kappa[t_{1},...,t_{n}]$
satisfying $0\not\in V$. In particular, $\kappa\otimes_{R}R(t_{1},...,t_{n})$
is an integral domain with $\dim(\kappa\otimes_{R}R(t_{1},...,t_{n}))\leq n$.

###### Proof.

Let us show (1). Note that $A[t_{1},...,t_{n}]\to A(t_{1},...,t_{n})$ is
injective (Lemma 3.3). Since both sides are localisations of
$A[t_{1},...,t_{n}]$, also $A\otimes_{R}R(t_{1},...,t_{n})\to
A(t_{1},...,t_{n})$ is injective (Lemma 2.3). Thus (1) holds. Let us show (2).
By Definition 3.1, we obtain $\kappa\otimes_{R}R(t_{1},...,t_{n})\simeq
V^{-1}(\kappa[t_{1},...,t_{n}])$ for some multiplicatively closed subset $V$
of $\kappa[t_{1},...,t_{n}]$. It follows from (1) that $0\not\in V$. Thus (2)
holds. ∎

######  Proposition 3.12.

Let $R$ be a noetherian ring, $X$ a scheme, and
$f:X\to{\operatorname{Spec}}\,R$ a morphism of finite type. Consider the
cartesian diagrams:

$\begin{CD}X_{{\operatorname{red}}}\times_{R}R(t_{1},...,t_{n})@>{\gamma}>{}>X_{{\operatorname{red}}}\\\
@V{}V{j^{\prime}}V@V{}V{j}V\\\ X\times_{R}R(t_{1},...,t_{n})@>{\beta}>{}>X\\\
@V{}V{f^{\prime}}V@V{}V{f}V\\\
{\operatorname{Spec}}\,R(t_{1},...,t_{n})@>{\alpha}>{}>{\operatorname{Spec}}\,R,\end{CD}$

where $j:X_{{\operatorname{red}}}\hookrightarrow X$ denotes the surjective
closed immersion from the reduced scheme $X_{{\operatorname{red}}}$. Then the
following induced morphism is an isomorphism:

$X_{{\operatorname{red}}}\times_{R}R(t_{1},...,t_{n})\xrightarrow{\simeq}(X\times_{R}R(t_{1},...,t_{n}))_{{\operatorname{red}}}.$

###### Proof.

Since $X_{{\operatorname{red}}}$ is reduced and $R\to R(t_{1},...,t_{n})$ is
essentially smooth (Definition 3.1), also
$X_{{\operatorname{red}}}\times_{R}R(t_{1},...,t_{n})$ is reduced. Then the
assertion follows from the fact that $j^{\prime}$ is a surjective closed
immersion from a reduced scheme
$X_{{\operatorname{red}}}\times_{R}R(t_{1},...,t_{n})$. ∎

######  Theorem 3.13.

Let $R$ be a noetherian ring, $X$ a scheme, and
$f:X\to{\operatorname{Spec}}\,R$ a morphism of finite type. Then the following
hold.

  1. (1)

$X$ is integral if and only if $X\times_{R}R(t_{1},...,t_{n})$ is integral.

  2. (2)

$X$ is irreducible if and only if $X\times_{R}R(t_{1},...,t_{n})$ is
irreducible.

  3. (3)

$X$ is connected if and only if $X\times_{R}R(t_{1},...,t_{n})$ is connected.

###### Proof.

The “if” parts for (1)–(3) follow from the fact that
$X\times_{R}R(t_{1},...,t_{n})\to X$ is faithfully flat (Proposition 3.6). In
what follows, let us consider the “only-if” parts.

Let us show (1). If $X$ is affine, then the assertion follows from Lemma 3.3
and Lemma 3.11(1). Set $Y:=X\times_{R}R[t_{1},...,t_{n}]$ and
$Z:=X\times_{R}R(t_{1},...,t_{n})$. We consider $Z$ as a subset of $Y$. Fix a
non-empty affine open subset $X_{0}$ of $X$. Set $Y_{0}\subset Y$ and
$Z_{0}\subset Z$ to be the inverse images of $X_{0}$. Let $\xi$ be the generic
point of $Y$. We have $\xi\in Y_{0}$ and $\xi\in Z_{0}\subset Z$. Therefore,
$Z$ coincides with the closure of $\\{\xi\\}$. Thus (1) holds.

The assertion (2) follows from (1) and Proposition 3.12.

Let us show (3). Let $X=\bigcup_{i\in I}X_{i}$ be the irreducible
decomposition. It follows from (2) that each
$Z_{i}:=X_{i}\times_{R}R(t_{1},...,t_{n})$ is irreducible. Hence it is enough
to show that $Z_{i}\cap Z_{j}\neq\emptyset$ if $X_{i}\cap X_{j}\neq\emptyset$.
This follows from the surjecitivity of
$\alpha:Z=X\times_{R}R(t_{1},...,t_{n})\to X$ (Proposition 3.6) and

$Z_{i}\cap
Z_{j}=\alpha^{-1}(X_{i})\cap\alpha^{-1}(X_{j})=\alpha^{-1}(X_{i}\cap X_{j}).$

Thus (3) holds. ∎

######  Proposition 3.14.

Let $R$ be an excellent ring, $X$ an integral scheme, and
$f:X\to{\operatorname{Spec}}\,R$ a morphism of finite type. For the
normalisation $\nu:X^{N}\to X$ of $X$, consider the cartesian diagrams:

$\begin{CD}X^{N}\times_{R}R(t_{1},...,t_{n})@>{\gamma}>{}>X^{N}\\\
@V{}V{\nu^{\prime}}V@V{}V{\nu}V\\\
X\times_{R}R(t_{1},...,t_{n})@>{\beta}>{}>X\\\ @V{}V{f^{\prime}}V@V{}V{f}V\\\
{\operatorname{Spec}}\,R(t_{1},...,t_{n})@>{\alpha}>{}>{\operatorname{Spec}}\,R.\end{CD}$

Then we have the following isomorphism as
$X\times_{R}R(t_{1},...,t_{n})$-schemes:

$X^{N}\times_{R}R(t_{1},...,t_{n})\simeq(X\times_{R}R(t_{1},...,t_{n}))^{N}.$

###### Proof.

Since $R$ is an excellent ring, all the schemes appearing above are excellent.
Note that both sides are integral normal schemes such that the induced
morphisms to $X\times_{R}R(t_{1},...,t_{n})$ are finite morphisms. Hence these
are isomorphic over $X\times_{R}R(t_{1},...,t_{n})$. ∎

####  3.2.2. Fibres

######  Lemma 3.15.

Let $R$ be a noetherian integral domain such that
$K:={\operatorname{Frac}}\,R$ is a finitely generated $R$-algebra. Then $R$ is
a semi-local ring with $\dim R\leq 1$.

###### Proof.

We have

$K=R\left[\frac{r^{\prime}_{1}}{r_{1}},...\frac{r^{\prime}_{m}}{r_{m}}\right]=R\left[\frac{1}{r_{1}},...\frac{1}{r_{m}}\right]=R\left[\frac{1}{f}\right]$

for some $r_{1},...,r_{m},r^{\prime}_{1},...,r^{\prime}_{m}\in R$ and the
product $f:=r_{1}\cdots r_{m}$. In other words, the equation $D_{f}=\\{(0)\\}$
holds in ${\operatorname{Spec}}\,R$. By taking the complements of both hands,
we get

${\operatorname{Spec}}R\setminus\\{(0)\\}=V(f)=V(\mathfrak{p}_{1})\cup\cdots\cup
V(\mathfrak{p}_{r}),$

where $\mathfrak{p}_{1},...,\mathfrak{p}_{r}$ are the minimal prime ideals of
$fR$, and hence of height one. In particular, $R$ has only finitely many
primes ideals of height one. We then obtain $\dim R\leq 1$ by Ratliff’s
existence theorem [Mat89, Theorem 31.2]. ∎

######  Lemma 3.16.

Let $R\subset S$ be an integral extension of noetherian integral domains. Then
the following induced ring homomorphism is an isomorphism:

$S\otimes_{R}R(t_{1},...,t_{n})\xrightarrow{\simeq}S(t_{1},...,t_{n}).$

###### Proof.

Let $\underline{t}:=(t_{1},...,t_{n})$ be the multi-variable. Note that both
sides are localisations of $S[\underline{t}]$. In particular, also
$S\otimes_{R}R(\underline{t})\to S(\underline{t})$ is a localisation.

We claim that the following hold.

  1. (1)

If $\mathfrak{n}^{\prime}$ is a maximal ideal of
$S\otimes_{R}R(\underline{t})$, then its pullback $\mathfrak{n}^{\prime}\cap
S$ is a maximal ideal of $S$, i.e. we have the following map

(3.16.1)
${\operatorname{Spm}}\,(S\otimes_{R}R(\underline{t}))\to{\operatorname{Spm}}\,S,\qquad\mathfrak{n}^{\prime}\mapsto\mathfrak{n}^{\prime}\cap
S.$

  2. (2)

The map (3.16.1) is bijective.

For now, we complete the proof under assuming (1) and (2). Since
${\operatorname{Spm}}\,S(\underline{t})\to{\operatorname{Spm}}\,S,\mathfrak{l}\mapsto\mathfrak{l}\cap
R$ is bijective (Theorem 3.7), we get the bijection by (1) and (2):

${\operatorname{Spm}}\,S(\underline{t})\to{\operatorname{Spm}}\,(S\otimes_{R}R(\underline{t})),\qquad\mathfrak{l}\mapsto\mathfrak{l}\cap
S\otimes_{R}R(\underline{t}).$

Since $S\otimes_{R}R(\underline{t})\to S(\underline{t})$ is a localisation,
this is faithfully flat, and hence an isomorphism.

Therefore, it is enough to show (1) and (2). Let us show (1). Fix
$\mathfrak{n}^{\prime}\in{\operatorname{Spm}}\,(S\otimes_{R}R(\underline{t}))$
and take its pullbacks:

(3.16.2) $\begin{CD}S@>{}>{}>S\otimes_{R}R(\underline{t})\\\
@A{}A{}A@A{}A{}A\\\
R@>{}>{}>R(\underline{t})\end{CD}\qquad\qquad\begin{CD}\mathfrak{n}@<{}<{}<\mathfrak{n}^{\prime}\\\
@V{}V{}V@V{}V{}V\\\ <EMAIL_ADDRESS>
All the arrows in the diagram (3.16.2) are injective. Indeed, since $R\to
R(\underline{t})$ is faithfully flat, also $S\to S\otimes_{R}R(\underline{t})$
is faithfully flat, and hence injective. Since $R(\underline{t})$ is reduced
and
${\operatorname{Spec}}\,(S\otimes_{R}R(\underline{t}))\to{\operatorname{Spec}}\,R(\underline{t})$
is surjective, $R(\underline{t})\to S\otimes_{R}R(\underline{t})$ is
injective. In what follows, we consider all the rings in the diagram (3.16.2)
as subrings of $S\otimes_{R}R(\underline{t})$. Furthermore, these rings are
integral domains since so is $S\otimes_{R}R(\underline{t})$ (Theorem 3.13(1)).
Since $R\subset S$ is an integral extension, so is $R(\underline{t})\subset
S\otimes_{R}R(\underline{t})$, and hence
$\mathfrak{m}^{\prime}:=\mathfrak{n}^{\prime}\cap R(\underline{t})$ is a
maximal ideal. Therefore, $\mathfrak{m}:=\mathfrak{m}^{\prime}\cap R$ is a
maximal ideal (Theorem 3.7). Since $R\to S$ is an integral extension of
integral domains, also $\mathfrak{n}:=\mathfrak{n}^{\prime}\cap S$ is a
maximal ideal. Thus (1) holds.

Let us show (2). It suffices to give the inverse map of the map (3.16.1). Pick
$\mathfrak{n}\in{\operatorname{Spm}}\,S$. Then $\mathfrak{m}:=\mathfrak{n}\cap
R$ and $\mathfrak{m}^{\prime}:=\mathfrak{m}R(\underline{t})$ are maximal
ideals (Theorem 3.7). We get a prime ideal
$\mathfrak{n}^{\prime}\in{\operatorname{Spec}}\,(S\otimes_{R}R(\underline{t}))$
such that $\mathfrak{n}^{\prime}\cap S=\mathfrak{n}$ and
$\mathfrak{n}^{\prime}\cap R(\underline{t})=\mathfrak{m}^{\prime}$. Since
$R(\underline{t})\to S\otimes_{R}R(\underline{t})$ is an integral extension of
integral domains, $\mathfrak{n}^{\prime}$ is a maximal ideal. Then the map
$\mathfrak{n}\mapsto\mathfrak{n}^{\prime}$ given here is nothing but the
inverse map of (3.16.1). Thus (2) holds. ∎

######  Lemma 3.17.

Let $R$ be a principal ideal domain. Set $K:={\operatorname{Frac}}\,R$. Then
the following induced ring homomorphism is an isomorphism:

$K\otimes_{R}R(t_{1},...,t_{n})\xrightarrow{\simeq}K(t_{1},...,t_{n})$

###### Proof.

Let $\underline{t}:=(t_{1},...,t_{n})$ be the multi-variable. If $\dim R=0$,
i.e. $R=K$, then the assertion is clear. Hence we may assume that $\dim R=1$.
Since the induced ring homomorphism $K\otimes_{R}R(\underline{t})\to
K(\underline{t})$ is injective (Lemma 3.11), we may consider
$K\otimes_{R}R(\underline{t})$ is a subring of $K(\underline{t})$. Inside the
field $K(\underline{t})$, we have that

$R(\underline{t})\otimes_{R}K=\left\\{\frac{f(\underline{t})}{ag(\underline{t})}\in
K(\underline{t})\,\middle|\,a\in R\setminus\\{0\\},f(\underline{t})\in
R[\underline{t}],g(\underline{t})\in\bigcap_{\mathfrak{m}\in{\operatorname{Spm}}\,R}R[\underline{t}]\setminus\mathfrak{m}R[t]\right\\}.$

Pick $h(\underline{t})\in R[\underline{t}]\setminus\\{0\\}$. For a greatest
common divisor $a\in R\setminus\\{0\\}$ of the coefficients of
$h(\underline{t})$, we have that $h(\underline{t})=ag(\underline{t})$ for some
$g(\underline{t})\in\bigcap_{\mathfrak{m}\in{\operatorname{Spm}}\,R}R[\underline{t}]\setminus\mathfrak{m}R[\underline{t}]$,
which implies $R(\underline{t})\otimes_{R}K=K(\underline{t})$. ∎

######  Theorem 3.18.

Let $R$ be an excellent ring and let $\kappa$ be a field which is a finitely
generated $R$-algebra. Then the following induced ring homomorphism is an
isomorphism:

$\kappa\otimes_{R}R(t_{1},...,t_{n})\xrightarrow{\simeq}\kappa(t_{1},...,t_{n}).$

###### Proof.

Let $\underline{t}:=(t_{1},...,t_{n})$ be the multi-variable. By Proposition
3.5, we have
$\kappa\otimes_{R}R(\underline{t})\simeq\kappa\otimes_{R^{\prime}}R^{\prime}(\underline{t})$
for the image $R^{\prime}:={\rm Im}(R\to\kappa)$. Hence, after replacing $R$
by $R^{\prime}$, we may assume that $R$ is a subring of $\kappa$. Furthermore,
by Lemma 3.16 and the assumption that $R$ is excellent, the problem is reduced
to the case when $R$ is integrally closed. For $K:={\operatorname{Frac}}\,R$,
we have

$R\subset K\subset\kappa.$

Since $\kappa$ is a finitely generated $R$-algebra, $\kappa$ is also a
finitely generated $K$-algebra. Then Hilbert’s Nullstellensatz implies that
$K\subset\kappa$ is a finite extension. By the Artin–Tate lemma [AM69,
Proposition 7.8], also $K$ is a finitely generated $R$-algebra. Then Lemma
3.15 implies that $R$ is a semi-local ring with $\dim R\leq 1$. Therefore, $R$
is either a field or a Dedekind domain with finitely many prime ideals, and
hence $R$ is a principal ideal domain. We then obtain

$\displaystyle\kappa\otimes_{R}R(\underline{t})\simeq\kappa\otimes_{K}(K\otimes_{R}R(\underline{t}))\simeq\kappa\otimes_{K}K(\underline{t})\simeq\kappa(\underline{t}),$

where the second isomorphism holds by Lemma 3.17 and the last one follows from
Lemma 3.16. ∎

######  Corollary 3.19.

Let $R$ be a noetherian ring, $X$ a scheme, and
$f:X\to{\operatorname{Spec}}\,R$ a morphism of finite type. Let $x\in X$ be a
point. For the induced morphism

$\alpha:X\times_{R}R(t_{1},...,t_{n})\to X,$

the following hold.

  1. (1)

The fibre $\alpha^{-1}(x)$ is an affine integral scheme.

  2. (2)

If $R$ is excellent and $x$ is a closed point, then $\alpha^{-1}(x)$ consists
of one point, i.e. $\alpha^{-1}(x)$ is an affine spectrum of a field.

###### Proof.

The assertion (1) follows from Lemma 3.11(2). The assertion (2) holds by
Theorem 3.18. ∎

####  3.2.3. Dimension

######  Lemma 3.20.

Let $A$ be a noetherian ring with $d:=\dim A<\infty$. Let $\mathfrak{n}$ be a
maximal ideal of $A[x]$ with ${\rm ht}(\mathfrak{n})=d+1$. Then
$\mathfrak{m}:=\mathfrak{n}\cap A$ is a maximal ideal of $A$ with ${\rm
ht}(\mathfrak{m})=d$.

###### Proof.

By $d=\dim A$, it suffices to prove that the prime ideal
$\mathfrak{m}=\mathfrak{n}\cap A$ satisfies ${\rm ht}(\mathfrak{m})=d$. We
have ${\rm ht}(\mathfrak{m})\leq\dim A=d$. Suppose that ${\rm
ht}(\mathfrak{m})<d$. It suffices to derive a contradiction. By
$\mathfrak{m}=\mathfrak{n}\cap A$, we have that $\mathfrak{n}\cap
S_{\mathfrak{m}}=\emptyset$ for $S_{\mathfrak{m}}:=A\setminus\mathfrak{m}$.
Therefore, $\mathfrak{n}$ comes from
$S^{-1}_{\mathfrak{m}}(A[x])=A_{\mathfrak{m}}[x]$, i.e.
$\mathfrak{n}=\mathfrak{n}S^{-1}_{\mathfrak{m}}(A[x])\cap A[x]$. We have

${\rm ht}(\mathfrak{n})={\rm
ht}(\mathfrak{n}S^{-1}_{\mathfrak{m}}(A[x]))\leq\dim
S^{-1}_{\mathfrak{m}}(A[x])=\dim A_{\mathfrak{m}}[x]=\dim
A_{\mathfrak{m}}+1={\rm ht}(\mathfrak{m})+1<d+1,$

which contradicts ${\rm ht}(\mathfrak{n})=d+1$. ∎

######  Lemma 3.21.

Let $R$ be an excellent ring and let $A$ be a finitely generated $R$-algebra.
Then $\dim A\otimes_{R}R(t)\leq\dim A$.

###### Proof.

If $\dim A=\infty$, then there is nothing to show. Hence we may assume that
$\dim A<\infty$. Fix a maximal ideal $\mathfrak{n}$ of $A\otimes_{R}R(t)$. It
suffices to show ${\rm ht}(\mathfrak{n})\leq\dim A$. Set
$\mathfrak{l}:=\mathfrak{n}\cap A[t]$ and $\mathfrak{m}:=\mathfrak{n}\cap A$:

$A\to A[t]\to
A\otimes_{R}R(t),\qquad\mathfrak{m}\leftarrow\mathfrak{l}\leftarrow\mathfrak{n}.$

We now treat the case when $\dim A_{\mathfrak{m}}<\dim A$. We have

${\rm ht}(\mathfrak{n})={\rm ht}(\mathfrak{l})\leq\dim
A_{\mathfrak{m}}[t]=\dim A_{\mathfrak{m}}+1<\dim A+1,$

where the first equality holds by
$A[t]_{\mathfrak{l}}\simeq(A\otimes_{R}R(t))_{\mathfrak{n}}$. This completes
the proof for the case when $\dim A_{\mathfrak{m}}<\dim A$.

We may assume that $\dim A_{\mathfrak{m}}=\dim A$. In particular,
$\mathfrak{m}$ is a maximal ideal of $A$. Since
$A_{\mathfrak{m}}\to(A\otimes_{R}R(t))_{\mathfrak{n}}$ is faithfually flat, it
holds by [Mat89, Theorem 15.1] that

${\rm ht}(\mathfrak{n})=\dim(A\otimes_{R}R(t))_{\mathfrak{n}}=\dim
A_{\mathfrak{m}}+\dim(A\otimes_{R}R(t)/\mathfrak{m}(A\otimes_{R}R(t)))_{\mathfrak{n}}$
$\leq\dim A+\dim((A/\mathfrak{m})\otimes_{R}R(t))=\dim A,$

where the last equality follows from Theorem 3.18. ∎

######  Theorem 3.22.

Let $R$ be an excellent ring, $X$ a scheme, and
$f:X\to{\operatorname{Spec}}\,R$ a morphism of finite type. Then it holds that

$\dim X=\dim(X\times_{R}R(t_{1},...,t_{n})).$

###### Proof.

By Proposition 3.8, we may assume that $n=1$. Set $t:=t_{1}$. Since $R\to
R(t)$ is faithfully flat (Proposition 3.6), we obtain $\dim
X\leq\dim(X\times_{R}R(t))$ [Mat89, Theorem 15.1(2)]. The opposite inequality
$\dim X\geq\dim(X\times_{R}R(t))$ follows from Lemma 3.21. ∎

###  3.3. Examples

######  Proposition 3.23.

Let $R$ be an excellent one-dimensional principal ideal domain. Set
$K:={\operatorname{Frac}}\,R$. Then the following hold.

  1. (1)

$R(t)$ is an excellent one-dimensional principal ideal domain.

  2. (2)

$R(t)\otimes_{R}(R/\mathfrak{m})\simeq(R/\mathfrak{m})(t)$ for any
$\mathfrak{m}\in{\operatorname{Spm}}\,R$.

  3. (3)

$R(t)\otimes_{R}K\simeq K(t)$.

  4. (4)

The induced morphism ${\operatorname{Spec}}\,R(t)\to{\operatorname{Spec}}\,R$
is bijective.

In particular, $\mathbb{Z}(t)$ is a one-dimensional principal ideal domain,
${\operatorname{Spec}}\,\mathbb{Z}(t)\to{\operatorname{Spec}}\,\mathbb{Z}$ is
bijective, $\mathbb{Z}(t)\otimes_{\mathbb{Z}}\mathbb{Q}\simeq\mathbb{Q}(t)$,
and $\mathbb{Z}(t)\otimes_{\mathbb{Z}}\mathbb{F}_{p}\simeq\mathbb{F}_{p}(t)$
for any prime number $p$.

###### Proof.

We have the induced ring homomorphisms

$R\to R[t]\to U_{R[t]/R}^{-1}(R[t])=R(t).$

Note that $\dim R(t)=\dim\,R=1$ (Theorem 3.22). Since $R[t]$ is a unique
factorisation domain, $R(t)$ is a one-dimensional unique factorisation domain,
and hence a principal ideal domain. Thus (1) holds. It follows from
Proposition 3.5 and Lemma 3.17 that (2) and (3) hold, respectively. Then (4)
holds by (2) and (3). ∎

As the following example shows, the localisation functor $A\mapsto S^{-1}A$
does not commute with the purely transcendental extension functor $R\mapsto
R(t)$.

######  Proposition 3.24.

Let $k$ be a field. Set

$R:=k[x,y]\quad{\rm and}\quad S:=R\setminus\\{0\\}.$

Then $S^{-1}(R(t))$ and $(S^{-1}R)(t)$ are not isomorphic as rings.

###### Proof.

Set $K:=S^{-1}R=k(x,y)$. We have $(S^{-1}R)(t)=(k(x,y))(t)=k(x,y,t)$. In
particular, it suffices to show that $S^{-1}(R(t))$ is not a field. As a
subring of ${\rm Frac}(R[t])=k(x,y,t)$, $S^{-1}(R(t))$ can be written as
follows:

$S^{-1}(R(t))=S^{-1}(U^{-1}_{k[x,y][t]/k[x,y]}k[x,y,t])$
$=\left\\{\frac{f(x,y,t)}{s(x,y)u(x,y,t)}\,\middle|\,f(x,y,t)\in
k[x,y,t],s(x,y)\in S,u(x,y,t)\in U_{k[x,y][t]/k[x,y]}\right\\}.$

We then have $S^{-1}(R(t))=V^{-1}k[x,y,t]$ for

$V:=\\{s(x,y)u(x,y,t)\,|\,s(x,y)\in S,u(x,y,t)\in U_{k[x,y][t]/k[x,y]}\\}.$

Set $\mathfrak{p}:=(x+ty)k[x,y,t]$, which is a nonzero prime ideal of
$k[x,y,t]$. It suffices to show that $\mathfrak{p}\cap V=\emptyset$.

Suppose $\mathfrak{p}\cap V\neq\emptyset$. Pick $\zeta\in\mathfrak{p}\cap V$.
We have

$\zeta=(x+ty)f(x,y,t)=s(x,y)u(x,y,t)\neq 0$

for some $f(x,y,t)\in k[x,y,t],s(x,y)\in S,u(x,y,t)\in U_{k[x,y][t]/k[x,y]}$.
Since $x+ty$ is a prime element of $k[x,y,t]$ and
$s(x,y)\not\in(x+ty)k[x,y,t]$, there exists $g(x,y,t)\in k[x,y,t]$ such that

$u(x,y,t)=(x+ty)g(x,y,t)\in\mathfrak{m}k[x,y,t],$

where $\mathfrak{m}:=(x,y)k[x,y]$. This contradicts

$u(x,y,t)\in
U_{k[x,y][t]/k[x,y]}=\bigcap_{\mathfrak{n}\in{\operatorname{Spm}}\,k[x,y]}k[x,y,t]\setminus\mathfrak{n}k[x,y,t]\subset
k[x,y,t]\setminus\mathfrak{m}k[x,y,t].$

∎

The following proposition shows that the equation $\dim
X=\dim(X\times_{R}R(t_{1},...,t_{n}))$ in Theorem 3.22 no longer holds if $X$
is essentially of finite type.

######  Proposition 3.25.

Let $k$ be a field. Then $\dim(B\otimes_{k}k(t))>\dim B$ for $B:=k(x)$.

###### Proof.

We have

$\displaystyle B\otimes_{k}k(t)$ $\displaystyle=$ $\displaystyle
k(x)\otimes_{k}k(t)$ $\displaystyle\simeq$
$\displaystyle\left\\{\frac{f(x,t)}{g(x)h(t)}\,\middle|\,f(x,t)\in
k[x,t],g(x)\in k[x]\setminus\\{0\\},h[t]\in k[t]\setminus\\{0\\}\right\\}$
$\displaystyle\simeq$ $\displaystyle V^{-1}k[x,t]$

for

$V:=\\{g(x)h(t)\in k[x,t]\,|\,g(x)\in k[x]\setminus\\{0\\},h[t]\in
k[t]\setminus\\{0\\}\\}.$

By $(x-t)k[x,t]\cap V=\emptyset$, we have that $\dim B\otimes_{k}k(t)=\dim
V^{-1}k[x,t]\neq 0=\dim B$. ∎

######  Proposition 3.26.

Let $k$ be a field. Set $R:=k[x]$. Then the following hold.

  1. (1)

$R(t)$ is an essentially finitely generated $k(t)$-algebra.

  2. (2)

$R(t)$ is not a finitely generated $k(t)$-algebra.

###### Proof.

Since (1) is obvious, let us show (2). Suppose that $R(t)=(k[x])(t)$ is a
finitely generated $k(t)$-algebra. We have the induced injective
$k(t)$-algebra homomorphism:

$\varphi:k(t)[x]\hookrightarrow(k[x])(t),$

which induces a $k(t)$-morphism:

$\varphi^{*}:{\operatorname{Spec}}\,((k[x])(t))\to{\operatorname{Spec}}\,k(t)[x].$

By Chevalley’s theorem, ${\rm Im}(\varphi^{*})$ is a constructible set. Since
$\varphi$ is injective, $\varphi^{*}$ is dominant. Therefore, ${\rm
Im}(\varphi^{*})$ is a cofinite set, i.e. its complement is a finite set.

We have morphisms.

$\psi^{*}\circ\varphi^{*}:{\operatorname{Spec}}\,((k[x])(t))\xrightarrow{\varphi^{*}}{\operatorname{Spec}}\,k(t)[x]\xrightarrow{\psi^{*}}{\operatorname{Spec}}\,k[x].$

By Proposition 3.23, the composition $\psi^{*}\circ\varphi^{*}$ is bijective
and

${\rm
Im}(\varphi^{*})=\\{\mathfrak{p}k(t)[x]\,|\,\mathfrak{p}\in{\operatorname{Spec}}\,k[x]\\}.$

In particular, we obtain

$(x-t^{m})k(t)[x]\not\in{\rm Im}(\varphi^{*})$

for any $m\in\mathbb{Z}_{>0}$. Hence ${\rm Im}(\varphi^{*})$ is not a cofinite
set. ∎

###  3.4. A generalisation

In order to treat a gluing problem of our construction (cf. Remark 4.2), we
give a slight generalisation of our purely transcendental extensions
(Definition 3.27) and establish a auxiliary result (Proposition 3.28).

######  Definition 3.27.

Let $R$ be a ring and let $S$ be an $R$-algebra such that $\mathfrak{m}S$ is a
prime ideal of $S$ for any $\mathfrak{m}\in{\operatorname{Spm}}\,R$. We set

$U_{S/R}:=\bigcap_{\mathfrak{p}\in{\operatorname{Spec}}\,R}(S\setminus\mathfrak{p}S)=\bigcap_{\mathfrak{m}\in{\operatorname{Spm}}\,R}(S\setminus\mathfrak{m}S)$

and

$R(S):=U_{S/R}^{-1}S.$

In particular, if $S=R[t_{1},...,t_{n}]$, then $R(S)=R(t_{1},...,t_{n})$
(Definition 3.1).

######  Proposition 3.28.

Let $R$ be a ring and let $S$ be an $R$-algebra such that $\mathfrak{m}S$ is a
prime ideal of $S$ for any $\mathfrak{m}\in{\operatorname{Spm}}\,R$. Take
$f\in U_{S/R}$. Then the following hold.

  1. (1)

For any $\mathfrak{m}\in{\operatorname{Spm}}\,R$, $\mathfrak{m}S_{f}$ is a
prime ideal of $S_{f}$.

  2. (2)

The composition $S\xrightarrow{\theta}S_{f}\xrightarrow{\beta}R(S_{f})$ of the
induced ring homomorphisms factors through the induced ring homomorphism
$\alpha:S\to R(S)$:

$\begin{CD}S@>{\theta}>{}>S_{f}\\\ @V{}V{\alpha}V@V{}V{\beta}V\\\
R(S)@>{\theta^{\prime}}>{}>R(S_{f}).\end{CD}$

Furthermore, $\theta^{\prime}:R(S)\to R(S_{f})$ is an isomorphism.

###### Proof.

The assertion (1) follows from $f\in U_{S/R}\subset S\setminus\mathfrak{m}S$.

Let us show (2). We first show that there exists a ring homomorphism
$\theta^{\prime}:R(S)\to R(S_{f})$ such that
$\theta^{\prime}\circ\alpha=\beta\circ\theta$. Take $u\in
U_{S/R}=\bigcap_{\mathfrak{m}\in{\operatorname{Spm}}\,R}(S\setminus\mathfrak{m}S)$.
It suffices to prove $\theta(u)\in
U_{S_{f}/R}=\bigcap_{\mathfrak{m}\in{\operatorname{Spm}}\,R}(S_{f}\setminus\mathfrak{m}S_{f})$.
Fix $\mathfrak{m}\in{\operatorname{Spm}}\,R$. Suppose
$\theta(u)\in\mathfrak{m}S_{f}$. Let us derive a contradiction. We have
$f^{r}u\in\mathfrak{m}S$ for some $r\in\mathbb{Z}_{>0}$. Since $\mathfrak{m}S$
is a prime ideal of $S$, we have $f^{r}\in\mathfrak{m}S$ or
$u\in\mathfrak{m}S$. By $f\not\in\mathfrak{m}S$, we obtain
$u\in\mathfrak{m}S$, which contradictis $u\in
U_{S/R}=\bigcap_{\mathfrak{m}\in{\operatorname{Spm}}\,R}(S\setminus\mathfrak{m}S)$.
Therefore, we get a ring homomorphism $\theta^{\prime}:R(S)\to R(S_{f})$ such
that $\theta^{\prime}\circ\alpha=\beta\circ\theta$.

We now prove that $\theta^{\prime}:R(S)\to R(S_{f})$ is injective. Take
$s/u\in R(S)$ with $\theta^{\prime}(s/u)=0$, where $s\in S$ and $u\in
U_{S/R}$. Then we have $\theta^{\prime}(s/1)=0$. Hence we obtain
$v\cdot(s/1)=0$ in $S_{f}$ for some $v\in U_{S_{f}/R}$. We can write
$v=t/f^{a}$ for some $t\in S$ and $a\in\mathbb{Z}_{>0}$. Hence, it holds that
$f^{b}ts=0$ in $S$ for some $b\in\mathbb{Z}_{>0}$. By $v\in
U_{S_{f}/R}=\bigcap_{\mathfrak{m}\in{\operatorname{Spm}}\,R}(S_{f}\setminus\mathfrak{m}S_{f})$,
we get
$t\in\bigcap_{\mathfrak{m}\in{\operatorname{Spm}}\,R}(S\setminus\mathfrak{m}S)=U_{S/R}$.
By $f\in U_{S/R}$, we have that

$\frac{s}{1}=\frac{f^{b}ts}{f^{b}t}=0\qquad{\rm in}\qquad R(S)=U_{S/R}^{-1}S.$

Thus $\theta^{\prime}$ is injective.

Let us prove that $\theta^{\prime}$ is surjective. Take $\zeta\in R(S_{f})$.
We can write $\zeta=\frac{s/f^{a}}{v}$ for some $s\in S,a\in\mathbb{Z}_{\geq
0},v\in U_{S_{f}/R}$. Furthermore, we have $v=t/f^{b}$ for some $t\in U_{S/R}$
and $b\in\mathbb{Z}_{>0}$. Hence it holds that

$\zeta=\frac{s/f^{a}}{t/f^{b}}.$

By $f^{b}s\in S$ and $f^{a}t\in U_{S/R}$, $\zeta$ is the image of
$f^{b}s/f^{a}t\in U_{S/R}^{-1}R=R(S)$, as required. ∎

##  4\. Bertini theorems for generic members

###  4.1. Universal members and generic members

The purpose of this subsection is to introduce the definitions and to
establish some fundamental properties of the universal members
$X^{{\operatorname{univ}}}_{\varphi}$ and the generic members
$X^{{\operatorname{gen}}}_{\varphi}$ of $\varphi:X\to\mathbb{P}^{n}_{R}$. We
use the word “members”, since the inverse image of a hyperplane of a
projective space $\mathbb{P}^{n}_{k}$ over an algebraically closed field $k$
is usually called a member of the corresponding linear system.

######  Definition 4.1.

Let $R$ be a noetherian ring and take $n\in\mathbb{Z}_{\geq 0}$. Fix

  1. (1)

an isomorphism
$\delta:\mathbb{P}^{n}_{R}\xrightarrow{\simeq}(\mathbb{P}^{n}_{R})^{*}$,

  2. (2)

an open immersion $j:\mathbb{A}^{n}_{R}\hookrightarrow\mathbb{P}^{n}_{R}$
which is obtained as the complement of a hyperplane of $\mathbb{P}^{n}_{R}$
over $R$, and

  3. (3)

a coordinate:
$\iota:{\operatorname{Spec}}\,R[t_{1},...,t_{n}]\xrightarrow{\simeq}\mathbb{A}^{n}_{R}$.

Consider $R$-morphisms

$\Theta:{\operatorname{Spec}}\,R(n):={\operatorname{Spec}}\,R(t_{1},...,t_{n})\xrightarrow{\theta^{*}}{\operatorname{Spec}}\,R[t_{1},...,t_{n}]\xrightarrow{\iota,\simeq}\mathbb{A}^{n}_{R}\overset{j}{\hookrightarrow}\mathbb{P}^{n}_{R}\xrightarrow{\delta,\simeq}(\mathbb{P}^{n}_{R})^{*},$

where $\theta^{*}$ is the morphism induced by the natural ring homomorphism
$\theta:R[t_{1},...,t_{n}]\to R(t_{1},...,t_{n})$. The $R$-algebra $R(n)$ is
independent of the choices of (1)–(3) up to isomorphisms (Remark 4.2).

######  Remark 4.2.

Let us prove that the $R$-algebra $R(n)$ is independent of (1)–(3) up to
$R$-algebra isomorphisms. It is clear that the $R$-algbera $R(n)$ is
independent of the choice of an isomorphism
$\delta:\mathbb{P}^{n}_{R}\xrightarrow{\simeq}(\mathbb{P}^{n}_{R})^{*}$ in
(1). By $R(t_{1},...,t_{n})=R(R[t_{1},...,t_{n}])$ (Definition 3.27), $R(n)$
is independent of the choice of a coordinate in (3).

Let $R(n)^{\prime}$ be another choice. Let $H$ and $H^{\prime}$ be the
hyperplanes of $\mathbb{P}^{n}_{R}$ over $R$ such that their complements
induce $R(n)$ and $R(n)^{\prime}$, respectively. For
$\mathbb{P}^{n}_{R}={\operatorname{Proj}}\,R[t_{0},...,t_{n}]$, we may assume
that $H$ is defined as $\\{t_{0}=0\\}$. We can write

$H^{\prime}=\\{a_{0}t_{0}+a_{1}t_{1}+\cdots+a_{n}t_{n}=0\\}\subset\mathbb{P}^{n}_{R}={\operatorname{Proj}}\,R[t_{0},t_{1},...,t_{n}].$

We then have

$\Gamma(\mathbb{P}^{n}_{R}\setminus(H\cup
H^{\prime}),\mathcal{O}_{\mathbb{P}^{n}_{R}})=\Gamma(\mathbb{A}^{n}_{R}\setminus
H^{\prime},\mathcal{O}_{\mathbb{A}^{n}_{R}})=R[t_{1},...,t_{n}]_{f}$

for $f:=a_{0}+a_{1}t_{1}+\cdots+a_{n}t_{n}\in R[t_{1},...,t_{n}]$. Given
$\mathfrak{m}\in{\operatorname{Spm}}\,R$, there exists $0\leq i\leq n$ such
that $a_{i}\not\in\mathfrak{m}$, because
$\\{a_{0}t_{0}+a_{1}t_{1}+\cdots+a_{n}t_{n}=0\\}\subset\mathbb{P}^{n}_{R}$ is
flat over $R$. This implies $f\in U_{R[t_{1},...,t_{n}]/R}$. It holds that

$R(n)=R(R[t_{1},...,t_{n}])=R(R[t_{1},...,t_{n}]_{f})=R(\Gamma(\mathbb{P}^{n}_{R}\setminus(H\cup
H^{\prime}),\mathcal{O}_{\mathbb{P}^{n}_{R}})),$

where the second equality follows from Proposition 3.28. By symmetry, we
obtain $R(n)^{\prime}=R(\Gamma(\mathbb{P}^{n}_{R}\setminus(H\cup
H^{\prime}),\mathcal{O}_{\mathbb{P}^{n}_{R}}))$.

######  Definition 4.3.

Let $R$ be a noetherian ring and fix $n\in\mathbb{Z}_{\geq 0}$. Let $X$ be an
$R$-scheme of finite type and let $\varphi:X\to\mathbb{P}^{n}_{R}$ be an
$R$-morphism.

  1. (1)

Let $(\mathbb{P}^{n}_{R})^{{\operatorname{univ}}}_{{\rm id}}$ be the universal
member of the hyperplanes, which is a closed subscheme of
$\mathbb{P}^{n}_{R}\times_{R}(\mathbb{P}^{n}_{R})^{*}$. We set
$(\mathbb{P}^{n}_{R})^{{\operatorname{gen}}}_{{\rm
id}}:=(\mathbb{P}^{n}_{R})^{{\operatorname{univ}}}_{{\rm
id}}\times_{(\mathbb{P}^{n}_{R})^{*}}{\operatorname{Spec}}\,R(n)$. In
particular, we have the following commutative diagram in which each square is
cartesian.

${(\mathbb{P}^{n}_{R})^{{\operatorname{gen}}}_{{\rm
id}}}$${(\mathbb{P}^{n}_{R})^{{\operatorname{univ}}}_{{\rm
id}}}$${\mathbb{P}^{n}_{R(n)}}$${\mathbb{P}^{n}_{R}\times_{R}(\mathbb{P}^{n}_{R})^{*}}$${\mathbb{P}^{n}_{R}}$${{\operatorname{Spec}}\,R(n)}$${(\mathbb{P}^{n}_{R})^{*}}$${{\operatorname{Spec}}\,R}$$\scriptstyle{\Theta}$$\scriptstyle{{\rm
pr}_{2}}$$\scriptstyle{{\rm pr}_{1}}$

  2. (2)

We define
$X^{{\operatorname{univ}}}_{\varphi}:=(\mathbb{P}^{n}_{R})^{{\operatorname{univ}}}_{{\rm
id}}\times_{\mathbb{P}^{n}_{R}}X$ and
$X^{{\operatorname{gen}}}_{\varphi}:=(\mathbb{P}^{n}_{R})^{{\operatorname{gen}}}_{{\rm
id}}\times_{\mathbb{P}^{n}_{R}}X$. We call
$X^{{\operatorname{univ}}}_{\varphi}$ and $X^{{\operatorname{gen}}}_{\varphi}$
the universal member and the generic member of $\varphi$, respectively. In
particular, taking the base changes of the above commutative diagram by
$\varphi:X\to\mathbb{P}^{n}_{R}$, we obtain the following commutative diagram
in which each square is cartesian.

${X^{{\operatorname{gen}}}_{\varphi}}$${X^{{\operatorname{univ}}}_{\varphi}}$${X\times_{R}R(n)}$${X\times_{R}(\mathbb{P}^{n}_{R})^{*}}$${X}$${{\operatorname{Spec}}\,R(n)}$${(\mathbb{P}^{n}_{R})^{*}}$${{\operatorname{Spec}}\,R}$$\scriptstyle{\Theta}$$\scriptstyle{{\rm
pr}_{2}}$$\scriptstyle{{\rm pr}_{1}}$

######  Remark 4.4.

We use the same notation as in Definition 4.3. We have the following cartesian
diagram in which each vertical arrow is a closed immersion.

${X^{{\operatorname{gen}}}_{\varphi}}$${(\mathbb{P}^{n}_{R})^{{\operatorname{gen}}}}$${X\times_{R}R(n)}$${\mathbb{P}^{n}_{R}\times_{R}R(n)=\mathbb{P}^{n}_{R(n)}}$$\scriptstyle{\varphi\times{\rm
id}}$

Then the following hold.

  1. (1)

$(\mathbb{P}^{n}_{R})^{{\operatorname{gen}}}$ is a hyperplane of
$\mathbb{P}^{n}_{R(n)}$ over $R(n)$ by Definition 4.3(1).

  2. (2)

By (1), the ideal sheaf on $X\times_{R}R(n)$ corresponding to
$X^{{\operatorname{gen}}}_{\varphi}$ is locally principal.

######  Remark 4.5.

Let $R$ be a noetherian ring and fix $n\in\mathbb{Z}_{\geq 0}$. Let $X$ and
$Y$ be $R$-schemes of finite type and let

$\varphi:X\xrightarrow{f}Y\xrightarrow{\psi}\mathbb{P}^{n}_{R}$

be $R$-morphisms. By Definition 4.3, it holds that

(4.5.1) $X^{{\operatorname{univ}}}_{\varphi}\simeq
Y^{{\operatorname{univ}}}_{\psi}\times_{Y}X\qquad{\rm and}\qquad
X^{{\operatorname{gen}}}_{\varphi}\simeq
Y^{{\operatorname{gen}}}_{\psi}\times_{Y}X.$

###  4.2. Bertini theorems for schemes

In this subsection, we prove a main theorem of this paper (Theorem 4.7).

######  Proposition 4.6.

Let $R$ be a noetherian ring and fix $n\in\mathbb{Z}_{>0}$. Set
$X:=\mathbb{P}^{n}_{R}={\rm Proj}\,R[x_{0},...,x_{n}]$ and let
$X=\bigcup_{i=0}^{n}X_{i}$ be the open cover with $X_{i}:=D_{+}(x_{i})$ (for
the definition of $D_{+}(x_{i})$, see [Har77, Ch. II, Proposition 2.5]). For
each $i\in\\{0,...,n\\}$, let $j_{i}:X_{i}\hookrightarrow X$ be the induced
open immersion. Then the following hold.

  1. (1)

For each $i\in\\{0,...,n\\}$, an isomorphism of $X_{i}$-schemes

$(X_{i})^{{\operatorname{univ}}}_{j_{i}}\simeq
X_{i}\times_{R}\mathbb{P}^{n-1}_{R}$

holds.

  2. (2)

The induced morphism $\beta:(X)^{{\operatorname{gen}}}_{{\rm id}}\to X$ is an
$\mathbb{A}^{n-1}$-localising morphism.

###### Proof.

Let us show (1). By symmetry, we may assume that $i=0$. For
$X=\mathbb{P}^{n}_{R,x}={\operatorname{Proj}}\,R[x_{0},...,x_{n}]$ and
$(\mathbb{P}^{n}_{R})^{*}\simeq\mathbb{P}^{n}_{R,s}={\operatorname{Proj}}\,R[s_{0},...,s_{n}]$,
$X^{{\operatorname{univ}}}_{{\rm id}}$ is the effective Cartier divisor on
$\mathbb{P}^{n}_{R,x}\times_{R}\mathbb{P}^{n}_{R,s}$ whose defining equation
is $\\{s_{0}x_{0}+\cdots+s_{n}x_{n}=0\\}$. In particular, for
$X_{0}=\mathbb{A}^{n}_{R,y}={\operatorname{Spec}}\,R[y_{1},...,y_{n}]$ with
$y_{i}=x_{i}/x_{0}$, the assertion (1) holds by the following computation:

$(X_{0})^{{\operatorname{univ}}}_{j_{0}}\simeq X^{{\operatorname{univ}}}_{{\rm
id}}\times_{X}X_{0}\simeq{\operatorname{Proj}}\,\frac{(R[y_{1},...,y_{n}])[s_{0},...,s_{n}]}{(s_{0}+s_{1}y_{1}\cdots+s_{n}y_{n})}$
$\simeq({\operatorname{Spec}}\,R[y_{1},...,y_{n}])\times_{R}\mathbb{P}^{n-1}_{R}=X_{0}\times_{R}\mathbb{P}^{n-1}_{R}.$

where the first isomorphism holds by Remark 4.5.

Let us show (2). Take the base change:

$\beta_{0}:=\beta\times_{X}X_{0}:(X_{0})^{{\operatorname{gen}}}_{j_{0}}\to
X_{0}.$

It suffices to show that $\beta_{0}$ is an $\mathbb{A}^{n-1}$-localising
morphism. For $t_{i}:=s_{i}/s_{n}$ and
$\mathbb{A}^{n}_{R,t}={\operatorname{Spec}}\,R[t_{0},...,t_{n-1}]$, we obtain
the following diagram in which all the squares are cartesian:

${(X_{0})^{{\operatorname{gen}}}_{j_{0}}}$${W}$${(X_{0})^{{\operatorname{univ}}}_{j_{0}}}$${\mathbb{A}^{n}_{R^{\prime},y}}$${\mathbb{A}^{n}_{R,y}\times_{R}\mathbb{A}^{n}_{R,t}}$${\mathbb{A}^{n}_{R,y}\times_{R}\mathbb{P}^{n}_{R,s}}$${\mathbb{A}^{n}_{R,y}=X_{0}}$${{\operatorname{Spec}}\,R^{\prime}}$${\mathbb{A}^{n}_{R,t}}$${\mathbb{P}^{n}_{R,s}}$${{\operatorname{Spec}}\,R}$$\scriptstyle{\iota_{3}}$$\scriptstyle{\rm
pr}$$\scriptstyle{\iota_{2}}$$\scriptstyle{{\rm
pr}_{2}}$$\scriptstyle{\iota_{1}}$$\scriptstyle{{\rm
pr}_{2}}$$\scriptstyle{\rm pr}$$\scriptstyle{h}$

where $R^{\prime}:=R(t_{0},...,t_{n-1})$ and $\iota_{1},\iota_{2}$, and
$\iota_{3}$ are closed immersions. We have that

$W={\operatorname{Spec}}\,\frac{R[y_{1},...,y_{n},t_{0},...,t_{n-1}]}{(t_{0}+t_{1}y_{1}\cdots+t_{n-1}y_{n-1}+y_{n})}\subset\mathbb{A}^{n}_{R,y}\times_{R}\mathbb{A}^{n}_{R,t}.$

It holds that

$\displaystyle\Gamma((X_{0})^{{\operatorname{gen}}}_{j_{0}},\mathcal{O}_{(X_{0})^{{\operatorname{gen}}}_{j_{0}}})$
$\displaystyle\simeq$
$\displaystyle\frac{R[y_{1},...,y_{n},t_{0},...,t_{n-1}]}{(t_{0}+t_{1}y_{1}\cdots+t_{n-1}y_{n-1}+y_{n})}\otimes_{R[t_{0},...,t_{n-1}]}R(t_{0},...,t_{n-1})$
$\displaystyle\simeq$
$\displaystyle\frac{R[y_{1},...,y_{n},t_{0},...,t_{n-1}]}{(t_{0}+t_{1}y_{1}\cdots+t_{n-1}y_{n-1}+y_{n})}\otimes_{R[t_{0},...,t_{n-1}]}S_{1}^{-1}(R[t_{0},...,t_{n-1}])$
$\displaystyle\simeq$ $\displaystyle
S_{2}^{-1}\left(\frac{R[y_{1},...,y_{n},t_{0},...,t_{n-1}]}{(t_{0}+t_{1}y_{1}\cdots+t_{n-1}y_{n-1}+y_{n})}\right)$
$\displaystyle\simeq$ $\displaystyle
S_{3}^{-1}\left(R[y_{1},...,y_{n},t_{1},...,t_{n-1}]\right)$

where each $S_{i}$ is a suitable multiplicatively closed subset. Thus (2)
holds. ∎

######  Theorem 4.7.

Let $R$ be a noetherian ring and fix $n\in\mathbb{Z}_{\geq 0}$. Let $X$ be an
$R$-scheme of finite type and let $\varphi:X\to\mathbb{P}^{n}_{R}$ be an
$R$-morphism. Then the following hold.

  1. (1)

There exists an open cover $X=\bigcup_{i=0}^{n}X_{i}$ such that an
$X_{i}$-isomorphism

$X^{{\operatorname{univ}}}_{\varphi}\times_{X}X_{i}\simeq\mathbb{P}^{n-1}_{R}\times_{R}X_{i}$

holds for each $i\in\\{0,...,n\\}$.

  2. (2)

The induced morphism $\beta:X^{{\operatorname{gen}}}_{\varphi}\to X$ is an
$\mathbb{A}^{n-1}$-localising morphism.

  3. (3)

Let (P) be a property for noetherian schemes which satisfies the following
properties (I)–(III).

    1. (I)

Let $Y$ be a noetherian scheme and let $Y=\bigcup_{j\in J}Y_{j}$ be an open
cover. Then $Y$ is (P) if and only if $Y_{j}$ is (P) for every $j\in J$.

    2. (II)

For a noetherian ring $A$, if ${\operatorname{Spec}}\,A$ is (P), then also
${\operatorname{Spec}}\,A[t]$ is (P), where $A[t]$ denotes the polynomial ring
over $A$ with one variable.

    3. (III)

For a noetherian ring $A$ and a multiplicatively closed subset $S$ of $A$, if
${\operatorname{Spec}}\,A$ is (P), then also ${\operatorname{Spec}}\,S^{-1}A$
is (P).

If $X$ is (P), then also $X^{{\operatorname{gen}}}_{\varphi}$ is (P).

###### Proof.

The assertion (1) follows from Remark 4.5 and Proposition 4.6(1).

Let us show (2). By Remark 4.5, we have the following cartesian diagram

$\begin{CD}X^{{\operatorname{gen}}}_{\varphi}@>{\beta}>{}>X\\\
@V{}V{}V@V{}V{\varphi}V\\\ (\mathbb{P}^{n}_{R})^{{\operatorname{gen}}}_{{\rm
<EMAIL_ADDRESS>
It holds by Proposition 4.6(2) that $\beta^{\prime}$ is an
$\mathbb{A}^{n-1}$-localising morphism. Then it follows from Proposition
2.7(1) that also $\beta$ is an $\mathbb{A}^{n-1}$-localising morphism. Thus
(2) holds. The assertion (3) directly follows from (2). ∎

######  Remark 4.8.

We use the same notation as in Theorem 4.7. As a stronger result of Theorem
4.7(1), it holds that the induced morphism

$\alpha:X^{{\operatorname{univ}}}_{\varphi}\to X$

is a $\mathbb{P}^{n-1}$-bundle, i.e., there exists a locally free sheaf $E$ on
$X$ of rank $n$ such that
$X^{{\operatorname{univ}}}_{\varphi}\simeq\mathbb{P}_{X}(E)$. In order to
prove this, it is enough to assume that $X=\mathbb{P}^{n}_{R}$ and
$\varphi:X\to\mathbb{P}^{n}_{R}$ is the identity morphism (Remark 4.5). By the
proof of Proposition 4.6(1), we obtain

$X^{{\operatorname{univ}}}_{{\rm
id}}=\\{s_{0}x_{0}+\cdots+s_{n}x_{n}=0\\}\subset\mathbb{P}^{n}_{R,x}\times\mathbb{P}^{n}_{R,s}$

for $X=\mathbb{P}^{n}_{R,x}={\operatorname{Proj}}\,R[x_{0},...,x_{n}]$ and
$(\mathbb{P}^{n}_{R})^{*}\simeq\mathbb{P}^{n}_{R,s}={\operatorname{Proj}}\,R[s_{0},...,s_{n}]$.
By using the Euler sequence

$0\to\mathcal{O}_{\mathbb{P}^{n}_{R}}\to\mathcal{O}_{\mathbb{P}^{n}_{R}}(1)^{\oplus
n+1}\to T_{\mathbb{P}^{n}_{R}/R}\to 0$

we can check
$X\simeq\mathbb{P}_{\mathbb{P}^{n}_{R,s}}(T_{\mathbb{P}^{n}_{R,s}/R})$.

######  Theorem 4.9.

Let $R$ be a noetherian ring and fix $n,m\in\mathbb{Z}_{\geq 0}$. Let $X$ be
an $R$-scheme of finite type and let $\varphi:X\to\mathbb{P}^{n}_{R}$ be an
$R$-morphism. Let (P) be one of the following properties.

  1. (1)

$R_{m}$.

  2. (2)

$S_{m}$.

  3. (3)

$G_{m}$.

  4. (4)

Regular.

  5. (5)

Cohen–Macaulay.

  6. (6)

Gorenstein.

  7. (7)

Reduced.

  8. (8)

Normal.

  9. (9)

Seminormal.

  10. (10)

Weakly normal.

  11. (11)

Irreducible or empty.

  12. (12)

Integral or empty.

  13. (13)

Resolution-rational.

  14. (14)

Du Bois (assume $R$ to be a field of characteristic zero).

  15. (15)

$F$-rational (assume $R$ to be an excellent $\mathbb{F}_{p}$-algebra).

  16. (16)

$F$-injective (assume $R$ to be an $\mathbb{F}_{p}$-algebra).

If $X$ is (P), then $X_{\varphi}^{{\operatorname{gen}}}$ is (P).

###### Proof.

By Theorem 4.7(1), the induced morphism
$\beta:X_{\varphi}^{{\operatorname{gen}}}\to X$ is an
$\mathbb{A}^{n-1}$-localising morphism. In particular,
$\beta:X_{\varphi}^{{\operatorname{gen}}}\to X$ is flat and any fibre of
$\beta$ is geometrically regular.

It follows from [Mat89, Theorem 23.9] that the cases (1) and (2) hold true. We
will handle the case (3) in the next paragraph. Then each of the conditions
(4)–(8) can be written as a combination of (1)–(3). The cases (9) and (10)
follow from [GT80, Proposition 5.1(a)] and [Man80, ii) of Proposition
(III.3)], respectively.

Let us show (3). It suffices to prove that if ${\operatorname{Spec}}\,A$ is
$G_{m}$, then also ${\operatorname{Spec}}\,A[x]$ is $G_{m}$ (although this is
probably a well-known fact). Fix $\mathfrak{q}\in{\operatorname{Spec}}\,A[x]$
with $\dim A[x]_{\mathfrak{q}}\leq m$ and set $\mathfrak{p}$ to be the
pullback of $\mathfrak{q}$ on $A$. Since $A_{\mathfrak{p}}\to
A[x]_{\mathfrak{q}}$ is a faithfully flat ring homomorphism, we have that
$\dim A_{\mathfrak{p}}\leq\dim A[x]_{\mathfrak{q}}\leq m$. As
${\operatorname{Spec}}\,A$ is $G_{m}$, $A_{\mathfrak{p}}$ is Gorenstein. We
have a ring isomorphism
$A[x]_{\mathfrak{q}}\simeq(A_{\mathfrak{p}}[x])_{\mathfrak{q}^{\prime}}$ for
some $\mathfrak{q}^{\prime}\in{\operatorname{Spec}}\,A_{\mathfrak{p}}[x]$.
Therefore, also $(A_{\mathfrak{p}}[x])_{\mathfrak{q}^{\prime}}$ is Gorenstein,
hence so is $A[x]_{\mathfrak{q}}$. Thus (3) holds true.

Let us treat the case (12). Assume that $X$ is an integral scheme. We have the
following diagram whose square is cartesian:

$\begin{CD}X@<{\alpha}<{}<X^{{\operatorname{univ}}}_{\varphi}@<{\gamma}<{}<X^{{\operatorname{gen}}}_{\varphi}\\\
@V{}V{}V@V{}V{}V\\\
(\mathbb{P}^{n}_{R})^{*}@<{}<{}<{\operatorname{Spec}}\,R(n).\end{CD}$

We have a factorisation:

$(\mathbb{P}^{n}_{R})^{*}\overset{\iota}{\hookleftarrow}\mathbb{A}^{n}_{R}\xleftarrow{\theta^{*}}{\operatorname{Spec}}\,R(n)$

where $\iota$ is an open immersion and $\theta^{*}$ is the morphism induced by
$\theta:R[t_{1},...,t_{n}]\to R(t_{1},...,t_{n})$ up to isomorphisms
(Definition 4.1). Hence it suffices to show that
$X^{{\operatorname{univ}}}_{\varphi}$ is either integral or empty. Recall that
$\alpha:X^{{\operatorname{univ}}}_{\varphi}\to X$ is a flat morphism of finite
type whose fibres are $\mathbb{P}^{n-1}$ (Theorem 4.7(1)). If $n=0$, then
$X^{{\operatorname{univ}}}_{\varphi}$ is empty. We may assume that $n>0$. Then
$\alpha$ is an open map whose fibres are connected. Since $X$ is connected,
$X^{{\operatorname{univ}}}_{\varphi}$ is connected. Thus (12) holds true.
Since the induced closed immersion $X_{{\operatorname{red}}}\hookrightarrow X$
is a universal homeomorphism, it follows from Proposition 4.5(2) that the case
(12) implies the case (11).

The remaining cases are (13)–(16). Let us treat (13). Let $f:W\to X$ be a
resolution. We have a cartesian diagram

$\begin{CD}W^{{\operatorname{gen}}}_{f\circ\varphi}\simeq
X^{{\operatorname{gen}}}_{\varphi}\times_{X}W@>{\beta_{W}}>{}>W\\\
@V{}V{f^{{\operatorname{gen}}}}V@V{}V{f}V\\\
<EMAIL_ADDRESS>
Then (13) holds true by the flat base change theorem. Concerning (14), we may
apply a similar argument to (13) by using [Sch07, Theorem 4.6].

As for (15), the assertion follows from [Vel95, Theorem 3.1] (note that a
smooth homomorphism in [Vel95] means a flat ring homomorphism with
geometrically regular fibres: [Vel95, the paragraph immdiately after Theorem
3.1]). Finally, (16) holds true by [DM, Corollary 4.2]. ∎

We prove the following lemma for a later usage.

######  Lemma 4.10.

Let $R$ be a noetherian ring and fix $n\in\mathbb{Z}_{\geq 0}$. Let $X$ be an
irreducible scheme of finite type over $R$ and let
$\varphi:X\to\mathbb{P}^{n}_{R}$ be an $R$-morphism. Let
$\beta:X^{{\operatorname{gen}}}_{\varphi}\to X$ be the induced morphism. Then
either $X^{{\operatorname{gen}}}_{\varphi}=\emptyset$ or
$\beta(X^{{\operatorname{gen}}}_{\varphi})$ contains the generic point
$\xi_{X}$ of $X$.

###### Proof.

We may assume that $X^{{\operatorname{gen}}}_{\varphi}\neq\emptyset$ and $X$
is an integral scheme (Remark 4.5). Note that
$X^{{\operatorname{univ}}}_{\varphi}$ is an integral scheme by the proof of
Theorem 4.9(12). We consider $X^{{\operatorname{gen}}}_{\varphi}$ as a subset
of $X^{{\operatorname{univ}}}_{\varphi}$ via the induced injection
$X^{{\operatorname{gen}}}_{\varphi}\hookrightarrow
X^{{\operatorname{univ}}}_{\varphi}$. Then we can find an affine open subset
${\operatorname{Spec}}\,A\subset X^{{\operatorname{univ}}}_{\varphi}$ whose
pullback to $X^{{\operatorname{gen}}}_{\varphi}$ can be written as
${\operatorname{Spec}}\,(S^{-1}A)$ for some multiplicatively closed subset $S$
of $A$ with $S^{-1}A\neq 0$. Since $X^{{\operatorname{univ}}}_{\varphi}$ is an
integral scheme, the image $X^{{\operatorname{gen}}}_{\varphi}$ contains the
generic point $\xi_{X^{{\operatorname{univ}}}_{\varphi}}$ of
$X^{{\operatorname{univ}}}_{\varphi}$. Since
$\xi_{X^{{\operatorname{univ}}}_{\varphi}}$ is mapped to the generic point
$\xi_{X}$ of $X$, $\beta(X^{{\operatorname{gen}}}_{\varphi})$ contains
$\xi_{X}$, as required. ∎

###  4.3. Avoidance property for generic members

######  Proposition 4.11.

Let $R$ be a noetherian ring, $X$ a non-empty $R$-scheme of finite type, and
$\varphi:X\to\mathbb{P}^{n}_{R}$ an $R$-morphism with $n\in\mathbb{Z}_{\geq
0}$. Then $X\times_{R}R(n)\neq X^{{\operatorname{gen}}}_{\varphi}$ holds as
sets.

###### Proof.

If $X^{{\operatorname{gen}}}_{\varphi}=\emptyset$, then there is nothing to
show. In what follows, we assume
$X^{{\operatorname{gen}}}_{\varphi}\neq\emptyset$. In particular, it holds
that $n\geq 1$.

######  Step 1.

Proposition 4.11 holds for the case when $R$ is an algebraically closed field.

###### Proof.

(of Step 1) Set $\kappa:=R$. Fix a closed point $x$ of $X$. We consider $x$ as
a reduced scheme. Recall that $X^{{\operatorname{univ}}}_{\varphi}$ is a
closed subscheme of $X\times_{\kappa}(\mathbb{P}^{n}_{\kappa})^{*}$ and
$X^{{\operatorname{gen}}}_{\varphi}$ is its generic fibre over
$K((\mathbb{P}^{n}_{\kappa})^{*})$ (Definition 4.3). Take the scheme-theoretic
inverse image of $x$ by the first projection
$b:X\times_{\kappa}(\mathbb{P}^{n}_{\kappa})^{*}\to X$:

$\Gamma:=b^{-1}(x)=\\{x\\}\times_{\kappa}(\mathbb{P}^{n}_{\kappa})^{*}\subset
X\times_{\kappa}(\mathbb{P}^{n}_{\kappa})^{*}.$

We have morphisms

$\pi:\Gamma\cap
X^{{\operatorname{univ}}}_{\varphi}\overset{j}{\hookrightarrow}X\times_{\kappa}(\mathbb{P}^{n}_{\kappa})^{*}\xrightarrow{{\rm
pr}_{2}}(\mathbb{P}^{n}_{\kappa})^{*},$

where $j$ is the induced closed immersion. Since $x\not\in\varphi^{-1}(H)$
holds for any general hyperplane $H$ of $\mathbb{P}^{n}_{\kappa}$, it holds
that

$\pi^{-1}(y)=\left(\Gamma\cap
X^{{\operatorname{univ}}}_{\varphi}\right)\cap(X\times_{\kappa}\\{y\\})=\emptyset$

for any general closed point $y\in(\mathbb{P}^{n}_{\kappa})^{*}$. By the
generic flatness, $\Gamma\cap X^{{\operatorname{univ}}}_{\varphi}$ does not
dominate $(\mathbb{P}^{n}_{\kappa})^{*}$, and hence the generic fibre of $\pi$
is empty, i.e. $\\{x^{\prime}\\}\cap
X^{{\operatorname{gen}}}_{\varphi}=\emptyset$ holds for
$x^{\prime}:=x\times_{\kappa}K((\mathbb{P}^{n}_{\kappa})^{*})$. This completes
the proof of Step 1. ∎

######  Step 2.

Proposition 4.11 holds for the case when $X$ is proper over $R$.

###### Proof.

(of Step 2) Fix a maximal ideal $\mathfrak{m}$ of $R$ that is contained in the
image of $X\to{\operatorname{Spec}}\,R$, whose existence is guaranteed by the
properness of $X\to{\operatorname{Spec}}\,R$. Set $\kappa:=R/\mathfrak{m}$ and
$X_{\kappa}:=X\times_{R}\kappa$. For the induced morphism
$\psi:=\varphi\times_{R}\kappa:X_{\kappa}\to\mathbb{P}^{n}_{\kappa}$, we have
the following cartesian diagram (Proposition 3.5, Definition 4.3):

${(X_{\kappa})^{{\operatorname{gen}}}_{\psi}}$${X^{{\operatorname{gen}}}_{\varphi}}$${X_{\kappa}\times_{\kappa}\kappa(n)}$${X\times_{R}R(n).}$closed
immersionclosed immersionclosed immersionclosed immersion

In order to show $X^{{\operatorname{gen}}}_{\varphi}\neq X\times_{R}R(n)$, it
suffices to prove $(X_{\kappa})^{{\operatorname{gen}}}_{\psi}\neq
X_{\kappa}\times_{\kappa}\kappa(n)$. Replacing $(R,X)$ by
$(\kappa,X_{\kappa})$, we may assume that $\kappa:=R$ is a field. Furthermore,
taking the base change to the algebraic clsoure $\overline{\kappa}$ of
$\kappa$, we may assume that $R$ is an algebraically closed field (Lemma 3.16,
Definition 4.3). By Step 1, we completes the proof of Step 2. ∎

######  Step 3.

Proposition 4.11 holds for the case when $X$ is an integral scheme.

###### Proof.

(of Step 3) Fix a non-empty affine open subset $X^{\prime}$ of $X$. For the
induced morphism $\varphi^{\prime}:X^{\prime}\hookrightarrow
X\xrightarrow{\varphi}\mathbb{P}^{n}_{R}$, we have the following cartesian
diagram (Remark 4.5):

${X^{\prime{\operatorname{gen}}}_{\varphi^{\prime}}}$${X^{{\operatorname{gen}}}_{\varphi}}$${X^{\prime}\times_{R}R(n)}$${X\times_{R}R(n).}$open
immersionclosed immersionclosed immersionopen immersion

Note that all the schemes in this diagram are integral (Theorem 3.13, Theorem
4.9). Therefore, $X^{{\operatorname{gen}}}_{\varphi}=X\times_{R}R(n)$ if and
only if
$X^{\prime{\operatorname{gen}}}_{\varphi^{\prime}}=X^{\prime}\times_{R}R(n)$.
Hence the problem is reduced to the case when $X$ is affine. Pick an open
immersion $X\hookrightarrow X^{\prime\prime}$ to an integral projective
$R$-scheme $X^{\prime\prime}$. Applying the above argument again, the problem
is reduced to the case when $X$ is projective over $R$, which has been settled
in Step 2. This completes the proof of Step 3. ∎

######  Step 4.

Proposition 4.11 holds without any additional assumptions.

###### Proof.

(of Step 4) Fix an irreducible component $Y$ of $X$, which we equip with the
reduced scheme structure. We have the following cartesian diagram (Remark
4.5):

(4.11.1)
${Y^{{\operatorname{gen}}}_{\psi}}$${X^{{\operatorname{gen}}}_{\varphi}}$${Y\times_{R}R(n)}$${X\times_{R}R(n),}$$\scriptstyle{\iota_{Y}}$$\scriptstyle{\iota_{X}}$

where $\psi:Y\hookrightarrow X\xrightarrow{\varphi}\mathbb{P}^{n}_{R}$ denotes
the induced morphism. Suppose the set-theoretic equation
$X\times_{R}R(n)=X^{{\operatorname{gen}}}_{\varphi}$ holds, i.e. the closed
immersion $\iota_{X}$ is surjective. Then $\iota_{Y}$ is surjective, and hence
we obtain the set-theoretic equation
$Y\times_{R}R(n)=Y^{{\operatorname{gen}}}_{\psi}$, which contradicts Step 3.
This completes the proof of Step 4. ∎

Step 4 completes the proof of Proposition 4.11. ∎

######  Theorem 4.12.

Let $R$ be a noetherian ring, $X$ a non-empty $R$-scheme of finite type, and
$\varphi:X\to\mathbb{P}^{n}_{R}$ an $R$-morphism with $n\in\mathbb{Z}_{\geq
0}$. Then the following hold.

  1. (1)

If $Y$ is a closed subscheme of $X$, then $Y\times_{R}R(n)\not\subset
X^{{\operatorname{gen}}}_{\varphi}$ holds as sets.

  2. (2)

$X^{{\operatorname{gen}}}_{\varphi}$ is an effective Cartier divisor on
$X\times_{R}R(n)$.

  3. (3)

If $R$ is an excellent ring, then $\dim
X^{{\operatorname{gen}}}_{\varphi}\leq\dim X-1$.

###### Proof.

The assertion (1) follows from the cartesian diagram (4.11.1) and Proposition
4.11.

Let us show (2). By (1), $X^{{\operatorname{gen}}}_{\varphi}$ contains no
irreducible components of $X\times_{R}R(n)$. Then
$X^{{\operatorname{gen}}}_{\varphi}$ is an effective Cartier divisor on
$X\times_{R}R(n)$, since $X^{{\operatorname{gen}}}_{\varphi}$ is a closed
subscheme on $X\times_{R}{R(n)}$ whose defining ideal sheaf is locally
principal (Remark 4.4). Thus (2) holds.

The assertion (3) follows from (2) and $\dim X=\dim X\times_{R}R(n)$ (Theorem
3.22). ∎

###  4.4. Restriction to generic members

Let $R$ be a noetherian ring and fix $n\in\mathbb{Z}_{\geq 0}$. Let $X$ be an
integral normal scheme of finite type over $R$ and let
$\varphi:X\to\mathbb{P}^{n}_{R}$ be an $R$-morphism. Then
$X_{R(n)}:=X\times_{R}R(n)$ is an integral normal scheme of finite type over
$R(n)$ (Proposition 3.6, Theorem 3.13(1)). Recall that the generic member
$X^{{\operatorname{gen}}}_{\varphi}$ is an effective Cartier divisor on
$X_{R(n)}$ (Theorem 4.12(2)). In particular, we have an exact sequence

(4.12.1)
$0\to\mathcal{O}_{X_{R(n)}}(-X^{{\operatorname{gen}}}_{\varphi})\to\mathcal{O}_{X_{R(n)}}\to\mathcal{O}_{X^{{\operatorname{gen}}}_{\varphi}}\to
0.$

Note that $X^{{\operatorname{gen}}}_{\varphi}$ might be the empty set,
although we are mainly interested in the case when
$X^{{\operatorname{gen}}}_{\varphi}\neq\emptyset$. Furthermore, we have $\dim
X_{R(n)}=\dim X$ when $R$ is excellent (Theorem 3.22).

######  Definition 4.13.

Let $R$ be a noetherian ring and fix $n\in\mathbb{Z}_{\geq 0}$. Let $X$ be an
integral normal scheme of finite type over $R$ and let
$\varphi:X\to\mathbb{P}^{n}_{R}$ be an $R$-morphism. Let
$\beta:X^{{\operatorname{gen}}}_{\varphi}\to X$ be the induced morphism.
Recall that

  1. (i)

$X^{{\operatorname{gen}}}_{\varphi}$ is empty or an integral normal scheme of
finite type over $R(n)$ (Theorem 4.9), and

  2. (ii)

if $P$ is a prime divisor on $X$, then $\beta^{-1}(P)\simeq
P^{{\operatorname{gen}}}_{j\circ\varphi}$ is either empty or a prime divisor
on $X^{{\operatorname{gen}}}_{\varphi}$ for the induced closed immersion
$j:P\hookrightarrow X$ (Remark 4.5 and Theorem 4.9).

  1. (1)

For a prime divisor $P$ on $X$, we set

$\beta^{*}P:=\begin{cases}\beta^{-1}(P)\qquad&\text{if }\beta^{-1}(P)\text{ is
a prime divisor}\\\ 0\qquad&\text{otherwise.}\\\ \end{cases}$

  2. (2)

For an $\mathbb{R}$-divisor $D$ on $X$ and its irreducible decomposition
$D=\sum_{i=1}^{r}a_{i}P_{i}$, we set

$\beta^{*}D:=\sum_{i=1}^{r}a_{i}\beta^{*}P_{i},$

which is an $\mathbb{R}$-divisor on $X^{{\operatorname{gen}}}_{\varphi}$.

######  Proposition 4.14.

Let $R$ be an excellent ring and fix $n\in\mathbb{Z}_{\geq 0}$. Let $X$ be an
integral normal scheme of finite type over $R$ and let
$\varphi:X\to\mathbb{P}^{n}_{R}$ be an $R$-morphism. Let $D$ be an
$\mathbb{R}$-divisor on $X$. Set $X_{R(n)}:=X\times_{R}R(n)$ and let
$D_{R(n)}$ be the pullback of $D$ to $X_{R(n)}$.

  1. (1)

If $P$ and $Q$ are distinct prime divisors on $X$ such that
$\beta^{-1}(P)\neq\emptyset$, then $\beta^{*}(P)\neq\beta^{*}(Q)$.

  2. (2)

The following hold:

$\llcorner\beta^{*}D\lrcorner=\beta^{*}(\llcorner
D\lrcorner),\quad\ulcorner\beta^{*}D\urcorner=\beta^{*}(\ulcorner
D\urcorner),\quad\text{and}\quad\\{\beta^{*}D\\}=\beta^{*}(\\{D\\}).$

  3. (3)

There exists an exact sequence of $\mathcal{O}_{X_{R(n)}}$-modules

$0\to\mathcal{O}_{X_{R(n)}}(D_{R(n)}-X^{{\operatorname{gen}}}_{\varphi})\to\mathcal{O}_{X_{R(n)}}(D_{R(n)})\to\mathcal{O}_{X^{{\operatorname{gen}}}_{\varphi}}(\beta^{*}D)\to
0$

whose restriction to the regular locus of $X_{R(n)}$ coincides with the
natural exact sequence obtained from (4.12.1).

  4. (4)

$\beta^{*}\mathcal{O}_{X}(D)\simeq\mathcal{O}_{X^{{\operatorname{gen}}}_{\varphi}}(\beta^{*}D)$.

###### Proof.

If $X^{{\operatorname{gen}}}_{\varphi}=\emptyset$, then all the assertions are
obvious. Hence we may assume that
$X^{{\operatorname{gen}}}_{\varphi}\neq\emptyset$.

Let us show (1). Assume that $\beta^{-1}(P)\neq\emptyset$ and
$\beta^{-1}(P)=\beta^{-1}(Q)$ (automatically, we have
$\beta^{-1}(Q)\neq\emptyset$). It suffices to show that $P=Q$. This follows
from

$P=\overline{\beta(P^{{\operatorname{gen}}}_{\varphi|_{P}})}=\overline{\beta(\beta^{-1}(P))}=\overline{\beta(\beta^{-1}(Q))}=\overline{\beta(Q^{{\operatorname{gen}}}_{\varphi|_{Q}})}=Q,$

where the second and fourth equalities hold by the surjectivity of $\beta$ and
the first and last equalities follow from Lemma 4.10. Thus (1) holds. The
assertion (2) follows from (1) and the fact that $\beta^{-1}(P)$ is either
empty or a prime divisor (cf. Definition 4.13).

Let us show (3). By (2), we may assume that $D$ is a $\mathbb{Z}$-divisor. For
the regular locus $U$ of $X$ and the induced open immersion
$i:U_{R(n)}\hookrightarrow X_{R(n)}$, we obtain
$i_{*}i^{*}\mathcal{F}=\mathcal{F}$ for

$\mathcal{F}\in\\{\mathcal{O}_{X_{R(n)}}(D_{R(n)}-X^{{\operatorname{gen}}}_{\varphi}),\mathcal{O}_{X_{R(n)}}(D_{R(n)}),\mathcal{O}_{X^{{\operatorname{gen}}}_{\varphi}}(\beta^{*}D)\\}.$

Therefore, we may assume that $X$ is regular, and hence $D$ is Cartier. Then
the exact sequence (4.12.1) induces the required exact sequence. Thus (3)
holds. The assertion (4) follows from (3). ∎

###  4.5. Bertini theorems for pairs

######  Theorem 4.15.

Let $R$ be an excellent ring admitting a dualising complex. Let $X$ be an
integral normal scheme of finite type over $R$, $\Delta$ an
$\mathbb{R}$-divisor such that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier, and
$\varphi:X\to\mathbb{P}^{n}_{R}$ an $R$-morphism. Let $(P)$ be one of the
following properties.

  1. (17)

Terminal.

  2. (18)

Canonical.

  3. (19)

Klt.

  4. (20)

Plt.

  5. (21)

Dlt.

  6. (22)

Lc.

If $(X,\Delta)$ is (P), then
$(X_{\varphi}^{{\operatorname{gen}}},\beta^{*}\Delta)$ is (P).

###### Proof.

We apply the same argument as in Theorem 4.9. Take an
$\mathbb{A}^{n}$-localising morphism $f:Y\to X$. It is enough to show that if
$(X,\Delta)$ satisfies (P), then $(Y,\Delta_{Y}:=f^{*}\Delta)$ satisfies (P).
Since taking blowups commutes with flat base changes, if $(X,\Delta)$
satisfies (P), then $(Y,\Delta_{Y})$ satisfies (P) (cf. [Kol13, Proposition
2.15 and Warning immidiately after Proposition 2.15]). ∎

######  Theorem 4.16.

Let $R$ be an $F$-finite noetherian $\mathbb{F}_{p}$-algebra. Let $f:X\to Y$
be an $R$-morphism of schemes which are of finite type over $R$. Assume that
$X$ is integral and normal. Let $\Delta$ an effective $\mathbb{R}$-divisor on
$X$. Let $\psi:Y\to\mathbb{P}^{n}_{R}$ be an $R$-morphism and set
$\varphi:=\psi\circ f$:

$\varphi:X\xrightarrow{f}Y\xrightarrow{\psi}\mathbb{P}^{n}_{R}.$

Recall that we have the following cartesian diagram (Remark 4.5):

$\begin{CD}X^{{\operatorname{gen}}}_{\varphi}@>{\alpha}>{}>X\\\
@V{}V{g}V@V{}V{f}V\\\ <EMAIL_ADDRESS>\end{CD}$

Let $(P)$ be one of the following properties.

  1. (i)

Globally $F$-regular.

  2. (ii)

Globally sharply $F$-split.

  3. (iii)

Globally $F$-split.

If $(X,\Delta)$ is (P) over $Y$, then $X^{{\operatorname{gen}}}_{\varphi}$ is
(P) over $Y^{{\operatorname{gen}}}_{\psi}$.

###### Proof.

We may replace $Y$ by a piece of an open cover of $Y$. In particular, we may
assume that

  1. (1)

$X$ is (P) (over ${\operatorname{Spec}}\,\mathbb{F}_{p}$), and

  2. (2)

$Y={\operatorname{Spec}}\,A$.

Since $\beta$ is an $\mathbb{A}^{n-1}$-localising morphism (Theorem 4.7(2)),
the problem is reduced to the case when there exists a sequence of morphisms
of affine schemes

$Y^{{\operatorname{gen}}}_{\psi}=:Y_{\ell+1}\xrightarrow{\beta_{\ell+1}}Y_{\ell}\xrightarrow{\beta_{\ell}}\cdots\xrightarrow{\beta_{2}}Y_{1}\xrightarrow{\beta_{1}}Y_{0}:=Y,\qquad
A_{j}:=\Gamma(Y_{j},\mathcal{O}_{Y_{j}})$

such that the following hold.

  1. (a)

For any $j\in\\{1,...,\ell\\}$, $A_{j}=A_{j-1}[t]$ and $\beta_{j}:Y_{j}\to
Y_{j-1}$ coincides with the projection
$Y_{j}=Y_{j-1}\times_{k}\mathbb{A}^{1}_{k}\to Y_{j-1}$.

  2. (b)

$A_{\ell+1}=S^{-1}A_{\ell}$ for some multiplicative subset $S$ of $A_{\ell}$
and $\beta_{\ell+1}:Y_{\ell+1}\to Y_{\ell}$ coincides with the morphism
induced from the natural ring homomorphism $A_{\ell}\to
S^{-1}A_{\ell},a\mapsto a/1$.

Set $X_{j}:=X\times_{Y}Y_{j}$ for each $j\in\\{1,...,\ell,\ell+1\\}$ and we
inductively define an effective $\mathbb{R}$-divisor $\Delta_{j}$ on $X_{j}$
as the pullback of $\Delta_{j-1}$ via the induced morphism
$\alpha_{j}:X_{j}\to X_{j-1}$. Fix $j\in\\{1,...,\ell,\ell+1\\}$ and assume
that $(X_{j-1},\Delta_{j-1})$ is (P). It suffices to show that
$(X_{j},\Delta_{j})$ is (P).

Assume $1\leq j\leq\ell$. Take compactifications
$Y^{\prime}_{j}:=Y_{j-1}\times_{k}\mathbb{P}^{1}_{k}$ and
$X^{\prime}_{j}:=X_{j-1}\times_{k}\mathbb{P}^{1}_{k}$ of $Y_{j}$ and $X_{j}$,
respectively. Set $\Delta^{\prime}_{j}$ to be the pullback of $\Delta_{j-1}$
to $X^{\prime}_{j}$. If (P) is (i) or (ii), then
$(X^{\prime}_{j},\Delta^{\prime}_{j})$ is (P) by the same argument as in
[GT16, Proposition 3.1], which in turn implies that $(X_{j},\Delta_{j})$ is
(P). Let us treat the case when (P) is (iii). By the case (ii), which we have
already settled, we may assume that $(X_{j},(1-\epsilon)\Delta_{j})$ is
globally $F$-split for any rational number $0<\epsilon<1$. It follows from
[CTW18, Theorem 3.2] that $(X_{j},\Delta_{j})$ is globally $F$-split. This
completes the proof for the case when $1\leq j\leq\ell$.

Assume $j=\ell+1$. In this case, for any $e>0$, the following diagram is
cartesian:

(4.16.1) $\begin{CD}X_{\ell+1}@>{\alpha_{\ell+1}}>{}>X_{\ell}\\\
@V{}V{F^{e}}V@V{}V{F^{e}}V\\\
X_{\ell+1}@>{\alpha_{\ell+1}}>{}>X_{\ell},\end{CD}$

where each $F^{e}$ denotes the $e$-th iterated absolute Frobenius morphism. We
first handle the case when (P) is (ii). We have a splitting

(4.16.2) ${\rm id}:\mathcal{O}_{X_{\ell}}\to
F^{e}_{*}\mathcal{O}_{X_{\ell}}(\ulcorner(p^{e}-1)\Delta_{\ell}\urcorner)\xrightarrow{\sigma}\mathcal{O}_{X_{\ell}}$

for some $e>0$ and $\sigma$. It holds that

$\displaystyle\alpha_{\ell+1}^{*}F^{e}_{*}\mathcal{O}_{X_{\ell}}(\ulcorner(p^{e}-1)\Delta_{\ell}\urcorner)$
$\displaystyle\simeq$ $\displaystyle
F^{e}_{*}\alpha_{\ell+1}^{*}\mathcal{O}_{X_{\ell}}(\ulcorner(p^{e}-1)\Delta_{\ell}\urcorner)$
$\displaystyle\simeq$ $\displaystyle
F^{e}_{*}\mathcal{O}_{X_{\ell}}(\alpha_{\ell+1}^{*}\ulcorner(p^{e}-1)\Delta_{\ell}\urcorner)$
$\displaystyle=$ $\displaystyle
F^{e}_{*}\mathcal{O}_{X_{\ell}}(\ulcorner(p^{e}-1)\Delta_{\ell+1}\urcorner),$

where the first isomorphism follows from the cartesian diagram (4.16.1).
Applying $\alpha_{\ell+1}^{*}(-)$ to (4.16.2), we obtain

${\rm id}:\mathcal{O}_{X_{\ell+1}}\to
F^{e}_{*}\mathcal{O}_{X_{\ell+1}}(\ulcorner(p^{e}-1)\Delta_{\ell+1}\urcorner)\xrightarrow{\alpha_{\ell+1}^{*}\sigma}\mathcal{O}_{X_{\ell+1}}.$

Hence, $(X_{\ell+1},\Delta_{\ell+1})$ is globally sharply $F$-split. This
completes the case when (P) is (ii). If (P) is (iii), then we can apply the
same argument as in the case (ii). Assume that (P) is (i). In this case, by
applying [SS10, Corollary 3.10], the problem is reduced to the case when (P)
is (ii). ∎

######  Theorem 4.17.

Let $R$ be an $F$-finite noetherian $\mathbb{F}_{p}$-algebra. Let $X$ be an
integral normal scheme of finite type over $R$, $\Delta$ an effective
$\mathbb{R}$-divisor, and $\varphi:X\to\mathbb{P}^{n}_{R}$ an $R$-morphism.
Let $(P)$ be one of the following properties

  1. (23)

Strongly $F$-regular.

  2. (24)

Sharply $F$-pure.

  3. (25)

$F$-pure.

If $(X,\Delta)$ is (P), then
$(X_{\varphi}^{{\operatorname{gen}}},\beta^{*}\Delta)$ is (P).

###### Proof.

The assertion follows from Theorem 4.16. ∎

######  Theorem 4.18.

Let $R$ be an excellent ring admitting a dualising complex. Let $X$ be an
integral normal scheme of finite type over $R$, $\Delta$ an
$\mathbb{R}$-divisor such that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier, and
$\varphi:X\to\mathbb{P}^{n}_{R}$ an $R$-morphism. Set
$X_{R(n)}:=X\times_{R}R(n)$. Let $\Delta_{R(n)}$ be the pullback of $\Delta$
on $X_{R(n)}$. Then the following hold.

  1. (1)

If $(X,\Delta)$ is lc, then
$(X_{R(n)},\Delta_{R(n)}+X^{{\operatorname{gen}}}_{\varphi})$ is lc.

  2. (2)

If $(X,\Delta)$ is klt, then
$(X_{R(n)},\Delta_{R(n)}+X^{{\operatorname{gen}}}_{\varphi})$ is plt.

###### Proof.

Recall that we have the following commutative diagram in which each square is
cartesian:

$\begin{CD}X_{R(n)}@>{}>{}>X\times_{R}(\mathbb{P}^{n}_{R})^{*}@>{}>{}>X\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\
{\operatorname{Spec}}\,R(n)@>{}>{}>(\mathbb{P}^{n}_{R})^{*}@>{}>{}>{\operatorname{Spec}}\,R.\end{CD}$

Set $\Delta_{(\mathbb{P}^{n}_{R})^{*}}$ to be the pullback of $\Delta$ on
$X\times_{R}(\mathbb{P}^{n}_{R})^{*}$. Note that
$X^{{\operatorname{univ}}}_{\varphi}\to X$ is a smooth morphism (Theorem
4.7(1)).

Let us show (1). It follows from [Kol13, (4) in page 47] that
$(X\times_{R}(\mathbb{P}^{n}_{R})^{*},\Delta_{(\mathbb{P}^{n}_{R})^{*}}+X^{{\operatorname{univ}}}_{\varphi})$
is lc. Therefore, also its localisation
$(X_{R(n)},\Delta_{R(n)}+X^{{\operatorname{gen}}}_{\varphi})$ is lc. Thus (1)
holds. The assertion (2) follows from the same argument as in (1). ∎

######  Theorem 4.19.

Let $R$ be an $F$-finite noetherian $\mathbb{F}_{p}$-algebra. Let $X$ be an
integral normal scheme of finite type over $R$, $\Delta$ an effective
$\mathbb{R}$-divisor, and $\varphi:X\to\mathbb{P}^{n}_{R}$ an $R$-morphism.
Set $X_{R(n)}:=X\times_{R}R(n)$. Let $\Delta_{R(n)}$ be the pullback of
$\Delta$ on $X_{R(n)}$. Then the following hold.

  1. (1)

If $(X,\Delta)$ is sharply $F$-pure, then
$(X_{R(n)},\Delta_{R(n)}+X^{{\operatorname{gen}}}_{\varphi})$ is sharply
$F$-pure.

  2. (2)

If $R$ is a field and $(X,\Delta)$ is strongly $F$-regular, then
$(X_{R(n)},\Delta_{R(n)}+X^{{\operatorname{gen}}}_{\varphi})$ is purely
$F$-regular.

###### Proof.

We use the same notation as in the proof of Theorem 4.18.

Let us show (1). Taking an affine open cover of $X$, we may assume that $X$ is
affine. Furthermore, the problem is reduced to the case when $\Delta$ is a
$\mathbb{Q}$-divisor. In this case, $(X,\Delta)$ is globally sharply
$F$-split. By [SS10, Theorem 4.3(ii)], we may assume that
$(p^{e}-1)(K_{X}+\Delta)$ is Cartier for some $e\in\mathbb{Z}_{>0}$.
Therefore,

  1. (a)

$(X\times_{R}(\mathbb{P}^{n}_{R})^{*},\Delta_{(\mathbb{P}^{n}_{R})^{*}})$ is
sharply $F$-pure and

  2. (b)

$(X^{{\operatorname{univ}}}_{\varphi},\Delta_{(\mathbb{P}^{n}_{R})^{*}}|_{X^{{\operatorname{univ}}}_{\varphi}})$
is sharply $F$-pure.

By (a),
$(X\times_{R}(\mathbb{P}^{n}_{R})^{*},\Delta_{(\mathbb{P}^{n}_{R})^{*}}+X^{{\operatorname{univ}}}_{\varphi})$
is sharply $F$-pure outside $X^{{\operatorname{univ}}}_{\varphi}$. By (b) and
inversion of adjunction [Sch09, Main Theorem(iii)],
$(X\times_{R}(\mathbb{P}^{n}_{R})^{*},\Delta_{(\mathbb{P}^{n}_{R})^{*}}+X^{{\operatorname{univ}}}_{\varphi})$
is sharply $F$-pure around $X^{{\operatorname{univ}}}_{\varphi}$. To
summarise,
$(X\times_{R}(\mathbb{P}^{n}_{R})^{*},\Delta_{(\mathbb{P}^{n}_{R})^{*}}+X^{{\operatorname{univ}}}_{\varphi})$
is sharply $F$-pure. Hence, its localisation
$(X_{R(n)},\Delta_{R(n)}+X^{{\operatorname{gen}}}_{\varphi})$ is sharply
$F$-pure. Thus (1) holds.

Let us show (2). By the same argument as in (1), we may assume that $X$ is
affine and $(p^{e}-1)(K_{X}+\Delta)$ is Cartier for some $e\in\mathbb{Z}_{>0}$
(use [SS10, Theorem 4.3(i)] instead of [SS10, Theorem 4.3(ii)]). Then the
following hold.

  1. (c)

$(X\times_{R}(\mathbb{P}^{n}_{R})^{*},\Delta_{(\mathbb{P}^{n}_{R})^{*}})$ is
strongly $F$-regular.

  2. (d)

$(X^{{\operatorname{univ}}}_{\varphi},\Delta_{(\mathbb{P}^{n}_{R})^{*}}|_{X^{{\operatorname{univ}}}_{\varphi}})$
is strongly $F$-regular.

By (c),
$(X\times_{R}(\mathbb{P}^{n}_{R})^{*},\Delta_{(\mathbb{P}^{n}_{R})^{*}}+X^{{\operatorname{univ}}}_{\varphi})$
is strongly $F$-regular outside $X^{{\operatorname{univ}}}_{\varphi}$. By (d)
and inversion of adjunction [Das15, Theorem A],
$(X\times_{R}(\mathbb{P}^{n}_{R})^{*},\Delta_{(\mathbb{P}^{n}_{R})^{*}}+X^{{\operatorname{univ}}}_{\varphi})$
is purely $F$-regular around $X^{{\operatorname{univ}}}_{\varphi}$. To
summarise,
$(X\times_{R}(\mathbb{P}^{n}_{R})^{*},\Delta_{(\mathbb{P}^{n}_{R})^{*}}+X^{{\operatorname{univ}}}_{\varphi})$
is purely $F$-regular. Hence, its localisation
$(X_{K},\Delta_{K}+X^{{\operatorname{gen}}}_{\varphi})$ is purely $F$-regular.
Thus, (2) holds. ∎

######  Corollary 4.20.

Fix $\mathbb{K}\in\\{\mathbb{Q},\mathbb{R}\\}$. Let $R$ be an excellent ring
admitting a dualising complex. Let $X$ be an integral normal scheme of finite
type over $R$, $\Delta$ an $\mathbb{R}$-divisor such that $K_{X}+\Delta$ is
$\mathbb{R}$-Cartier, and $D$ a semi-ample $\mathbb{K}$-Cartier
$\mathbb{K}$-divisor. Let (P) be one of the following properties.

  1. (1)

Lc.

  2. (2)

Klt.

If $(X,\Delta)$ is (P), then there exist $n\in\mathbb{Z}_{>0}$ and an
effective $\mathbb{K}$-Cartier $\mathbb{K}$-divisor $D^{\prime}$ such that
$D^{\prime}\sim_{\mathbb{K}}D_{R(n)}$ and
$(X_{R(n)},\Delta_{R(n)}+D^{\prime})$ is (P), where
$X_{R(n)}:=X\times_{R}R(n)$ and we denote by $\Delta_{R(n)}$ and $D_{R(n)}$
the pullbacks of $\Delta$ and $D$ to $X_{R(n)}$, respectively.

###### Proof.

We first treat the case when $\mathbb{K}=\mathbb{Q}$. In this case, $|nD|$ is
a base point free Cartier divisor for some $n\in\mathbb{Z}_{>0}$. Hence we may
assume that $D$ itself is a base point free Cartier divisor. Then the case (1)
directly follows from Theorem 4.18(1). Since $|2D|$ is base point free, there
exists an $R$-morphism $\varphi:X\to\mathbb{P}^{n}_{R}$ such that
$\mathcal{O}_{X}(2D)\simeq\varphi^{*}\mathcal{O}_{\mathbb{P}^{n}_{R}}(1)$. For
the generic member $X^{{\operatorname{gen}}}_{\varphi}\sim 2D_{R(n)}$, we set
$D^{\prime}:=\frac{1}{2}X^{{\operatorname{gen}}}_{\varphi}\sim_{\mathbb{Q}}D_{R(n)}$.
Let us treat the case (2). Assume that $(X,\Delta)$ is klt. Since
$(X_{R(n)},\Delta_{R(n)})$ is klt and
$(X_{R(n)},\Delta_{R(n)}+X^{{\operatorname{gen}}}_{\varphi})$ is lc, their
midpoint $(X_{R(n)},\Delta_{R(n)}+D^{\prime})$ is klt. Thus the case (2) holds
true.

Let us consider the case when $\mathbb{K}=\mathbb{R}$. By definition, we have
$D=\sum_{i=1}^{\ell}a_{i}D_{i}$ for some
$a_{1},...,a_{\ell}\in\mathbb{R}_{>0}$ and semi-ample $\mathbb{Q}$-Cartier
$\mathbb{Q}$-divisors $D_{1},...,D_{\ell}$. Then the problem is reduced to the
case when $\mathbb{K}=\mathbb{Q}$. ∎

######  Corollary 4.21.

Fix $\mathbb{K}\in\\{\mathbb{Q},\mathbb{R}\\}$. Let $R$ be an $F$-finite
noetherian $\mathbb{F}_{p}$-algebra. Let $X$ be an integral normal scheme of
finite type over $R$, $\Delta$ an $\mathbb{R}$-divisor such that
$K_{X}+\Delta$ is $\mathbb{R}$-Cartier, and $D$ a semi-ample
$\mathbb{K}$-Cartier $\mathbb{K}$-divisor. Let (P) be one of the following
properties.

  1. (1)

Sharply $F$-pure.

  2. (2)

Strongly $F$-regular.

If $(X,\Delta)$ is (P), then there exist $n\in\mathbb{Z}_{>0}$ and an
effective $\mathbb{K}$-Cartier $\mathbb{K}$-divisor $D^{\prime}$ such that
$D^{\prime}\sim_{\mathbb{K}}D_{R(n)}$ and
$(X_{R(n)},\Delta_{R(n)}+D^{\prime})$ is (P), where
$X_{R(n)}:=X\times_{R}R(n)$ and we denote by $\Delta_{R(n)}$ and $D_{R(n)}$
the pullbacks of $\Delta$ and $D$ to $X_{R(n)}$, respectively.

###### Proof.

We can apply the same argument as in Corollary 4.20 after replacing Theorem
4.18 by Theorem 4.19. ∎

######  Remark 4.22.

Assume that (P) is (1) or (2) in Corollary 4.20. If $R$ is a field $k$ of
characteristic zero, Corollary 4.20 is known to hold for (P) even without
taking the base change $R\subset R(n)$ [KM98, Lemma 5.17]. If $R$ is a field
$k$ of positive characteristic, the same assertion holds under the following
assumptions [Tan17, Theorem 1].

  1. (a)

$k$ is an $F$-finite field containing an infinite perfect field of
characteristic $p>0$.

  2. (b)

$X$ is projective over $k$.

  3. (c)

There exists a log resolution of $(X,\Delta)$.

###  4.6. Examples

In this subsection, we study the following three examples.

  1. (1)

The Frobenius morphism of $\mathbb{P}^{n}_{k}$ in positive characteristic
(Example 4.23).

  2. (2)

Fibrations over $\mathbb{P}^{1}$ in positive characteristic (Example 4.24).

  3. (3)

Counterexamples to the Bertini theorem for base point free linear systems in
mixed characteristic (Proposition 4.25, Example 4.26).

The author was not able to find a similar example to (3) in the literature.

####  4.6.1. Positive characteristic

######  Example 4.23.

Let $k$ be an algebraically closed field of characteristic $p>0$ and set
$X:=X^{\prime}:=\mathbb{P}^{n}_{k}$. Let $F_{X/k}:X=\mathbb{P}^{n}_{k}\to
X^{\prime}=\mathbb{P}^{n}_{k}$ be the relative Frobenius morphism

$F_{X/k}:\mathbb{P}^{n}_{k}\to\mathbb{P}^{n}_{k},\qquad[a_{0}:\cdots:a_{n}]\mapsto[a_{0}^{p}:\cdots:a_{n}^{p}],$

which is a $k$-morphism. In what follows, we use the identifications via the
following isomorphisms

  * •

$(\mathbb{P}^{n}_{k})^{*}\simeq\mathbb{P}^{n}_{k}={\operatorname{Proj}}\,k[s_{0},...,s_{n}]$,
and

  * •

${\operatorname{Frac}}(\mathbb{P}^{n}_{k})^{*}\simeq{\operatorname{Frac}}({\operatorname{Proj}}\,k[s_{0},...,s_{n}])=k(t_{1},...,t_{n})$
with $t_{i}:=s_{i}/s_{0}$.

We have the following diagram in which each square is cartesian (Remark 4.5):

$\begin{CD}X^{{\operatorname{gen}}}_{F_{X/k}}@>{}>{}>X^{\prime{\operatorname{gen}}}_{{\rm
id}}@>{}>{}>{\operatorname{Spec}}\,k(n)={\operatorname{Spec}}\,k(t_{1},...,t_{n})\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\
X^{{\operatorname{univ}}}_{F_{X/k}}@>{}>{}>X^{\prime{\operatorname{univ}}}_{{\rm
id}}@>{}>{}>(\mathbb{P}^{n}_{k})^{*}\\\ @V{}V{}V@V{}V{}V\\\
X\times_{k}(\mathbb{P}^{n}_{k})^{*}@>{F_{X/k}\times{\rm
id}}>{}>X^{\prime}\times_{k}(\mathbb{P}^{n}_{k})^{*}\\\ @V{}V{}V@V{}V{}V\\\
<EMAIL_ADDRESS>
By

$X^{\prime{\operatorname{univ}}}_{{\rm
id}}=\\{s_{0}x_{0}+\cdots+s_{n}x_{n}=0\\}\subset{\operatorname{Proj}}\,k[x_{0},...,x_{n}]\times_{k}{\operatorname{Proj}}\,k[s_{0},...,s_{n}]=\mathbb{P}^{n}_{k}\times_{k}(\mathbb{P}^{n}_{k})^{*},$

we obtain

$X^{{\operatorname{univ}}}_{F_{X/k}}=\\{s_{0}x_{0}^{p}+\cdots+s_{n}x_{n}^{p}=0\\}\subset{\operatorname{Proj}}\,k[x_{0},...,x_{n}]\times_{k}{\operatorname{Proj}}\,k[s_{0},...,s_{n}]=\mathbb{P}^{n}_{k}\times_{k}(\mathbb{P}^{n}_{k})^{*}$

and

$X^{{\operatorname{gen}}}_{F_{X/k}}=\\{x_{0}^{p}+t_{1}x_{1}^{p}+\cdots+t_{n}x_{n}^{p}=0\\}\subset{\operatorname{Proj}}\,k[x_{0},...,x_{n}]\times_{k}k(t_{1},...,t_{n})=\mathbb{P}^{n}_{k}\times_{k}k(t_{1},...,t_{n}).$

Note that $X^{{\operatorname{univ}}}_{F_{X/k}}$ is a smooth projective variety
over $k$ (Theorem 4.7(1)) and $X^{{\operatorname{gen}}}_{F_{X/k}}$ is a
regular projective variety over $k(n)=k(t_{1},...,t_{n})$ (Theorem 4.9(4)).
Although the generic member $X^{{\operatorname{gen}}}_{F_{X/k}}$ is regular,
any member of (the linear system corresponding to) the morphism