Comptes Rendus Mathématique

Timo Keller

Michael Stoll

Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces

Yu Wang

Umamaheswaran Arunachalam

Bernhard Keller

Comptes Rendus
Mathématique
Yu Wang, Umamaheswaran Arunachalam and Bernhard Keller
On the Hochschild homology of singularity categories
Volume 360 (2022), p. 491-496
<https://doi.org/10.5802/crmath.318>
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
2022, Vol. 360, p. 491-496
https://doi.org/10.5802/crmath.318
Algebra, Combinatorics / Algèbre, Combinatoire
On the Hochschild homology of singularity
categories
YuWanga, b, Umamaheswaran Arunachalamc and Bernhard Keller∗, d
a Department of Mathematics, Nanjing University, Nanjing 210093, PR China
b Université Paris Cité, UFR de mathématiques, CNRS IMJ–PRG, 8 place Aurélie
Nemours, 75013 Paris, France
c Department of Mathematics, National Institute of Technology (NIT) Warangal,
Warangal, Telangana, 506004, India
URL: https://sites.google.com/site/ruthreswaranuma/home
d Université Paris Cité, Sorbonne Université, CNRS IMJ–PRG 8 place Aurélie
Nemours, 75013 Paris, France
URL: https://webusers.imj-prg.fr/~bernhard.keller/
E-mails: dg1621017@smail.nju.edu.cn (Y. Wang), yu.wang@imj-prg.fr,
ruthreswaran@gmail.com (U. Arunachalam), bernhard.keller@imj-prg.fr (B. Keller)
Abstract. Let k be an algebraically closed field and A a finite-dimensional k-algebra. In this note, we deter-
mine complexes which compute the Hochschild homology of the canonical dg enhancement of the bounded
derived category of A and of the canonical dg enhancement of the singularity category of A. As an application,
we obtain a new approach to the computation of Hochschild homology of Leavitt path algebras.
2020 Mathematics Subject Classification. 16E35, 16E40, 16E45, 18G80.
Funding. The second author thanks the Abel visiting scholar program for financing a visit to the third
author in September 2019 during which this collaboration started. The third author thanks the Sherbrooke
representation theory team for organizing the VirtARTA 2020, where the material of this note was first
presented.
Manuscript received 9 October 2021, revised and accepted 21 December 2021.
1. Reminder on Hochschild homology of algebras and categories
Let k be a field. We write ⊗ for ⊗k . Let A be a k-algebra (associative, with 1). We write Mod A for
the category of all (right) A-modules and D A =D(Mod A) for its unbounded derived category. Let
Ae = A ⊗ Aop be the enveloping algebra of A so that Ae -modules identify with A-bimodules. The
Hochschild homology of A is defined by
H Hp (A) = TorA
e
p (A, A) , p ∈Z.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
492 Yu Wang, Umamaheswaran Arunachalam and Bernhard Keller
Alternatively, we may define it as the pth homology group of the Hochschild chain complex
H H(A) of A, i.e. the complex C∗A concentrated in homological degrees ≥ 0
A A⊗ A . . . A⊗p A⊗(p+1) . . .
with Cp A = A⊗(p+1), p ≥ 0, and differential given by
d
(
a0, . . . , ap
)= p−1∑
i=0
(−1)i (a0, . . . , ai ai+1, . . . , ap)+ (−1)p (ap a0, . . . , ap−1) , (1)
where we write (a0, . . . , ap ) for a0 ⊗ ·· · ⊗ ap . Notice that the first differential takes a ⊗ b to the
commutator ab −ba.
We see that H H0(A) is the quotient A/[A, A] of the vector space A by its subspace generated
by all commutators and that H Hp (A) and H H(A) ∈ Dk are functorial in the algebra A. The def-
initions extend from k-algebras to small k-categories A . For example, the Hochschild complex
then becomes the complex⊕
A
(
X0, X0
) ⊕
A
(
X1, X0
)⊗A (X0, X1)oo . . .oo
whose pth term (p ≥ 0) is the sum⊕
A
(
Xp , X0
)⊗A (Xp−1, Xp)⊗·· ·⊗A (X0, X1)
taken over all sequences of objects X0, X1, . . . , Xp of A and whose horizontal differential is given
by formula (1). One then shows that the inclusion A → proj(A) of the one-object category given
by A into the category proj(A) of finitely generated projective right A-modules induces a quasi-
isomorphism
H H(A) ∼−→ H H (proj A) .
In particular, this yields Morita invariance of Hochschild homology. The definitions further ex-
tend to small differential graded (=dg) categories A , for example the dg category C bd g (proj A) of
bounded complexes over proj(A). We refer the reader to [10] for more information on this exam-
ple and dg categories in general. The inclusion proj(A) →C bd g (proj A) yields an isomorphism
H H
(
proj A
) ∼−→ H H (C bd g (proj A))
and this yields the invariance of Hochschild homology under derived equivalences. We will need
the following localization theorem.
Theorem 1 ([9]). Let
A
F // B
G // C
be a sequence of dg categories such that the induced sequence of derived categories
0 // DA
F∗ // DB
G∗ // DC // 0
is exact. Then there is a canonical triangle
H H(A )
H H(F ) // H H(B)
H H(G) // H H(C ) // ΣH H(A )
in Dk and hence long exact sequences in Hochschild (and cyclic) homology.
Let Q be a finite quiver and I an admissible ideal in kQ, i.e. a two-sided ideal contained in the
square of the ideal generated by the arrows and such that the quotient kQ/I is finite-dimensional.
Let R be the quotient of A by its radical. Thus, as an A-module, the algebra R is the direct sum of
the simple A-modules. Following [8], we define the Koszul dual of A to be the dg algebra
A! = RHomA (R,R) .
Yu Wang, Umamaheswaran Arunachalam and Bernhard Keller 493
Thus, if P is a projective resolution of the A-module R, then the Koszul dual is quasi-isomorphic
to the dg endomorphism algebra HomA(P,P ) of P . The following theorem is a special case of
Corollary D.2 of Van den Bergh’s [2]. We write D for the dual Homk (?,k) over the ground field.
Theorem 2 (Van den Bergh). We have a canonical isomorphism
H H
(
A!
) ∼−→ D H H(A).
We refer to [7] for a comparison taking into account much more structure.
2. Hochschild homology of derived categories and singularity categories
Let Q be a finite quiver and I an admissible ideal in kQ. Let mod A be the category of k-finite-
dimensional right A-modules. Denote by Db(A) = Db(mod A) the bounded derived category of
A and by per(A) the perfect derived category, i.e. the thick subcategory generated by the free A-
module of rank 1. Following Buchweitz [3] and Orlov [11], one defines the singularity category of
A as the Verdier quotient
sg(A) =Db(A)/per(A).
Using the canonical dg enhancements of Db(A) and per(A), cf. [10], we obtain a canonical exact
sequence of dg categories
0 // perd g (A) // D
b
d g (A)
// sgd g (A) // 0 ,
where the dg quotient sgd g (A) yields a canonical dg enhancement for sg(A). It is not hard to
see that, in the homotopy category of dg categories, it is functorial with respect to bimodule
complexes X ∈ D(Aop ⊗B) such that XB is perfect over B and A X is perfect over A. From the
localization Theorem 1, we deduce a triangle
H H
(
perd g (A)
)
// H H
(
Dbd g (A)
)
// H H
(
sgd g (A)
)
// ΣH H
(
perd g (A)
)
(2)
in the derived category of vector spaces.
Theorem 3. We have a canonical isomorphism H H(Dbd g (A))
∼−→ D H H(A).
Proof. Recall that we have defined R to be the quotient of A by its radical and the Koszul dual
A! as RHomA(R,R). Since the module R is a classical generator of the bounded derived category
Db(A), we deduce from the results of [8] that we have a triangle equivalence
RHomA(R, ?) : Db(A)
∼ // per
(
A!
)
.
This lifts to a quasi-equivalence
Dbd g (A)
∼−→ perd g
(
A!
)
.
By Morita invariance of Hochschild homology, we have
H H
(
A!
) ∼−→ H H (perd g (A!)) .
By Van den Bergh’s Theorem 2, we have
H H
(
A!
) ∼−→ D H H(A).
The claim follows if we combine these isomorphisms. 
Define a linear map τ : A → D A by sending an element a ∈ A to the linear form which takes
b ∈ A to the trace of the linear map
λaρb : A → A , x 7→ axb ,
494 Yu Wang, Umamaheswaran Arunachalam and Bernhard Keller
where λa is left multiplication by a and ρb right multiplication by b. Notice that since A is finite-
dimensional, this is well-defined. Moreover, the value of 〈a,b〉 = (τ(a))(b) only depends on the
classes of a and b in H H0(A), which is canonically isomorphic to R. It is not hard to check that in
the basis formed by the ei , the matrix of the induced bilinear form
H H0(A)×H H0(A) → k
is the Cartan matrix of A, whose (i , j )-entry is the dimension of ei Ae j . Define the double
Hochschild complex of A to be the complex
. . .
b // A⊗ A b // A τ // D A Db // D(A⊗ A) Db // . . . ,
where D A sits in degree 0, the differentials b are those of the Hochschild complex and the Db
their duals.
Let us abbreviate S = sgd g (A).
Theorem 4. In Dk, we have a canonical isomorphism between H H(S ) and the double
Hochschild complex of A.
Notice that this implies in particular that H Hn(S ) is finite-dimensional for all n. This is sur-
prising since the singularity category sg(A) is usually not Hom-finite (except if A is Gorenstein),
cf. for example [4].
Proof. We use the triangle
H H
(
perd g (A)
)
// H H
(
Dbd g (A)
)
// H H(S ) // ΣH H
(
perd g (A)
)
obtained from the localization Theorem 1. We have already seen that it is isomorphic to a triangle
H H(A) → H H (A!)→ H H(S ) →ΣH H(A) ,
where the first morphism is induced by the inclusion perd g (A) → Dbd g (A). Thus, the complex
H H(S ) identifies with the mapping cone over the morphism H H(A) → H H(A!). Let us deter-
mine this morphism explicitly. Recall that the functor H H , considered as a functor on the ho-
motopy category of small dg categories with values in the derived category Dk, commutes with
tensor products. We have the following commutative square
perd g (A
op )⊗perd g (A) perd g (k)
perd g (A)
op ⊗Dbd g (A) perd g (k)
Here, a pair (P1,P2), P1 ∈ proj(Aop ), P2 ∈ proj(A) is taken to P2 ⊗A P1 by the top arrow and
to (HomA(P1, A),P2) by the left vertical arrow. It follows from Appendix D in [2] that the lower
horizontal arrow induces a non degenerate pairing
H H(A)⊗H H
(
Dbd g (A)
)
→ H H(k) = k.
A direct computation now shows that the morphism
H H(A) → D H H(A)
is the composition
H H(A) → H H0(A) → D H H0(A) → D H H(A)
where the middle morphism is induced by the map τ. 
Yu Wang, Umamaheswaran Arunachalam and Bernhard Keller 495
Corollary 5. For n ≥ 2, we have canonical isomorphisms
H Hn(S )
∼−→ H Hn−1(A) ∼−→ D H H1−n(S ).
Moreover, we have
H H1(S )
∼−→ ker
(
H H0(A)
τ→ D H H0(A)
)
∼−→ D H H0(S ).
3. Application: Hochschild homology of dg Leavitt path algebras
Let Q be a finite quiver, for example a quiver with one vertex and a unique loop α. Let A be the
associated radical square zero algebra, i.e. the quotient of kQ by the square of the ideal generated
by the arrows. So for the one-loop quiver, we have A = k[ε]/(ε2). Let Q∗ be the graded quiver
obtained from the opposite quiver of Q by assigning each arrow α∗ : j → i corresponding to an
arrow α : i → j of Q the degree +1. For each vertex i of Q, consider the arrows α∗s : i → t (α∗s ),
1 ≤ s ≤ ti , starting in Q∗ at i . Let
ϕi : Pi →
ti⊕
s=1
ΣPt(α∗s )
be the morphism with components α∗s , where Pi = ei kQ∗. For example, for the one-loop quiver,
we just have ϕ(1) =α∗ : P1 →ΣP1. Note that if i is a sink of Q, then
ti⊕
s=1
Pt(α∗s ) = 0.
For each vertex i ∈Q0, let
ϕ(i )−1 = [βi ,1, . . . , βi , ti ] : ti⊕
s=1
ΣPt(α∗s ) → Pi
be the formal inverse of ϕ(i ). The graded Leavitt path algebra of Q is obtained from kQ∗ by
adjoining all coefficients βi j of all formal inverses ϕ(i )−1, i ∈ Q0. We endow LQ with the grading
inherited from Q∗ and with d = 0.
Theorem 6 (Smith [12], Chen–Yang [6]). We have a triangle equivalence per(LQ )
∼−→ sg(A) taking
ei LQ to the simple Si .
Corollary 7. The Hochschild homology H H∗(LQ ) of the Leavitt path algebra is computed by the
double Hochschild complex
. . .
b // A⊗ A b // A τ // D A Db // D(A⊗ A) Db // . . . ,
(with D A in degree 0). In particular, we have
dim H Hp
(
LQ
)= 0 <∞
for all p ∈Z.
A different description of the Hochschild homology of Leavitt path algebras is due to Ara–
Cortiñas [1].
496 Yu Wang, Umamaheswaran Arunachalam and Bernhard Keller
4. Beyond radical square zero
Let Q be a finite quiver and A = kQ/I the quotient of its path algebra by an admissible ideal. Let J
be the radical of A and R = kQ0 so that we have A = R ⊕ J as R-bimodules. Let A0 = (TR J )/(J ⊗R J )
be the radical square zero algebra associated with A. Thus, we have A0 = R⊕J = A as R-bimodules
but we have x y = 0 in A0 for any two elements of J . We view A0 as a degeneration of A and
A as a deformation of A0. As pointed out by Chen–Wang [5], this suggests that the singularity
category sg(A) is a deformation of the singularity category sg(A0), which is equivalent to the
perfect derived category per(L A0 ) of the graded Leavitt path algebra L A0 . Hence we can hope for
the existence of a dg algebra L A obtained from L A0 by deformation such that per(L A) is equivalent
to sg(A). We sum up the situation in the following diagram
A0 Adeformation
sg(A0) sg(A)
per
(
L A0
)
per(L A)
deformation
∼ ∼
deformation?
L A0 L Adeformation? ?
?
The following theorem confirms this hope.
Theorem 8 (Chen–Wang [5]). The graded algebra L A0 admits a canonical differential dA such
that for L A = (L A0 ,dA), we have a triangle equivalence
per(L A)
∼−→ sg(A).
Corollary 9. The Hochschild homology of the dg Leavitt path algebra L A is computed by the double
Hochschild complex of A.
References
[1] P. Ara, G. Cortiñas, “Tensor products of Leavitt path algebras”, Proc. Am. Math. Soc. 141 (2013), no. 8, p. 2629-2639.
[2] M. Van den Bergh, “Calabi–Yau algebras and superpotentials”, Sel. Math., New Ser. 21 (2015), no. 2, p. 555-603.
[3] R.-O. Buchweitz, Maximal Cohen–Macaulay modules and Tate cohomology, Mathematical Surveys and Monographs,
vol. 262, American Mathematical Society, 2021, with appendices by Luchezar L. Avramov, Benjamin Briggs, Srikanth
B. Iyengar and Janina C. Letz.
[4] X.-W. Chen, “The singularity category of a quadratic monomial algebra”, Q. J. Math 69 (2018), no. 3, p. 1015-1033.
[5] X.-W. Chen, Z. Wang, “The dg Leavitt algebra, singular Yoneda category and singularity category”, https://arxiv.org/
abs/2109.11278, with an appendix by B. Keller and Yu Wang, 2021.
[6] X.-W. Chen, D. Yang, “Homotopy categories, Leavitt path algebras, and Gorenstein projective modules”, Int. Math.
Res. Not. 2015 (2015), no. 10, p. 2597-2633.
[7] E. Herscovich, “Hochschild (co)homology of Koszul dual pairs”, J. Noncommut. Geom. 13 (2019), no. 1, p. 59-85.
[8] B. Keller, “Deriving DG categories”, Ann. Sci. Éc. Norm. Supér. 27 (1994), no. 1, p. 63-102.
[9] ——— , “Invariance and localization for cyclic homology of DG algebras”, J. Pure Appl. Algebra 123 (1998), no. 1-3,
p. 223-273.
[10] ——— , “On differential graded categories”, in Proceedings of the international congress of mathematicians (ICM),
Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures, European Mathematical Society, 2006, p. 151-190.
[11] D. O. Orlov, “Triangulated categories of singularities and D-branes in Landau–Ginzburg models”, Tr. Mat. Inst.
Steklova 246 (2004), p. 240-262.
[12] S. P. Smith, “Category equivalences involving graded modules over path algebras of quivers”, Adv. Math. 230 (2012),
no. 4-6, p. 1780-1810.


On the Hochschild homology of singularity categories

Mikael de la Salle

Orthogonalization of Positive Operator Valued Measures

Federico Binda

Doosung Park

Paul Arne Østvær

Motives and homotopy theory in logarithmic geometry

Yuchen Ding

Comptes Rendus
Mathématique
Yuchen Ding
A counterexample of two Romanov type conjectures
Volume 360 (2022), p. 1183-1185
https://doi.org/10.5802/crmath.425
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
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2022, Vol. 360, p. 1183-1185
https://doi.org/10.5802/crmath.425
Number theory / Théorie des nombres
A counterexample of two Romanov type
conjectures
Yuchen Ding∗, a
a School of Mathematical Science, Yangzhou University, Yangzhou 225002, People’s
Republic of China
E-mail: ycding@yzu.edu.cn
Abstract. In this note, we disprove two Romanov type conjectures posed by Chen.
Mathematical subject classification (2010). 11A41, 11A67.
Funding. The author is supported by the Natural Science Foundation of Jiangsu Province of China, Grant
No. BK20210784, China Postdoctoral Science Foundation, Grant No. 2022M710121, the foundations of the
projects “Jiangsu Provincial Double–Innovation Doctor Program”, Grant No. JSSCBS20211023 and “Golden
Phoenix of the Green City–Yang Zhou” to excellent PhD, Grant No. YZLYJF2020PHD051.
Manuscript received 21 May 2022, revised 3 August 2022, accepted 19 September 2022.
For any subset A of natural numbers setN, let A(x) = |A∩ [1, x]|. In 2008, Chen communicated
to the authors of [3] and made the following two conjectures therein.
Conjecture 1. Let A and B be two sets of positive integers. If there exists a constant c > 0 such
that A (log x/log2)B(x) > cx for all sufficiently large x, then the set {2a +b : a ∈ A , b ∈ B} has
positive lower asymptotic density.
Conjecture 2. Let A and B be two sets of positive integers. If there exists a constant c > 0 such
that A (log x/log2)B(x) > cx for infinitely many x, then the set {2a +b : a ∈A , b ∈B} has positive
upper asymptotic density.
Romanov’s theorem [6] offers a positive answer to Chen’s conjectures when B is the set
of primes. However, it will not be the case in general. We construct a counterexample which
disproves simultaneously Chen’s conjectures.
Let pi be the i th odd prime and dt = p1p2 · · ·pt for t ∈Z+. For any t ∈Z+, define
Bt = {n : n ∈N, dt |n}∩
[
22
t2
,22
(t+1)2
)
, B =
∞⋃
t=1
Bt
and
A =N, C = 2A +B = {2a +b : a ∈A , b ∈B} .
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1184 Yuchen Ding
Let x be a large number and j be the number such that
22
j 2 ≤ x < 22( j+1)
2
, (1)
which means that √
loglog x < j ≤ 2√
log2
√
loglog x. (2)
From the Chebyshev estimate, we have
d j = exp
( ∑
2<p ≤p j
log p
)
≤ exp(2 j log j )
≤ exp
(
3
√
loglog x logloglog x
)
.
(3)
In view of equation (1), we know 22
( j−1)2 <px. So by the construction of B and equation (3), we
have
B(x) ≥ x −2
2 j
2
d j
+ 2
2 j
2
−22( j−1)
2
d j−1
−2
≥ x −2
2( j−1)2
d j
−2
À x
exp
(
3
√
loglog x logloglog x
) .
(4)
It follows that
A
(
log x/log2
)
B(x) À x log x
exp
(
3
√
loglog x logloglog x
) > x.
It remains to prove C (x) = o(x) as x →∞. It is clear that
C (x) = #{c ≤ x : c = 2a +b, a ∈A ,b ∈B}≤ S1(x)+S2(x), (5)
where
S1(x) = #
{
c ≤ x : c = 2a +b, a ∈A ,b ∈B j
}
and
S2(x) = #
{
c ≤ x : c = 2a +b, a ∈A ,b ∉B j
}
.
Note that if c = 2a +b for some b ∈B j , then pi - c for any 1 ≤ i ≤ j . This fact leads to the following
bound
S1(x) ≤
∑
c ≤x(
c,
∏
p ≤p j p
)
=1
1
= ∑
`|∏p ≤p j pµ(`)
⌊ x
`
⌋
≤ x ∏
p ≤p j
(
1− 1
p
)
+2 j .
(6)
The same observation yields the following estimate
S2(x) ≤ #
{
c ≤ x : c = 2a +b, a ∈A ,b ∈B j−1
}+22( j−1)2 log x
log2
¿ x ∏
p ≤p j−1
(
1− 1
p
)
+px log x +2 j .
(7)
Yuchen Ding 1185
It can be seen that
2 j ¿ exp
(
3
√
loglog x
)
¿px
from equation (2). Therefore, combing equations (5), (6) and (7) we have
C (x) ¿ x ∏
p ≤p j−1
(
1− 1
p
)
+px log x
¿ x (log p j−1)−1 +px log x
¿ x (logloglog x)−1 ,
where the last but one step follows from the Mertens estimate, which is surely more to expectation
than our requirement.
We remark that the Romanov type problems start from the remarkable paper [6], where it is
proved that there is a positive lower asymptotic density of odd numbers which can be represented
by the sum of a prime and a power of 2. In the opposite direction, van der Corput [1] showed
that there is a positive lower asymptotic density of odd numbers none of whose members can
be represented by the sum of a prime and a power of 2. Subsequently, Erdős [2] constructed an
arithmetic progression of odd numbers having the same property required in the paper of van
der Corput. The results of van der Corput and Erdős give a negative answer to an old conjecture
of de Polignac [4, 5].
At present, the author of this note has no answer to the following question. Let B be a set of
positive integers satisfying B(x) =O(x/log x), then is it true that
limsup
x→∞
C (x)
B(x)
=∞,
where
C = 2N+B =
{
2k +b : k ∈N, b ∈B
}
.
Acknowledgments
The author would like to thank the anonymous referee for his/her helpful suggestions which
improved the quality of this note greatly.
References
[1] J. G. van der Corput, “On de Polignac’s conjecture”, Simon Stevin 27 (1950), p. 99-105.
[2] P. Erdős, “On the integers of the form 2k +p and some related problems”, Summa Brasil. Math. 2 (1950), p. 113-123.
[3] H. Pan, H. Li, “The Romanoff theorem revisited”, Acta Arith. 135 (2008), no. 2, p. 137-142.
[4] A. de Polignac, “Recherches nouvelles sur les nombres prEmiers”, C. R. Acad. Sci. Paris 29 (1849), p. 738-739.
[5] ——— , “Six propositions arithmologiques déduites du crible d’Eratosthène”, Nouv. Ann. Math. 8 (1849), p. 423-429.
[6] N. P. Romanov, “Über einige Sätze der additiven Zahlentheorie”, Math. Ann. 109 (1934), p. 668-678.


A counterexample of two Romanov type conjectures

Kenta Higuchi

Résonances Semiclassiques Engendrées par des Croisements de Trajectoires Classiques

Andrei S. Rapinchuk

Igor A. Rapinchuk

Comptes Rendus
Mathématique
Andrei S. Rapinchuk and Igor A. Rapinchuk
Finiteness theorems for algebraic tori over function fields
Volume 359, issue 8 (2021), p. 939-944
<https://doi.org/10.5802/crmath.248>
© Académie des sciences, Paris and the authors, 2021.
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
2021, 359, n 8, p. 939-944
https://doi.org/10.5802/crmath.248
Géométrie algébrique, Théorie des nombres / Algebraic geometry, Number theory
Finiteness theorems for algebraic tori over
function fields
Andrei S. Rapinchuk∗, a and Igor A. Rapinchukb
a Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA
b Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
E-mails: asr3x@virginia.edu, rapinchu@msu.edu
Abstract. We present a number of finiteness results for algebraic tori (and, more generally, for algebraic
groups with toric connected component) over two classes of fields: finitely generated fields and function
fields of algebraic varieties over fields of type (F), as defined by J.-P. Serre.
Résumé. Nous présentons plusieurs résultats de finitude pour les tores (et, plus généralement, pour les
groupes algébriques dont la composante connexe est un tore) définis sur les corps de type fini et les corps
de fonctions des variétés algébriques définies sur les corps satisfaisant la condition (F) de Serre.
Manuscript received 9th March 2021, revised 31st July 2021, accepted 23rd July 2021.
The goal of this note is to formulate a series of finiteness results of a local-global nature for
algebraic tori over two classes of fields that are more general than classical local and global fields.
First, we will consider fields that are finitely over their prime subfields (these will be referred to
simply as finitely generated fields). For our purposes, it is important to point out that any such
field K comes equipped with an almost canonical set V of discrete valuations (places) called
divisorial. More precisely, one can choose a model X for K (i.e., a normal separated irreducible
scheme of finite type overZ or over a finite field such that K is the function field of X), and then V
consists of the discrete valuations corresponding to the prime divisors ofX. We recall that any two
divisorial sets of places of K corresponding to different models are commensurable (cf. [11, 5.3]).
Second, we will also consider function fields K = k(X ) of normal geometrically integral varieties
X over a base field k. Again, K is naturally equipped with an almost canonical set V of discrete
valuations that correspond to the prime divisors of X ; we will call such a set V geometric. For most
of our finiteness results to hold in this set-up, we need to assume that k is of type (F). This notion
goes back to Serre [13, Ch. III, 4.2], who defined a profinite group G to be of type (F) if it satisfies
the following condition:
(F) For every integer n, the group G has only finitely many open subgroups of index n.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
940 Andrei S. Rapinchuk and Igor A. Rapinchuk
Then a field k is said to be of type (F) if its absolute Galois group Gal(ksep/k), where ksep is a
separable closure, is of type (F). (It should pointed out that Serre also requires a field of type (F) to
be perfect, but this requirement is not necessary in our context.) We note that finitely generated
fields of positive characteristic can be treated as function fields over a finite field, which of course
is a field of type (F) (cf. Section 1 below).
1. Tori with good reduction
Let G be a (connected) reductive algebraic group over a field K . Given a discrete valuation v of
K , we let Kv and Ov ⊂ Kv denote the corresponding completion and valuation ring, respectively.
We say that G has good reduction at v if there exists a reductive group scheme G over Ov whose
generic fiber G×Ov Kv is isomorphic to G×K Kv . A recent finiteness conjecture predicts that given
a reductive group G over a finitely generated field K equipped with a divisorial set of places V , the
set of K -isomorphism classes of K -forms that have good reduction at all v ∈ V is finite provided
that the characteristic of K is “good” for the type of G (cf. [11, Conj. 5.7]). For algebraic tori
over finitely generated fields of characteristic zero, this is confirmed by the following Hermite–
Minkowski type theorem.
Theorem 1. Let K be a finitely generated field of characteristic 0, and let V be a divisorial set of
places of K . Then for any integer d ≥ 1, the set of K -isomorphism classes of d-dimensional K -tori
that have good reduction at all v ∈V is finite.
This result essentially reduces the proof of the finiteness conjecture over fields of characteris-
tic zero to semi-simple groups. On the other hand, Theorem 1 is no longer valid in positive char-
acteristic without additional assumptions, as one can see by considering Artin–Schreier exten-
sions already of global fields. The following theorem describes appropriate assumptions in the
more general context of function fields over fields k of type (F). We let p denote the characteristic
exponent of k, i.e. p = 1 if k is of characteristic zero and p = chark otherwise.
Theorem 2. Let K = k(X ) be the function field of a normal geometrically integral variety X defined
over a field k of type (F), and let V be the corresponding geometric set of places of K .
(i) If X is complete, then for each d ≥ 1, the set of K -isomorphism classes of d-dimensional
K -tori that have good reduction at all v ∈V is finite.
(ii) In the general case, for each d ≥ 1, the set of K -isomorphism classes of d-dimensional
K -tori T that have good reduction at all v ∈ V and for which the degree [KT : K ] of the
minimal splitting field is prime p is finite.
Any finitely generated field K of characteristic p > 0 can be realized as the function field k(X )
of a geometrically integral normal variety X over a finite field k. The choice of such a realization
gives rise to a divisorial/geometric set V of discrete valuations of K . If X is chosen to be complete,
the corresponding V is also called complete. Since finite fields are of type (F), we obtain the
following.
Corollary 3. Let K be a finitely generated field of characteristic p > 0, and let V be a divisorial set
of places of K .
(i) If V is complete, then for any d ≥ 1, the set of K -isomorphism classes of d-dimensional
K -tori that have good reduction at all v ∈V is finite.
(ii) In the general case, for any d ≥ 1, the set of K -isomorphism classes of d-dimensional K -tori
T that have good reduction at all v ∈ V and for which the degree [KT : K ] of the minimal
splitting field is prime to p is finite.
C. R. Mathématique — 2021, 359, n 8, 939-944
Andrei S. Rapinchuk and Igor A. Rapinchuk 941
The proofs of Theorems 1 and 2 rely on the information of when the étale fundamental group
of a flat scheme of finite type over Z / a variety, or its maximal prime-to p quotient, is a group of
type (F) (see [5], [7, Exp. IX and X], [8], [12]).
2. Class sets and Condition (T)
Let K be a field equipped with a set V of discrete valuations, and let G be a linear algebraic K -
group with a fixed matrix realization G ⊂ GLn . For each v ∈ V , we set G(Ov ) = G(Kv )∩GLn(Ov )
and then define the corresponding adelic group as
G(A(K ,V )) =
{
(gv ) ∈
∏
v∈V
G(Kv )
∣∣∣∣∣ gv ∈G(Ov ) for almost all v ∈V
}
.
The product G(A∞(K ,V )) = ∏v∈V G(Ov ) is called the subgroup of integral adeles. Let us now
assume that V satisfies the following condition (which holds automatically for all divisorial (resp.,
geometric) sets of places of finitely generated (resp., function) fields):
(∗) For any a ∈ K ×, the set V (a) := {v ∈V |v(a) 6= 0} is finite.
Then the group G(A(K ,V )) does not depend on the choice of a faithful representation G ,→ GLn
(while the group G(A∞(K ,V )) does). Furthermore, in this case there is a diagonal embedding
G(K ) ,→G(A(K ,V )), whose image is called the subgroup of principal adeles and which will still be
denoted simply by G(K ). The set of double cosets
cl(G ,K ,V ) :=G(A∞(K ,V ))\G(A(K ,V ))/G(K )
is called the class set of G . It was proved by A. Borel [1] that for any algebraic group G over a
number field K and the set V of all nonarchimedean places of K , the class set is always finite,
and then its cardinality is called the class number. The reader can find some interpretations and
computations of the class number in this case in [10, Ch. VIII]. Over fields more general than
global fields, the class set may be infinite. For example, for the 1-dimensional split torus G =Gm ,
the class set can be identified with the Picard group Pic(V ), defined as the quotient of the free
abelian group on the set V by the subgroup of “principal divisors”, which can be considered in
view of condition (∗) (cf. [3, §2]). Then it is easy to find examples of finitely generated fields K
and their divisorial sets of places V for which Pic(V ) is infinite, although in this case it is known
to be finitely generated [9]. When G is commutative, the class set has a group structure and is
then called the class group, and one can again ask if/when it is finitely generated; however, in
the general case, no natural group structure can be introduced. So, in [4] we proposed a weaker
finiteness condition (which holds automatically if either the class set is finite or G is commutative
and the class group is finitely generated) that turned out to be instrumental in the analysis of
unramified cohomology.
Condition (T). There exists a finite subset S ⊂V such that |cl(G ,K ,V \ S)| = 1.
We note that in the presence of (∗), the fact whether or not Condition (T) holds for a given
algebraic K -group G does not depend on the choice of its matrix realization. Referring to [11, §6]
for a general discussion of Condition (T), in this note we will limit ourselves to the following
statement that treats the case of groups whose connected component is a torus and will play a
key role in the analysis of the global-to-local map for Galois cohomology in the next section.
Theorem 4. Let K be a finitely generated field and V be a divisorial set of places of K . Then any
algebraic K -group D whose connected component D◦ is a torus satisfies Condition (T).
One reduces the general case to the case where D is a torus, where the required fact is derived
from a result of B. Kahn [9].
C. R. Mathématique — 2021, 359, n 8, 939-944
942 Andrei S. Rapinchuk and Igor A. Rapinchuk
3. Global-to-local map in Galois cohomology: finitely generated fields
Let G be an algebraic group over a field K , and let L/K be a Galois extension. As usual, the 1-
cohomology set H 1(Gal(L/K ),G(L)) will be denoted H 1(L/K ,G). If G is commutative, the groups
H i (L/K ,G) are defined for all i ≥ 1. We will write H i (K ,G) for H i (K sep/K ,G). Suppose now that
K is equipped with a set V of discrete valuations. Then one considers the global-to-local map
λiG ,V ,L/K : H
i (L/K ,G) −→ ∏
v∈V
H i (Lw /Kv ,G), w |v,
which will be denoted λiG ,V when L = K sep. One says that the Hasse principle holds if λiG ,V ,L/K is
injective (this term is particularly used for i = 1, cf. [10, Ch. 6], [13, Ch. III, §4]). This may or may
not be the case, but Borel and Serre [2] showed that when K is a number field and V consists of
almost all places of K , the map λ1G ,V is proper (i.e., has finite fibers) for any algebraic K -group G .
We note that when G is commutative, then the properness of λ1G ,V is equivalent to the finiteness
of the Tate–Shafarevich group X(G ,V ) := kerλ1G ,V . Currently, it is expected that λ1G ,V should be
proper for any reductive group G over a finitely generated field K for any divisorial set V of places
of K (cf. [11, §6]). The results below show that this is indeed the case for algebraic groups whose
connected component is a torus.
Theorem 5. Let K be a finitely generated field and V be a divisorial set of places of K . Then for
any linear algebraic K -group D whose connected component D◦ is a torus and any finite Galois
extension L/K , we have:
(i) if D is commutative, then the map λiD,V ,L/K is proper for any i ≥ 1;
(ii) in the general case, the map λ1D,V ,L/K is proper.
For the proof, one first re-interprets the statement in terms of adelic groups and then uses The-
orem 4 to complete the argument. It should be noted that the standard proof of the finiteness of
the Tate–Shafarevich group for tori over global fields (cf. [15, 11.3]) makes use of Tate–Nakayama
duality, while our argument in this case only requires the finiteness of the class number and the
finite generation of the unit group.
Theorem 6. For K , V , and D as in Theorem 5, the map λ1D,V is proper. In particular, for any K -
torus T , the Tate–Shafarevich group X(T,V ) is finite.
Here one begins by considering the case of a finite group D . This enables one to reduce the
proof to the statement proved in Theorem 5(ii). We also observe that for a semi-simple K -group
G , the truth of the finiteness conjecture for the K -forms of G having good reduction at all v ∈V \S
for an arbitrary finite subset S ⊂V (see Section 1) automatically implies the properness ofλ1
G ,V
for
the corresponding adjoint group G (cf. [11, §6]); for tori, however, these two finiteness properties
do not appear to be related.
As an application, we obtain the following finiteness result on the local-global conjugacy
problem for maximal tori in any reductive group.
Theorem 7. Let G be a connected reductive group over a finitely generated field K , and let V be a
divisorial set of places of K . Fix a maximal K -torus T of G and let C(T ) be the set of all maximal
K -tori T ′ of G such that T and T ′ are G(Kv )-conjugate for all v ∈ V . Then C(T ) consists of finitely
many G(K )-conjugacy classes.
It was pointed out to us by the anonymous referee that over global fields, this result is due to
P. Gille and L. Moret-Bailly [6, Thm. 7.9]; it was used by E. Ullmo and A. Yafaev [14] in their work
on the André–Oort conjecture. Yet another consequence of Theorem 6 is the following.
Theorem 8. Let T be a K -torus, and let X2(T,V ) = kerλ2T,V . Then for any integer ` > 0, the `-
torsion subgroup `X
2(T,V ) is finite.
C. R. Mathématique — 2021, 359, n 8, 939-944
Andrei S. Rapinchuk and Igor A. Rapinchuk 943
4. Global-to-local map in Galois cohomology: function fields
In this section, we take K to be the function field k(X ) of a normal geometrically integral variety
X defined over a field k, and V will be the associated set of geometric places. We also denote by
p the characteristic exponent of k.
Theorem 9. If k has characteristic zero and is of type (F), then for any K -torus T the Tate–
Shafarevich group X(T,V ) is finite.
The proof uses finiteness theorems for étale cohomology and purity results. We also have the
following properness result for finite Galois modules that does not require any assumptions on
the base field k.
Proposition 10. Let Ω be a finite (but not necessarily commutative) étale K -group. Then in each
of the following situations
(1) dim X ≥ 2,
(2) X is a projective curve,
(3) X is an arbitrary curve, but the order of Ω is prime to p,
the map
H 1(K ,Ω) −→ ∏
v∈V
H 1(Kv ,Ω)
is proper.
This proposition has the following consequence in the spirit of Theorem 7.
Proposition 11. Let G be a connected reductive K -group. Fix a maximal K -torus T of K and let
C(T ) be the set of all maximal K -tori T ′ of G such that T and T ′ are G(Kv )-conjugate for all v ∈V .
Then, with the exception of the following case
(•) X is an affine curve and the order of the Weyl group W (G ,T ) is divisible by p,
C(T ) consists of finitely many K -isomorphism classes.
Using Artin–Schreier extensions of the field K = k(x) of rational functions on X = A1k over
an algebraically closed field k of characteristic p > 0, one can construct an infinite family of
pairwise non-isomorphic maximal K -tori of the group G = SLp such that any two of these are
G(Kv )-conjugate for all v in the corresponding geometric set of places V . So, the situation in (•)
is an honest exception in Proposition 11.
Acknowledgements
We are grateful to M. Rapoport for raising questions about general reductive groups, and in
particular, tori with good reduction, and to J.-L. Colliot-Thélène for suggesting to investigate the
properness of the global-to-local map for groups of multiplicative type rather than just tori. We
also thank the anonymous referee for their comments.
References
[1] A. Borel, “Some finiteness properties of adele groups over number fields”, Publ. Math., Inst. Hautes Étud. Sci. 16
(1963), p. 101-126.
[2] A. Borel, J.-P. Serre, “Théorèmes de finitude en cohomologie galoisienne”, Comment. Math. Helv. 39 (1964), p. 111-
164.
[3] V. I. Chernousov, A. S. Rapinchuk, I. A. Rapinchuk, “Spinor groups with good reduction”, Compos. Math. 155 (2019),
no. 3, p. 484-527.
C. R. Mathématique — 2021, 359, n 8, 939-944
944 Andrei S. Rapinchuk and Igor A. Rapinchuk
[4] ——— , “The finiteness of the genus of a finite-dimensional division algebra, and some generalizations”, Isr. J. Math.
236 (2020), no. 2, p. 747-799.
[5] H. Esnault, M. Shusterman, V. Srinivas, “Finite presentation of the tame fundamental groups”, preprint, 2021,
https://arxiv.org/abs/2102.13424.
[6] P. Gille, L. Moret-Bailly, “Actions algébriques de groupes arithmétiques”, in Torsors, étale homotopy and applications
to rational points, London Mathematical Society Lecture Note Series, vol. 405, Cambridge University Press, 2013,
p. 231-249.
[7] A. Grothendieck, M. Raynaud (eds.), Revêtements étales et groupe fondamentale (SGA 1), Lecture Notes in Mathemat-
ics, vol. 224, Springer, 1971.
[8] S. Harada, T. Hiranouchi, “Smallness of fundamental groups of arithmetic schemes”, J. Number Theory 129 (2009),
no. 11, p. 2702-2712.
[9] B. Kahn, “Sur le groupe des classes d’un schéma arithmétique”, Bull. Soc. Math. Fr. 134 (2006), no. 3, p. 395-415.
[10] V. Platonov, A. S. Rapinchuk, Algebraic Groups and Number Theory, Pure and Applied Mathematics, vol. 139,
Academic Press Inc., 1994.
[11] A. S. Rapinchuk, I. A. Rapinchuk, “Linear algebraic groups with good reduction”, Res. Math. Sci. 7 (2020), no. 3, article
no. 28 (65 pages).
[12] M. Raynaud, “Propriétés de finitude du groupe fondamental”, in SGA 7 I, Lecture Notes in Mathematics, vol. 288,
Springer, 1972, p. 25-31.
[13] J.-P. Serre, Galois cohomology, Springer, 1997.
[14] E. Ullmo, A. Yafaev, “Galois orbits and equidistribution of special subvarieties: towards the André–Oort conjecture”,
Ann. Math. 180 (2014), no. 3, p. 823-865.
[15] V. Voskresenskĭı, Algebraic Groups and Their Birational Invariants, Translations of Mathematical Monographs, vol.
179, American Mathematical Society, 1998.
C. R. Mathématique — 2021, 359, n 8, 939-944


Finiteness theorems for algebraic tori over function fields

David Gontier

Spectral properties of periodic systems cut at an angle

Ronan Le Guével

Exponential inequalities for the supremum of some counting processes and their square martingales

Chandranandan Gangopadhyay

Ronnie Sebastian

Nef cones of some Quot schemes on a Smooth Projective Curve

Not a member yet