Comptes Rendus Mathématique

Marco Bravin

On the 2D “viscous incompressible fluid + rigid body” system with Navier conditions and unbounded energy

Gil Alon

François Legrand

Elad Paran

Comptes Rendus
Mathématique
Gil Alon, François Legrand and Elad Paran
Galois groups over rational function fields over skew fields
Volume 358, issue 7 (2020), p. 785-790.
<https://doi.org/10.5802/crmath.20>
© Académie des sciences, Paris and the authors, 2020.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2020, 358, n 7, p. 785-790
https://doi.org/10.5802/crmath.20
Number Theory / Théorie des nombres
Galois groups over rational function fields
over skew fields
Gil Alona, François Legrandb and Elad Paran∗, a
a Department of Mathematics and Computer Science, the Open University of Israel,
Ra’anana 4353701, Israel
b Institut für Algebra, Fachrichtung Mathematik, TU Dresden, 01062 Dresden,
Germany
E-mails: gilal@openu.ac.il, francois.legrand@tu-dresden.de, paran@openu.ac.il
Abstract. Let H be a skew field of finite dimension over its center k. We solve the Inverse Galois Problem over
the field of fractions H(X ) of the ring of polynomial functions over H in the variable X , if k contains an ample
field.
Résumé. Soit H un corps gauche de dimension finie sur son centre k. Nous résolvons le Problème Inverse de
Galois sur le corps des fractions H(X ) de l’anneau des fonctions polynomiales en la variable X et à coefficients
dans H , si k contient un corps ample.
Manuscript received 17th December 2019, accepted 28th May 2020.
1. Introduction
The Inverse Galois Problem over a field k asks whether every finite group occurs as the Galois
group of a Galois field extension of k. Hilbert showed in 1892, via his celebrated irreducibility
theorem, that this problem over the fieldQ of rational numbers is equivalent to the same problem
over the field Q(t ) of rational functions over Q. While the problem is wide open over Q(t ), it is
known to have an affirmative answer over many other function fields, e.g., over the field C(t ) of
complex rational functions, as a consequence of Riemann’s Existence Theorem.
The aim of this note is to contribute to inverse Galois theory over skew fields, following a first
work on this topic by Deschamps and Legrand (see [4]). In this more general context, given skew
fields (equivalently, division rings) H ⊆ M , the extension M/H is said to be Galois if every element
of M which is fixed under any automorphism of M fixing H pointwise lies in H . See [3, Section 3.3]
for more on Galois theory over skew fields.
Let H be a skew field, and let H [X ] denote the ring of all polynomial functions over H in the
variable X . That is, H [X ] is the ring of all functions from H to H that can be expressed by sums
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
786 Gil Alon, François Legrand and Elad Paran
and products of the variable X and elements of H . We observe that, if H is of finite dimension
over its center k and if k is infinite, then H [X ] has a classical (right) field of fractions, denoted by
H(X ). See Section 2 for more details.
In the sequel, we solve the Inverse Galois Problem over the skew field H(X ), if the center of H
contains an ample field:
Theorem 1. Let H be a skew field of finite dimension over its center k. If k contains an ample field,
then every finite group is the Galois group of a Galois extension of H(X ).
Recall that a field k is ample (or large) if every smooth geometrically irreducible k-curve has
either zero or infinitely many k-rational points. Ample fields, which were introduced by Pop in [9]
(and which are necessarily infinite), include algebraically closed fields, some complete valued
fields (e.g.,Qp ,R,κ((T ))), the fieldQtr of all totally real algebraic numbers, etc. See [7], [2], and [10]
for more details. Consequently, a special (but fundamental) case of Theorem 1 is that the Inverse
Galois Problem has an affirmative answer overH(X ), whereHdenotes the skew field of Hamilton’s
quaternions.
Given a skew field H of finite dimension over its center k, with k infinite, the ring H [X ] is
one possible natural generalization of the usual polynomial ring in one variable over an infinite
field. Another one is the polynomial ring Hc [t ], where t is a central indeterminate, commuting
with the coefficients1 2. While these rings are isomorphic in the special case H = k, it is not clear
that such an isomorphism exists if H is non-commutative. This suggests that the Inverse Galois
Problem over the field of fractions Hc (t ) of Hc [t ], which is studied by Deschamps and Legrand,
and the same problem over H(X ) are a priori independent. In particular, although the Inverse
Galois Problem over Hc (t ) has a positive answer if k contains an ample field (see [4, Théorème B]),
Theorem 1 has its own merits and, as [4, Théorème B], extends the deep result of Pop solving the
Inverse Galois Problem over the field k(t ), if k contains an ample field.
We prove Theorem 1 in Section 3, by reducing it to the case settled by Deschamps and Legrand.
The main observation needed is that the ring H [X ] is isomorphic to the ring Hc [t1, . . . , tn] of
polynomials over H in n central variables, where n denotes the dimension of H over its center
(see Proposition 4). This follows from a theorem of Wilczynski [12, Theorem 4.1]. We also make
use of the general observation that the Inverse Galois Problem over skew fields is “algebraic”; see
Proposition 6.
2. Polynomial rings and fields of fractions
2.1. Polynomial rings
For this subsection, let H be a skew field.
The polynomial ring Hc [t ] in the central variable t is the set of all sequences (an)n∈N of
elements of H such that an = 0 for all but finitely many n. As in the commutative setting, the
addition is defined componentwise and the multiplication is defined by (an)n · (bn)n = (cn)n ,
where cn = ∑l+m=n al bm for every n ∈ N. Setting (an)n = ∑n an t n , one has at = t a for every
a ∈ H , thus justifying the terminology “central”. If H is a field, then Hc [t ] is nothing but the usual
polynomial ring in the variable t over H . In the sense of Ore [8], Hc [t ] is the skew polynomial ring
1We adopt a different notation from that of [4], where this ring is denoted by H [t ], in order to distinguish between the
cases of central variables and non-central ones. We note that there are alternative notations for this ring in the literature,
such as H [x, id,0] in [8], or HL [t ] in [6].
2Throughout this note, we use upper-case letters to denote non-central variables and lower case-letter to denote
central ones, to add a visual distinction between the two.
C. R. Mathématique, 2020, 358, n 7, 785-790
Gil Alon, François Legrand and Elad Paran 787
H [t ,α,δ] in the variable t , where the automorphism α is the identity of H and the derivation δ
is 0. One can iteratively construct rings of polynomials in several central variables over H , by
putting Hc [t1, t2] = (Hc [t1])c [t2], Hc [t1, t2, t3] = (Hc [t1, t2])c [t3], and so on. Since the variables
are all central, the order in which they are added does not change the ring obtained, up to
isomorphism.
On the other hand, let H〈X 〉 be the free algebra in one symbol X over H . That is, H〈X 〉 is
the algebra spanned by all words whose letters are elements of H or X . For an element a ∈ H
and f (X ) ∈ H〈X 〉, the substitution f (a) ∈ H is defined in the obvious way, by replacing each
occurrence of X in f (X ) by a, and computing the resulting value in H . For a fixed a ∈ H , the map
f (X ) 7→ f (a) is a homomorphism from H〈X 〉 to H . We say that f vanishes at a if f (a) = 0. Let I
be the (two-sided) ideal of H〈X 〉 which consists of all f (X ) ∈ H〈X 〉 that vanish at all a ∈ H . Then
the ring H [X ] is defined as the quotient H〈X 〉/I , and it is isomorphic to the ring of polynomial
functions over H . Note that, if H is an infinite field, then this definition coincides with the usual
definition of the polynomial ring in the variable X over H .
2.2. Classical fields of fractions
For this subsection, let R be a non-zero ring, not necessarily commutative. Recall that R is an
integral domain if, for all r ∈ R \ {0} and s ∈ R \ {0}, one has sr 6= 0 and r s 6= 0. From now on,
suppose R is an integral domain.
A classical right quotient ring for R is an overring S ⊇ R such that every non-zero element of
R is invertible in S, and such that every element of S can be written as ab−1 for some a ∈ R and
some b ∈ R \ {0}. We say that R is a right Ore domain if, for all non-zero elements x and y of R,
there exist r and s in R such that xr = y s 6= 0. By [5, Theorem 6.8], if R is a right Ore domain, then
R has a classical right quotient ring H which is a skew field and, by [3, Proposition 1.3.4], H is
unique up to isomorphism. We then say that H is the classical right field of fractions of R.
If H denotes an arbitrary skew field, then the polynomial ring Hc [t ] in the central variable t
over H is an integral domain, since the degree is additive on products. Moreover, Hc [t ] is a right
Ore domain, by [5, Theorem 2.6 and Corollary 6.7]. By an easy induction, given a positive integer
n, the polynomial ring Hc [t1, . . . , tn] in n central variables over H has a classical right field of
fractions, which we denote by Hc (t1, . . . , tn).
Proposition 2. Let H be a skew field and n ≥ 2. Then the equality Hc (t1, . . . , tn) =
(Hc (t1, . . . , tn−1))c (tn) holds.
Proof. First, it is clear that the inclusion Hc [t1, . . . , tn] ⊆ (Hc (t1, . . . , tn−1))c (tn) holds. As every
element of Hc (t1, . . . , tn) can be written as f g−1 with f and g in Hc [t1, . . . , tn], we actually have
Hc (t1, . . . , tn) ⊆ (Hc (t1, . . . , tn−1))c (tn).
For the converse, take a polynomial f = ∑ml=0 al t ln with al ∈ Hc (t1, . . . , tn−1) for every l ∈
{0, . . . ,m}. As before, we can write al = bl c−1l with bl ∈ Hc [t1, . . . , tn−1] and cl ∈ Hc [t1, . . . , tn−1] \
{0}, for every l ∈ {0, . . . ,m}. Since Hc [t1, . . . , tn−1] ⊆ Hc [t1, . . . , tn] ⊆ Hc (t1, . . . , tn) and tn ∈
Hc (t1, . . . , tn), we get that f = ∑ml=0 bl c−1l t ln is in Hc (t1, . . . , tn). This shows the desired inclusion
(Hc (t1, . . . , tn−1))c (tn) ⊆ Hc (t1, . . . , tn), since every element of (Hc (t1, . . . , tn−1))c (tn) can be written
as f g−1 with f and g in (Hc (t1, . . . , tn−1))c [tn]. 
Proposition 3. Let H be a skew field of center k and let n be a positive integer. The center of
Hc (t1, . . . , tn) equals k(t1, . . . , tn). Moreover, if the dimension of H over k is finite, then the equality
dimk(t1,...,tn ) Hc (t1, . . . , tn) = dimk H holds.
C. R. Mathématique, 2020, 358, n 7, 785-790
788 Gil Alon, François Legrand and Elad Paran
Proof. By, e.g., [3, Proposition 2.1.5], if K is an arbitrary skew field of center C , then C (t ) is the
center of Kc (t ). Hence, by iterating Proposition 2, the center of Hc (t1, . . . , tn) equals k(t1, . . . , tn).
Now, suppose dimk H is finite. Then, by [4, Proposition 9], we have Hc (t1) ∼= H ⊗k k(t1). Conse-
quently, dimk(t1) Hc (t1) is finite and equals dimk H . As before, it remains to iterate Proposition 2
to conclude the proof. 
Proposition 4. Let H be a skew field of finite dimension n over its center k. Assume k is infinite.
Then the ring H [X ] is isomorphic to Hc [t1, . . . , tn].
Proof. The existence of such an isomorphism follows from [12, Theorem 4.1]. See also [1,
Theorem 5] for a different, more explicit proof. For the convenience of the reader, we include an
elementary proof in the special case H =H, where H is the skew field of Hamilton’s quaternions.
One has the following classical identity for each a ∈H:
Re(a) = 1
4
(a − i ai − j a j −kak),
where Re(a) is the real component of a. More generally, putting
y1 = 1
4
(X − i X i − j X j −k X k),
y2 = 1
4
( j X k −X i − i X −k X j ),
y3 = 1
4
(k X i −X j − j X − i X k),
y4 = 1
4
(i X j −X k −k X − j X i ),
the functions y1, y2, y3, y4 ∈H[X ] obtain real values only, and one has X = y1 + i y2 + j y3 +k y4. In
particular, y1, y2, y3, y4 belong to the center of H[X ], and we may then define a homomorphism
φ : Hc [t1, t2, t3, t4] → H[X ] by φ(tl ) = yl , 1 ≤ l ≤ 4, and φ(a) = a for all a ∈ H. The equality
X = y1 + i y2 + j y3 +k y4 implies that φ is surjective.
Let p = p(t1, t2, t3, t4) ∈Hc [t1, t2, t3, t4]. By decomposing the coefficients of p into their real, i ,
j , and k components, we may present p in the form p = p1 +p2i +p3 j +p4k with p1, p2, p3, p4 ∈
R[t1, t2, t3, t4]. If p 6= 0, then pl 6= 0 for some 1 ≤ l ≤ 4. Then there exists a non-zero tuple
a = (a1, a2, a3, a4) ∈R4 such that pl (a) 6= 0. Hence,φ(p) does not vanish at X = a1+a2i+a3 j+a4k,
thus showing that φ is also injective. 
Corollary 5. Let H be a skew field of finite dimension n over its center k. Assume k is infinite.
Then the ring H [X ] has a classical right field of fractions, denoted by H(X ), which is isomorphic to
Hc (t1, . . . , tn).
Proof. As recalled, the ring Hc [t1, . . . , tn] is a right Ore domain. Since H [X ] is isomorphic to
Hc [t1, . . . , tn] by Proposition 4, H [X ] is also a right Ore domain and so has a classical right
field of fractions. Finally, [3, Section 1.3] shows that the isomorphism H [X ] ∼= Hc [t1, . . . , tn] from
Proposition 4 extends to an isomorphism H(X ) ∼= Hc (t1, . . . , tn). 
3. Proof of Theorem 1
We first make the general observation that the Inverse Galois Problem over skew fields is an
“algebraic problem”. More precisely:
Proposition 6. Let H1 and H2 be isomorphic skew fields and let G be a finite group. Then there
exists a Galois extension of H1 of group G if and only if there exists a Galois extension of H2 of
group G.
C. R. Mathématique, 2020, 358, n 7, 785-790
Gil Alon, François Legrand and Elad Paran 789
Proof. Let ϕ : H1 → H2 be an isomorphism. Suppose there exists a Galois extension K1/H1 of
group G .
By the exchange principle3, there exists a set C such that C ∩ H2 = ; and |C | = |K1 \ H1|. Let
f : K1\H1 →C be a bijection. Then set K2 =C∪H2 and consider the well-defined mapψ : K1 → K2
given by ψ(x) = ϕ(x) if x ∈ H1 and ψ(x) = f (x) if x ∈ K1 \ H1. The map ψ is surjective and, as
C ∩H2 =;, it is also injective. Now, define the ring operations on K2 as inherited from K1 via ψ:
∀ x, y ∈ K2, x · y =ψ(ψ−1(x) ·ψ−1(y)), x + y =ψ(ψ−1(x)+ψ−1(y)).
Then K2 is isomorphic to K1 via ψ and, in particular, K2 is a skew field containing H2.
It remains to show that K2/H2 is Galois of group G . To that end, note that the isomorphism ψ :
K1 → K2, whose restriction to H1 equalsϕ, induces an isomorphismφ : Aut(K1/H1) → Aut(K2/H2)
(namely, φ(σ) =ψ◦σ◦ψ−1 for every σ ∈ Aut(K1/H1)). Finally, if x is any element of K2 such that
σ(x) = x for every σ ∈ Aut(K2/H2), then we have τ(ψ−1(x)) =ψ−1(x) for every τ ∈ Aut(K1/H1). As
K1/H1 is Galois, we then haveψ−1(x) ∈ H1, and so x ∈ H2, thus showing that K2/H2 is Galois. This
concludes the proof. 
Proof of Theorem 1. By Corollary 5, we have H(X ) ∼= Hc (t1, . . . , tn), where n denotes the dimen-
sion of H over k. Moreover, by Proposition 3, the center of Hc (t1, . . . , tn−1) equals k(t1, . . . , tn−1)
and the dimension of Hc (t1, . . . , tn−1) over k(t1, . . . , tn−1) is finite. Finally, k(t1, . . . , tn−1) contains
an ample field. Hence, by [4, Théorème B], the Inverse Galois Problem has an affirmative answer
over the skew field (Hc (t1, . . . , tn−1))c (tn), that is, over Hc (t1, . . . , tn) by Proposition 2. It then re-
mains to apply Proposition 6 to get that the Inverse Galois Problem also has an affirmative an-
swer over H(X ), thus concluding the proof. 
Remark 7. Similarly, we have this result, which follows from [4, Proposition 12] as Theorem 1
follows from [4, Théorème B]:
Let G be a finite group and H a skew field of finite dimension n over its center k. In each of the
following cases, G occurs as the Galois group of a Galois extension of H(X ):
(1) G is abelian and k is infinite,
(2) G = Sm (m ≥ 3) and k is infinite,
(3) G = Am (m ≥ 4) and k has characteristic zero,
(4) G is solvable, n ≥ 2, and k has positive characteristic.
References
[1] G. Alon, E. Paran, “A quaternionic Nullstellensatz”, J. Pure Appl. Algebra 225 (2020), no. 4, article ID 106572.
[2] L. Bary-Soroker, A. Fehm, “Open problems in the theory of ample fields”, in Geometric and differential Galois theories,
Séminaires et Congrès, vol. 27, Société Mathématique de France, 2013, p. 1-11.
[3] P. M. Cohn, Skew fields. Theory of general division rings, Encyclopedia of Mathematics and Its Applications, vol. 57,
Cambridge University Press, 1995, xvi+500 pages.
[4] B. Deschamps, F. Legrand, “Le problème inverse de Galois sur les corps des fractions tordus à indéterminée centrale”,
J. Pure Appl. Algebra 224 (2020), no. 5, article ID 106240 (13 pages).
[5] K. R. Goodearl, R. B. Warfield, Jr., An Introduction to noncommutative Noetherian rings, 2nd ed., London Mathemat-
ical Society Student Texts, vol. 61, Cambridge University Press, 2004, xxiv+344 pages.
[6] B. Gordon, T. S. Motzkin, “On the zeros of polynomials over division rings”, Trans. Am. Math. Soc. 116 (1965), p. 218-
226.
[7] M. Jarden, Algebraic patching, Springer Monographs in Mathematics, Springer, 2011, xxiv+290 pages.
[8] O. Ore, “Theory of non-commutative polynomials”, Ann. Math. 34 (1933), no. 3, p. 480-508.
[9] F. Pop, “Embedding problems over large fields”, Ann. Math. 144 (1996), no. 1, p. 1-34.
3which asserts that, given two sets A and B , there exists a set C such that A∩C =; and |C | = |B | (see, e.g., [11, p. 31]).
C. R. Mathématique, 2020, 358, n 7, 785-790
790 Gil Alon, François Legrand and Elad Paran
[10] ——— , “Little survey on large fields - old & new”, in Valuation theory in interaction, EMS Series of Congress Reports,
European Mathematical Society, 2014, p. 432-463.
[11] R. L. Vaught, Set theory. An introduction, 2nd ed., Birkhäuser, 1995, x+167 pages.
[12] D. M. Wilczynski, “On the fundamental theorem of algebra for polynomial equations over real composition algebras”,
J. Pure Appl. Algebra 218 (2014), no. 7, p. 1195-1205.
C. R. Mathématique, 2020, 358, n 7, 785-790


Galois groups over rational function fields over skew fields

Alin Bostan

Antonio Jiménez-Pastor

Comptes Rendus
Mathématique
Alin Bostan and Antonio Jiménez-Pastor
On the exponential generating function of labelled trees
Volume 358, issue 9-10 (2020), p. 1005-1009.
<https://doi.org/10.5802/crmath.108>
© Académie des sciences, Paris and the authors, 2020.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2020, 358, n 9-10, p. 1005-1009
https://doi.org/10.5802/crmath.108
Combinatorics / Combinatoire
On the exponential generating function
of labelled trees
Sur la série génératrice des arbres étiquetés
Alin Bostana and Antonio Jiménez-Pastor∗, b
a Inria, 1 rue Honoré d’Estienne d’Orves 91120 Palaiseau, France
b Johannes Kepler University Linz, Doctoral Program Computational Mathematics,
DK15
E-mails: Alin.Bostan@inria.fr (A. Bostan), ajpastor@risc.uni-linz.ac.at
(A. Jiménez-Pastor)
Abstract. We show that the generating function of labelled trees is not D∞-finite.
Résumé. Nous montrons que la série génératrice des arbres étiquetés n’est pas D∞-finie.
2020 Mathematics Subject Classification. 05A15, 12H05, 34A34.
Funding. This research was partially funded by the Austrian Science Fund (FWF): W1214-N15, project DK15.
It was also supported in part by the ANR DeRerumNatura project, grant ANR-19-CE40-0018 of the French
Agence Nationale de la Recherche.
Manuscript received 29th June 2020, revised and accepted 20th August 2020.
Version française abrégée
Nous montrons que la série génératrice exponentielle des arbres étiquetés, T (x) = ∑n≥1 nn−1n! xn ,
n’est pas D∞-finie. En particulier, cela implique que, bien que T (x) vérifie des équations différen-
tielles non-linéaires, ces dernières ne peuvent pas être « trop simples ». En particulier, T (x) n’est
pas un quotient de deux fonctions D-finies (vérifiant des équations différentielles à coefficients
polynomiaux), et plus généralement, T (x) ne vérifie aucune équation différentielle linéaire à co-
efficients des fonctions D-finies. La preuve repose ultimement sur un résultat de théorie de Ga-
lois différentielle. Plusieurs questions ouvertes sont proposées, dont une sur la nature de la série
génératrice ordinaire des arbres étiquetés,
∑
n≥1 nn−1xn .
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1006 Alin Bostan and Antonio Jiménez-Pastor
1. Context and main result
A formal power series f (x) = ∑n≥0 an xn in C[[x]] is called differentially finite, or simply D-
finite [23], if it satisfies a linear differential equation with polynomial coefficients in C[x]. Many
generating functions in combinatorics and many special functions in mathematical physics are
D-finite [2, 9].
DD-finite series and more generally Dn-finite series are larger classes of power series, recently
introduced in [13]. DD-finite power series satisfy linear differential equations, whose coefficients
are themselves D-finite power series. One of the simplest examples is tan(x), which is DD-finite
(because it satisfies cos(x) f (x)−sin(x) = 0), but is not D-finite (because it has an infinite number
of complex singularities, a property which is incompatible with D-finiteness). Another basic
example is the exponential generating function of the Bell numbers Bn , which count partitions
of {1,2, . . . ,n}, namely:
B(x) := ∑
n≥0
Bn
n!
xn . (1)
Indeed, it is classical [9, p. 109] that B(x) = eex−1, therefore B(x) is DD-finite. On the other hand,
B(x) is not D-finite: this can be proved either analytically (using the too fast growth of B(x) as
x →∞), or purely algebraically (using [22], and the fact that the power series ex is not algebraic).
More generally, given a differential ring R, the set of differentially definable functions over R,
denoted by D(R), is the differential ring of formal power series satisfying linear differential
equations with coefficients in R. In particular, D(C[x]) is the ring of D-finite power series,
D2(C[x]) := D(D(C[x])) is the ring of DD-finite power series, and Dn(C[x]) := D(Dn−1(C[x])) is
the ring of Dn-finite power series. We say that a power series f (x) ∈ C[[x]] is D∞-finite if there
exists an n such that f (x) is Dn-finite.
It is known [14] that Dn-finite power series form a strictly increasing sequence of sets and
that any D∞-finite power series is differentially algebraic, in short D-algebraic, that is, it satisfies
a differential equation, possibly non-linear, with polynomial coefficients in C[x]. This class, as
well as its complement (of D-transcendental series), are quite well studied [11, 21].
Let now (tn)n≥0 = (0,1,2,9,64,625,7776, . . .) be the sequence whose general term tn counts
labelled rooted trees with n nodes. It is well known that tn = nn−1, for any n ≥ 1. This beautiful
and non-trivial result is usually attributed to Cayley [6], although an equivalent result had been
proved earlier by Borchardt [4], and even earlier by Sylvester, see [3, Ch. 4]. Due to the importance
of the combinatorial class of trees, and to the simplicity of the formula, Cayley’s result has
attracted a lot of interest over the time, and it admits several different proofs, see e.g., [16, §4]
and [1, §30]. One of the more conceptual proofs goes along the following lines (see [9, §II. 5.1] for
details). Let
T (x) := ∑
n≥0
tn
n!
xn (2)
be the exponential generating function of the sequence (tn)n . The class T of all rooted labelled
trees is definable by a symbolic equation T = Z ? SET(T ) reflecting their recursive definition,
where Z represents the atomic class consisting of a single labelled node, and ? denotes the la-
belled product on combinatorial classes. This symbolic equation provides, by syntactic transla-
tion, an implicit equation on the level of exponential generating functions:
T (x) = x eT (x), (3)
which can be solved using Lagrange inversion
tn = n! · [xn]T (x) = n! ·
(
1
n
[zn−1](ez )n
)
= nn−1. (4)
C. R. Mathématique, 2020, 358, n 9-10, 1005-1009
Alin Bostan and Antonio Jiménez-Pastor 1007
From (3), it follows easily that T (x) is D-algebraic and satisfies the non-linear equation
x(1−T (x))T ′(x) = T (x),
and from there, that the sequence (tn)n≥0 satisfies the non-linear recurrence relation
tn+1 = n +1
n
·
n∑
i=1
(
n
i
)
ti tn−i+1, for all n ≥ 1.
This recurrence can also be proved using (4), by taking y = n, x = w = 1 in Abel’s identity [12]
(x + y)n =
n∑
k=0
(
n
k
)
x(x +wk)k−1(y −wk)n−k ,
and then by isolating the term k = n in the resulting equality.
On the other hand, it is known that the power series T (x) is not D-finite, see [10, Thm. 7],
or [8, Thm. 2]. This raises the natural question whether T (x) is DD-finite, or Dn-finite for some
n ≥ 2. Our main result is that this is not the case:
Theorem 1. The power series T (x) =∑n≥1 nn−1n! xn in (2) is not D∞-finite.
To our knowledge, this is the first explicit example of a natural combinatorial generating
function which is provably D-algebraic but not D∞-finite. In particular, Theorem 1 implies that
T (x) is not equal to the quotient of two D-finite functions, and more generally, that it does not
satisfy any linear differential equation with D-finite coefficients.
2. Proof of the main result
Our proof of Theorem 1 builds upon the following recent result by Noordman, van der Put
and Top.
Theorem 2 ([18]). Assume that u(x) ∈ C[[x]] \C is a solution of u′ = u3 −u2. Then u is not D∞-
finite.
The proof of Theorem 2 is based on two ingredients. The first one is a result by Rosenlicht [20]
stating that any set of non-constant solutions (in any differential field) of the differential equation
u′ = u3 −u2 is algebraically independent over C (see also [18, Prop. 7.1]); the proof is elementary.
The second one [18, Prop. 7.1] is that any non-constant power series solution of an autonomous
first-order differential equation with this independence property cannot be D∞-finite; the proof
is based on differential Galois theory.
Proof of Theorem 1. We will use Theorem 2 and a few facts about the (principal branch of the)
Lambert W function, satisfying W (x) ·eW (x) = x for all x ∈C.
Recall [7] that the Taylor series of W around 0 is given by
W (x) =
∞∑
n=1
(−n)n−1
n!
xn = x −x2 + 32 x3 − 83 x4 + 12524 x5 −·· · .
In other words, our T (x) and W (x) are simply related by W (x) =−T (−x).
The function defined by this series can be extended to a holomorphic function defined on
all complex numbers with a branch cut along the interval (−∞,− 1e ]; this holomorphic function
defines the principal branch of the Lambert W function.
We can substitute x 7→ ex+1 in the functional equation for W (x) obtaining then
W (ex+1)eW (e
x+1) = ex+1,
or, renaming Y (x) = W (ex+1), we have a new functional equation: Y (x)eY (x)−1 = ex . From this
equality it follows by logarithmic differentiation that Y ′(x) · (1+Y (x)) = Y (x).
C. R. Mathématique, 2020, 358, n 9-10, 1005-1009
1008 Alin Bostan and Antonio Jiménez-Pastor
Take now U (x) := 11+Y (x) = 12 − 18 x + 164 x2 + 1768 x3 +·· · . We have that
U ′(x) = −Y
′(x)
(1+Y (x))2 =
−Y (x)
(1+Y (x))3 =U (x)
3 −U (x)2.
By Theorem 2, U (x) is not D∞-finite. By closure properties of D∞-finite functions (see [14,
Thm. 4] and [13, §3]), it follows that Y (x) is not D∞-finite either.
To conclude, note that by definition, for real x in the neighborhood of 0, we have W (x) =
Y (log(x)−1), and by Theorem 10 in [14], it follows that W (x) and T (x) are not D∞-finite either,
proving Theorem 1. 
3. Open questions
The class of D-finite power series is closed under Hadamard (term-wise) product. This is false
for D∞-finite power series; for instance, Klazar showed in [15] that the ordinary generating
function
∑
n≥0 Bn xn of the Bell numbers is not differentially algebraic, contrary to its exponential
generating function (1), which is DD-finite.
Moreover, it was conjectured by Pak and Yeliussizov [19, Open Problem 2.4] that this is an
instance of a more general phenomenon.
Conjecture 3 ([19, Open Problem 2.4]). If for a sequence (an)n≥0 both ordinary and exponential
generating functions
∑
n≥0 an xn and
∑
n≥0 an x
n
n! are D-algebraic, then both are D-finite. (Equiva-
lently, (an)n≥0 satisfies a linear recurrence with polynomial coefficients in n.)
This conjecture has been recently proven for large (infinite) classes of generating functions [5].
However, the very natural example of the generating function for labelled trees escapes the
method in [5].
We therefore leave the following as an open question.
Open question 4. Is the power series
∑
n≥1 nn−1xn D∞-finite? Is it at least differentially algebraic?
According to Conjecture 3, the answer should be “no” for both questions in Open question 4.
Another natural question concerns the generating function for partition numbers:∑
n≥0
pn x
n := ∏
n≥1
1
1−xn = 1+x +2 x
2 +3 x3 +5 x4 +7 x5 +11 x6 +·· · ,
which is known to be differentially algebraic [17].
Open question 5. Is it true that
∑
n≥0 pn xn is not D∞-finite?
One may also ask for the nature of the exponential variant of the generating function for
partition numbers.
Open question 6. Is the power series
∑
n≥0
pn
n! x
n D∞-finite, or at least differentially algebraic?
Acknowledgements
We would like to warmly thank the referees for their careful reading and for their constructive
remarks.
C. R. Mathématique, 2020, 358, n 9-10, 1005-1009
Alin Bostan and Antonio Jiménez-Pastor 1009
References
[1] M. Aigner, G. M. Ziegler, Proofs from The Book, fourth ed., Springer, 2010, viii+274 pages.
[2] G. E. Andrews, R. Askey, R. Roy, Special functions, Encyclopedia of Mathematics and Its Applications, vol. 71,
Cambridge University Press, 1999, xvi+664 pages.
[3] N. L. Biggs, E. K. Lloyd, R. J. Wilson, Graph theory. 1736–1936, second ed., Clarendon Press, 1986, xii+239 pages.
[4] C. W. Borchardt, “Ueber eine der Interpolation entsprechende Darstellung der Eliminations-Resultante”, J. Reine
Angew. Math. 57 (1860), p. 111-121.
[5] A. Bostan, L. Di Vizio, K. Raschel, “Differential transcendence of Bell numbers and relatives — a Galois theoretic
approach”, 2020, In preparation.
[6] A. Cayley, “A theorem on trees”, Q. J. Math 23 (1889), p. 376-378.
[7] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, D. E. Knuth, “On the Lambert W function”, Adv. Comput.
Math. 5 (1996), no. 4, p. 329-359.
[8] P. Flajolet, S. Gerhold, B. Salvy, “On the non-holonomic character of logarithms, powers, and the nth prime function”,
Electron. J. Comb. 11 (2005), no. 2, article ID A2 (16 pages).
[9] P. Flajolet, R. Sedgewick, Analytic combinatorics, Cambridge University Press, 2009, xiv+810 pages.
[10] S. Gerhold, “On some non-holonomic sequences”, Electron. J. Comb. 11 (2004), no. 1, article ID 87 (8 pages).
[11] J. van der Hoeven, “Computing with D-algebraic power series”, Appl. Algebra Eng. Commun. Comput. 30 (2019),
no. 1, p. 17-49.
[12] J. L. W. V. Jensen, “Sur une identité d’Abel et sur d’autres formules analogues”, Acta Math. 26 (1902), no. 1, p. 307-318.
[13] A. Jiménez-Pastor, V. Pillwein, “A computable extension for D-finite functions: DD-finite functions”, J. Symb. Comput.
94 (2019), p. 90-104.
[14] A. Jiménez-Pastor, V. Pillwein, M. F. Singer, “Some structural results on Dn -finite functions”, Adv. Appl. Math. 117
(2020), article ID 102027 (29 pages).
[15] M. Klazar, “Bell numbers, their relatives, and algebraic differential equations”, J. Comb. Theory, Ser. A 102 (2003),
no. 1, p. 63-87.
[16] L. Lovász, Combinatorial problems and exercises, second ed., North-Holland, 1993, 635 pages.
[17] A. M. Mian, S. Chowla, “The differential equations satisfied by certain functions”, J. Indian Math. Soc., New Ser. 8
(1944), p. 27-28.
[18] M. P. Noordman, M. van der Put, J. Top, “Combinatorial Autonomous first order differential equations”, 2019,
preprint, https://arxiv.org/abs/1904.08152v1.
[19] I. Pak, “Complexity problems in enumerative combinatorics”, in Proceedings of the International Congress of Mathe-
maticians — Rio de Janeiro 2018. Vol. IV. Invited lectures, World Scientific, 2018, p. 3153-3180.
[20] M. Rosenlicht, “The nonminimality of the differential closure”, Pac. J. Math. 52 (1974), p. 529-537.
[21] L. A. Rubel, “A survey of transcendentally transcendental functions”, Am. Math. Mon. 96 (1989), no. 9, p. 777-788.
[22] M. F. Singer, “Algebraic relations among solutions of linear differential equations”, Trans. Am. Math. Soc. 295 (1986),
no. 2, p. 753-763.
[23] R. P. Stanley, “Differentiably finite power series”, Eur. J. Comb. 1 (1980), no. 2, p. 175-188.
C. R. Mathématique, 2020, 358, n 9-10, 1005-1009


On the exponential generating function  of labelled trees

Abita Rahmoune

Upper bound estimate for the blow-up time of a class of integrodifferential equation of parabolic type involving variable source

Chimere Stanley Anabanti

Alexander Moretó

Mohammad Zarrin

Comptes Rendus
Mathématique
Chimere Stanley Anabanti, Alexander Moretó and Mohammad
Zarrin
Influence of the number of Sylow subgroups on solvability of finite
groups
Volume 358, issue 11-12 (2020), p. 1227-1230.
<https://doi.org/10.5802/crmath.146>
© Académie des sciences, Paris and the authors, 2020.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2020, 358, n 11-12, p. 1227-1230
https://doi.org/10.5802/crmath.146
Group Theory / Théorie des groupes
Influence of the number of Sylow subgroups
on solvability of finite groups
Chimere Stanley Anabantia, Alexander Moretó∗, b andMohammad Zarrinc
a Institut für Analysis und Zahlentheorie, Technische Universität Graz (TU Graz),
Graz 8010 Austria.
b Departament de Matemàtiques, Universitat de València, 46100, Burjassot, València
Spain.
c Department of Mathematics, University of Kurdistan, P.O. Box: 416, Sanandaj, Iran.
E-mails: anabanti@math.tugraz.at; chimere.anabanti@unn.edu.ng (C.S. Anabanti),
Alexander.Moreto@uv.es (A. Moretó), M.Zarrin@uok.ac.ir (M. Zarrin)
Abstract. Let G be a finite group. We prove that if the number of Sylow 3-subgroups of G is at most 7 and
the number of Sylow 5-subgroups of G is at most 1455, then G is solvable. This is a strong form of a recent
conjecture of Robati.
2020 Mathematics Subject Classification. 20D10, 20D20, 20F16, 20F19.
Funding. The first author is supported by both TU Graz (R-1501000001) and partial funding from the Austrian
Science Fund (FWF): P30934–N35, F05503, F05510. He is also at the University of Nigeria, Nsukka (UNN). The
research of the second author is supported by Ministerio de Ciencia e Innovación PID−2019−103854GB−100,
Generalitat Valenciana AICO/2020/298 and FEDER funds.
Manuscript received 4th October 2020, revised and accepted 5th November 2020.
1. Introduction
Given a prime p and a finite group G , we write νp (G) to denote the number of Sylow p-subgroups
of G . The study of Sylow numbers has played an important role in group theory since the
beginning of the work on the classification of finite simple groups by Burnside and others. It
was proved in the main theorem of the recent paper [5], that if G is a finite non-solvable group
and νp (G) 6 p2 + 1 for every prime p, then G/K = A5 × N with K = O2(G) × O3(G) × O5(G)
and π(N )∩ {2,3,5} = ;. The proof of this result is unassumingly complicated and relies on the
classification of finite simple groups. We start with a slightly stronger result that just depends on
Feit–Thompson theorem.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1228 Chimere Stanley Anabanti, Alexander Moretó and Mohammad Zarrin
Theorem A. Let G be a non-solvable finite group and let R be the solvable radical of G. If K is a
Hall {2,3,5}-subgroup of R, then the following are equivalent:
(1) νp (G)6 p2 +1 for p ∈ {2,3,5};
(2) K is nilpotent, normal in G and G/K = A5 ×R.
Robati conjectured in [5] that “if G is a finite group and νp (G)6 p2 −p +1 for each odd prime
p, then G is solvable”. We prove that it suffices to assume this hypothesis for the primes p = 3 and
p = 5.
Theorem B. Let G be a finite group. Assume that νp (G)6 p2−p+1 for p ∈ {3,5}. Then G is solvable.
In fact, we can obtain an even stronger theorem (see Theorem 4 below). Our proof of Theo-
rem B does not depend on the classification of finite simple groups, but it relies on Thompson’s
classification of minimal simple groups. Our results suggest that the Sylow numbers for small
primes are relevant to determine solvability and even to determine some simple groups. For in-
stance, we have observed that, as a consequence of Guralnick’s [1] classification of maximal sub-
groups of prime power index of simple groups (see also [4]), the number of Sylow 2-subgroups of
a simple group G is a prime p if and only if G = PSL(2, p −1), where p is a Fermat prime. We have
not pursued this further, however.
2. Proofs
We start by recalling some standard group theory exercises that will sometimes be used without
further explicit mention. See also [2] for a strong form of Lemma 2(iii), that we will not need here.
Lemma 1(ii) can be checked with GAP [7].
Lemma 1. Let G be a nonabelian finite simple group and H < G with |G : H | = n. Then G is
isomorphic to a subgroup of An . In particular, n> 5 and we have the following:
(i) if n = 5, then G ∼= A5;
(ii) if n = 7, then G ∼= A5, A6, A7 or PSL(2,7);
(iii) νp (G)> 5 for every prime p that divides |G|. Moreover, if νp (G) = 5 for some prime p, then
p = 2 and G ∼= A5.
Lemma 2. Let G be a finite group, and let p be a prime. The following hold:
(i) if G = H ×K , then νp (G) = νp (H)×νp (K );
(ii) if H 6G, then νp (H)6 νp (G);
(iii) if N EG, then νp (G/N )νp (N ) divides νp (G);
(iv) In particular, if N EG and N 6H 6G, then νp (H/N )6 νp (G).
Recall that if G is a finite group then the generalized Fitting subgroup of G is F∗(G) = F (G)E(G),
where F (G) is the Fitting subgroup and E(G) is the subgroup generated by the components of G .
We will use that C (F∗(G))6 F∗(G) (see [3, Theorem 6.5.8], for instance). Notice that if F (G) = 1,
then F∗(G) = E(G) coincides with the socle of G . Now, we are ready to prove Theorem A.
Proof of Theorem A. It is clear that (ii) implies (i), so we assume (i) and want to prove (ii). Let
P ∈ Syl2(G). Let F /R be the generalized Fitting subgroup of G/R, which coincides with the socle
of G/R, so it is a direct product of nonabelian simple groups. We have that
5> ν2(G)> ν2(F /R)
and it follows from Lemma 2(i) and Lemma 1(iii) that F /R = A5. Since CG/R (F /R) 6 F /R, we
deduce that CG/R (F /R) = 1 so
A5 = F /R 6G/R 6Aut(F /R) = S5
C. R. Mathématique, 2020, 358, n 11-12, 1227-1230
Chimere Stanley Anabanti, Alexander Moretó and Mohammad Zarrin 1229
and we deduce that G/R ∼= A5 or S5. In the later case, ν2(G/R) = 15 > 5, a contradiction. It follows
that G/R ∼= A5. By Lemma 2(iii), νp (G/R)νp (R) divides νp (G) 6 p2 + 1. Since νp (G/R) = νp (A5)
= p2 +1 for p = 2 and p = 3, this implies that ν2(R) = ν3(R) = 1. Also, when p = 5 we have 6ν5(R)
divides ν5(G)6 26. Since ν5(R) ≡ 1 (mod 5), this clearly implies that ν5(R) = 1, so R has normal
Sylow 2,3 and 5–subgroups. Hence, we have seen that K is nilpotent and normal in G
Now we work in G =G/K and use the bar convention. Note that R is the normal Hall {2,3,5}′-
subgroup of G and let H ∼= A5 be a Hall {2,3,5}-subgroup of G , which exists by the Schur–
Zassenhaus theorem. It remains to see that the action of H on R is trivial. Recall that
5 = ν2(G) =
∣∣∣G : NG (P )∣∣∣
so R 6 NG (P ). Thus P acts trivially on R. Since H is generated by conjugates of P , the result
follows. 
Note that an analysis of the proof shows that we just need to assume that ν5(G) < 36 for this
argument to work. Next, we recall Thompson’s classification of minimal simple groups ; i.e., finite
non-abelian simple groups G such that all proper subgroups of G are solvable.
Theorem 3. [8, Corollary 1] Let G be a minimal simple group. Then G is one of the following
groups:
(1) PSL(2, p) with p > 7 an odd prime and p ≡±2 (mod 5);
(2) PSL(2,2p ) with p a prime;
(3) PSL(2,3p ) with p an odd prime;
(4) PSL(3,3);
(5) Sz(2p ) with p an odd prime.
In particular, if G is a minimal simple group and 3 does not divide |G|, then G is isomorphic to a
Suzuki simple group.
We will just use the statement on minimal simple 3′-groups. Finally, we obtain the promised
strong version of Theorem B.
Theorem 4. Let G be a finite group. Assume that ν3(G)6 7 and ν5(G)6 1455. Then G is solvable.
Proof. By way of contradiction, assume that G is not solvable. Let S = K /L be a nonabelian
composition factor of G . Assume first that 3 divides |S|. Then ν3(S) = 7 and S is isomorphic to
a subgroup of A7. Looking at the number of Sylow 3-subgroups for the groups in Lemma 1(ii), we
obtain a contradiction.
Let H 6 S be a (nonabelian) minimal simple subgroup of G . Since 3 does not divide |H |, using
Theorem 3, we deduce that H is isomorphic to a Suzuki simple group. Now it follows from [6,
Theorem 9] that the index of a Sylow 5-normalizer is, at least, 1456. In particular, ν5(H) > 1455.
This is the final contradiction. 
Acknowledgement
The authors are grateful to Professor Sarah Hart for useful discussion during this writing. They
also thank the referee for helpful comments.
References
[1] R. M. Guralnick, “Subgroups of prime power index in a simple group”, J. Algebra 81 (1983), p. 312-319.
[2] M. j. Hall, “On the number of Sylow subgroups in a finite group”, J. Algebra 73 (1967), p. 363-371.
[3] H. Kurzweil, B. Stellmacher, The Theory of Finite Groups: An Introduction, Universitext, Springer, 2004.
[4] R. A. Liebler, J. E. Yellen, “In search of non-solvable groups of central type”, Pac. J. Math. 82 (1979), no. 2, p. 485-492.
C. R. Mathématique, 2020, 358, n 11-12, 1227-1230
1230 Chimere Stanley Anabanti, Alexander Moretó and Mohammad Zarrin
[5] S. M. Robati, “A solvability criterion for finite goups related to the number of Sylow subgroups”, Commun. Algebra 48
(2020), no. 12, p. 5176-5180.
[6] M. Suzuki, “On a class of doubly transitive groups”, Ann. Math. 75 (1962), p. 105-145.
[7] The GAP Group, “GAP – Groups, Algorithms, and Programming”, 2019, Version 4.10.2, https://www.gap-system.org.
[8] J. G. Thompson, “Nonsolvable finite groups all of whose local subgroups are solvable”, Bull. Am. Math. Soc. 74 (1968),
p. 383-437.
C. R. Mathématique, 2020, 358, n 11-12, 1227-1230


Influence of the number of Sylow subgroups on solvability of finite groups

Jiafu Ning

Xiangyu Zhou

Comptes Rendus
Mathématique
Jiafu Ning and Xiangyu Zhou
On the boundedness of invariant hyperbolic domains
Volume 358, issue 3 (2020), p. 321-326.
<https://doi.org/10.5802/crmath.42>
© Académie des sciences, Paris and the authors, 2020.
Some rights reserved.
This article is licensed under the
CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2020, 358, n 3, p. 321-326
https://doi.org/10.5802/crmath.42
Complex Analysis, Analytic Geometry / Analyse complexe, Géométrie analytique
On the boundedness of invariant hyperbolic
domains
Sur le caractère borné des domaines hyperboliques
invariants
Jiafu Ninga and Xiangyu Zhoub
a Department of Mathematics, Central South University, Changsha, Hunan 410083,
China.
b Institute of Mathematics, Academy of Mathematics and Systems Science, and Hua
Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing
100190, China.
E-mails: jfning@csu.edu.cn, xyzhou@math.ac.cn.
Abstract. In this paper, we generalize a theorem of A. Kodama about boundedness of hyperbolic circular
domains. We will prove that if K is a compact Lie group which acts linearly on Cn with O (Cn )K =C, and Ω is
a K -invariant orbit convex domain in Cn which contains 0, then Ω is bounded if and only if Ω is Kobayashi
hyperbolic.
Résumé. Dans cet article, nous généralisons un théorème de A. Kodama sur le caractère borné des domaines
circulaires hyperboliques. Nous démontrons que si K est un groupe de Lie compact qui agit linéairement sur
Cn et vérifie O (Cn )K =C, et siΩ est un domaine K -invariant orbitalement convexe deCn qui contient 0, alors
Ω est borné si et seulement s’il est hyperbolique au sens de Kobayashi.
2020 Mathematics Subject Classification. 32M05, 32T99, 32A07.
Funding. The authors were partially supported by NSFC.
Manuscript received 28th October 2019, accepted 24th March 2020.
1. Introduction
In this note, we investigate the boundedness of invariant hyperbolic domains. In the following, by
hyperbolicity we always mean Kobayashi hyperbolicity. A domain Ω ⊂ Cn is said to be a starlike
circular domain or balanced domain, if for any z = (z1, z2, . . . , zn) ∈Ω and t ∈ C with |t | ≤ 1, one
has t z := (t z1, t z2, . . . , t zn) ∈Ω. A. Kodama [7] proved that:
Theorem 1. For any starlike circular domainΩ⊂Cn ,Ω is hyperbolic if and only if it is bounded.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
322 Jiafu Ning and Xiangyu Zhou
In the language of group actions, a circular domain is a special S1-invariant domain. We want
to study more general invariant domains with respect to compact Lie group actions.
Let G be a Lie group, act linearly onCn by ρ, i.e. ρ : G →GL(Cn) is a Lie group homomorphism.
We may write g z or g ·z for ρ(g )z for g ∈G and z = (z1, . . . , zn)′ ∈Cn as a column vector. A domain
Ω⊂Cn is called G-invariant if for any z ∈Ω and g ∈G , then g z ∈Ω.
For a compact Lie group K , denote by K C the universal complexification of K . If K acts linearly
on Cn , then the K action induces a K C linear action on Cn . If k is the Lie algebra of K , and
exp : k → K is the exponential map, then the first Cartan decomposition theorem says that,
K C = K ·exp(ik) = exp(ik) ·K , and K ∩exp(ik) = {e}, where e is the identity of G .
We need to recall the following definition (see [3] or [13]).
Definition 2. Let K be a compact Lie group with Lie algebra k, and let K C be the complexification
of K . Let X be a K C-space. A K -invariant subset U of X is said to be orbit convex if for each z ∈U
and v ∈ ik, such that exp(v) · z ∈U , it follows that exp(t v) · z ∈U for all t ∈ [0,1].
Let O (Cn)K = { f ∈ O (Cn) : f ◦ρ(k) = f ,∀ k ∈ K }, i.e. O (Cn)K is the set of K -invariant entire
functions.
Let ρ : S1 → GL(Cn), ρ(t ) = diag{t , t , . . . , t }, then S1 acts linearly on Cn and O (Cn)S1 = C. A
domain Ω ⊂ Cn is said to be circular if it is S1-invariant, i.e. for any z = (z1, z2, . . . , zn) ∈ Ω and
t ∈ S1, we have t z := (t z1, t z2, . . . , t zn) ∈Ω. It is easy to check that a domainΩ is a starlike circular
domain if and only ifΩ is an orbit convex S1-invariant domain containing 0.
Studying problems in several complex variables from a point view of group actions is fre-
quently very fruitful. In this note, we discuss the boundedness problem of hyperbolic invari-
ant domains in a setting of group actions. For prerequisites about Lie groups and several com-
plex variables, one is referred to [4, 8]. We consider a compact Lie group K acting linearly on Cn
with O (Cn)K = C. Let Ω be a K -invariant domain containing 0. There are many interesting re-
sults about some problems in several complex variables related to group actions on Ω, see [2]–
[3], [5], [10], [12]– [14] et al.
We will prove the following theorem which is a generalization of Kodama’s theorem.
Theorem 3. Let K be a compact Lie group which acts linearly on Cn with O (Cn)K = C. Let Ω
be a K -invariant orbit convex domain in Cn containing 0. Then Ω is bounded if and only if Ω
is hyperbolic.
In the last section, we deal with the case for the Brody hyperbolic domains.
2. Some preliminary results
Let G be a Hermitian closed subgroup of GL(n;Cn) acting on Cn by multiplication.
Theorem 4 (The Hilbert–Mumford theorem (see [9])). For z ∈ Cn , the closure of the orbit Gz
contains 0 if and only if there exist an algebraic group homomorphism ϕ : C∗ → G such that
limλ→0ϕ(λ)z = 0.
Remark 5. For compact Lie group K which acts linearly on Cn , we may identify K (and K C)
with the image of K (and K C). After a linear transformation, we can assume that K (and K C) is a
subgroup of U (n) (and GL(n;Cn)). Let k be the Lie algebra of K , then k ⊂ M(n,n,C), and we can
give the usual norm on ik, i.e. ‖υ‖2 = Tr (υυt ). For any υ ∈ ik, we have υ= υt . There is a g ∈U (n),
such that υ= g diag{λ1,λ2, . . . ,λn}g−1 with λ j ∈R for j = 1,2, . . . ,n.
We need the following lemma (see [11, Theorem 12]), which is another form of the Hilbert–
Mumford theorem, in order to suit our case. We give a proof in detail for completeness.
C. R. Mathématique, 2020, 358, n 3, 321-326
Jiafu Ning and Xiangyu Zhou 323
Lemma 6. Let K be a compact Lie group, which acts linearly on Cn . Let b be the Cartan sub-
algebra of k, where k is the Lie algebra of K . For z ∈Cn , if 0 ∈ K Cz, then there is a k ∈ K and υ ∈ ib,
such that limt→+∞ exp(tυ)kz = 0.
Proof. As the lemma is obviously true for z = 0, in the following, we assume z 6= 0. We may assume
that K is a closed subgroup of U (n). As b is the Cartan subalgebra of k, there is a g ∈U (n), such
that for any υ ∈ ib,
υ= gΛ(υ)g−1
where Λ(υ) = diag{λ1(υ),λ2(υ), . . . ,λn(υ)}, λ j (υ) ∈ R for j = 1,2, . . . ,n. By a theorem of Lie algebra,
k=∪k∈K kbk−1, and since K C = exp(ik) ·K and 0 ∈ K Cz, there is a sequence {al }, {υl } and {kl } with
al > 0, υl ∈ ib, ‖υl‖ = 1 and kl ∈ K for l = 1,2, . . . , such that
lim
l→∞
exp(alυl )kl z = 0.
We may write υl = gΛl g−1, where Λl = diag{λl1,λl2, . . . ,λln} with λl j ∈ R for j = 1,2, . . . ,n
and l = 1,2, . . . . Since ‖υl‖ = 1 and U (n) and K are compact, after a subsequence, we may
assume that liml→∞Λl = Λ0 and liml→∞ kl = k. So k ∈ K and υ0 := gΛ0g−1 ∈ ik. Set wl =
(wl1, wl2, . . . , wl n) := g−1kl z, then liml→∞ wl = g−1kz. Put w0 = (w01, w02, . . . , w0n) := g−1kz and
Λ0 = diag{λ1,λ2, . . . ,λn}.
If w0 j 6= 0 for some j ∈ {1,2, . . . ,n},
‖exp(alυl )kl z‖ = ‖exp(alΛl )wl‖ ≥ ealλl j |wl j |.
As liml→∞ exp(alυl )kl z = 0 and al > 0, then we have λ j ≤ 0; furthermore, if λ j = 0, then λl j < 0
for l > N j . So, take v = v0 + vN ∈ ib for some N large enough,
v = g diag{λ1 +λN 1,λ2 +λN 2, . . . ,λn +λN n}g−1,
satisfying λ j +λN j < 0, whenever w0 j 6= 0 for j ∈ {1,2, . . . ,n}. Therefore,
lim
t→+∞exp(tυ)kz = 0. 
Lemma 7. Let K be a compact Lie group, which acts linearly onCn . Let b be the Cartan subalgebra
of k, where k is the Lie algebra of K . If O (Cn)K = C, there are υ1,υ2, . . . ,υs ∈ ib, such that for any
z ∈Cn , z 6= 0, there is an lz ∈ {1,2, . . . , s} and kz ∈ K , satisfying limt→+∞ exp(tυlz )kz z = 0.
Proof. We may assume that K is a closed subgroup of U (n). As b is the Cartan subalgebra of k,
there is a g ∈U (n), such that for any υ ∈ ib,
υ= gΛ(υ)g−1
whereΛ(υ) = diag{λ1(υ),λ2(υ), . . . ,λn(υ)}, λ j (υ) ∈R for j = 1,2, . . . ,n.
As O (Cn)K =C, by [10, Fact 2.1], we can get 0 ∈ K Cz for any z ∈Cn . By Lemma 6, for any z ∈Cn ,
z 6= 0, there is a kz ∈ K and υ ∈ ib, such that limt→+∞ exp(tυ)kz z = 0. As exp(tυ) = g exp(tΛ(υ))g−1,
therefore,
lim
t→+∞g exp(tΛ(υ))g
−1kz z = lim
t→+∞exp(tυ)kz z = 0.
Let
wz = (w1, w2, . . . , wn) := g−1kz z.
Then set az := (a1, a2, . . . , an) with
a j :=
{
0 if w j = 0
1 if w j 6= 0
and A = {az : z ∈Cn , z 6= 0}. As ]A < 2n , and for each a ∈ A, we may choose
υa ∈ ib,υa = g diag{λ1(υa),λ2(υa), . . . ,λn(υa)}g−1
C. R. Mathématique, 2020, 358, n 3, 321-326
324 Jiafu Ning and Xiangyu Zhou
such that λ j (υa) < 0 if a j = 1 for j ∈ {1,2, . . . ,n}. So we get
lim
t→+∞exp(tυaz )kz z = 0.
Then taking the index of A as {1,2, . . . , s}, we prove the lemma. 
3. Hyperbolicity and boundedness
The aim of this subsection is to prove Theorem 3. For convenience, we restate it here.
Theorem 8 (= Theorem 3). Let K be a compact Lie group, which acts linearly on Cn with
O (Cn)K =C. LetΩ be a K -invariant orbit convex domain in Cn containing 0. ThenΩ is bounded if
and only if Ω is hyperbolic.
Proof. It is well known (see [6]) that ifΩ is bounded, thenΩ is hyperbolic.
In the following, we prove the ’if’ part. We may assume the unit ball B ⊂Ω and K is a closed
subgroup of U (n). Suppose thatΩ is unbounded and hyperbolic, and we will get a contradiction.
Let zl ∈ Ω for l = 1,2, . . . such that ‖zl‖ > 1 and liml→∞ ‖zl‖ = ∞. By Lemma 7, we can find
a subsequence {xl }
∞
l=1 of {zl }
∞
l=1 such that there is a υ ∈ {υ j : j = 1,2, . . . , s}, satisfying for each xl ,
there is a kl ∈ K , such that limt→+∞ exp(tυ)kl xl = 0 for any l = 1,2, . . . .
Choose bl > 0 satisfying ‖exp(blυ)kl xl‖ = 1/3 for each l . As liml→∞ ‖xl‖ = ∞, we have
liml→∞ bl =∞. SinceΩ is orbit convex, so
fl (λ) = exp((1−λ)blυ)kl xl :∆→Ω
is holomorphic, where ∆ = {λ ∈ C : |λ| < 1}. Notice ‖ fl (0)‖ = 13 and ‖ fl (1)‖ > 1, we
can choose tl ∈ (0,1), such that ‖ fl (tl )‖ = 12 . As in the proof of Lemma 7, we write υ =
g diag{λ1(υ),λ2(υ), . . . ,λn(υ)}g−1 with g ∈U (n). Let
wl = (wl1, wl2, . . . , wln) := g−1kl xl ,
then λ j (υ) < 0, if wl j 6= 0 for some l ∈N, j ∈ {1,2, . . . ,n}. Set
hl (t ) := ‖ fl (t )‖2 =Σnj=1e2(1−t )blλ j (υ)|wl j |2, for t ∈ [0,1],
then hl (0) = 1/9, hl (tl ) = 1/4. We have h′′l (t ) ≥ 0, and
h′l (0) = 2blΣnj=1(−λ j (υ))e2blλ j (υ)|wl j |2
≥ 2blλΣnj=1e2blλ j (υ)|wl j |2
= 2blλhl (0)
→∞, as l →∞,
(1)
where λ= min{−λ j (υ) :λ j (υ) < 0 for j ∈ {1,2, . . . ,n}} > 0. Since
hl (tl ) ≥ hl (0)+h′l (0)tl ,
we have
tl ≤
hl (tl )−hl (0)
h′l (0)
= 5
36h′l (0)
→ 0 as l →∞ by (1).
(2)
By the compactness of {z ∈Cn : ‖z‖ = 1/3} and {z ∈Cn : ‖z‖ = 1/2}, we have a subsequence { flp (0)}
of { fl (0)} and { flp (tlp )} of { fl (tl )} with limp→∞ flp (0) = y0 ∈Ω and limp→∞ flp (tlp ) = y1 ∈Ω. By the
distance decreasing property of holomorphic mappings with respect to the Kobayashi distances,
kΩ( flp (0), flp (tlp )) ≤ k∆(0, tlp ). (3)
C. R. Mathématique, 2020, 358, n 3, 321-326
Jiafu Ning and Xiangyu Zhou 325
Taking p →∞, by (2) and (3), we have kΩ(y0, y1) = 0. Since y0 6= y1, we get a contradiction. Hence,
Ωmust be bounded. 
Corollary 9. Let K be a compact Lie group, which acts linearly on Cn with O (Cn)K = C. Let Ω be
a K -invariant pseudoconvex domain in Cn containing 0. Then Ω is bounded if and only if Ω is
hyperbolic.
Proof. It is proved in [10, Theorem 2.4] that Ω is orbit convex, so Corollary is followed by the
above theorem. 
4. Some results about Brody hyperbolicity
In this section, we discuss Brody hyperbolicity. A domain without nonconstant holomorphic
mappings from C into it is said to be Brody hyperbolic. It is well known that hyperbolic domains
are Brody hyperbolic, the reverse is true for compact complex manifolds but is false in generally.
We focus on invariant domains.
Theorem 10. Let K be a compact Lie group acting linearly on Cn with O (Cn)K = C. Let Ω
be an unbounded K -invariant taut domain in Cn containing 0. Then there is a nonconstant
holomorphic mapping f :C→Ω.
Proof. We may assume that the closed unit ball B is contained in Ω and that K is a closed
subgroup of U (n).
AsΩ is taut, thenΩ is pseudoconvex, therefore, it is orbit convex (see [10, Theorem 2.4]). Since
Ω is unbounded, by the proof of Theorem 8, we may find zl ∈ Ω for l = 1,2, . . . satisfying the
following two conditions:
(1) ‖zl‖ > 0 and liml→∞ ‖zl‖ =∞,
(2) there is a υ ∈ k, such that for each l , limt→+∞ exp(tυ)zl = 0.
We may write υ= g diag{λ1,λ2, . . . ,λn}g−1, where g ∈U (n) andλ j ∈R for j = 1,2, . . . ,n. Let wl =
(wl1, wl2, . . . , wl n) := g−1zl . By condition (2), we can get that if wl j 6= 0 for some j ∈ {1,2, . . . ,n} and
some l ∈N, then λ j < 0. Since
exp(tυ)zl = g exp(t ·diag{λ1,λ2, . . . ,λn})g−1zl
= g (e tλ1 wl 1,e tλ2 wl2, . . . ,e tλn wln),
the equation ‖exp(tυ)zl‖ = 1 for t > 0, has only one solution which is denoted by al . As
liml→∞ ‖zl‖ =∞, we get liml→∞ al =+∞. SinceΩ is orbital convex,
fl (ξ) = exp(ξυ)exp(alυ)zl : Hl →Ω
is holomorphic on Hl , where Hl = {ξ = x + i y ∈ C : x > −al }. As {exp(alυ)zl } ⊂ B , we may find its
subsequence {exp(alpυ)zlp }, such that
lim
p→∞exp(alpυ)zlp = z0 ∈Ω.
So
exp(ξυ)z0 = lim
p→∞ flp (ξ).
Let f (ξ) = exp(ξυ)z0. By the tautness of Ω, and ⋃p Hlp = C, we get a holomorphic mapping
f :C→Ω. As
‖z0‖ = lim
p→∞‖exp(alpυ)zlp ‖ = 1,
C. R. Mathématique, 2020, 358, n 3, 321-326
326 Jiafu Ning and Xiangyu Zhou
write g−1z0 = (ζ1,ζ2, . . . ,ζn), then there is a j0, such that ζ j0 6= 0. Since
(ζ1,ζ2, . . . ,ζn) = g−1 lim
p→∞exp(alpυ)zlp
= lim
p→∞(e
alp λ1 wlp 1,e
alp λ2 wlp 2, . . . ,e
alp λn wlp n)
and ζ j0 6= 0, we get that wlp j0 6= 0 for p >> 1, hence λ j0 < 0. Note that
f (ξ) = exp(ξυ)z0 = g (eξλ1ζ1,eξλ2ζ2, . . . ,eξλnζn),
we can get that f is nonconstant. 
K. Azukawa [1] gives an example showing that, there is a pseudoconvex circular domain in
C2 containing 0, which is Brody hyperbolic but not hyperbolic. Hence, his domain must be
unbounded. However, one has the following reformulation of Theorem 10:
Corollary 11 (= Theorem 10). Let K be a compact Lie group, which acts linearly on Cn with
O (Cn)K = C. Let Ω be a K -invariant taut domain in Cn containing 0. Then Ω is Brody hyperbolic
if and only if it is bounded.
References
[1] K. Azukawa, “Hyperbolicity of circular domains”, Tôhoku Math. J. 35 (1983), p. 259-265.
[2] E. Casadio Tarabusi, S. Trapani, “Envelopes of holomorphy of Hartogs and circular domains”, Pac. J. Math. 149
(1991), no. 2, p. 231-249.
[3] P. Heinzner, “Geometric Invariant Theory on Stein spaces”, Math. Ann. 289 (1991), no. 4, p. 631-662.
[4] S. Helgason, Differential geometry, Lie groups and symmetric spaces, Pure and Applied Mathematics, vol. 80, Aca-
demic Press Inc., 1978.
[5] W. Kaup, “Über das Randverhalten von Holomorphen Automorphismen beschränkter Gebiete”, Manuscr. Math. 3
(1970), p. 257-270.
[6] S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings: an introduction, World Scientific, 2005.
[7] A. Kodama, “Boundedness of circular domains”, Proc. Japan Acad., Ser. A 58 (1982), p. 227-230.
[8] S. Krantz, Function Theory of Several Complex Variables, American Mathematical Society, 2001.
[9] D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete.
3. Folge., vol. 34, Springer, 1993.
[10] J. Ning, H.-P. Zhang, X.-Y. Zhou, “Proper holomorphic mappings between invariant domains inCn ”, Trans. Am. Math.
Soc. 369 (2017), no. 1, p. 517-536.
[11] N. R. Wallach, “Hilbert-Mumford type theorems”, an unpublished lecture, www.math.ucsd.edu/~nwallach/
lectures-3-math207.pdf.
[12] X.-W. Wu, F.-S. Deng, X.-Y. Zhou, “Rigidity and regularity in group actions”, Sci. China, Ser. A 51 (2008), no. 4, p. 819-
826.
[13] X.-Y. Zhou, “On orbit connectedness, orbit convexity, and envelopes of holomorphy”, Izv. Ross. Akad. Nauk, Ser. Mat.
58 (1994), no. 2, p. 196-205.
[14] ——— , “Some results related to group actions in several complex variables”, in Proceedings of the international
congress of mathematicians (ICM 2002). Vol. II: Invited lectures, Higher Education Press, 2002, p. 743-753.
C. R. Mathématique, 2020, 358, n 3, 321-326


On the boundedness of invariant hyperbolic domains

Thibaut Le Gouic

Comptes Rendus
Mathématique
Thibaut Le Gouic
A note on flatness of non separable tangent cone at a barycenter
Volume 358, issue 4 (2020), p. 489-495.
<https://doi.org/10.5802/crmath.66>
© Académie des sciences, Paris and the authors, 2020.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2020, 358, n 4, p. 489-495
https://doi.org/10.5802/crmath.66
Geometry, Probability Theory / Géométrie, Probabilité
A note on flatness of non separable tangent
cone at a barycenter
Une note sur la platitude du cône tangeant à un
barycentre
Thibaut Le Gouic a
a Massachusetts Institute of Technology, Department of Mathematics and Centrale
Marseille, I2M, UMR 7373, CNRS, Aix-Marseille univ., Marseille, 13453, France.
E-mail: thibaut.le_gouic@math.cnrs.fr.
Abstract. Given a probability measure P on an Alexandrov space S with curvature bounded below, we prove
that the support of the pushforward of P on the tangent cone Tb?S at its (exponential) barycenter b
? is a
subset of a Hilbert space, without separability of the tangent cone.
Résumé. Étant donné une mesure de probabilité P sur un espace d’Alexandrov S avec courbure minorée,
nous prouvons que le support de la mesure poussée de P sur le cône tangent Tb?S à son barycentre
(exponentiel) b? est un sous-ensemble d’un espace de Hilbert, sans condition de séparabilité du cône
tangent.
Funding. Supported by ONR grant N00014-17- 1-2147.
Manuscript received 16th September 2019, revised 31st December 2019 and 20th April 2020, accepted
29th April 2020.
1. Introduction
Barycenter of a probability measure P (a.k.a. Fréchet means) provides an extension of expectation
on Euclidean space to arbitrary metric spaces. We present here a useful tool for the study of
barycenters on Alexandrov spaces with curvature bounded below: the support of logb? #P in the
tangent cone at the barycenter is included in a Hilbert space. This rigidity result has been stated
in [9] as Theorem 45, however the proof is not written. Moreover, there is an extra assumption
of support of logb? #P being separable, which does not even seem to be a consequence of the
support of P being separable. As pointed out by [7], it is not clear if even S being proper ensures
that the tangent cone is separable. This paper presents a proof of this rigidity result, without this
extra separable assumption on the tangent cone. For measurability purposes (see Lemma 6), we
suppose however that S is separable. The proof is essentially the one of Theorem 45 of [9], with
needed approximations dealt with a bit differently.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
490 Thibaut Le Gouic
2. Setting and main result
We use a classical notion of curvature bounded below for geodesic spaces, referred to as Alexan-
drov curvature. We recall several notions whose formal definitions can be found for instance in [3]
or in the work in progress [2].
For a metric space (S,d), we denote by P1(S) the set of probability measures on S with finite
moment of order 1 (i.e. such that there exists x ∈ S such that ∫ d(x, y)dP(y) <∞). The support of
a measure P will be denoted by suppP. We use both notation
∫
f dP and P f for the integral of f
w.r.t. P.
A geodesic space is a metric space (S,d) such that every two points x, y ∈ S at distance is
connected by a curve of length d(x, y). Such shortest curves are called geodesics. For κ ∈ R, the
model space (Mκ,dκ) denotes the 2-dimensional simply connected complete surface of constant
Gauss curvature κ. A geodesic space (S,d) is an Alexandrov space with curvature bounded below
by κ ∈ R if for every triangle (3-uple) (x0, x1, y) ∈ S, and a constant speed geodesic (xt )t∈[0;1]
there exists an isometric triangle (x̃0, x̃1, ỹ) ∈ Mκ, such that the geodesic (x̃ t )t∈[0;1] satisfies for
all t ∈ [0;1],
d(y, xt ) ≥ dκ(ỹ , x̃ t ).
For such spaces, angles between two unit-speed geodesics γ1,γ2 starting at the same point p ∈ S
can be defined as follows:
cos∠p (γ1,γ2) = lim
t→0
d 2(γ1(t ), p)+d 2(γ2(t ), p)−d 2(γ1(t ),γ2(t ))
2d(p,γ1(t ))d(p,γ2(t ))
,
where angle ∠p (γ1,γ2) ∈ [0;π]. Denote by Γp the set of all unit-speed geodesics emanating from
p. Using angles, we can define the tangent cone Tp S at p ∈ S as follows. First define T ′p S as the
(quotient) set Γp ×R+, equipped with the (pseudo-)metric defined by
‖(γ1, t )− (γ2, s)‖2p := s2 + t 2 −2s.t cos∠p (γ1,γ2).
Then, the tangent cone Tp S is defined as the completion of T ′p S equipped with the metric ‖ · ‖p .
We will use the notation for u, v ∈ Tp S,
〈u, v〉p := 1
2
(‖u‖2p +‖v‖2p −‖u − v‖2p ),
We will often identify a point γ(t ) ∈ S with (γ, t ) ∈ Tp S. Although such γ might not be unique, we
will assume a choice of a map logp : S → Tp S, called logarithmic map, such that for all x ∈ S, there
exists a unit-speed geodesic γ emanating from p such that, for some t > 0, γ(t ) = x and
logp (x) = (γ, t ).
This map can be chosen to be SB -measurable, where SB denotes the σ-algebra generated by
open balls on the tangent cone Tp S (see Lemma 6) and this weak measurability is enough for our
results to hold and will be assumed for the rest of the paper. Then the pushforward of P by logp
will be denoted by logp #P.
The tangent cone is not necessarily a geodesic space (see [4]), however, it is included in a
geodesic space, namely the ultratangent space (see for instance Theorem 14.4.2 and 14.4.1 in [2])
that is an Alexandrov space with curvature bounded below by 0.
The tangent cone Tp S contains the subspace Linp of all points with an opposite, formally
defined as follows. A point u belongs to Linp ⊂ Tp S if and only if there exists v ∈ Tp S such that
‖u‖p = ‖v‖p and
〈u, v〉p =−‖u‖2p .
Our main result is based on the following Theorem.
Theorem (Theorem 14.5.4 in [2]). The set Linp equipped with the induced metric of Tp S is a
Hilbert space.
C. R. Mathématique, 2020, 358, n 4, 489-495
Thibaut Le Gouic 491
A point b? ∈ S is a barycenter of the probability measure P ∈P1(S) if for all b ∈ S
0 ≤
∫
d 2(x,b)−d 2(x,b?)dP(x).
Such barycenter might not be unique, neither exists. However, when they exist, they satisfy∫
〈x, y〉b?dP⊗P(x, y) = 0. (1)
A point b? ∈ S satisfying (1) is called an exponential barycenter of P.
We can now state our main result.
Theorem 1. Let (S,d) be an Alexandrov space with curvature bounded below by some κ ∈ R and
P ∈ P1(S). If b? ∈ S is an exponential barycenter of P, then supplogb? #P ⊂ Linb? S. In particular,
supplogb? #P is included in a Hilbert space.
This result allows to prove the following Corollary, that has been implicitly used in [1].
Corollary 2 (Linearity). Let b ∈ Tb?S. Then, the map 〈 · ,b〉b? : Linb? → R is continuous and linear.
In particular, if b? is an exponential barycenter of P, then∫
〈x,b〉b?dP(x) = 0.
3. Proofs
Recall that we always identify a point in S and its image in the tangent cone Tp S by the logp map.
Proof of Corollary 2. Linearity is obvious from the definition of 〈 · ,b〉b? . We check that x 7→
〈x,b〉b? is a convex and concave function in Linb? S. Let t ∈ (0,1), x0, x1 in Linb? S, and set
xt = (1− t )x0 + t x1. Since the tangent cone is included in an Alexandrov space with curvature
bounded below by 0 on one hand, and Linb? is a Hilbert space on the other hand,
〈xt ,b〉b? =
1
2
(‖xt‖2b? +‖b‖2b? −‖xt −b‖2)
≤ 1
2
(
(1− t )(‖x0‖2b? −‖x0 −b‖2b? )+ t (‖x1‖2b? −‖x1 −b‖2)+‖b‖2b?
)
= (1− t )〈x0,b〉b? + t〈x1,b〉b? .
The same lines applied to −x0 and −x1 gives the converse inequality
〈−xt ,b〉b? ≤ (1− t )〈−x0,b〉b? + t〈−x1,b〉b? .
The second statement follows from the fact that b? is a Pettis integral of the pushforward of P
onto Linb? ⊂ Tb?S, as a direct consequence of Theorem 1. 
Proof of Theorem 1. Let L ⊂ {x ∈ S|∫ 〈x, · 〉b?dP = 0} be a measurable set such that P(L) = 1 given
by Lemma 3. Let x ∈ L. For U = {x}, use Lemma 5 with Q = P and Bδ a ball of radius δ around x in
Tb?S, to get a sequence (y
n
δ
)n ⊂ Tb?S such that,
limsup
n
cos∠(↑xb? ,↑
yn
δ
b?
) = limsup
n
〈x, yn
δ
〉b?
d(b?, x)d(b?, yn
δ
)
= 1
d(b?, x)
limsup
n
〈x, ynδ 〉b?
1
limn d(b?, ynδ )
≤ 1
d(b?, x)
∫
B c
δ
〈x, y〉b?dP(x)
P(Bδ)
P(Bδ)(∫
Bδ
∫
Bδ
〈x, y〉b?dP⊗P(x, y)
)1/2 .
C. R. Mathématique, 2020, 358, n 4, 489-495
492 Thibaut Le Gouic
Then, since
∫ 〈x, y〉b?dP(y) = 0, letting δ→ 0, one gets
1
P(Bδ)
∫
B c
δ
〈x, y〉b?dP(y) =−
1
P(Bδ)
∫
Bδ
〈x, y〉b?dP(y) →−d 2(b?, x).
and (∫
Bδ
∫
Bδ
〈x, y〉b?dP⊗P(x, y)
)1/2
P(Bδ)
→ d(b?, x)
Thus,
lim
δ→0+
limsup
n
cos∠(↑xb? ,↑
yn
δ
b?
) =−1
One can thus choose (ȳn)n a sequence in (ynδ )n,δ such that cos∠(↑xb? ,↑
ȳn
b?
) →−1. Since Tb?S is a
subspace of an Alexandrov space of curvature bounded below by 0, we also have
∠(↑ȳn
b?
,↑ȳk
b?
) ≤ 2π−∠(↑ȳn
b?
,↑xb? )−∠(↑xb? ,↑
ȳk
b?
)
→ 0,
as n,k → ∞. Thus (ȳn)n corresponds to a Cauchy sequence in the space of direction, and
thus admits a limit in Tb?S, since its “norm” also admits a limit d(b
?, x). Its limit ȳ satisfies
cos∠(↑x
b?
,↑ȳ
b?
) =−1, and therefore, it is the opposite ȳ =−x.
Finally, by definition of the support, for x ∈ supp(logb? P), every ball centered at x have a
positive probability, and thus there exists a sequence (xn)n≥1 ⊂ L such that xn → x. We conclude
with the completeness of Linb? . 
Lemma 3 (Proposition 1.7 of [8] for non separable metric space). Suppose (S,d) is an Alexan-
drov space with curvature bounded below. Then, for any probability measure Q ∈ P1(S), and any
b? ∈ S, ∫
〈x, y〉b?dQ⊗Q(x, y) ≥ 0.
Moreover, if b? is an exponential barycenter of Q, then for Q-almost all x ∈ S,∫
〈x, y〉b?dQ(y) = 0.
Proof. For brevity, we will adopt the notation Qg for
∫
g dQ.
The result for Q finitely supported is the Lang–Schroeder inequality (Proposition 3.2 in [5]).
Thus, we just need to approximate Q⊗Q〈 · , · 〉b? by some Qn ⊗Qn〈 · , · 〉b? for some finitely sup-
ported Qn .
To approximate Q⊗Q〈 · , · 〉b? , draw two independent sequences of i.i.d. random variables (X 1i )i
and (X 2i )i of common law Q, and denote Q
1
n and Q
2
n the corresponding empirical measures. In
particular, Q1n⊗Q2n and Q2n⊗Q1n are both empirical measures of Q⊗Q. Since S is not separable, we
can not apply the fundamental theorem of statistics that ensures almost sure weak convergence
of Q1n ⊗Q1n to Q⊗Q. However, for a measurable function f : S ×S → R, such that Q⊗Q f <∞, the
law of large number ensures that almost surely
Q1n ⊗Q2n f → Q⊗Q f
and
Q2n ⊗Q1n f → Q⊗Q f .
Since the sequence (X 11 , X
2
1 , X
1
2 , X
2
2 , X
1
3 , X
2
3 , . . . ) is also an i.i.d. sequence of random variables of
common law Q, the subsequence of the associated empirical measures (Q3n)n defined by
Q3n :=
1
2
(Q1n +Q2n)
also satisfies the almost sure convergence
Q3n ⊗Q3n f → Q⊗Q f .
C. R. Mathématique, 2020, 358, n 4, 489-495
Thibaut Le Gouic 493
Now, since
Q3n ⊗Q3n =
1
4
(Q1n ⊗Q2n +Q1n ⊗Q1n +Q2n ⊗Q1n +Q2n ⊗Q2n),
we proved that almost surely
Q1n ⊗Q1n f +Q2n ⊗Q2n f → 2Q⊗Q f .
And since (Q1n)n and (Q
2
n)n are independent and with same law, it implies that both Q
1
n ⊗Q1n f
and Q2n ⊗ Q2n f converge to Q ⊗ Q f almost surely. In particular, since Q1n is supported on n
points, there exists a sequence of finitely supported measures (that we rename (Qn)n) such that
Qn ⊗Qn f → Q⊗Q f . We thus proved the first result applying f = 〈· , · 〉b? .
Now, for any x ∈ S, applying this first result to the measure Qε := 11+εQ+ ε1+εδx , we get
0 ≤ (1+ε)Qε⊗Qε〈 · , · 〉b?
= Q⊗Q〈 · , · 〉b? +2εQ〈x, .〉b? +ε2‖x‖2b? .
Letting ε→ 0+, we get
Q〈x, · 〉b? ≥ 0.
Then equality follows from the hypothesis Q⊗Q〈 · , · 〉b? = 0 meaning that b? is an exponential
barycenter. 
Lemma 4 (Subadditivity, Lemma A.6 of [5]). Let (S,d) be an Alexandrov space with curvature
bounded below. Take b? ∈ S. Let x1, . . . , xn ∈ T ′b?S and U ⊂ Tb?S finite. Then, for all ε > 0, there
exists y ∈ Tb?S such that for all u ∈U ,
〈y,u〉b? ≤
n∑
i=1
〈xi ,u〉b? +ε,
and
‖y‖2 ≤
n∑
i , j=1
〈xi , x j 〉b? +ε.
Lemma 5 (Approximation). Let U ⊂ Tb?S finite. Take B ⊂ S measurable and a probability
measure P ∈ P1(S) such that P⊗P〈 · , · 〉b? = 0 and P(B) > 0. Then, there exists a sequence (yn)n
such that for all u ∈U
1
P(B)
∫
B c
〈u, x〉b?dP(x) ≥ limsup
n
〈u, yn〉b? (2)
and
1
P(B)2
∫
B
∫
B
〈x, y〉b?dP⊗P(x, y) = limn d
2(b?, yn). (3)
Proof. Using the same arguments as in Lemma 3, we see that the empirical measures (Pn)n
satisfy
Pn ⊗Pn f → P⊗P f ,
almost surely, for any f : S × S → R ∈ L1(S × S, P ⊗ P). In particular, taking f (x, y) =
〈x, y〉b?1B×B (x, y), the following convergence holds in L2(P⊗∞),∫
B
∫
B
〈 · , · 〉b?dPn ⊗Pn →
∫
B
∫
B
〈 · , · 〉b?dP⊗P, (4)
and similarly for B c . Also, the law of large number ensures that almost surely, for all u ∈U ,∫
B
〈 · ,u〉b?dPn →
∫
B
〈 · ,u〉b?dP, (5)
and again, the same for B c . Thus, there exists a subsequence (of a deterministic realization of) Pn
(that we rename Pn) such that (4) and (5) both hold for all u ∈U .
C. R. Mathématique, 2020, 358, n 4, 489-495
494 Thibaut Le Gouic
Then, applying Lemma 4 to finite sum
1
P(B)
∫
B c
〈 · ,u〉b?dPn ,
shows that there exists a sequence (yn)n ∈ T ′b?S such that (2) holds and for a sequence (εn)n s.t.
εn → 0,
‖yn‖2b? ≤
1
P(B)2
∫
B c
∫
B c
〈 · , · 〉b?dPn ⊗Pn +εn .
Then, applying the same Lemma 4 again shows that there exists a sequence (zn)n ⊂ T ′b?S, such
that
0 ← 1
P(B)2
∫ ∫
〈x, y〉b?dPn ⊗Pn(x, y)
= 1
P(B)2
(∫
B
∫
B
+
∫
B c
∫
B c
+2
∫
B
∫
B c
)
〈x, y〉b?dPn ⊗Pn(x, y)
≥ ‖zn‖2b? +‖yn‖2b? +2〈yn , zn〉b? −εn . (6)
Letting n →∞, one obtains
0 ≥ lim
n
‖zn‖2b? +2〈yn , zn〉b? +‖yn‖2b?
≥ lim
n
‖zn‖2b? −2‖yn‖b?‖zn‖b? +‖yn‖2b?
= lim
n
(‖zn‖b? −‖yn‖b? )2 ≥ 0.
and which shows limn ‖yn‖ = limn ‖zn‖ and also that (6) becomes an equality at the limit and
therefore
lim
n
‖zn‖2b? =
1
P(B)2
∫
B
∫
B
〈x, y〉b?dP⊗P(x, y) 
This Lemma appears in a remark of [6].
Lemma 6 (Measurability of the log map). Let (S,d) be a separable Alexandrov space. Let p ∈ S.
Then logp : S → Tp S can be chosen to be SB -measurable.
Proof. Denote Gp the space of all constant speed geodesics emanating from p equipped with
the sup distance ‖ · ‖∞. Then (Gp ,‖ · ‖∞) is separable and complete too. Using Kuratowski and
Ryll-Nardzewski measurable selection theorem, one can choose a Borel map g : S → Gp such
that g maps x to a geodesic from p to x. Then, using the (proof of) Lemma 4.2 of [7], the map
l : Gp → Tp S is measurable Tp S is equipped with the σ-algebra S generated by open balls.
Therefore, logp := l ◦ g is SB -measurable. 
Acknowledgements
This note originated from technical issues emanating from the problem considered in [1] and [6].
I would like to thank Quentin Paris and Philippe Rigollet for fruitful discussions that lead to this
article. I am also indebted to the anonymous referee for their careful review and comments.
References
[1] A. Ahidar-Coutrix, T. Le Gouic, Q. Paris, “On the rate of convergence of empirical barycentres in metric spaces:
curvature, convexity and extendible geodesics”, https://arxiv.org/abs/1806.02740v1, 2018.
[2] S. Alexander, V. Kapovitch, A. Petrunin, “Alexandrov geometry”, http://arxiv.org/abs/1903.08539, 2019.
[3] D. Burago, Y. Burago, S. Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American
Mathematical Society, 2001.
[4] S. Halbeisen, “On tangent cones of Alexandrov spaces with curvature bounded below”, Manuscr. Math. 103 (2000),
no. 2, p. 169-182.
C. R. Mathématique, 2020, 358, n 4, 489-495
Thibaut Le Gouic 495
[5] U. Lang, V. Schroeder, “Kirszbraun’s theorem and metric spaces of bounded curvature”, Geom. Funct. Anal. 7 (1997),
no. 3, p. 535-560.
[6] T. Le Gouic, Q. Paris, P. Rigollet, A. J. Stromme, “Fast convergence of empirical barycenters in Alexandrov spaces and
the Wasserstein space”, https://arxiv.org/abs/1908.00828, 2019.
[7] S.-I. Ohta, “Barycenters in Alexandrov spaces of curvature bounded below”, Adv. Geom. 12 (2012), no. 4, p. 571-587.
[8] K.-T. Sturm, “Metric spaces of lower bounded curvature”, Expo. Math. 17 (1999), no. 1, p. 35-47.
[9] T. Yokota, “A rigidity theorem in Alexandrov spaces with lower curvature bound”, Math. Ann. 353 (2012), no. 2, p. 305-
331.
C. R. Mathématique, 2020, 358, n 4, 489-495


A note on flatness of non separable tangent cone at a barycenter

Fritz Hörmann

Descent for coherent sheaves along formal/open coverings

Rodrigo A. Freire

An axiomatic approach to forcing and generic extensions

Makoto Sakagaito

Comptes Rendus
Mathématique
Makoto Sakagaito
A note on Gersten’s conjecture for étale cohomology over
two-dimensional henselian regular local rings
Volume 358, issue 1 (2020), p. 33-39.
<https://doi.org/10.5802/crmath.9>
© Académie des sciences, Paris and the authors, 2020.
Some rights reserved.
This article is licensed under the
CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2020, 358, n 1, p. 33-39
https://doi.org/10.5802/crmath.9
Number Theory, Homological Algebra / Théorie des nombres, Algèbre homologique
A note on Gersten’s conjecture for étale
cohomology over two-dimensional henselian
regular local rings
Une note sur la conjecture de Gersten pour la
cohomologie étale sur des anneaux locaux réguliers
henséliens à deux dimensions
Makoto Sakagaitoa
a Indian Institute of Science Education and Research, Mohali.
E-mail: makoto@iiserb.ac.in.
Abstract. We prove Gersten’s conjecture for étale cohomology over two dimensional henselian regular local
rings without assuming equi-characteristic. As an application, we obtain the local-global principle for Galois
cohomology over mixed characteristic two-dimensional henselian local rings.
Résumé. Nous montrons la conjecture de Gersten pour la cohomologie étale sur des anneaux locaux régu-
liers henséliens sans supposer de caractère équicaractéristique. En application, nous obtenons le principe
local-global pour la cohomologie de Galois sur des anneaux locaux henséliens à deux dimensions de carac-
téristique mixte.
Manuscript received 30th April 2019, revised 17th September 2019 and 24th November 2019, accepted 17th De-
cember 2019.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
34 Makoto Sakagaito
1. Introduction
Let R be an equi-characteristic regular local ring, k(R) the field of fractions of R, l a positive
integer which is invertible in R and µl the étale sheaf of l -th roots of unity. Then the sequence
of étale cohomology groups
0 −→ Hn+1ét
(
R,µ⊗nl
)−→Hn+1ét (k(R),µ⊗nl ) (1)
−→ ⊕
p∈SpecR
ht(p)=1
Hnét
(
κ(p),µ⊗(n−1)l
)
−→ ⊕
p∈SpecR
ht(p)=2
Hn−1ét
(
κ(p),µ⊗(n−2)l
)
−→ ·· ·
is exact by Bloch–Ogus ([2]) and Panin ([10]). Here κ(p) is the residue field of p ∈ SpecR.
By using the exactness of the complex (1) at the first two terms, Harbater–Hartmann–Krashen
([7]) and Hu ([8]) proved the local-global principle as follows.
Let K be a field of one of the following types:
(a) (semi-global case) The function field of a connected regular projective curve over the
field of fractions of a henselian excellent discrete valuation ring A.
(b) (local case) The function field of a two-dimensional henselian excellent normal local
domain A.
Then the following question was raised by Colliot-Thélène ([3]):
Let n ≥ 1 be an integer and l a positive integer which is invertible in R. Is the natural map
Hn+1ét
(
K ,µ⊗nl
)−→ ∏
v∈ΩK
Hn+1ét
(
Kv ,µ
⊗n
l
)
(2)
injective ?
Here ΩK is the set of normalized discrete valuations on K and Kv is the corresponding
henselization of K for each v ∈ΩK .
Suppose that A is equi-characteristic. Harbater–Hartmann–Krashen ([7, Theorem 3.3.6])
proved that the local-global map (2) is injective in the semi-global case. Later, Hu ([8, Theo-
rem 2.5]) proved that the local-global map (2) is injective in both the semi-global case and the
local case by an alternative method.
If the sequence (1) is exact (at the first two terms) in the case where R is a mixed characteristic
two-dimensional excellent henselian local ring, then the local-global map (2) is injective even
without assuming equi-characteristic (cf. [7, Remark 3.3.7] and [8, Remark 2.6(2)]).
In the case where R is a local ring of a smooth algebra over a (mixed characteristic) discrete
valuation ring, the sequence (1) is exact (cf. [6, Theorem 1.2 and Theorem 3.2b)]).
In this paper, we show the following result:
Theorem 1 (Theorem 9). Let R be a mixed characteristic two-dimensional excellent henselian
local ring and l a positive integer which is invertible in R. Then Gersten’s conjecture for étale
cohomology with µ⊗nl coefficients holds over SpecR. That is, the sequence (1) is exact.
See Remark 8(iii) for the reason why we assume dim(R) = 2 in Theorem 1. We obtain the
following result as an application of Theorem 1 :
Theorem 2. With notations as above, assume that A is mixed characteristic and l is a positive
integer which is invertible in A.
In both the semi-global case and the local case, the local-global principle for the Galois coho-
mology group Hn+1(K ,µ⊗nl ) holds for n ≥ 1. That is, the local-global map (2) is injective for n ≥ 1.
V. Suresh also proved Theorem 2 by an alternative method (cf. [8, Remark in Theorem 1.2]).
C. R. Mathématique, 2020, 358, n 1, 33-39
Makoto Sakagaito 35
1.1. Notations
For a scheme X , X (i ) is the set of points of codimension i , k(X ) is the ring of rational functions on
X and κ(p) is the residue field of p ∈ X . If X = SpecR, k(SpecR) is abbreviated as k(R). The symbol
µl denotes the étale sheaf of l -th roots of unity.
2. Proof of the main result (Theorem 1)
In this section, we use the following results (Theorem 3 and Theorem 4) repeatedly:
Theorem 3 (cf. [4, Theorem B.2.1 and Examples B.1.1.(2)]). Let A be a discrete valuation ring, K
the function field of A and l a positive integer which is invertible in A. Then the homomorphism
Hiét(A,µ
⊗n
l ) −→ Hiét(K ,µ⊗nl )
is injective for any i ≥ 0.
Theorem 4 (The absolute purity theorem [5, p. 159, Theorem 2.1.1]). Let Y
i
,→ X be a closed
immersion of noetherian regular schemes of pure codimension c. Let n be an integer which is
invertible on X , and let Λ= Z/n. Then the cycle class (cf. [5, 1.1]) give an isomorphism
ΛY
∼−→ Ri !Λ(c)[2c]
in D+(Yét,Λ). Here D+(Yét,Λ) is the derived category of complexes bounded below of étale sheaves
of Λ-modules on Y .
In this section, we use Theorem 4 in the case where dim X ≤ 2. In this case, Theorem 4 was
proved much earlier by Gabber in 1976. See also [11, §5, Remark 5.6] for a published proof.
Proposition 5. Let R be a henselian regular local ring,m the maximal ideal of R and K the function
field of R. Let l be a positive integer such that l ∉m. Then the homomorphism
Hiét
(
SpecR,µ⊗nl
)−→ Hiét (SpecK ,µ⊗nl ) (3)
is injective for any i ≥ 0.
Proof. We prove the statement by induction on dim(R). Let R be a discrete valuation ring (which
does not need to be henselian). Then the homomorphism (3) is injective by Theorem 3.
Assume that the statement is true for a henselian regular local ring of dimension d .
Let R be a henselian regular local ring of dimension d +1, a ∈m\m2 and p= (a). Then R/p is a
henselian regular local ring of dimension d and
k(R/p) = Rp/pRp
where k(R/p) is the function field of R/p.
Therefore the diagram
Hiét
(
SpecR,µ⊗nl
) //

Hiét
(
SpecRp,µ⊗nl
)

Hiét
(
SpecR/p,µ⊗nl
) // Hiét (Speck(R/p),µ⊗nl )
(4)
is commutative. Then the left vertical map in the diagram (4) is an isomorphism by [1, p. 93,
Theorem (4.9)] and the bottom horizontal map in the diagram (4) is injective by the induction
hypothesis. Hence the homomorphism
Hiét
(
SpecR,µ⊗nl
)−→ Hiét (SpecRp,µ⊗nl )
C. R. Mathématique, 2020, 358, n 1, 33-39
36 Makoto Sakagaito
is injective. Moreover the homomorphism
Hiét
(
SpecRp,µ
⊗n
l
)−→ Hiét (SpecK ,µ⊗nl )
is injective by Theorem 3. Therefore the statement follows. 
Proposition 6 (cf. [12, Proposition 4.7]). Let R be a regular local ring and l a positive integer which
is invertible in R. Suppose that dim(R) = 2. Then the sequence
Hiét(R,µ
⊗n
l ) −→ Hiét(k(R),µ⊗nl )
(∗)−−→ ⊕
p∈(SpecR)(1)
Hi−1ét (κ(p),µ
⊗(n−1)
l )
is exact for any i ≥ 0.
Proof. Let A be a Dedekind ring, q a maximal ideal of A. Then
Hi+1q ((Spec A)ét,µ
⊗n
l ) = Hi−1ét (κ(q),µ⊗(n−1)l )
by Theorem 4. Hence the sequence
Hiét(A,µ
⊗n
l ) −→ Hiét(U ,µ⊗nl ) −→
⊕
q∈Z (1)
Hi−1ét (κ(q),µ
⊗(n−1)
l )
is exact where Z is a closed subscheme of Spec A and U = SpecR \ Z . Since
lim→
U
Hiét(U ,µ
⊗n
l ) = Hiét(k(A),µ⊗nl )
by [9, pp. 88–89, III, Lemma 1.16], the sequence
Hiét(A,µ
⊗n
l ) −→ Hiét(k(A),µ⊗nl ) −→
⊕
q∈(Spec A)(1)
Hi−1ét (κ(q),µ
⊗(n−1)
l ) (5)
is exact.
Let m be the maximal ideal of R. Let g ∈m\m2, p= (g ) and Z = SpecR/p. Then R/p is a regular
local ring and we have
Hi+1Z ((SpecR)ét,µ
⊗n
l ) = Hi−1ét (R/p,µ⊗(n−1)l )
by Theorem 4.
We consider the commutative diagram
Hiét(R,µ
⊗n
l )
//

Hiét(Rg ,µ
⊗n
l )
//

Hi−1ét (R/p,µ
⊗(n−1)
l )
′

// 0
0 // Ker(∗) // Hiét(Rg ,µ⊗nl )′ // Hi−1ét (k(R/p),µ⊗(n−1)l )
(6)
where
Hi−1ét (R/p,µ
⊗(n−1)
l )
′ = Im
(
Hiét(Rg ,µ
⊗n
l ) −→ Hi−1ét (R/p,µ⊗(n−1)l )
)
and
Hiét(Rg ,µ
⊗n
l )
′ = Ker
Hiét(k(Rg ),µ⊗nl ) −→ ⊕
q∈(SpecRg )(1)
Hi−1ét (κ(q)),µ
⊗(n−1)
l )
 .
Then the rows in the diagram (6) are exact by Theorem 4. Since Rg is a Dedekind domain, the
middle map in the diagram (6) is surjective by (5). Moreover, since
Hi−1ét (R/p,µ
⊗(n−1)
l )
′ ⊂ Hi−1ét (R/p,µ⊗(n−1)l )
and R/p is a discrete valuation ring, the right map in the diagram (6) is injective by Theorem 3.
Therefore the statement follows from the snake lemma. 
C. R. Mathématique, 2020, 358, n 1, 33-39
Makoto Sakagaito 37
Corollary 7. Let R be the henselization of a regular local ring which is essentially of finite type over
a mixed characteristic discrete valuation ring. Suppose that dim(R) = 2. Then
Hn+1Zar (R,Z/l (n)) = 0
for a positive integer l which is invertible in R. Here Z(n) is Bloch’s cycle complex and Z/l (n) =
Z(n)⊗Z/l (cf. [6, p. 779]).
Proof. Let m be the maximal ideal of R. Let g ∈m\m2 and p= (g ). Then the homomorphism
Hn+1ét (R,µ
⊗n
l ) −→ Hn+1ét (Rg ,µ⊗nl )
is injective by Proposition 5. Hence the homomorphism
Hnét(Rg ,µ
⊗n
l ) −→ Hn−1ét (R/p,µ⊗n−1l )
is surjective by Theorem 4. Therefore the homomorphism
HnZar(Rg ,Z/l (n)) −→ Hn−1Zar (R/p,Z/l (n −1))
is surjective by [6, p. 774, Theorem 1.2] and [14]. Moreover the homomorphism
Hn+1Zar (R,Z/l (n)) −→ Hn+1Zar (Rg ,Z/l (n))
is injective by the localization theorem [6, p. 779, Theorem 3.2]. We consider the commutative
diagram
Hn+1Zar (Rg ,Z/l (n)) //

Hn+1ét (Rg ,Z/l (n))

Hn+1Zar (k(Rg ),Z/l (n)) // H
n+1
ét (k(Rg ),Z/l (n)).
(7)
Then the upper map in the commutative diagram (7) is injective by the Beilinson–Lichenbaum
conjecture ([6, p. 774, Theorem 1.2], [14]) and the right map in the commutative diagram (7) is in-
jective by the commutative diagram (6) in the proof of Proposition 6. Hence the homomorphism
Hn+1Zar (Rg ,Z/l (n)) −→ Hn+1Zar (k(Rg ),Z/l (n))
is injective and the homomorphism
Hn+1Zar (R,Z/l (n)) −→ Hn+1Zar (k(Rg ),Z/l (n))
is also injective. Since
Hn+1Zar (k(Rg ),Z/l (n)) = 0,
we have
Hn+1Zar (R,Z/l (n)) = 0.
This completes the proof. 
Remark 8.
(i) If R is a local ring of a smooth algebra over a discrete valuation ring, then
HiZar(R,Z/m(n)) = 0
for i > n and any positive integer m (cf. [6, p. 786, Corollary 4.4]).
(ii) If we have
Hn+1Zar (R,Z/l (n)) = 0
for any regular local ring R which is finite type over a discrete valuation ring and a positive
integer l which is invertible in R, we can show that the homomorphism
Hn+1ét (R,µ
⊗n
l ) −→ Hn+1ét (k(R),µ⊗nl )
is injective by a similar argument as in the proof of [13, Theorem 4.2].
C. R. Mathématique, 2020, 358, n 1, 33-39
38 Makoto Sakagaito
(iii) The reason why we assume dim(R) = 2 in Propositin 6 and Theorem 9 is that we have to
show that the middle map in the diagram (6), i.e., the homomorphism
Hn+1ét (Rg ,µ
⊗n
l ) −→ Hn+1ét (Rg ,µ⊗nl )′
is surjective for an element g of m\m2. Here m is the maximal ideal of R and
Hn+1ét (Rg ,µ
⊗n
l )
′ = Ker
Hn+1ét (k(R),µ⊗nl ) −→ ⊕
q∈(SpecRg )(1)
Hnét(κ(q),µ
⊗(n−1)
l )
 .
If we have
Hn+1Zar (R,Z/l (n)) = Hn+2Zar (R,Z/l (n)) = 0
for any regular local ring R which is finite type over a discrete valuation ring and a positive
integer l which is invertible in R, then
Hn+1Zar (Rg ,Z/l (n)) = Hn+2Zar (Rg ,Z/l (n)) = 0
by the localization theorem ([6, p. 779, Theorem 3.2]) and we can show that
Hn+1ét (Rg ,µ
⊗n
l ) = Γ(SpecRg ,Rn+1ε∗(µ⊗nl )) = Hn+1ét (Rg ,µ⊗nl )′
and Proposition 6 holds. Here ε : (SpecRg )ét → (SpecRg )Zar is the change of site maps.
Theorem 9. Let R be a henselian regular local ring with dim(R) = 2 and l a positive integer which
is invertible in R. Then the sequence
0 −→ Hiét
(
R,µ⊗nl
)−→ Hiét (k(R),µ⊗nl )−→ ⊕
p∈(SpecR)(1)
Hi−1ét
(
κ(p),µ⊗(n−1)l
)
−→ ⊕
p∈(SpecR)(2)
Hi−2ét
(
κ(p),µ⊗(n−2)l
)
−→ 0 (8)
is exact for any i ≥ 0.
Proof. The exactness of the complex (8) at the first two terms follows from Proposition 5 and
Proposition 6.
We consider the coniveau spectral sequence
Hp,q1 =
∐
x∈(SpecR)(p)
Hp+qx
(
SpecR,µ⊗nl
)⇒ Hp+qét (R,µ⊗nl )= Hp+q
(cf. [4, §1]). Then we have a filtration
0 ⊂ Hp+qp+q ⊂ ·· · ⊂ Hp+q1 ⊂ H
p+q
0 = Hp+q ,
such that
Hp+qp /H
p+q
p+1 ' H
p,q
∞ .
By Theorem 4, it suffices to show that
H1,i−12 = H2,i−22 = 0.
By Proposition 5, the morphism
Hi −→ H0,i∞
is injective and
Hi1 = Hi2 = 0.
Hence we have
H1,i−1∞ = H2,i−2∞ = 0.
Since
Hp,i−p+1r = 0
C. R. Mathématique, 2020, 358, n 1, 33-39
Makoto Sakagaito 39
for p ≥ 3 and
H1−r,i+r−2r = 0
for r ≥ 2, we have
H1,i−12 = H1,i−1∞ = 0.
By the exactness of the complex (8) at the second term, we have
H0,i−12 = H0,i−1∞ = Hi−1
and
Im
(
H0,i−12
d0,i−12−−−−→ H2,i−22
)
= 0.
Hence we have
H2,i−22 = H2,i−23 .
Moreover, since
H2−r,i+r−3r = 0
for r ≥ 3, we have
H2,i−2r+1 =
Ker(d2,i−2r )
Im(d2−r,i+r−3r )
= H2,i−2r
for r ≥ 3. Therefore
H2,i−22 = H2,i−23 = H2,i−2∞ = 0.
This completes the proof. 
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C. R. Mathématique, 2020, 358, n 1, 33-39


A note on Gersten’s conjecture for étale cohomology over two-dimensional henselian regular local rings

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