Séminaire Arithmétique et géométrie algébrique
organisé par l'équipe Arithmétique et géométrie algébrique
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Tess Bouis
Motivic cohomology of mixed characteristic schemes
11 janvier 2024 - 14:00Salle de séminaires IRMA
I will present a new theory of motivic cohomology for general (qcqs) schemes. It is related to non-connective algebraic K-theory via an Atiyah-Hirzebruch spectral sequence. In particular, it is non-A^1-invariant in general, but it recovers classical motivic cohomology on smooth schemes over a Dedekind domain after A^1-localisation. The construction relies on the syntomic cohomology of Bhatt-Morrow-Scholze and the cdh-local motivic cohomology of Bachmann-Elmanto-Morrow, and generalises the construction of Elmanto-Morrow in the case of schemes over a field. -
Enrica Mazzon
A non-archimedean approach to the SYZ conjecture
18 janvier 2024 - 14:00Salle de séminaires IRMA
The SYZ conjecture concerns degenerations of complex Calabi-Yau manifolds and was proposed as a geometric explanation of mirror symmetry. Kontsevich and Soibelman introduced a non-archimedean approach to this conjecture, and more recently, Yang Li's work has connected the non-archimedean approach with the original SYZ conjecture.
In this talk, I will explain the key concepts of the non-archimedean approach and present recent developments in the context of hypersurfaces. This is based on a project in collaboration with Jakob Hultgren, Mattias Jonsson and Nick McCleerey. -
Mirko Mauri
Hodge-to-singular correspondence
25 janvier 2024 - 14:00Salle de séminaires IRMA
We show that the cohomology of moduli spaces of Higgs bundles decomposes in elementary summands depending on the topology of the symplectic singularities on a (fixed!) master object and/or the combinatorics of certain posets and lattice polytopes. This is based on a joint work with Luca Migliorini and Roberto Pagaria. -
Mingmin Shen
A degeneration argument and the integral Hodge conjecture
1 février 2024 - 14:00Salle de séminaires IRMA
An algebraic variety has the special property that its Zariski closure has no boundary in topological sense. I will explain how this can leads to non-algebraicity results via a degeneration argument. Then I will explain how this was recently extended by Kees Kok using refined unramified cohomology. -
Lucas Mann
Duality in p-Adic Cohomology of Rigid Varieties
8 février 2024 - 14:00None
In joint work with Anschütz and Le Bras we study pro-étale Q_p-cohomology on rigid varieties (i.e. "analytic spaces over C_p") and in particular investigate duality results. Examples show that even on proper smooth varieties, Poincaré duality for Q_p-line bundles does not hold in the naive sense and instead requires one to replace the coefficient category by so-called Banach-Colmez spaces. In joint work in progress we construct a general sheaf theory and associated six functors for a "relative" version of BC spaces. This will formally recover many existing duality results for p-adic geometry in the literature. -
Quentin Gazda
Quelques réflexions autour de la conjecture de Zagier — des Fonctions aux Nombres
15 février 2024 - 14:00Salle de séminaires IRMA
La conjecture de Zagier est une certaine formulation du slogan suivant : les relations linéaires entre les polylogarithmes évalués en des nombres algébriques proviennent de relations entre symboles de K-théorie. En identifiant les différents acteurs de cette conjecture, on peut énoncer et démontrer une version analogue en arithmétique des corps de fonctions. Les polylogarithmes classiques sont alors remplacés par ceux de Carlitz. La preuve, très différente des techniques développées jusqu’ici, utilise des ingrédients de la théorie des équations aux différences. Elle fait intervenir des déformations des polylogarithmes de Carlitz où figure une variable t. Dans cet exposé, je présenterai ces résultats, essentiellement fruits d’un travail en commun avec A. Maurischat. S’il on ne dispose pas aujourd’hui de la technologie nécessaires pour reproduire cet argument en théorie des nombres, il est amusant de spéculer sur une hypothétique transcription. Des q-déformations des polylogarithmes remplacent ces «t-déformations». Avec T. Bouis, on a récemment observé un q-Li_1 dans la classe de Chern syntomique introduite par Bhatt—Lurie. Encourageant…! Je mentionnerai ces travaux dans une deuxième partie d’exposé. -
Fangzhou Jin
The limit and boundary characteristic classes in Borel-Moore motivic homology
22 février 2024 - 11:00Salle de séminaires IRMA
Abstract: We define limit and boundary characteristic classes in Borel-Moore motivic homology, and compare them with Aluffi's pro-Chern-Schwartz-MacPherson class and Kato-Saito's Swan class respectively. This is a joint work with P. Sun and E. Yang. -
Marco D'addezio
Cohomologie cristalline bordée
22 février 2024 - 14:00Salle de séminaires IRMA
Je parlerai d'une nouvelle théorie cohomologique pour les variétés algébriques en caractéristique positive, appelée cohomologie cristalline bordée. Il s'agit d'une généralisation de la cohomologie cristalline qui dépend du choix d'une "fonction de décroissance à l'infini". Les fonctions de décroissance linéaires correspondent à des versions entières de la cohomologie rigide, tandis que les fonctions de décroissance logarithmiques produisent la famille conjecturée des théories de cohomologie cristalline à décroissance logarithmique. Au cours de l'exposé, j'expliquerai la construction de cette théorie après un bref rappel des théories classiques de cohomologie cristalline et rigide. -
Annalisa Grossi
Automorphisms of OG10 towards Enriques manifolds
29 février 2024 - 14:00Salle de séminaires IRMA
Automorphisms of HK manifolds have been studied for many different reasons: construct symplectic quotients or study fixed loci in order to find examples of irreducible symplectic varieties, define maps among different deformation families of HK manifolds, find new examples of Enriques manifolds, that are higher dimensional analogue of Enriques surfaces, and for which Pacienza and Sarti recently proved the Morrison-Kawamata cone conjecture. In the first part of the talk, I will show a recent result about symplectic rigidity of HK manifolds of OG10 type. Then I will show how to construct examples of Enriques manifolds considering nonysmplectic automorphisms of a Laza—Saccà—Voisin manifold that are induced by a nonysmplectic automorphism of the underlying cubic fourfold. The talk is based on a joint work with L. Giovenzana, Onorati, and Veniani and on a joint work in progress with Billi, F and L. Giovenzana. -
Johannes Sprang
Irrationality and linear independence of p-adic zeta values
14 mars 2024 - 14:00Salle de séminaires IRMA
The celebrated theorem of Ball-Rivoal shows that there are infinitely many irrational odd zeta values. More precisely, it provides asymptotically lower bounds on the dimension of the Q-vector space spanned by these numbers. Although we are still far from fully understanding the structure of odd zeta values, the corresponding question for p-adic zeta values is even more difficult. For example, the question of the non-vanishing of these numbers is still open today, although it has interesting consequences e.g. for the non-vanishing of p-adic regulators. In this talk we would like to report recent progress on the irrationality and linear independence of p-adic zeta values. -
Vonk Jan
Around the class number one problem
21 mars 2024 - 14:00Salle de séminaires IRMA
As part of a systematic computational study of equivalence classes of binary quadratic forms, Gauss stated several conjectures on their class numbers. In this talk, I will discuss some recent results related to these conjectures. We will begin with the case of negative discriminants, and its intimate connection with the important but challenging problem of finding rational points on modular curves with non-split Cartan level structures. We then turn to the more mysterious case of positive discriminants, where much less is known. The key missing notion is that of singular moduli for real quadratic fields, which is the subject of recent joint work with Darmon. -
Junliang Shen
Cohomology and motives for Hitchin systems
27 mars 2024 - 09:00Salle de séminaires IRMA
In the last a few decades, rich cohomological structures have been found for the topology of the Hitchin system from various perspectives. For example, the work of Ngô reduces the fundamental lemma of the Langlands program to a study of the topology of the Hitchin system; the P=W conjecture (now a theorem) connecting the topology of the Hitchin system to Hodge theory of the character variety via non-abelian Hodge correspondence. I will first discuss some new cohomological structures led by these progress. Then I will explain that tools in these developments further yield a proof of Corti-Hanamura’s motivic decomposition conjecture for the Hitchin system, which states that the decomposition theorem associated with the Hitchin system is induced by algebraic cycles. Based on joint work in progress with Davesh Maulik and Qizheng Yin. -
Yagna Dutta
Twists of intermediate Jacobian fibrations
28 mars 2024 - 14:00Salle de séminaires IRMA
Given an elliptic fibration of a K3 surface, one can reglue the fibres of the elliptic fibration differently to obtain different K3 surfaces. The data of such regluing are dictated either by degree twists or by Brauer twists. A similar story exists for other curves on K3 surfaces. Moving from curves to 3-folds, I will report on a joint work in progress with Mattei and Shinder where we look at the cubic 3-folds obtained as the hyperplane sections of a fixed smooth cubic 4-fold. The relative intermediate Jacobians of the universal hyperplanes induces a well behaved abelian group scheme over the projective space of dimension 5. The total space this time is a hyperKähler manifold of dimension 10. This group scheme gives rise to a Brauer type group associated to the K3-type Hodge structure of the cubic 4-fold. -
Matilde Maccan
Variétés homogènes en petite caractéristique
4 avril 2024 - 14:00Salle de séminaires IRMA
Toute variété homogène, projective et rationnelle peut s’écrire comme quotient d’un groupe algébrique semi-simple par un sous-groupe dit parabolique. Dans cet exposé, on généralisera les résultats de Wenzel, Haboush et Lauritzen en traitant le cas des sous-groupes paraboliques sur un corps algébriquement clos de caractéristique petite (deux et trois), achevant ainsi leur classification en toute caractéristique. Si le temps le permet, on mentionnera quelques consequences géométriques. -
Anna Bot
Real forms and automorphism groups: some open questions
18 avril 2024 - 14:00Salle de séminaires IRMA
A real form of a complex variety X is a real variety whose complexification is isomorphic to X. Many varieties, for example curves or abelian varieties, are known to only have finitely many isomorphism classes of real forms. Recently, there has been some development in the field: now, there are examples of projective varieties (in fact, even surfaces) with infinitely many nonisomorphic real forms, and of affine surfaces with uncountably many. There are still some interesting questions one can pose, especially in connection with the automorphism group of the variety. I will give an overview of the field and relevant notions, and survey some open problems. -
Nirvana Coppola
On perfect powers that are sums of cubes of a nine term arithmetic progression
25 avril 2024 - 14:00Salle de séminaires IRMA
Solving Diophantine equations has always been one of the most fascinating problems in number theory, since even if it can be easily formulated, it almost always requires advanced techniques. In this talk, I will focus on equations that relate sums of powers to perfect powers. After showing some examples that are in the literature, I will discuss the (non-)existence of perfect powers that are sums of cubes of a nine-term arithmetic progression. The proof involves a battery of techniques and both theoretical and computational tools. This is joint work with Mar Curcó-Iranzo, Maleeha Khawaja, Vandita Patel, Özge Ülkem. -
Dustin Clausen
Poincaré duality for p-adic Lie groups
16 mai 2024 - 14:00Salle de séminaires IRMA
Poincaré duality is a familiar phenomenon from real manifold topology, but it shows up in many other settings well. One such is in the theory of (continuous) group cohomology of p-adic Lie groups G. Indeed, Lazard and Serre showed that if G is a compact p-torsionfree p-adic Lie group, then the group cohomology of G with p-power torsion coefficients satisfies Poincaré duality. A subtle part of this theory is in identifying the "dualizing object": the analog to the orientation sheaf in real manifold topology. One wants to know that the dualizing object is controlled by the adjoint representation of G, thereby "linearizing" the problem, analogously to how orientations of a real manifold are controlled by the tangent bundle. Lazard's proof of this involved passing to Q_p-coefficients and using a comparison with Lie algebra cohomology. I will explain a different proof, which works directly in the torsion context. This allows to extend the statement to spectrum coefficients, yielding applications in stable homotopy theory following Beaudry-Goerss-Hopkins-Stojanoska. -
Zhixin Xie
Courants rigides et géométrie birationnelle
23 mai 2024 - 14:00Salle de séminaires IRMA
Un courant rigide est un courant positif fermé dont la classe de cohomologie contient un unique courant positif fermé. Cette notion a été initiée dans le domaine de dynamique complexe et apparaît dans des contextes variés. Dans cet exposé, je présenterai nombreux d’exemples de courants rigides et j'expliquerai comment cette notion intervient naturellement dans l’étude de la conjecture d’abondance en géométrie birationnelle. Il s’agit d’un travail en collaboration avec Vladimir Lazić. -
Margherita Pagano
The wild Brauer-Manin obstruction on K3 surfaces
30 mai 2024 - 14:00Salle de séminaires IRMA
A way to study rational points on a variety is by looking at their image in the p-adic points. Some natural questions that arise are the following: are the rational points dense inside the p-adic points? If not, can we have control on the set of primes that cause the failure of the density of the rational points inside the product of the p-adic points? I will explain how primes of good reduction can play a role in the Brauer-Manin obstruction to weak approximation, with particular emphasis on the case of K3 surfaces. I will then explain how the reduction type (in particular, ordinary or non-ordinary good reduction) plays a role. -
Luca Terenzi
The six functor formalism for perverse Nori motives
6 juin 2024 - 14:00Salle de séminaires IRMA
Let k be a field of characteristic 0. As envisioned by Grothendieck, Beilinson, Deligne, and others, there should exist an abelian category of mixed motives over k defining the universal Q-linear cohomology theory for algebraic k-varieties. The existence of the category of mixed motives is still conjectural; however, in the 1990's an abelian category carrying (in a suitable sense) a universal cohomology theory for k-varieties was constructed unconditionally by M. Nori. In the last decade, there have been several attempts at extending Nori's construction to a theory of motivic sheaves endowed with a six functor formalism. After reviewing Nori's theory in some detail, I will present the theory of perverse Nori motives introduced by F. Ivorra and S. Morel. By work of Ivorra--Morel and of myself, a complete six functor formalism is now available in this setting; the final goal of my talk is to sketch the main ideas behind its construction. -
Ferdinand Wagner
q-Hodge cohomology and Efimov's refined TC^-
13 juin 2024 - 14:00Salle de séminaires IRMA
q-de Rham cohomology, constructed by Bhatt and Scholze, is an important invariant. In many cases, it computes prismatic cohomology, but it can be already defined in a global setting (for rings/schemes over Z). It's a natural question whether the Hodge filtration on de Rham cohomology can be deformed to a filtration on q-de Rham cohomology. In this talk, I'll explain why this fails in general -- but I'll also explain a recent workaround: On a certain full sub-category of rings, q-de Rham cohomology does admit a "derived q-Hodge filtration". This has applications to Efimov's refinement of TC^-. If time permits, I'll also sketch how to descend the resulting "q-Hodge cohomology" from the power series ring Z[[q-1]] to the Habiro ring. -
Lue Pan
Some vanishing results for rational completed cohomology of Shimura varieties
20 juin 2024 - 14:00Salle de séminaires IRMA
Let p be a prime number. Emerton introduced the p-adically completed cohomology, which admits a representation of some p-adic group and can be thought of as some spaces of p-adic automorphic forms. In this talk, I want to explain that for Shimura varieties, sufficiently regular infinitesimal characters of the p-adic group can only show up in the middle degree of the completed cohomology. The proof is based on a very recent result of Bhatt on Kodaira vanishing in mixed characteristic and an old idea of using translation functors. This is joint work in progress with Kai-Wen Lan. -
Mauro Porta
Variations d'algèbres de Hall cohomologiques
27 juin 2024 - 14:00Salle de séminaires IRMA
Dans cet exposé je vais donner une vision d'ensemble de mes travaux en théorie géométrique des représentations. En commençant par rappeler la notion d'algèbre de Hall cohomologique et sa relevance dans l'étude de la géométrie des espaces de modules de faisceaux, je vais ensuite expliquer comment un approche uniforme à la construction à l'aide de la géométrie dérivée à amené à une compréhension plus profonde de sa structure. Je conclurai en esquissant deux travaux en cours de rédaction ; dans le premier, en collaboration avec E. D. Diaconescu, F. Sala, O. Schiffmann et E. Vasserot nous utilisons l'algèbre de Hall des faisceaux cohérents nilpotents pour donner une nouvelle réalisation des groupes quantiques de type ADE affines ; et dans un deuxième, en collaboration avec L. Hennecart, F. Sala et O. Schiffmann, nous étudions la variation d'algèbres de Hall de type Higgs d'une courbe sur M_{g,n}, avec le but de démontrer une conjecture de positivité forte du polynôme de Kac-Schiffmann. -
Wenfei Liu
On the Iitaka volumes of log canonical surfaces and threefolds
4 juillet 2024 - 14:00Salle de séminaires IRMA
For projective log canonical pairs (X,B) with nonnegative Iitaka—Kodaira dimension κ, the Iitaka volume vol_κ (K_X+B) measures the asymptotic growth of the pluricanonical systems. In the general type case, it is just the usual notion of volume, and plays a key role in the classification theory. In this talk, I will report on some results about the distribution of the Iitaka volumes of log canonical surfaces and threefolds with intermediate Kodaira dimensions. In particular, we have found the minimal possible Iitaka volume of a projective log canonical surface X with κ(K_X)=1. The talk is based on joint work in progress with Guodu Chen and Jingjun Han. -
Lucien Hennecart
Intégralité cohomologique pour les représentations des groupes réductifs
12 septembre 2024 - 14:00Salle de séminaires IRMA
Résumé : La cohomologie équivariante d'une représentation d'un groupe réductif est une algèbre de polynômes qui est indépendante de la représentation elle-même. Ce travail vise à comprendre cet espace à l'aide des morphismes d'induction parabolique, en particulier pour les représentations symétriques. Le résultat principal est une décomposition en un nombre fini d'espaces vectoriels de dimension finie, qui rappelle la théorie de Springer et dépend de la représentation choisie. La dimension graduée de ces espaces fournit de nouveaux invariants énumératifs, que l'on cherche à interpréter d'un point de vue géométrique. L'isomorphisme d'intégralité cohomologique obtenu constitue une première étape dans l'étude de la géométrie énumérative des champs généraux. Ce travail est motivé par l'étude de la conjecture P=W pour les groupes réductifs ainsi que par une conjecture de pureté due à Halpern-Leistner concernant l'homologie de Borel–Moore des champs dérivés possédant un complexe cotangent auto-dual et un bon espace de module propre. -
Yefeng Shen
Quantum spectrum and Gamma structures for flips
3 octobre 2024 - 14:00Salle de séminaires IRMA
Quantum spectrum and Gamma structures play key roles in Katzarkov-Kontsevich-Pantev-Yu’s proposal to extract birational invariants from quantum cohomology and Iritani's decomposition theorem for quantum D-modules of blowups. In this talk, we investigate these structures for standard flips in a much simpler setup, by restricting the quantum multiplication to a fiber curve direction. In this setup, we can show that both quantum spectrum and asymptotic behavior can be reduced from the local models, where the small I/J-functions are known explicitly. Using the asymptotic behavior of cohomology-valued Meijer G-functions, we obtain a decomposition of the cohomology of standard flips into asymptotic Gamma classes. This decomposition is compatible with the semi-orthogonal decomposition for standard flips constructed by Bondal-Orlov and Belmans-Fu-Raedschelders. The talk is based on work in progress joint with Mark Shoemaker. -
Remy Van Dobben De Bruyn
Constructible sheaves on toric varieties
17 octobre 2024 - 14:00Salle de séminaires IRMA
Given a reasonable topological space or algebraic variety, its covering spaces are classified by the fundamental group, via the monodromy correspondence. Recently, this was upgraded to an exodromy correspondence, classifying constructible sheaves as representations of a 'stratified fundamental category'. I will show how to prove this for toric varieties over an arbitrary field, including an explicit determination of the fundamental category. Over the complex numbers, this is a slight simplification of a result of Braden and Lunts from 2006. -
Haohao Liu
Fourier-Mukai transform on complex tori
24 octobre 2024 - 14:00Salle de séminaires IRMA
Classical Fourier transform occupies a major part of analysis. An analog on abelian varieties was introduced by S. Mukai in 1981, known as the Fourier-Mukai transform. Mukai proved a duality theorem similar to the Fourier inversion formula. This reveals the phenomenon that, the derived category of coherent sheaves of two non-isomorphic projective varieties can be equivalent. We will present the Fourier-Mukai transform on complex tori and emphasise the difference between the algebraic and the analytic cases. -
Emanuel Reinecke
Unipotent homotopy theory of schemes
7 novembre 2024 - 14:00Salle de séminaires IRMA
I will present a notion of unipotent homotopy theory for schemes, which is based on Toën's work on affine stacks. After discussing some general properties of the resulting unipotent homotopy group schemes, I will explain how over a field of characteristic p>0, they often recover the unipotent completion of the Nori fundamental group scheme, p-adic etale homotopy groups, and Artin-Mazur formal groups. As examples, we will see computations in the case of curves, abelian varieties, and Calabi-Yau varieties. Joint work with Shubhodip Mondal. -
Christine Huyghe
Transformation de Fourier des D-modules coadmissibles
14 novembre 2024 - 14:00Salle de séminaires IRMA
La transformation de Fourier pour les D-modules en géométrie algébrique complexe a été construite par Malgrange. Une telle transformation existe aussi dans le contexte des D-modules arithmétiques et est un outil important pour la formule du produit de Abe-Marmora. J'expliquerai dans cet exposé comment construire une transformation de Fourier pour les D-modules coadmissibles sur les espaces analytiques rigides. -
Sean Howe
Tangent Bundles in p-adic geometry
21 novembre 2024 - 14:00Salle de séminaires IRMA
Many important moduli spaces in modern p-adic Hodge theory mix the theory of rigid analytic geometry, which is a p-adic analog of the theory of complex analytic spaces, with the topological theory of profinite sets. In the usual approaches based on perfectoid rings, the profinite directions often eliminate the possibility of obtaining an interesting differential theory for these objects. In this talk, we discuss some progress towards constructing a better differential theory, and how it connects to other recent advances in p-adic geometry and p-adic Hodge theory. -
Nobuo Tsuzuki
Towards construction of $p$-adic representations associated to the minimal slope of $F$-isocrystals
28 novembre 2024 - 14:00Salle de séminaires IRMA
The minimal slope conjecture of $F$-isocrystals, which was proposed by K.Kedlaya, insists an irreducible overconvergent $F$-isocrystal is characterized by the minimal slope part of the associated slope filtration. Since an isoclinic $F$-isocrystal of slope $0$ corresponds to a $p$-adic representation of etale fundamental group, there exists a fully faithful subcategory of the category of $p$-adic representations such that the subcategory consists of objects coming from minimal slope parts of overconvergent $F$-isocrystals. Our final goal is to understand this subcategory well. In this talk we will discuss a method to recover such $p$-adic representations in the case of smooth varieties over a finite field. -
Amos Turchet
Vojta’s exceptional set and hypertangency of curves
5 décembre 2024 - 14:00Salle de séminaires IRMA
The distribution of integral points over number fields is conjecturally described by the far reaching Vojta conjecture. In the case of surfaces of log general type, it predicts the existence of a proper closed subset, the “exceptional set”, that contains all but finitely many integral points, on every finite extension of the ground field. In this talk we will discuss the exceptional set for the complement in the projective plane of a curve with at least three components. We will completely describe the “algebraic" exceptional set, how it relates to so-called hypertangent curves, and some generalization to arbitrary surfaces. This is joint work with Lucia Caporaso. -
Salvatore Floccari
K3 surfaces associated to varieties of generalized Kummer type
12 décembre 2024 - 14:00Salle de séminaires IRMA
Varieties of generalized Kummer type (Kum^n-type) are one of the two infinite series of known hyper-Kähler varieties, the other being given by deformations of Hilbert schemes of points of K3 surfaces. Via Hodge theory and the Torelli theorem it is possible to associate a K3 surface S_K to any variety K of Kum^n type. It is defined as the K3 surface whose transcendental lattice is Hodge isometric to that of K with the form rescaled by a factor 2. In my talk, I will explain how K and the associated K3 surface are related geometrically through a moduli space of sheaves on S_K. Building upon the work of O'Grady, Markman, Voisin and Varesco, we are able to prove the Hodge conjecture for all powers of these K3 surfaces. As a corollary we obtain the Hodge conjecture for all powers of any abelian fourfold of Weil type with discriminant 1, strenghtening a result of Markman. -
Qin Yanshuai
Boundedness of the p-primary torsion of geometric Brauer groups
19 décembre 2024 - 14:00Salle de séminaires IRMA
Let $X$ be a smooth projective integral variety over a finitely generated field $k$ of characteristic $p>0$. We show that the finiteness of the exponent of the $p$-primary part of $Br(X_{k^s})^{G_k}$ is equivalent to the Tate conjecture for divisors, generalizing D'Addezio's theorem for abelian varieties to arbitrary smooth projective varieties. In combination with the Leray spectral sequence for rigid cohomology derived from the Berthelot conjecture recently proved by Ertl-Vezzani, we show that the cokernel of $Br_{nr}(K(X)) \rightarrow Br(X_{k^s})^{G_k}$ has finite exponent. This completes the $p$-primary part of the generalization of Artin-Grothendieck's theorem on relations between Brauer groups and Tate-Shafarevich groups to higher relative dimensions. This is joint with Zhenghui Li