# A survey on algebraic dilatations

###### Abstract

Adrien Dubouloz, Arnaud Mayeux and João Pedro dos Santos

###### Contents

  1. I Algebraic dilatations
    1. 1 Dilatations of rings
    2. 2 Dilatations of schemes
    3. 3 Dilatations of group schemes or Néron blowups
  2. II Some applications
    1. 4 Models of group schemes, representation categories and Tannakian groups
    2. 5 Congruent isomorphisms and relations with Bruhat-Tits buildings, the Moy-Prasad isomorphism and admissible representations of $p$-adic groups
    3. 6 Torsors, level structures and shtukas
    4. 7 The “topology”of dilatations of affine schemes

######  Acknowledgement.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme (A.
M. grant agreement No 101002592).

###### :

13A99, 14A20, 14L15

###### keywords:

survey, algebraic geometry, commutative algebra, Grothendieck schemes, group
schemes, algebraic dilatations, dilatations of schemes, multi-centered
dilatations, localizations of rings, mono-centered dilatations, affine
modifications, affine blowups, formal blowups, Kaliman-Zaidenberg
modifications, Néron blowups, Tannakian groups, differential Galois groups,
congruent isomorphisms, Moy-Prasad isomorphism, representations of $p$-adic
groups, torsors, level structures, shtukas, affine geometry,
${\mathbb{A}}^{1}$-homotopy theory

Adrien Dubouloz, Arnaud Mayeux and João Pedro dos Santos

###### Contents

  1. I Algebraic dilatations
    1. 1 Dilatations of rings
    2. 2 Dilatations of schemes
    3. 3 Dilatations of group schemes or Néron blowups
  2. II Some applications
    1. 4 Models of group schemes, representation categories and Tannakian groups
    2. 5 Congruent isomorphisms and relations with Bruhat-Tits buildings, the Moy-Prasad isomorphism and admissible representations of $p$-adic groups
    3. 6 Torsors, level structures and shtukas
    4. 7 The “topology”of dilatations of affine schemes

######  Acknowledgement.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme (A.
M. grant agreement No 101002592).

### Introduction

What is the concept of algebraic dilatations about ?

Dilatation of rings is a basic construction of commutative algebra, like
localization or tensor product. It can be globalized so that it also make
sense on schemes or algebraic spaces. In fact dilatations generalize
localizations. Let $A$ be a ring and let $S$ be a multiplicative subset of
$A$. Recall that the localization $S^{-1}A$ is an $A$-algebra such that for
any $A$-algebra $A\to B$ such that the image of $s$ is invertible for any
$s\in S$, then $A\to B$ factors through $A\to S^{-1}A$. Intuitively, $S^{-1}A$
is the $A$-algebra obtained from $A$ adding all fractions $\frac{a}{s}$ with
$a\in A$ and $s\in S$. Formally, $S^{-1}A$ is made of classes of fractions
$\frac{a}{s}$ where $a\in A$ and $s\in S$ (two representative $\frac{a}{s}$
and $\frac{b}{t}$ are identified if $atr=bsr$ for some $r\in S$), addition and
multiplication are given by usual formulas. Now let us give for any element
$s\in S$ an ideal $M_{s}$ of $A$ containing $s$. The dilatation of $A$
relatively to the data $S,\\{M_{s}\\}_{s\in S}$ is an $A$-algebra $A^{\prime}$
obtained intuitively by adding to $A$ only the fractions $\frac{m}{s}$ with
$s\in S$ and $m\in M_{s}$. The dilatation $A^{\prime}$ satisfies that for any
$s\in S$, we have $sA^{\prime}=M_{s}A^{\prime}$ (intuitively any $m\in M_{s}$
belongs to $sA^{\prime}$, i.e. becomes a multiple of $s$, so that we have an
element $\frac{m}{s}$ such that $m=s\frac{m}{s}$). As a consequence of the
construction, the elements $s\in S$ become non-zero-divisor in $A^{\prime}$ so
that $\frac{m}{s}$ is well-defined (i.e. unique). It turns out that it is
convenient, with dilatations of schemes in mind, to make a bit more flexible
the above framework, namely to remove the conditions that $S$ is
multiplicative and that $s\in M_{s}$, so we use the following definition.

Definition. Let $A$ be a ring. Let $I$ be an index set. A multi-center in $A$
indexed by $I$ is a set of pairs $\\{[M_{i},a_{i}]\\}_{i\in I}$ where for each
$i$, $M_{i}$ is an ideal of $A$ and $a_{i}$ is an element of $A$.

To each multi-center $\\{[M_{i},a_{i}]\\}_{i\in I}$, one has the dilatation
$A[\\{\frac{M_{i}}{a_{i}}\\}_{i\in I}]$, it is an $A$-algebra. We will define
and study in details dilatations of rings in Section 1, in particular we will
state formally the universal property they enjoy. We will also see that
$A[\\{\frac{M_{i}}{a_{i}}\\}_{i\in I}]$ is generated, as $A$-algebra, by
$\\{\frac{M_{i}}{a_{i}}\\}_{i\in I}$. We will also see that if $M_{i}=A$ for
all $i$, then $A[\\{\frac{M_{i}}{a_{i}}\\}_{i\in I}]=S^{-1}A$ where $S$ is the
multiplicative subset generated by $\\{a_{i}\\}_{i\in I}$. Reciprocally, we
will see that any sub-$A$-algebra of a localization $S^{-1}A$ for a certain
$S$ is isomorphic to a dilatation of $A$. Dilatations of schemes and algebraic
spaces are obtained from dilatations of rings via glueing. We introduce the
following definition.

Definition. Let $X$ be a scheme. Let $I$ be an index set. A multi-center in
$A$ indexed by $I$ is a set of pairs $\\{[Y_{i},D_{i}]\\}_{i\in I}$ such that
$Y_{i}$ and $D_{i}$ are closed subschemes for each $i$ and such that locally,
all $D_{i}$ are principal for $i\in I$.

Associated to each multi-center, one has the dilatation
${\operatorname{Bl}}\big{\\{}{}^{D_{i}}_{Y_{i}}\big{\\}}_{i\in I}X$, it is a
scheme endowed with a canonical affine morphism
$f:{\operatorname{Bl}}\big{\\{}{}^{D_{i}}_{Y_{i}}\big{\\}}_{i\in I}X\to X$. It
satisfies, in a universal way, that $f^{-1}(D_{i})$ is a cartier divisor (i.e.
is locally defined by a non-zero-divisor) and that
$f^{-1}(D_{i})=f^{-1}(Y_{i})$ for all $i\in I$. If $\\#I=1$, we use the
terminology mono-centered dilatation. We will study several facets of this
construction and show that it enjoys many wonderfull properties in Sections 2
and 3.

Where does this construction come from ?

As we saw in the previous section, dilatations are a basic construction which
can be easily encountered in specific situations. As a consequence it was used
for a very long time. As the reader may well know, the theory of dilatations
has deep and distinguished roots, even though not formulated in the language
which we use. Right from the start, we warn the reader that we do not mean to,
and probably could not, present a comprehensive historical account. As soon as
Cremona and Bertini started using quadratic transformations (or blowups) in
the framework of algebraic geometry over fields, “substitutions” of the form
$x^{\prime}=x$ and $y^{\prime}=y/x$ started being made by algebraic geometers,
see for example equation (8) in [No1884, Section 11] and Noether’s
acknowledgement, at the start of [No1884, Section 12], that these
manipulations come from Cremona’s point of view. Examples of dilatations
appear frequently in some works of Zariski and Abhyankar, cf. [AZ55,
Definition, p. 86] and [Za43, p499 proof of Th.4, case (b)]. Other forerunner
examples of dilatations play a central role in several independent and
unrelated works later, cf. [Da67], [Ner64, Section 25] and [Ar69, Section 4].
As far as we know, the terminology dilatations emerged in [BLR90, §3.2], where
a section is devoted to study dilatations of schemes over discrete valuation
rings systematically. In the context of schemes over a discrete valuation
ring, we draw the reader’s attention to [Ana73], [WW80] and [PY06]. The paper
[KZ99] studies dilatations (under the name affine modifications)
systematically in the framework of algebraic geometry over fields. Over two-
dimensional base schemes dilatations also appear in [PZ13, p. 175]. Set aside
localizations, mono-centered dilatations have been the main focus of
mathematicians in the past. However, in the context of group schemes over
discrete valuation rings, examples of multi-centered dilatations of rings and
schemes that are not localizations or mono-centered dilatations appeared and
were used in [SGA3, Exp. VIB Ex. 13.3], [PY06] and [DHdS18]. In recent times,
the authors of [Du05], [MRR20] and [Ma23d] have set out to accommodate all
these constructions in a larger and unified frame, namely for arbitrary
schemes and algebraic spaces and arbitrary multi-center. The paper [MRR20]
introduces dilatations of arbitrary schemes in the mono-centered case and
provides a systematic treatment of mono-centered dilatations of general
schemes. An equivalent definition of mono-centered dilatations of general
schemes, under the name affine modifications, was introduced earlier in [Du05,
Définition 2.9] under few assumptions. The paper [Ma23d] introduces and
studies dilatations of arbitrary rings, schemes and algebraic spaces for
arbitrary multi-centers. Allowing multi-centers also leads naturally to the
formulation of combinatorial isomorphisms on dilatations and gives birth to
refined universal properties. Nevertheless, the mono-centered case remains a
fundamental case that is frequently the ’atom’ for some aspects of the theory.
The first part (Sections 1-2-3) of this survey is devoted to theoretical and
formal results on dilatations of rings, schemes and group schemes following
[Du05], [MRR20] and [Ma23d]. Sections 4-5-6-7 of the second part will deal
with several concrete situations were specific kind of dilatations play a
role, also providing complementary inceptions on this construction. To finish,
beyond rings and schemes, the concept of dilatations makes sense for other
structures and geometric settings. Let us indicate some constructions already
available. Some dilatation constructions in the framework of complex analytic
spaces were introduced in [Ka94], these are used and discussed in Section 7.
Dilatations also make sense for general algebraic spaces [Ma23d]. Similarly,
for many other structures than rings, dilatations also make sense (e.g.
categories, monoids and semirings) as noticed in [Ma23c]. It is possible that
dilatations in other settings will be explored and find a significant role
since, at the end, these are a basic mathematical concept.

Terminology

Recall that dilatations have distinguished roots, as a consequence, several
other terminologies are used to call certains dilatations in literature. For
examples the constructions named affine blowups, affine modifications,
automatic blowups, formal blowups, Kaliman modifications, localizations and
Néron blowups are examples of (eventually multi-centered) dilatations.

Some simple examples

We provided an intuition on dilatations of rings before. Let us now provide
some simple examples of dilatations of schemes. If $S$ is a scheme, we denote
by $e_{S}$ the trivial group scheme over $S$, as scheme it is isomorphic to
$S$. If $G$ is a separated group scheme over $S$, we denote by $e_{G}$ the
trivial closed group scheme, $e_{G}$ is isomorphic to $e_{S}$ as group schemes
over $S$.

  1. 1.

We consider, once given a prime number $p$, the multiplicative group scheme

$G=\mathbb{G}_{m,{\mathbb{Z}}_{p}}$

over the ring $\mathbb{Z}_{p}$; its Hopf algebra is
$A=\mathbb{Z}_{p}[x,x^{-1}]$ while the morphism $\Delta:A\to
A\otimes_{{\mathbb{Z}}_{p}}A$ induced from multiplication
$G\times_{{\mathbb{Z}}_{p}}G\to G$ is defined by $\Delta(x)=x\otimes x$. Now,
consider the couple $e_{G}\text{ and
}G\times_{{\mathbb{Z}}_{p}}{\mathbb{Z}}_{p}/{\mathfrak{p}}^{r}\text{ of closed
subschemes of }G$ for any $r>0$ (${\mathfrak{p}}^{r}$ denotes
$p^{r}{\mathbb{Z}}_{p}$). These are cut out, respectively, by the ideals
$I=(x-1)$ and $(p^{r})$ of $A$.

    1. (a)

For any $r>0$, the dilatation $A^{\prime}$ of $A$ centered at
$[e_{G},G\times_{{\mathbb{Z}}_{p}}{\mathbb{Z}}_{p}/{\mathfrak{p}}^{r}]$, or at
$[I,(p^{r})]$, is the sub-$A$-algebra of $A[1/p]$ generated by all the
elements $p^{-r}f$, where $f\in I$. The dilatation
$G^{\prime}:=\mathrm{Spec}\,A^{\prime}$ is a group scheme of finite type over
${\mathbb{Z}}_{p}$. The base change
$G^{\prime}\times_{{\mathbb{Z}}_{p}}{\mathbb{Z}}_{p}/{\mathfrak{p}}^{r}$ is
isomorphic to the additive group $\mathbb{G}_{a}$ over
${\mathbb{Z}}_{p}/{\mathfrak{p}}^{r}$, while
$G^{\prime}\times_{{\mathbb{Z}}_{p}}{\mathbb{Q}}_{p}$ is the multiplicative
group $\mathbb{G}_{m}$ over $\mathbb{Q}_{p}$. Furthermore, on the level of
points, $G^{\prime}({\mathbb{Z}}_{p})=1+{\mathfrak{p}}^{r}$ is a congruence
subgroup.

    2. (b)

The dilatation $A^{\prime}$ of $A$ centered at
$\\{[e_{G},G\times_{{\mathbb{Z}}_{p}}{\mathbb{Z}}_{p}/{\mathfrak{p}}^{r}]\\}_{\\{r>0\\}}$,
or at $\\{[I,(p^{r})]\\}_{\\{r>0\\}}$, is the sub-$A$-algebra of $A[1/p]$
generated by all the elements $p^{-r}f$, where $f\in I$ and $r>0$. The
dilatation $G^{\prime}:=\mathrm{Spec}\,A^{\prime}$ is a group scheme over
${\mathbb{Z}}_{p}$, it is not of finite type. The base change
$G^{\prime}\times_{{\mathbb{Z}}_{p}}{\mathbb{Z}}_{p}/{\mathfrak{p}}^{r}$ is
isomorphic to the trivial group scheme over
${\mathbb{Z}}_{p}/{\mathfrak{p}}^{r}$, while
$G^{\prime}\times_{{\mathbb{Z}}_{p}}{\mathbb{Q}}_{p}$ is the multiplicative
group $\mathbb{G}_{m}$ over $\mathbb{Q}_{p}$. Furthermore, on the level of
points, $G^{\prime}({\mathbb{Z}}_{p})=\\{1\\}$.

  2. 2.

Let $G$ be $GL_{3}$ over ${\mathbb{Z}}_{p}$ and let $H$ be $GL_{2}\times
e_{{\mathbb{Z}}_{p}}$ diagonally inside $G$ and let
$e_{G}\cong(e_{{\mathbb{Z}}_{p}})^{3}$ be the trivial closed subgroup of $G$.
For any $r>0$, let
$G\times_{{\mathbb{Z}}_{p}}{\mathbb{Z}}_{p}/{\mathfrak{p}}^{r}$ be the base
change of $G$ to ${\mathbb{Z}}/p^{r}{\mathbb{Z}}$. The dilatation
$G^{\prime}={\operatorname{Bl}}\big{\\{}{}^{G\times_{{\mathbb{Z}}_{p}}{\mathbb{Z}}_{p}/{\mathfrak{p}}^{5}}_{~{}~{}H},{}^{G\otimes{\mathbb{Z}}_{p}/{\mathfrak{p}}^{2}}_{~{}e_{G}}\big{\\}}G$
is a group scheme over ${\mathbb{Z}}_{p}$. On the level of points, we have

$\displaystyle GL_{3}({\mathbb{Z}}_{p})\supset G^{\prime}({\mathbb{Z}}_{p})$
$\displaystyle=\begin{pmatrix}1+{\mathfrak{p}}^{2}&{\mathfrak{p}}^{2}&{\mathfrak{p}}^{5}\\\
{\mathfrak{p}}^{2}&1+{\mathfrak{p}}^{2}&{\mathfrak{p}}^{5}\\\
{\mathfrak{p}}^{5}&{\mathfrak{p}}^{5}&1+{\mathfrak{p}}^{5}\end{pmatrix}$
$\displaystyle=\left\\{\begin{pmatrix}1+a&b&e\\\ c&1+d&f\\\
g&h&1+k\end{pmatrix}\middle|a,b,c,d\in{\mathfrak{p}}^{2}~{}~{}e,f,g,h,k\in{\mathfrak{p}}^{5}\right\\}.$

  3. 3.

Let $X={\mathbb{A}}^{1}={\operatorname{Spec}}({\mathbb{Z}}[T])$ be the affine
line over ${\mathbb{Z}}$, let $0\subset{\mathbb{A}}^{1}$ be the closed
subscheme defined by the ideal $(T)$ and let
$\emptyset\subset{\mathbb{A}}^{1}$ be the closed subscheme defined by the
ideal ${\mathbb{Z}}[T]$. Then the dilatation
${\operatorname{Bl}}_{\emptyset}^{D}X$ identifies with the open subscheme
${\mathbb{G}}_{m}$ of ${\mathbb{A}}^{1}$, indeed
${\mathbb{Z}}[T][\frac{{\mathbb{Z}}[T]}{T}]\cong{\mathbb{Z}}[T,T^{-1}]$. This
is an example of localization.

  4. 4.

Let $X={\mathbb{A}}^{2}={\operatorname{Spec}}({\mathbb{Z}}[A,B])$ be the
affine plane over ${\mathbb{Z}}$, let $D\subset{\mathbb{A}}^{2}$ be the line
defined by the ideal $(A)$ and let $0\subset{\mathbb{A}}^{2}$ be the origin
defined by the ideal $(A,B)$. Then ${\operatorname{Bl}}^{D}_{0}X$ identifies
with ${\operatorname{Spec}}({\mathbb{Z}}[A,B,C]/(AC-B))$. Indeed, one has an
isomorphism (e.g. by Proposition 1.4)

${\mathbb{Z}}[A,B][\frac{(A,B)}{A}]\cong{\mathbb{Z}}[A,B][\frac{(B)}{A}]\cong{\mathbb{Z}}[A,B,C]/AC=B.$

The morphism ${\operatorname{Bl}}_{0}^{D}X\to X$ is given by
${\mathbb{Z}}[A,B]\to{\mathbb{Z}}[A,B,C]/(AC-B),A,B\mapsto A,B$. At the level
of points $\big{(}{\operatorname{Bl}}_{0}^{D}X\big{)}({\mathbb{Z}})$ is made
of pairs $(a,b)\in{\mathbb{Z}}^{2}$ such that $b$ is a multiple of $a.$

  5. 5.

More advanced examples of dilatations in contextual situations are available
in the second part of this survey.

What is the aim of this survey ?

Recall that we wish to provide the reader with a short guide to recent works
on the theory of dilatations. We do not mean to present a comprehensive
account. We rather concentrate on the contributions that ourselves were
responsible for [Du05, DF18, DHdS18, Ma19t, HdS21, MRR20, ADØ21, Ma23d] and
those which were our starting points [Ar69, Ana73, And01, WW80, BLR90, KZ99,
Yu15, PY06, PZ13, HKØ16]. Part I is devoted to an exposition of general
definitions and results around the concept of algebraic dilatations introduced
and proved in [MRR20] and [Ma23d]. Section 1 discusses dilatations of
commutative unital rings. Section 2 summarizes general results on dilatations
of schemes. Section 3 focuses on dilatations of group schemes. Part II
provides an overview on some applications of dilatations in various
mathematical contexts. In Section 4, we explain some recent applications of
dilatations to the theory of affine group schemes and their representation
categories in the case where objects are defined over a discrete valuation
ring $R$. These were developed mainly in order to advance the study of
Tannakian categories defined over $R$ and appearing in geometry, such as the
case of group schemes associated to $\mathcal{D}$-modules [And01, dS09, DH18,
DHdS18, HdS21] and principal bundles with finite structure groups [HdS23].
After a brief introduction to Tannakian categories over $R$ (Section 4.1), we
go on to explain how to filter these categories by smaller ones and produce in
this way the “Galois-Tannaka” group schemes 4.2. We show why Néron blowups are
a fundamental tool for studying these groups and explain what has been done so
far in order to exhaust Galois-Tannaka groups by means of Néron blowups, both
in the case of mono-centered and multi-centered Néron blowups (cf. Section 4.2
and Section 4.3). In Section 5, congruent isomorphisms, formulated and stated
using the language of dilatations, are discussed in relation with the Moy-
Prasad isomorphism, Bruhat-Tits buildings and representations of $p$-adic
groups. Section 6 shows that many level structures on moduli stacks of
$G$-bundles are encoded in torsors under dilatations and that this can be used
to obtain integral models of shtukas. Section 7 discusses dilatations in
affine geometry and related progress in ${\mathbb{A}}^{1}$-homotopy theory.
All results stated in this paper are proved in indicated references. This
survey is an expository text and does not contain any new mathematical result.
What is perhaps new is to summarize some aspects of several independent works
involving dilatations in a single text. We hope this could be a source of
inspiration for future works.

### Some conventions and notations

  1. (1)

All rings are unital and commutative, unless otherwise mentioned.

  2. (2)

Let $(M,+)$ be a monoid. A submonoid $F$ is a face of $M$ if whenever $x+y\in
F$, then both $x$ and $y$ belong to $F$.

  3. (3)

If $R$ is a discrete valuation ring with field of fractions $K$, then, for
each $R$-scheme, we call $X\otimes_{R}K$ the generic fibre of $X$.

  4. (4)

If $A$ is a ring, then $A\text{-}\mathbf{mod}$ is the category of finitely
presented $A$-modules.

  5. (5)

If $G$ is a group scheme over a noetherian ring $R$, or an abstract group, we
denote by $\mathrm{Rep}_{R}(G)$ the category of all $R$–modules of finite type
affording a representation of $G$ as explained in [Ja03].

##  Part I Algebraic dilatations

In this part, we introduce formally dilatations of rings and schemes. Locally,
dilatations of schemes will be studied through dilatations of rings.

###  1 Dilatations of rings

We summarily present basic results on dilatations of rings following the more
general path given in [Ma23d]. Let $A$ be a ring. A center in $A$ is a pair
$[M,a]$ consisting of an ideal $M\subset A$ and an element $a\in A$. A multi-
center is a family of center indexed by some set. Let $I$ be an index set and
let $\\{[M_{i},a_{i}]\\}_{i\in I}$ be a multi-center. For $i\in I$, we put
$L_{i}=M_{i}+(a_{i})$, an ideal of $A$. Let ${\mathbb{N}}_{I}$ be the monoid
$\bigoplus_{i\in I}{\mathbb{N}}$. If
$\nu=(\nu_{1},\ldots,\nu_{i},\ldots)\in{\mathbb{N}}_{I}$ we put
$L^{\nu}=L_{1}^{\nu_{1}}\cdots L_{i}^{\nu_{i}}\cdots$ (product of ideals of
$A$) and $a^{\nu}=a_{1}^{\nu_{1}}\cdots a_{i}^{\nu_{i}}\cdots$ (product of
elements of $A$). We also put
$a^{{\mathbb{N}}_{I}}=\\{a^{\nu}|\nu\in{\mathbb{N}}_{I}\\}$.

######  Definition and Proposition 1.1 ([Ma23d]).

The dilatation of $A$ with multi-center $\\{[M_{i},a_{i}]\\}_{i\in I}$ is the
unital commutative ring $A[\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in I}]$
defined as follows: $\bullet$ The underlying set of
$A[\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in I}]$ is the set of equivalence
classes of symbols $\frac{m}{a^{\nu}}$ where $\nu\in{\mathbb{N}}_{I}$ and
$m\in L^{\nu}$ under the equivalence relation

$\frac{m}{a^{\nu}}\equiv\frac{p}{a^{\lambda}}\Leftrightarrow\exists\beta\in{\mathbb{N}}_{I}\text{
such that }ma^{\beta+\lambda}=pa^{\beta+\nu}\text{ in }A.$

From now on, we abuse notation and denote a class by any of its representative
$\frac{m}{a^{\nu}}$ if no confusion is likely. $\bullet$ The addition law is
given by
$\frac{m}{a^{\nu}}+\frac{p}{a^{\beta}}=\frac{ma^{\beta}+pa^{\nu}}{a^{\beta+\nu}}$.
$\bullet$ The multiplication law is given by
$\frac{m}{a^{\nu}}\times\frac{p}{a^{\beta}}=\frac{mp}{a^{\nu+\beta}}$.
$\bullet$ The additive neutral element is $\frac{0}{1}$ and the multiplicative
neutral element is $\frac{1}{1}$.

From now on, we also use the notation $A[\frac{M}{a}]$ to denote
$A[\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in I}]$. We have a canonical
morphism of rings $A\to A[\frac{M}{a}]$ given by $a\mapsto\frac{a}{1}$.

The element $\frac{a}{1}$ of $A[\frac{M}{a}]$ will be denoted by $a$ if no
confusion is likely.

######  Fact 1.2 ([Ma23d]).

  1. 1.

Let $\\{N_{i}\\}_{i\in I}$ be ideals in $A$ such that $N_{i}+(a_{i})=L_{i}$
for all $i\in I$. Then we have identifications of $A$-algebras
$A[\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in
I}]=A[\big{\\{}\frac{N_{i}}{a_{i}}\big{\\}}_{i\in
I}]=A[\big{\\{}\frac{L_{i}}{a_{i}}\big{\\}}_{i\in I}].$

  2. 2.

Dilatations of rings generalize entirely localizations of rings. Indeed, let
$A$ be a ring and let $S$ be a multiplicative subset of $A$. Then
$S^{-1}A=A[\big{\\{}\frac{A}{s}\big{\\}}_{s\in S}]$.

  3. 3.

Any sub-$A$-algebra of a localization $S^{-1}A$ for a subset $S\subset A$ can
be obtained as a multi-centered dilatation.

  4. 4.

Note that we did not used substraction to define dilatations of rings. In fact
Definition 1.1 makes sense for arbitrary unital commutative semirings, cf.
[Ma23d, §2] or more generally for categories (e.g. monoids) cf. [Ma23c].

This construction enjoys the following properties, cf. [Ma23d, §2]. If
$\\#I=1$, most results appear in [StP, Tag 052P].

######  Proposition 1.3 ([Ma23d]).

The following assertions hold.

  1. 1.

As $A$-algebra, $A[\frac{M}{a}]$ is generated by
$\big{\\{}\frac{L_{i}}{a_{i}}\big{\\}}_{i\in I}$. Since $L_{i}=M_{i}+(a_{i})$,
this implies that $A[\frac{M}{a}]$ is generated by
$\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in I}$.

  2. 2.

If $A$ is a domain and $a_{i}\neq 0$ for all $i$, then $A[\frac{M}{a}]$ is a
domain.

  3. 3.

If $A$ is reduced, then $A[\frac{M}{a}]$ is reduced.

  4. 4.

The following assertions are equivalent.

    1. (a)

There exists $\nu\in{\mathbb{N}}_{I}$ such that $a^{\nu}=0$ in $A$.

    2. (b)

The ring $A[\frac{M}{a}]$ equals to the zero ring.

  5. 5.

Let $\nu$ be in $\mathbb{N}_{I}$. The image of $a^{\nu}$ in $A[\frac{M}{a}]$
is a non-zero-divisor.

  6. 6.

Let $f:A\to B$ be a morphism of rings. Let $\\{[N_{i},b_{i}]\\}_{i\in I}$ be
centers of $B$ such that $f(M_{i})\subset N_{i}$ and $f(a_{i})=b_{i}$ for all
$i\in I$. Then we have a canonical morphism of $A$-algebras

$\phi:A[\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in I}]\to
B[\big{\\{}\frac{N_{i}}{b_{i}}\big{\\}}_{i\in I}]$ .

  7. 7.

Let $c$ be a non-zero-divisor element in $A$. Then $\frac{c}{1}$ is a non-
zero-divisor in $A[\frac{M}{a}]$.

  8. 8.

Let $K\subset I$ put $J=I\setminus K$. Then we have a canonical morphism of
$A$-algebras

$\varphi:A[\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in K}]\longrightarrow
A[\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in I}].$

Moreover

    1. (a)

if $M_{i}\subset(a_{i})$ for all $i\in J$, then $\varphi$ is surjective, and

    2. (b)

if $a_{i}$ is a non-zero-divisor in $A$ for all $i\in J$, then $\varphi$ is
injective.

  9. 9.

Let $K\subset I$. Then we have a canonical isomorphism of
$A[\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in K}]$-algebras

$A[\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in
I}]=A[\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in
K}][\big{\\{}\frac{A[\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in
I}]\frac{M_{j}}{1}}{\frac{a_{j}}{1}}\big{\\}}_{j\in I\setminus K}],$

where ${A[\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in I}]\frac{M_{j}}{1}}$ is
the ideal of $A[\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in I}]$ generated by
$\frac{M_{j}}{1}\subset A[\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in I}]$.

  10. 10.

Assume that $a_{i}=a_{j}=:b$ for all $i,j\in I$, then

$A[\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in I}]=A[\frac{\sum_{i\in
I}M_{i}}{b}]$

  11. 11.

Let $\nu\in{\mathbb{N}}_{I}$. We have
$L^{\nu}A[\frac{M}{a}]=a^{\nu}A[\frac{M}{a}]$.

  12. 12.

(Universal property) If $\chi:A\to B$ is a morphism of rings such that
$\chi(a_{i})$ is a non-zero-divisor and generates $\chi(L_{i})B$ for all $i\in
I$, then there exists a unique morphism $\chi^{\prime}$ of $A$-algebras
$A[\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in I}]\to B$. The morphism
$\chi^{\prime}$ sends $\frac{l}{a^{\nu}}$ ($\nu\in{\mathbb{N}}_{I},l\in
L^{\nu}$) to the unique element $b\in B$ such that $\chi(a^{\nu})b=\chi(l)$.

  13. 13.

Assume that $I=\\{1,\ldots,k\\}$ is finite. Then we have a canonical
identification of $A$-algebras

$A[\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in I}]=A[\frac{\sum_{i\in
I}(M_{i}\cdot\prod_{j\in I\setminus{\\{i\\}}}a_{j})}{a_{1}\cdots a_{k}}].$

  14. 14.

Write $I={\operatorname{colim}}_{J\subset I}J$ as a filtered colimit of sets.
We have a canonical identification of $A$-algebras

$A[\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in
I}]={\operatorname{colim}}_{J\subset
I}A[\big{\\{}\frac{M_{j}}{a_{i}}\big{\\}}_{i\in J}]$.

  15. 15.

Let $f:A\to B$ be an $A$-algebra. Put $N_{i}=f(M_{i})B$ and $b_{i}=f(a_{i})$
for $i\in I$. Then $B[\big{\\{}\frac{N_{i}}{b_{i}}\big{\\}}_{i\in I}]$ is the
quotient of $B\otimes_{A}A[\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in I}]$ by
the ideal $T_{b}$ of elements annihilated by some element in
$b^{\mathbb{N}_{I}}:=\\{b^{\nu}|\nu\in{\mathbb{N}}_{I}\\}$. If moreover
$f:A\to B$ is flat, then $T_{b}=0$ and we have a canonical isomorphism

$B[\\{\frac{N_{i}}{b_{i}}\\}_{i\in
I}]=B\otimes_{A}A[\\{\frac{M_{i}}{a_{i}}\\}_{i\in I}].$

  16. 16.

Let $f:R\to A$ be a morphism of rings and let $\\{r_{i}\\}_{i\in I}\subset R$.
Let $R^{\prime}=R[\big{\\{}\frac{R}{r_{i}}\big{\\}}_{i\in I}]$; this is a
localization of $R$ and hence $R\to R^{\prime}$ is a flat morphism. Let
$A^{\prime}=A\otimes_{R}R^{\prime}$,
$M_{i}^{\prime}=M_{i}\otimes_{R}R^{\prime}\subset A^{\prime}$. Then, if
$a_{i}:=f(r_{i})$, the dilatation
$A[\big{\\{}\frac{M_{i}}{a_{i}}\big{\\}}_{i\in I}]$ is isomorphic to the
$A$-subalgebra of $A^{\prime}=A\otimes_{R}R^{\prime}$ generated by
$\\{M_{i}\otimes r_{i}^{-1}\\}_{i\in I}$ and $A$.

We finish with an important description of dilatations in a particular case,
cf. [Ma23d, Proposition 5.5] and [StP, Tag 0BIQ].

######  Proposition 1.4.

Let $A$ be a ring. Let $a,g_{1},\ldots,g_{n}$ be a $H_{1}$-regular sequence in
$A$ (cf. [StP, Tag 062E] for $H_{1}$-regularity). Let $d_{1},\ldots,d_{n}$ be
positive integers. The dilatation algebra identifies with a quotient of a
polynomial algebra as follows

$A[\frac{(g_{1})}{a^{d_{1}}},\ldots,\frac{(g_{n})}{a^{d_{n}}}]=A[x_{1},\ldots,x_{n}]/(g_{1}-a^{d_{1}}x_{1},\ldots,g_{n}-a^{d_{n}}x_{n})$.

###  2 Dilatations of schemes

This section is an introduction to dilatations of schemes, the main references
are [MRR20] and [Ma23d]. Dilatations of schemes involve operations on closed
subschemes that we recall at the beginning of this section. We suggest readers
to be familiar with §2.1 before reading other subsections of Section 2. Note
that [Ma23d] deals with general algebraic spaces, in fact most results of
Section 2 extend to this setting.

####  2.1 Definitions

Let $X$ be a scheme. Let $Clo(X)$ be the set of closed subschemes of $X$.
Recall that $Clo(X)$ corresponds to quasi-coherent ideals of
${\mathcal{O}}_{X}$. Let $IQCoh({\mathcal{O}}_{X})$ denote the set of quasi-
coherent ideals of ${\mathcal{O}}_{X}$. It is clear that
$(IQCoh({\mathcal{O}}_{X}),+,\times,0,{\mathcal{O}}_{X})$ is a semiring. So we
obtain a semiring structure on $Clo(X)$, usually denoted by
$(Clo(X),\cap,+,X,\emptyset)$. For clarity, we now recall directly operations
on $Clo(X)$. Given two closed subschemes $Y_{1},Y_{2}$ given by ideals
${\mathcal{J}}_{1},{\mathcal{J}}_{2}$, their sum $Y_{1}+Y_{2}$ is defined as
the closed subscheme given by the ideal ${\mathcal{J}}_{1}{\mathcal{J}}_{2}$.
Moreover, if $n\in{\mathbb{N}}$, we denote by $nY_{1}$ the $n$-th multiple of
$Y_{1}$. The set of locally principal closed subschemes of $X$ (cf. [StP, Tag
01WR]), denoted $Pri(X)$, forms a submonoid of $(Clo(X),+)$. Effective Cartier
divisors of $X$ [StP, Tag 01WR], denoted $Car(X)$, form a submonoid of
$(Pri(X),+)$. Note that $Car(X)$ is a face of $Pri(X)$. We have an other
monoid structure on $Clo(X)$ given by intersection, this law is denoted
$\cap$. The operation $\cap$ corresponds to the sum of quasi-coherent sheaves
of ideals . The set $Clo(X)$ endowed with $\cap,+$ is a semiring whose neutral
element for $+$ is $\emptyset$ and whose neutral element for $\cap$ is $X$.
Let $C\in Car(X)$, a non-zero-divisor (for $+$) in the semiring $Clo(X)$. Let
$Y,Y^{\prime}\in Clo(X)$. If $C+Y$ is a closed subscheme of $C+Y^{\prime}$,
then $Y$ is a closed subscheme of $Y^{\prime}$. Moreover if
$C+Y=C+Y^{\prime}$, then $Y=Y^{\prime}$. Let $f:X^{\prime}\to X$ be a morphism
of schemes, then $f$ induces a morphism of semirings $Clo(f):Clo(X)\to
Clo(X^{\prime}),Y\mapsto Y\times_{X}X^{\prime}$, moreover $Clo(f)$ restricted
to $(Pri(X),+)$ factors through $(Pri(X^{\prime}),+)$, this morphism of
monoids is denoted $Pri(f)$. In general the image of the map
$Pri(f)|_{Car(X)}$ is not included in $Car(X^{\prime}).$ Let $Y_{1},Y_{2}\in
Clo(X)$, we write $Y_{1}\subset Y_{2}$ if $Y_{1}$ is a closed subscheme of
$Y_{2}$. We obtain a poset $(Clo(X),\subset)$. Let $Y_{1},Y_{2},Y_{3}\in
Clo(X)$, if $Y_{1}\subset Y_{2}$ and $Y_{1}\subset Y_{3}$ then $Y_{1}\subset
Y_{2}\cap Y_{3}$. Let $Y_{1},Y_{2}\in Clo(X)$, then $(Y_{1}\cap Y_{2})\subset
Y_{1}$ and $Y_{1}\subset(Y_{1}+Y_{2})$. Finally, if $Y=\\{Y_{e}\\}_{e\in E}$
is a subset of $Clo(X)$ and if $\nu\in{\mathbb{N}}^{E}$, we put
$Y^{\nu}=\\{\nu_{e}Y_{e}\\}_{e\in E}$ and if moreover
$\nu\in{\mathbb{N}}_{E}$, we put $\nu Y=\sum_{e\in E}\nu_{e}Y_{e}$.

######  Definition 2.1 ([MRR20, §2.3] [Ma23d]).

Let $D=\\{D_{i}\\}_{i\in I}$ be a subset of $Clo(X)$. Let
${\operatorname{Sch}}_{X}^{D\text{-reg}}$ be the full subcategory of schemes
$f:T\to X$ over $X$ such that $T\times_{X}D_{i}$ is an effective Cartier
divisor of $T$ for each $i$.

If $T^{\prime}\to T$ is flat and $T\to X$ is an object in
${\operatorname{Sch}}_{X}^{D\text{-reg}}$, so is the composition
$T^{\prime}\to T\to X$. In particular, the category
${\operatorname{Sch}}_{X}^{D\text{-reg}}$ can be equipped with the
fpqc/fppf/étale/Zariski Grothendieck topology so that the notion of sheaves is
well-defined.

######  Fact 2.2 ([Ma23d]).

Let $D=\\{D_{i}\\}_{i\in I}$ be a subset of $Clo(X)$.

  1. 1.

Let $f:T\to X$ be an object in ${\operatorname{Sch}}_{X}^{D\text{-reg}}$. Then
for any $\nu\in{\mathbb{N}}_{I}$, the scheme $T\times_{X}\nu D$ is an
effective Cartier divisor of $T$, namely $\nu(T\times_{X}D)$.

  2. 2.

Assume that $\\#I$ is finite, then ${\operatorname{Sch}}_{X}^{D\text{-reg}}$
equals ${\operatorname{Sch}}_{X}^{\sum_{i\in I}D_{i}}$.

######  Definition 2.3 ([Ma23d]).

A multi-center in $X$ is a set $\\{[Y_{i},D_{i}]\\}_{i\in I}$ such that

  1. 1.

$Y_{i}$ and $D_{i}$ belong to $Clo(X)$,

  2. 2.

there exists an affine open covering $\\{U_{\gamma}\to X\\}_{\gamma\in\Gamma}$
of $X$ such that $D_{i}|_{U_{\gamma}}$ is principal for all $i\in I$ and
$\gamma\in\Gamma$ (in particular $D_{i}$ belongs to $Pri(X)$ for all $i$).

In other words a multi-center $\\{[Y_{i},D_{i}]\\}_{i\in I}$ is a set of pairs
of closed subschemes such that locally each $D_{i}$ is principal.

######  Remark 2.4.

Let $\\{Y_{i},D_{i}\\}_{i\in I}$ such that $Y_{i}\in Clo(X)$ and $D_{i}\in
Pri(X)$ for any $i\in I$. Assume that $I$ is finite, then
$\\{[Y_{i},D_{i}]\\}_{i\in I}$ is a multi-center in $X$, i.e. the second
condition in Definition 2.3 is satisfied.

We now fix a multi-center $\\{[Y_{i},D_{i}]\\}_{i\in I}$ in $X$. Denote by
${\mathcal{M}}_{i}$, respectively ${\mathcal{J}}_{i}$, the quasi-coherent
sheaf of ideals of ${\mathcal{O}}_{X}$ defining $Y_{i}$, respectively $D_{i}$.
We put $Z_{i}=Y_{i}\cap D_{i}$ and
${\mathcal{L}}_{i}={\mathcal{M}}_{i}+{\mathcal{J}}_{i}$ so that $Z_{i}$ is
defined by ${\mathcal{L}}_{i}$. We put $Y=\\{Y_{i}\\}_{i\in I}$,
$D=\\{D_{i}\\}_{i\in I}$ and $Z=\\{Z_{i}\\}_{i\in I}$. We now introduce
dilatations ${\mathcal{O}}_{X}$-algebras by glueing.

######  Definition and Proposition 2.5.

The dilatation of ${\mathcal{O}}_{X}$ with multi-center
$\\{[{\mathcal{M}}_{i},{\mathcal{J}}_{i}]\\}_{i\in I}$ is the quasi-coherent
${\mathcal{O}}_{X}$-algebra
${\mathcal{O}}_{X}\Big{[}\Big{\\{}\frac{{\mathcal{M}}_{i}}{{\mathcal{J}}_{i}}\Big{\\}}_{i\in
I}\Big{]}$ obtained by glueing as follows. The quasi-coherent
${\mathcal{O}}_{X}$-algebra
${\mathcal{O}}_{X}\Big{[}\Big{\\{}\frac{{\mathcal{M}}_{i}}{{\mathcal{J}}_{i}}\Big{\\}}_{i\in
I}\Big{]}$ is characterized by the fact that its restriction, on any open
subscheme $U\subset X$ such that $U$ is an affine scheme and each $D_{i}$ is
principal on $U$ and generated by $a_{iU}$, is given by

$\Big{(}{\mathcal{O}}_{X}\Big{[}\Big{\\{}\frac{{\mathcal{M}}_{i}}{{\mathcal{J}}_{i}}\Big{\\}}_{i\in
I}\Big{]}\Big{)}_{\big{|}_{U}}=\widetilde{\Gamma(U,{\mathcal{O}}_{X})\Big{[}\Big{\\{}\frac{\Gamma(U,{\mathcal{M}}_{i})}{a_{iU}}\Big{\\}}_{i\in
I}}\Big{]}$

where $\widetilde{~{}~{}~{}~{}~{}~{}~{}~{}~{}}$ is the associated sheaf of
algebras on $U$.

######  Definition 2.6 ([Ma23d]).

The dilatation of $X$ with multi-center $\\{[Y_{i},D_{i}]\\}_{i\in I}$ is the
$X$-affine scheme

${\operatorname{Bl}}_{Y}^{D}X\hskip
2.84544pt\stackrel{{\scriptstyle\text{\tiny\rm def}}}{{=}}\hskip
2.84544pt{\operatorname{Spec}}_{X}\big{(}{\mathcal{O}}_{X}\Big{[}\Big{\\{}\frac{{\mathcal{M}}_{i}}{{\mathcal{J}}_{i}}\Big{\\}}_{i\in
I}\Big{]}\big{)}.$

The terminologies affine blowups and affine modifications are also used.

######  Remark 2.7.

In the mono-centered case, this definition is the one of [MRR20]. If moreover
$D$ is a Cartier divisor, one has another equivalent definition (cf.
Proposition 2.14) that goes back to [KZ99] and [Du05].

######  Remark 2.8.

We always have ${\operatorname{Bl}}_{Y}^{D}X={\operatorname{Bl}}_{Z}^{D}X$.

######  Notation 2.9.

We will also use the notation
${\operatorname{Bl}}\big{\\{}_{Y_{i}}^{D_{i}}\big{\\}}_{i\in I}X$ and
${\operatorname{Bl}}_{\\{Y_{i}\\}_{i\in I}}^{\\{D_{i}\\}_{i\in I}}X$ to denote
${\operatorname{Bl}}_{Y}^{D}X$. If $I=\\{i\\}$ is a singleton we also use the
notation ${\operatorname{Bl}}_{Y_{i}}^{D_{i}}X$. If $K\subset I$, we sometimes
use the notation ${\operatorname{Bl}}_{\\{Y_{i}\\}_{i\in K},\\{Y_{i}\\}_{i\in
I\setminus K}}^{\\{D_{i}\\}_{i\in K},\\{D_{i}\\}_{i\in I\setminus K}}X$. If
$I=\\{1,\ldots,k\\}$, we use the notation
${\operatorname{Bl}}_{Y_{1},\ldots,Y_{k}}^{D_{1},\ldots,D_{k}}X$. Etc.

######  Definition 2.10.

We say that a morphism $f:X^{\prime}\to X$ is a dilatation morphism if $f$ is
equal to ${\operatorname{Bl}}\big{\\{}_{Y_{i}}^{D_{i}}\big{\\}}_{i\in I}X\to
X$ for some multi-center $\\{[Y_{i},D_{i}]\\}_{i\in I}.$

######  Fact 2.11.

[Du05, Ma23d] The dilatation morphism
${\operatorname{Bl}}\big{\\{}_{\emptyset}^{D_{i}}\big{\\}}_{i\in I}X\to X$ is
an open immersion. In other words, if $Y_{i}$ is the empty closed subscheme
defined by the ideal ${\mathcal{O}}_{X}$ for all $i\in I$, then
${\operatorname{Bl}}_{Y}^{D}X$ identifies with an open subscheme of $X$. In
this case, we say that
${\operatorname{Bl}}\big{\\{}_{\emptyset}^{D_{i}}\big{\\}}_{i\in I}X\to X$ is
a localization.

####  2.2 Exceptional divisors

We proceed with the notation from §2.1.

######  Proposition 2.12 ([MRR20, Ma23d]).

As closed subschemes of ${\operatorname{Bl}}_{Y}^{D}X$, one has, for all
$\nu\in\mathbb{N}_{I}$,

${\operatorname{Bl}}_{Y}^{D}X\times_{X}\nu
Z={\operatorname{Bl}}_{Y}^{D}X\times_{X}\nu D,$

which is an effective Cartier divisor on ${\operatorname{Bl}}_{Y}^{D}X$.

####  2.3 Relation to affine projecting cone

We proceed with the notation from §2.1 and assume that $\\{D_{i}\\}_{i\in I}$
belong to $Car(X)$. In this case, we can also realize
${\operatorname{Bl}}_{Y}^{D}X$ as a closed subscheme of the multi-centered
affine projecting cone associated to $X,Z$ and $D$.

######  Definition 2.13.

The affine projecting cone ${\mathcal{O}}_{X}$-algebra  with multi-center
$\\{[Z_{i}=V({\mathcal{L}}_{i}),D_{i}=V({\mathcal{J}}_{i})]\\}_{i\in I}$ is

${\operatorname{C}}_{{\mathcal{L}}}^{{\mathcal{J}}}{\mathcal{O}}_{X}\hskip
2.84544pt\stackrel{{\scriptstyle\text{\tiny\rm def}}}{{=}}\hskip
2.84544pt\bigoplus_{\nu\in\mathbb{N}_{I}}{\mathcal{L}}^{\nu}\otimes{\mathcal{J}}^{-\nu}.$

The affine projecting cone  of $X$ with multi-center
$\\{[Z_{i},D_{i}]\\}_{i\in I}$ is

${\operatorname{C}}_{Z}^{D}X\hskip
2.84544pt\stackrel{{\scriptstyle\text{\tiny\rm def}}}{{=}}\hskip
2.84544pt{\operatorname{Spec}}\big{(}{\operatorname{C}}_{{\mathcal{L}}}^{{\mathcal{J}}}{\mathcal{O}}_{X}\big{)}.$

######  Proposition 2.14 ([KZ99, Du05, MRR20, Ma23d]).

The dilatation ${\operatorname{Bl}}_{Z}^{D}X$ is the closed subscheme of the
affine projecting cone ${\operatorname{C}}_{Z}^{D}X$ defined by the equations
$\\{\varrho_{i}-1\\}_{i\in I}$, where for all $i\in I$,
$\varrho_{i}\in{\operatorname{C}}_{{\mathcal{L}}}^{{\mathcal{J}}}{\mathcal{O}}_{X}$
is the image of $1\in{\mathcal{O}}_{X}$ under the map

${\mathcal{O}}_{X}\cong{\mathcal{J}}_{i}\otimes{\mathcal{J}}_{i}^{-1}\subset{\mathcal{L}}_{i}\otimes{\mathcal{J}}_{i}^{-1}\subset{\operatorname{C}}_{{\mathcal{L}}}^{{\mathcal{J}}}{\mathcal{O}}_{X}.$

####  2.4 Description of the exceptional divisor in the mono-centered case

We proceed with the notation from §2.1 and assume $I=\\{i\\}$ is a singleton
and we ommit the subscripts $i$ in notation. We saw in Lemma 2.12 that the
preimage of the center
${\operatorname{Bl}}_{Z}^{D}X\times_{X}Z={\operatorname{Bl}}_{Z}^{D}X\times_{X}D$
is an effective Cartier divisor in ${\operatorname{Bl}}_{Z}^{D}X$. In order to
describe it following [MRR20], as before we denote by ${\mathcal{L}}$ and
${\mathcal{J}}$ the sheaves of ideals of $Z$ and $D$ in ${\mathcal{O}}_{X}$.
Also we let
${\mathcal{C}}_{Z/D}={\mathcal{L}}/({\mathcal{L}}^{2}+{\mathcal{J}})$ and
${\mathcal{N}}_{Z/D}={\mathcal{C}}_{Z/D}^{\vee}$ be the conormal and normal
sheaves of $Z$ in $D$.

######  Proposition 2.15 ([MRR20, Proposition 2.9]).

Assume that $X$ is a scheme. Assume that $D\subset X$ is an effective Cartier
divisor, and $Z\subset D$ is a regular immersion. Write
${\mathcal{J}}_{Z}:={\mathcal{J}}|_{Z}$.

  1. (1)

The exceptional divisor ${\operatorname{Bl}}_{Z}^{D}X\times_{X}Z\to Z$ is an
affine bundle (i.e. a torsor under a vector bundle), Zariski locally over $Z$
isomorphic to
${\mathbb{V}}({\mathcal{C}}_{Z/D}\otimes{\mathcal{J}}_{Z}^{-1})\to Z$.

  2. (2)

If $H^{1}(Z,{\mathcal{N}}_{Z/D}\otimes{\mathcal{J}}_{Z})=0$ (for example if
$Z$ is affine), then ${\operatorname{Bl}}_{Z}^{D}X\times_{X}Z\to Z$ is
globally isomorphic to
${\mathbb{V}}({\mathcal{C}}_{Z/D}\otimes{\mathcal{J}}_{Z}^{-1})\to Z$.

  3. (3)

If $Z$ is a transversal intersection in the sense that there is a cartesian
square of closed subschemes whose vertical maps are regular immersions

${\ignorespaces W}$ |  | ${\ignorespaces X}$ |
---|---|---|---
${\ignorespaces Z}$ |  | ${\ignorespaces D}$ |
${\ignorespaces\square}$

then ${\operatorname{Bl}}_{Z}^{D}X\times_{X}Z\to Z$ is globally and
canonically isomorphic to
${\mathbb{V}}({\mathcal{C}}_{Z/D}\otimes{\mathcal{J}}_{Z}^{-1})\to Z$.

####  2.5 Universal property

We proceed with the notation from §2.1. As ${\operatorname{Bl}}_{Y}^{D}X\to X$
defines an object in
${\operatorname{Sch}}_{X}^{D\text{-}{\operatorname{reg}}}$ by Proposition
2.12, the contravariant functor

${\operatorname{Sch}}_{X}^{D\text{-}{\operatorname{reg}}}\to{\operatorname{Sets}},\;\;\;(T\to
X)\mapsto{\operatorname{Hom}}_{X\text{-Schemes}}\big{(}T,{\operatorname{Bl}}_{Y}^{D}X\big{)}$
(2.1)

together with ${\operatorname{id}}_{{\operatorname{Bl}}_{Y}^{D}X}$ determines
${\operatorname{Bl}}_{Y}^{D}X\to X$ uniquely up to unique isomorphism. The
next proposition gives the universal property of dilatations.

######  Proposition 2.16 ([MRR20, Ma23d]).

The dilatation ${\operatorname{Bl}}_{Y}^{D}X\to X$ represents the
contravariant functor
${\operatorname{Sch}}_{X}^{D\text{-}{\operatorname{reg}}}\to{\operatorname{Sets}}$
given by

$(f{\colon}T\to X)\;\longmapsto\;\begin{cases}\\{*\\},\;\text{if
$f|_{T\times_{X}D_{i}}$ factors through $Y_{i}\subset X$ for $i\in I$;}\\\
\varnothing,\;\text{else.}\end{cases}$ (2.2)

######  Proposition 2.17 ([Ma23d]).

Put $f:{\operatorname{Bl}}_{Y}^{D}X\to X$. Then the morphism of monoids
$Clo(f)|_{Car(X)}$ factors through $Car({\operatorname{Bl}}_{Y}^{D}X)$. In
other words, any effective Cartier divisor $C\subset X$ is defined for $f$,
i.e. the fiber product
$C\times_{X}{\operatorname{Bl}}_{Y}^{D}X\subset{\operatorname{Bl}}_{Y}^{D}X$
is an effective cartier divisor.

####  2.6 Dilatations or affine blowups

We proceed with the notation from §2.1 and assume $I=\\{i\\}$ is a singleton
and we ommit the subscripts $i$ in notation.

######  Proposition 2.18 ( [MRR20]).

The dilatation ${\operatorname{Bl}}_{Z}^{D}X$ is the open subscheme of the
blowup
${\operatorname{Bl}}_{Z}X={\operatorname{Proj}}({\operatorname{Bl}}_{\mathcal{I}}{\mathcal{O}}_{X})$
defined by the complement of $V_{+}(\overline{\mathcal{J}})$ where
$\overline{\mathcal{J}}$ is the sheaf of ideals generated by
${\mathcal{J}}\subset{\mathcal{I}}$, where ${\mathcal{I}}$ is the degree 1
part of ${\operatorname{Bl}}_{\mathcal{I}}{\mathcal{O}}_{X}$.

For this reason dilatations are also called affine blowups. A similar
description holds in the multi-centered case, cf [Ma23d].

####  2.7 Combinatorial and arithmetic relations

We proceed with the notation from §2.1.

######  Proposition 2.19 ([Ma23d]).

Let $J$ be a subset of $I$ and put $K=I\setminus J$. Then

${\operatorname{Bl}}\big{\\{}_{Y_{i}}^{D_{i}}\big{\\}}_{i\in
I}X={\operatorname{Bl}}\Big{\\{}_{Y_{k}\times_{X}{\operatorname{Bl}}\big{\\{}_{Y_{i}}^{D_{i}}\big{\\}}_{i\in
J}X}^{D_{k}\times_{X}{\operatorname{Bl}}\big{\\{}_{Y_{i}}^{D_{i}}\big{\\}}_{i\in
J}X}\Big{\\}}_{k\in
K}{\operatorname{Bl}}\big{\\{}_{Y_{i}}^{D_{i}}\big{\\}}_{i\in J}X.$

In particular, there is a unique $X$-morphism

${\operatorname{Bl}}\big{\\{}_{Y_{i}}^{D_{i}}\big{\\}}_{i\in
I}X\to{\operatorname{Bl}}\big{\\{}_{Y_{i}}^{D_{i}}\big{\\}}_{i\in J}X.$

######  Proposition 2.20 ([Ma23d]).

Write $I={\operatorname{colim}}_{J\subset I}J$ as a filtered colimit of sets
where transition maps are given by inclusions of subsets. We have a canonical
identification

${\operatorname{Bl}}\big{\\{}^{D_{i}}_{Y_{i}}\big{\\}}_{i\in
I}X=\lim_{J\subset
I}{\operatorname{Bl}}\big{\\{}^{D_{i}}_{Y_{i}}\big{\\}}_{i\in J}X$.

######  Proposition 2.21 ([Ma23d]).

Assume that $\\#I=k$ is finite. We fix an arbitrary bijection
$I=\\{1,\ldots,k\\}$. We have a canonical isomorphism of $X$-schemes

${\operatorname{Bl}}_{\\{Y_{i}\\}_{i\in I}}^{\\{D_{i}\\}_{i\in
I}}X\cong{\operatorname{Bl}}^{({\operatorname{Bl}}\cdots)\times_{X}D_{k}}_{({\operatorname{Bl}}\cdots)\times_{X}Y_{k}}\Biggl{(}\cdots{\operatorname{Bl}}_{({\operatorname{Bl}}\cdots)\times_{X}Y_{3}}^{({\operatorname{Bl}}\cdots)\times_{X}D_{3}}\biggl{(}{\operatorname{Bl}}_{({\operatorname{Bl}}_{Y_{1}}^{D_{1}}X)\times_{X}Y_{2}}^{({\operatorname{Bl}}_{Y_{1}}^{D_{1}}X)\times_{X}D_{2}}\bigl{(}{\operatorname{Bl}}_{Y_{1}}^{D_{1}}X\bigl{)}\biggl{)}\Biggl{)}.$

######  Proposition 2.22 ([Ma23d]).

Assume that $\\#I=k$ is finite. We fix an arbitrary bijection
$I=\\{1,\ldots,k\\}$. We have a canonical isomorphism of $X$-schemes

${\operatorname{Bl}}_{\\{Y_{i}\\}_{i\in I}}^{\\{D_{i}\\}_{i\in
I}}X\cong{\operatorname{Bl}}_{\bigcap_{i\in
I}(Y_{i}+D_{1}+\ldots+D_{i-1}+D_{i+1}+\ldots+D_{k})}^{D_{1}+\ldots+D_{k}}X.$

####  2.8 Functoriality

We proceed with the notation from §2.1. Let $X^{\prime}$ and
$\\{[Y^{\prime}_{i},D^{\prime}_{i}]\\}_{i\in I}$ be another datum as in §2.1.
As usual, put $Z^{\prime}_{i}=Y_{i}^{\prime}\cap D^{\prime}_{i}$. A morphism
$f:X^{\prime}\to X$ such that, for all $i\in I$, its restriction to
$D^{\prime}_{i}$ (resp. ${Z^{\prime}_{i}}$) factors through $D_{i}$ (resp.
$Z_{i}$), and such that $f^{-1}(D_{i})=D_{i}^{\prime}$, induces a unique
morphism
${\operatorname{Bl}}_{Y^{\prime}}^{D^{\prime}}X^{\prime}\to{\operatorname{Bl}}_{Y}^{D}X$
such that the following diagram of schemes commutes

${\ignorespaces{\operatorname{Bl}}_{Y^{\prime}}^{D^{\prime}}X^{\prime}}$ |  | ${\ignorespaces{\operatorname{Bl}}_{Y}^{D}X}$ |
---|---|---|---
${\ignorespaces X^{\prime}}$ |  | ${\ignorespaces X.}$ |

####  2.9 Base change

We proceed with the notation from §2.1. Let $X^{\prime}\to X$ be a map of
schemes, and denote by $Y^{\prime}_{i},Z^{\prime}_{i},D^{\prime}_{i}\subset
X^{\prime}$ the preimage of $Y_{i},Z_{i},D_{i}\subset X$. Then
$D^{\prime}_{i}\subset X^{\prime}$ is locally principal for any $i$ so that
the dilatation ${\operatorname{Bl}}_{Y^{\prime}}^{D^{\prime}}X^{\prime}\to
X^{\prime}$ is well-defined. By §2.8 there is a canonical morphism of
$X^{\prime}$-schemes

${\operatorname{Bl}}_{Y^{\prime}}^{D^{\prime}}X^{\prime}\;\longrightarrow\;{\operatorname{Bl}}_{Y}^{D}X\times_{X}X^{\prime}.$
(2.3)

######  Lemma 2.23 ([MRR20, Ma23d]).

If ${\operatorname{Bl}}_{Y}^{D}X\times_{X}X^{\prime}\to X^{\prime}$ is an
object of ${\operatorname{Sch}}_{X^{\prime}}^{D\text{-}{\operatorname{reg}}}$,
then (2.3) is an isomorphism.

######  Corollary 2.24 ([MRR20, Ma23d]).

If the morphism $X^{\prime}\to X$ is flat and satisfies a property
${\mathcal{P}}$ which is stable under base change, then
${\operatorname{Bl}}_{Y^{\prime}}^{D^{\prime}}X^{\prime}\to{\operatorname{Bl}}_{Y}^{D}X$
is flat and satisfies ${\mathcal{P}}$.

####  2.10 Iterated multi-centered dilatations

We proceed with the notation from § 2.1. Let $\nu,\theta\in{\mathbb{N}}^{I}$
such that $\theta\leqslant\nu$, i.e. $\theta_{i}\leqslant\nu_{i}$ for all
$i\in I$.

######  Proposition 2.25 ([Ma23d]).

There is a unique $X$-morphism

$\varphi_{\nu,\theta}:{\operatorname{Bl}}_{Y}^{D^{\nu}}X\longrightarrow{\operatorname{Bl}}_{Y}^{D^{\theta}}X.$

Assume now moreover that
$\nu,\theta\in{\mathbb{N}}_{I}\subset{\mathbb{N}}^{I}.$ We will prove that,
under some assumptions, $\varphi_{\nu,\theta}$ is a dilatation morphism with
explicit descriptions. We need the following observation.

######  Proposition 2.26 ([MRR20, Ma23d]).

Assume that we have a commutative diagram of schemes

${\ignorespaces B}$ |  |  |  | ${\ignorespaces{\operatorname{Bl}}_{Y}^{D}X}$ |
---|---|---|---|---|---
|  | ${\ignorespaces X}$ |
$\scriptstyle{\ignorespaces f^{\prime}}$$\scriptstyle{\ignorespaces f}$

where the right-hand side morphism is the dilatation map. Assume that $f$ is a
closed immersion. Then $f^{\prime}$ is a closed immersion.

We now assume that $Z_{i}\subset Y_{i}$ is a Cartier divisor inclusion for all
$i\in I$. Let $\mathcal{D}_{i}$ be the canonical diagram of closed immersions

${\ignorespaces Y_{i}}$ |  | ${\ignorespaces{\operatorname{Bl}}_{Y_{i}}^{\nu_{i}D_{i}}X}$ |
---|---|---|---
${\ignorespaces Z_{i}}$ |  | ${\ignorespaces D_{i}}$ |
${\ignorespaces\square}$

obtained by Proposition 2.26. Let $f_{i}$ be the canonical morphism (e.g. cf.
2.19 or 2.25)

${\operatorname{Bl}}_{Y}^{D^{\nu}}X\to{\operatorname{Bl}}_{Y_{i}}^{{\nu_{i}}D_{i}}X.$

We denote by
$Y_{i}\times_{{\operatorname{Bl}}_{Y_{i}}^{{\nu_{i}}D_{i}}X}{\operatorname{Bl}}_{Y}^{D^{\nu}}X$
the fiber product obtained via the arrows given by $f_{i}$ and
$\mathcal{D}_{i}$. We use similarly the notation
$D_{i}\times_{{\operatorname{Bl}}_{Y_{i}}^{{\nu_{i}}D_{i}}X}{\operatorname{Bl}}_{Y}^{D^{\nu}}X$.

######  Proposition 2.27 ([PY06, §7.2] [MRR20, Ma23d]).

Recall that $\theta\leqslant\nu$. Put $\gamma=\nu-\theta$. Put $K=\\{i\in
I|\gamma_{i}>0\\}.$ We have an identification

${\operatorname{Bl}}_{Y}^{D^{\nu}}X={\operatorname{Bl}}^{\\{{{\gamma_{i}}D_{i}\times_{{\operatorname{Bl}}_{Y_{i}}^{\theta_{i}D_{i}}X}{\operatorname{Bl}}_{Y}^{D^{\theta}}X\\}_{i\in
K}}}_{\\{Y_{i}\times_{{\operatorname{Bl}}_{Y_{i}}^{\theta_{i}D_{i}}X}{\operatorname{Bl}}_{Y}^{D^{\theta}}X\\}_{i\in
K}}{\operatorname{Bl}}_{Y}^{D^{\theta}}X.$

In particular the unique $X$-morphism

$\varphi_{\nu,\theta}:{\operatorname{Bl}}_{Y}^{D^{\nu}}X\longrightarrow{\operatorname{Bl}}_{Y}^{D^{\theta}}X$

of Proposition 2.25 is a dilatation map.

It is now natural to introduce the following terminology.

######  Definition 2.28.

For any $\nu\in{\mathbb{N}}^{k},$ let us consider

${\operatorname{Bl}}_{Y}^{D^{\nu}}X={\operatorname{Bl}}\big{\\{}_{Y_{i}}^{\nu_{i}D_{i}}\big{\\}}_{i\in
I}X$

and call it the $\nu$-th iterated dilatation of $X$ with multi-center
$\\{Y_{i},D_{i}\\}_{i\in I}$.

####  2.11 Some flatness and smoothness results

We proceed with the notation from §2.1 and assume $I=\\{i\\}$ is a singleton
and we ommit the subscripts $i$ in notation. We assume further that there
exists a scheme $S$ under $X$ together with a locally principal closed
subscheme $S_{0}\subset S$ fitting into a commutative diagram of schemes

${\ignorespaces Z}$ |  | ${\ignorespaces D}$ |  | ${\ignorespaces X}$ |
---|---|---|---|---|---
|  | ${\ignorespaces S_{0}}$ |  | ${\ignorespaces S,}$ |
(2.4)

where the square is cartesian, that is $D\to X_{0}:=X\times_{S}S_{0}$ is an
isomorphism.

######  Proposition 2.29 ([MRR20]).

Assume that $S_{0}$ is an effective Cartier divisor on $S$.

  1. (1)

If $Z\subset D$ is regular, then ${\operatorname{Bl}}_{Z}^{D}X\to X$ is of
finite presentation.

  2. (2)

If $Z\subset D$ is regular, the fibers of
${\operatorname{Bl}}_{Z}^{D}X\times_{S}S_{0}\to S_{0}$ are connected (resp.
irreducible, geometrically connected, geometrically irreducible) if and only
if the fibers of $Z\to S_{0}$ are.

  3. (3)

If $X\to S$ is flat and if moreover one of the following holds:

     * (i)

$Z\subset D$ is regular, $Z\to S_{0}$ is flat and $S,X$ are locally
noetherian,

     * (ii)

$Z\subset D$ is regular, $Z\to S_{0}$ is flat and $X\to S$ is locally of
finite presentation,

     * (iii)

the local rings of $S$ are valuation rings,

then ${\operatorname{Bl}}_{Z}^{D}X\to S$ is flat.

  4. (4)

If both $X\to S$, $Z\to S_{0}$ are smooth, then
${\operatorname{Bl}}_{Z}^{D}X\to S$ is smooth.

######  Remark 2.30.

Complementary smoothness and flatness results for multi-centered dilatations
can be found in [Ma23d, §6].

####  2.12 Remarks

Dilatations commute with algebraic attractors [Ma23a, Proposition 13.1].

###  3 Dilatations of group schemes or Néron blowups

One of the key properties allowed by dilatations is that it preserves the
structure of group schemes in many cases. Dilatations of group schemes are
also called Néron blowups and we also often use this terminology.

####  3.1 Definitions of multi-centered Néron blowups

Let $S$ be a scheme and $G\to S$ a group scheme. Let $C=\\{C_{i}\\}_{i\in I}$
be a set of locally principal closed subschemes of $S$. Put
$G|_{C_{i}}=G\times_{S}C_{i}$ and ${G}|_{C}=\\{G|_{C_{i}}\\}$. Let
$H_{i}\subset G|_{C_{i}}$ be a closed subgroup scheme over $C_{i}$ for all
$i\in I$ and let $H=\\{H_{i}\\}$. The multi-centered dilatation

${\mathcal{G}}:={\operatorname{Bl}}_{H}^{{G}|_{C}}G\longrightarrow G$

is called the Néron blowup of $G$ with multi-center $H,{G}|_{C}$. We also use
the notation ${\operatorname{Bl}}_{H}^{C}G$ to denote ${\mathcal{G}}$. In the
case $I$ has a single element, we shall refer to
${\operatorname{Bl}}_{H}^{C}G$ as mono-centred Néron blowups By Proposition
2.12 the structural morphism ${\mathcal{G}}\to S$ defines an object in
${\operatorname{Sch}}_{S}^{C\text{-}{\operatorname{reg}}}$.

######  Proposition 3.1 ([MRR20, Ma23d]).

Let ${\mathcal{G}}\to S$ be the above multi-centered Néron blowup.

  1. (1)

The $S$-scheme ${\mathcal{G}}$ represents the contravariant functor
${\operatorname{Sch}}_{S}^{C\text{-}{\operatorname{reg}}}\to{\operatorname{Sets}}$

$T\,\,\longmapsto\,\,\left\\{T\to
G\,\,:\,\,\begin{array}[]{c}\text{$T|_{C_{i}}\to G|_{C_{i}}$ factors
through}\\\ \text{$H_{i}\subset G|_{C_{i}}$ for all $i$}\end{array}\right\\}.$

  2. (2)

Let $T\to S$ be an object in
${\operatorname{Sch}}_{S}^{C\text{-}{\operatorname{reg}}}$, then as subsets of
$G(T)$

${\mathcal{G}}(T)=\bigcap_{i\in
I}\big{(}{\operatorname{Bl}}_{H_{i}}^{C_{i}}G\big{)}(T).$

  3. (3)

The map ${\mathcal{G}}\to G$ is affine. Its restriction over $C_{i}$ factors
as ${\mathcal{G}}_{i}\to H_{i}\subset G|_{C_{i}}$ for all $i$

  4. (4)

If the Néron blowup ${\mathcal{G}}\to S$ is flat, then it is equipped with the
structure of a group scheme such that ${\mathcal{G}}\to G$ is a morphism of
$S$-group schemes.

######  Remark 3.2.

We saw that in favorable cases, dilatations preserve group scheme structures.
In fact dilatations preserve similarly monoid scheme structures and Lie
algebra schemes structures, or more generally structures defined by products,
cf. [Ma23d, §7] for details.

######  Remark 3.3.

Dilatations commute with the formation of Lie algebra schemes in a natural
sense

${\mathbb{L}}ie({\operatorname{Bl}}_{H}^{G|_{C}}G)\cong{\operatorname{Bl}}\big{\\{}{}^{{\mathbb{L}}ie(G)\times_{S}C_{i}}_{{\mathbb{L}}ie(H_{i})}\big{\\}}_{i\in
I}{\mathbb{L}}ie(G)$

cf. [Ma23d, §7] for precise flatness assumptions.

####  3.2 Mono-centered Néron blowups

We proceed with the notation from §3.1 and now deal with the mono-centered
case, so now $k=1$. We put $S_{0}=C_{1}$ and $H=H_{1}$. We also put
$G_{0}=G\times_{S}S_{0},H_{0}=H\times_{S}S_{0}$ and $K_{0}=K\times_{S}S_{0}$.

######  Proposition 3.4 ([Ma23d]).

Assume that $S_{0}$ is a Cartier divisor in $S$ and $G\to S$ is flat. Let
$\eta:K\to G$ be a morphism of group schemes over $S$ such that $K\to S$ is
flat. Assume that $H\subset G$ is a closed subgroup scheme over $S$ such that
$H\to S$ is flat and ${\operatorname{Bl}}_{H}^{S_{0}}G\to S$ is flat (and in
particular a group scheme). Assume that $K_{0}$ commutes with $H_{0}$ in the
sense that the morphism $K_{0}\times_{S_{0}}H_{0}\to G_{0}$,
$(k,h)\mapsto\eta(k)h\eta(k)^{-1}$ equals the composition morphism
$K_{0}\times_{S_{0}}H_{0}\to H_{0}\subset G_{0}$, $(k,h)\mapsto h$. Then $K$
normalizes ${\operatorname{Bl}}_{H}^{S_{0}}G$, more precisely the solid
composition map

${\ignorespaces K\times_{S}{\operatorname{Bl}}_{H}^{S_{0}}G}$ |  |  | ${\ignorespaces K\times_{S}G}$ |  |  |  | ${\ignorespaces G}$ |
---|---|---|---|---|---|---|---|---
|  |  | ${\ignorespaces{\operatorname{Bl}}_{H}^{S_{0}}G}$ |  |  |
$\scriptstyle{\ignorespaces
Id\times{\operatorname{Bl}}}$$\scriptstyle{\ignorespaces
k{,}g\mapsto\eta(k)g\eta(k)^{-1}}$$\scriptstyle{\ignorespaces{\operatorname{Bl}}}$

factors uniquely through ${\operatorname{Bl}}_{H}^{S_{0}}G$.

######  Theorem 3.5 ([WW80, MRR20]).

Assume that $G\to S$ is flat, locally finitely presented and $H\to S_{0}$ is
flat, regularly immersed in $G_{0}$. Let ${\mathcal{G}}\to G$ be the
dilatation ${\operatorname{Bl}}_{H}^{S_{0}}G$ with exceptional divisor
${\mathcal{G}}_{0}:={\mathcal{G}}\times_{S}S_{0}$. Let ${\mathcal{J}}$ be the
ideal sheaf of $G_{0}$ in $G$ and ${\mathcal{J}}_{H}:={\mathcal{J}}|_{H}$. Let
$V$ be the restriction of the normal bundle
${\mathbb{V}}({\mathcal{C}}_{H/G_{0}}\otimes{\mathcal{J}}_{H}^{-1})\to H$
along the unit section $e_{0}{\colon}S_{0}\to H$.

  1. (1)

Locally over $S_{0}$, there is an exact sequence of $S_{0}$-group schemes
$1\to V\to{\mathcal{G}}_{0}\to H\to 1$.

  2. (2)

Assume given a lifting of $H$ to a flat $S$-subgroup scheme of $G$. Then there
is globally an exact, canonically split sequence $1\to
V\to{\mathcal{G}}_{0}\to H\to 1$.

  3. (3)

If $G\to S$ is smooth, separated and ${\mathcal{G}}\to G$ is the dilatation of
the unit section of $G$, there is a canonical isomorphism of smooth
$S_{0}$-group schemes ${\mathcal{G}}_{0}\begin{array}[b]{c}{}_{\sim}\\\
\longrightarrow\end{array}{\operatorname{Lie}}(G_{0}/S_{0})\otimes{\operatorname{N}}_{S_{0}/S}^{-1}$
where ${\operatorname{N}}_{S_{0}/S}$ is the normal bundle of $S_{0}$ in $S$.

######  Remark 3.6.

In the situation of Theorem 3.5 (2), the group $H$ acts by conjugation on
$V={\mathbb{V}}(e_{0}^{*}{\mathcal{C}}_{H/G_{0}}\otimes{\mathcal{J}}_{S_{0}}^{-1})$.
It is expected that this additive action is linear, and is in fact none other
than the “adjoint” representation of $H$ on its normal bundle as in [SGA3,
Exp. I, Prop. 6.8.6]. When the base scheme is the spectrum of a discrete
valuation ring this is proved in [DHdS18, Prop. 2.7].

Assume now that $j{\colon}S_{0}\hookrightarrow S$ is an effective Cartier
divisor, that $G\to S$ is a flat, locally finitely presented group scheme and
that $H\subset G_{0}:=G\times_{S}S_{0}$ is a flat, locally finitely presented
closed $S_{0}$-subgroup scheme. In this context, there is another viewpoint on
the dilatation ${\mathcal{G}}$ of $G$ in $H$, namely as the kernel of a
certain map of syntomic sheaves. To explain this, let $f{\colon}G_{0}\to
G_{0}/H$ be the morphism to the fppf quotient sheaf, which by Artin’s theorem
([Ar74, Cor. 6.3] and [StP, 04S6]) is representable by an algebraic space. By
the structure theorem for algebraic group schemes (see [SGA3, Exp.
VII${}_{\operatorname{B}}$, Cor. 5.5.1]) the morphisms $G\to S$ and $H\to
S_{0}$ are syntomic. Since $f{\colon}G_{0}\to G_{0}/H$ makes $G_{0}$ an
$H$-torsor, it follows that $f$ is syntomic also.

######  Proposition 3.7 ([MRR20, Lemma 3.8]).

Let $S_{{\operatorname{syn}}}$ be the small syntomic site of $S$. Let
$\eta{\colon}G\to j_{*}j^{*}G$ be the adjunction map in the category of
sheaves on $S_{{\operatorname{syn}}}$ and consider the composition
$v=(j_{*}f)\circ\eta$:

${\ignorespaces G}$ |  | ${\ignorespaces j_{*}j^{*}G=j_{*}G_{0}}$ |  | ${\ignorespaces j_{*}(G_{0}/H).}$ |
---|---|---|---|---|---
$\scriptstyle{\ignorespaces\eta}$$\scriptstyle{\ignorespaces j_{*}f}$

Then the dilatation ${\mathcal{G}}\to G$ is the kernel of $v$. More precisely,
we have an exact sequence of sheaves of pointed sets in
$S_{{\operatorname{syn}}}$:

${\ignorespaces 1}$ |  | ${\ignorespaces{\mathcal{G}}}$ |  | ${\ignorespaces G}$ |  | ${\ignorespaces j_{*}(G_{0}/H)}$ |  | ${\ignorespaces 1.}$ |
---|---|---|---|---|---|---|---|---|---
$\scriptstyle{\ignorespaces v}$

If $G\to S$ and $H\to S_{0}$ are smooth, then the sequence is exact as a
sequence of sheaves on the small étale site of $S$.

As a corollary, one has the useful and typical following result.

######  Corollary 3.8.

[MRR20] Let ${\mathcal{O}}$ be a ring and $\pi\subset{\mathcal{O}}$ an
invertible ideal such that $({\mathcal{O}},\pi)$ is a henselian pair. Let $G$
be a smooth, separated ${\mathcal{O}}$-group scheme and ${\mathcal{G}}\to G$
the dilatation of the trivial subgroup over ${\mathcal{O}}/\pi$. If either
${\mathcal{O}}$ is local or $G$ is affine, then the exact sequence of
Proposition 3.7 induces an exact sequence of groups:

$1\longrightarrow{\mathcal{G}}({\mathcal{O}})\longrightarrow
G({\mathcal{O}})\longrightarrow G({\mathcal{O}}/\pi)\longrightarrow 1.$

##  Part II Some applications

###  4 Models of group schemes, representation categories and Tannakian groups

In several mathematical theories, one finds the structure of a category with a
tensor product, and one of the main goals of categorical Tannakian theory is
to realize the latter categories as representations of group schemes. If we
deal with categories over a field, and this is a somewhat well-known area with
[DM82] being a fundamental reference, dilatations have not played a role. In
the case we deal with categories which are linear over a discrete valuation
ring, a Dedekind domain, or more complicated rings, the outputs are much
scarcer and the main reference is the beautiful, yet arid, monograph [S72].
But in this situation, dilatations have played a role. Following [dS09], N. D.
Duong and P. H. Hai [DH18] went into technical aspects of [S72] and produced a
more contemporaneous text to study tensor categories over Dedekind domain.
This prompted further study [DHdS18, HdS21]; in these papers, the authors
begin to look at Néron blowups (in the sense of Section 3) and the resulting
categories systematically. It is also useful to mention here the paper [CK10],
where the idea of looking at representation categories of Néron blowups
already appears. In this section we fix a discrete valuation ring $R$ with
uniformizer $\pi$, residue field $k$ and fraction field $K$. We put
$S={\operatorname{Spec}}(R)$ and $S_{i}={\operatorname{Spec}}(R/(\pi^{i+1}))$
for $i\in{\mathbb{N}}$.

####  4.1 Group schemes from categories

Let $\mathcal{T}$ be a neutral Tannakian category over $R$ in the sense of
[DH18, Definition 1.2.5]. The reader having encountered only (neutral)
Tannakian categories over fields [DM82, Section 2] should note that the
distinctive property of $\mathcal{T}$ is a weakening of the existence of
“duals” [DM82, Definition 1.7]. This is to be replaced by the property that
every object is a quotient of an “object having a dual.” That this property
holds for representation categories of group schemes is [Se68, Proposition 3].
(For a higher dimensional bases, see [Th87, Lema 2.5].) But we face a non-
trivial requirement: for example,
$\mathrm{Rep}_{W(\overline{\mathbb{F}}_{p})}(\overline{\mathbb{F}}_{p})$ fails
to satisfy it [HdS21, Example 4.7]. Once this definition of neutral Tannakian
category is given, the main theorem of [DM82, Theorem 2.11] has his analogue
in the present context: If $\omega:\mathcal{T}\to R\text{-}\mathbf{mod}$ is a
faithful, $R$–linear and exact tensor functor, then there exists an affine and
flat group scheme $\Pi_{\mathcal{T}}$ over $R$ and an equivalence

$\overline{\omega}:\mathcal{T}\longrightarrow\mathrm{Rep}_{R}(\Pi_{\mathcal{T}})$

such that composing $\overline{\omega}$ with the forgetful functor
$\mathrm{Rep}_{R}(\Pi_{\mathcal{T}})\to R\text{-}\mathbf{mod}$ renders us
$\omega$ back. See [S72, II.4.1.1] and [DH18, Theorem 1.2.2]. Let us present
some examples of categories to which the theory can be applied.

######  Example 4.1.

Let $\Gamma$ be an abstract group and suppose that
$R=k\llbracket\pi\rrbracket$. Then, the category of $R[\Gamma]$-modules which
are of finite type over $R$ together with the forgetful functor is a neutral
Tannakian category [HdS21, 4.1].

######  Example 4.2.

Let $X$ be a smooth and connected scheme over $R$, $\mathcal{D}_{X/R}$ the
ring of differential operators [EGA, IV.16.8], and $\mathcal{T}^{+}$ the
category of $\mathcal{D}_{X/R}$-modules which, as $\mathcal{O}_{X}$–modules,
are coherent. Using the fibre-by-fibre flatness criterion and [BO78], one
proves that an object $E\in\mathcal{T}^{+}$ is locally free if and only if it
is $R$-flat. Let now $\mathcal{T}$ be the full subcategory of
$\mathcal{T}^{+}$ having

$\left\\{M\in\mathcal{T}^{+}\,:\,\begin{array}[]{c}\text{There exists
$E\in\mathcal{T}^{+}$ which }\\\ \text{is $R$-flat and a surjection $E\to
M$}\end{array}\right\\}$

as objects. Once we give ourselves an $R$-point $x_{0}\in X(R)$, it follows
that

$\mathcal{T}\longrightarrow R\text{-}\mathbf{mod},\qquad
E\longmapsto\text{(global sections of) $x_{0}^{*}(E)$}$

defines a neutral Tannakian category. For more details, see [And01] and
[DHdS18].

######  Example 4.3.

We assume that $R$ is Henselian and Japanese, e.g. $R$ is complete. Let $X$ be
an irreducible, proper and flat $R$-scheme with geometrically reduced fibres.
Let $x_{0}\in X(R)$. Given a coherent sheaf $E$ on $X$, we say that $E$ is
trivialized by a proper morphism if there exists a surjective and proper
morphism $\psi:Y\to X$ such that $\psi^{*}E$ “comes from
$S=\mathrm{Spec}\,R$”, by which we mean that $\psi^{*}E$ is the pull-back of a
module via the structural morphism $Y\to S$. Let $\mathcal{T}^{+}$ be the full
subcategory of the category of coherent modules on $X$ having as objects those
sheaves which are trivialized by a proper morphism. Proceeding along the lines
of Example 4.2, it is possible to construct a smaller full subcategory
$\mathcal{T}$ of $\mathcal{T}^{+}$ such that, endowing $\mathcal{T}$ with the
tensor product of sheaves, the functor

$\mathcal{T}\longrightarrow R\text{-}\mathbf{mod},\qquad
E\longmapsto\text{(global sections of) $x_{0}^{*}(E)$}$

defines a neutral Tannakian category. Details are in [HdS23]. This is the
analogue theory of Nori’s theory for the fundamental group scheme [Nor76] in
the relative setting, and one objective is to show that the group scheme
associated to $\mathcal{T}$ is pro-finite. See [HdS23, Theorem 8.8].

####  4.2 Galois-Tannaka group schemes

One obvious strategy to study Tannakian categories is to filter them by
categories “generated” by a single object, just as in studying Galois groups
it is fundamental to study finite extensions. Let $\omega:\mathcal{T}\to
R\text{-}\mathbf{mod}$ be as in the previous section so that $\mathcal{T}$ is
equivalent to $\mathrm{Rep}_{R}(\Pi)$ for some affine and flat group scheme
$\Pi$. We shall take this equivalence as an equality, but we warn the reader
that the structure of $\Pi$ should be considered as being very complicated
(just as is that of an absolute Galois group).

######  Definition 4.4.

Let $M\in\mathcal{T}$ be an object possessing a dual $M^{\vee}$ and for each
couple of non-negative integers $a,b$, define $\mathbf{T}^{a,b}M$ as
$M^{\otimes a}\otimes M^{\vee\,\otimes b}$. Then, $\langle M\rangle_{\otimes}$
is the full subcategory of $\mathcal{T}$ having as objects those which are
quotients of subobjects of elements of the form

$\mathbf{T}^{a_{1},b_{1}}M\oplus\cdots\oplus\mathbf{T}^{a_{r},b_{r}}M,$

for varying $r$, $a_{1},\ldots,a_{r}$, $b_{1},\ldots,b_{r}$. The Tannakian
group scheme associated to $\langle M\rangle_{\otimes}$ via $\omega$ shall
will be called here the (full) Galois-Tannaka group (scheme) of $M$.

As we concentrate on a neutral Tannakian category, it is instructive to note
that the splicing of $\mathcal{T}$ by various $\langle M\rangle_{\otimes}$
amounts to looking at various “images” of $\Pi$. Before entering this topic,
recall that, given a base field $F$ and a morphism $\varphi:G^{\prime}\to G$
of affine group schemes over $F$, the closed image $\mathrm{Im}_{\varphi}$
[EGA, I.9.5] is a closed subgroup scheme of $G$ such that the natural morphism
$G^{\prime}\to\mathrm{Im}_{\varphi}$ is faithfully flat [Wa79, Theorem on
15.1]. In this case, $\mathrm{Im}_{\varphi}$ enjoys both “desirable
properties” of and image.

######  Definition 4.5.

Let $\rho:\Pi\to G$ be a morphism of flat and affine group schemes over $R$.
Define the restricted image of $\rho$, denoted $\mathrm{Im}_{\rho}$, as the
affine scheme associated to the algebra

$B_{\rho}$ = Image of $\mathcal{O}(G)\to\mathcal{O}(\Pi)$.

(In other words, $\mathrm{Im}_{\rho}$ is the “closed” image of $\rho$ [EGA,
I.9.5].) Define its full image $\mathrm{Im}^{\prime}_{\rho}$ as being the
affine scheme associated to

$B_{\rho}^{\prime}=\\{f\in K\otimes\mathcal{O}(\Pi)\,:\,\text{ $\pi^{m}f\in
B_{\rho}$, for some $m\geqslant 0$}\\}.$

It is not difficult to see that $\mathrm{Im}_{\rho}$ and
$\mathrm{Im}^{\prime}_{\rho}$ are affine group schemes. With these
definitions, $\rho$ factors as

$\Pi\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}\mathrm{Im}^{\prime}_{\rho}\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\mathrm{Im}_{\rho}\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}G,$

where $\iota$ is a closed immersion and $u$ induces an isomorphism between
generic fibres. A fundamental result [DH18, Theorem 4.1.1] now assures that
$\psi$ is faithfully flat, so that the the terms “images” are justified and
the factorization in $(\dagger)$ is called the diptych of $\rho$. In addition,
if

$\rho_{K}:\Pi\otimes K\longrightarrow G\otimes K$

stands for the morphism obtained from $\rho$ by base-change to $K$, we have

$\mathrm{Im}^{\prime}_{\rho}\otimes K=\mathrm{Im}_{\rho}\otimes
K=\mathrm{Im}(\rho_{K}).$

######  Proposition 4.6 ([DHdS18, Proposition 4.10]).

Let $M$ be a finite and free $R$-module affording a representation of $\Pi$
and let $\rho:\Pi\to\mathrm{GL}(M)$ be the associated homomorphism. Then the
obvious functor
$\mathrm{Rep}_{R}(\mathrm{Im}^{\prime}_{\rho})\to\mathrm{Rep}_{R}(\Pi)$
defines an equivalence between $\mathrm{Rep}_{R}(\mathrm{Im}^{\prime}_{\rho})$
and $\langle M\rangle_{\otimes}$. Put differently,
$\mathrm{Im}^{\prime}_{\rho}$ is the Galois-Tannaka group of $M$ (in
$\mathrm{Rep}_{R}(\Pi)$).

######  Remark 4.7.

Let $\mathrm{Rep}_{R}^{\circ}(\mathrm{Im}_{\rho})$ be the full subcategory of
$\mathrm{Rep}_{R}(\mathrm{Im}_{\rho})$ consisting of objects having a dual; it
is possible to show that $\mathrm{Rep}_{R}^{\circ}(\mathrm{Im}_{\rho})$ is
equivalent to a full subcategory of $\langle M\rangle_{\otimes}$. On the other
hand, the functor
$\mathrm{Rep}_{R}(\mathrm{Im}_{\rho})\to\mathrm{Rep}_{R}(\Pi)$ may easily fail
to be full.

From now on, we give ourselves a representation $\rho:\Pi\to\mathrm{GL}(M)$ as
in Proposition 4.6. It is at this point that the theory over $R$ parts from
the theory over a field in a significant way. Indeed, in the case of a base-
field, Galois-Tannaka group schemes are known to be of finite type [DM82,
Proposition 2.20]. This is not unconditionally true over $R$ since in order to
construct $\mathrm{Im}^{\prime}_{\rho}$, it was required to “saturate” the
ring $B_{\rho}$ in (§). On the other hand, the morphism
$\mathrm{Im}_{\rho}\to\mathrm{GL}(M)$ is a closed immersion and
$\mathrm{Im}_{\rho}$ is of finite type.

######  Definition 4.8.

A model of a group scheme of finite type $G$ over $K$ is a flat group scheme
${\mathbb{G}}$ over $R$ such that ${\mathbb{G}}\otimes_{R}K\cong G$, as
$K$-group schemes. We often identify $G$ and the generic fibre
${\mathbb{G}}\otimes_{R}K$. A morphism of models
${\mathbb{G}}\to{\mathbb{G}}^{\prime}$ of $G$ is a morphism
${\mathbb{G}}\to{\mathbb{G}}^{\prime}$ of group schemes over $R$ which induces
the identity on $G$ once unravelled the proper identifications.

######  Remark 4.9.

The definition of model used here differs from the one used in [StP, Tag 0C2R]
and [WW80] namely, we do not assume our models to be of finite type over $R$.

With this terminology, $\mathrm{Im}_{\rho}$ and $\mathrm{Im}^{\prime}_{\rho}$
are models of $\mathrm{Im}(\rho_{K})$. A well-known result of Waterhouse-
Weisfeler about the relations between models is the following.

######  Theorem 4.10 ([WW80, Theorem 1.4], [DHdS18, Theorem 2.11]).

Let $v:G^{\prime}\to G$ be a morphism of flat $S$-group schemes such that $v$
is an isomorphism on generic fibres. Then $v$ is a composite of mono-centered
Néron blowups (along the divisor defined by $\pi$). In other words, a morphism
of models of finite type is a composite of mono-centered Néron blowups. If $G$
and $G^{\prime}$ are of finite type, then the number of Néron blowups is
finite. More precisely: Define $v_{0}=v$ and $G_{0}=G$. Suppose that
$v_{n}:G^{\prime}\to G_{n}$ is obtained and put

$G_{n+1}=\mathrm{Bl}_{\mathrm{Im}_{v_{n}\otimes k}}^{G_{n}\otimes k}(G_{n}).$

(Recall that $k$ is the residue field.) Letting $v_{n+1}:G^{\prime}\to
G_{n+1}$ be the morphism deduced from the universal property of
$\mathrm{Bl}_{\mathrm{Im}_{v_{n}\otimes k}}^{G_{n}\otimes k}(G_{n})$ (cf.
Proposition 3.1), then

$\varprojlim_{n}v_{n}:G^{\prime}\longrightarrow\varprojlim_{n}G_{n}$

is an isomorphism. In particular, if for some $n\in\mathbb{N}$ the
homomorphism $v_{n}\otimes k$ is faithfully flat, then $G^{\prime}\simeq
G_{n}$.

As was mentioned before, it is possible that $\mathrm{Im}^{\prime}_{\rho}$
fails to be of finite type and hence the number of Néron blowups proposed by
Theorem 4.10 to describe $u:\mathrm{Im}_{\rho}^{\prime}\to\mathrm{Im}_{\rho}$
may be infinite. But in some cases, it does happen that the number of Néron
blowups is finite and a condition for this situation is described in Theorem
4.10. At this point, we remind the reader that in the situations we have in
mind, the group scheme $\Pi$ is usually extremely complicated and the
determination of the image of a morphism $\Pi\otimes k\to G\otimes k$, so that
it is possible to apply the last claim in Theorem 4.10, can only be achieved
on the side of $\mathrm{Rep}_{R}(\Pi)$. It then becomes relevant to determine
faithful representations of Néron blowups. (Here, we say that a representation
is faithful if the morphism to the associated general linear group is a closed
immersion. This is not universally adopted.) The next result explains how to
proceed in certain cases.

######  Theorem 4.11 ([DHdS18, Corollary 3.6]).

Let $G$ be an affine and flat group scheme of finite type over $S$. Let $M$ be
a finite and free $R$-module affording a faithful representation of $G$. Given
$m\in M$, let

$H_{0}=\text{stabilizer of $m\otimes 1\in M\otimes k$}$

in $G\otimes k$. Let $G^{\prime}=\mathrm{Bl}_{H_{0}}^{G\otimes k}(G)$. Then,
letting $\mathbbm{1}=R$ stand for the trivial representation of $G^{\prime}$,
the obvious map $\mathbbm{1}\to M\otimes k$ determined by $1\mapsto v\otimes
1$ is $G^{\prime}$-equivariant and the fibered product

$M^{\prime}:=M\underset{M\otimes k}{\times}\mathbbm{1}$

now affords a faithful representation of $G^{\prime}$.

Let us illustrate the above result with a simple example showing how to
compute a Galois-Tannaka group.

######  Example 4.12.

Let $k$ be of characteristic zero and $\mathcal{T}$ be the category of
representations of the abstract group $\mathbb{Z}$ on finite $R$-modules. It
is not difficult to see that $\mathcal{T}$ is neutral Tannakian [HdS21,
Corollary 4.5]. Let $\mathbb{Z}$ act on $M=R$ by $\gamma\cdot
r=(1+\pi)^{\gamma}r$ and write
$\rho:\Pi\to\mathrm{GL}(M)(\simeq\mathbb{G}_{m,R})$ for the associated
morphism of group schemes. It is not difficult to see that
$\mathrm{Im}_{\rho}=\mathbb{G}_{m,R}$ and we wish to compute
$\mathrm{Im}_{\rho}^{\prime}$. As mentioned above, the construction ($\S$) is
of little use. On the other hand, we know that $\Pi$ will act trivially on
$M\otimes k$ because $\mathbb{Z}$ does. We then need to perform the
“dilatation” $M^{\prime}$ of $M$ as in $(\P)$, which is a faithful
representation of the Néron blowup
$\mathrm{Bl}_{\\{e\\}}^{\mathbb{G}_{m}\otimes k}(\mathbb{G}_{m})$. The
elements $m_{1}:=(\pi,0)$ and $m_{2}:=(1,1)$ obviously form a basis for
$M^{\prime}$ and hence the resulting representation of $\mathbb{Z}$ is defined
by

$\gamma\longmapsto\begin{pmatrix}1+\pi&1\\\ 0&1\end{pmatrix}^{\gamma}.$

If $\rho^{\prime}:\Pi\to\mathrm{GL}(M^{\prime})(\simeq\mathrm{GL}_{2})$ stands
for the associated representation of $\Pi$, we can say that
$\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}\in\mathrm{GL}_{2}(k)$ belongs to the
image of $\rho^{\prime}\otimes k$ and therefore
$\mathrm{Im}_{\rho}^{\prime}\simeq\mathrm{Bl}_{\\{e\\}}^{\mathbb{G}_{m}\otimes
k}(\mathbb{G}_{m})$ because $\mathrm{Bl}_{\\{e\\}}^{\mathbb{G}_{m}\otimes
k}(\mathbb{G}_{m})\otimes\simeq\mathbb{G}_{a,k}$, and any element of
$k\setminus\\{0\\}$ generates a dense subgroup.

On the other hand, when the number of Néron blowups envisaged by Theorem 4.10
is infinite, a general principle behind [DHdS18, HdS21] is that the Galois-
Tannaka groups can be obtained from group schemes of finite type via certain
special types of (what we now call) multi-centered Néron blowups. This is
treated in the next section.

####  4.3  Néron blowups of formal subgroup schemes

Multi-centered dilatations having divisors which are supported on the same
space have been studied more closely. For an affine group scheme $G$ over $R$,
we shall write $\widehat{G}$ for the completion $G_{/G_{0}}$ of $G$ along its
closed fiber [EGA, I.10].

######  Definition 4.13 ([DHdS18, Definition 5.6]).

Let $G\to S$ be an affine flat group scheme of finite type. For each
$i\in\mathbb{N}$, let $G_{i}$ be the $S_{i}$-group scheme $G\times_{S}S_{i}$,
and let $H_{i}\to S_{i}$ be a closed, $S_{i}$–flat, subgroup-scheme of
$G_{i}$. Assume, in addition, that the natural base-change morphism

$H_{i+1}\times_{S_{i+1}}S_{i}\longrightarrow
G_{i+1}\times_{S_{i+1}}S_{i}=G_{i}$

defines an isomorphism $H_{i+1}\times_{S_{i+1}}S_{i}\simeq H_{i}$ of group
schemes. Said differently, the family $\\{H_{i}\\}$ induces a formal closed
subgroup scheme $\mathfrak{H}$ of $\widehat{G}$. We define the Néron blowup of
$G$ along $\mathfrak{H}$, call it $\mathrm{Bl}^{\widehat{G}}_{\mathfrak{H}}G$,
as being $\mathrm{Bl}_{\\{H_{i}\\}}^{\\{G_{i}\\}}G\to G$.

######  Remark 4.14.

If the formal scheme $\mathfrak{H}$ is “algebraizable”, meaning that it comes
from a closed and flat subgroup scheme $H\subset G$, this is mentioned in
[PY06, §7.2].

######  Example 4.15.

Let $p$ be a prime number, $R=\mathbb{Z}_{p}$ and $G=\mathbb{G}_{a,R}$. It
then follows that the completion of $G$ along its closed fibre is
$\mathrm{Spf}\,\mathbb{Z}_{p}\langle x\rangle$, where $\mathbb{Z}_{p}\langle
x\rangle$ is the subring of $\mathbb{Z}_{p}\llbracket x\rrbracket$ consisting
of power series $\sum_{n}a_{n}x^{n}$ such that $\lim a_{n}=0$. Let
$\mathfrak{H}$ be the closed formal subscheme of $\widehat{G}$ determined by
the ideal $(x)\subset\mathbb{Z}_{p}\langle x\rangle$. Then, it is not
difficult to see that $\mathrm{Bl}_{\mathfrak{H}}^{\widehat{G}}G$ is the group
scheme determined by the Hopf subalgbra
$A=\\{P\in\mathbb{Q}_{p}[x]\,:\,P(0)\in\mathbb{Z}_{p}\\}$. Note that
$\mathrm{Bl}_{\mathfrak{H}}^{\widehat{G}}G\otimes\mathbb{F}_{p}$ is the
trivial group scheme, while
$\mathrm{Bl}_{\mathfrak{H}}^{\widehat{G}}G\otimes\mathbb{Q}_{p}$ is
$\mathbb{G}_{a,\mathbb{Q}_{p}}$. In particular, the dimension of the generic
and special fibres is distinct, even though
$\mathrm{Bl}_{\mathfrak{H}}^{\widehat{G}}G$ is itself flat over
$\mathbb{Z}_{p}$. Note, on the other hand, that the $\mathbb{Z}_{p}$-module
$A$ contains a copy of $\mathbb{Q}_{p}$ and hence fails to be projective over
$\mathbb{Z}_{p}$. This seemingly harmless property is the cause of
complications in the category of representations [HdS21, Proposition 6.19] as
the inexistence of intersections of subrepresentations.

######  Example 4.16 ([HdS21, 4.3]).

Let $R=k\llbracket\pi\rrbracket$, where $k$ is a field of characteristic zero
and let $G=\mathbb{G}_{a,R}\times_{R}\mathbb{G}_{m,R}$. Letting $x$ stand for
“the” coordinate of $\mathbb{G}_{a,R}$ and $y$ for “the” coordinate of
$\mathbb{G}_{m,R}$, we define

$e^{\pi x}=\sum_{i=0}^{\infty}\frac{\pi^{i}}{i!}x^{i};$

this is an element of $\widehat{\mathcal{O}(G)}$. It is not difficult to see
that $y-e^{\pi x}$ cuts out a closed and formal subgroup scheme of
$\widehat{G}$, call it $\mathfrak{H}$, and hence we obtain a model
$\mathrm{Bl}_{\mathfrak{H}}^{\widehat{G}}\to G$. Note that $\mathfrak{H}$ is
not algebraizable. Differently from the situation in Exemple 4.15, the
$R$-module $\mathcal{O}(\mathrm{Bl}_{\mathfrak{H}}^{\widehat{G}})$ is
_projective_.

One important consequence of the procedure of taking formal blowups is the
following. It says that, in some contexts, all the information concerning a
model of a group scheme can be encoded in a formal Néron blowup (Theorem
4.17).

######  Theorem 4.17 ([HdS21, Corollary 3.3]).

Suppose that the $R$ is complete and of residual characteristic zero. Let
$\mathcal{G}\to G$ be a morphism of affine and flat $R$-group schemes inducing
an isomorphism on the generic fibres, and suppose in addition that $G$ is of
finite type. Then, there exists a group scheme $G^{\prime}$ over $R$, flat and
of finite type, and a morphism of group schemes $G^{\prime}\to G$ which is an
isomorphism on generic fibres, a closed and formal subgroup scheme
$\mathfrak{H}^{\prime}$ of $\widehat{G}^{\prime}$, and an isomorphism

$\mathcal{G}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{Bl}^{\widehat{G}^{\prime}}_{\mathfrak{H}^{\prime}}G^{\prime}.$

######  Remark 4.18.

Under the assumptions of Theorem 4.17, Theorems 4.10 and 4.17 together say
that any morphism of models $G^{\prime}\to G$ with $G$ of finite type over $R$
is obtained as a composite of multi-centered Néron blowups, and more precisely
as a formal Néron blowup composed by several mono-centered Néron blowups.

###  5 Congruent isomorphisms and relations with Bruhat-Tits buildings, the
Moy-Prasad isomorphism and admissible representations of $p$-adic groups

In this section we report on congruent isomorphisms. Let $({\mathcal{O}},\pi)$
be a henselian pair where $\pi\subset{\mathcal{O}}$ is an invertible ideal.
Let us start with the following result proved in [MRR20].

######  Theorem 5.1 ([MRR20]).

(Congruent isomorphism) Let $r,s$ be integers such that $0\leqslant
r/2\leqslant s\leqslant r$. Let $G$ be a smooth, separated
${\mathcal{O}}$-group scheme. Let $G_{r}$ be the $r$-th iterated dilatation of
the unit section (i.e.
$G_{r}={\operatorname{Bl}}_{e_{G}}^{{\mathcal{O}}/\pi^{r}}G$) and
${\mathfrak{g}}_{r}$ be its Lie algebra. If ${\mathcal{O}}$ is local or $G$ is
affine, there is a canonical and functorial isomorphism of groups:

$G_{s}({\mathcal{O}})/G_{r}({\mathcal{O}})\begin{array}[b]{c}{}_{\sim}\\\
\longrightarrow\end{array}{\mathfrak{g}}_{s}({\mathcal{O}})/{\mathfrak{g}}_{r}({\mathcal{O}}).$

######  Remark 5.2.

We comment on works prior to Theorem 5.1.

  1. 1.

In the case of an affine, smooth group scheme over a discrete valuation ring,
the isomorphism of Theorem 5.1 appears without proof in [Yu15, proof of Lemma
2.8].

  2. 2.

The proof of Theorem 5.1 relies on Proposition 2.15 and Theorem 3.5 whose
proofs (given in [MRR20, Prop. 2.9] and [MRR20, Th. 3.5]) basically consist in
playing and computing with quasi-coherent ideals. These computations on quasi-
coherent ideals in [MRR20] were partly motivated by related computations on
ideals done in the affine case in [Ma19t, Appendix A] to understand the
congruent isomorphism. The statement of [MRR20, Th. 3.5] is moreover partly
inspired by [WW80, Th. 1.5, Th. 1.7].

  3. 3.

If $G=\mathbb{G}_{m}/{\mathbb{Z}}_{p}$, isomorphism ($\star$) follows from the
multiplicative structure of ${\mathbb{Z}}_{p}$ cf. e.g. [He1913], [Ha50] and
[Ha80, Chap. 15]. Similar isomorphisms for matrix groups over non-Archimedean
local fields were used in [Ho77, p. 442 line 1], [Mo91, 2.13], [BK93, p. 22],
[Sec04, p. 337] and many other references to study admissible representations
of $p$-adic classical groups. In the matrix case, the filtrations involved are
defined using matrix theoretic descriptions and avoiding scheme theoretic
tools. For general reductive groups over non-Archimedean local fields, such
kind of isomorphisms were introduced and used in [PR84, §2], [MP94, §2],
[MP96], [Ad98, §1], [Yu01, §1] to study admissible representations. In the
reductive case, the filtrations involved are the Moy-Prasad filtrations
[MP94], [MP96] and the isomorphism is called the Moy-Prasad isomorphism. These
filtrations are defined for points in the Bruhat-Tits building using the
associated valued root datum [BT72] [BT84]. The Moy-Prasad isomorphim in these
references was defined using somehow ad hoc formulas and the valued root
datum, in particular avoiding the congruent isomorphism. However it is known
that one has to modify the original Moy-Prasad filtrations to ensure the
validity of the Moy-Prasad isomorphism in full generality, cf. [Yu15, §0.3]
and [KP22, §13].

If $G=\mathrm{GL}_{2}/{\mathbb{Z}}_{p}$,
$G_{n}({\mathbb{Z}}_{p})=\begin{pmatrix}1+{\mathfrak{p}}^{n}&{\mathfrak{p}}^{n}\\\
{\mathfrak{p}}^{n}&1+{\mathfrak{p}}^{n}\end{pmatrix}\subset\mathrm{GL}_{2}({\mathbb{Z}}_{p})$
and
$\mathfrak{g}_{n}({\mathbb{Z}}_{p})=\begin{pmatrix}{\mathfrak{p}}^{n}&{\mathfrak{p}}^{n}\\\
{\mathfrak{p}}^{n}&{\mathfrak{p}}^{n}\end{pmatrix}\subset
M_{2}({\mathbb{Z}}_{p})$ for any $n>0$. The isomorphism $(\star)$ gives us,
for pairs $(r,s)$ such that $0<\frac{r}{2}\leqslant s\leqslant r$,
isomorphisms

$\begin{pmatrix}1+{\mathfrak{p}}^{s}&{\mathfrak{p}}^{s}\\\
{\mathfrak{p}}^{s}&1+{\mathfrak{p}}^{s}\end{pmatrix}/\begin{pmatrix}1+{\mathfrak{p}}^{r}&{\mathfrak{p}}^{r}\\\
{\mathfrak{p}}^{r}&1+{\mathfrak{p}}^{r}\end{pmatrix}\cong\begin{pmatrix}{\mathfrak{p}}^{s}&{\mathfrak{p}}^{s}\\\
{\mathfrak{p}}^{s}&{\mathfrak{p}}^{s}\end{pmatrix}/\begin{pmatrix}{\mathfrak{p}}^{r}&{\mathfrak{p}}^{r}\\\
{\mathfrak{p}}^{r}&{\mathfrak{p}}^{r}\end{pmatrix}.$

These maps are given by $[1+M]\mapsto[M]$. Using the formula
$[1+M]\mapsto[M]$, it is elementary to check that we have other isomorphisms
of abstract groups

$\displaystyle\begin{pmatrix}1+{\mathfrak{p}}^{3}&{\mathfrak{p}}^{3}\\\
{\mathfrak{p}}^{3}&1+{\mathfrak{p}}^{3}\end{pmatrix}/\begin{pmatrix}1+{\mathfrak{p}}^{5}&{\mathfrak{p}}^{6}\\\
{\mathfrak{p}}^{6}&1+{\mathfrak{p}}^{5}\end{pmatrix}\cong\begin{pmatrix}{\mathfrak{p}}^{3}&{\mathfrak{p}}^{3}\\\
{\mathfrak{p}}^{3}&{\mathfrak{p}}^{3}\end{pmatrix}/\begin{pmatrix}{\mathfrak{p}}^{5}&{\mathfrak{p}}^{6}\\\
{\mathfrak{p}}^{6}&{\mathfrak{p}}^{5}\end{pmatrix},$ ($**$)
$\displaystyle\begin{pmatrix}1+{\mathfrak{p}}^{3}&{\mathfrak{p}}^{9}\\\
{\mathfrak{p}}^{3}&1+{\mathfrak{p}}^{3}\end{pmatrix}/\begin{pmatrix}1+{\mathfrak{p}}^{6}&{\mathfrak{p}}^{9}\\\
{\mathfrak{p}}^{6}&1+{\mathfrak{p}}^{6}\end{pmatrix}\cong\begin{pmatrix}{\mathfrak{p}}^{3}&{\mathfrak{p}}^{9}\\\
{\mathfrak{p}}^{3}&{\mathfrak{p}}^{3}\end{pmatrix}/\begin{pmatrix}{\mathfrak{p}}^{6}&{\mathfrak{p}}^{9}\\\
{\mathfrak{p}}^{6}&{\mathfrak{p}}^{6}\end{pmatrix}.$ ($***$)

These isomorphisms are obtained as follows from the point of view of
dilatations.

######  Theorem 5.3 ([Ma23d]).

(Multi-centered congruent isomorphism) Let $G$ be a separated and smooth group
scheme over $S$. Let $H_{0}\subset H_{1}\subset\ldots\subset H_{k}$ be closed
subgroup schemes of $G$ such that $H_{i}$ is smooth over $S$ for $0\leqslant
i\leqslant d$ and $H_{0}=e_{G}$. Let $s_{0},s_{1},\ldots,s_{k}$ and
$r_{0},r_{1},\ldots,r_{k}$ be in ${\mathbb{N}}$ such that

  1. 1.

$s_{i}\geqslant s_{0}$ and $r_{i}\geqslant r_{0}$ for all
$i\in\\{0,\ldots,k\\}$,

  2. 2.

$r_{i}\geqslant s_{i}$ and $r_{i}-s_{i}\leqslant s_{0}$ for all
$i\in\\{0,\ldots,k\\}$.

Assume that $G$ is affine or ${\mathcal{O}}$ is local. Then we have a
canonical isomorphism of groups

${\operatorname{Bl}}_{H_{0},H_{1},\ldots,H_{k}}^{s_{0},~{}s_{1},~{}\ldots,s_{k}}G/{\operatorname{Bl}}_{H_{0},H_{1},\ldots,H_{k}}^{r_{0},~{}r_{1},~{}\ldots,r_{k}}G\cong{\operatorname{Lie}}({\operatorname{Bl}}_{H_{0},H_{1},\ldots,H_{k}}^{s_{0},~{}s_{1},~{}\ldots,s_{k}}G)/{\operatorname{Lie}}({\operatorname{Bl}}_{H_{0},H_{1},\ldots,H_{k}}^{r_{0},~{}r_{1},~{}\ldots,r_{k}}G)$

where ${\operatorname{Bl}}_{H_{0},\ldots,H_{k}}^{t_{0},~{}\ldots,t_{k}}G$
denotes
${\operatorname{Bl}}_{H_{0},~{}\ldots~{},H_{k}}^{{\mathcal{O}}/\pi^{t_{0}},\ldots,{\mathcal{O}}/\pi^{t_{k}}}G$
for any $t_{0},\ldots,t_{k}\in{\mathbb{N}}$.

Now let $G$ be $GL_{2}/{\mathbb{Z}}_{p}$. Let $e_{G}\subset G$ be the trivial
subgroup. Let $T$ be the diagonal split torus in $G$. Let $B$ be the lower
triangular Borel in $G$ over ${\mathbb{Z}}_{p}$.

  1. 1.

The isomorphism $(**)$ above is given by Theorem 5.3 with
$({\mathcal{O}},\pi)=({\mathbb{Z}}_{p},\mathfrak{p})$, $H_{0}=e_{G}$,
$H_{1}=T$, $s_{0}=3,s_{1}=3,r_{0}=5$ and $r_{1}=6$.

  2. 2.

The isomorphism $(***)$ above is given by Theorem 5.3 with
$({\mathcal{O}},\pi)=({\mathbb{Z}}_{p},\mathfrak{p})$, $H_{0}=e_{G}$,
$H_{1}=B$, $s_{0}=3,s_{1}=9,r_{0}=6$ and $r_{1}=9$.

######  Remark 5.4.

We comment Theorem 5.3.

  1. 1.

Theorem 5.3 corresponds to [Ma23d, Corollary 8.3], a slightly more general
result is given by [Ma23d, Theorem 8.1].

  2. 2.

The proof of Theorem 5.3 given in [Ma23d] relies on Theorem 5.1 and the study
of multi-centered dilatations.

  3. 3.

Note that [Yu01, Lemma 1.3] provides a comparable ”multi-centered”
isomorphism, in the framework of reductive groups over non-Archimedean local
field.

Recall that dilatations of schemes over discrete valuation rings are used in
Yu’s approach [Yu15] on Bruhat-Tits theory for reductive groups over henselian
discrete valuation field with perfect residue field. We refer to the monograph
by Kaletha and Prasad [KP22] that include among other things a detailled
exposition of [Yu15]. The congruent isomorphism (Theorem 5.1) and its proof
(relying on several results of [MRR20]) are now used as foundation to prove
the Moy-Prasad isomorphism for reductive groups mentioned in Remark 5.2, cf.
[KP22, Theorem 13.5.1 and its proof, Proposition A.5.19 (3) and its proof]. As
a consequence, dilatations and congruent isomorphisms are now part of the
foundation to study admissible representations of reductive $p$-adic groups.
Furthermore, other connections between dilatations and groups used to study
admissible representations can be found in [Yu15, §10] and [Ma23d, Example
1.4]. Reciprocally, the problem of constructing supercuspidal representations
of $p$-adic groups (cf. e.g. [Ma19o, Remark/Conclusion], [Ma19t, MY22]), or
more generally types in the sense of [BK98], could continue to be a source of
inspiration to expend the theory of dilatations.

######  Remark 5.5.

As we explained before, the book [KP22] provides a carefully written new
approach to Bruhat-Tits theory in the case of discrete valuations. This
beautiful monograph uses the theory of dilatations to deal with integral
models whereas the original Bruhat-Tits theory [BT84] did not. Let us quote
[KP22, Introduction]:

“Next we turn to the construction of integral models […]. Instead of using the
approach of Bruhat–Tits via schematic root data, we employ a simpler and more
direct method due to Jiu-Kang Yu [Yu15], based on the systematic use of Néron
dilatations.”

The book [KP22] offers an appendix on dilatations. Though [KP22, Appendix A.5]
takes into account the treatment of dilatations in [MRR20], it restricts to
the framework of discrete valuations. Originally, Bruhat-Tits theory [BT84]
deals also with non discrete valuations, it is natural to ask whether the
modern and general approach to dilatations of schemes initiated in [MRR20]
could help to provide a more conceptual treatment (in the spirit of [Yu15] and
[KP22]) of some parts of [BT84]. Bruhat-Tits theory and dilatations over non
discrete valuations were used in [RTW10] and [Ma22] to study Berkovich’s point
of view [Be90, Chap. 5] on Bruhat-Tits buildings of reductive groups over
discrete and non-discrete valuations (cf. e.g. [RTW10, 1.3.4] for precise
assumptions).

###  6 Torsors, level structures and shtukas

In this subsection, we explain that many level structures on moduli stacks of
$G$-bundles are encoded in torsors under Néron blowups of $G$ following
[MRR20]. Assume that $X$ is a smooth, projective, geometrically irreducible
curve over a field $k$ with a Cartier divisor $N\subset X$, that $G\to X$ is a
smooth, affine group scheme and that $H\to N$ is a smooth closed subgroup
scheme of $G|_{N}$. In this case, the Néron blowup ${\mathcal{G}}\to X$ is a
smooth, affine group scheme. Let ${\operatorname{Bun}}_{G}$ (resp.
${\operatorname{Bun}}_{\mathcal{G}}$) denote the moduli stack of $G$-torsors
(resp. ${\mathcal{G}}$-torsors) on $X$. This is a quasi-separated, smooth
algebraic stack locally of finite type over $k$ (cf. e.g. [He10, Prop. 1] or
[AH19, Thm. 2.5]). Pushforward of torsors along ${\mathcal{G}}\to G$ induces a
morphism ${\operatorname{Bun}}_{\mathcal{G}}\to{\operatorname{Bun}}_{G}$,
${\mathcal{E}}\mapsto{\mathcal{E}}\times^{\mathcal{G}}G$. We also consider the
stack ${\operatorname{Bun}}_{(G,H,N)}$ of $G$-torsors on $X$ with
level-($H$,$N$)-structures, cf. [MRR20, Definition 4.5]. Its $k$-points
parametrize pairs $({\mathcal{E}},{\beta})$ consisting of a $G$-torsor
${\mathcal{E}}\to X$ and a section ${\beta}$ of the fppf quotient
$({\mathcal{E}}|_{N}/H)\to N$, i.e., ${\beta}$ is a reduction of
${\mathcal{E}}|_{N}$ to an $H$-torsor.

######  Proposition 6.1 ([MRR20]).

There is an equivalence of $k$-stacks

${\operatorname{Bun}}_{\mathcal{G}}\;\overset{\cong}{\longrightarrow}\;{\operatorname{Bun}}_{(G,H,N)},\;\;{\mathcal{E}}\longmapsto({\mathcal{E}}\times^{\mathcal{G}}G,\,{\beta}_{\operatorname{can}}),$

where ${\beta}_{\operatorname{can}}$ denotes the canonical reduction induced
from the factorization ${\mathcal{G}}|_{N}\to H\subset G|_{N}$.

Thus, many level structures are encoded in torsors under Néron blowups. This
construction is also compatible with the adelic viewpoint as follows. Let
$|X|\subset X$ be the set of closed points, and let $\eta\in X$ be the generic
point. We denote by $F=\kappa(\eta)$ the function field of $X$. For each
$x\in|X|$, we let ${\mathcal{O}}_{x}$ be the completed local ring at $x$ with
fraction field $F_{x}$ and residue field
$\kappa(x)={\mathcal{O}}_{x}/{\mathfrak{m}}_{x}$. Let
${\mathbb{A}}:=\bigsqcap^{\prime}_{x\in|X|}F_{x}$ be the ring of adeles with
subring of integral elements
${\mathbb{O}}=\bigsqcap_{x\in|X|}{\mathcal{O}}_{x}$.

######  Proposition 6.2.

Assume either that $k$ is a finite field and $G\to X$ has connected fibers, or
that $k$ is a separably closed field. The Néron blowup ${\mathcal{G}}\to X$ is
smooth, affine with connected fibers, and there is a commutative diagram of
groupoids

${\ignorespaces{\operatorname{Bun}}_{\mathcal{G}}(k)}$ |  | ${\ignorespaces\bigsqcup_{\gamma}G_{\gamma}(F)\big{\backslash}\big{(}G_{\gamma}({\mathbb{A}})/{\mathcal{G}}({\mathbb{O}})\big{)}}$ |
---|---|---|---
${\ignorespaces{\operatorname{Bun}}_{G}(k)}$ |  | ${\ignorespaces\bigsqcup_{\gamma}G_{\gamma}(F)\big{\backslash}\big{(}G_{\gamma}({\mathbb{A}})/G({\mathbb{O}})\big{)},}$ |
$\simeq$$\simeq$

identifying the vertical maps as the level maps.

Now assume that $k$ is a finite field. As a consequence of Proposition 6.1 one
naturally obtains integral models for moduli stacks of $G$-shtukas on $X$ with
level structures over $N$ via an isomorphism
$\operatorname{Sht}_{{\mathcal{G}},I_{\bullet}}\;\overset{\cong}{\longrightarrow}\;\operatorname{Sht}_{(G,H,N),I_{\bullet}}$
(cf. [MRR20, §4.2.2] for precise definitions and details).

###  7 The “topology”of dilatations of affine schemes

####  7.1 Constructing smooth complex affine varieties with controlled
topology

Dilatations have played an important role in complex affine algebraic geometry
during the nineties in connection to the construction and study of exotic
complex affine spaces [KZ99, Za00], that is, smooth algebraic
$\mathbb{C}$-varieties $X$ of dimension $n$ whose analytifications
$X^{\mathrm{an}}$ are homeomorphic to the Euclidean space $\mathbb{R}^{2n}$
endowed with its standard structure of topological manifold but which are not
isomorphic to the affine space $\mathbb{A}_{\mathbb{C}}^{n}$ as
$\mathbb{C}$-varieties. In this context, dilatations appeared under the name
_affine modifications_ and were used as a powerful tool to produce from a
given smooth complex affine variety $X$ a new smooth complex affine variety
$X^{\prime}=\mathrm{Bl}_{Z}^{D}X$ for which the homology or homotopy type of
the underlying topological manifold of the analytification of $X^{\prime}$ can
determined under suitable hypotheses in terms of those of $X$ and of the
center $\\{[Z,D]\\}$ of the dilatation. The study of the strong topology of
affine modifications was initiated in this context mainly by Kaliman through
an analytic counter part of the notion of dilatation:

######  Definition 7.1 ([Ka94]).

Given a triple $(M,H,C)$ consisting of a complex analytic manifold $M$, a
closed submanifold $C$ of $M$ of codimension at least $2$ and a complex
analytic hypersurface $H$ of $M$ containing $C$ in its smooth locus, the
_Kaliman modification $M$ along $H$ with center at $C$_ is the complex
analytic manifold defined as the complement $M^{\prime}$ of the proper
transform $H^{\prime}$ of $H$ in the blow-up $\sigma_{C}:\hat{M}\to M$ of $M$
with center at $C$.

In the case where $(M,H,C)$ is the analytification of a triple $(X,D,Z)$
consisting of a smooth algebraic $\mathbb{C}$-variety $X$, a smooth algebraic
sub-variety $Z$ of $X$ of codimenion at least two and of a reduced effective
Cartier divisor $D$ on $X$ containing $Z$ in its regular locus, the Kaliman
modification of $(M,H,C)$ coincides with the analytification of the dilatation
$X^{\prime}=\mathrm{Bl}_{Z}^{D}X$ of $X$ with center $\\{[Z,D]\\}$ of Section
2. Kaliman and Zaidenberg [KZ99] developed a series of tools to describe the
topology of the analytications of affine modifications of smooth affine
$\mathbb{C}$-varieties along principal divisors $D$ with non-necessarily
smooth centers. One of these provides in particular a control on the
preservation of the topology of the analytification under affine
modifications:

######  Theorem 7.2 ([KZ99, Proposition 3.1 and Theorem 3.1]).

Let $X$ be a smooth affine $\mathbb{C}$-variety and let $\\{[Z,D]\\}$ be a
center on $X$ consisting of closed sub-scheme $Z$ of codimension at least $2$
and of a principal effective divisor $D$ containing $Z$ as a closed subscheme.
Let $\sigma:\tilde{X}=\mathrm{Bl}_{Z}^{D}(X)\to X$ be the dilatation of $X$
with center $\\{[Z,D]\\}$ and let $E$ be the exceptional divisor of $\sigma$.
Assume that the following conditions are satisfied: (i) The
$\mathbb{C}$-variety $\tilde{X}=\mathrm{Bl}_{Z}^{D}(X)$ is smooth; (ii) The
divisors $E$ and $D$ are irreducible, $E=\sigma^{*}D$, and the
analytifications of $E_{\mathrm{red}}$ and $D_{\mathrm{red}}$ are topological
manifolds. Then the following properties hold: (a) The homomorphism
$\sigma_{*}^{\mathrm{an}}:\pi_{1}(\tilde{X}^{\mathrm{an}})\to\pi_{1}(X^{\mathrm{an}})$
induced by $\sigma^{\mathrm{an}}$ is an isomorphism; (b) The homomorphism
$\sigma_{*}^{\mathrm{an}}:H_{*}(\tilde{X}^{\mathrm{an}};\mathbb{Z})\to
H_{*}(X^{\mathrm{an}};\mathbb{Z})$ induced by $\sigma^{\mathrm{an}}$ is an
isomorphism if and only if the homomorphism
$\sigma|_{E,*}^{\mathrm{an}}:H_{*}(E_{\mathrm{red}}^{\mathrm{an}};\mathbb{Z})\to
H_{*}(D_{\mathrm{red}}^{\mathrm{an}},\mathbb{Z})$ is.

######  Corollary 7.3.

In the setting of Theorem 7.2, assume that $X^{\mathrm{an}}$ is a contractible
smooth manifold, that $Z_{\mathrm{red}}^{\mathrm{an}}$ is a topological
manifold and that the homomorphism

$j_{*}^{\mathrm{an}}:H_{*}(Z_{\mathrm{red}}^{\mathrm{an}};\mathbb{Z})\to
H_{*}(D_{\mathrm{red}}^{\mathrm{an}};\mathbb{Z})$

induced by the closed immersion $j:Z\hookrightarrow D$ is an isomorphisms.
Then the analytification of $\tilde{X}=\mathrm{Bl}_{Z}^{D}(X)$ is a
contractible smooth manifold.

Having the flexibility to use as centers or divisors of modifications schemes
which are either non-reduced or whose analytifications are not necessarily
smooth manifolds but only topological manifolds is particularly relevant for
applications to the construction smooth $\mathbb{C}$-varieties with
contractible analytifications, as illustrated by the following examples.

######  Example 7.4 (The tom Dieck - Petrie surfaces).

Let $p,q\geqslant 2$ be a pair of relatively prime integers and let
$C_{p,q}\subset\mathbb{A}_{\mathbb{C}}^{2}=\mathrm{Spec}(\mathbb{C}[x,y])$ be
an irreducible rational cuspidal curve with equation $x^{p}-y^{q}=0$. The
underlying topological space of $C_{p,q}^{\mathrm{an}}$ is a contractible real
topological surface, and hence, Corollary 7.3 applies to conclude that the
analytification of the dilatation
$S_{p,q}=\mathrm{Bl}_{(1,1)}^{C_{p,q}}\mathbb{A}_{\mathbb{C}}^{2}$ of
$\mathbb{A}_{\mathbb{C}}^{2}$ along the principal Cartier divisor $D=C_{p,q}$
with center at the closed point $Z=(1,1)\in C_{p,q}$ is a smooth contractible
real $4$-manifold. The smooth affine surface $S_{p,q}$, which can be described
explicitly as the hypersurface in
$\mathbb{A}_{\mathbb{C}}^{3}=\mathrm{Spec}(\mathbb{C}[x,y,z])$ with equation

$\frac{(xz+1)^{p}-(yz+1)^{q}}{z}=1,$

is not isomorphic to $\mathbb{A}_{\mathbb{C}}^{2}$ since, for instance, it has
non negative logarithmic Kodaira dimension [Za00, Example 2.4]. Moreover, the
underlying real $4$-manifold of $S_{p,q}^{\mathrm{an}}$ is an example of a
contractible $4$-manifold with non-trivial fundamental group at infinity,
hence non-homeomorphic to the standard euclidean space $\mathbb{R}^{4}$.

######  Example 7.5 (Some Koras-Russell threefolds).

Let again $p,q\geqslant 2$ be a pair of relative prime integers and consider
for every $n\geqslant 2$ the smooth hypersurface $X_{p,q,n}$ in
$\mathbb{A}_{\mathbb{C}}^{4}=\mathrm{Spec}(\mathbb{C}[x,y,z,w])$ with equation

$x^{n}y+z^{p}+w^{q}+x=0.$

The restriction $\sigma_{p,q,n}:X_{p,q,n}\to\mathbb{A}_{\mathbb{C}}^{3}$ of
the projection to the coordinates $x$, $z$ and $w$ expresses $X_{p,q,n}$ as
the dilatation of $\mathbb{A}_{k}^{3}$ along the principal divisor
$D_{n}=\mathrm{div}x^{n}$ and with center at the non-reduced condimension
closed sub-scheme $Z=Z_{p,q,n}$ with defining ideal

$I_{p,q,n}=(z^{p}+w^{q}+x,x^{n})\subset\mathbb{C}[x,z,w].$

The analytification of $Z_{\mathrm{red}}$ is a topological manifold
homeomorphic to the underlying topological space of the curve
$C_{p,q}^{\mathrm{an}}$ of the previous example. Thus, Corollary 7.3 applies
to conclude that $X_{p,q,n}^{\mathrm{an}}$ is a contractible real
$6$-manifold, hence, by a result of Dimca–Ramanujam, is diffeomorphic to the
standard euclidean space $\mathbb{R}^{6}$, see [Za00, Theorem 3.2]. The
interest in these affine threefolds $X_{p,q,n}$ was motivated in the nineties
by their appearance in the course of the study of the linearization problem
for actions of the multiplicative group $\mathbb{G}_{m,\mathbb{C}}$ on
$\mathbb{A}_{\mathbb{C}}^{3}$ by Koras and Russell [KR97]. One crucial
question at that time was to decide whether these threefolds were isomorphic
to $\mathbb{A}_{\mathbb{C}}^{3}$ or not. The fact that none of them is
isomorphic to $\mathbb{A}_{\mathbb{C}}^{3}$ was finally established by Makar-
Limanov [ML96] by commutative algebra techniques. An interesting by-product of
his proof is that the dilatation morphism
$\sigma_{p,q,n}:X_{p,q,n}\to\mathbb{A}_{\mathbb{C}}^{3}$ is equivariant with
respect to the natural action of the group of $\mathbb{C}$-automorphism of
$X_{p,q,n}$, more precisely, $\sigma_{p,q,n}$ induces an isomorphism

$\sigma_{p,q,n}^{*}:\mathrm{Aut}_{\mathbb{C}}(\mathbb{A}_{\mathbb{C}}^{3},\\{[Z_{p,q,n},D_{n}]\\})\to\mathrm{Aut}_{\mathbb{C}}(X_{p,q,n})$

between the subgroup
$\mathrm{Aut}_{\mathbb{C}}(\mathbb{A}_{\mathbb{C}}^{3},\\{[Z_{p,q,n},D_{n}]\\})$
of $\mathrm{Aut}_{\mathbb{C}}(\mathbb{A}_{\mathbb{C}}^{3})$ consisting of
$\mathbb{C}$-automorphisms preserving the divisor and the center of the
dilatation $\sigma_{p,q,n}$ and the group
$\mathrm{Aut}_{\mathbb{C}}(X_{p,q,n})$, see [MJ11].

####  7.2 Deformation to the normal cone

A very natural class of dilatations which plays a fundamental role in
intersection theory is given by the affine version of the deformation space
$D(X,Y)$ of a closed immersion $Y\hookrightarrow X$ of schemes of finite type
over a fixed base scheme $S$ to its normal cone, [Fu98], [Ro96, §10]. Indeed,
$D(X,Y)$ is simply the dilatation of $X\times_{S}\mathbb{A}_{S}^{1}$ with
divisor $D=X\times_{S}\\{0\\}_{S}$, where $\\{0\\}_{S}$ denotes the zero
section, and center $Z=Y\times_{S}\\{0\\}_{S}$. In the affine setting, say
$X=\mathrm{Spec}(A)$ and $Y=\mathrm{Spec}(A/I)$ for some ideal $I\subset A$,
$D(X,Y)$ is the spectrum of the sub-algebra $A[t]$-algebra

$A[\frac{(I,t)}{t}]\cong\sum_{n}I^{n}t^{-n}\subset A[t,t^{-1}].$

The composition $f:D(X,Y)\to\mathbb{A}_{S}^{1}$ of the dilatation morphism
$\sigma:D(X,Y)\to X\times_{S}\mathbb{A}_{S}^{1}$ with the projection
$\mathrm{p}_{2}:X\times_{S}\mathbb{A}_{S}^{1}\to\mathbb{A}_{S}^{1}$ is a flat
morphism restricting to the trivial bundle
$X\times_{S}(\mathbb{A}_{S}^{1}\setminus\\{0\\}_{S})$ over
$\mathbb{A}_{S}^{1}\setminus\\{0\\}_{S}$ and whose fiber over $\\{0\\}_{S}$
equals the normal cone $N_{Y/X}=\mathrm{Spec}(\bigoplus_{n\geqslant
0}I^{n}/I^{n+1})$ of the closed embedding $Y\hookrightarrow X$ (see
Proposition 2.15). For regular immersions $Y\hookrightarrow X$ between smooth
schemes of dimension $n$ and $m$ over a field $k$, the deformation space
$D(X,Y)$ étale locally looks like the deformation space

$\mathrm{Spec}(k[x_{1},\ldots,x_{m}][t][u_{1},\ldots,u_{m-n}]/(tu_{i}-x_{i})_{i=1,\ldots,m-n})\cong\mathbb{A}_{k}^{m+1}$

of the immersion of $\mathbb{A}_{k}^{n}$ as the linear subspace
$\\{x_{1}=\ldots=x_{m-n}=0\\}$ of
$\mathbb{A}_{k}^{m}=\mathrm{Spec}(k[x_{1},\ldots,x_{m}])$.
Deformation spaces of closed immersions between smooth affine
$\mathbb{C}$-varieties provide an endless source of smooth affine
$\mathbb{C}$-varieties whose analytifications are contractible smooth
manifolds:

######  Example 7.6.

Given a smooth affine $\mathbb{C}$-variety $X$ such that $X^{\mathrm{an}}$ is
contractible and a smooth subvariety $Y\subset X$ such that the induced
inclusion $Y^{\mathrm{an}}\subset X^{\mathrm{an}}$ is a topological homotopy
equivalence, Theorem 7.2 implies that the analytification of the deformation
space $D(X,Y)$ is a contractible smooth manifold. For instance, the
deformation spaces
$D(\mathbb{A}_{\mathbb{C}}^{3},S_{p,q})\subset\mathbb{A}_{\mathbb{C}}^{5}$ of
the tom Dieck - Petrie surfaces $S_{p,q}$ of Example 7.4 are smooth affine
$\mathbb{C}$-varieties of dimension $4$ whose analytifications are all
diffeomorphic to $\mathbb{R}^{8}$. In the same way, for every Koras-Russell
threefold $X_{p,q,n}\subset\mathbb{A}_{\mathbb{C}}^{4}$ in Example 7.5, the
deformation space

$D(\mathbb{A}_{\mathbb{C}}^{4},X_{p,q,n})\cong\\{tu=x^{n}y+z^{p}+w^{q}+x\\}\subset\mathbb{A}_{\mathbb{C}}^{6}$

is a smooth affine $\mathbb{C}$-variety whose analytification is diffeomorphic
to $\mathbb{R}^{10}$. It is not known whether these deformation spaces are
algebraically isomorphic to affine spaces.

More general versions of Kaliman and Zaidenberg techniques allow to fully
describe the singular homology of the analytification of the deformation space
$D(\mathbb{A}_{\mathbb{C}}^{n},Z)$ of a smooth hypersurface
$Z=Z(p)=\mathrm{Spec}(\mathbb{C}[x_{1},\ldots,x_{n}]/(p))$ of
$\mathbb{A}_{\mathbb{C}}^{n}$ in terms of that of the analytification of $Z$,
namely

######  Proposition 7.7 ([KZ99, Proposition 4.1]).

For a smooth hypersurface $Z\subset\mathbb{A}_{\mathbb{C}}^{n}$,
$D(\mathbb{A}_{\mathbb{C}}^{n},Z)^{\mathrm{an}}$ is simply connected and the
inclusion $N_{Z/\mathbb{A}_{\mathbb{C}}^{n}}\hookrightarrow
D(\mathbb{A}_{\mathbb{C}}^{n},Z)$ induces an isomorphism of reduced homology
groups
$\tilde{H}_{*}(D(\mathbb{A}_{\mathbb{C}}^{n},Z)^{\mathrm{an}};\mathbb{Z})\cong\tilde{H}_{*-2}(Z^{\mathrm{an}};\mathbb{Z})$.
In particular, $D(\mathbb{A}_{\mathbb{C}}^{n},Z)^{\mathrm{an}}$ has the
reduced homology type of the $S^{2}$-suspension of $Z^{\mathrm{an}}$.

####  7.3 Contractible affine varieties in motivic $\mathbb{A}^{1}$-homotopy
theory

The possibility to import Kaliman and Zaidenberg techniques in the framewok of
Morel-Voevodsky $\mathbb{A}^{1}$-homotopy theory of schemes [MV99] has focused
quite a lot of attention recently, especially in the direction of the
construction of $\mathbb{A}^{1}$-contractible smooth affine varieties,
motivated in part by possible applications to the Zariski Cancellation
Problem, see [AØ21] and the reference therein for a survey. Very informally,
one views in this context smooth schemes over a fixed base field $k$ as
analogous to topological manifolds, with the affine line $\mathbb{A}_{k}^{1}$
playing the role of the unit interval, and consider the corresponding homotopy
category. More rigorously, the $\mathbb{A}^{1}$-homotopy category
$\mathrm{H}_{\mathbb{A}^{1}}(k)$ of $k$-schemes is defined as the left
Bousfield localization of the injective Nisnevich-local model structure on the
category of simplicial presheaves of sets on the category $\mathrm{Sm}_{k}$ of
smooth $k$-schemes, with respect to the class of maps generated by projections
from the affine line $\mathcal{X}\times_{k}\mathbb{A}_{k}^{1}\to\mathcal{X}$.
Isomorphisms in the homotopy category $\mathrm{H}_{\mathbb{A}^{1}}(k)$ are
called $\mathbb{A}^{1}$-_weak equivalences_ , and a smooth $k$-scheme $X$ is
called $\mathbb{A}^{1}$-_contractible_ if the structure morphism
$X\to\mathrm{Spec}(k)$ is an isomorphism in $\mathrm{H}_{\mathbb{A}^{1}}(k)$.
The affine space $\mathbb{A}_{k}^{n}$ is by definition
$\mathbb{A}^{1}$-contractible. Since the analytification of an
$\mathbb{A}^{1}$-contractible smooth $\mathbb{C}$-variety is a contractible
smooth manifold, smooth algebraic $\mathbb{C}$-varieties with contractible
analytifications provided conversely a first natural framework to seek for
interesting $\mathbb{A}^{1}$-contractible affine varieties non isomorphic to
affine spaces. A first step in this direction was accomplished by Hoyois,
Krishna and Østvær [HKØ16, Theorem 4.2] who used the underlying geometry
associated to the dilatations morphisms
$\sigma_{p,q,n}:X_{p,q,n}\to\mathbb{A}_{\mathbb{C}}^{3}$ to verify that the
Koras-Russell threefolds of Example 7.5 were $\mathbb{A}^{1}$-contractible
possibly up to a finite number of $\mathbb{P}^{1}$-suspensions, in the sense
that for some $n\geqslant 0$, the suspension
$(X_{p,q,n},o)\wedge(\mathbb{P}^{1})^{\wedge n}$ is an
$\mathbb{A}^{1}$-contractible object in
$\mathrm{H}_{\mathbb{A}^{1}}(\mathbb{C})$, where here,
$X_{p,q,n}\subset\mathbb{A}_{\mathbb{C}}^{4}$ is considered as a pointed
smooth $\mathbb{C}$-scheme with distinguished point $o=(0,0,0,0)$. These first
constructions motivated a more systematic study to obtain
$\mathbb{A}^{1}$-homotopic analogues of Kaliman and Zaidenberg’s topological
comparison results for affine modifications. The best counterparts of Theorem
7.2 and Corollary 7.3 available so far are the following:

######  Theorem 7.8 ([DPØ19, Theorem 2.17]).

Let $(X,D,Z)$ be a triple in $\mathrm{Sm}_{k}$ where $D$ is a Cartier divisor
on $X$ and $Z\subset D$ is a closed subscheme and let
$\sigma:\tilde{X}=\mathrm{Bl_{D}^{Z}X\to X}$ be the dilatation of $X$ along
$D$ with center at $Z$. Assume that the following conditions are satisfied:
(i) The supports of $D$ and of the exceptional divisor $E$ of $\sigma$ are
irreducible; (ii) The closed immersion $Z\hookrightarrow D$ is an
$\mathbb{A}^{1}$-weak equivalence. Then there is a naturally induced
$\mathbb{A}^{1}$-weak equivalence
$\Sigma_{s}\sigma:\Sigma_{s}\tilde{X}\to\Sigma_{s}X$ between the simplicial
$1$-suspensions of $\tilde{X}$ and $X$ respectively. In particular, if $X$ is
$\mathbb{A}^{1}$-contractible then $\Sigma_{s}\tilde{X}$ is
$\mathbb{A}^{1}$-contractible. Moreover, a stronger conclusion holds in the
reverse direction: if $\tilde{X}$ is $\mathbb{A}^{1}$-contractible then $X$ is
$\mathbb{A}^{1}$-contractible.

######  Corollary 7.9.

Let $i:Y\hookrightarrow X$ be a closed immersion between
$\mathbb{A}^{1}$-contractible smooth $k$-schemes. Then the simplicial
$1$-suspension $\Sigma_{s}D(X,Y)$ of the deformation space $D(Y,X)$ of $Y$ in
$X$ is $\mathbb{A}^{1}$-contractible.

In contrast with the results of subsection 7.1 which can be applied to
possibly singular triples $(X,D,Z)$, Theorem 7.8 and its corollary
fundamentally depend on smoothness hypotheses. In particular, Theorem 7.8 is
not applicable to tom Dieck -Petrie surfaces and Koras-Russell threefolds over
$\mathbb{C}$ and their natural generalization over other fields. It was
nevertheless verified in [DF18] by different geometric methods that over any
base field $k$ of characteristic zero, the Koras-Russell threefolds
$X_{p,q,n}=\\{x^{n}y+z^{p}+w^{q}+x=0\\}$ are indeed all
$\mathbb{A}^{1}$-contractible. These provide in turn when combined with
Theorem 7.8 and Corollary 7.9 the building blocks for the construction of many
other new examples of smooth affine $k$-varieties whose simplicial
$1$-suspensions are $\mathbb{A}^{1}$-contractible, among which some can be
further verified by additional methods to be genuinely
$\mathbb{A}^{1}$-contractible, see [DPØ19, Section 4]. A more detailed re-
reading of the notion of deformation to the normal cone of a closed immersion
$Y\hookrightarrow X$ between smooth $k$-schemes gives rise to a notion of
“parametrized” deformation space over a smooth base $k$-scheme $W$, which is
defined as a dilatation of the scheme $Y\times_{k}W$ with appropriate center,
see [ADØ21, Construction 2.1.2]. This leads to the following counterpart and
extension of Proposition 7.7 in the $\mathbb{A}^{1}$-homotopic framework:

######  Theorem 7.10 ([ADØ21, Theorem 2]).

Let $X$ be a smooth $k$-scheme, let $\pi:X\to\mathbb{A}_{k}^{n}$ be a smooth
morphism with a section $s$. Assume that
$\pi|_{\pi^{-1}(\mathbb{A}_{k}^{n}\setminus\\{0\\})}:\pi^{-1}(\mathbb{A}_{k}^{n}\setminus\\{0\\})\to\pi^{-1}(\mathbb{A}_{k}^{n}\setminus\\{0\\})$
is an $\mathbb{A}^{1}$-weak equivalence. Then there exists an induced pointed
$\mathbb{A}^{1}$-weak equivalence

$(X,s(0))\sim(\mathbb{P}^{1})^{\wedge n}\wedge(\pi^{-1}(0),s(0)).$

In particular, the deformation space $D(X,Y)$ of a closed immersion
$(Y,\star)\hookrightarrow(X,\star)$ between pointed smooth $k$-schemes is
$\mathbb{A}^{1}$-weakly equivalent to $\mathbb{P}^{1}\wedge(Y,\star)$.

######  Example 7.11.

Let
$Q_{2n}\subset\mathbb{A}_{k}^{2n+1}=\mathrm{Spec}(k[u_{1},\ldots,u_{n},v_{1},\ldots,v_{n},z])$
be the smooth $2n$-dimensional split quadric with equation
$\sum_{i=1}^{n}u_{i}v_{i}=z(z+1)$. The projection
$\pi=\mathrm{pr}_{u_{1},\ldots,u_{,n}}:Q_{2n}\to\mathbb{A}_{k}^{n+1}$ is a
smooth morphism restricting to a Zariski locally trivial
$\mathbb{A}^{n}$-bundle over $\mathbb{A}_{k}^{n}\setminus\\{0\\},$ hence to an
$\mathbb{A}^{1}$-weak equivalence over $\mathbb{A}_{k}^{n}\setminus\\{0\\}$,
and having the morphism

$s:\mathbb{A}_{k}^{n}\to
Q_{2n},\,(u_{1},\ldots,u_{n})\mapsto(u_{1},\ldots,u_{n},0,\ldots,0,0)$

as a natural section. On the other hand, $\pi^{-1}(0)$ is
$\mathbb{A}^{1}$-weakly equivalent to the disjoint union of $s(0)$ and of the
point $p=(0,\ldots,0\ldots,-1)$. Theorem 7.10 thus renders the conclusion that
$(Q_{2n},s(0))$ is $\mathbb{A}^{1}$-weakly equivalent to
$(\mathbb{P}^{1})^{\wedge n}\wedge(p\sqcup s(0))\sim(\mathbb{P}^{1})^{\wedge
n}$. In particular $Q_{2n}$ provides a smooth $k$-scheme model of the motivic
sphere $(\mathbb{P}^{1})^{\wedge n}=S^{n}\wedge\mathbb{G}_{m,k}^{\wedge n}$.

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IMB UMR5584, CNRS, Université de Bourgogne Franche-Comté, Dijon, France E-mail
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LMBP, F-63000 Clermont-Ferrand, France E-mail address<EMAIL_ADDRESS>Institut Montpelliérain Alexander Grothendieck, Université de Montpellier.
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