154. The motivic Thom Sebastiani theorem for regular and formal functions

J. reine angew. Math., Ahead of Print
DOI 10.1515 / crelle-2015-0022

Journal für die reine und angewandte Mathematik
© De Gruyter 2015

The motivic Thom–Sebastiani theorem
for regular and formal functions
By Quy Thuong Lê at Hanoi

Abstract. Thanks to the work of Hrushovski and Loeser on motivic Milnor fibers, we
give a model-theoretic proof for the motivic Thom–Sebastiani theorem in the case of regular
functions. Moreover, slightly extending Hrushovski–Loeser’s construction adjusted to Sebag,
Loeser and Nicaise’s motivic integration for formal schemes and rigid varieties, we formulate
and prove an analogous result for formal functions. The latter is meaningful as it has been a
crucial element of constructing Kontsevich–Soibelman’s theory of motivic Donaldson–Thomas
invariants.

1. Introduction
Let f and g be holomorphic functions on complex manifolds of dimensions d1 and d2 ,
having isolated singularities at x and y, respectively. Define f ˚ g by
f ˚ g.x; y/ D f .x/ C g.y/:
Let Ff;x be the (topological) Milnor fiber of .f; x/, the same for .g; y/ and .f ˚ g; .x; y//.
The original Thom–Sebastiani theorem [24] states that there exists an isomorphism between
the cohomology groups
H d1 Cd2

1

.Ff ˚g;.x;y/ ; Q/ Š H d1

1

.Ff;x ; Q/ ˝ H d2

1

.Fg;y ; Q/

compatible with the monodromies. Steenbrink in [25] refined a conjecture on the Thom–
Sebastiani theorem for the mixed Hodge structures, which was fulfilled later and independently
by Varchenko [27] and Saito [22]. In the letters to A’Campo (1972) and to Illusie (1999), Pierre
Deligne discussed the `-adic version for an arbitrary field (rather than complex numbers), in
which he replaced the Milnor fibers by the nearby cycles and used Laumon’s construction of
convolution product (cf. [16, Définition 2.7.2]); this work recently has been fully realized by
The author is partially supported by the Centre Henri Lebesgue (program “Investissements d’avenir”, ANR11-LABX-0020-01) and by ERC under the European Community’s Seventh Framework Programme (FP7/20072013), ERC Grant Agreement no. 246903/NMNAG.

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Lê, Regular and formal motivic Thom–Sebastiani theorem

Fu [8]. Furthermore, Denef–Loeser [5] and Looijenga [19] also provided proofs of the motivic
version for motivic vanishing cycles in the case of fields of characteristic zero, from which the
classical results were recovered without the hypothesis that x and y are isolated singularities.
We come back to the problem on the motivic Thom–Sebastiani theorem in the framework for the motivic Milnor fibers of formal functions. It has been likely a formally unsolved
problem, but already used in Kontsevich–Soibelman’s theory of motivic Donaldson–Thomas

invariants for non-commutative Calabi–Yau threefolds (see [15]). Using Temkin’s results on
resolution of singularities of an excellent formal scheme [26] and Denef–Loeser’s formulas for
the motivic Milnor fiber of a regular function [3, 6], Kontsevich and Soibelman introduce in
[15] the motivic Milnor fiber of a formal function. The motivic Thom–Sebastiani theorem for
formal functions that concerns this notion is a key to construct the motivic Donaldson–Thomas
invariants. In fact, it has the same interpretation as Denef–Loeser’s and Looijenga’s local version (cf. [5, 19]) and a complete proof for it should be required. This is the main purpose of the
present article.
The motivic Milnor fiber of a regular function may be described in terms of resolution
of singularity, after the works of Denef–Loeser [3, 6, 7] and of Guibert–Loeser–Merle [10–
12]. In particular, Guibert–Loeser–Merle had the refinement when applying this method to
further extensions of the motivic Thom–Sebastiani theorem (see [10–12]). Recently, with the
help of Hrushovski–Kazhdan’s motivic integration, Hrushovski and Loeser [14] have given
an even more flexible manner to describe the motivic Milnor fiber in terms of the data of
the corresponding analytic Milnor fiber (introduced by Nicaise–Sebag [21]). An important
application of this approach is our proof of the integral identity conjecture in [17]. There it
is also shown that a slight generalization of Hrushovski–Loeser’s construction [14] combined
with Nicaise’s formula on volume Poincaré series [20] allows to interpret the motivic Milnor
fiber of a formal function in the same way as in [14]. However, this method requires the
restriction to studying the motivic Milnor fibers over algebraically closed fields of characteristic
zero (hence the hypothesis in the present work).
Our article is organized as follows. In Section 2, we recall some basic and essential
backgrounds on the motivic Milnor fiber of a regular function, in which the local form of
Denef–Loeser’s and Looijenga’s motivic Thom–Sebastiani theorem is included (Theorem 2.1),
using the main references [3–7] and [19]. The local form states that
Sf ˚g;.x;y/ D Sf;x

Sg;y ;

where Sf;x is the motivic Milnor fiber of .f; x/, Sf;x WD . 1/d1 1 .Sf;x 1/, the same for
.g; y/ and .f ˚g; .x; y//, and is the convolution product (cf. Section 2.3). Here, one does not

need to assume that x and y are isolated singular points. Using the tools from [13,14], recalled
partly in Section 4 below, we introduce a new proof for this formula in Section 5. Notice that
the previous formula lives in the monodromic Grothendieck ring MkO , by a technical reason,
however, our proof only runs in a localization of MkO .
In Section 3, we mark the highlights and the essences of motivic integration for special
formal schemes, following [17, 18, 20, 21, 23]. In particular, by [17], we show that Kontsevich–
Soibelman’s motivic Milnor fiber of a formal function and Nicaise’s volume Poincaré series
mention the same thing, which can be also read off from the corresponding analytic Milnor
fiber. Furthermore, we can use the model-theoretic tools recalled in Section 4 to describe
the volume Poincaré series, hence the motivic Milnor fiber of a formal function. The formal

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version of the motivic Thom–Sebastiani theorem has the same form as the regular one but f
and g replaced by formal functions f and g, respectively (Theorem 3.4). This is proved in
Section 6 using the development of tools in Section 4 as well as some analogous techniques in
the proof of the regular version in Section 5.

2. Preliminaries
Throughout the present article, we always assume that k is an algebraically closed field
of characteristic zero.
2.1. Grothendieck rings of algebraic varieties. By definition, an algebraic k-variety
is a separated reduced k-scheme of finite type. Let Vark be the category of algebraic k-varieties,

its morphisms are morphisms of algebraic k-varieties. The Grothendieck group K0 .Vark /
is an abelian group generated by symbols ŒX for objects X in Vark subject to the relations
ŒX D ŒY if X and Y are isomorphic in Vark , ŒX D ŒY C ŒX n Y if Y is Zariski closed
in X. Moreover, K0 .Vark / is also a ring with unit with respect to the cartesian product. Set
L WD ŒA1k  and denote by Mk the localization of K0 .Vark / with respect to the multiplicative
system ¹Li j i 2 Nº.
Let m (or m .k/) be the group scheme of mth roots of unity in k. Varying m 1 in N,
such schemes give rise to a projective system with respect to morphisms mn ! m given
by 7! n , and its limit will be denoted by O . A good m -action on an object X of Vark is
a group action of m on X such that each orbit is contained in an affine k-subvariety of X. A
good O -action on X is a O -action which factors through a good m -action for some m
1
in N.
The O -equivariant Grothendieck group K0O .Vark / is an abelian group generated by the
iso-equivariant classes of varieties ŒX; , with X an algebraic k-variety, a good O -action
on X, modulo the conditions ŒX;  D ŒY; jY  C ŒX n Y; jXnY  if Y is Zariski closed in X
and ŒX Ank ;  D ŒX Ank ; 0  if , 0 lift the same O -action on X to an affine action on
X Ank . In the present article we shall denote ŒX;  simply by ŒX when the O -action
is clear. Similarly, K0O .Vark / has a natural ring structure due to the cartesian product. Let
O
MkO denote K0O .Vark /ŒL 1 , it is the O -equivariant version of Mk above. Let Mk;loc
be the
O
localization of Mk with respect to the multiplicative family generated by the elements 1 Li ,
O
with i 1 in N. We shall also write loc for the localization morphism MkO ! Mk;loc
.
2.2. Motivic Milnor fiber. Let X be a pure d -dimensional smooth k-variety, f a nonconstant regular function on X, and x a closed point in the zero locus of f . Denote by Xx;m (or
Xx;m .f /) the set of arcs '.t / in X.kŒt =.t mC1 // originated at x with f .'.t // Á t m mod t mC1 ,
which is a locally closed subvariety of k-variety X.kŒt =.t mC1 //. Since Xx;m is invariant by

the good m -action on X.kŒt =.t mC1 // given by '.t / D '. t /, it defines an iso-equivariant
class ŒXx;m  in MkO . The motivic zeta function of f at x is the formal series
X
Zf;x .T / D
ŒXx;m L md T m
m 1

with coefficients in MkO .

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Lê, Regular and formal motivic Thom–Sebastiani theorem

By Denef–Loeser [3], Zf;x .T / is a rational function, i.e., an MkO -linear combination of 1
and finite products (possibly empty) of La T b =.1 La T b / with .a; b/ in Z N>0 . Remark
that we can take by [6] the limit limT !1 for rational functions such that
La T b
D
T !1 1
La T b
lim

Then the motivic Milnor fiber of f at x is defined as
This is a virtual variety in MkO .

1:
limT !1 Zf;x .T / and denoted by Sf;x .

2.3. The motivic Thom–Sebastiani theorem for regular functions. In this subsection, we restate the motivic Thom–Sebastiani theorem for motivic Milnor fibers.
Let us recall from [5, 11, 19] the concept of convolution product. Consider the Fermat
2
defined by the equations um C v m D 0 and um C v m D 1,
varieties F0m and F1m in Gm;k
respectively. We endow with the standard . m
m /-action on these varieties. If X and Y are
algebraic k-varieties with m -action, one defines
ŒX

ŒY D

ŒF1m

m

m

.X

Y/ C ŒF0m

m

m

.X

Y/;

where, for i 2 ¹0; 1º,
Fim

m

m

.X

Y/ D Fim

.X

Y/=

with .au; bv; x; y/ .u; v; ax; by/ for any a, b in m . The group scheme m acts diagonally
m
m .X
on Fim
Y/. Passing to the projective limit that MkO equals limMk m , we get the
convolution product on MkO . This product is commutative and associative (see for example
[11]).
Let f and g be regular functions on smooth algebraic k-varieties X and Y, respectively.
Define f ˚ g.x; y/ D f .x/ C g.y/. For closed points x in X0 and y in Y0 , we set
Sf;x D . 1/dim X

1

.Sf;x

1/;

Sg;y D . 1/dim Y

1

.Sg;y

1/

and similarly for Sf ˚g;.x;y/ .
Theorem 2.1 ([5, 19]). The identity Sf ˚g;.x;y/ D Sf;x

Sg;y holds in MkO .

Remark 2.2. In fact, in [5] and [19], the motivic Thom–Sebastiani theorem is proved
in the framework of motivic vanishing cycles, which implies Theorem 2.1.

3. The motivic Thom–Sebastiani formula for formal functions
Let X be a generically smooth special formal kŒŒt -scheme of relative dimension d , with
reduction X0 and structural morphism f. Let x be a closed point of X0 .
3.1. The motivic Milnor fiber of a formal function. By [26] (see also [20]), there
exists a resolution of singularities h W Y ! X of X0 . Let Ei , i 2 J , be the irreducible

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components of .Ys /red . Let Ni be the multiplicity of Ei in Ys . We set Ei D .Ei /0 for i 2 J ,
and
\
[
EI D
Ei and EIı D EI n
Ej
i 2I

j 62I

for a nonempty subset I of J . Let ¹U º be a covering of Y by affine open subschemes with
U \ EIı 6D ; such that, on this piece,
Y N
f ı h D uQ
yi i ;
i 2I

where uQ is a unit, yi is a local coordinate defining Ei . Set mI WD gcd.Ni /i 2I . One can
construct as in [7] an unramified Galois covering I W EQIı ! EIı with Galois group mI , which
is given over U \ EIı by
®
¯
1

.z; y/ 2 A1k .U \ EIı / j z mI D u.y/
Q
:
Note that EQIı is endowed with a natural mI -action good over EIı obtained by multiplying the
z-coordinate with elements of mI . We also restrict this covering over EIı \ h 1 .x/ and obtain
a class, written as ŒEQIı \ h 1 .x/, in MkO . The motivic Milnor fiber of the formal germ .X; x/,
or of f at x, is defined to be the quantity
X
.1 L/jI j 1 ŒEQIı \ h 1 .x/
;6DI J

in MkO . We denote it by S.X; x/ or by Sf;x . By [17, Lemma 5.7], using volume Poincaré series,
Sf;x is well defined, i.e., independent of the choice of the resolution of singularities h.
Remark 3.1. Let b
Xx denote the formal completion of X at x, and let fx be the structural
morphism of b
Xx , which is induced by f. We are able to use a resolution of singularity of X
at x to define the motivic Milnor fiber Sfx ;x . Then, it is clear that Sf;x D Sfx ;x .
3.2. Integral of a gauge form and volume Poincaré series.
Stft formal schemes. Assume that X is a separated generically smooth formal kŒŒt scheme topologically of finite type and that the relative dimension of X is d . One may regard
X as the inductive limit of the kŒt =.t mC1 /-schemes locally of finite type
Xm WD X; OX ˝kŒŒt  kŒt =.t mC1 /
in the category of formal kŒŒt -schemes. By Greenberg [9], there exists a unique k-scheme
Grm .Xm / topologically of finite type, up to isomorphism, which for any k-scheme Y admits a
natural bijection
Homk .Y; Grm .Xm // ! HomSpec.k/ Y

k

kŒt =.t mC1 /; Xm :

These k-schemes Grm .Xm / together with the natural translation give rise to a projective system, we denote its limit by Gr.X/ (cf. [18, 23]). We denote by m the canonical projection
Gr.X/ ! Grm .Xm /. See more in [9] for some basic properties of the functor Gr. Notice that
the notion of stable cylinder of Gr.X/ in this context was already introduced in [18, 23].

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Lê, Regular and formal motivic Thom–Sebastiani theorem

By [18, 23], the motivic measure of a stable cylinder A in Gr.X/ is
.A/ D Œ

(3.1)

` .A/L

.`C1/d

for ` 2 N large enough. Let ˛ W A ! Z [ ¹1º be a function on A that takes only a finite
number of values such that every fiber ˛ 1 .m/ is a stable cylinder in Gr.X/. Let ! be a gauge
form on XÁ . By [2, Proposition 1.5] (see also [18]), there exists a canonical isomorphism
d
XÁ .XÁ /

Š

d
XjkŒŒt  .X/

˝kŒŒt  k..t //;

Q Let '
thus there exist an n in N and a differential form !Q in dXjkŒŒt  .X/ such that ! D t n !.
be a point of Gr.X/ outside Gr.Xsing /. Then, we can regard it as a morphism of formal schemes
Spf.kŒŒt / ! X, or as a morphism of rings OX .X/ ! kŒŒt . Thus it induces a morphism of
rings 'Q W ' dXjkŒŒt  .X/ ! kŒŒt , which is a surjection. One defines
ord.!/.'/
Q
D ord t .'.'
Q
!//
Q
and

(3.2)

ordX .!/ D ord.!/
Q

n:

The latter is independent of the choice of !Q (cf. [18]). Since ! is a gauge form, it follows
from the proof of [18, Theorem-Definition 4.1.2] that ordX .!/ is an integer-valued function
taking only a finite number of values and that its fibers are stable cylinder. Then one defines
(cf. [18, 23])

Z
X ®
¯
j!j WD
' 2 Gr.X/ j ordX .!/.'/ D m L m 2 Mk :
XÁ

m2Z

Special formal schemes. We consider the more general case where X is a generically
smooth special formal kŒŒt -scheme (see [1] for definition). Let Y ! X be a Néron smoothening for X, i.e., a morphism of special formal kŒŒt -schemes, with Y being adic smooth over
kŒŒt, inducing an open embedding YÁ ! XÁ with
b k..t // K D XÁ ˝
b k..t // K
YÁ ˝
for any finite unramified extension K of k..t //. Such a Néron smoothening exists by [20].
Furthermore, we are able to (and we shall from now on) choose Y to be a separated generically
smooth formal kŒŒt -scheme topologically of finite type. Using [20, Propositions 4.7, 4.8], for
any gauge form ! on XÁ , we define
Z
Z
j!j WD
j!j 2 Mk :
XÁ

YÁ

For any m in N>0 , let
b kŒŒt  kŒŒt 1=m ;
X.m/ WD X ˝

b k..t // k..t 1=m //;
XÁ .m/ WD XÁ ˝

and let !.m/ be the pullback of ! via the natural morphism XÁ .m/ ! XÁ . The Néron
smoothening Y ! X for X induces a Néron smoothening Y.m/ ! X.m/ for X.m/, and
Y.m/ is also topologically of finite type, like Y. The canonical
-action on Gr.Y.m/ is
R
1=m / D '.at 1=m /. It induces a
given
by
a'.t
-action
on
j!.m/j,
thus we regard
m
XÁ .m/
R
O
XÁ .m/ j!.m/j as an element of Mk .

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Volume Poincaré series. Let X be a generically smooth special formal kŒŒt -scheme,
x a closed point of X0 , and b
Xx the formal completion of X at x. Denoting by xŒ the tube
of x, namely the analytic Milnor fiber of f at x (cf. [21]), we have the canonical isomorphism
xŒ Š .b
Xx /Á . Set
xŒm WD xŒ k..t // k..t 1=m //:
Let us consider the volume Poincaré series of .xŒ; !/, where ! is a gauge form on xŒ
(cf. [20]):
Ã
X ÂZ
j!.m/j T m 2 MkO ŒŒT :
S.xŒ; !I T / WD
m 1

xŒm

Remark 3.2. More generally, the volume Poincaré series of separated generically
smooth formal schemes topologically of finite type (resp. separated quasi-compact smooth
rigid varieties) were introduced and studied first by Nicaise–Sebag in [21]. After that, Nicaise
[20] studied these objects in the framework of generically smooth special formal schemes (resp.
bounded smooth rigid varieties).
In practice, one may assume that ! is b
Xx -bounded, i.e., ! lies in the image of the natural
map (cf. [20, Definition 2.11])
d

b

Xx jkŒŒt 

˝kŒŒt  k..t // .b
Xx / !

d
xŒjk..t // .xŒ/:

Since k is an algebraically closed field, S.xŒ; !I T / is independent of the choice of the
uniformizing parameter t. Indeed, let t 0 be another uniformizing parameter for kŒŒt . Then
t 0 D ˛t, where ˛ D ˛.t / 2 kŒŒt  and ˛.0/ 2 k . Since k contains all roots, the mth roots of
˛ are again in kŒŒt . This induces a canonical isomorphism of k..t //-fields
k..t 1=m // ! k..t 01=m //;
which implies the previous claim. By Nicaise [20, Corollary 7.13], if the gauge form ! is b
Xx bounded, this series S.xŒ; !I T / is a rational function.
Proposition 3.3. With the notation and the hypotheses as previous, the following identity holds in MkO :
Ã
X ÂZ
d
Sf;x D L lim
j!.m/j T m :
T !1

m 1

xŒm

Proof. The identity is true in Mk because of the definition of Sf;x as well as Nicaise’s
formula for limT !1 S.xŒ; !I T / in [20, Proposition 7.36]. To see that it is true in MkO , we
refer to the proof of [17, Lemma 5.7].

3.3. Statement of result for formal functions. Let d1 ; d2 be integers with d1 1 and
d2 1. Let f be a formal power series in kŒŒx with f .0/ D 0 and g in kŒŒy with g.0/ D 0.
Here x D .x1 ; : : : ; xd1 /, y D .y1 ; : : : ; yd2 / and we use the same symbol 0 for the origin of
Adk1 , Adk2 or A1k (whenever necessary, e.g., in Section 6, however, we shall write 0di for the
origin of Adki , i 2 ¹1; 2º). Let us consider the following special formal kŒŒt -schemes:

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Lê, Regular and formal motivic Thom–Sebastiani theorem

X WD Spf kŒŒt; x=.f .x/

t/ ;

Y WD Spf kŒŒt; y=.g.y/

t/ ;

X ˚ Y WD Spf kŒŒt; x; y=.f .x/ C g.y/

t/ ;

with structural morphisms f, g and f ˚ g induced by f , g and f ˚ g, respectively. Set
Sf;0 WD . 1/d1

1

.Sf;0

1/

for f and the same for g and f ˚g. We now set up the statement of the motivic Thom–Sebastiani
O
, using Hrushovski–
theorem for formal schemes and then prove it in the setting of Mk;loc
Kazhdan’s integration [13] via the work of Hrushovski–Loeser [14].
Theorem 3.4. The identity loc.Sf˚g;.0;0/ / D loc.Sf;0

O
.
Sg;0 / holds in Mk;loc

The complete proof is given in Section 6.

4. Extension of Hrushovski–Loeser’s morphism
4.1. The theory ACVFk..t// .0; 0/. Let us consider the theory ACVFk..t // .0; 0/ of algebraically closed valued fields of equal characteristic zero that extend k..t // (cf. [13]). Its sort
VF admits the language of rings, while the sort RV is endowed with abelian group operations
and =, a unary predicate k for a subgroup, and a binary operation C on k D k [ ¹0º. We
also have an imaginary sort € that is with a uniquely divisible abelian group. For a model L of
this theory, let RL (resp. mL ) denote its valuation ring (resp. the maximal ideal of RL ). The
following are the “elementary” L-definable sets of ACVFk..t // .0; 0/:
VF.L/ D L;

RV.L/ D L =.1 C mL /;

€.L/ D L =RL ;

k.L/ D RL =mL :

In general, a definable subset of VFn .L/ is a finite Boolean combination of set of the forms
val.f1 / Ä val.f2 / or f3 D 0, where fi are polynomials with coefficients in k..t //. The same
definition may apply to definable subsets of RVn .L/, € n .L/ or kn .L/. Correspondingly, there
are the following natural maps between these sets:
rv W VF ! RV;

val W VF ! €;

valrv W RV ! €;

res W RL ! k.L/:

There is an exact sequence of groups:
1 ! k ! RV

valrv

! € ! 0:

4.2. Measured categories (following [13]).
VF-categories. Let € VF be the category of k..t //-definable sets (or definable sets,
for short) endowed with definable volume forms, up to €-equivalence. One may show that
it is graded via the following subcategories € VFŒn, n 2 N. An object of € VFŒn is a
0
triple .X; f; "/ with X a definable subset of VF` RV` , for some `, `0 in N, f W X ! VFn a
definable map with finite fibers, and " W X ! € a definable function; a morphism from .X; f; "/

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to .X 0 ; f 0 ; "0 / is a definable essential bijection F W X ! X 0 such that
" D "0 ı F C val.JacF /
away from a proper closed subvariety of X (the measure preserving property). Here, that
F W X ! X 0 is an essential bijection means that there exists a proper closed subvariety Y of X
such that F jXnY W X n Y ! X 0 n F .Y / is a bijection (see [13, Section 3.8]).
Let € VFbdd Œn be the full subcategory of € VFŒn whose objects are bounded definable
sets with bounded definable forms ". If considering " W X ! € as the zero function, we obtain
the categories volVF and volVFŒn as well as volVFbdd and volVFbdd Œn. In this case, the
measure preserving property of a morphism F is characterized by the condition val.JacF / D 0,
outside a proper closed subvariety.
Convention. For simplicity, we shall omit the symbol f in the triple .X; f; "/ when no
confusion can arise.
RV-categories. Similarly, we consider the category € RV graded by € RVŒn, n 2 N.
By definition, an object of € RVŒn is a triple .X; f; "/ with X a definable subset of RV` , for
some ` in N, f W X ! RVn a definable map with finite fibers, and " W X ! € a definable
function; a morphism .X; f; "/ ! .X 0 ; f 0 ; "0 / is a definable bijection F W X ! X 0 such that
" C jvalrv .f /j D "0 ı F C jvalrv .f 0 ı F /j
away from a proper closed subvariety. Here, for x D .x1 ; : : : ; xn / 2 € n , we define jxj as the
P
sum niD1 xi . The category € RESŒn is defined as the full subcategory of € RVŒn such

that, for each object .X; f; "/, valrv .X / is a finite set. The category € RVbdd is defined as
€ RV with valrv -image of objects bounded below. In the case where, for each object .X; f; "/
of one of the previous categories, " is the zero function, we obtain the subcategories volRV,
volRVbdd and volRES.
In the present article, we also consider RES, a category defined exactly as volRES but
the measure preserving property is not required for morphisms.
€ -categories. The category €Œn consists of pairs .; l/ with  a definable subset of
€ n and l W  ! € a definable map. A morphism .; l/ ! .0 ; l 0 / is a definable bijection
W  ! 0 which is liftable to a definable bijection valrv1  ! valrv1 0 such that
jxj C l.x/ D j .x/j C l 0 . .x//:
The category € bdd Œn is the full subcategory of €Œn such that, for each object .; l/ of
€ bdd Œn, there exists a 2 € with  Œ ; 1/n . By definition, the categories € and € bdd
L
L
are the direct sums n 1 €Œn and n 1 € bdd Œn, respectively. The subcategories whose
objects are of the form .; 0/ will be denoted by vol€ and vol€ bdd .
4.3. Structure of K. € VFbdd /. Let C be one of the categories in Section 4.2. Then, as
in [13], we denote the Grothendieck semiring of C by KC .C/ and the associated ring by K.C/.
By [13], there is a natural morphism of semirings
(4.1)

N W KC . € bdd / ˝ KC .

€ RES/

! KC .

bdd
/
€ VF

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Lê, Regular and formal motivic Thom–Sebastiani theorem

constructed as follows. Note that two objects admitting a morphism in € bdd Œn define
the same element in KC . € bdd Œn/, hence lifts to a morphism in € VFbdd Œn between their
pullbacks. Thus there exists a natural morphism KC . € bdd Œn/ ! KC . € VFbdd / mapping the
class of .; l/ to the class of .val 1 ./; l ı val/. Also, for each object .X; f; "/ in € RESŒn,
we may consider an étale map ` W X ! kn . By this, we have the natural morphism
KC .

€ RESŒn/

! KC .

bdd
/
€ VF

by sending the class of .X; f; "/ to the class of .X `;res Rn ; pr1 ı "/. In particular, if X is
Zariski open in kn , then X `;res Rn is simply res 1 .X /.
Theorem 4.1 (Hrushovski–Kazhdan [13]). The morphism N is a surjection. Moreover,
it also induces a surjective morphism N between the associated rings.
There is a more intrinsic description of N , which follows from [13, Theorem 8.29, Proposition 10.10]. More precisely, one first constructs the natural morphism

KC . € bdd / ˝ KC .

€ RES/

! KC .

bdd
/
€ RV

due to the inclusion RES
RV and the valuation map valrv . This morphism is a surjection,
its kernel is generated by 1 ˝ Œvalrv1 . /1 Œ 1 ˝ 1, with definable in €. The subscript 1
means that the classes are in degree 1. Second, the canonical morphism
KC .

bdd
Œn/
€ VF

! KC .

bdd
Œn/=Œ11 ŒRV>0 1
€ RV

induced by the map Ob € RVŒn ! Ob € VFŒn sending .X; f; "/ to .LX; Lf; L"/, where
LX D X f;rv .VF /n , Lf .a; b/ D f .a; rv.b// and L".a; b/ D ".a; rv.b//, is an isomorphism. Then the composition of the first morphism with the natural projection
KC .

bdd
Œn/
€ RV

! KC .

bdd
Œn/=Œ11 ŒRV>0 1
€ RV

and with the inverse morphism of the second morphism (with all n) yields the morphism N .
Remark 4.2. According to [13, Proposition 10.10], an element of KC . € RVbdd / may
be written as a finite sum of elements of the form Œ.X valrv1 ./; f; "/. Furthermore, an
argument in the proof of [13, Proposition 10.10] implies
Œ.X

valrv1 ./; f; "/ D Œ.X; f0 ; 1/ ˝ Œ.; l/;

where f0 W X ! RVn and l W  ! € are some definable functions.
4.4. Extending Hrushovski–Loeser’s construction.
The morphisms hm and hQ m . From now on, we shall denote by ŠK.RES/ the quotient
of K.RES/ subject to the relations Œvalrv1 .a/ D Œvalrv1 .0/ for a in €, and by ŠK.RES/ŒL 1 loc
the localization of ŠK.RES/ŒL 1  with respect to the multiplicative family generated by 1 Li ,
i 1. Let m; n be in N, m 1, .; l/ in € bdd Œn, and e in € with me 2 Z. Set
.m/ WD  \ .1=mZ/n ;

l;e WD l

1

.e/

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11

and
X

˛m .; l/ WD

X

e2€;me2Z

D

X
e2Z

L

m.j jCe/

.L

1/n

2l;e .m/

X

L

mj j e

1/n :

.L

2l;e=m .m/

It is clear that ˛m .; l/ is an element of ŠK.RES/ŒL 1 loc , and moreover, ˛m is independent
of the choice of coordinates for € n . Indeed, let be the morphism in € bdd from .l;e ; ljl;e /
to .0 ; l 0 /. Then j . /j C l 0 . . // D j j C l. / D j j C e and the claim follows. Thus ˛m
defines a natural morphism of rings
˛m W K. € bdd / ! ŠK.RES/ŒL
Q in vol€ bdd , one sets
By using [14], for any 
X
Q WD
˛Q m ./
L

mj j

.L

1

loc :

1/n

Q
2.m/

and obtains a morphism of rings
˛Q m W K.vol€ bdd / ! ŠK.RES/ŒL
Thus we can consider ˛m as an extension of ˛Q m ; moreover,
X
(4.2)
˛m .; l/ D
˛Q m .l;e=m /L

1

loc :

e

:

1

loc

e2Z

We are able to construct a morphism
ˇm W K.

€ RES/

! ŠK.RES/ŒL

by using Hrushovski–Loeser’s method. Thanks to Remark 4.2, however, it suffices to define ˇm at elements of the form Œ.X; f; 1/ with .X; f; 1/ an object in € RES. Assume that
f .X/ V 1
V n , i.e., valrv .fi .x// D i for every x in X. We set
´
ŒX .L 1 Œ11 /mj j if m 2 Zn ;
ˇm .X; f; 1/ WD
0
otherwise:
There are two steps to check that ker.˛Q m ˝ ˇm / contains ker.N0 /, where N0 is N reduced to the volume version (for the structure of K.volVFbdd /). These steps correspond to the
factorization of N0 into
K.vol€ bdd / ˝ K.volRES/ ! K.volRVbdd /
and
K.volVFbdd Œn/ ! K.volRVbdd Œn/=Œ11

ŒRV>0 1 :

Hrushovski and Loeser [14] passed these by direct computation. This can be applied to show
that ker.˛m ˝ ˇm / contains ker.N /. Consequently, from the tensor products ˛Q m ˝ ˇm and

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Lê, Regular and formal motivic Thom–Sebastiani theorem

˛m ˝ ˇm we obtain morphisms of rings
hQ m W K.volVFbdd / ! ŠK.RES/ŒL
hm W K.

€ VF

bdd

/ ! ŠK.RES/ŒL

1

loc ;

1

loc :

Moreover, there is a presentation of hm in terms of hQ m induced from (4.2). Namely, we have
the following lemma whose proof is trivial and left to the reader.
Lemma 4.3. hm .Œ.X; "// D

Q

P

1 .e=m//L e

e2Z hm .Œ"

(in ŠK.RES/ŒL

1  ).
loc

The morphism h. We also use the morphisms from [14, Section 8.5] with their restriction, namely,
˛ W K.vol€ bdd / ! ŠK.RES/ŒL

1

;

ˇ W K.volRES/ ! ŠK.RES/ŒL

1

:

By definition, ˇ.ŒX / D ŒX  and ˛.Œ/ D ./.L 1/n if  is a definable subset of € n ,
where is the o-minimal Euler characteristic in the sense of [13, Lemma 9.5]. Since ker.˛˝ˇ/
contains ker.N0 / (cf. [14]), it gives rise to a morphism of rings

K.volVFbdd / ! ŠK.RES/ŒL

1

:

The composition of it with the localization morphism ŠK.RES/ŒL
will be denoted by h.

1

! ŠK.RES/ŒL

1
loc

P
Proposition 4.4. The formal series Z 0 .X; "/.T / WD m 1 hm .Œ.X; "//T m is a rational function. Moreover, we have limT !1 Z 0 .X; "/.T / D h.ŒX /.
Proof. It is similar to the proof of [14, Proposition 8.5.1].
4.5. Endowing with a O -action and the morphisms hm , hQ m and h. First, let us recall
m D t , n
[14, Section 4.3]. Define a series ¹tm ºm 1 by setting t1 D t , tnm
1. For a k..t //n
definable set X over RES, we may assume X Vi1 =m
Vin =m for some n, m and ij . It
is endowed with a natural action ı of m . Now the k..t 1=m //-definable function
i1
in
.x1 ; : : : ; xn / 7! .x1 =rv.tm
/; : : : ; xn =rv.tm

//

maps X to a constructible subset Y of Ank , where Y is endowed with a
ı. The correspondence X 7! Y in its turn defines a morphism of rings
ŠK.RES/ŒL

1

 ! ŠK0O .Vark /ŒL

1

m -action induced from



(cf. [13, Lemma 10.7], [14, Proposition 4.3.1]). Here, by definition, ŠK0O .Vark / is the quotient
of K0O .Vark / by identifying all the classes ŒGm ; , where is a O -action on Gm induced by
multiplication by roots of 1. The previous morphism together with the natural one
ŠK0O .Vark /ŒL

1

 ! MkO

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Lê, Regular and formal motivic Thom–Sebastiani theorem

induces the following morphisms of rings, both are denoted by ‚:
ŠK.RES/ŒL

1

 ! MkO

and ŠK.RES/ŒL

1

O
loc ! Mk;loc
:

Q m WD ‚ ı hQ m and h WD ‚ ı h with
We now define ring morphisms hm WD ‚ ı hm , h
O
Q m and h starts from
the same target Mk;loc . In fact, while hm has the source K. € VFbdd /, h
bdd
K.volVF /. Similarly to Lemma 4.3 and Proposition 4.4, we get
Lemma 4.5. hm .Œ.X; "// D

P

Q

e2Z hm .Œ"

1 .e=m//L e

O
).
(in Mk;loc

P
Proposition 4.6. The formal series Z.X; "/.T / WD m 1 hm .Œ.X; "//T m is a rational function. Moreover, we have limT !1 Z.X; "/.T / D h.ŒX /.
4.6. Description of the motivic Milnor fibers.
0

Regular case. Let be in €. A definable subset X of VF` RV` is -invariant if, for
0
0
any .x; x 0 / 2 VF` RV` and any .y; y 0 / 2 VF` RV` with val.y/
, both .x; x 0 / and
0
.x; x 0 / C .y; y 0 / simultaneously belong to either X or the complement of X in VF` RV` .
By [14, Lemma 3.1.1], any bounded definable subset of VF` that is closed in the valuation
topology is -invariant for some in €.
0
Assume that X is a -invariant definable subset of VFn RV` with 2 .1=m/Z €.
By [13], the set X.k..t 1=m /// of k..t 1=m //-points of X is the pullback of some definable subset
0
XŒmI  of .kŒt 1=m =t /n RV` and the projection X ŒmI  ! VFn is a finite-to-one map. If
0 is in € with 0

, the equality
ŒX ŒmI

0

 D ŒX ŒmI Lnm.

0

/

holds in ŠK.volRESŒn/, thus ŒX ŒmI L nm in ŠK.RES/ŒL 1  is independent of the choice
of large enough. For brevity, we shall write XQ Œm for the quantity ŒX ŒmI L nm Cn as
well as for its image under ‚.
Proposition 4.7.

(i) For X as previous, hQ m .ŒX / D loc.XQ Œm/.

(ii) Let f be a nonzero function on a d -dimensional smooth connected k-variety X, x a point
of f 1 .0/. Let be the reduction map X.R/ ! X.k/. Set
®
¯
X WD x 2 X.R/ j .x/ D x; rv.f .x// D rv.t / :
Then h.ŒX / D loc.Sf;x /.
(iii) For any in €, h.Œ 1 / D 1 and h.Œ 1 / D L. (Note that Œ 1 and Œ 1 are the open and
closed disks of valuative radius .)
Proof. (i) See Hrushovski–Loeser [14].
(ii) We use [14, Corollary 8.4.2] for proving (ii). Since X is 2-invariant (it is in fact
-invariant for any > 1 in €), we have
®

¯
X ŒmI 2 D ' 2 X.kŒt 1=m =.t 2 // j '.0/ D x; rv.f .'// D rv.t / :

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Lê, Regular and formal motivic Thom–Sebastiani theorem

The condition rv.f .'// D rv.t / is equivalent to f .'/ Á t mod t .mC1/=m , thus X ŒmI 2 is
definably isomorphic via the map t 1=m 7! t to
.m 1/d1

®
¯
' 2 X.kŒt =.t mC1 // j '.0/ D x; f .'/ Á t m mod t mC1

Ak

:

We get hQ m .ŒX / D loc.ŒX0;m L md1 / and the conclusion follows.
(iii) Assume D a=b with a; b in Z and .a; b/ D 1. Then
´
ma if n D mb;
Q n .Œa=b1 / D L
h

0
otherwise;
thus h.Œa=b1 / D 1. Also,
´

L
hQ n .Œa=b1 / D
0

maC1

if n D mb;
otherwise;

thus h.Œa=b1 / D L.
Formal case. Let X be a rigid k..t //-variety which is the generic fiber of a special
formal kŒŒt -scheme X, let ! be a gauge form on X . We set
b kŒŒt  kŒŒt alg
X WD X ˝

and

X WD X ˝k..t // k..t //alg :

The integer-valued function ordX .!/ on X was already recalled in (3.2). Using the same way,
one may define a rational-valued function ordX .!/ on X , where ! is the pullback of ! via the
natural morphism X ! X . We denote this rational-valued function by val! .
Theorem 4.8. Let X be a relatively d -dimensional special formal kŒŒt -scheme with
structural morphism f. Let XÁ;rv (resp. XÁ .m/rv ) be a version of XÁ (resp. XÁ .m/) in which
fÁ .x/ D t is replaced by rvfÁ .x/ D rv.t / (resp. fÁ .x/ Á t mod t .mC1/=m ). Then, for any

gauge form ! on XÁ ,
R
(i) hm .Œ.XÁ ; val! // D loc.Ld XÁ .m/ j!.m/j/,
R
(ii) hm .Œ.XÁ;rv ; val! // D loc.Ld XÁ .m/rv j!.m/j/,
(iii) h.ŒXÁ;rv / D h.ŒXÁ /.
As a consequence, for a closed point x of X0 and a gauge form ! 0 on xŒ,
R
(iv) hm .Œ.xŒ; val! 0 // D loc.Ld xŒ j! 0 .m/j/,
m
R
d
(v) hm .Œ.xŒrv ; val! 0 // D loc.L xŒ
j! 0 .m/j/,
m;rv

(vi) h.xŒrv / D h.xŒ/ D loc.Sf;x /.
Proof. We prove (i). By Lemma 4.5,
X
Q m .Œval! 1 .e=m//L
h
(4.3)
hm .Œ.XÁ ; val! // D

e

:

e2Z

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Lê, Regular and formal motivic Thom–Sebastiani theorem

By [14, Lemma 3.1.1], for each e in Z, there exists a
invariant. Thus it follows from Proposition 4.7 (i) that
(4.4)

e;m

in € such that val! 1 .e=m/ is

e;m -

E

Q m .Œval! 1 .e=m// D loc val! 1 .e=m/Œm
h
D loc val! 1 .e=m/ŒmI

0

md

L

0 Cd

for any 0
€. Let Y ! X be a Néron smoothening for X, where Y is
e;m in .1=m/Z
a relatively d -dimensional kŒŒt -formal scheme topologically of finite type. Then, XÁ D YÁ ,
since kŒŒt  is henselian, so we can regard val! as a function on YÁ . As val! is induced by the
gauge form !, thus val! 1 .e=m/.m/ is a stable cylinder in Gr.Y.m//. Moreover,
ordY.m/ .!.m//
By definition of the measure
(4.5)

1

.e/ D val! 1 .e=m/.m/:

(cf. (3.1)), we have

ordY.m/ .!.m//

1

.e/ D

val! 1 .e=m/.m/

D val! 1 .e=m/ŒmI
for

0

0

L

in N large enough. From (4.3), (4.4) and (4.5), it follows that
 Z
Ã
 Z
d
hm .Œ.XÁ ; val! // D loc L
j!.m/j D loc Ld
YÁ .m/

XÁ .m/

md

0

Ã
j!.m/j :

This identity is also compatible with the canonical m -action by definition, thus it holds in
O
Mk;loc
.
The identities (ii)–(vi) are direct consequences of the first one.
Remark 4.9. In [17], we define the motivic nearby cycles of a formal function f and

denote it by Sf . This is a virtual variety in the Grothendieck ring MXO 0 of X0 -varieties with
R good
/
is
nothing
but
loc.
O -action.
In
the
context
of
Theorem
4.8
(iii),
the
quantity
h.ŒX
Á
X0 Sf /,
R
O
O
where X0 is the forgetful (or pushforward) morphism MX0 ! Mk .

5. A new proof for the motivic Thom–Sebastiani theorem
In this section, we give a model-theoretic proof for Theorem 2.1 by using the morphisms
Q m and h. For notational simplicity, we let f and g be regular functions on Ad1 and
of rings h
k

Adk2 , vanishing at their origins, respectively. Then, we shall prove that the following identity
O
holds in Mk;loc
:
(5.1)

loc.Sf ˚g;.0;0/ / D loc

Sf;0

Sg;0 C Sf;0 C Sg;0 :

5.1. Decomposition of the analytic Milnor fiber. Consider the analytic Minor fiber of
f ˚ g at the origin of Adk1 Adk2 :
®
¯
Z WD .x; y/ 2 md1 Cd2 j rv.f .x/ C g.y// D rv.t / :

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Lê, Regular and formal motivic Thom–Sebastiani theorem

This is a bounded 2-invariant definable subset of VFd1 Cd2 . By Proposition 4.7 (ii), we have
O
h.ŒZ/ D loc.Sf ˚g;.0;0/ / in Mk;loc

. Let us decompose Z into a disjoint union of sets X, Y
and Z subject to conditions valf .x/ < valg.y/, valf .x/ > valg.y/ and valf .x/ D valg.y/,
respectively. In the sequel, we are going to compute h.ŒX /, h.ŒY /, h.ŒZ / and conclude.
Write
®
¯
X D .x; y/ 2 md1 Cd2 j rv.f .x// D rv.t /
as the product of the definable sets X 0 WD ¹x 2 md1 j rvf .x/ D rv.t /º and md2 D Œ0d1 2 .
Statements (ii) and (iii) of Proposition 4.7 give h.ŒX1 / D loc.Sf;0 / and h.Œmd2 / D 1, thus
O
O
.
. Similarly, we also have h.ŒY / D loc.Sg;0 / in Mk;loc
h.ŒX/ D loc.Sf;0 / in Mk;loc
Set
®
¯
®
¯
Z1 D .x; y/ 2 Z j valf .x/ D 1 ; Z<1 D .x; y/ 2 Z j 0 < valf .x/ < 1 ;
then Z D Z1 t Z<1 . For our goal we introduce the following definable set:
®
Z0 WD .x; y/ 2 md1 Cd2 j val.f .x/ C g.y// > 1;
We shall consider the identity ŒZ  D .ŒZ1 
Proposition 5.1. For m
hQ m .ŒZ1 

¯
rvf .x/ D rvg.y/ D rv.t / :

ŒZ0 / C .ŒZ<1  C ŒZ0 / in K.volVFbdd /.

1, the equality

ŒZ0 / D

loc ŒX0;m .f /

ŒX0;m .g/L

m.d1 Cd2 /

O
O
holds in Mk;loc
. Moreover, also in this ring Mk;loc
, we have

h.ŒZ1 

ŒZ0 / D

loc.Sf;0

Sg;0 /:

O
Lemma 5.2. The following identities hold in Mk;loc
:

(5.2)
(5.3)

hQ m .ŒZ1 / D loc Xm;0 .f /
Q m .ŒZ0 / D loc Xm;0 .f /
h

Xm;0 .g/
Xm;0 .g/

m
m

m

F1m L

m.d1 Cd2 /

;

m

F0m

m.d1 Cd2 /

:

L

Proof. Since Z1 is 2-invariant, we consider Z1 ŒmI 2 which equals
ˇ
²
³
kŒt 1=m  Ád1 Cd2 ˇˇ .'.0/; .0// D .0; 0/; valf .'/ D valg. / D 1;
.'; / 2
ˇ f .'/ C g. / Á t mod t .mC1/=m
t2
ˇ
³
²
t kŒt  Ád1 Cd2 ˇˇ ordf .'/ D ordg. / D m;
Š .'; / 2
ˇ f .'/ C g. / Á t m mod t mC1
t 2m
ˇ
²
³
t kŒt  Ád1 Cd2 ˇˇ ordf .'/ D ordg. / D m;
.m 1/.d1 Cd2 /
Š .'; / 2 mC1
Ak
:
ˇ
m
mC1
t
f .'/ C g. / Á t mod t
We claim that there is a

m -equivariant

²
V WD .'; / 2

canonical isomorphism between

ˇ
³
t kŒt  Ád1 Cd2 ˇˇ ordf .'/ D ordg. / D m;
ˇ f .'/ C g. / Á t m mod t mC1
t mC1

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Lê, Regular and formal motivic Thom–Sebastiani theorem

and
X0;m .f /

X0;m .g/

m

m

F1m :

Indeed, we define a map X0;m .f / X0;m .g/ F1m ! V that sends .'.t /; .t /I a; b/ to
.'.at/; .bt //. It then induces a well-defined morphism on the quotient
W X0;m .f /

X0;m .g/

m

m

F1m ! V:

We also define a morphism
Á W V ! X0;m .f /

X0;m .g/

m

m

F1m

given by
Á.'.t /; .t // D '..acf '/

1=m

t /; ..acg /

1=m

t /I .acf '/1=m ; .acg /1=m :

It is clear that and Á are m -equivariant and inverse of each other, and the existence of the
claimed isomorphism follows. Consequently, Proposition 4.7 (i) implies (5.2).
We prove (5.3) in much the same way. Since Z0 is 2-invariant, Z0 ŒmI 2 is isomorphic to
ˇ
²
³
t kŒt  Ád1 Cd2 ˇˇ ord.f .'/ C g. // > m;
.m 1/.d1 Cd2 /
.'; / 2 mC1
Ak
:
ˇ f .'/ Á g. / Á t m mod t mC1
t
Also as above, we are able to prove that the constructible set
ˇ
²
³
t kŒt  Ád1 Cd2 ˇˇ ord.f .'/ C g. // > m;
.'; / 2 mC1
ˇ f .'/ Á g. / Á t m mod t mC1
t
is isomorphic to X0;m .f /

X0;m .g/

m

m

F0m . Thus (5.3) is proved.

Proof of Proposition 5.1. By Lemma 5.2 and by definition of the convolution product
(cf. Section 2.3) we get
hQ m .ŒZ1 

ŒZ0 / D

loc ŒX0;m .f /

ŒX0;m .g/L

m.d1 Cd2 /

:

By a property of the Hadamard product, namely,
X
lim
ŒX0;m .f / ŒX0;m .g/L m.d1 Cd2 / T m
T !1

D

D

m 1

X

lim

T !1

Sf;0

ŒX0;m .f /L

md1

m 1

Tm

Á

lim

T !1

X

ŒX0;m .g/L

md2

Tm

Á

m 1

Sg;0 ;

we deduce that h.ŒZ1 

ŒZ0 / D

loc.Sf;0

O
Sg;0 / in Mk;loc
.

O
5.2. Integral over € . Let D be a definable subset of €. A function W D ! Mk;loc
is called definable if D may be partitioned into finitely many disjoint definable subsets Di ,
O
i 2 I , such that jDi is constant ci 2 Mk;loc
for every i in I . Then, we define the integral of
O
over D, which takes value in Mk;loc , as follows:
Z
Z

X
. /D
. /d WD
ci .Di /:
2D

2D

i 2I

Here, is the o-minimal Euler characteristic defined in [13, Lemma 9.5] followed by the
localization morphism.

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Lê, Regular and formal motivic Thom–Sebastiani theorem

5.3. Completion of the proof of Theorem 2.1. In this subsection, we shall prove that
O
and thus finish the proof of (5.1).
h.ŒZ<1  C ŒZ0 / D 0 in Mk;loc
Computation of h.ŒZ<1 / and h.ŒZ0 /. Let
Z<1 ! .0; 1/

<1

denote the definable function

€

mapping .x; y/ to valf .x/, and let
O
W .0; 1/ ! Mk;loc

be the function defined by . / D h.Œ
Lemma 5.3. The function

1
<1 .

//.

is definable.

Proof. Via the definable injection .x; y/ 7! .x; y; valf .x//, we may regard Z<1 as a
definable subset of md1 Cd2 .0; 1/. Consider the surjective morphism of rings
N0 W K.vol€ bdd / ˝ K.volRES/ ! K.volVFbdd /
induced by N in (4.1). There exist definable subsets Wi of RESi and i of € i
0 Ä i Ä d1 C d2 , such that
N0

 d1X
Cd2

Œi  ˝ ŒWd1 Cd2

.0; 1/,

Ã
i  D ŒZ<1 :

i D0

By definition of ˛, ˇ (cf. Section 4.4), we have
.˛ ˝ ˇ/

 d1X
Cd2

Œi  ˝ ŒWd1 Cd2

à d1X
Cd2
.i /wd1 Cd2 i ;
i D

i D0

i D0

where wd1 Cd2 i WD ŒWd1 Cd2 i .L 1/i . Similarly, for
W ;i of RESi ,  ;i of € i ¹ º such that
N0

 d1X

Cd2

Œ

;i 

˝ ŒW

;d1 Cd2

2 .0; 1/, there are definable subsets

Ã
i D Œ

1
<1 .

/:

i D0

Also,
.˛ ˝ ˇ/

 d1X
Cd2
i D0

Œ

;i 

˝ ŒW

;d1 Cd2

à d1X
Cd2
.
i D

;i /w ;d1 Cd2 i ;

i D0

where w ;d1 Cd2 i WD ŒW ;d1 Cd2 i .L 1/i .
We claim that wi D w ;i in ŠK.RES/ for any 0 Ä i Ä d1 C d2 and any 2 .0; 1/.
Indeed, the image of Wi (resp. W ;i ) in K.volVFbdd / is ŒWi `;res Ri  (resp. ŒW ;i ` ;res Ri ),
where ` W Wi ! ki and ` W W ;i ! ki are étale maps and R D ¹ 2 VF j val. /
0º.
The unique difference between Wi `;res Ri and W ;i ` ;res Ri is that the former admits the
condition 0 < valf .x/ < 1 while the latter satisfies valf .x/ D . Thus ŒWi  D ŒW ;i  in

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Lê, Regular and formal motivic Thom–Sebastiani theorem

ŠK.RES/. Consequently,
.˛ ˝ ˇ/

 d1X
Cd2

Œ

;i 

˝ ŒW

;d1 Cd2

à d1X
Cd2
.
i D

i D0

;i /wd1 Cd2 i :

i D0

Since h is induced by ˛ ˝ ˇ, we have
(5.4)

h.ŒZ<1 / D

d1X
Cd2

.i /‚i

h.Œ

and

1
<1 .

// D

i D0

d1X
Cd2

.

;i /‚i ;

i D0

where ‚i WD ‚.wd1 Cd2 i /.
For 0 Ä i Ä d1 C d2 , we identify € i .0; 1/ with a subset of € d1 Cd2 .0; 1/ in an

obvious manner. Let pr2 be the second projection € d1 Cd2 .0; 1/ ! .0; 1/ and Di WD pr2 .i /.
F 1 Cd2
Then .0; 1/ D di D0
Di . Moreover, for any in .0; 1/,  ;i is a fiber of the definable map
i ! Di , all the fibers of this map are definably isomorphic. Thus .i / D .Di / . ;i /.
This identity and (5.4) show that, on Di ,
. / D h.Œ

(5.5)

1
<1 .

// D

d1X
Cd2

.i / .Di /

1

‚i ;

i D0

which proves the definability.
Z
Corollary 5.4.

h.Œ
2.0;1/

1
<1 .

// D h.ŒZ<1 /.

Proof. By definition as well as by (5.4) and (5.5),
Z
2.0;1/

. /D

d1X
Cd2

jDi .Di / D

i D0

d1X
Cd2

.i / .Di /

1

‚i .Di / D h.ŒZ<1 /:

i D0

Let 0 be the function Z0 ! .1; 1/ € that sends .x; y/ to val.f .x/ C g.y//. In the
same manner as above we can prove the following corollary.
O
Corollary 5.5. The function .1; 1/ ! Mk;loc
given by
Moreover,
Z

h.Œ
2.1;1/

0

1

7! h.Œ

0

1

. // is definable.

. // D h.ŒZ0 /:

Conclusion. Let A be the annulus ¹ 2 VF j 0 < val. / < 1º and p<1 the function
Z<1 ! A mapping .x; y/ to f .x/. Then <1 D p<1 ı val. The fiber over 2 A of p<1 is
®

¯
(5.6)
p<11 . / D .x; y/ 2 md1 Cd2 j f .x/ D ; g.y/ D
Ct :
Thanks to the description (5.6) we see that, for each in .0; 1/, all the fibers p<11 . /, in
val 1 . /, are definably isomorphic. Thus we get the following equalities in K.volVFbdd /:
Z
1
Œ <11 . / D
Œp<11 . / D Œval 1 . /Œp<1;
;
2val

1

. /

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Lê, Regular and formal motivic Thom–Sebastiani theorem
1

1
 is the constant function Œp<11 . / on val
where Œp<1;

Z
(5.7)

h.ŒZ<1 / D

h.Œ
2.0;1/

. /. By Corollary 5.4, we have

Z

1
<1 .

// D

h.Œval

1
/ is independent of the choice of
Lemma 5.6. h.Œp<1;

Proof. Using (5.6), namely,
®
¯
p<11 . / D x 2 md1 j f .x/ D

®

1

2.0;1/

1
. //h.Œp<1;
/:

in .0; 1/.

y 2 md2 j g.y/ D

¯
Ct ;

it suffices to prove
h

®

x 2 md j f .x/ D t

¯

®
¯
x 2 md j f .x/ D t

Dh

for any regular function f W Adk ! A1k vanishing at the origin of Adk and for any
Equivalently, it suffices to prove
®
¯
®
¯
h x 2 md j rvf .x/ D rv.t /
D h x 2 md j rvf .x/ D rv.t /

in .0; 1/.

for D a=b in .0; 1/ with a and b coprime integers, a < b. Indeed, if m is not divisible by b,
then hm .Œ¹x 2 md j rvf .x/ D rv.t a=b /º/ D 0. Otherwise, say, m D bs, we have
¯
®
¯
®
Q as x 2 md j rvf .x/ D rv.t / ;
Q bs x 2 md j rvf .x/ D rv.t a=b /
Dh
h
because, by a simple geometric computation, both sides are equal to ŒX0;as .f /L
equality then implies the lemma.
Using (5.7) and Lemma 5.6, we get
ÂZ
h.ŒZ<1 / D

1

h.Œval

2.0;1/

asd .

This

Ã
1
. // h.Œp<1;
/:

As in the proofs of Lemma 5.3 and Corollary 5.4, we can easily show that
Z
h.Œval 1 . // D h.ŒA/ D 1:
2.0;1/

Thus
(5.8)

h.ŒZ<1 / D

1
h.Œp<1;
/

. 2 .0; 1//:

Denote by B the set ¹ 2 VF j val. / > 1º and consider the function p0 W Z0 ! B

defined by p0 .x; y/ D f .x/ C g.y/. Then, we have 0 D p0 ı val. Moreover, the fiber over
2 B of p0 equals
®
¯
p0 1 . / D .x; y/ 2 md1 Cd2 j f .x/ C g.y/ D ; rvf .x/ D rvg.y/ D rv.t /
®
¯
D .x; y/ 2 md1 Cd2 j f .x/ D ct; g.y/ D ct C ; c 2 1 C m :

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Lê, Regular and formal motivic Thom–Sebastiani theorem

Similarly to the proof of Lemma 5.6, we can show that h.Œp0 1 . // is independent of
and, moreover, that
1
h.Œp<1;
/ D h.Œp0 1 . //

(5.9)
for any in .0; 1/ and any
the formula
(5.10)

2B

in B. An analogue of Lemma 5.3 and Corollary 5.4 gives rise to

h.ŒZ0 / D h.ŒB/h.Œp0 1 . // D h.Œp0 1 . //

. 2 B/:

O
. This
Finally, it follows from (5.8), (5.10) and (5.9) that h.ŒZ<1  C ŒZ0 / D 0 in Mk;loc
together with Proposition 5.1 proves (5.1).

6. Proof of Theorem 3.4
It is Theorem 4.8 that completely interprets the role of the morphisms hm and h in understanding the motivic Milnor fiber of a formal function from the non-archimedean geometry
point of view. Motivated by this, to prove Theorem 3.4, also as the proof of the regular version
(Section 5), we work on analytic Milnor fibers (in the sense of [21]) considered as definable
sets in the theory ACVFk..t // .0; 0/.
6.1. Using arguments from Section 5. Let Z be the analytic Minor fiber .0; 0/Œ of
f ˚ g at the origin .0; 0/ of Adk1 Adk2 , namely,
®
¯
Z WD .x; y/ 2 md1 Cd2 j f .x/ C g.y/ D t :
(To indicate precisely the origin of Akdi , if necessary, we write 0di instead of 0.)
O
Theorem 4.8 immediately induces that h.ŒZ/ D loc.Sf˚g;.0;0/ / in Mk;loc
.
Write Z as a disjoint union of definable subsets X, Y and Z defined respectively by
valf .x/ < valg.y/, valf .x/ > valg.y/ and valf .x/ D valg.y/. Again by Theorem 4.8, we
O
have h.ŒX/ D loc.Sf;0 / and h.ŒY/ D loc.Sg;0 / in Mk;loc

.
Ž
To continue, we modify slightly Z into Z , where
®
¯
Z Ž WD .x; y/ 2 md1 Cd2 j rv.f .x/ C g.y// D rv.t /; valf .x/ D valg.y/ ;
and note that
h.ŒZ Ž / D h Œ01d1 Cd2 ŒZ  D h.ŒZ /
O
in Mk;loc
, since h.Œ01 / D 1. We now decompose Z Ž into a disjoint union of
®
¯
Ž
Z1 D .x; y/ 2 Z Ž j valf .x/ D 1

and

®
¯
Ž
Z<1 D .x; y/ 2 Z Ž j 0 < valf .x/ < 1 :

Similarly to Section 5, we use the definable set
®
Ž
Z0 WD .x; y/ 2 md1 Cd2 j val.f .x/ C g.y// > 1;
Ž

Ž

Ž

rvf .x/ D rvg.y/ D rv.t /

¯

Ž

and present ŒZ Ž  as the sum .ŒZ1  ŒZ0 / C .ŒZ<1  C ŒZ0 / in K.volVFbdd /. As in Section 5.3,
Ž
Ž
we also obtain h.ŒZ<1  C ŒZ0 / D 0. In the sequel, we shall prove that
Ž

h.ŒZ1 

Ž

ŒZ0 / D

loc.Sf;0

Sg;0 /

and the proof of Theorem 3.4 is completed.

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Lê, Regular and formal motivic Thom–Sebastiani theorem

6.2. Using motivic integration via Sections 3.2, 4.4–4.6 and 5. In this subsection, we
prove the following proposition.
Proposition 6.1. With the previous notation, we have
Ž

h.ŒZ1 

Ž

ŒZ0 / D

loc.Sf;0

Sg;0 /:

Let Z1 (resp. Z2 ) be a Néron smoothening for the formal completion of X at 0d1 (resp.
for the formal completion of Y at 0d2 ) with Z1 and Z2 smooth, topologically of finite type over
kŒŒt. For any integer m 1, let Gr.Z.m//rv be the space defined as Gr.Z.m// but f.x/ D t
replaced by f.x/ Á t mod t .mC1/=m , where f is the structural morphism of Z. For i 2 ¹1; 2º,
let !i be a bounded gauge form on 0di Œ (remark that 0d1Œ D Z1;Á and 0d2Œ D Z2;Á ), and, for
any integer ei , set
®
¯
ˆ.Zi .m/; !i .m/; ei / WD

' 2 Gr.Zi .m//rv j ordZi .m/ .!i .m//.'/ D ei ;
which is an element of MkO , by the m -action a'.t / WD '.at /. By definition,
Z
X
j!i .m/j D
ˆ.Zi .m/; !i .m/; ei /L ei
0di Œm;rv

ei 2Z

for i 2 ¹1; 2º, where the sum runs over a finite set as !i is a gauge form (see [18]). One thus
deduces that
ÂZ
à ÂZ
Ã
(6.1)
j!1 .m/j
j!2 .m/j
0d1Œm;rv

D

0d2Œm;rv

X

ˆ.Z1 .m/; !1 .m/; e1 /

ˆ.Z2 .m/; !2 .m/; e2 /L

.e1 Ce2 /

:

e1 ;e2 2Z
Ž

Ž

Ž

Ž

For e1 , e2 in €, let Z1;e1 ;e2 (resp. Z0;e1 ;e2 ) be the subset of Z1 (resp. Z0 ) such that
val!1 .x/ D e1 and val!2 .y/ D e2 . For e in €, set
[
[
Ž
Ž
Ž
Ž
Z1;e WD
Z1;e1 ;e2 and Z0;e WD
Z0;e1 ;e2 :
e1 Ce2 De

Lemma 6.2. For any integer m
Q m ŒZ Ž
h
1;e1 ;e2 

D

e1 Ce2 De

1 and any e1 , e2 in € with me1 ; me2 2 Z,

Ž

ŒZ0;e1;e2 

loc Ld1 Cd2 ˆ.Z1 .m/; !1 .m/; me1 /

ˆ.Z2 .m/; !2 .m/; me2 / :

Proof. Since Z1 is topologically of finite type, there exists a convergent power series fQ
in k¹xº vanishing at 0 (hence a kŒŒt -scheme X D Spec.kŒŒt Œx=.fQ.x/ t //) such that
®
¯
Gr` .Z1 /.k/ D ' 2 X ˝kŒŒt  .kŒt =t `C1 / .kŒt =t `C1 / j val.'/ > 0
®
¯
Š ' 2 .t kŒt =t `C1 /d1 j fQ.'/ D t
for ` in N. Similarly,
®
¯
Gr` .Z2 /.k/ Š ' 2 .t kŒt =t `C1 /d2 j g.'/
Q
Dt

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Lê, Regular and formal motivic Thom–Sebastiani theorem

for some gQ in k¹yº with g.0/
Q
D 0. Thus, Ld1 ˆ.Z1 .m/; !1 .m/; me1 / equals L `md1 times
®
¯
' 2 .t kŒt =t `mC1 /d1 j fQ.'/ Á t m mod t mC1 ; ordZ1 .!1 /.'/ D me1 ;
and Ld2 ˆ.Z2 .m/; !2 .m/; me2 / equals L `md2 times
®
¯
2 .t kŒt =t `mC1 /d2 j g.
Q / Á t m mod t mC1 ; ordZ2 .!2 /. / D me2 ;
for ` 2 N large enough. The lemma follows using the arguments in the proof of Lemma 5.2
and Proposition 5.1.
Lemma 6.3. For any integer m
hm

0d1Œrv ; val!1
Ž
Z1 ; val!1

D hm

1,
hm

0d2Œrv ; val!2
Ž

˚ val!2

Z0 ; val!1 ˚ val!2

;

where, by definition, val!1 ˚ val!2 .x; y/ D val!1 .x/ C val!2 .y/.
Proof. Applying (6.1) and Lemmas 6.2, 4.5, we get
0d1Œrv ; val!1
Â
Z
d1
D loc L

0d1Œm;rv

Â

D

0d2Œrv ; val!2
Ã
Â
Z

d2
j!1 .m/j
loc L

hm

hm

loc Ld1 Cd2

0d2Œm;rv

X

ˆ.Z1 .m/; !1 .m/; e1 /

Ã
j!2 .m/j

ˆ.Z2 .m/; !2 .m/; e2 /L

.e1 Ce2 /

Ã

e1 ;e2 2Z

D

X

Q m ŒZ Ž
h

1;e=m

Ž

ŒZ0;e=m  L

e

e2Z

D hm

Ž

Z1 ; val!1 ˚ val!2

Ž

Z0 ; val!1 ˚ val!2

:

Proof of Proposition 6.1 and Theorem 3.4. Thanks to Lemma 6.3, Proposition 4.6 and
Ž
Ž
Theorem 4.8, we have h.ŒZ1  ŒZ0 / D loc.Sf;0 Sg;0 / as desired. The proof of Theorem

3.4 is deduced from Proposition 6.1 and Section 6.1.
Acknowledgement. The author is grateful to François Loeser, Julien Sebag and Michel
Raibaut for useful discussions. He would like to thank the Centre Henri Lebesgue and the
Université de Rennes 1 for awarding him a postdoctoral fellowship and an excellent atmosphere
during his stay there.

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Lê, Regular and formal motivic Thom–Sebastiani theorem

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Quy Thuong Lê, Department of Mathematics, Vietnam National University,
334 Nguyen Trai Street, Thanh Xuan, Hanoi, Vietnam
e-mail:
Eingegangen 28. Mai 2014

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