# Inversion of adjunction for quotient singularities II: Non-linear actions

Yusuke Nakamura  Graduate School of Mathematical Sciences, the University of
Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan.
<EMAIL_ADDRESS>and  Kohsuke Shibata  Department of Mathematics,
College of Humanities and Sciences, Nihon University, 3-25-40 Sakurajosui,
Setagaya-ku, Tokyo 156-8550, Japan<EMAIL_ADDRESS>
###### Abstract.

We prove the precise inversion of adjunction formula for quotient
singularities. As an application, we prove the semi-continuity of minimal log
discrepancies for hyperquotient singularities. This paper is a continuation of
[NS], and we generalize the previous results to non-linear group actions.

###### Key words and phrases:

minimal log discrepancy, arc space, hyperquotient singularity, LSC conjecture,
PIA conjecture

###### 2020 Mathematics Subject Classification:

Primary 14E18; Secondary 14E30, 14B05

###### Contents

  1. 1 Introduction
  2. 2 Sheaves of special differentials
  3. 3 Log pairs
  4. 4 Arc spaces of $k[[x_{1},\ldots,x_{N}]]$-schemes
  5. 5 Arc spaces of $k[t]$-schemes
    1. 5.1 Arc spaces of $k[t][[x_{1},\ldots,x_{N}]]$-schemes
      1. 5.1.1 Arc spaces
      2. 5.1.2 Cylinders and Codimension
      3. 5.1.3 Thin and very thin cylinders
    2. 5.2 Arc spaces of affine formal $k[[t]]$-schemes
    3. 5.3 Codimension formulae
  6. 6 Denef and Loeser’s theory for quotient singularities
  7. 7 Arc spaces of hyperquotient singularities
    1. 7.1 Minimal log discrepancies of hyperquotient singularities
    2. 7.2 Properties on $\overline{B}^{\prime(\gamma)}$
  8. 8 PIA formula for quotient singularities of non-linear action
  9. 9 Proof of the main theorems

##  1\. Introduction

The minimal log discrepancy is an invariant of singularities defined in
birational geometry. The importance of this invariant is that two conjectures
on the invariant, the LSC (lower semi-continuity) conjecture and the ACC
(ascending chain condition) conjecture, imply the conjecture of termination of
flips [Sho04]. This paper is a continuation of [NS], and we focus on the LSC
conjecture. We always work over an algebraically closed field $k$ of
characteristic zero.

The LSC conjecture is proposed by Ambro [Amb99] and the conjecture predicts
that the minimal log discrepancies satisfy the lower semi-continuity.

######  Conjecture 1.1 (LSC conjecture, [Amb99]).

Let $(X,\mathfrak{a})$ be a log pair with an $\mathbb{R}$-ideal
$\mathfrak{a}$, and let $|X|$ be the set of all closed points of $X$ with the
Zariski topology. Then the function

$|X|\to\mathbb{R}_{\geq 0}\cup\\{-\infty\\};\quad
x\mapsto\operatorname{mld}_{x}(X,\mathfrak{a})$

is lower semi-continuous.

The LSC conjecture is proved when $\dim X\leq 3$ by Ambro [Amb99]. Ein,
Mustaţǎ, and Yasuda in [EMY03] prove the conjecture when $X$ is smooth. Ein
and Mustaţǎ in [EM04] generalize the argument to the case where $X$ is a
locally complete intersection variety. In [Nak16], the first author proves the
conjecture when $X$ has quotient singularities, more generally when $X$ has a
crepant resolution in the category of the Deligne-Mumford stacks. In [NS], the
authors study the conjecture when $X$ has hyperquotient singularities and
prove the conjecture for the following $X$:

  * •

Suppose that a finite subgroup $G\subset{\rm GL}_{N}(k)$ acts linearly on
$\mathbb{A}_{k}^{N}$ freely in codimension one. Let $Y:=\mathbb{A}_{k}^{N}/G$
be the quotient variety. Let $X\subset Y$ be a klt subvariety of codimension
$c$ that is locally defined by $c$ equations in $Y$.

The main purpose of this paper is to generalize the result in [NS] to non-
linear group actions. See Definition 8.1 for the definition of quotient
singularity in this paper.

######  Theorem 1.2 ($=$ Theorem 9.2).

Let $Y$ be a variety with only quotient singularities. Let $X$ be a klt
subvariety of $Y$ of codimension $c$ that is locally defined by $c$ equations
in $Y$. Let $\mathfrak{a}$ be an $\mathbb{R}$-ideal sheaf on $X$. Then the
function

$|X|\to\mathbb{R}_{\geq 0}\cup\\{-\infty\\};\quad
x\mapsto\operatorname{mld}_{x}(X,\mathfrak{a})$

is lower semi-continuous.

In this paper, we also treat the PIA (precise inversion of adjunction)
conjecture.

######  Conjecture 1.3 (PIA conjecture, [92, 17.3.1]).

Let $(X,\mathfrak{a})$ be a log pair and let $D$ be a normal Cartier prime
divisor. Let $x\in D$ be a closed point. Suppose that $D$ is not contained in
the cosupport of the $\mathbb{R}$-ideal sheaf $\mathfrak{a}$. Then

$\operatorname{mld}_{x}\bigl{(}X,\mathfrak{a}\mathcal{O}_{X}(-D)\bigr{)}=\operatorname{mld}_{x}(D,\mathfrak{a}\mathcal{O}_{D})$

holds.

Ein, Mustaţǎ, and Yasuda in [EMY03] prove the PIA conjecture when $X$ is
smooth. Ein and Mustaţǎ in [EM04] generalize the argument to the case where
$X$ is a locally complete intersection variety. The authors in [NS] prove the
conjecture for the following $X$ and $D$:

  * •

Suppose that a finite subgroup $G\subset{\rm GL}_{N}(k)$ acts linearly on
$\mathbb{A}_{k}^{N}$ freely in codimension one. Let $Y:=\mathbb{A}_{k}^{N}/G$
be the quotient variety, and let $x\in Y$ be the image of the origin. Let $X$
be a subvariety of $Y$ of codimension $c$ that has only klt singularities and
is locally defined by $c$ equations in $Y$ at $x$. Let $D$ be a Cartier prime
divisor on $X$ through $x$ with a klt singularity at $x\in D$.

In this paper, this result in [NS] is generalized to non-linear group actions.

######  Theorem 1.4 ($=$ Corollary 9.1).

Suppose that a variety $Y$ has a quotient singularity at a closed point $x\in
Y$. Let $X$ be a subvariety of $Y$ of codimension $c$ that is locally defined
by $c$ equations at $x$. Suppose that $X$ is klt at $x$. Let $\mathfrak{a}$ be
an $\mathbb{R}$-ideal sheaf on $X$. Let $D$ be a prime divisor on $X$ through
$x$ that is klt and Cartier at $x$. Suppose that $D$ is not contained in the
cosupport of the $\mathbb{R}$-ideal sheaf $\mathfrak{a}$. Then it follows that

$\operatorname{mld}_{x}\bigl{(}X,\mathfrak{a}\mathcal{O}_{X}(-D)\bigr{)}=\operatorname{mld}_{x}(D,\mathfrak{a}\mathcal{O}_{D}).$

Due to Theorem 1.4, Theorem 1.2 can be reduced to the known case where $X$ has
quotient singularities. Hence, this paper is mainly devoted to proving Theorem
1.4. If $X$ has a quotient singularity at a closed point $x\in X$, then the
completion $\widehat{\mathcal{O}}_{X,x}$ is isomorphic to
$k[[x_{1},\ldots,x_{N}]]^{G}$ for some linear group action
$G\subset\operatorname{GL}_{N}(k)$. Therefore, Theorem 1.4 can be proved by
extending the proofs in [NS] to the case of the formal power series ring. In
what follows, we shall explain the main differences from the proofs in [NS]
and the difficulties that arise when dealing with formal power series rings.

The key ingredient of the proofs in [NS] is the theory of arc spaces of
quotient singularities established by Denef and Loeser in [DL02]. Suppose that
a finite group $G\subset\operatorname{GL}_{N}(k)$ of order $d$ acts on
$\overline{A}=\operatorname{Spec}k[x_{1},\ldots,x_{N}]$. Let
$\overline{X}\subset\overline{A}$ be a $G$-invariant closed subvariety and let
$X:=\overline{X}/G$ be its quotient. Let $I\subset k[x_{1},\ldots,x_{N}]^{G}$
be the defining ideal of $X\subset\overline{A}/G$. For each $\gamma\in G$,
$\gamma$ can be diagonalized with some new basis
$x^{(\gamma)}_{1},\ldots,x^{(\gamma)}_{N}$. Let
$\operatorname{diag}(\xi^{e_{1}},\ldots,\xi^{e_{N}})$ be the diagonal matrix
with $0\leq e_{i}\leq d-1$, where $\xi$ is a primitive $d$-th root of unity in
$k$. Then we define the ring homomorphism $\overline{\lambda}_{\gamma}^{*}$ by

$\overline{\lambda}_{\gamma}^{*}:k[x_{1},\ldots,x_{N}]^{G}\to
k[t][x_{1},\ldots,x_{N}];\quad x^{(\gamma)}_{i}\mapsto
t^{\frac{e_{i}}{d}}x^{(\gamma)}_{i},$

and define a $k[t]$-scheme $\overline{X}^{(\gamma)}$ by

$\overline{X}^{(\gamma)}=\operatorname{Spec}\bigl{(}k[t][x_{1},\ldots,x_{N}]/\overline{I}^{(\gamma)}\bigr{)},$

where $\overline{I}^{(\gamma)}$ is the ideal generated by the elements of
$\overline{\lambda}_{\gamma}^{*}(I)$. Then the theory of Denef and Loeser in
[DL02] allows us to compare the spaces $X_{\infty}$ and $\bigsqcup_{\gamma\in
G}\overline{X}^{(\gamma)}_{\infty}$. In [NS], using this theory, $X_{\infty}$
is studied through each $\overline{X}^{(\gamma)}_{\infty}$.

In this paper, we deal with the case of formal power series rings, i.e. when
$\overline{A}=\operatorname{Spec}k[[x_{1},\ldots,x_{N}]]$. Let $G$, $\gamma$,
$\overline{X}$, and $X$ be as above. In this case, we take $e_{i}$’s above to
satisfy $1\leq e_{i}\leq d$. Then we can define the following two natural ring
homomorphisms

$\displaystyle\overline{\lambda}_{\gamma}^{*}$
$\displaystyle:k[[x_{1},\ldots,x_{N}]]^{G}\to k[x_{1},\ldots,x_{N}][[t]];$
$\displaystyle x^{(\gamma)}_{i}\mapsto t^{\frac{e_{i}}{d}}x^{(\gamma)}_{i},$
$\displaystyle\overline{\lambda}_{\gamma}^{\prime*}$
$\displaystyle:k[[x_{1},\ldots,x_{N}]]^{G}\to k[t][[x_{1},\ldots,x_{N}]];$
$\displaystyle x^{(\gamma)}_{i}\mapsto t^{\frac{e_{i}}{d}}x^{(\gamma)}_{i},$

and we define $k[t]$-schemes

$\overline{X}^{(\gamma)}=\operatorname{Spec}\bigl{(}k[x_{1},\ldots,x_{N}][[t]]/\overline{I}^{(\gamma)}\bigr{)},\quad\overline{X}^{\prime(\gamma)}=\operatorname{Spec}\bigl{(}k[t][[x_{1},\ldots,x_{N}]]/\overline{I}^{\prime(\gamma)}\bigr{)},$

where $\overline{I}^{\prime(\gamma)}$ is the ideal generated by
$\overline{\lambda}_{\gamma}^{\prime*}(I)$. In this paper, we will use both
arc spaces $\overline{X}^{(\gamma)}_{\infty}$ and
$\overline{X}^{\prime(\gamma)}_{\infty}$ to study $X_{\infty}$. We shall also
explain below how to use $\overline{X}^{(\gamma)}_{\infty}$ and
$\overline{X}^{\prime(\gamma)}_{\infty}$ differently.

First, the theory of Denef and Loeser in [DL02] can be generalized to the
formal power series rings, and $X_{\infty}$ can be compared with
$\overline{X}^{(\gamma)}_{\infty}$. Indeed, we shall see in Proposition 6.3
that $\overline{\lambda}_{\gamma}^{*}$ gives a map $\bigsqcup_{\gamma\in
G}\overline{X}^{(\gamma)}_{\infty}\to X_{\infty}$ that is surjective outside a
thin set. We note that it is not enough to consider
$\overline{X}^{\prime(\gamma)}_{\infty}$ in this respect (see Remark 6.4 for
the detail).

On the other hand, $\overline{X}^{\prime(\gamma)}_{\infty}$ will be used with
the following motivation. In [NS], a $k$-arc
$\beta\in\overline{X}^{(\gamma)}_{\infty}$ is called the “trivial arc” when it
corresponds to the $k[t]$-ring homomorphism
$\beta^{*}:k[t][x_{1},\ldots,x_{N}]\to k[[t]]$ satisfying $\beta^{*}(x_{i})=0$
for each $i$. Another key point of the argument in [NS] is to show the fact
that the trivial arc always has a lift on a resolution $W$ of
$\overline{X}^{(\gamma)}$. The existence of such a lift is proved by combining
the result by Graber, Harris, and Starr [GHS03] and the rational chain
connectedness of the fibers of the resolution proved by Hacon and Mckernan
[HM07]. In our formal power series ring setting, this argument does not work
directly on $\overline{X}^{(\gamma)}$ because each closed fiber
$\overline{X}^{(\gamma)}\to\operatorname{Spec}k[t]$ over $t=a\in k^{\times}$
is empty. Whereas, the same argument works on $\overline{X}^{\prime(\gamma)}$
and proves that the trivial arc has a lift on a desingularization $W^{\prime}$
of $\overline{X}^{\prime(\gamma)}$ (see Claim 8.4). It should be noted that it
is not clear whether the results [GHS03] and [HM07] can be applied to the
formal power series ring setting. However, this difficulty can be avoided by
using the functorial desingularization due to Temkin [Tem12].

A large part of this paper (Sections 2, 4 and 5) is devoted to proving and
listing the basic facts for dealing with the arc spaces
$\overline{X}^{(\gamma)}_{\infty}$ and
$\overline{X}^{\prime(\gamma)}_{\infty}$. Firstly,
$\overline{X}^{(\gamma)}_{\infty}$ can be seen as the arc space (Greenberg
scheme) of a formal $k[[t]]$-scheme, and the theory developed by Sebag in
[Seb04] can be applied (cf. [CLNS, Yas]). On the other hand, as far as we
know, the arc spaces of the form $\overline{X}^{\prime(\gamma)}$ have not been
fully discussed so far. In Subsection 5.1, we discuss the basic facts on the
arc spaces of $k[t][[x_{1},\ldots,x_{N}]]$-schemes of finite type.
Furthermore, we discuss in Section 2 the theory of sheaves of special
differentials introduced by de Fernex, Ein, and Mustaţǎ in [dFEM11], and the
theory of derivations which are needed in Section 5. The theory of the arc
spaces of $k[t][[x_{1},\ldots,x_{N}]]$-schemes has the following technical
difficulties (see Remark 5.16 for the detail). For a $k$-variety $X$, it is
almost trivial that $Z_{\infty}$ is a thin set of $X_{\infty}$ for the closed
subscheme $Z\subset X$ defined by the Jacobian ideal $\operatorname{Jac}_{X}$.
This fact is also valid for $k[t]$-schemes $X$ of finite type dealt with in
[NS] and for formal $k[[t]]$-schemes of finite type dealt with in [Seb04].
However, it is not clear to us whether the same statement is valid for
$k[t][[x_{1},\ldots,x_{N}]]$-schemes of finite type. For avoiding this
difficulty, many propositions in Subsection 5.1.3 are proved under stronger
assumptions.

The paper is organized as follows. In Section 2, we discuss the theory of
sheaves of special differentials introduced by de Fernex, Ein, and Mustaţǎ in
[dFEM11] and the theory of derivations. In Section 3, we review some
definitions on log pairs. In Section 4, we discuss the theory of arc spaces of
$k[[x_{1},\ldots,x_{N}]]$-schemes and see that the formula in [EMY03] and
[EM09] representing the minimal log discrepancies of $k$-varieties in terms of
arc spaces can be generalized to the formal power series ring setting (Theorem
4.11). In Section 5, we discuss the theory of arc spaces for
$k[t][[x_{1},\ldots,x_{N}]]$-schemes and affine formal $k[[t]]$-schemes. In
Section 6, we review the theory of arc spaces of quotient varieties
established by Denef and Loeser in [DL02] in the formal power series ring
setting. In Section 7, we discuss the minimal log discrepancies of quotient
singularities and describe them by the codimension of cylinders in arc spaces
of the $k[t]$-schemes using the theories in Sections 5 and 6. In Section 8, we
prove the PIA conjecture for quotient singularities, where the group action
may be non-linear (Theorem 8.2). In Section 9, we prove the main theorems
Corollary 9.1 and Theorem 9.2.

######  Acknowledgements.

The first author is partially supported by JSPS KAKENHI No. 18K13384,
16H02141, 17H02831, 18H01108, and JPJSBP120219935. The second author is
partially supported by the Grant-in-Aid for Young Scientists (KAKENHI No.
19K14496).

## Notation

  * •

We basically follow the notations and the terminologies in [Har77] and
[Kol13].

  * •

Throughout this paper, $k$ is an algebraically closed field of characteristic
zero. We say that $X$ is a variety over $k$ or a $k$-variety if $X$ is an
integral scheme that is separated and of finite type over $k$.

##  2\. Sheaves of special differentials

Let $R_{0}$ be a ring. In this section, following [dFEM11, Appendix A], we
define the sheaf $\Omega^{\prime}_{X/R_{0}}$ of special differentials for a
scheme $X$ of finite type over
$\operatorname{Spec}R_{0}[[x_{1},\ldots,x_{N}]]$. In [dFEM11, Appendix A], the
sheaf $\Omega^{\prime}_{X/R_{0}}$ of special differentials is defined for
$R_{0}=k$. This definition can be generalized to an arbitrary ring $R_{0}$. We
are interested in the case where $R_{0}=k$ or $R_{0}=k[t]$ for our later use.

Let $R=R_{0}[[x_{1},\ldots,x_{N}]]$.

######  Definition 2.1 (cf. [dFEM11, Appendix A]).

  1. (1)

Let $M$ be an $R$-module. Then an $R_{0}$-derivation $D:R\to M$ is called a
special $R_{0}$-derivation if $D$ satisfies

$D(f)=\sum_{i=1}^{N}\frac{\partial f}{\partial x_{i}}D(x_{i})$

for any $f\in R$.

  2. (2)

For an $R$-algebra $A$ and an $A$-module $M$, an $R_{0}$-derivation $D:A\to M$
is called a special $R_{0}$-derivation if its restriction to $R$ is a special
$R_{0}$-derivation. We denote by $\operatorname{Der}^{\prime}_{R_{0}}(A,M)$
the set of all special $R_{0}$-derivations. Then
$\operatorname{Der}^{\prime}_{R_{0}}(A,M)$ is an $A$-submodule of
$\operatorname{Der}_{R_{0}}(A,M)$.

######  Lemma 2.2.

Let $M$ be an $A$-module that is separated in the $(x_{1},\ldots,x_{N})$-adic
topology, i.e. $M$ satisfies $\bigcap_{\ell\geq
1}(x_{1},\ldots,x_{N})^{\ell}M=0$. Then

$\operatorname{Der}^{\prime}_{R_{0}}(A,M)=\operatorname{Der}_{R_{0}}(A,M)$

holds. In particular, it holds in the following two cases.

  1. (1)

When $M$ is a finite $R$-module.

  2. (2)

When $M=A$ and $A$ is an integral domain such that
$(x_{1},\ldots,x_{N})A\not=A$.

###### Proof.

By the definition of special derivations, it is sufficient to show that
$\operatorname{Der}^{\prime}_{R_{0}}(R,M)=\operatorname{Der}_{R_{0}}(R,M)$.
Let $D\in\operatorname{Der}_{R_{0}}(R,M)$. If we set $D^{\prime}:R\to M$ by
$D^{\prime}(f):=\sum_{i=1}^{N}\frac{\partial f}{\partial x_{i}}D(x_{i})$, then
$D^{\prime}\in\operatorname{Der}^{\prime}_{R_{0}}(R,M)$ holds. Hence it is
sufficient to show that $D=0$ if $D(x_{i})=0$ holds for all $i$. Let $f\in R$.
For any $\ell\geq 0$, we may write $f=f_{1}+f_{2}$ with $f_{1}\in
R_{0}[x_{1},\ldots,x_{n}]$ and $f_{2}\in(x_{1},\ldots,x_{n})^{\ell+1}R$. Then
we have

$D(f)=D(f_{1})+D(f_{2})=D(f_{2})\in(x_{1},\ldots,x_{n})^{\ell}M.$

It shows that $D(f)\in\bigcap_{\ell\geq 1}(x_{1},\ldots,x_{N})^{\ell}M=0$.

In both cases (1) and (2), $M$ is separated by [Mat89, Theorem 8.9]. ∎

######  Proposition 2.3 (cf. [dFEM11, Appendix A]).

For any $R$-algebra $A$, there exists an $A$-module
$\Omega^{\prime}_{A/R_{0}}$ with a special $R_{0}$-derivation
$d^{\prime}_{A/R_{0}}:A\to\Omega^{\prime}_{A/R_{0}}$ such that the induced map

$\operatorname{Hom}_{A}(\Omega^{\prime}_{A/R_{0}},M)\to\operatorname{Der}^{\prime}_{R_{0}}(A,M);\quad
f\mapsto f\circ d^{\prime}_{A/R_{0}}$

is an isomorphism for any $A$-module $M$.

###### Proof.

The same proof as in [dFEM11, A.1-A.4] works. ∎

An $A$-module $\Omega^{\prime}_{A/R_{0}}$ satisfying Proposition 2.3 is unique
up to an isomorphism commuting with $d^{\prime}_{A/R_{0}}$, and
$\Omega^{\prime}_{A/R_{0}}$ is called the module of special differentials. We
sometimes abbreviate $d^{\prime}_{A/R_{0}}$ to $d^{\prime}$ when no confusion
can arise. We note that $\Omega^{\prime}_{A/R_{0}}$ depends on the choice of
$R$. We list some properties on $\Omega^{\prime}_{A/R_{0}}$ from [dFEM11,
Appendix A].

######  Proposition 2.4 (cf. [dFEM11, Appendix A]).

  1. (1)

If $A=R[y_{1},\ldots,y_{m}]$, then $\Omega^{\prime}_{A/R_{0}}$ is a free
$A$-module of rank $N+m$ with basis

$d^{\prime}_{A/R_{0}}(x_{1}),\ldots,d^{\prime}_{A/R_{0}}(x_{N}),d^{\prime}_{A/R_{0}}(y_{1}),\ldots,d^{\prime}_{A/R_{0}}(y_{m}).$

  2. (2)

Let $f:A\to B$ be a homomorphism of $R$-algebras. Then we have an exact
sequence

$\Omega^{\prime}_{A/R_{0}}\otimes_{A}B\xrightarrow{\alpha}\Omega^{\prime}_{B/R_{0}}\xrightarrow{\beta}\Omega_{B/A}\to
0$

of $B$-modules, where the maps $\alpha$ and $\beta$ are defined by
$\alpha\bigl{(}d^{\prime}_{A/R_{0}}(g)\otimes
1\bigr{)}=d^{\prime}_{B/R_{0}}(f(g))$ for $g\in A$ and
$\beta\bigl{(}d^{\prime}_{B/R_{0}}(g)\bigr{)}=d_{B/A}(g)$ for $g\in B$.

  3. (3)

Let $f:A\to B$ be a surjective homomorphism of $R$-algebras with
$I=\operatorname{Ker}f$. Then we have an exact sequence

$I/I^{2}\xrightarrow{\delta}\Omega^{\prime}_{A/R_{0}}\otimes_{A}B\xrightarrow{\alpha}\Omega^{\prime}_{B/R_{0}}\to
0$

of $B$-modules, where the map $\delta$ is defined by
$\delta(\overline{g})=d^{\prime}_{A/R_{0}}(g)\otimes 1$ for $g\in I$.

  4. (4)

If $S\subset A$ is a multiplicative system of $A$, then we have a canonical
isomorphism

$\Omega^{\prime}_{S^{-1}A/R_{0}}\simeq\Omega^{\prime}_{A/R_{0}}\otimes_{A}S^{-1}A.$

###### Proof.

The same proofs as in [dFEM11, Lemmas A.1-3, A.6-7] work. The exact sequences
in (2) and (3) are derived from the following corresponding exact sequences

$\displaystyle
0\to\operatorname{Der}_{A}(B,M)\to\operatorname{Der}^{\prime}_{R_{0}}(B,M)\to\operatorname{Der}^{\prime}_{R_{0}}(A,M),\hskip
8.53581pt$ $\displaystyle
0\to\operatorname{Der}^{\prime}_{R_{0}}(B,M)\to\operatorname{Der}^{\prime}_{R_{0}}(A,M)\to\operatorname{Hom}_{B}(I/I^{2},M),$

for any $B$-module $M$. The isomorphism in (4) is derived from the following
isomorphisms

$\displaystyle\operatorname{Hom}_{S^{-1}A}(\Omega^{\prime}_{A/R_{0}}\otimes_{A}S^{-1}A,M)$
$\displaystyle\simeq\operatorname{Hom}_{A}(\Omega^{\prime}_{A/R_{0}},M)$
$\displaystyle\simeq\operatorname{Der}^{\prime}_{R_{0}}(A,M)\simeq\operatorname{Der}^{\prime}_{R_{0}}(S^{-1}A,M)$

for any $S^{-1}A$-module $M$. ∎

######  Remark 2.5.

  1. (1)

The usual module $\Omega_{A/R_{0}}$ of differentials is not a finite
$A$-module in general when $A$ is an $R$-algebra of finite type. However, the
module $\Omega^{\prime}_{A/R_{0}}$ of special differentials becomes a finite
$A$-module and has similar properties as the module of differentials defined
for $R_{0}$-algebras of finite type.

  2. (2)

The universal finite module of differentials is also a module defined with the
same motivation (see [Kun86, Section 11]). For an $R_{0}$-algebra $A$, a
finite $A$-module $\widetilde{\Omega}_{A/R_{0}}$ with an $R_{0}$-derivation
$\widetilde{d}_{A/R_{0}}:A\to\widetilde{\Omega}_{A/R_{0}}$ is called the
universal finite module of differentials if it satisfies the following
universal property.

     * •

For any $R_{0}$-derivation $D:A\to M$ to a finite $A$-module $M$, there exists
a unique homomorphism $\alpha:\widetilde{\Omega}_{A/R_{0}}\to M$ of
$A$-modules satisfying $D=\alpha\circ\widetilde{d}_{A/R_{0}}$.

In other words, $\widetilde{\Omega}_{A/R_{0}}$ and $\widetilde{d}_{A/R_{0}}$
satisfy

$\operatorname{Hom}_{A}(\widetilde{\Omega}_{A/R_{0}},M)\xrightarrow{\simeq}\operatorname{Der}_{R_{0}}(A,M);\quad\alpha\mapsto\alpha\circ\widetilde{d}_{A/R_{0}}$

for any finite $A$-module $M$.

In contrast to the module $\Omega^{\prime}_{A/R_{0}}$ of special
differentials, the universal finite module $\widetilde{\Omega}_{A/R_{0}}$ of
differentials does not necessarily exist.

  3. (3)

$A$ is called an analytic $R_{0}$-algebra if there exists
$R=R_{0}[[x_{1},\ldots,x_{N}]]$ for some $N\geq 0$ such that $A$ is a finite
$R$-algebra. If $A$ is an analytic $R_{0}$-algebra, then the universal finite
module $\widetilde{\Omega}_{A/R_{0}}$ of differentials exists. Furthermore, if
$A$ is a finite $R_{0}[[x_{1},\ldots,x_{N}]]$-algebra and if
$\Omega^{\prime}_{A/R_{0}}$ is the module of special differentials with
respect to $R=R_{0}[[x_{1},\ldots,x_{N}]]$, then we have
$\widetilde{\Omega}_{A/R_{0}}\simeq\Omega^{\prime}_{A/R_{0}}$. This is because
we have

$\operatorname{Der}^{\prime}_{R_{0}}(A,M)=\operatorname{Der}_{R_{0}}(A,M)$

for any finite $A$-module $M$ by Lemma 2.2 in this case. Therefore, we can
also see that $\Omega^{\prime}_{A/R_{0}}$ does not depend on the choice of $R$
as long as $A$ is finite as an $R$-module.

  4. (4)

Even if $A$ is an algebra of finite type over $R=R_{0}[[x_{1},\ldots,x_{N}]]$,
$\widetilde{\Omega}_{A/R_{0}}$ does not necessarily exist despite the fact
that $\Omega^{\prime}_{A/R_{0}}$ is a finite $A$-module. We shall see that
$\widetilde{\Omega}_{A/k}$ and $\widetilde{\Omega}_{B/k}$ do not exist for

$A=k[[x]][y],\qquad B=k[[x]][y]/(1-xy)\simeq k((x)).$

In fact, since

$\displaystyle\dim_{k((x))}\operatorname{Der}_{k}\bigl{(}k((x))\bigr{)}$
$\displaystyle=\dim_{k((x))}\operatorname{Hom}_{k((x))}\bigl{(}\Omega_{k((x))/k},k((x))\bigr{)}$
$\displaystyle=\operatorname{trdeg}_{k}k((x))=\infty,$

$\operatorname{Der}_{k}(B)$ is not a finite $B$-module and hence
$\widetilde{\Omega}_{B/k}$ does not exist. Furthermore, since there is a
natural injective map
$\operatorname{Der}_{k}(B,B)\to\operatorname{Der}_{k}(A,B)$,
$\operatorname{Der}_{k}(A,B)$ is not a finite $A$-module and hence
$\widetilde{\Omega}_{A/k}$ does not exist (cf. [Kun86, Corollary 11.10]).

######  Remark 2.6.

  1. (1)

Let $A$ be a ring. For a non-negative integer $\ell$, and for subsets
$F\subset A$ and $\Delta\subset\operatorname{Der}(A)$, we denote by
$\mathcal{J}_{\ell}(F;\Delta)$ the ideal of $A$ generated by the determinants
$\det\bigl{(}D_{i}(f_{j})\bigr{)}_{1\leq i,j\leq\ell}$ of all the matrices
$\bigl{(}D_{i}(f_{j})\bigr{)}_{1\leq i,j\leq\ell}$ of size $\ell$ with
$D_{i}\in\Delta$ and $f_{j}\in F$.

If $I$ is an ideal of $A$ generated by $f_{1},\ldots,f_{t}$ and the
$A$-submodule $A\Delta$ of $\operatorname{Der}(A)$ is generated by
$D_{1},\ldots,D_{s}$ as an $A$-module, then we have

$\mathcal{J}_{\ell}(I;\Delta)+I=\mathcal{J}_{\ell}(\\{f_{1},\ldots,f_{t}\\};\\{D_{1},\ldots,D_{s}\\})+I.$

  2. (2)

Let $A$ be a regular ring and let $P$ be a prime ideal. For an ideal $I$ of
$A$ such that $I\subset P$, the following hold (cf. [Mat89, Theorem 30.4]).

    1. (a)

$\mathcal{J}_{\ell}(I;\operatorname{Der}(A))\subset P$ holds for any
$\ell>\operatorname{ht}(IA_{P})$.

    2. (b)

$A_{P}/IA_{P}$ is regular if
$\mathcal{J}_{\ell}(I;\operatorname{Der}(A))\not\subset P$ holds for
$\ell=\operatorname{ht}(IA_{P})$.

Some regular rings satisfy the inverse implication of (b), and such rings are
said to satisfy the weak Jacobian condition (WJ) (cf. [Mat89, Section 30]).
Rings of finite type over $k$ are classically known to satisfy (WJ), and this
is known as the Jacobian criterion for regularity. Matsumura proved in [Mat77]
that $R$-algebras of finite type satisfy (WJ)k when
$R=k[[x_{1},\ldots,x_{N}]]$ (see [Mat77, Theorem 9] for more general result):

    1. (c)

Let $A=k[[x_{1},\ldots,x_{N}]][y_{1},\ldots,y_{m}]$ and let $P$ and $Q$ be
prime ideals of $A$ such that $Q\subset P$. Then $A/Q$ is regular at $P$ if
and only if $\mathcal{J}_{\ell}(Q;\operatorname{Der}_{k}(A))\not\subset P$
holds for $\ell=\operatorname{ht}Q$.

Note that $\operatorname{Der}_{k}(A)=\operatorname{Der}^{\prime}_{k}(A)$ holds
for $A=k[[x_{1},\ldots,x_{N}]][y_{1},\ldots,y_{m}]$ (cf. Lemma 2.2), and this
is a free $A$-module generated by $\frac{\partial}{\partial x_{i}}$’s and
$\frac{\partial}{\partial y_{i}}$’s.

In [dFEM11, Proposition A.8], the local freeness of $\Omega^{\prime}_{A/k}$ is
proved for regular rings $A$ when $R_{0}=k$.

######  Proposition 2.7 ([dFEM11, Proposition A.8]).

Suppose that $R_{0}=k$. Let $A$ be an $R$-algebra of finite type, and let
$\mathfrak{q}$ be a prime ideal of $A$. If $A_{\mathfrak{q}}$ is regular, then
$\Omega^{\prime}_{A_{\mathfrak{q}}/k}$ is a free $A_{\mathfrak{q}}$-module of
rank $\dim
A_{\mathfrak{q}}+\dim_{k(\mathfrak{q})}(\Omega^{\prime}_{k(\mathfrak{q})/k})$.

In the proof of [dFEM11, Proposition A.8], the following statement is proved.

######  Proposition 2.8 ([dFEM11, Proposition A.8]).

Suppose that $R_{0}=k$. Let $S:=R[y_{1},\ldots,y_{m}]$, and let $P$ and $Q$ be
prime ideals of $S$ with $P\subset Q$. Let $A:=S/P$ and let $\mathfrak{q}$ be
the prime ideal of $A$ corresponding to $Q$. If $A_{\mathfrak{q}}$ is regular,
then the sequence

$0\to
PS_{Q}/P^{2}S_{Q}\to\Omega^{\prime}_{S/k}\otimes_{S}A_{\mathfrak{q}}\to\Omega^{\prime}_{A_{\mathfrak{q}}/k}\to
0$

obtained by Proposition 2.4(3) is exact and splits.

######  Remark 2.9.

  1. (1)

By Propositions 2.7 and 2.8, we have

$\dim S-\operatorname{ht}P=\dim
A_{\mathfrak{q}}+\dim_{k(\mathfrak{q})}(\Omega^{\prime}_{k(\mathfrak{q})/k}).$

It shows that $\operatorname{coht}P=\dim S-\operatorname{ht}P$ is independent
of the choice of $S$. Lemma 2.11(1) below also proves this independence. Lemma
2.11(1) also gives a ring-theoretic interpretation of this value without using
$\Omega^{\prime}$.

  2. (2)

In Proposition 2.7, we note that the rank of
$\Omega^{\prime}_{A_{\mathfrak{q}}/k}$ is not equal to $\dim A$ in general.

If we set $S=k[[x]][y]$ and $P=Q=(xy-1)$ in Proposition 2.8, then we have
$A_{\mathfrak{q}}=A=k((x))$. (1) shows that the rank of
$\Omega^{\prime}_{A_{\mathfrak{q}}/k}$ is equal to one even though $\dim A=0$.

######  Definition 2.10.

Suppose that $R_{0}$ is a Noetherian domain. Let $X$ be an irreducible scheme
of finite type over $R$ and let $X_{\rm red}$ be its underlying reduced
subscheme. Let $X_{\rm red}\to\operatorname{Spec}R$ be the structure morphism,
and let $\mathfrak{p}\in\operatorname{Spec}R$ be the image of the generic
point of $X_{\rm red}$. Then we define

$\dim^{\prime}X:=\operatorname{trdeg}_{k(\mathfrak{p})}K(X_{\rm red})+\dim
R-\operatorname{ht}\mathfrak{p},$

where $k(\mathfrak{p}):=R_{\mathfrak{p}}/\mathfrak{p}R_{\mathfrak{p}}$ and
$K(X_{\rm red})$ is the function field of $X_{\rm red}$. When
$X=\operatorname{Spec}A$ is an affine scheme, we also write
$\dim^{\prime}A:=\dim^{\prime}X$.

######  Lemma 2.11.

Suppose that $R_{0}$ is a Noetherian domain. Let $X=\operatorname{Spec}A$ be
an irreducible affine scheme of finite type over $R$. Then the following hold.

  1. (1)

If $A$ is a domain, then $\dim S-\operatorname{ht}P=\dim^{\prime}A$ holds for
any representation $A\simeq S/P$ with $S=R[y_{1},\ldots,y_{m}]$ and a prime
ideal $P$ of $S$.

  2. (2)

$\dim A\leq\dim^{\prime}A$ holds.

  3. (3)

Suppose that $R_{0}=k$ and $A$ is a domain. Then
$\dim^{\prime}A=\dim_{K}(\Omega^{\prime}_{K/k})$ holds for
$K=\operatorname{Frac}A$.

  4. (4)

Suppose that $R_{0}=k$ or $R_{0}=k[t]$. Then $\dim^{\prime}A=\dim A$ holds if
$A/\mathfrak{m}=k$ holds for some maximal ideal $\mathfrak{m}$ of $A$.

  5. (5)

Suppose that $R$ is a universally catenary ring. If $I=(f_{1},\ldots,f_{c})$
is an ideal of $A$ generated by $c$ elements, then
$\dim^{\prime}(A/Q)\geq\dim^{\prime}A-c$ holds for any minimal prime $Q$ of
$I$.

  6. (6)

Suppse that $R$ is a universally catenary ring. Let $\mathfrak{p}$ and
$\mathfrak{q}$ be prime ideals of $A$ such that
$\mathfrak{p}\subset\mathfrak{q}$. Let $B:=A/\mathfrak{p}$ and
$\overline{\mathfrak{q}}:=\mathfrak{q}/\mathfrak{p}\in\operatorname{Spec}B$.
Then we have $\operatorname{ht}\mathfrak{p}=\dim A_{\mathfrak{q}}-\dim
B_{\overline{\mathfrak{q}}}=\dim^{\prime}A-\dim^{\prime}B$.

###### Proof.

First, we prove (1). Let $\mathfrak{p}$ be the image of $P$ according to the
map $\operatorname{Spec}S\to\operatorname{Spec}R$. Then by [Mat89, Theorem
15.5] (and the remark below in [Mat89]), it follows that

$\operatorname{trdeg}_{\operatorname{Frac}R}(\operatorname{Frac}S)-\operatorname{ht}P=\operatorname{trdeg}_{k(\mathfrak{p})}(\operatorname{Frac}(S/P))-\operatorname{ht}\mathfrak{p}.$

Therefore, the assertion follows from
$\operatorname{trdeg}_{\operatorname{Frac}R}(\operatorname{Frac}S)=m=\dim
S-\dim R$.

(2) follows from (1) and the inequality $\dim(S/P)+\operatorname{ht}P\leq\dim
S$. (3) follows from (1) and Remark 2.9(1).

Next, we prove (4) for $R_{0}=k[t]$ (the case when $R_{0}=k$ is similar). We
may assume that $A$ is a domain. Take $S$ and $P$ as in (1). Then it is
sufficient to prove

$\dim S-\operatorname{ht}P=\dim(S/P).$

Let $M$ be the maximal ideal of $S$ corresponding to $\mathfrak{m}$. Since
$S/M=k$, $M$ is of the form

$M=(t-a,x_{1},\ldots,x_{N},y_{1}-b_{1},\ldots,y_{m}-b_{m})$

with $a,b_{i}\in k$. Therefore we have $\dim S=N+m+1=\operatorname{ht}M$.
Since $S$ is a catenary ring, we also have
$\dim(S/P)=\operatorname{ht}M-\operatorname{ht}P$, which proves $\dim
S-\operatorname{ht}P=\dim(S/P)$.

(5) follows from (1) and Krull’s height theorem. (6) also follows from (1). ∎

######  Proposition 2.12.

Suppose that $R_{0}=k$. Let $A$ be a domain of finite type over $R$, and let
$\mathfrak{q}_{1}$ and $\mathfrak{q}_{2}$ be prime ideals of $A$ such that
$\mathfrak{q}_{1}\subset\mathfrak{q}_{2}$. Let $B:=A/\mathfrak{q}_{1}$ and
$\overline{\mathfrak{q}}_{2}:=\mathfrak{q}_{2}/\mathfrak{q}_{1}\in\operatorname{Spec}B$.
If both $A_{\mathfrak{q}_{2}}$ and $B_{\overline{\mathfrak{q}}_{2}}$ are
regular, then the sequence

$0\to\mathfrak{q}_{1}A_{\mathfrak{q}_{2}}/\mathfrak{q}_{1}^{2}A_{\mathfrak{q}_{2}}\to\Omega^{\prime}_{A_{\mathfrak{q}_{2}}/k}\otimes_{A_{\mathfrak{q}_{2}}}B_{\overline{\mathfrak{q}}_{2}}\to\Omega^{\prime}_{B_{\overline{\mathfrak{q}}_{2}}/k}\to
0$

induced by Proposition 2.8(3) is exact and splits.

###### Proof.

By Proposition 2.8(3), the sequence

$\mathfrak{q}_{1}A_{\mathfrak{q}_{2}}/\mathfrak{q}_{1}^{2}A_{\mathfrak{q}_{2}}\to\Omega^{\prime}_{A_{\mathfrak{q}_{2}}/k}\otimes_{A_{\mathfrak{q}_{2}}}B_{\overline{\mathfrak{q}}_{2}}\xrightarrow{\delta}\Omega^{\prime}_{B_{\overline{\mathfrak{q}}_{2}}/k}\to
0$

is exact. Since $A_{\mathfrak{q}_{2}}$ and $B_{\overline{\mathfrak{q}}_{2}}$
are regular, it follows from Proposition 2.7 that

$\Omega^{\prime}_{A_{\mathfrak{q}_{2}}/k}\otimes_{A_{\mathfrak{q}_{2}}}B_{\overline{\mathfrak{q}}_{2}}\simeq
B_{\overline{\mathfrak{q}}_{2}}^{\oplus\dim^{\prime}A},\quad\Omega^{\prime}_{B_{\overline{\mathfrak{q}}_{2}}/k}\simeq
B_{\overline{\mathfrak{q}}_{2}}^{\oplus\dim^{\prime}B}.$

Therefore $\operatorname{Ker}(\delta)$ is a free
$B_{\overline{\mathfrak{q}}_{2}}$-module of rank equal to
$\dim^{\prime}A-\dim^{\prime}B$. On the other hand, since
$A_{\mathfrak{q}_{2}}$ and $B_{\overline{\mathfrak{q}}_{2}}$ are regular,
$\mathfrak{q}_{1}A_{\mathfrak{q}_{2}}/\mathfrak{q}_{1}^{2}A_{\mathfrak{q}_{2}}$
is also a free $B_{\overline{\mathfrak{q}}_{2}}$-module of rank equal to
$\operatorname{ht}(\mathfrak{q}_{1})$, which is equal to
$\dim^{\prime}A-\dim^{\prime}B$ by Lemma 2.11(6). Hence, the induced
surjective map
$\mathfrak{q}_{1}A_{\mathfrak{q}_{2}}/\mathfrak{q}_{1}^{2}A_{\mathfrak{q}_{2}}\to\operatorname{Ker}(\delta)$
should be an isomorphism. We complete the proof. ∎

######  Definition 2.13.

  1. (1)

Let $X$ be a scheme over $\operatorname{Spec}R$. Then due to Proposition
2.4(4), there exists a quasi-coherent sheaf $\Omega^{\prime}_{X/R_{0}}$
satisfying
$\Omega^{\prime}_{X/R_{0}}(U)\simeq\Omega^{\prime}_{\mathcal{O}_{X}(U)/R_{0}}$
for any affine open subset $U\subset X$. Note that $\Omega^{\prime}_{X/R_{0}}$
is coherent by Proposition 2.4(1)(3) when $X$ is of finite type over
$\operatorname{Spec}R$. The sheaf $\Omega^{\prime}_{X/R_{0}}$ is called the
sheaf of special differentials. We denote $\Omega^{\prime
n}_{X/R_{0}}:=\bigwedge^{n}\Omega^{\prime}_{X/R_{0}}$ for a non-negative
integer $n$.

  2. (2)

Suppose that $R_{0}=k$ and $X$ is a scheme of finite type over
$\operatorname{Spec}R$. Let $n$ be a non-negative integer. Suppose that any
irreducible component $X_{i}$ of $X$ satisfies $\dim^{\prime}X_{i}=n$, where
$\dim^{\prime}X$ is defined in Definition 2.10. Then we denote
$\operatorname{Jac}^{\prime}_{X/k}:=\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k})$
and it is called the special Jacobian ideal of $X$ (see [Eis95, Section 20.2]
for the definition of the Fitting ideal). We note that
$\dim^{\prime}X_{i}=\dim X_{i}$ holds if $X_{i}$ contains a $k$-point by Lemma
2.11(4).

  3. (3)

Suppose that $R_{0}=k$ and $X$ is an integral normal scheme of finite type
over $\operatorname{Spec}R$. Let $n=\dim^{\prime}X$ and let $i:X_{\rm reg}\to
X$ be the inclusion map from the regular locus $X_{\rm reg}$ of $X$. Then the
special canonical sheaf $\omega^{\prime}_{X/k}$ is defined by
$\omega^{\prime}_{X/k}=i_{*}(\Omega^{\prime n}_{X_{\rm reg}/k})$.

  4. (4)

Under the same setting in (3), a Weil divisor $K_{X}$ on $X$ satisfying
$\mathcal{O}_{X}(K_{X})|_{X_{\mathrm{reg}}}\simeq\Omega^{\prime n}_{X_{\rm
reg}/k}$ is called the canonical divisor on $X$. The canonical divisor $K_{X}$
is defined up to linear equivalence. Note that
$\omega^{\prime}_{X/k}\simeq\mathcal{O}_{X}(K_{X})$ holds as usual. In fact,
since we have $\operatorname{codim}_{X}(X\setminus X_{\rm reg})\geq 2$ by the
normality of $X$, it follows that

$\Gamma\bigl{(}V,\mathcal{O}_{X}(K_{X})\bigr{)}=\Gamma\bigl{(}V\cap X_{\rm
reg},\mathcal{O}_{X}(K_{X})\bigr{)}=\Gamma\bigl{(}V,i_{*}i^{*}\mathcal{O}_{X}(K_{X})\bigr{)}$

for any open subset $V\subset X$.

  5. (5)

Under the same setting in (3), we say that $X$ is $\mathbb{Q}$-Gorenstein if
$\omega_{X/k}^{\prime[r]}:=(\omega_{X/k}^{\prime\otimes
r})^{**}\simeq\mathcal{O}_{X}(rK_{X})$ is an invertible sheaf for some
$r\in\mathbb{Z}_{>0}$. In this case, we have a canonical map

$\eta_{r}\colon(\Omega_{X/k}^{\prime n})^{\otimes
r}\to\omega_{X/k}^{\prime[r]}.$

Since $\omega_{X/k}^{\prime[r]}$ is an invertible sheaf, an ideal sheaf
$\mathfrak{n}_{r,X}\subset\mathcal{O}_{X}$ is uniquely determined by ${\rm
Im}(\eta_{r})=\mathfrak{n}_{r,X}\otimes_{\mathcal{O}_{X}}\omega_{X/k}^{\prime[r]}$.
The ideal sheaf $\mathfrak{n}_{r,X}$ is called the $r$-th Nash ideal of $X$.

######  Remark 2.14.

As with the usual Jacobian ideals for varieties over $k$, the special Jacobian
ideal can be locally described by the Jacobian matrix. Let
$S=R[y_{1},\ldots,y_{m}]$ with $R=R_{0}[[x_{1},\ldots,x_{N}]]$ and let $A=S/I$
for some ideal $I=(f_{1},\ldots,f_{r})\subset S$. Then by Proposition 2.4(3),
we have an exact sequence

$I/I^{2}\to\Omega^{\prime}_{S/R_{0}}\otimes_{S}A\to\Omega^{\prime}_{A/R_{0}}\to
0.$

Here, we have $\Omega^{\prime}_{S/R_{0}}\simeq S^{\oplus N+m}$ with basis
$d^{\prime}x_{i}$’s and $d^{\prime}y_{j}$’s. Furthermore, for $f\in I$, we
have

$d^{\prime}f=\sum_{i=1}^{N}\frac{\partial f}{\partial
x_{i}}d^{\prime}x_{i}+\sum_{j=1}^{m}\frac{\partial f}{\partial
y_{j}}d^{\prime}y_{j}.$

Therefore, we have

$\operatorname{Fitt}^{n}(\Omega^{\prime}_{A/R_{0}})=\bigl{(}\mathcal{J}_{N+m-n}(I;\operatorname{Der}_{R_{0}}(S))+I\bigr{)}/I.$

Note here that
$\operatorname{Der}_{R_{0}}(S)=\operatorname{Der}^{\prime}_{R_{0}}(S)$ is a
free $S$-module generated by $\partial/\partial x_{i}$’s and
$\partial/\partial y_{j}$’s.

This observation shows that if $R_{0}=k$ and $X$ is an integral scheme of
finite type over $R$, then $\operatorname{Jac}^{\prime}_{X/k}$ defines the
singular locus of $X$ by Remark 2.6(2)(c). Note here that
$\operatorname{ht}I=N+m-\dim^{\prime}X$ holds by Lemma 2.11(1).

##  3\. Log pairs

A log pair $(X,\mathfrak{a})$ is a normal $\mathbb{Q}$-Gorenstein $k$-variety
$X$ and an $\mathbb{R}$-ideal sheaf $\mathfrak{a}$ on $X$. Here, an
$\mathbb{R}$-ideal sheaf $\mathfrak{a}$ on $X$ is a formal product
$\mathfrak{a}=\prod_{i=1}^{s}\mathfrak{a}_{i}^{r_{i}}$, where
$\mathfrak{a}_{1},\ldots,\mathfrak{a}_{s}$ are non-zero coherent ideal sheaves
on $X$ and $r_{1},\ldots,r_{s}$ are positive real numbers. For a morphism
$Y\to X$ and an $\mathbb{R}$-ideal sheaf
$\mathfrak{a}=\prod_{i=1}^{s}\mathfrak{a}_{i}^{r_{i}}$ on $X$, we denote by
$\mathfrak{a}\mathcal{O}_{Y}$ the $\mathbb{R}$-ideal sheaf
$\prod_{i=1}^{s}(\mathfrak{a}_{i}\mathcal{O}_{Y})^{r_{i}}$ on $Y$.

Let $\bigl{(}X,\mathfrak{a}=\prod_{i=1}^{s}\mathfrak{a}_{i}^{r_{i}}\bigr{)}$
be a log pair. Let $f:X^{\prime}\to X$ be a proper birational morphism from a
normal variety $X^{\prime}$ and let $E$ be a prime divisor on $X^{\prime}$. We
denote by $K_{X^{\prime}/X}:=K_{X^{\prime}}-f^{*}K_{X}$ the relative canonical
divisor. Then the log discrepancy of $(X,\mathfrak{a})$ at $E$ is defined as

$a_{E}(X,\mathfrak{a}):=1+\operatorname{ord}_{E}(K_{X^{\prime}/X})-\operatorname{ord}_{E}\mathfrak{a},$

where we define
$\operatorname{ord}_{E}\mathfrak{a}:=\sum_{i=1}^{s}r_{i}\operatorname{ord}_{E}\mathfrak{a}_{i}$.
The image $f(E)$ is called the center of $E$ on $X$ and we denote it by
$c_{X}(E)$. For a closed point $x\in X$, we define the minimal log discrepancy
at $x$ as

$\operatorname{mld}_{x}(X,\mathfrak{a}):=\inf_{c_{X}(E)=\\{x\\}}a_{E}(X,\mathfrak{a})$

if $\dim X\geq 2$, where the infimum is taken over all prime divisors $E$ over
$X$ with center $c_{X}(E)=\\{x\\}$. It is known that
$\operatorname{mld}_{x}(X,\mathfrak{a})\in\mathbb{R}_{\geq
0}\cup\\{-\infty\\}$ in this case (cf. [KM98, Corollary 2.31]). When $\dim
X=1$, we define
$\operatorname{mld}_{x}(X,\mathfrak{a}):=\inf_{c_{X}(E)=\\{x\\}}a_{E}(X,\mathfrak{a})$
if the infimum is non-negative, and we define
$\operatorname{mld}_{x}(X,\mathfrak{a}):=-\infty$ otherwise.

Let $R=k[[x_{1},\ldots,x_{N}]]$. By [dFEM11, Appendix A], we can extend the
definition above to normal $R$-schemes of finite type. Let $X$ be an integral
normal scheme of finite type over $R$. Then the canonical divisor $K_{X}$ is
defined in Definition 2.13(4). Suppose that $X$ is $\mathbb{Q}$-Gorenstein,
that is, $rK_{X}$ is Cartier for some positive integer $r$. Let $f:Y\to X$ be
a proper birational morphism over $R$ from a regular scheme $Y$. Then the
relative canonical divisor $K_{Y/X}$ of $f$ is defined as the
$\mathbb{Q}$-divisor supported on the exceptional locus of $f$ such that
$rK_{Y}-f^{*}(rK_{X})$ and $rK_{Y/X}$ are linearly equivalent. We note that
$K_{Y/X}$ is uniquely defined as a $\mathbb{Q}$-divisor (cf. [dFEM11, Lemma
A.11(ii)]). Therefore, the log discrepancies and the minimal log discrepancies
for $k$-varieties defined above can be extended to $\mathbb{Q}$-Gorenstein
normal schemes of finite type over $R$, and we use the same notation.

######  Remark 3.1.

Let $X$ be a normal $k$-variety and let $x\in X$ be a closed point. Let
$\widehat{\mathcal{O}}_{X,x}$ be the completion of the local ring
$\mathcal{O}_{X,x}$ at its maximal ideal. Let
$\widehat{X}:=\operatorname{Spec}\bigl{(}\widehat{\mathcal{O}}_{X,x}\bigr{)}$
and let $\widehat{x}\in\widehat{X}$ be the closed point. Then for the induced
flat morphism $f:\widehat{X}\to X$, it follows that

$f^{*}(K_{X})=K_{\widehat{X}},\qquad
f^{*}(\omega_{X})=\omega^{\prime}_{\widehat{X}/k}$

by [dFEM11, Proposition A.14]. Furthermore, for an integer $r$, if $rK_{X}$ is
Cartier, then so is $rK_{\widehat{X}}$.

Suppose further that $X$ is $\mathbb{Q}$-Gorenstein. Let $\mathfrak{a}$ be an
$\mathbb{R}$-ideal sheaf on $X$. Then it follows from [Kaw21, Remark 2.6] (cf.
[dFEM11, Proposition 2.11]) that

$\operatorname{mld}_{\widehat{x}}(\widehat{X},\widehat{\mathfrak{a}})=\operatorname{mld}_{x}(X,\mathfrak{a}),$

where $\widehat{\mathfrak{a}}:=\mathfrak{a}\mathcal{O}_{\widehat{X}}$.

##  4\. Arc spaces of $k[[x_{1},\ldots,x_{N}]]$-schemes

In this section, we suppose $R_{0}=k$ and $R=k[[x_{1},\ldots,x_{N}]]$, and we
discuss the jet schemes and the arc spaces of $R$-schemes of finite type. We
refer the reader to [EM09] and [CLNS] for the theory of jet schemes and arc
spaces of $k$-varieties. In this section, we see that the codimensions of
cylinders of arc spaces can be defined in the same way as with $k$-varieties.

Let $X$ be a scheme over $k$. Let $({\sf Sch}/k)$ be the category of
$k$-schemes and $({\sf Sets})$ the category of sets. For a non-negative
integer $m$, we define a contravariant functor $F^{X}_{m}:({\sf
Sch}/k)\to({\sf Sets})$ by

$F^{X}_{m}(Y)=\operatorname{Hom}_{k}\left(Y\times_{\operatorname{Spec}k}\operatorname{Spec}k[t]/(t^{m+1}),X\right).$

It is known that the functor $F^{X}_{m}$ is always represented by a scheme
$X_{m}$ over $k$ (cf. [CLNS, Ch.3. Proposition 2.1.3]).

For $m\geq n\geq 0$, the canonical surjective ring homomorphism
$k[t]/(t^{m+1})\to k[t]/(t^{n+1})$ induces a morphism $\pi^{X}_{mn}:X_{m}\to
X_{n}$, which is called the truncation morphism. There exist the projective
limit and the projections

$X_{\infty}:=\mathop{\varprojlim}\limits_{m}X_{m},\qquad\psi^{X}_{m}:X_{\infty}\to
X_{m},$

and $X_{\infty}$ is called the arc space of $X$. Then there is a bijective map

$\operatorname{Hom}_{k}(\operatorname{Spec}K,X_{\infty})\simeq\operatorname{Hom}_{k}(\operatorname{Spec}K[[t]],X)$

for any field $K$ with $k\subset K$. For $m\in\mathbb{Z}_{\geq
0}\cup\\{\infty\\}$, we denote by $\pi^{X}_{m}:X_{m}\to X$ the canonical
truncation morphism. For $m\in\mathbb{Z}_{\geq 0}\cup\\{\infty\\}$ and a
morphism $f:Y\to X$ of schemes over $k$, we denote by $f_{m}:Y_{m}\to X_{m}$
the morphism induced by $f$. We often abbreviate $\pi^{X}_{mn}$, $\pi^{X}_{m}$
and $\psi^{X}_{m}$ to $\pi_{mn}$, $\pi_{m}$ and $\psi_{m}$, respectively when
no confusion can arise.

If $X$ is a scheme of finite type over $k$, then so is $X_{m}$ (cf. [EM09,
Proposition 2.2]). In this paper, we deal with a scheme of finite type over
$R=k[[x_{1},\ldots,x_{N}]]$.

######  Proposition 4.1 (cf. [Ish09, Corollary 4.2]).

Let $X$ be a scheme of finite type over $R=k[[x_{1},\ldots,x_{N}]]$. Then the
following hold.

  1. (1)

$X_{m}$ is a scheme of finite type over $R$.

  2. (2)

For any $m\geq n\geq 0$, the truncation map $\pi_{mn}:X_{m}\to X_{n}$ is a
morphism of finite type.

###### Proof.

We omit the proof because we will give a complete proof for Proposition 5.4,
which deals with a more complicated case. See also Remark 4.2 below. ∎

######  Remark 4.2.

The same arguments in Lemmas 5.1 and 5.2 give a local description of $X_{m}$
as follows.

  1. (1)

Let $S:=k[[x_{1},\ldots,x_{N}]][y_{1}\ldots,y_{M}]$ and
$A:=\operatorname{Spec}S$. Then we have $A_{m}\simeq\operatorname{Spec}S_{m}$,
where

$S_{m}:=k\bigl{[}\bigl{[}x_{1}^{(0)},\ldots,x_{N}^{(0)}\bigr{]}\bigr{]}\bigl{[}x_{j}^{(1)},\dots,x_{j}^{(m)},y_{j^{\prime}}^{(0)},\dots,y_{j^{\prime}}^{(m)}\
\big{|}\ 1\leq j\leq N,1\leq j^{\prime}\leq M\bigr{]}.$

Furthermore, for $m\geq n\geq 0$, the truncation map $\pi_{mn}:A_{m}\to A_{n}$
is induced by the ring inclusion $S_{n}\hookrightarrow S_{m}$.

  2. (2)

Let $X=\operatorname{Spec}(S/I)$ be the closed subscheme of $A$ defined by an
ideal $I=(f_{1},\ldots,f_{r})\subset S$. For $1\leq i\leq r$ and
$0\leq\ell\leq m$, we define $F_{i}^{(\ell)}\in S_{m}$ as follows:

$f_{i}\biggl{(}\sum_{\ell=0}^{m}x_{1}^{(\ell)}t^{\ell},\dots,\sum_{\ell=0}^{m}x_{N}^{(\ell)}t^{\ell},\sum_{\ell=0}^{m}y_{1}^{(\ell)}t^{\ell},\dots,\sum_{\ell=0}^{m}y_{M}^{(\ell)}t^{\ell}\biggr{)}\equiv\sum_{\ell=0}^{m}F_{i}^{(\ell)}t^{\ell}\pmod{t^{m+1}}.$

Let

$I_{m}:=\bigl{(}F_{i}^{(s)}\ \big{|}\ 1\leq i\leq r,\ 0\leq s\leq
m\bigr{)}\subset S_{m}$

be the ideal of $S_{m}$ generated by $F_{i}^{(s)}$’s. Then we have
$X_{m}\simeq\operatorname{Spec}(S_{m}/I_{m})$. Furthermore, for $m\geq n\geq
0$, the truncation map $\pi_{mn}:X_{m}\to X_{n}$ is induced by the ring
homomorphism $S_{n}/I_{n}\to S_{m}/I_{m}$.

A subset $C\subset X_{\infty}$ is called a cylinder if $C=\psi_{m}^{-1}(S)$
holds for some $m\geq 0$ and a constructible subset $S\subset X_{m}$. Typical
examples of cylinders appearing in this paper are the contact loci
$\operatorname{Cont}^{m}(\mathfrak{a})$ and $\operatorname{Cont}^{\geq
m}(\mathfrak{a})$ defined as follows.

######  Definition 4.3.

  1. (1)

For an arc $\gamma\in X_{\infty}$ and an ideal sheaf
$\mathfrak{a}\subset\mathcal{O}_{X}$, the order of $\mathfrak{a}$ measured by
$\gamma$ is defined as follows:

$\operatorname{ord}_{\gamma}(\mathfrak{a}):=\sup\\{r\in\mathbb{Z}_{\geq
0}\mid\gamma^{*}(\mathfrak{a})\subset(t^{r})\\},$

where $\gamma^{*}:\mathcal{O}_{X}\to K[[t]]$ is the induced ring homomorphism
by $\gamma$. Here $K$ is the field extension of $k$.

  2. (2)

For $m\in\mathbb{Z}_{\geq 0}$, we define
$\operatorname{Cont}^{m}(\mathfrak{a}),\operatorname{Cont}^{\geq
m}(\mathfrak{a})\subset X_{\infty}$ as follows:

$\displaystyle\operatorname{Cont}^{m}(\mathfrak{a})$
$\displaystyle:=\\{\gamma\in
X_{\infty}\mid\operatorname{ord}_{\gamma}(\mathfrak{a})=m\\},$
$\displaystyle\operatorname{Cont}^{\geq m}(\mathfrak{a})$
$\displaystyle:=\\{\gamma\in
X_{\infty}\mid\operatorname{ord}_{\gamma}(\mathfrak{a})\geq m\\}.$

By definition, we have

$\operatorname{Cont}^{\geq
m}(\mathfrak{a})=\psi_{m-1}^{-1}(Z(\mathfrak{a})_{m-1}),$

where $Z(\mathfrak{a})$ is the closed subscheme of $X$ defined by the ideal
sheaf $\mathfrak{a}$. Therefore, $\operatorname{Cont}^{m}(\mathfrak{a})$ and
$\operatorname{Cont}^{\geq m}(\mathfrak{a})$ are cylinders.

For $m\leq n+1$, we also define the subsets
$\operatorname{Cont}^{m}(\mathfrak{a})_{n}$ and $\operatorname{Cont}^{\geq
m}(\mathfrak{a})_{n}$ of $X_{n}$ in the same way.

We denote by $\mathfrak{o}_{X}\subset\mathcal{O}_{X}$ the ideal sheaf

$\mathfrak{o}_{X}:=(x_{1},\ldots,x_{N})\mathcal{O}_{X}\subset\mathcal{O}_{X}$

generated by $x_{1},\ldots,x_{N}\in R$. In this paper, we are interested in
arcs contained in the contact locus $\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{X})$. Due to the following lemma, the contact locus
$\operatorname{Cont}^{\geq 1}(\mathfrak{o}_{X})_{m}$ is a scheme of finite
type over $k$.

######  Lemma 4.4.

Let $X$ be a scheme of finite type over $R=k[[x_{1},\ldots,x_{N}]]$. Then for
each $m\geq 0$, the contact locus $\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{X})_{m}\subset X_{m}$ is a scheme of finite type over $k$.

###### Proof.

The assertion follows from Proposition 4.1(2). ∎

For the proof of Lemma 4.7, we state Hensel’s lemma in several variables.

######  Lemma 4.5.

Let $K$ be a field. Let $N,s$ and $r$ be non-negative integers with $N+s\geq
r$. Let $f_{1},\ldots,f_{r}\in
K[[t]][[x_{1},\ldots,x_{N}]][x_{N+1},\ldots,x_{N+s}]$ and let
$a_{1},\ldots,a_{N+s}\in K[[t]]$. Let $\mathcal{S}\subset\\{1,\ldots,N+s\\}$
be a subset with cardinality $\\#\mathcal{S}=r$. Let $m$ and $e$ be non-
negative integers with $m\geq e$. Suppose that

  * •

$a_{1},\ldots,a_{N}\in(t)$,

  * •

$f_{i}(a_{1},\ldots,a_{N+s})\in(t^{m+e+1})$ for each $1\leq i\leq r$, and

  * •

$\det\left(\frac{\partial f_{i}}{\partial
x_{j}}(a_{1},\ldots,a_{N+s})\right)_{1\leq i\leq r,\
j\in\mathcal{S}}\not\in(t^{e+1})$.

Then the following hold.

  1. (1)

There exist $b_{1},\ldots,b_{N+s}\in K[[t]]$ such that

     * •

$f_{i}(b_{1},\ldots,b_{N+s})=0$ for each $1\leq i\leq r$, and

     * •

$a_{j}-b_{j}\in(t^{m+1})$ for each $1\leq j\leq N+s$.

Furthermore, for $b_{1},\ldots,b_{N+s}\in K[[t]]$ and
$b^{\prime}_{1},\ldots,b^{\prime}_{N+s}\in K[[t]]$ with the above two
conditions, if

     * •

$b_{j}-b^{\prime}_{j}\in(t^{m+2})$ holds for each
$j\in\\{1,\ldots,N+s\\}\setminus\mathcal{S}$,

then

     * •

$b_{j}-b^{\prime}_{j}\in(t^{m+2})$ holds also for each $j\in\mathcal{S}$.

  2. (2)

Moreover, for any sequence $\bigl{(}a^{\prime}_{j}\in K[[t]]\ \big{|}\
j\in\\{1,\ldots,N+s\\}\setminus\mathcal{S}\bigr{)}$ satisfying
$a^{\prime}_{j}-a_{j}\in(t^{m+1})$, there exist $b_{1},\ldots,b_{N+s}\in
K[[t]]$ satisfying the following conditions:

     * •

$f_{i}(b_{1},\ldots,b_{N+s})=0$ for each $1\leq i\leq r$,

     * •

$a^{\prime}_{j}-b_{j}\in(t^{m+2})$ for each
$j\in\\{1,\ldots,N+s\\}\setminus\mathcal{S}$, and

     * •

$a_{j}-b_{j}\in(t^{m+2})$ for each $j\in\mathcal{S}$.

###### Proof.

When $N=0$ and $f_{1},\ldots,f_{r}\in K[x_{1},\ldots,x_{s}]$, the assertions
are proved in the proof of [DL99, Lemma 4.1] (cf. [EM09, Proposition 4.1]).
The same proof works in our setting. ∎

######  Remark 4.6.

The same statement as in Lemma 4.5 holds even when we replace
$K[[t]][[x_{1},\ldots,x_{N}]][x_{N+1},\ldots,x_{N+s}]$ with
$K[x_{N+1},\ldots,x_{N+s}][[t]]$. This version will be used in the proof of
Proposition 5.31.

######  Lemma 4.7.

Let $N,s$ and $r$ be non-negative integers with $N+s\geq r$. Let
$R=k[[x_{1},\ldots,x_{N}]]$ and let $S=R[y_{1},\ldots,y_{s}]$. Let
$I=(F_{1},\ldots,F_{r})$ be the ideal generated by elements
$F_{1},\ldots,F_{r}\in S$, and let $M=\operatorname{Spec}(S/I)$. Let
$\mathfrak{o}_{M}\subset\mathcal{O}_{M}$ be the ideal sheaf generated by
$x_{1},\ldots,x_{N}\in R$. Let
$\overline{J}=\operatorname{Fitt}^{N+s-r}(\Omega^{\prime}_{M/k})$. Then, for
non-negative integers $m$ and $e$ with $m\geq e$, the following hold.

  1. (1)

It follows that

$\psi_{m}\left(\operatorname{Cont}^{e}(\overline{J})\cap\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{M})\right)=\pi_{m+e,m}\left(\operatorname{Cont}^{e}(\overline{J})_{m+e}\cap\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{M})_{m+e}\right).$

  2. (2)

$\pi_{m+1,m}:M_{m+1}\to M_{m}$ induces a piecewise trivial fibration

$\psi_{m+1}\left(\operatorname{Cont}^{e}(\overline{J})\cap\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{M})\right)\to\psi_{m}\left(\operatorname{Cont}^{e}(\overline{J})\cap\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{M})\right)$

with fiber $\mathbb{A}^{N+s-r}$.

###### Proof.

Let $J:=\mathcal{J}_{r}\bigl{(}I;\operatorname{Der}_{k}(S)\bigr{)}\subset S$.
Then we have $\overline{J}=(J+I)/I$ by Remark 2.14. Note here that
$\operatorname{Der}_{k}(S)=\operatorname{Der}^{\prime}_{k}(S)$ is generated by
$\partial/\partial x_{i}$’s and $\partial/\partial y_{j}$’s. Therefore, (1)
follows from the first assertion of Lemma 4.5(1). Furthermore, (2) follows
from Lemma 4.5(2) and the second assertion of Lemma 4.5(1). ∎

The following proposition is a formal power series ring version of [EM09,
Proposition 4.1].

######  Proposition 4.8.

Let $X$ be an integral scheme of finite type over $R=k[[x_{1},\ldots,x_{N}]]$
of $\dim X=n$. Then there exists a positive integer $c$ such that the
following hold for non-negative integers $m$ and $e$ with $m\geq ce$.

  1. (1)

We have

$\displaystyle\psi_{m}\left(\operatorname{Cont}^{e}(\operatorname{Jac}^{\prime}_{X/k})\cap\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{X})\right)$
$\displaystyle=\pi_{m+e,m}\left(\operatorname{Cont}^{e}(\operatorname{Jac}^{\prime}_{X/k})_{m+e}\cap\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{X})_{m+e}\right).$

  2. (2)

$\pi_{m+1,m}:X_{m+1}\to X_{m}$ induces a piecewise trivial fibration

$\psi_{m+1}\left(\operatorname{Cont}^{e}(\operatorname{Jac}^{\prime}_{X/k})\cap\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{X})\right)\to\psi_{m}\left(\operatorname{Cont}^{e}(\operatorname{Jac}^{\prime}_{X/k})\cap\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{X})\right)$

with fiber $\mathbb{A}^{n}$.

###### Proof.

We omit the proof. See the proof of Proposition 5.9 to see how it can be
reduced to the complete intersection case proved in Lemma 4.7. Note that we
may assume that $X$ has a $k$-point and hence we have $\dim^{\prime}X=\dim
X=n$ by Lemma 2.11(4). Otherwise, we have $\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{X})=\emptyset$ (cf. Lemma 4.4), and the assertions are clear.
∎

######  Remark 4.9.

Proposition 4.8 is a formal power series ring version of [EM09, Proposition
4.1]. Note that in [EM09, Proposition 4.1], they prove that $c=1$ satisfies
the statement. However, the weaker statement as in Proposition 4.8 using $c$
is enough for our later use.

By Proposition 4.8, the codimension of cylinder contained in
$\operatorname{Cont}^{\geq 1}(\mathfrak{o}_{X})$ is well-defined as follows.

######  Definition 4.10.

Let $X$ be an integral scheme of finite type over $R=k[[x_{1},\ldots,x_{N}]]$
and let $C\subset X_{\infty}$ be a cylinder contained in
$\operatorname{Cont}^{\geq 1}(\mathfrak{o}_{X})$.

  1. (1)

Assume that
$C\subset\operatorname{Cont}^{e}(\operatorname{Jac}^{\prime}_{X/k})$ for some
$e\in\mathbb{Z}_{\geq 0}$. Then we define the codimension of $C$ in
$X_{\infty}$ as

$\operatorname{codim}(C):=(m+1)\operatorname{dim}X-\operatorname{dim}(\psi_{m}(C))$

for any sufficiently large $m$. This definition is well-defined by Proposition
4.8.

  2. (2)

In general, we define the codimension of $C$ in $X_{\infty}$ as follows:

$\operatorname{codim}(C):=\min_{e\in\mathbb{Z}_{\geq
0}}\operatorname{codim}\bigl{(}C\cap\operatorname{Cont}^{e}(\operatorname{Jac}^{\prime}_{X/k})\bigr{)}.$

By convention, $\operatorname{codim}(C)=\infty$ if
$C\cap\operatorname{Cont}^{e}(\operatorname{Jac}^{\prime}_{X/k})=\emptyset$
for any $e\geq 0$.

The following theorem is a formal power series ring version of [EM09, Theorem
7.4].

######  Theorem 4.11.

Let $X$ be a $\mathbb{Q}$-Gorenstein integral normal scheme of finite type
over $R=k[[x_{1},\ldots,x_{N}]]$. Let $x$ be a $k$-point of $X$ and let
$\mathfrak{m}_{x}\subset\mathcal{O}_{X}$ be the corresponding maximal ideal
sheaf. Let $r$ be a positive integer such that $rK_{X}$ is Cartier. Let
$\mathfrak{a}$ be a non-zero ideal sheaf on $X$ and $\delta$ a positive real
number. Then we have

$\displaystyle\operatorname{mld}_{x}(X,\mathfrak{a}^{\delta})$
$\displaystyle=\inf_{w,m\in\mathbb{Z}_{\geq
0}}\Bigl{\\{}\operatorname{codim}(C_{w,m})-\frac{m}{r}-\delta w\Bigr{\\}}$
$\displaystyle=\inf_{w,m\in\mathbb{Z}_{\geq
0}}\Bigl{\\{}\operatorname{codim}(C^{\prime}_{w,m})-\frac{m}{r}-\delta
w\Bigr{\\}},$

where

$\displaystyle C_{w,m}$
$\displaystyle:=\operatorname{Cont}^{w}(\mathfrak{a})\cap\operatorname{Cont}^{m}(\mathfrak{n}_{r,X})\cap\operatorname{Cont}^{\geq
1}(\mathfrak{m}_{x}),$ $\displaystyle C^{\prime}_{w,m}$
$\displaystyle:=\operatorname{Cont}^{\geq
w}(\mathfrak{a})\cap\operatorname{Cont}^{m}(\mathfrak{n}_{r,X})\cap\operatorname{Cont}^{\geq
1}(\mathfrak{m}_{x}).$

###### Proof.

The assertions are formal power series ring versions of Theorem 7.4 and Remark
7.5 in [EM09], and their proofs also work in this setting by making the
following modifications:

  * •

Replacing $\Omega_{\text{--}/k}$ with $\Omega^{\prime}_{\text{--}/k}$, and
$\operatorname{Jac}_{\text{--}/k}$ with
$\operatorname{Jac}^{\prime}_{\text{--}/k}$.

  * •

Considered cylinders $C$ are contained in $\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{X})$.

Theorem 7.4 in [EM09] is a consequence of Lemma 7.3 in [EM09]. The key
ingredients of the proof of Lemma 7.3 in [EM09] are

  * •

Theorem 6.2 and Corollary 6.4 in [EM09], and

  * •

Proposition 5.11 in [EM09].

Theorem 6.2 and Corollary 6.4 in [EM09] are the codimension formula as in
Proposition 5.43, and they are formal consequences of Proposition 4.4(i) in
[EM09]. Proposition 4.4(i) in [EM09] is still valid in our setting by
replacing $\Omega_{X}$ with $\Omega^{\prime}_{X/k}$ due to Lemma 2.2 (see
Lemma 5.13 for the detailed argument).

Proposition 5.11 in [EM09] is a proposition on codimensions as in Proposition
5.36, and it is a consequence of Lemma 5.1 and Corollary 5.2 in [EM09]
(Corollary 5.2 is a corollary of Lemma 5.1). The proof of Lemma 5.1 in [EM09]
still works in our setting by replacing $\operatorname{Jac}_{\text{--}/k}$
with $\operatorname{Jac}^{\prime}_{\text{--}/k}$. The only important point is
that the ideal $\operatorname{Jac}^{\prime}_{X/k}$ defines the singular locus
of $X$ even when $X$ is an integral scheme of finite type over $R$ (cf. Remark
2.14).

Besides, Lemma 6.1 in [EM09] is used in the proof of Lemma 7.3 in [EM09], and
Proposition 3.2 in [EM09] is used in the proof of Corollary 5.2 in [EM09].
Proposition 3.2 and Lemma 6.1 in [EM09] are formal consequences of the
valuative criterion of properness, and their proofs work in our setting. ∎

##  5\. Arc spaces of $k[t]$-schemes

In this section, we deal with the arc spaces of $k[t]$-schemes. Let $X$ be a
scheme over $k[t]$. For a non-negative integer $m$, we define a contravariant
functor $F^{X}_{m}:({\sf Sch}/k)\to({\sf Sets})$ by

$F^{X}_{m}(Y)=\operatorname{Hom}_{k[t]}\left(Y\times_{\operatorname{Spec}k}\operatorname{Spec}k[t]/(t^{m+1}),X\right).$

By the same argument as in [CLNS, Ch.4. Theorem 3.2.3], we can see that the
functor $F^{X}_{m}$ is always represented by a scheme $X_{m}$ over $k$. We
shall use the same symbols $X_{\infty}$, $\pi_{mn}$, $\psi_{m}$ and $\pi_{m}$
as in Section 4 also for this setting.

In this section, we deal with the following two categories of $k[t]$-schemes:

  1. (1)

$X$ is a scheme of finite type over $k[t][[x_{1},\ldots,x_{N}]]$.

  2. (2)

$X$ is an affine scheme of the form
$X=\operatorname{Spec}\bigl{(}k[x_{1},\ldots,x_{N}][[t]]/I\bigr{)}$.

We note that in [DL02], Denef and Loeser extend the theory of arc spaces of
$k$-varieties to the case where

  1. (3)

$X$ is a scheme of finite type over $k[t]$.

In [NS], we dealt with the arc spaces of $X$ in (3). However, in this paper,
we need to work on the arc spaces of $X$ in (1) and (2). We also note that
Sebag in [Seb04] deals with formal $k[[t]]$-schemes of finite type, and this
theory can be applied to (2) and (3) (see also [CLNS] for this theory).

In Subsection 5.1, we discuss case (1). In Subsection 5.2, we discuss case
(2), where we will not deal with formal $k[[t]]$-schemes in general but deal
with only affine schemes in a minimum way.

###  5.1. Arc spaces of $k[t][[x_{1},\ldots,x_{N}]]$-schemes

In this subsection, we suppose $R_{0}=k[t]$ and
$R=k[t][[x_{1},\ldots,x_{N}]]$, and we discuss the arc spaces of $R$-schemes
of finite type.

####  5.1.1. Arc spaces

First, we prove that if $X$ is a scheme of finite type over $R$, then so is
$X_{m}$.

######  Lemma 5.1.

Let $S:=k[t][[x_{1},\ldots,x_{N}]][y_{1},\ldots,y_{M}]$ and let
$A:=\operatorname{Spec}S$. Then we have $A_{m}\simeq\operatorname{Spec}S_{m}$,
where

$S_{m}:=k\bigl{[}\bigl{[}x_{1}^{(0)},\ldots,x_{N}^{(0)}\bigr{]}\bigr{]}\bigl{[}x_{j}^{(1)},\dots,x_{j}^{(m)},y_{j^{\prime}}^{(0)},\dots,y_{j^{\prime}}^{(m)}\
\big{|}\ 1\leq j\leq N,1\leq j^{\prime}\leq M\bigr{]}.$

Furthermore, for $m\geq n\geq 0$, the truncation map $\pi_{mn}:A_{m}\to A_{n}$
is induced by the ring inclusion $S_{n}\hookrightarrow S_{m}$.

###### Proof.

Let $Y:=\operatorname{Spec}C$ be an affine scheme over $k$. We shall give a
natural bijective map

$\Phi:\operatorname{Hom}_{k[t]}\bigl{(}S,C[t]/(t^{m+1})\bigr{)}\to\operatorname{Hom}_{k}(S_{m},C).$

For each $0\leq i\leq m$, we denote by $p_{i}$ the projection

$p_{i}:C[t]/(t^{m+1})\to C;\quad c_{0}+c_{1}t+\cdots+c_{m}t^{m}\mapsto c_{i}.$

For $\alpha\in\operatorname{Hom}_{k[t]}\left(S,C[t]/(t^{m+1})\right)$, we
define $\Phi(\alpha)\in\operatorname{Hom}_{k}(S_{m},C)$ as follows. First, we
define the ring homomorphism $\alpha^{\prime}_{0}:S_{0}\to C$ as the
composition

$S_{0}\hookrightarrow
S\xrightarrow{\alpha}C[t]/(t^{m+1})\xrightarrow{p_{0}}C.$

Then we define $\alpha^{\prime}:S_{m}\to C$ as the ring homomorphism uniquely
determined by the following conditions:

  * •

$\alpha^{\prime}(f)=\alpha^{\prime}_{0}(f)$ holds for any $f\in S_{0}$.

  * •

$\alpha^{\prime}\bigl{(}x_{j}^{(s)}\bigr{)}=p_{s}\bigl{(}\alpha(x_{j})\bigr{)}$
holds for each $1\leq j\leq N$ and $1\leq s\leq m$.

  * •

$\alpha^{\prime}\bigl{(}y_{j}^{(s)}\bigr{)}=p_{s}\bigl{(}\alpha(y_{j})\bigr{)}$
holds for each $1\leq j\leq M$ and $1\leq s\leq m$.

Then we define $\Phi(\alpha)=\alpha^{\prime}$.

Next, we define the inverse map

$\Psi:\operatorname{Hom}_{k}(S_{m},C)\to\operatorname{Hom}_{k[t]}\bigl{(}S,C[t]/(t^{m+1})\bigr{)}.$

We set

$\displaystyle S^{\prime}_{m}$
$\displaystyle:=k\bigl{[}x_{j}^{(1)},\dots,x_{j}^{(m)}\ \big{|}\ 1\leq j\leq
N\bigr{]}\bigl{[}\bigl{[}x^{(0)}_{1},\ldots,x^{(0)}_{N}\bigr{]}\bigr{]},$
$\displaystyle S^{\prime\prime}_{m}$
$\displaystyle:=k\bigl{[}\bigl{[}x^{(0)}_{1},\ldots,x^{(0)}_{N}\bigr{]}\bigr{]}\bigl{[}x_{j}^{(1)},\dots,x_{j}^{(m)}\
\big{|}\ 1\leq j\leq N\bigr{]}.$

We define a ring homomorphim $\Lambda_{1}:k[t][x_{1},\ldots,x_{N}]\to
S^{\prime}_{m}[t]/(t^{m+1})$ by

$\Lambda_{1}(t)=t,\qquad\Lambda_{1}(x_{j})=x_{j}^{(0)}+x_{j}^{(1)}t+\cdots+x_{j}^{(m)}t^{m}.$

Since
$\Lambda_{1}\bigl{(}(x_{1},\ldots,x_{N})\bigr{)}\subset\bigl{(}x_{1}^{(0)},\ldots,x_{N}^{(0)},t\bigr{)}$
holds, $\Lambda_{1}$ induces a ring homomorphism
$\Lambda_{2}:k[t][[x_{1},\ldots,x_{N}]]\to S^{\prime}_{m}[t]/(t^{m+1})$. Note
here that its image is contained in $S^{\prime\prime}_{m}[t]/(t^{m+1})$.
Furthermore, $S^{\prime\prime}_{m}[t]/(t^{m+1})$ is a subring of
$S_{m}[t]/(t^{m+1})$. Therefore, we have a ring homomorphism
$\Lambda_{3}:k[t][[x_{1},\ldots,x_{N}]]\to S_{m}[t]/(t^{m+1})$. Then we define
$\Lambda:S\to S_{m}[t]/(t^{m+1})$ as the ring homomorphism uniquely determined
by the following conditions:

  * •

$\Lambda(f)=\Lambda_{3}(f)$ holds for any $f\in k[t][[x_{1},\ldots,x_{N}]]$.

  * •

$\Lambda(y_{j})=y_{j}^{(0)}+y_{j}^{(1)}t+\cdots+y_{j}^{(m)}t^{m}$ holds for
each $1\leq j\leq M$.

Then $\Lambda$ is a $k[t]$-ring homomorphism.

For $\beta\in\operatorname{Hom}_{k}(S_{m},C)$, we define
$\Psi(\beta)\in\operatorname{Hom}_{k[t]}\bigl{(}S,C[t]/(t^{m+1})\bigr{)}$ as
the composition

$S\xrightarrow{\Lambda}S_{m}[t]/(t^{m+1})\xrightarrow{\beta}C[t]/(t^{m+1}),$

where $S_{m}[t]/(t^{m+1})\to C[t]/(t^{m+1})$ is the $k[t]$-ring homomorphism
induced by $\beta:S_{m}\to C$.

By the construction of $\Phi$ and $\Psi$, if $\alpha^{\prime}=\Phi(\alpha)$
and $\beta^{\prime}=\Psi(\beta)$, then they satisfy the following:

  * •

$p_{s}\bigl{(}\alpha(x_{j})\bigr{)}=\alpha^{\prime}\bigl{(}x_{j}^{(s)}\bigr{)}$
holds for each $1\leq j\leq N$ and $0\leq s\leq m$.

  * •

$p_{s}\bigl{(}\alpha(y_{j})\bigr{)}=\alpha^{\prime}\bigl{(}y_{j}^{(s)}\bigr{)}$
holds for each $1\leq j\leq M$ and $0\leq s\leq m$.

  * •

$p_{s}\bigl{(}\beta^{\prime}(x_{j})\bigr{)}=\beta\bigl{(}x_{j}^{(s)}\bigr{)}$
holds for each $1\leq j\leq N$ and $0\leq s\leq m$.

  * •

$p_{s}\bigl{(}\beta^{\prime}(y_{j})\bigr{)}=\beta\bigl{(}y_{j}^{(s)}\bigr{)}$
holds for each $1\leq j\leq M$ and $0\leq s\leq m$.

Therefore, we have $\Psi\circ\Phi=\operatorname{id}$ and
$\Phi\circ\Psi=\operatorname{id}$. Hence, $F_{m}^{A}$ is represented by
$\operatorname{Spec}S_{m}$. The second assertion follows from the construction
of $A_{m}$. ∎

######  Lemma 5.2.

We take over the notation in Lemma 5.1. Let $X=\operatorname{Spec}(S/I)$ be
the closed subscheme of $A$ defined by an ideal $I=(f_{1},\ldots,f_{r})\subset
S$. For $1\leq i\leq r$ and $0\leq\ell\leq m$, we define $F_{i}^{(\ell)}\in
S_{m}$ as follows:

$f_{i}\biggl{(}\sum_{\ell=0}^{m}x_{1}^{(\ell)}t^{\ell},\dots,\sum_{\ell=0}^{m}x_{N}^{(\ell)}t^{\ell},\sum_{\ell=0}^{m}y_{1}^{(\ell)}t^{\ell},\dots,\sum_{\ell=0}^{m}y_{M}^{(\ell)}t^{\ell}\biggr{)}\equiv\sum_{\ell=0}^{m}F_{i}^{(\ell)}t^{\ell}\pmod{t^{m+1}}.$

Let

$I_{m}:=\bigl{(}F_{i}^{(s)}\ \big{|}\ 1\leq i\leq r,\ 0\leq s\leq
m\bigr{)}\subset S_{m}$

be the ideal of $S_{m}$ generated by $F_{i}^{(s)}$’s. Then we have
$X_{m}\simeq\operatorname{Spec}(S_{m}/I_{m})$. Furthermore, for $m\geq n\geq
0$, the truncation map $\pi_{mn}:X_{m}\to X_{n}$ is induced by the ring
homomorphism $S_{n}/I_{n}\to S_{m}/I_{m}$.

###### Proof.

Let $Y:=\operatorname{Spec}C$ be an affine scheme over $k$. We can see that
the bijective map $\Phi$ in the proof of Lemma 5.1 induces the bijective map

$\operatorname{Hom}_{k[t]}\bigl{(}S/I,C[t]/(t^{m+1})\bigr{)}\to\operatorname{Hom}_{k}(S_{m}/I_{m},C).$

Therefore, the functor $F_{m}^{X}$ is represented by
$\operatorname{Spec}(S_{m}/I_{m})$. The second assertion follows from the
construction of $X_{m}$. ∎

######  Remark 5.3.

More precisely,

$f_{i}\biggl{(}\sum_{\ell=0}^{m}x_{1}^{(\ell)}t^{\ell},\dots,\sum_{\ell=0}^{m}x_{N}^{(\ell)}t^{\ell},\sum_{\ell=0}^{m}y_{1}^{(\ell)}t^{\ell},\dots,\sum_{\ell=0}^{m}y_{M}^{(\ell)}t^{\ell}\biggr{)}$

in Lemma 5.2 is defined as $\Lambda(f_{i})\in S_{m}[t]/(t^{m+1})$, where
$\Lambda$ is defined within the proof of Lemma 5.1.

######  Proposition 5.4.

If $X$ is a scheme of finite type over $R=k[t][[x_{1},\ldots,x_{N}]]$. Then
the following hold.

  1. (1)

$X_{m}$ is a scheme of finite type over $R$.

  2. (2)

For any $m\geq n\geq 0$, the truncation map $\pi_{mn}:X_{m}\to X_{n}$ is a
morphism of finite type.

###### Proof.

Take an affine cover $X=U_{1}\cup\cdots\cup U_{s}$. Then, $F_{m}^{X}$ is
represented by the scheme obtained by gluing the schemes $(U_{i})_{m}$
constructed in Lemma 5.2 (cf. [EM09, Proposition 2.2]). Therefore, the
assertions follow from Lemma 5.2. ∎

Cylinders and the contact loci

$\operatorname{Cont}^{m}(\mathfrak{a}),\ \operatorname{Cont}^{\geq
m}(\mathfrak{a})\subset
X_{\infty},\qquad\operatorname{Cont}^{m}(\mathfrak{a})_{n},\
\operatorname{Cont}^{\geq m}(\mathfrak{a})_{n}\subset X_{n}$

are also defined in this setting in the same way.

We denote by $\mathfrak{o}_{X}\subset\mathcal{O}_{X}$ the ideal sheaf

$\mathfrak{o}_{X}:=(x_{1},\ldots,x_{N})\mathcal{O}_{X}\subset\mathcal{O}_{X}$

generated by $x_{1},\ldots,x_{N}\in R$. From the next subsection, we basically
work on arcs contained in the contact locus $\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{X})$. Due to the following lemma, the contact locus
$\operatorname{Cont}^{\geq 1}(\mathfrak{o}_{X})_{m}$ is a scheme of finite
type over $k$.

######  Lemma 5.5.

Let $X$ be a scheme of finite type over $R=k[t][[x_{1},\ldots,x_{N}]]$. Then
the following hold.

  1. (1)

For each $m\geq 0$, the contact locus $\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{X})_{m}\subset X_{m}$ is a scheme of finite type over $k$.

  2. (2)

Any $k$-arc of $X$ is contained in $\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{X})$.

###### Proof.

(1) follows from Proposition 5.4.

We shall prove (2). Let $\gamma\in X_{\infty}$ be a $k$-arc. We may assume
that $X$ is affine, and we may write $X=\operatorname{Spec}A$ with $A=S/I$,
where

$S:=k[t][[x_{1},\ldots,x_{N}]][y_{1},\ldots,y_{m}]$

and $I$ is an ideal of $S$. Let $\gamma^{*}:A\to k[[t]]$ be the corresponding
$k[t]$-ring homomorphism. Let $M$ be the kernel of the composite map $S\to
A\xrightarrow{\gamma^{*}}k[[t]]\to k$. Since $S/M=k$, $M$ is of the form

$(t,x_{1},\ldots,x_{N},y_{1}-a_{1},\ldots,y_{m}-a_{m})$

for some $a_{i}\in k$. It shows that $\gamma^{*}(\mathfrak{o}_{X})\subset(t)$
and hence $\gamma\in\operatorname{Cont}^{\geq 1}(\mathfrak{o}_{X})$. ∎

######  Lemma 5.6.

Let $n$ be a non-negative integer and let $X$ be a scheme of finite type over
$R=k[t][[x_{1},\ldots,x_{N}]]$. Suppose that each irreducible component
$X_{i}$ of $X$ has $\dim^{\prime}X_{i}\geq n+1$ (see Definition 2.10). Let
$\gamma\in X_{\infty}$ be a $k$-arc with
$\operatorname{ord}_{\gamma}\bigl{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}<\infty$.
Then we have

$\gamma^{*}\Omega^{\prime}_{X/k[t]}/T\simeq k[[t]]^{\oplus n},$

where $T$ is the torsion part of $\gamma^{*}\Omega^{\prime}_{X/k[t]}$.

###### Proof.

We may assume that $X$ is affine, and we may write $X=\operatorname{Spec}A$
with $A=S/I$, where

$S:=k[t][[x_{1},\ldots,x_{N}]][y_{1},\ldots,y_{m}]$

and $I$ is an ideal of $S$. If $P$ is a minimal prime of $I$, then we have

$\operatorname{ht}P=\dim S-\dim^{\prime}(S/P)\leq(N+m+1)-(n+1)=N+m-n$

by Lemma 2.11(1). Therefore we have $\operatorname{ht}I\leq N+m-n$.

Let $\gamma^{*}:A\to k[[t]]$ be the corresponding $k[t]$-ring homomorphism,
and let $\overline{\gamma}^{*}:A\to k((t))$ be its composition with $k[[t]]\to
k((t))$. Let $\mathfrak{q}\subset A$ be the kernel of $\gamma^{*}$ and
$Q\subset S$ the corresponding prime ideal. Since $\overline{\gamma}^{*}$
factors through $A_{\mathfrak{q}}$, it is sufficient to show that
$\Omega^{\prime}_{A_{\mathfrak{q}}/k[t]}\otimes_{A_{\mathfrak{q}}}k((t))$ has
dimension $n$ as a $k((t))$-vector space.

Let $w_{1},\ldots,w_{\ell}\in I$ be generators of $I$. Let $M\in
M_{N+m,\ell}(A_{\mathfrak{q}})$ be the Jacobian matrix with respect to
$w_{1},\ldots,w_{\ell}\in I$ and derivations $\frac{\partial}{\partial
x_{i}}$’s and $\frac{\partial}{\partial y_{i}}$’s. Then $M$ defines a map
$M:A_{\mathfrak{q}}^{\ell}\to A_{\mathfrak{q}}^{N+m}$ and its cokernel is
isomorphic to $\Omega^{\prime}_{A_{\mathfrak{q}}/k[t]}$ by Proposition
2.4(3)(4). Since
$\operatorname{ord}_{\gamma}\bigl{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}<\infty$,
$M$ has an $(N+m-n)$-minor which is not contained in
$\mathfrak{q}A_{\mathfrak{q}}$ (cf. Remark 2.14). Furthermore, since we have

$\operatorname{ht}(IS_{Q})\leq\operatorname{ht}I\leq N+m-n,$

any $(N+m-n+1)$-minor of $M$ is contained in $\mathfrak{q}A_{\mathfrak{q}}$ by
Remark 2.6(2)(a) (cf. [Mat89, Theorem 30.4]). Therefore, the image of $M$ in
$M_{N+m,\ell}(k((t)))$ has rank $N+m-n$, and it follows that
$\Omega^{\prime}_{A_{\mathfrak{q}}/k[t]}\otimes_{A_{\mathfrak{q}}}k((t))$ has
dimension $n$ as a $k((t))$-vector space. ∎

######  Lemma 5.7.

Let $n$ and $e$ be non-negative integers and let $X$ be a scheme of finite
type over $R=k[t][[x_{1},\ldots,x_{N}]]$. Suppose that each irreducible
component $X_{i}$ of $X$ has $\dim^{\prime}X_{i}\geq n+1$. Let
$\gamma\in\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}$
be a $k$-arc. Then we have

$\gamma^{*}\Omega^{\prime}_{X/k[t]}\simeq k[[t]]^{\oplus
n}\oplus\bigoplus_{i}k[t]/(t^{e_{i}})$

as $k[[t]]$-modules with $\sum_{i}e_{i}=e$.

###### Proof.

The same proof as in [NS, Lemma 2.13(1)] works due to Lemma 5.6. ∎

####  5.1.2. Cylinders and Codimension

In this subsection, we define and discuss the codimensions of cylinders of the
arc space of an $R$-scheme $X$ of finite type. We define the codimension only
for cylinders contained in the contact locus $\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{X})$, where $\mathfrak{o}_{X}\subset\mathcal{O}_{X}$ is the
ideal sheaf generated by $x_{1},\ldots,x_{N}\in R$. Due to Lemma 5.5, the
contact locus $\operatorname{Cont}^{\geq 1}(\mathfrak{o}_{X})_{m}\subset
X_{m}$ is a scheme of finite type over $k$, and hence cylinders contained in
$\operatorname{Cont}^{\geq 1}(\mathfrak{o}_{X})$ are easier to handle than the
general cylinders.

First, we prove Proposition 5.9, which is necessary for defining the
codimension of cylinders.

######  Lemma 5.8.

Let $N,s$ and $r$ be non-negative integers with $N+s\geq r$. Let
$R=k[t][[x_{1},\ldots,x_{N}]]$ and let $S=R[y_{1},\ldots,y_{s}]$. Let
$I=(F_{1},\ldots,F_{r})$ be the ideal generated by elements
$F_{1},\ldots,F_{r}\in S$, and let $M=\operatorname{Spec}(S/I)$. Let
$\mathfrak{o}_{M}\subset\mathcal{O}_{M}$ be the ideal sheaf generated by
$x_{1},\ldots,x_{N}\in R$. Let
$\overline{J}=\operatorname{Fitt}^{N+s-r}(\Omega^{\prime}_{M/k[t]})$. For non-
negative integers $m$ and $e$ with $m\geq e$, the following hold.

  1. (1)

It follows that

$\psi_{m}\left(\operatorname{Cont}^{e}(\overline{J})\cap\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{M})\right)=\pi_{m+e,m}\left(\operatorname{Cont}^{e}(\overline{J})_{m+e}\cap\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{M})_{m+e}\right).$

  2. (2)

$\pi_{m+1,m}:M_{m+1}\to M_{m}$ induces a piecewise trivial fibration

$\psi_{m+1}\left(\operatorname{Cont}^{e}(\overline{J})\cap\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{M})\right)\to\psi_{m}\left(\operatorname{Cont}^{e}(\overline{J})\cap\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{M})\right)$

with fiber $\mathbb{A}^{N+s-r}$.

###### Proof.

Let $J:=\mathcal{J}_{r}\bigl{(}I;\operatorname{Der}_{k[t]}(S)\bigr{)}\subset
S$. Then we have $\overline{J}=(J+I)/I$ by Remark 2.14. Note here that
$\operatorname{Der}_{k[t]}(S)=\operatorname{Der}^{\prime}_{k[t]}(S)$ is
generated by $\partial/\partial x_{i}$’s and $\partial/\partial y_{j}$’s.
Therefore, (1) follows from the first assertion of Lemma 4.5(1). Furthermore,
(2) follows from Lemma 4.5(2) and the second assertion of Lemma 4.5(1). ∎

######  Proposition 5.9.

Let $n$ be a non-negative integer and let $X$ be a scheme of finite type over
$R=k[t][[x_{1},\ldots,x_{N}]]$. Suppose that each irreducible component
$X_{i}$ of $X$ has $\dim^{\prime}X_{i}\geq n+1$. Then there exists a positive
integer $c$ such that the following hold for non-negative integers $m$ and $e$
with $m\geq ce$.

  1. (1)

It follows that

$\displaystyle\psi_{m}\left(\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}\cap\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{X})\right)$
$\displaystyle=\pi_{m+e,m}\left(\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}_{m+e}\cap\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{X})_{m+e}\right).$

  2. (2)

$\pi_{m+1,m}:X_{m+1}\to X_{m}$ induces a piecewise trivial fibration

$\displaystyle\psi_{m+1}$
$\displaystyle\left(\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}\cap\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{X})\right)$
$\displaystyle\to\psi_{m}\left(\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}\cap\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{X})\right)$

with fiber $\mathbb{A}^{n}$.

###### Proof.

The same proof as in [NS, Proposition 2.17] works. We shall give a sketch of
the proof.

We may assume that $X$ is affine, and we may write
$X=\operatorname{Spec}(S/I_{X})$, where

$S:=k[t][[x_{1},\ldots,x_{N}]][y_{1},\ldots,y_{m}]$

and $I_{X}$ is an ideal of $S$. By the assumption and Lemma 2.11(1), we have

$\operatorname{ht}P=\dim S-\dim^{\prime}(S/P)\leq(N+m+1)-(n+1)=N+m-n$

for any minimal prime $P$ of $I_{X}$. We set $r:=N+m-n$.

Let $f_{1},\ldots,f_{d}$ be generators of $I_{X}$. For $1\leq i\leq r$, we set
$F_{i}:=\sum_{j=1}^{d}a_{ij}f_{j}$ for general $a_{ij}\in k$. Let
$M\subset\operatorname{Spec}S$ be the closed subscheme defined by the ideal
$I_{M}:=(F_{1},\ldots,F_{r})$. We denote

$I_{X^{\prime}}:=(I_{M}:I_{X})\subset S,\qquad
J:=\mathcal{J}_{r}\bigl{(}I_{M};\operatorname{Der}_{k[t]}(S)\bigr{)}\subset
S.$

Here, we claim that

  * ($\spadesuit$)

$J\subset\sqrt{I_{X}+I_{X^{\prime}}}$ holds.

We note that if ($\spadesuit$) is true, then the assertions for $X$ can be
reduced to those for $M$ by the same argument as in [NS, Proposition 2.17].
Therefore, the assertions follow from Lemma 5.8.

Let $\mathfrak{p}$ be a prime ideal satisfying
$I_{X}+I_{X^{\prime}}\subset\mathfrak{p}$. To prove ($\spadesuit$), it is
sufficient to show that $S/I_{M}$ is not regular at $\mathfrak{p}$. Indeed, if
$S/I_{M}$ is not regular at $\mathfrak{p}$, then we have

$J=\mathcal{J}_{r}\bigl{(}I_{M};\operatorname{Der}_{k[t]}(S)\bigr{)}\subset\mathcal{J}_{r}\bigl{(}I_{M};\operatorname{Der}_{k}(S)\bigr{)}\subset\mathfrak{p}$

by $\operatorname{ht}(I_{M}S_{\mathfrak{p}})\leq r$ and the Jacobian criterion
of regularity (Remark 2.6(2)(a)(b)).

Suppose the contrary that $M$ is regular at $\mathfrak{p}$. Since any minimal
prime $P$ of $I_{X}$ satisfies $\operatorname{ht}P\leq r$ and $a_{ij}\in k$
are general, for any irreducible component $X_{0}$ of $X$, there exists an
irreducible component $M_{0}$ of $M$ such that $X_{0}\subset M_{0}$ and
$(X_{0})_{\rm red}=(M_{0})_{\rm red}$. Therefore, since $I_{M}\subset
I_{X}\subset\mathfrak{p}$ and $M$ is regular at $\mathfrak{p}$, we have
$(I_{M})_{\mathfrak{p}}=(I_{X})_{\mathfrak{p}}$. It shows that

$(I_{X^{\prime}})_{\mathfrak{p}}=(I_{M}:I_{X})_{\mathfrak{p}}=\bigl{(}(I_{M})_{\mathfrak{p}}:(I_{X})_{\mathfrak{p}}\bigr{)}=S_{\mathfrak{p}},$

which contradicts $I_{X^{\prime}}\subset\mathfrak{p}$. We complete the proof
of ($\spadesuit$). ∎

For an $R$-scheme $X$, a subset $C\subset X_{\infty}$ is called a cylinder if
$C=\psi_{m}^{-1}(S)$ holds for some $m\geq 0$ and a constructible subset
$S\subset X_{m}$.

######  Proposition 5.10.

Let $n$ be a non-negative integer and let $X$ be a scheme of finite type over
$R=k[t][[x_{1},\ldots,x_{N}]]$. Suppose that each irreducible component
$X_{i}$ of $X$ has $\dim^{\prime}X_{i}\geq n+1$. Let $C$ be a cylinder in
$X_{\infty}$ which is contained in $\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{X})\cap\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}$
for some $e\geq 0$. Then its image $\psi_{m}(C)\subset X_{m}$ is a
constructible subset for any $m\geq 0$.

###### Proof.

Let $S\subset X_{\ell}$ be a constructible subset such that
$\psi^{-1}_{\ell}(S)=C$. For $m\geq\ell$, we have

$\pi_{m,\ell}^{-1}(S)\cap\psi_{m}(C)=\psi_{m}(C)=\pi_{m,\ell}^{-1}(S)\cap\psi_{m}(X_{\infty}).$

By the assumption $C\subset\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{X})\cap\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}$,
we also have

$\psi_{m}(C)=\pi_{m,\ell}^{-1}(S)\cap\psi_{m}\bigl{(}\operatorname{Cont}^{\geq
1}(\mathfrak{o}_{X})\cap\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}\bigr{)}.$

Let $c$ be the positive integer appearing in Proposition 5.9. Then the
constructibility of $\psi_{m}(C)$ follows from Proposition 5.9(1) if
$m\geq\max\\{ce,\ell\\}$. When $m<\max\\{ce,\ell\\}$, the constructibility
follows from that for $m=\max\\{ce,\ell\\}$ and Chevalley’s theorem. ∎

We define the codimensions of cylinders $C$ when they satisfy
$C\subset\operatorname{Cont}^{\geq 1}(\mathfrak{o}_{X})$.

######  Definition 5.11.

Let $n$ be a non-negative integer and let $X$ be a scheme of finite type over
$R=k[t][[x_{1},\ldots,x_{N}]]$. Suppose that each irreducible component
$X_{i}$ of $X$ has $\dim^{\prime}X_{i}\geq n+1$. Let $C\subset X_{\infty}$ be
a cylinder contained in $\operatorname{Cont}^{\geq 1}(\mathfrak{o}_{X})$.

  1. (1)

Assume that
$C\subset\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}$
for some $e\in\mathbb{Z}_{\geq 0}$. Then we define the codimension of $C$ in
$X_{\infty}$ as

$\operatorname{codim}(C):=(m+1)n-\operatorname{dim}(\psi_{m}(C))$

for any sufficiently large $m$. This definition is well-defined by Proposition
5.9(2).

  2. (2)

In general, we define the codimension of $C$ in $X_{\infty}$ as follows:

$\operatorname{codim}(C):=\min_{e\in\mathbb{Z}_{\geq
0}}{\operatorname{codim}\bigl{(}C\cap\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}\bigr{)}}.$

By convention, $\operatorname{codim}(C)=\infty$ if
$C\cap\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}=\emptyset$
for any $e\geq 0$.

######  Remark 5.12.

The definition of the codimension above depends on the choice of $n$. In
Subsection 5.1.3, we fix a non-negative integer $n$, and we use the
codimension defined for this $n$.

######  Lemma 5.13.

Let $X$ be a scheme of finite type over $R=k[t][[x_{1},\ldots,x_{N}]]$. Let
$p$ and $m$ be non-negative integers with $2p+1\geq m\geq p$. Let $\gamma\in
X_{p}(k)$ be a jet. If $\pi_{m,p}^{-1}(\gamma)\not=\emptyset$, it follows that

$\pi_{m,p}^{-1}(\gamma)\simeq\operatorname{Hom}_{k[t]/(t^{p+1})}\bigl{(}\gamma^{*}\Omega^{\prime}_{X/k[t]},(t^{p+1})/(t^{m+1})\bigr{)}.$

###### Proof.

We may assume that $X$ is affine, and we may write $X=\operatorname{Spec}A$
with an $R$-algebra $A$ of finite type. Let $\gamma^{*}:A\to k[t]/(t^{p+1})$
be the corresponding $k[t]$-ring homomorphism to $\gamma$. Take any
$\alpha\in\pi_{m,p}^{-1}(\gamma)$. Let $\alpha^{*}:A\to k[t]/(t^{m+1})$ be the
corresponding $k[t]$-ring homomorphism. Then for the same reason as in the
case of $k$-schemes (cf. [EM09, Proposition 4.4]), we have an isomorphism

$\pi_{m,p}^{-1}(\gamma)\simeq\operatorname{Der}_{k[t]}\left(A,(t^{p+1})/(t^{m+1})\right);\quad\beta\mapsto\beta^{*}-\alpha^{*}.$

Here, $(t^{p+1})/(t^{m+1})$ in the right-hand side has an $A$-module structure
via $\gamma^{*}$. Then the assertion follows from the isomorphisms

$\displaystyle\operatorname{Der}_{k[t]}\left(A,(t^{p+1})/(t^{m+1})\right)$
$\displaystyle=\operatorname{Der}^{\prime}_{k[t]}\left(A,(t^{p+1})/(t^{m+1})\right)$
$\displaystyle\simeq\operatorname{Hom}_{A}\bigl{(}\Omega^{\prime}_{A/k[t]},(t^{p+1})/(t^{m+1})\bigr{)}.$

Here, the first equality follows from Lemma 2.2. ∎

####  5.1.3. Thin and very thin cylinders

We fix a non-negative integer $n$ throughout this subsection.

######  Definition 5.14.

Let $X$ be a scheme of finite type over $R=k[t][[x_{1},\ldots,x_{N}]]$.
Suppose that each irreducible component $X_{i}$ of $X$ has
$\dim^{\prime}X_{i}\geq n+1$. A subset $A\subset X_{\infty}$ is called thin if
$A\subset Z_{\infty}$ holds for some closed subscheme $Z$ of $X$ with $\dim
Z\leq n$. $A$ is called very thin if $A\subset Z_{\infty}$ holds for some
closed subscheme $Z$ of $X$ with $\dim Z\leq n-1$.

The term “very thin” is used only in this paper. In Question 5.15(1) and
Remark 5.16(1), we shall explain the motivation to introduce this terminology.

######  Question 5.15.

Let $R$ and $X$ be as in Definition 5.14.

  1. (1)

Suppose that $C$ is a thin cylinder of $X_{\infty}$. Then, does
$C\cap\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}=\emptyset$
hold for any $e\geq 0$?

  2. (2)

Suppose that $X$ is an integral scheme and $Y\subset X$ is the closed
subscheme defined by the ideal
$\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})$. Then, is $Y_{\infty}$ a
thin set of $X_{\infty}$?

  3. (3)

Let $S=k[t][[x_{1},\ldots,x_{N}]][y_{1},\ldots,y_{m}]$, and let $P$ be a prime
ideal of $S$ of height $r$. Suppose that $P\cap k[t]=(0)$. Then, does
$\mathcal{J}_{r}\bigl{(}P;\operatorname{Der}_{k[t]}(S)\bigr{)}\not\subset P$
hold?

######  Remark 5.16.

  1. (1)

Note that Question 5.15(1) is true for the arc spaces of $k$-varieties $X$:

     * •

If $C$ is a thin cylinder of $X_{\infty}$, then
$C\cap\operatorname{Cont}^{e}(\operatorname{Jac}_{X/k})=\emptyset$ holds for
any $e\geq 0$ (cf. [EM09, Lemma 5.1]).

The same statement is true for schemes $X$ of finite type over
$k[[x_{1},\ldots,x_{N}]]$ by replacing $\operatorname{Jac}_{X/k}$ with
$\operatorname{Jac}^{\prime}_{X/k}$. Furthermore, Question 5.15(1) is true
also for schemes $X$ of finite type over $k[t]$ (cf. [NS, Lemma 2.23]).
However, the same proofs do not work for schemes $X$ of finite type over
$R=k[t][[x_{1},\ldots,x_{N}]]$, and hence it is not clear to us whether
Question 5.15(1) is true for this setting (see also the discussion in (3)
below). This is why we introduce the term “very thin” and we will prove weaker
statements instead in Lemma 5.18 for very thin cylinders and Proposition 5.23
for $X$ with an additional assumption.

  2. (2)

Due to the proof of [EM09, Lemma 5.1], Question 5.15(1) can be reduced to
Question 5.15(2) by the Noetherian induction on dimension. Furthermore,
Question 5.15(3) implies Question 5.15(2).

  3. (3)

Question 5.15(3) is related to the weak Jacobian condition (WJ) explained in
Remark 2.6(2). Indeed, if $N=0$, then Question 5.15(3) can be proved using
Remark 2.6(2)(c) as follows. We denote

$S^{\prime}:=(k[t]\setminus\\{0\\})^{-1}S=k(t)[y_{1},\ldots,y_{m}]$

the localization. Then by the assumption $P\cap k[t]=(0)$, we have
$PS^{\prime}\not=S^{\prime}$ and hence $PS^{\prime}$ is a prime ideal of
height $r$. Since $S^{\prime}$ satisfies (WJ)k(t), we have

$\mathcal{J}_{r}\bigl{(}P;\operatorname{Der}_{k[t]}(S)\bigr{)}S^{\prime}+PS^{\prime}=\mathcal{J}_{r}\bigl{(}PS^{\prime};\operatorname{Der}_{k(t)}(S^{\prime})\bigr{)}+PS^{\prime}\not\subset
PS^{\prime},$

which proves
$\mathcal{J}_{r}\bigl{(}P;\operatorname{Der}_{k[t]}(S)\bigr{)}\not\subset P$.
Note that the same proof does not work when $N>0$ because we have

$S^{\prime}:=(k[t]\setminus\\{0\\})^{-1}S\not=k(t)[[x_{1},\ldots,x_{N}]][y_{1},\ldots,y_{m}],$

and it is not clear whether
$\mathcal{J}_{r}\bigl{(}PS^{\prime};\Delta\bigr{)}\not\subset PS^{\prime}$
holds for $\Delta=\bigl{\\{}\partial/\partial x_{i},\partial/\partial y_{j}\
\big{|}\ i,j\bigr{\\}}$.

  4. (4)

Question 5.15(1) is also true for the arc spaces (Greenberg schemes) of formal
$k[[t]]$-schemes of finite type, which will be dealt with in Subsection 5.2
(see [CLNS, Ch.6. Proposition 2.4.3]). Actually, Question 5.15(3) is true for
this setting:

     * •

Let $S=k[x_{1},\ldots,x_{N}][[t]]$, and let $P$ be a prime ideal of $S$ of
height $r$. Suppose that $P\cap k[[t]]=(0)$. Then we have
$\mathcal{J}_{r}\bigl{(}P;\operatorname{Der}_{k[[t]]}(S)\bigr{)}\not\subset
P$.

We denote

$S^{\prime}:=S_{t}=k[x_{1},\ldots,x_{N}]((t))$

the localization. We note that the assumption $P\cap k[[t]]=(0)$ is equivalent
to $t\not\in P$, and hence $PS^{\prime}$ is a prime ideal of height $r$. Since
$S^{\prime}$ satisfies (WJ)k((t)) by [Nag62, Theorem 46.3], the same proof as
in (3) above works and we have
$\mathcal{J}_{r}\bigl{(}P;\operatorname{Der}_{k[[t]]}(S)\bigr{)}\not\subset
P$. Note here that both $\operatorname{Der}_{k[[t]]}(S)$ and
$\operatorname{Der}_{k((t))}(S^{\prime})$ are generated by $\partial/\partial
x_{i}$’s.

In Lemmas 5.17 and 5.18 below, for a scheme $X$ over
$R=k[t][[x_{1},\ldots,x_{N}]]$, we also consider the jet schemes and the arc
space in the sense of Section 4. To avoid confusion, we denote them by
$\mathcal{L}_{m}(X)$ and $\mathcal{L}_{\infty}(X)$, that is,
$\mathcal{L}_{m}(X)$ is the scheme representing the functor

$F_{m}:({\sf Sch}/k)\to({\sf Set});\quad
Y\mapsto\operatorname{Hom}_{k}\left(Y\times_{\operatorname{Spec}k}\operatorname{Spec}k[t]/(t^{m+1}),X\right)$

and
$\mathcal{L}_{\infty}(X)=\mathop{\varprojlim}\limits_{m}\mathcal{L}_{m}(X)$ is
the projective limit.

######  Lemma 5.17.

Let $X$ be a scheme of finite type over $R=k[t][[x_{1},\ldots,x_{N}]]$. Then
the following hold.

  1. (1)

There exist natural closed immersions $X_{m}\to\mathcal{L}_{m}(X)$ for $m\geq
0$ which commute with the truncation morphisms.

  2. (2)

Let $x\in X$ be a $k$-point and let $\widehat{\mathcal{O}}_{X,x}$ be the
completion of the local ring $\mathcal{O}_{X,x}$ at its maximal ideal. Let
$X^{\prime}:=\operatorname{Spec}(\widehat{\mathcal{O}}_{X,x})$ and let
$x^{\prime}\in X^{\prime}$ be the corresponding $k$-point. Then for the
truncation morphisms $\pi^{X}_{m}:\mathcal{L}_{m}(X)\to X$ and
$\pi^{X^{\prime}}_{m}:\mathcal{L}_{m}(X^{\prime})\to X^{\prime}$, we have
$\bigl{(}\pi^{X}_{m}\bigr{)}^{-1}(x)\simeq\bigl{(}\pi^{X^{\prime}}_{m}\bigr{)}^{-1}(x^{\prime})$.

###### Proof.

We may assume that $X$ is affine, and we may write
$X=\operatorname{Spec}(S/I)$ with

$S:=k[t][[x_{1},\ldots,x_{N}]][y_{1},\ldots,y_{M}]$

and $I:=(f_{1},\ldots,f_{r})$ an ideal of $S$. We set
$A:=\operatorname{Spec}S$.

Then, by the same argument as in Lemma 5.1, we have
$\mathcal{L}_{m}(A)\simeq\operatorname{Spec}T_{m}$, where

$T_{m}:=k\bigl{[}u^{(0)}\bigr{]}\bigl{[}\bigl{[}x_{1}^{(0)},\ldots,x_{N}^{(0)}\bigr{]}\bigr{]}\biggl{[}u^{(s)},x_{j}^{(s)},y_{j^{\prime}}^{(s^{\prime})}\
\bigg{|}\ \begin{array}[]{l}1\leq j\leq N,\ 1\leq j^{\prime}\leq M,\\\ 1\leq
s\leq m,\ 0\leq s^{\prime}\leq m\end{array}\biggr{]}.$

In the same way as in Lemma 5.1, we can define a ring homomorphism
$\Lambda^{\prime}:S\to T_{m}[t]/(t^{m+1})$ that satisfies

  * •

$\Lambda^{\prime}(t)=u^{(0)}+u^{(1)}t+\cdots+u^{(m)}t^{m}$,

  * •

$\Lambda^{\prime}(x_{j})=x_{j}^{(0)}+x_{j}^{(1)}t+\cdots+x_{j}^{(m)}t^{m}$ for
each $0\leq j\leq N$, and

  * •

$\Lambda^{\prime}(y_{j})=y_{j}^{(0)}+y_{j}^{(1)}t+\cdots+y_{j}^{(m)}t^{m}$ for
each $0\leq j\leq M$.

For $1\leq i\leq r$ and $0\leq\ell\leq m$, we define $G_{i}^{(\ell)}\in T_{m}$
as

$\Lambda^{\prime}(f_{i})=\sum_{\ell=0}^{m}G_{i}^{(\ell)}t^{\ell}\pmod{t^{m+1}}.$

Let

$J_{m}:=\bigl{(}G_{i}^{(s)}\ \big{|}\ 1\leq i\leq r,\ 0\leq s\leq
m\bigr{)}\subset T_{m}$

be the ideal generated by $G^{(s)}_{i}$’s. Then by the same argument as in
Lemma 5.2, we have $\mathcal{L}_{m}(X)\simeq\operatorname{Spec}(T_{m}/J_{m})$.

Let $S_{m}$ be the ring defined in Lemma 5.1. Let $\Xi:T_{m}\to S_{m}$ be a
surjective ring homomorphism defined by

  * •

$\Xi(u^{(1)})=1$, and $\Xi(u^{(s)})=0$ for each $s=0$ and $2\leq s\leq m$.

  * •

$\Xi(x^{(s)}_{j})=x^{(s)}_{j}$ for each $1\leq j\leq N$ and $0\leq s\leq m$.

  * •

$\Xi(y^{(s)}_{j})=y^{(s)}_{j}$ for each $1\leq j\leq M$ and $0\leq s\leq m$.

We note that $\Lambda:S\to S_{m}[t]/(t^{m+1})$ defined in the proof of Lemma
5.1 coincides with the composition

$S\xrightarrow{\Lambda^{\prime}}T_{m}[t]/(t^{m+1})\xrightarrow{\Xi}S_{m}[t]/(t^{m+1}),$

where $T_{m}[t]/(t^{m+1})\to S_{m}[t]/(t^{m+1})$ is the $k[t]$-ring
homomorphism induced by $\Xi$. Therefore, $F^{(s)}_{i}$ in Lemma 5.2 coincides
with $\Xi(G^{(s)}_{i})$ for each $i$ and $s$.

Let $I_{m}\subset S_{m}$ be the ideal defined in Lemma 5.2. Then $\Xi$ induces
a surjective ring homomorphism $T_{m}/J_{m}\to S_{m}/I_{m}$. It gives a closed
immersion

$X_{m}\simeq\operatorname{Spec}(S_{m}/I_{m})\hookrightarrow\operatorname{Spec}(T_{m}/J_{m})\simeq\mathcal{L}_{m}(X),$

which completes the proof of (1).

Since $x\in X$ is a $k$-point, the corresponding maximal ideal of $S$ is of
the form

$\bigl{(}t-a,x_{1},\ldots,x_{N},y_{1}-b_{1},\ldots,y_{M}-b_{M}\bigr{)}$

with $a,b_{1},\ldots,b_{M}\in k$. For simplicity, we assume that
$a=b_{1}=\cdots=b_{M}=0$. Then
$\bigl{(}\pi^{X}_{m}\bigr{)}^{-1}(x)\subset\mathcal{L}_{m}(X)$ is isomorphic
to the closed subscheme of $\operatorname{Spec}T_{m}$ defined by

$J_{m}+\bigl{(}u^{(0)},x_{1}^{(0)},\ldots,x_{N}^{(0)},y_{1}^{(0)},\ldots,y_{M}^{(0)}\bigr{)}.$

On the other hand, we have

$\widehat{\mathcal{O}}_{A,x}\simeq
k[[t,x_{1},\ldots,x_{N},y_{1},\ldots,y_{M}]].$

We set $A^{\prime}:=\operatorname{Spec}(\widehat{\mathcal{O}}_{A,x})$ and

$T^{\prime}_{m}:=k\bigl{[}\bigl{[}u^{(0)},x_{1}^{(0)},\ldots,x_{N}^{(0)},y_{1}^{(0)},\ldots,y_{M}^{(0)}\bigr{]}\bigr{]}\left[u^{(s)},x_{j}^{(s)},y_{j^{\prime}}^{(s)}\
\middle|\ \begin{array}[]{l}1\leq j\leq N,\\\ 1\leq j^{\prime}\leq M,\\\ 1\leq
s\leq m\end{array}\right].$

Then by [Ish09, Proposition 4.1], we have
$\mathcal{L}_{m}(A^{\prime})\simeq\operatorname{Spec}T^{\prime}_{m}$.
Furthermore, by [Ish09, Corollary 4.2], we have
$\mathcal{L}_{m}(X^{\prime})\simeq\operatorname{Spec}(T^{\prime}_{m}/J_{m}T^{\prime}_{m})$.
Therefoere,
$\bigl{(}\pi^{X^{\prime}}_{m}\bigr{)}^{-1}(x^{\prime})\subset\mathcal{L}_{m}(X^{\prime})$
is isomorphic to the closed subscheme of $\operatorname{Spec}T^{\prime}_{m}$
defined by

$J_{m}T^{\prime}_{m}+\bigl{(}u^{(0)},x_{1}^{(0)},\ldots,x_{N}^{(0)},y_{1}^{(0)},\ldots,y_{M}^{(0)}\bigr{)}.$

Therefore we have
$\bigl{(}\pi^{X}_{m}\bigr{)}^{-1}(x)\simeq\bigl{(}\pi^{X^{\prime}}_{m}\bigr{)}^{-1}(x^{\prime})$,
which completes the proof of (2). ∎

######  Lemma 5.18.

Let $X$ be a scheme of finite type over $R=k[t][[x_{1},\ldots,x_{N}]]$.
Suppose that each irreducible component $X_{i}$ of $X$ has
$\dim^{\prime}X_{i}\geq n+1$. Let $C\subset X_{\infty}$ be a cylinder
contained in $\operatorname{Cont}^{\geq 1}(\mathfrak{o}_{X})$. If $C$ is very
thin, then
$C\cap\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}=\emptyset$
holds for any $e\geq 0$.

###### Proof.

Suppose the contrary that
$C\cap\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}\not=\emptyset$
for some $e\geq 0$. By replacing $C$ with
$C\cap\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}$,
we may assume that
$\emptyset\not=C\subset\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}$.
Pick a $k$-arc $\gamma\in C$. Let $x:=\pi^{X}_{\infty}(\gamma)\in X$ be the
$k$-point of $X$. Then, by replacing $C$ with
$C\cap(\pi^{X}_{\infty})^{-1}(x)$, we may assume that
$C\subset(\pi^{X}_{\infty})^{-1}(x)$.

Since $C$ is a very thin set, there exists a closed subscheme $Z\subset X$
such that $C\subset Z_{\infty}$ and $\dim Z\leq n-1$. Since $\gamma\in
Z_{\infty}$, it follows that $x\in Z$. Let $\widehat{\mathcal{O}}_{Z,x}$ be
the completion of the local ring $\mathcal{O}_{Z,x}$ at its maximal ideal. Let
$Z^{\prime}:=\operatorname{Spec}(\widehat{\mathcal{O}}_{Z,x})$, and let
$x^{\prime}\in Z^{\prime}$ be the corresponding $k$-point. Then, since
$C\subset(\pi_{\infty}^{Z})^{-1}(x)$, we may identify $C$ with a subset of
$\mathcal{L}_{\infty}(Z^{\prime})$ by Lemma 5.17(1)(2). Note that
$C\subset\mathcal{L}_{\infty}(Z^{\prime})$ is not necessarily a cylinder of
$\mathcal{L}_{\infty}(Z^{\prime})$ under this identification.

Let $\mathcal{S}$ be the set of the closed subschemes $Y^{\prime}$ of
$Z^{\prime}$ with the following condition:

  * •

There exists a cylinder $C^{\prime}$ of $X_{\infty}$ such that
$\emptyset\not=C^{\prime}\subset C$ and
$C^{\prime}\subset\mathcal{L}_{\infty}(Y^{\prime})$.

Here, the inclusion $C^{\prime}\subset\mathcal{L}_{\infty}(Y^{\prime})$ is
considered by the identifications
$\mathcal{L}_{\infty}(Y^{\prime})\subset\mathcal{L}_{\infty}(Z^{\prime})$ and
$C^{\prime}\subset C\subset\mathcal{L}_{\infty}(Z^{\prime})$. Let $Y^{\prime}$
be a minimal element of $\mathcal{S}$, and let $C^{\prime}$ be a corresponding
cylinder of $X_{\infty}$. Then $Y^{\prime}$ is reduced by the minimality.

We shall prove that $Y^{\prime}$ is irreducible. Suppose the contrary that
$Y^{\prime}=Y^{\prime}_{1}\cup\cdots\cup Y^{\prime}_{\ell}$ is the irreducible
decomposition with $\ell\geq 2$. By the minimality of $Y^{\prime}$, it follows
that $C^{\prime}\not\subset\mathcal{L}_{\infty}(Y^{\prime}_{1})$ and hence we
have

$C^{\prime\prime}:=C^{\prime}\cap\bigl{(}(\psi^{Z^{\prime}}_{q})^{-1}(\mathcal{L}_{q}(Y^{\prime}_{1}))\setminus(\psi^{Z^{\prime}}_{q+1})^{-1}(\mathcal{L}_{q+1}(Y^{\prime}_{1}))\bigr{)}\not=\emptyset$

for some $q\geq-1$, where we set
$(\psi^{Z^{\prime}}_{q})^{-1}(\mathcal{L}_{q}(Y^{\prime}_{1}))=\mathcal{L}_{\infty}(Z^{\prime})$
for $q=-1$ by abuse of notation. Here, we have taken the intersection in the
space $\mathcal{L}_{\infty}(Z^{\prime})$. Since
$C^{\prime\prime}\cap\mathcal{L}_{\infty}(Y^{\prime}_{1})=\emptyset$, we have

$C^{\prime\prime}\subset
C\setminus\mathcal{L}_{\infty}(Y^{\prime}_{1})\subset\mathcal{L}_{\infty}(Y^{\prime})\setminus\mathcal{L}_{\infty}(Y^{\prime}_{1})\subset\mathcal{L}_{\infty}(Y^{\prime}_{2}\cup\cdots\cup
Y^{\prime}_{\ell}).$

To get a contradiction by the minimality of $Y^{\prime}$, it is sufficient to
show that $C^{\prime\prime}$ is a cylinder of $X_{\infty}$. For this purpose,
we shall see that

$C^{\prime}_{q}:=C^{\prime}\cap(\psi^{Z^{\prime}}_{q})^{-1}(\mathcal{L}_{q}(Y^{\prime}_{1}))$

is a cylinder of $X_{\infty}$. Under the following identifications

$\textstyle{X_{\infty}\cap(\pi^{X}_{\infty})^{-1}(x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{X}_{q}}$$\textstyle{X_{q}\cap(\pi^{X}_{q})^{-1}(x)}$$\textstyle{Z_{\infty}\cap(\pi^{Z}_{\infty})^{-1}(x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\subset$
$\textstyle{Z_{q}\cap(\pi^{Z}_{q})^{-1}(x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\subset$
$\textstyle{\mathcal{L}_{\infty}(Z)\cap(\pi^{Z}_{\infty})^{-1}(x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\subset$
$\textstyle{\mathcal{L}_{q}(Z)\cap(\pi^{Z}_{q})^{-1}(x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\subset$ $\textstyle{C^{\prime}\ \
\subset}$$\textstyle{\mathcal{L}_{\infty}(Z^{\prime})\cap(\pi^{Z^{\prime}}_{\infty})^{-1}(x^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{Z^{\prime}}_{q}}$$\scriptstyle{\simeq}$$\textstyle{\mathcal{L}_{q}(Z^{\prime})\cap(\pi^{Z^{\prime}}_{q})^{-1}(x^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\mathcal{L}_{\infty}(Y^{\prime}_{1})\cap(\pi^{Y^{\prime}_{1}}_{\infty})^{-1}(x^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\subset$
$\textstyle{\mathcal{L}_{q}(Y^{\prime}_{1})\cap(\pi^{Y^{\prime}_{1}}_{q})^{-1}(x^{\prime}),\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\subset$

we can consider the intersection $F=\mathcal{L}_{q}(Y^{\prime}_{1})\cap
Z_{q}\cap(\pi^{Z}_{q})^{-1}(x)$ and it can be identified with a closed subset
of $X_{q}\cap(\pi^{X}_{q})^{-1}(x)$. Then we have
$C^{\prime}_{q}=C^{\prime}\cap(\psi_{q}^{X})^{-1}(F)$ since $C^{\prime}\subset
Z_{\infty}$. Therefore, $C^{\prime}_{q}$ turns out to be a cylinder of
$X_{\infty}$, and hence so is $C^{\prime\prime}=C^{\prime}_{q}\setminus
C^{\prime}_{q+1}$. We have proved that $Y^{\prime}$ is integral.

Let $Y^{\prime\prime}\subset Y^{\prime}$ be the subscheme defined by
$\operatorname{Jac}^{\prime}_{Y^{\prime}/k}$. Since $Y^{\prime}$ is reduced,
we have $Y^{\prime\prime}\subsetneq Y^{\prime}$ by the Jacobian criterion of
regularity (cf. Remark 2.6(2)(c)). By the minimality of $Y^{\prime}$, we have
$C^{\prime}\not\subset Y^{\prime\prime}_{\infty}$ and hence
$C^{\prime}\cap\operatorname{Cont}^{e^{\prime}}\bigl{(}\operatorname{Jac}^{\prime}_{Y^{\prime}/k}\bigr{)}\not=\emptyset$
holds for some $e^{\prime}\geq 0$. Take a $k$-arc $\beta\in
C^{\prime}\cap\operatorname{Cont}^{e^{\prime}}\bigl{(}\operatorname{Jac}^{\prime}_{Y^{\prime}/k}\bigr{)}$.
For $m\geq 0$, we set

$D_{m,\beta}:=(\psi^{Y^{\prime}}_{m})^{-1}(\psi^{Y^{\prime}}_{m}(\beta))\subset\mathcal{L}_{\infty}(Y^{\prime}),\quad
E_{m,\beta}:=(\psi^{X}_{m})^{-1}(\psi^{X}_{m}(\beta))\subset X_{\infty}.$

Then by applying Proposition 4.8(2) to the map

$\psi^{Y^{\prime}}_{m+1}(D_{m,\beta})\to\psi^{Y^{\prime}}_{m}(D_{m,\beta})=\bigl{\\{}\psi^{Y^{\prime}}_{m}(\beta)\bigr{\\}},$

we have

$\dim\bigl{(}\psi^{Y^{\prime}}_{m+1}(D_{m,\beta})\bigr{)}=\dim Y^{\prime}$

for sufficiently large $m$. On the other hand, by applying Proposition 5.9(2)
to the map

$\psi^{X}_{m+1}(E_{m,\beta})\to\psi^{X}_{m}(E_{m,\beta})=\bigl{\\{}\psi^{X}_{m}(\beta)\bigr{\\}},$

we have

$\dim\bigl{(}\psi^{X}_{m+1}(E_{m,\beta}))\bigr{)}=n$

for sufficiently large $m$.

Since $\dim Y^{\prime}\leq\dim Z\leq n-1$, to get a contradiction, it is
enough to show $E_{m,\beta}\subset D_{m,\beta}$ for sufficiently large $m$.
Since $C^{\prime}$ is a cylinder of $X_{\infty}$, there exists a constructible
subset $V\subset X_{p}$ for some $p\geq 0$ such that
$C^{\prime}=(\psi_{p}^{X})^{-1}(V)$. We shall prove the inclusion
$E_{m,\beta}\subset D_{m,\beta}$ for any $m\geq p$.

Let $X^{\prime}:=\operatorname{Spec}(\widehat{\mathcal{O}}_{X,x})$. Then by
Lemma 5.17(1)(2), we have the following diagram:

$\textstyle{X_{\infty}\cap(\pi^{X}_{\infty})^{-1}(x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{X}_{m}}$$\subset$$\textstyle{\mathcal{L}_{\infty}(X^{\prime})\cap(\pi^{X^{\prime}}_{\infty})^{-1}(x^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{X^{\prime}}_{m}}$$\textstyle{\mathcal{L}_{\infty}(Y^{\prime})\cap(\pi^{Y^{\prime}}_{\infty})^{-1}(x^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{Y^{\prime}}_{m}}$$\supset$$\textstyle{X_{m}\cap(\pi^{X}_{m})^{-1}(x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\subset$$\textstyle{\mathcal{L}_{m}(X^{\prime})\cap(\pi^{X^{\prime}}_{m})^{-1}(x^{\prime})}$$\textstyle{\mathcal{L}_{m}(Y^{\prime})\cap(\pi^{Y^{\prime}}_{m})^{-1}(x^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\supset$

Let $\beta_{m}:=\psi^{X}_{m}(\beta)$. For $m\geq p$, we have

$E_{m,\beta}=(\psi^{X}_{m})^{-1}(\beta_{m})=(\psi^{X^{\prime}}_{m})^{-1}(\beta_{m})\cap
X_{\infty}\cap(\pi^{X}_{\infty})^{-1}(x)=(\psi^{X^{\prime}}_{m})^{-1}(\beta_{m})\cap
C^{\prime}.$

On the other hand, we have

$D_{m,\beta}=(\psi^{Y^{\prime}}_{m})^{-1}(\beta_{m})=(\psi^{X^{\prime}}_{m})^{-1}(\beta_{m})\cap\mathcal{L}_{\infty}(Y^{\prime})\cap(\pi^{Y^{\prime}}_{m})^{-1}(x^{\prime}).$

Since
$C^{\prime}\subset\mathcal{L}_{\infty}(Y^{\prime})\cap(\pi^{Y^{\prime}}_{m})^{-1}(x^{\prime})$,
we have $E_{m,\beta}\subset D_{m,\beta}$ for $m\geq p$. We complete the proof.
∎

We prove a much weaker version of [NS, Lemma 2.26(1)].

######  Lemma 5.19.

Let $X$ be a scheme of finite type over $R=k[t][[x_{1},\ldots,x_{N}]]$. For
each $a\in k$, we denote by $X_{a}$ the closed subscheme of $X$ defined by
$(t-a)\mathcal{O}_{X}$. Suppose for any $a\in k^{\times}$ that $X_{a}$ is an
integral regular scheme and has $\dim^{\prime}X_{a}=n$. Then there exists a
positive integer $\ell$ such that
$\operatorname{ord}_{\gamma}\bigl{(}\mathfrak{o}_{X}+\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}\leq\ell$
holds for any $k$-arc $\gamma\in X_{\infty}$. In particular, if $\gamma$
satisfies $\operatorname{ord}_{\gamma}(\mathfrak{o}_{X})=\infty$, then
$\operatorname{ord}_{\gamma}\bigr{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}\leq\ell$.

###### Proof.

We may assume that $X$ is affine, and we may write
$X=\operatorname{Spec}(S/I)$, where

$S:=k[t][[x_{1},\ldots,x_{N}]][y_{1},\ldots,y_{m}]$

and $I$ is an ideal of $S$. We set $I_{a}:=(I+(t-a))/(t-a)$, which is an ideal
of the ring $S/(t-a)\simeq k[[x_{1},\ldots,x_{N}]][y_{1},\ldots,y_{m}]$. We
have $\operatorname{ht}(I_{a})=N+m-n$ for any $a\in k^{\times}$ since
$\dim^{\prime}X_{a}=n$.

We set

$J:=\mathcal{J}_{N+m-n}\bigl{(}I;\operatorname{Der}_{k[t]}(S)\bigr{)}\subset
S.$

Note that $\operatorname{Der}_{k[t]}(S)=\operatorname{Der}^{\prime}_{k[t]}(S)$
is generated by $\frac{\partial}{\partial x_{i}}$’s and
$\frac{\partial}{\partial y_{i}}$’s. Then we have
$\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})=(J+I)/I$ by Remark 2.14.
Let $a\in k^{\times}$. Since $\operatorname{ht}(I_{a})=N+m-n$, we have

$\operatorname{Jac}^{\prime}_{X_{a}/k}=(J+I+(t-a))/(I+(t-a)).$

Since $X_{a}$ is regular, we have

$J+I+(t-a)=S$

by the Jacobian criterion of regularity (cf. Remark 2.6(2)(c)). Therefore, for
any $a\in k^{\times}$, we have

$(J+I+(t-a))S^{\prime}=S^{\prime},$

where $S^{\prime}:=S/(x_{1},\ldots,x_{N})\simeq k[t][y_{1},\ldots,y_{m}]$.
Then by Hilbert’s nullstellensatz, we have $t^{\ell}\in
J+I+(x_{1},\ldots,x_{N})$ for some $\ell\geq 0$, which proves the assertions.
∎

######  Lemma 5.20.

Let $P$ be a prime ideal of $S=k[[x_{1},\ldots,x_{N}]]$ of height $r$, and let
$I$ be an ideal of $S$ satisfying $P\subsetneq I$. If $S/P$ is regular, then
$\mathcal{J}_{r+1}(I;\operatorname{Der}_{k}(S))\not\subset I$.

###### Proof.

Note that $\operatorname{Der}_{k}(S)=\operatorname{Der}^{\prime}_{k}(S)$ is
generated by $\partial/\partial x_{i}$’s. First, we prove the assertion when
$r=0$. Let $f\in I\setminus\\{0\\}$ be an element with the minimum order $a$.
Suppose that $x_{i}$ appears in the lowest order term of $f$. Then it follows
from the minimality of $a$ that $\frac{\partial f}{\partial x_{i}}\not\in I$,
which proves the assetion when $r=0$.

Suppose $r>0$. Since $S/P$ is regular, by the Jacobian criterion of regularity
(cf. Remark 2.6(2)(c)), there exist
$D_{1},\ldots,D_{r}\in\operatorname{Der}_{k}(S)$ and $f_{1},\ldots,f_{r}\in P$
such that

$u:=\det(D_{i}(f_{j}))_{1\leq i,j\leq r}\not\in(x_{1},\ldots,x_{N}).$

Since $S/P$ is a complete regular local ring with the coefficient field $k$,
$S/P$ is isomorphic to $k[[y_{1},\ldots,y_{N-r}]]$. Therefore, by what we have
already proved, there exist $D^{\prime}\in\operatorname{Der}_{k}(S/P)$ and
$f^{\prime}\in I/P$ such that $D^{\prime}(f^{\prime})\not\in I/P$. Let
$f_{r+1}\in I$ be a lift of $f^{\prime}$. By [Mat89, Theorem 30.8], there
exists a lift $D_{r+1}\in\operatorname{Der}_{k}(S)$ of $D^{\prime}$ too. Since
$D_{r+1}(P)=0$, we have

$\det(D_{i}(f_{j}))_{1\leq i,j\leq r+1}=uD_{r+1}(f_{r+1})\not\in I,$

which shows that $\mathcal{J}_{r+1}(I;\operatorname{Der}_{k}(S))\not\subset
I$. ∎

######  Remark 5.21.

We are interested in the case where $\operatorname{ht}I=r+1$. If $I$ is a
prime ideal, then it is true more generally that
$\mathcal{J}_{\ell}(I;\operatorname{Der}_{k}(S))\not\subset I$ for
$\ell=\operatorname{ht}I$. It is true because $S$ satisfies the weak Jacobian
condition (WJ)k (cf. Remark 2.6(2)). If $I$ is not a prime ideal, then
$\mathcal{J}_{\ell}(I;\operatorname{Der}_{k}(S))\not\subset I$ does not hold
in general.

######  Lemma 5.22.

Let $P$ be a prime ideal of $S=k[[x_{1},\ldots,x_{N}]][y_{1},\ldots,y_{m}]$ of
height $r$, and let $I$ be an ideal of $S$ satisfying $P\subsetneq I$. Suppose
that $S/P$ is regular and $I+(x_{1},\ldots,x_{N})\not=S$. Then it follows that
$\mathcal{J}_{r+1}(I;\operatorname{Der}_{k}(S))\not\subset I$.

###### Proof.

Note that $\operatorname{Der}_{k}(S)=\operatorname{Der}^{\prime}_{k}(S)$ is
generated by $\partial/\partial x_{i}$’s and $\partial/\partial y_{i}$’s.
Since $I+(x_{1},\ldots,x_{N})\not=S$, there exists a maximal ideal
$\mathfrak{m}$ containing $I$ of the form

$\mathfrak{m}=(x_{1},\ldots,x_{N},y_{1}-a_{1},\ldots,y_{m}-a_{m}),$

where $a_{i}\in k$. Let $\widehat{S}$ be the completion of $S$ at
$\mathfrak{m}$. Let $Y_{i}\in\widehat{S}$ be the image of $y_{i}-a_{i}$. Then
we have $\widehat{S}\simeq k[[x_{1},\ldots,x_{N},Y_{1},\ldots,Y_{m}]]$, and
$\operatorname{Der}_{k}(\widehat{S})$ is generated by $\partial/\partial
x_{i}$’s and $\partial/\partial Y_{i}$’s. Therefore we have

$\mathcal{J}_{r+1}\bigl{(}I\widehat{S};\operatorname{Der}_{k}(\widehat{S})\bigr{)}+I\widehat{S}=\mathcal{J}_{r+1}(I;\operatorname{Der}_{k}(S))\widehat{S}+I\widehat{S}.$

We also note that $\widehat{S}/P\widehat{S}$ is regular and
$P\widehat{S}\subsetneq I\widehat{S}$. Therefore by Lemma 5.20, we have
$\mathcal{J}_{r+1}\bigl{(}I\widehat{S};\operatorname{Der}_{k}(\widehat{S})\bigr{)}\not\subset
I\widehat{S}$, which shows the assertion
$\mathcal{J}_{r+1}(I;\operatorname{Der}_{k}(S))\not\subset I$. ∎

######  Proposition 5.23.

Let $X$ be a scheme of finite type over $R=k[t][[x_{1},\ldots,x_{N}]]$.
Suppose that each irreducible component $X_{i}$ of $X$ has
$\dim^{\prime}X_{i}\geq n+1$. For each $a\in k$, we denote by $X_{a}$ the
closed subscheme of $X$ defined by $(t-a)\mathcal{O}_{X}$. Suppose for any
$a\in k^{\times}$ that $X_{a}$ is an integral regular scheme and has
$\dim^{\prime}X_{a}=n$. Then, there is no thin cylinder $C$ of $X_{\infty}$
containing a $k$-arc $\gamma$ with
$\operatorname{ord}_{\gamma}(\mathfrak{o}_{X})=\infty$.

###### Proof.

We may assume that $X$ is affine, and we may write $X=\operatorname{Spec}A$
with $A=S/I$, where

$S:=k[t][[x_{1},\ldots,x_{N}]][y_{1},\ldots,y_{m}]$

and $I$ is an ideal of $S$.

Suppose the contrary that there exists a thin cylinder $C$ containing a
$k$-arc $\gamma$ with $\operatorname{ord}_{\gamma}(\mathfrak{o}_{X})=\infty$.
Replacing $C$ with $C\cap\operatorname{Cont}^{\geq 1}(\mathfrak{o}_{X})$, we
may assume that $C\subset\operatorname{Cont}^{\geq 1}(\mathfrak{o}_{X})$. By
Lemma 5.19, it follows that
$e:=\operatorname{ord}_{\gamma}\bigr{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}<\infty$.
By replacing $C$ with
$C\cap\operatorname{Cont}^{e}\bigr{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}$,
we may assume that
$\emptyset\not=C\subset\operatorname{Cont}^{e}\bigr{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}$.

Let $\mathcal{S}$ be the set of the closed subschemes $W$ of $X$ with the
following condition:

  * •

There exists a cylinder $C^{\prime}$ of $X_{\infty}$ such that $\gamma\in
C^{\prime}\subset C$ and $C^{\prime}\subset W_{\infty}$.

Let $W$ be a minimal element of $\mathcal{S}$, and let $C^{\prime}$ be a
corresponding cylinder of $X$. Then $W$ is reduced by the minimality. Let
$W=W_{1}\cup\cdots\cup W_{\ell}$ be its irreducible decomposition. Since $C$
is thin, we may assume $\dim W\leq n$. Here, we claim as follows:

######  Claim 5.24.

  1. (1)

$\gamma\in(W_{i})_{\infty}$ holds for each $1\leq i\leq\ell$.

  2. (2)

$\dim W_{i}=n$ holds for each $1\leq i\leq\ell$.

  3. (3)

We denote by $Z_{i}\subset W_{i}$ the closed subscheme defined by
$\operatorname{Fitt}^{n-1}\bigl{(}\Omega^{\prime}_{W_{i}/k[t]}\bigr{)}$. Then
$Z_{i}\subsetneq W_{i}$ holds for each $1\leq i\leq\ell$.

First, we assume this claim and finish the proof. By Claim 5.24(3) and the
minimality of $W$, we have

$C^{\prime}\not\subset(Z_{1}\cup\cdots\cup
Z_{\ell})_{\infty}=(Z_{1})_{\infty}\cup\cdots\cup(Z_{\ell})_{\infty}.$

Take a $k$-arc $\beta\in
C^{\prime}\setminus\bigl{(}(Z_{1})_{\infty}\cup\cdots\cup(Z_{\ell})_{\infty}\bigr{)}$.
For each $i$, we denote by $I_{Z_{i}}\subset A$ the ideal corresponding to
$Z_{i}$, and we set $q_{i}:=\operatorname{ord}_{\beta}(I_{Z_{i}})<\infty$.
Then

$C^{\prime\prime}:=C^{\prime}\cap\bigcap_{i}\operatorname{Cont}^{q_{i}}(I_{Z_{i}})$

is a non-empty cylinder of $X_{\infty}$. By applying Proposition 5.9(2) to
$W_{i}$ and its cylinder

$C^{\prime\prime}\cap(W_{i})_{\infty}\subset\operatorname{Cont}^{q_{i}}\bigl{(}\operatorname{Fitt}^{n-1}\bigl{(}\Omega^{\prime}_{W_{i}/k[t]}\bigr{)}\bigr{)},$

it follows that the truncation map

$\psi_{m+1}\bigl{(}C^{\prime\prime}\cap(W_{i})_{\infty}\bigr{)}\to\psi_{m}\bigl{(}C^{\prime\prime}\cap(W_{i})_{\infty}\bigr{)}$

has $(n-1)$-dimensional fibers for sufficiently large $m$. Therefore,

$\psi_{m+1}(C^{\prime\prime})=\bigcup_{i}\psi_{m+1}\bigl{(}C^{\prime\prime}\cap(W_{i})_{\infty}\bigr{)}\to\psi_{m}(C^{\prime\prime})=\bigcup_{i}\psi_{m}\bigl{(}C^{\prime\prime}\cap(W_{i})_{\infty}\bigr{)}$

also has $(n-1)$-dimensional fibers for sufficiently large $m$. However, by
Proposition 5.9(2), it should have $n$-dimensional fibers because
$\emptyset\not=C^{\prime\prime}\subset\operatorname{Cont}^{e}\bigr{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}$.
We get a contradiction. Therefore, it is sufficient to prove Claim 5.24(3).

###### Proof of Claim 5.24.

We shall prove (1). Suppose the contrary that $\gamma\not\in(W_{1})_{\infty}$.
Let $I_{W_{1}}\subset A$ be the ideal corresponding to $W_{1}$ and let
$q:=\operatorname{ord}_{\gamma}(I_{W_{1}})<\infty$. Then the cylinder

$C^{\prime\prime}:=C^{\prime}\cap\operatorname{Cont}^{q}(I_{W_{1}})$

contains $\gamma$ and satisfies

$C^{\prime\prime}\subset
W_{\infty}\setminus(W_{1})_{\infty}\subset(W_{2}\cup\cdots\cup
W_{\ell})_{\infty},$

which contradicts the minimality of $W$.

We shall prove (2). Suppose the contrary that $\dim W_{1}\leq n-1$. Let
$W^{\prime}:=W_{2}\cup\cdots\cup W_{\ell}$ and let $I_{W^{\prime}}\subset A$
be the ideal corresponding to $W^{\prime}$. By the minimality of $W$, it
follows that $C^{\prime}\not\subset W^{\prime}_{\infty}$. Therefore, we have

$C^{\prime\prime}:=C^{\prime}\cap\operatorname{Cont}^{q}(I_{W^{\prime}})\not=\emptyset$

for some $q\geq 0$. Since $C^{\prime\prime}\cap
W^{\prime}_{\infty}=\emptyset$, we have
$C^{\prime\prime}\subset(W_{1})_{\infty}$ and $C^{\prime\prime}$ turns out to
be a very thin cylinder of $X_{\infty}$. It contradicts
$\emptyset\not=C^{\prime\prime}\subset\operatorname{Cont}^{e}\bigr{(}\operatorname{Fitt}^{n}(\Omega^{\prime}_{X/k[t]})\bigr{)}$
by Lemma 5.18.

We shall prove (3). Let $H$ be one of $W_{i}$’s. Let $Q$ be the prime ideal of
$S$ corresponding to $H$. Since $H_{\infty}$ contains a $k$-arc, $H$ contains
a $k$-point. Therefore, by Lemma 2.11(4), we have

$\operatorname{ht}Q=\dim S-\dim^{\prime}H=\dim S-\dim H=N+m-n+1.$

First, we prove that

$Q+(x_{1},\ldots,x_{N})+(t-a)\not=S$

for some $a\in k^{\times}$. Suppose the contrary that
$Q+(x_{1},\ldots,x_{N})+(t-a)=S$ holds for any $a\in k^{\times}$. Then by
Hilbert’s nullstellensatz, it follows that $t^{\ell}\in
Q+(x_{1},\ldots,x_{N})$ for some $\ell\geq 0$. This contradicts $\gamma\in
H_{\infty}$ and $\operatorname{ord}_{\gamma}(\mathfrak{o}_{X})=\infty$, and we
get ($\clubsuit$) for some $a\in k^{\times}$.

($\clubsuit$) implies $Q+(t-a)\not=S$. Furthermore, we have $t-a\not\in Q$
because $H_{\infty}$ contains a $k$-arc. Therefore, we have
$\operatorname{ht}(Q+(t-a))=\operatorname{ht}Q+1=N+m-n+2$.

We set

$\displaystyle S_{a}$ $\displaystyle:=S/(t-a)\simeq
k[[x_{1},\ldots,x_{N}]][y_{1},\ldots,y_{m}],$ $\displaystyle I_{a}$
$\displaystyle:=(I+(t-a))/(t-a),\quad Q_{a}:=(Q+(t-a))/(t-a).$

Then we have $\operatorname{ht}(Q_{a})=N+m-n+1$. Furthermore, we have
$\operatorname{ht}(I_{a})=N+m-n$ by the assumption $\dim^{\prime}X_{a}=n$.
Therefore it follows that $Q_{a}\supsetneq I_{a}$. Let
$J:=\mathcal{J}_{N+m-n+1}\bigl{(}Q;\operatorname{Der}_{k[t]}(S)\bigr{)}$. Then
the ideal $(J+(t-a))/(t-a)$ of $S_{a}$ coincides with
$\mathcal{J}_{N+m-n+1}\bigl{(}Q_{a};\operatorname{Der}_{k}(S_{a})\bigr{)}$.
Note that $S_{a}/I_{a}$ is regular, $\operatorname{ht}(I_{a})=N+m-n$, and
$Q_{a}\supsetneq I_{a}$. Therefore by Lemma 5.22, we have

$(J+(t-a))/(t-a)=\mathcal{J}_{N+m-n+1}\bigl{(}Q_{a};\operatorname{Der}_{k}(S_{a})\bigr{)}\not\subset
Q_{a}=(Q+(t-a))/(t-a).$

In particular, we have $J\not\subset Q$.

Since $\operatorname{Fitt}^{n-1}(\Omega^{\prime}_{H/k[t]})=(J+Q)/Q$, we
complete the proof of (3). ∎

∎

######  Lemma 5.25.

Let $R=k[t][[x_{1},\ldots,x_{N}]]$. Let $e_{1},\ldots,e_{N}$ and $d$ be
integers satisfying $0<e_{i}\leq d$ for each $i$. For each $c\in k^{\times}$,
let $T_{c}:R\to R$ be the ring isomorphism defined by $T_{c}(t)=c^{-d}t$ and
$T_{c}(x_{i})=c^{e_{i}}x_{i}$. Let $I$ be an ideal of $R$ that is
$T_{c}$-invariant (i.e. $T_{c}(I)=I$ holds) for any $c\in k^{\times}$. Let $P$
be a minimal prime of $I$. Then $P$ satisfies one of the following conditions.

  1. (1)

$P\cap k[t]\not=(0)$ and $t\in P$.

  2. (2)

$P\cap k[t]=(0)$, and $P+(t-a)\not=R$ holds for any $a\in k^{\times}$.

  3. (3)

$P\cap k[t]=(0)$, and there exists $f\in P$ such that

$f-t^{\ell}\in(t^{\ell+1}x_{1},\ldots,t^{\ell+1}x_{N})+(x_{1},\ldots,x_{N})^{\ell+1}$

holds for some $\ell\geq 0$.

###### Proof.

First, we prove that $P$ is also $T_{c}$-invariant for any $c\in k^{\times}$.
Let $P_{1},\ldots,P_{m}$ be the minimal primes of $I$. Since $T_{c}$ is an
isomorphism, $T_{c}$ induces a permutation on $P_{1},\ldots,P_{m}$. Let
$p:k^{\times}\to\mathfrak{S}_{m}$ be the induced group homomorphism, where
$\mathfrak{S}_{m}$ is the symmetric group of degree $m$. For any $c\in
k^{\times}$, we can take $b\in k^{\times}$ such that $c=b^{m!}$. Therefore, we
have $p(c)=p(b^{m!})=(p(b))^{m!}=1$. It shows that $T_{c}(P_{i})=P_{i}$ for
any $c\in k^{\times}$ and $1\leq i\leq m$.

Suppose $P\cap k[t]\not=(0)$. Then $t-a\in P$ for some $a\in k$. Since $P$ is
$T_{c}$-invariant for any $c\in k^{\times}$, it follows that $a=0$. Therefore,
$P$ satisfies (1).

Suppose that $P\cap k[t]=(0)$ and $P+(t-a)=R$ holds for some $a\in
k^{\times}$. We shall prove that $P$ satisfies (3). Since $P$ is
$T_{c}$-invariant for any $c\in k^{\times}$, it follows that $P+(t-a)=R$ holds
for any $a\in k^{\times}$. Then by Hilbert’s nullstellensatz, it follows that

  * •

$t^{\ell}\in P+(x_{1},\ldots,x_{N})$ for some $\ell\geq 0$.

Therefore there exists $g\in P$ such that $g-t^{\ell}\in(x_{1},\ldots,x_{N})$.

We denote $M:=(x_{1},\ldots,x_{N})\subset k[t][[x_{1},\ldots,x_{N}]]$. Since
$M$ is $T_{c}$-invariant, $T_{c}$ induces an automorphism on
$k[t][[x_{1},\ldots,x_{N}]]/M^{\ell+1}$. Hence,
$k[t][[x_{1},\ldots,x_{N}]]/M^{\ell+1}$ has a graded ring structure satisfying
$\deg t=-d$ and $\deg x_{i}=e_{i}$. Then $(P+M^{\ell+1})/M^{\ell+1}$ is a
homogeneous ideal. Therefore, the term $g_{-d\ell}$ of $g$ with degree
$-d\ell$ is contained in $P+M^{\ell+1}$. We may write $g_{-d\ell}=f-h$ with
$f\in P$ and $h\in M^{\ell+1}$. On the other hand, since
$g_{-d\ell}-t^{\ell}\in M$, we have
$g_{-d\ell}-t^{\ell}\in(t^{\ell+1}x_{1},\ldots,t^{\ell+1}x_{N})$ by looking at
the degrees of its terms. Therefore, the condition (3) holds for this $f$. ∎

######  Remark 5.26.

Let $I$ and $P$ be as in Lemma 5.25. Then the following hold for
$Y:=\operatorname{Spec}(R/P)$:

  * •

$Y_{\infty}=\emptyset$ if $P$ is of the form (1).

  * •

$Y_{\infty}\cap\operatorname{Cont}^{\geq 1}(\mathfrak{o}_{Y})=\emptyset$ holds
if $P$ is of the form (3).

###  5.2. Arc spaces of affine formal $k[[t]]$-schemes

In this subsection, we discuss the arc space of $X$ of the form
$X=\operatorname{Spec}\bigl{(}k[x_{1},\ldots,x_{N}][[t]]/I\bigr{)}$. As we
will mention in Remark 5.27, the arc space of $X$ can be seen as the Greenberg
scheme of the corresponding affine formal scheme. In this subsection, we do
not deal with general formal $k[[t]]$-schemes.

######  Remark 5.27.

Sebag in [Seb04] investigates the theory of arc spaces of formal
$k[[t]]$-schemes with $k$ a perfect field, and the theory can be applied to
$X=\operatorname{Spec}\bigl{(}k[x_{1},\ldots,x_{N}][[t]]/I\bigr{)}$ dealt with
in this subsection. The reader is also referred to [CLNS] to this theory.

For a scheme
$X=\operatorname{Spec}\bigl{(}k[x_{1},\ldots,x_{N}][[t]]/I\bigr{)}$, we can
associate the formal affine scheme
$\mathcal{X}=\operatorname{Spf}\bigl{(}k[x_{1},\ldots,x_{N}][[t]]/I\bigr{)}$.
Then the Greenberg schemes $\operatorname{Gr}_{m}(\mathcal{X})$ and
$\operatorname{Gr}(\mathcal{X})$ defined in [Seb04] are isomorphic to $X_{m}$
and $X_{\infty}$, respectively. Therefore, the theory of Greenberg schemes
developed in [Seb04] and [CLNS] can be applied to the arc space $X_{\infty}$
of $X$.

######  Definition 5.28 (cf. [CLNS, Appendix 3.3]).

Let $I$ be an ideal of $S=k[x_{1},\ldots,x_{N}][[t]]$ and let $A:=S/I$. Then
we denote by $\widehat{\Omega}_{A/k[[t]]}$ the completion of the $A$-module
$\Omega_{A/k[[t]]}$ with respect to the $(t)$-adic topology, i.e.

$\widehat{\Omega}_{A/k[[t]]}:=\varprojlim_{n}\left(\Omega_{A/k[[t]]}/(t^{n})\Omega_{A/k[[t]]}\right).$

The canonical derivation $d_{A/k[[t]]}:A\to\Omega_{A/k[[t]]}$ induces a
derivation

$\widehat{d}_{A/k[[t]]}:A\to\widehat{\Omega}_{A/k[[t]]}.$

We sometimes abbreviate $\widehat{d}_{A/k[[t]]}$ to $\widehat{d}$.

When $X=\operatorname{Spec}A$, we denote by $\widehat{\Omega}_{X/k[[t]]}$ the
sheaf on $X$ associated to the $A$-module $\widehat{\Omega}_{A/k[[t]]}$.

######  Remark 5.29 (cf. [CLNS, Example 3.3.5 in Appendix]).

  1. (1)

$\widehat{\Omega}_{S/k[[t]]}$ is a free $S$-module of rank $N$ with basis

$\widehat{d}_{S/k[[t]]}(x_{1}),\ldots,\widehat{d}_{S/k[[t]]}(x_{N}).$

Furthermore, we have an exact sequence

$I/I^{2}\xrightarrow{\delta}\widehat{\Omega}_{S/k[[t]]}\otimes_{S}A\xrightarrow{\alpha}\widehat{\Omega}_{A/k[[t]]}\to
0$

of $A$-modules, where $\alpha$ is the map satisfying
$\alpha\bigl{(}\widehat{d}_{S/k[[t]]}(g)\otimes
1\bigr{)}=\widehat{d}_{A/k[[t]]}(\overline{g})$ for $g\in S$, and $\delta$ is
the map satisfying $\delta(\overline{g})=\widehat{d}_{S/k[[t]]}(g)\otimes 1$
for $g\in I$. In particular, $\widehat{\Omega}_{A/k[[t]]}$ is a finite
$A$-module.

  2. (2)

The canonical derivation $\widehat{d}:A\to\widehat{\Omega}_{A/k[[t]]}$ has the
following universal property:

     * (2-1)

The induced map

$\operatorname{Hom}_{A}\bigl{(}\widehat{\Omega}_{A/k[[t]]},M\bigr{)}\to\operatorname{Der}_{k[[t]]}\left(A,M\right);\quad
f\mapsto f\circ\widehat{d}$

is an isomorphism for any $A$-module $M$ that is complete with respect to the
$(t)$-adic topology. In particular, this map is an isomorphism for any finite
$A$-module (cf. [Mat89, Theorem 8.7]).

This follows from the following general fact from [EGAIV1, 20.4.8.2]:

     * (2-2)

Let $B$ be a topological ring and $C$ a topological $B$-algebra. Let $N$ be a
topological $C$-module. Then we have an isomorphism

$\operatorname{Hom}^{\rm
c}_{C}(\Omega_{C/B},N)\xrightarrow{\simeq}\operatorname{Der}^{\rm
c}_{B}(C,N);\quad f\mapsto f\circ d_{C/B}.$

Here, $\operatorname{Hom}^{\rm c}_{C}(\Omega_{C/B},N)$ denotes the set of the
continuous homomorphisms $\Omega_{C/B}\to N$ of $C$-modules, and
$\operatorname{Der}^{\rm c}_{B}(C,N)$ denotes the set of the continuous
$B$-derivations $C\to N$. The topology on
$\Omega_{C/B}=\mathcal{I}/\mathcal{I}^{2}$ is defined as the quotient
topology, where $\mathcal{I}$ is the kernel of the augmentation map
$C\otimes_{B}C\to C;\ a\otimes b\mapsto ab$.

In our case, the topology on $\Omega_{A/k[[t]]}$ coincides with the $(t)$-adic
topology (cf. [EGAIV1, 20.4.5]). Therefore, we have

$\displaystyle\operatorname{Hom}^{\rm
c}_{A}\bigl{(}\Omega_{A/k[[t]]},M\bigr{)}$
$\displaystyle=\operatorname{Hom}_{A}\bigl{(}\Omega_{A/k[[t]]},M\bigr{)},$
$\displaystyle\operatorname{Der}^{\rm c}_{k[[t]]}(A,M)$
$\displaystyle=\operatorname{Der}_{k[[t]]}(A,M)$

for any $A$-module $M$ with the $(t)$-adic topology. Hence, by (2-2), we have
an isomorphism

$\operatorname{Hom}_{A}\bigl{(}\Omega_{A/k[[t]]},M\bigr{)}\xrightarrow{\simeq}\operatorname{Der}_{k[[t]]}(A,M)$

for any $A$-module $M$. Moreover, if $M$ is complete with respect to the
$(t)$-adic topology, then we have an isomorphism

$\operatorname{Hom}_{A}\bigl{(}\widehat{\Omega}_{A/k[[t]]},M\bigr{)}\xrightarrow{\simeq}\operatorname{Hom}_{A}\bigl{(}\Omega_{A/k[[t]]},M\bigr{)},$

which proves (2-1).

  3. (3)

By (2), it follows that $\operatorname{Der}_{k[[t]]}(S)$ is a free $S$-module
of rank $N$ generated by $\partial/\partial x_{i}$’s. Therefore, by the exact
sequence in (1), we have

$\operatorname{Fitt}^{n}\bigl{(}\widehat{\Omega}_{A/k[[t]]}\bigr{)}=\bigl{(}\mathcal{J}_{N-n}\bigl{(}I;\operatorname{Der}_{k[[t]]}(S)\bigr{)}+I\bigr{)}/I.$

######  Lemma 5.30.

Let $n$ and $e$ be non-negative integers, and let $X$ be an affine scheme of
the form $X=\operatorname{Spec}\bigl{(}k[x_{1},\ldots,x_{N}][[t]]/I\bigr{)}$.
Suppose that each irreducible component $X_{i}$ of $X$ has $\dim X_{i}\geq
n+1$. Let
$\gamma\in\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}(\widehat{\Omega}_{X/k[[t]]})\bigr{)}$
be a $k$-arc. Then we have

$\gamma^{*}\widehat{\Omega}_{X/k[[t]]}\simeq k[[t]]^{\oplus
n}\oplus\bigoplus_{i}k[t]/(t^{e_{i}})$

as $k[[t]]$-modules with $\sum_{i}e_{i}=e$.

###### Proof.

The same proofs as in Lemmas 5.6 and 5.7 work. Note that any minimal prime $P$
of $I$ satisfies $\operatorname{ht}P\leq N-n$. This is because

$\operatorname{ht}P=\dim S-\dim(S/P)\leq(N+1)-(n+1)=N-n,$

where we set $S:=k[x_{1},\ldots,x_{N}][[t]]$. The first equality follows from
the facts that any maximal ideal $M$ of $S$ has $\operatorname{ht}M=N+1$ and
$S$ is a catenary ring. ∎

######  Proposition 5.31.

Let $n$ be a non-negative integer, and let $X$ be an affine scheme of the form
$X=\operatorname{Spec}\bigl{(}k[x_{1},\ldots,x_{N}][[t]]/I\bigr{)}$. Suppose
that each irreducible component $X_{i}$ of $X$ has $\dim X_{i}\geq n+1$. Then,
there exists a positive integer $c$ such that the following hold for non-
negative integers $m$ and $e$ with $m\geq ce$.

  1. (1)

We have

$\displaystyle\psi_{m}\Bigl{(}\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}\bigl{(}\widehat{\Omega}_{X/k[[t]]}\bigr{)}\bigr{)}\Bigr{)}=\pi_{m+e,m}\Bigl{(}\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}\bigl{(}\widehat{\Omega}_{X/k[[t]]}\bigr{)}\bigr{)}_{m+e}\Bigr{)}.$

  2. (2)

$\pi_{m+1,m}:X_{m+1}\to X_{m}$ induces a piecewise trivial fibration

$\psi_{m+1}\Bigl{(}\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}\bigl{(}\widehat{\Omega}_{X/k[[t]]}\bigr{)}\bigr{)}\Bigr{)}\to\psi_{m}\Bigl{(}\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}\bigl{(}\widehat{\Omega}_{X/k[[t]]}\bigr{)}\bigr{)}\Bigr{)}$

with fiber $\mathbb{A}^{n}$.

###### Proof.

The same proof as in Proposition 5.9 works (cf. Remark 4.6). ∎

######  Remark 5.32.

When $X$ is flat over $k[[t]]$, Proposition 5.31(1) is proved in [CLNS, Ch.5.
Proposition 2.3.4], and Proposition 5.31(2) is proved in [Seb04, Lemme 4.5.4]
(cf. [CLNS, Ch.5. Theorem 2.3.11]). We also note that Proposition 5.31(2) can
be reduced to the flat case by the argument in [NS, Remark 2.14(3)].

We define cylinders and their codimensions.

######  Definition 5.33.

Let $n$ be a non-negative integer, and let $X$ be an affine scheme of the form
$X=\operatorname{Spec}\bigl{(}k[x_{1},\ldots,x_{N}][[t]]/I\bigr{)}$. Suppose
that each irreducible component $X_{i}$ of $X$ has $\dim X_{i}\geq n+1$. A
subset $C\subset X_{\infty}$ is called a cylinder if $C=\psi_{m}^{-1}(S)$
holds for some $m\geq 0$ and a constructible subset $S\subset X_{m}$. We
define the codimension of $C$ as follows:

  1. (1)

Assume that
$C\subset\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}\bigl{(}\widehat{\Omega}_{X/k[[t]]}\bigr{)}\bigr{)}$
for some $e\in\mathbb{Z}_{\geq 0}$. Then we define the codimension of $C$ in
$X_{\infty}$ as

$\operatorname{codim}(C):=(m+1)n-\operatorname{dim}(\psi_{m}(C))$

for any sufficiently large $m$. This definition is well-defined by Proposition
5.31.

  2. (2)

In general, we define the codimension of $C$ in $X_{\infty}$ as follows:

$\operatorname{codim}(C):=\min_{e\in\mathbb{Z}_{\geq
0}}\operatorname{codim}\Bigl{(}C\cap\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}\bigl{(}\widehat{\Omega}_{X/k[[t]]}\bigr{)}\bigr{)}\Bigr{)}.$

By convention, $\operatorname{codim}(C)=\infty$ if
$C\cap\operatorname{Cont}^{e}\bigl{(}\operatorname{Fitt}^{n}\bigl{(}\widehat{\Omega}_{X/k[[t]]}\bigr{)}\bigr{)}=\emptyset$
for any $e\geq 0$.

######  Remark 5.34.

As in Remark 5.12, the definition of the codimension above depends on the
choice of $n$.

######  Definition 5.35.

Let $n$ be a non-negative integer, and let $X$ be an affine scheme of the form
$X=\operatorname{Spec}\bigl{(}k[x_{1},\ldots,x_{N}][[t]]/I\bigr{)}$. Suppose
that each irreducible component $X_{i}$ of $X$ has $\dim X_{i}\geq n+1$. A
subset $A\subset X_{\infty}$ is called thin if $A\subset Z_{\infty}$ holds for
some closed subscheme $Z$ of $X$ with $\dim Z\leq n$.

######  Proposition 5.36 (cf. [Seb04, Théorème 6.3.5]).

Let $n$ be a non-negative integer, and let $X$ be an affine scheme of the form
$X=\operatorname{Spec}\bigl{(}k[x_{1},\ldots,x_{N}][[t]]/I\bigr{)}$. Suppose
that each irreducible component $X_{i}$ of $X$ has $\dim X_{i}\geq n+1$. Let
$C$ be a cylinder in $X_{\infty}$. Let $\\{C_{\lambda}\\}_{\lambda\in\Lambda}$
be a set of countably many disjoint subcylinders $C_{\lambda}\subset C$. If
$C\setminus(\bigsqcup_{\lambda\in\Lambda}C_{\lambda})\subset X_{\infty}$ is a
thin set, then it follows that

$\operatorname{codim}(C)=\min_{\lambda\in\Lambda}\operatorname{codim}(C_{\lambda}).$

###### Proof.