Pub Date : 2018-01-09DOI: 10.2140/tunis.2019.1.373
Takeshi Saito
We give a purely scheme theoretic construction of the filtration by ramification groups of the Galois group of a covering. The valuation need not be discrete but the normalizations are required to be locally of complete intersection.
{"title":"Ramification groups of coverings and valuations","authors":"Takeshi Saito","doi":"10.2140/tunis.2019.1.373","DOIUrl":"https://doi.org/10.2140/tunis.2019.1.373","url":null,"abstract":"We give a purely scheme theoretic construction of the filtration by ramification groups of the Galois group of a covering. The valuation need not be discrete but the normalizations are required to be locally of complete intersection.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2019.1.373","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47710803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-12-28DOI: 10.2140/tunis.2020.2.733
Tetsushi Ito, Kazuya Kato, Chikara Nakayama, S. Usui
We define the categories of log motives and log mixed motives. The latter gives a new formulation for the category of mixed motives. We prove that the former is a semisimple abelian category if and only if the numerical equivalence and homological equivalence coincide, and that it is also equivalent to that the latter is a Tannakian category. We discuss various realizations, formulate Tate and Hodge conjectures, and verify them in curve case.
{"title":"On log motives","authors":"Tetsushi Ito, Kazuya Kato, Chikara Nakayama, S. Usui","doi":"10.2140/tunis.2020.2.733","DOIUrl":"https://doi.org/10.2140/tunis.2020.2.733","url":null,"abstract":"We define the categories of log motives and log mixed motives. The latter gives a new formulation for the category of mixed motives. We prove that the former is a semisimple abelian category if and only if the numerical equivalence and homological equivalence coincide, and that it is also equivalent to that the latter is a Tannakian category. We discuss various realizations, formulate Tate and Hodge conjectures, and verify them in curve case.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2020.2.733","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48837359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-12-06DOI: 10.2140/tunis.2020.2.841
Rita Gitik
Let H and K be quasiconvex subgroups of a negatively curved torsion-free group G. We give an algorithm which decides whether an element of H is conjugated in G to an element of K.
设H和K是负弯曲无扭群G的拟凸子群，给出了判定G中H的一个元素是否共轭于K中的一个元素的算法。
{"title":"A generalization of a power-conjugacy problem in torsion-free negatively curved groups","authors":"Rita Gitik","doi":"10.2140/tunis.2020.2.841","DOIUrl":"https://doi.org/10.2140/tunis.2020.2.841","url":null,"abstract":"Let H and K be quasiconvex subgroups of a negatively curved torsion-free group G. We give an algorithm which decides whether an element of H is conjugated in G to an element of K.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2020.2.841","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46828093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-12-06DOI: 10.2140/tunis.2019.1.347
P. Gérard, S. Grellier
This paper explores the regularity properties of an inverse spectral transform for Hilbert--Schmidt Hankel operators on the unit disc. This spectral transform plays the role of action-angles variables for an integrable infinite dimensional Hamiltonian system -- the cubic Szeg"o equation. We investigate the regularity of functions on the tori supporting the dynamics of this system, in connection with some wave turbulence phenomenon, discovered in a previous work and due to relative small gaps between the actions. We revisit this phenomenon by proving that generic smooth functions and a G $delta$ dense set of irregular functions do coexist on the same torus. On the other hand, we establish some uniform analytic regularity for tori corresponding to rapidly decreasing actions which satisfy some specific property ruling out the phenomenon of small gaps.
{"title":"Generic colourful tori and inverse spectral transform for Hankel operators","authors":"P. Gérard, S. Grellier","doi":"10.2140/tunis.2019.1.347","DOIUrl":"https://doi.org/10.2140/tunis.2019.1.347","url":null,"abstract":"This paper explores the regularity properties of an inverse spectral transform for Hilbert--Schmidt Hankel operators on the unit disc. This spectral transform plays the role of action-angles variables for an integrable infinite dimensional Hamiltonian system -- the cubic Szeg\"o equation. We investigate the regularity of functions on the tori supporting the dynamics of this system, in connection with some wave turbulence phenomenon, discovered in a previous work and due to relative small gaps between the actions. We revisit this phenomenon by proving that generic smooth functions and a G $delta$ dense set of irregular functions do coexist on the same torus. On the other hand, we establish some uniform analytic regularity for tori corresponding to rapidly decreasing actions which satisfy some specific property ruling out the phenomenon of small gaps.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2019.1.347","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45105064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-10-24DOI: 10.2140/tunis.2019.1.127
R. Danchin, P. Mucha
This paper is concerned with the mathematical derivation of the inhomoge-neous incompressible Navier-Stokes equations (INS) from the compressible Navier-Stokes equations (CNS) in the large volume viscosity limit. We first prove a result of large time existence of regular solutions for (CNS). Next, as a consequence, we establish that the solutions of (CNS) converge to those of (INS) when the volume viscosity tends to infinity. Analysis is performed in the two dimensional torus, for general initial data. In particular, we are able to handle large variations of density.
{"title":"From compressible to incompressible inhomogeneous flows in the case of large data","authors":"R. Danchin, P. Mucha","doi":"10.2140/tunis.2019.1.127","DOIUrl":"https://doi.org/10.2140/tunis.2019.1.127","url":null,"abstract":"This paper is concerned with the mathematical derivation of the inhomoge-neous incompressible Navier-Stokes equations (INS) from the compressible Navier-Stokes equations (CNS) in the large volume viscosity limit. We first prove a result of large time existence of regular solutions for (CNS). Next, as a consequence, we establish that the solutions of (CNS) converge to those of (INS) when the volume viscosity tends to infinity. Analysis is performed in the two dimensional torus, for general initial data. In particular, we are able to handle large variations of density.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2019.1.127","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46576824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-10-13DOI: 10.2140/tunis.2020.2.359
Paul Balmer
We prove nilpotence theorems in tensor-triangulated categories using suitable Gabriel quotients of the module category, and discuss examples.
利用模范畴的Gabriel商证明了张量三角范畴中的幂零性定理，并讨论了例子。
{"title":"Nilpotence theorems via homological residue fields","authors":"Paul Balmer","doi":"10.2140/tunis.2020.2.359","DOIUrl":"https://doi.org/10.2140/tunis.2020.2.359","url":null,"abstract":"We prove nilpotence theorems in tensor-triangulated categories using suitable Gabriel quotients of the module category, and discuss examples.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2020.2.359","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43985492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let G be a special orthogonal group SO(2n+1) defined over a p-adic field F. Let $pi$ be an admissible irreducible representation of G(F) which is tempered and of unipotent reduction. We prove that $pi$ has a wave front set. In some particular cases, for instance if $pi$ is of the discrete series, we give a method to compute this wave front set.
{"title":"Fronts d’onde des représentations tempérées\u0000et de réduction unipotente pour SO(2n + 1)","authors":"J. Waldspurger","doi":"10.2140/tunis.2020.2.43","DOIUrl":"https://doi.org/10.2140/tunis.2020.2.43","url":null,"abstract":"Let G be a special orthogonal group SO(2n+1) defined over a p-adic field F. Let $pi$ be an admissible irreducible representation of G(F) which is tempered and of unipotent reduction. We prove that $pi$ has a wave front set. In some particular cases, for instance if $pi$ is of the discrete series, we give a method to compute this wave front set.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2020.2.43","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49297461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-10-06DOI: 10.2140/TUNIS.2019.1.295
R. Guralnick, N. M. Katz, P. Tiep
In earlier work, Katz exhibited some very simple one parameter families of exponential sums which gave rigid local systems on the affine line in characteristic p whose geometric (and usually, arithmetic) monodromy groups were SL(2,q), and he exhibited other such very simple families giving SU(3,q). [Here q is a power of the characteristic p with p odd]. In this paper, we exhibit equally simple families whose geometric monodromy groups are the alternating groups Alt(2q). $. We also determine their arithmetic monodromy groups. By Raynaud's solution of the Abhyankar Conjecture, any finite simple group whose order is divisible by p will occur as the geometric monodromy group of some local system on the affine line in characteristic p; the interest here is that it occurs in our particularly simple local systems. In the earlier work of Katz, he used a theorem to Kubert to know that the monodromy groups in question were finite, then work of Gross to determine which finite groups they were. Here we do not have, at present, any direct way of showing this finiteness. Rather, the situation is more complicated and more interesting. Using some basic information about these local systems, a fundamental dichotomy is proved: The geometric monodromy group is either Alt(2q) or it is the special orthogonal group SO(2q-1). An elementary polynomial identity is used to show that the third moment is 1. This rules out the SO(2q-1) case. This roundabout method establishes the theorem. It would be interesting to find a "direct" proof that these local systems have integer (rather than rational) traces; this integrality is in fact equivalent to their monodromy groups being finite, Even if one had such a direct proof, it would still require serious group theory to show that their geometric monodromy groups are the alternating groups.
{"title":"Rigid local systems and alternating groups","authors":"R. Guralnick, N. M. Katz, P. Tiep","doi":"10.2140/TUNIS.2019.1.295","DOIUrl":"https://doi.org/10.2140/TUNIS.2019.1.295","url":null,"abstract":"In earlier work, Katz exhibited some very simple one parameter families of exponential sums which gave rigid local systems on the affine line in characteristic p whose geometric (and usually, arithmetic) monodromy groups were SL(2,q), and he exhibited other such very simple families giving SU(3,q). [Here q is a power of the characteristic p with p odd]. In this paper, we exhibit equally simple families whose geometric monodromy groups are the alternating groups Alt(2q). $. We also determine their arithmetic monodromy groups. By Raynaud's solution of the Abhyankar Conjecture, any finite simple group whose order is divisible by p will occur as the geometric monodromy group of some local system on the affine line in characteristic p; the interest here is that it occurs in our particularly simple local systems. In the earlier work of Katz, he used a theorem to Kubert to know that the monodromy groups in question were finite, then work of Gross to determine which finite groups they were. Here we do not have, at present, any direct way of showing this finiteness. Rather, the situation is more complicated and more interesting. Using some basic information about these local systems, a fundamental dichotomy is proved: The geometric monodromy group is either Alt(2q) or it is the special orthogonal group SO(2q-1). An elementary polynomial identity is used to show that the third moment is 1. This rules out the SO(2q-1) case. This roundabout method establishes the theorem. It would be interesting to find a \"direct\" proof that these local systems have integer (rather than rational) traces; this integrality is in fact equivalent to their monodromy groups being finite, Even if one had such a direct proof, it would still require serious group theory to show that their geometric monodromy groups are the alternating groups.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/TUNIS.2019.1.295","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47662995","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-09-19DOI: 10.2140/tunis.2020.2.633
A. Gholampour, Richard P. Thomas
We express nested Hilbert schemes of points and curves on a smooth projective surface as "virtual resolutions" of degeneracy loci of maps of vector bundles on smooth ambient spaces. We show how to modify the resulting obstruction theories to produce the virtual cycles of Vafa-Witten theory and other sheaf-counting problems. The result is an effective way of calculating invariants (VW, SW, local PT and local DT) via Thom-Porteous-like Chern class formulae.
{"title":"Degeneracy loci, virtual cycles and nested Hilbert schemes, I","authors":"A. Gholampour, Richard P. Thomas","doi":"10.2140/tunis.2020.2.633","DOIUrl":"https://doi.org/10.2140/tunis.2020.2.633","url":null,"abstract":"We express nested Hilbert schemes of points and curves on a smooth projective surface as \"virtual resolutions\" of degeneracy loci of maps of vector bundles on smooth ambient spaces. \u0000We show how to modify the resulting obstruction theories to produce the virtual cycles of Vafa-Witten theory and other sheaf-counting problems. The result is an effective way of calculating invariants (VW, SW, local PT and local DT) via Thom-Porteous-like Chern class formulae.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2020.2.633","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47625346","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-08-31DOI: 10.2140/tunis.2020.2.567
B. Guillou, M. Hill, Daniel Isaksen, D. Ravenel
We compute the cohomology of the subalgebra $A^{C_2}(1)$ of the $C_2$-equivariant Steenrod algebra $A^{C_2}$. This serves as the input to the $C_2$-equivariant Adams spectral sequence converging to the $RO(C_2)$-graded homotopy groups of an equivariant spectrum $ko_{C_2}$. Our approach is to use simpler $mathbb{C}$-motivic and $mathbb{R}$-motivic calculations as stepping stones.
{"title":"The cohomology of C2-equivariant 𝒜(1) and the\u0000homotopy of koC2","authors":"B. Guillou, M. Hill, Daniel Isaksen, D. Ravenel","doi":"10.2140/tunis.2020.2.567","DOIUrl":"https://doi.org/10.2140/tunis.2020.2.567","url":null,"abstract":"We compute the cohomology of the subalgebra $A^{C_2}(1)$ of the $C_2$-equivariant Steenrod algebra $A^{C_2}$. This serves as the input to the $C_2$-equivariant Adams spectral sequence converging to the $RO(C_2)$-graded homotopy groups of an equivariant spectrum $ko_{C_2}$. Our approach is to use simpler $mathbb{C}$-motivic and $mathbb{R}$-motivic calculations as stepping stones.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2020.2.567","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47393765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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