An adelic construction of Chern classes

Budylin, Roman Ya

doi:10.1070/sm2011v202n11abeh004202

Cited by 5 publications

(10 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The reader can also find similar or related constructions in the work of Hübl-Yekutieli [31], Morrow [40] and Budylin [6]. Let us remark here that although our main interest in this paper is to the case of bundles for a group ring, our techniques can also provide interesting new results when the group G is trivial and even in the context of these references.…”

Section: H(r((t))[g]) → Gl ′ (R((t))[g]) →mentioning

confidence: 84%

“…Statement (2) is stated, without proof, in [4]. Statement (1) follows easily from the definition of the Steinberg symbol and the calculations in [38, §9] (see the last line of p. 1654 in [6]). To show (2), observe that we can write A * × GL(A) as a semi-direct product (A * × A * ) ⋉ SL(A).…”

mentioning

confidence: 97%

“…We first have the adelic Chow groups CH (A similar construction was given by Budylin [6]. ) We are now ready to state our main result.…”

Section: Introductionmentioning

confidence: 99%

“…6.4 can be constructed for any group G as in [50,Appendix] by using the theorems of Moret-Bailly and Rumely on the existence of integral points.) 6.c. The second adelic Chern class.…”

Section: Elementary Structures and The Second Chern Classmentioning

confidence: 99%

“…In this appendix, we assume that G is trivial. We explain how our results, when combined with the Deligne-Riemann-Roch theorem [15], and some standard constructions ( [6], [4]) imply, in this case, a general adelic Riemann-Roch theorem for all bundles. 2) The restriction of (10.1) to the subgroup…”

mentioning

confidence: 99%

See 4 more Smart Citations

Higher adeles and non-abelian Riemann–Roch

Chinburg

Pappas

Taylor

2015

Advances in Mathematics

View full text Add to dashboard Cite

Abstract. We show a Riemann-Roch theorem for group ring bundles over an arithmetic surface; this is expressed using the higher adeles of Beilinson-Parshin and the tame symbol via a theory of adelic equivariant Chow groups and Chern classes. The theorem is obtained by combining a group ring coefficient version of the local Riemann-Roch formula as in Kapranov-Vasserot with results on K-groups of group rings and an explicit description of group ring bundles over P 1 . Our set-up provides an extension of several aspects of the classical Fröhlich theory of the Galois module structure of rings of integers of number fields to arithmetic surfaces. ContentsIntroduction §1. Beilinson-Parshin adeles on a surface §2. Equivariant adelic Chow groups §3. Lattices, determinant functors and determinant theories §4. Pushdown maps and reciprocity laws §5. Transition matrices and the first Chern class §6. Elementary structures and the second Chern class §7. Equivariant Euler characteristics and the Riemann-Roch theorem §8. The proof of the theorem; reduction to the case of P 1 Z §9. The proof of the theorem; bundles over P 1 Z §10. Appendix: Adelic Riemann-Roch for general bundles when G = 1 References

show abstract

Section: H(r((t))[g]) → Gl ′ (R((t))[g]) →mentioning

confidence: 84%

mentioning

confidence: 97%

“…We first have the adelic Chow groups CH (A similar construction was given by Budylin [6]. ) We are now ready to state our main result.…”

Section: Introductionmentioning

confidence: 99%

“…6.4 can be constructed for any group G as in [50,Appendix] by using the theorems of Moret-Bailly and Rumely on the existence of integral points.) 6.c. The second adelic Chern class.…”

Section: Elementary Structures and The Second Chern Classmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Higher adeles and non-abelian Riemann–Roch

Chinburg

Pappas

Taylor

2015

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

The unramified two-dimensional Langlands correspondence

Osipov¹

2013

Izv. Math.

View full text Add to dashboard Cite

In this paper we describe the unramified Langlands correspondence for twodimensional local fields, we construct a categorical analogue of the unramified principal series representations and study its properties. The main tool for this description is the construction of a central extension. For this (and other) central extension we prove noncommutative reciprocity laws (i.e. the splitting of the central extensions over some subgroups) for arithmetic surfaces and projective surfaces over a finite field. These reciprocity laws connect central extensions which are constructed locally and globally.

show abstract

Неразветвленное Двумерное Соответствие Ленглендса

Осипов¹,

Osipov²

2013

Известия Российской академии наук. Серия математическая

View full text Add to dashboard Cite

Описано неразветвленное соответствие Ленглендса для двумерных ло-кальных полей, построен категорный аналог неразветвленных представ-лений основной серии и изучены его свойста. Для этого используется конструкция некоторого центрального расширения, для которого (и дру-гих центральных расширений) доказаны некоммутативные законы взаим-ности (т. е. расщепление центральных расширений над некоторыми под-группами) для арифметических поверхностей и проективных поверхно-стей над конечным полем, связывающие центральные расширения, по-строенные локально и глобально.Библиография: 24 наименования.Ключевые слова: 2-векторные пространства, двумерные локаль-ные поля, высшие адели, обобщение программы Ленглендса, двумерные некоммутативные законы взаимности.

show abstract

An adelic construction of Chern classes

Cited by 5 publications

References 7 publications

Higher adeles and non-abelian Riemann–Roch

Higher adeles and non-abelian Riemann–Roch

The unramified two-dimensional Langlands correspondence

Неразветвленное Двумерное Соответствие Ленглендса

Contact Info

Product

Resources

About