# Arithmetic surfaces and adelic quotient groups

@article{Osipov2018ArithmeticSA, title={Arithmetic surfaces and adelic quotient groups}, author={Denis Vasilievich Osipov}, journal={Izvestiya: Mathematics}, year={2018}, volume={82}, pages={817 - 836} }

We explicitly calculate an arithmetic adelic quotient group for a locally free sheaf on an arithmetic surface when the fibre over the infinite point of the base is taken into account. The result is stated in the form of a short exact sequence. We relate the last term of this sequence to the projective limit of groups which are finite direct products of copies of the one-dimensional real torus and are connected with the first cohomology groups of locally free sheaves on the arithmetic surface.

## 2 Citations

### Adelic quotient group for algebraic surfaces

- MathematicsSt. Petersburg Mathematical Journal
- 2018

We calculate explicitly an adelic quotient group for an excellent Noetherian normal integral two-dimensional separated scheme. An application to an irreducible normal projective algebraic surface…

### Grothendieck–Serre Duality and Theta-Invariants on Arithmetic Surfaces

- MathematicsDoklady Mathematics
- 2019

In the paper, a description of the Grothendieck–Serre duality on an arithmetic surface by means of fixing a horizontal divisor is given and this description is applied to the generalization of…

## 27 References

### Arithmetic Cohomology Groups

- Mathematics
- 2015

We first introduce global arithmetic cohomology groups for quasi-coherent sheaves on arithmetic varieties, adopting an adelic approach. Then, we establish fundamental properties, such as topological…

### Intersections of adelic groups on a surface

- Mathematics
- 2013

We solve a technical problem related to adeles on an algebraic surface. Given a finite set of natural numbers, one can associate with it an adelic group. We show that this operation commutes with…

### Representations of Higher Adelic Groups and Arithmetic

- Mathematics
- 2010

We discuss the following topics: n-dimensional local fields and adelic groups; harmonic analysis on local fields and adelic groups for two-dimensional schemes (function spaces, Fourier transform,…

### Second Chern numbers of vector bundles and higher adeles

- Mathematics
- 2017

We give a construction of the second Chern number of a vector bundle over a smooth projective surface by means of adelic transition matrices for the vector bundle. The construction does not use an…

### Harmonic analysis on local fields and adelic spaces. II

- Mathematics
- 2008

We develop harmonic analysis in certain categories of filtered Abelian groups and vector spaces. The objects of these categories include local fields and adelic spaces arising from arithmetic…

### Harmonic analysis on local fields and adelic spaces. I

- Mathematics
- 2008

We develop harmonic analysis on the objects of a category of infinite-dimensional filtered vector spaces over a finite field. This category includes two-dimensional local fields and adelic spaces of…

### Algebraic Geometry and Arithmetic Curves

- Mathematics
- 2002

Introduction 1. Some topics in commutative algebra 2. General Properties of schemes 3. Morphisms and base change 4. Some local properties 5. Coherent sheaves and Cech cohmology 6. Sheaves of…

### A holomorphic version of the Tate-Iwasawa method for unramified -functions. I

- Mathematics
- 2014

Using the Tate-Iwasawa method the problem of meromorphic continuation and of the existence of a functional equation can be solved for the zeta and -functions of one-dimensional arithmetical schemes.…

### Higher dimensional local fields and L–functions

- Mathematics
- 2000

Example. Let X be an algebraic projective irreducible surface over a field k and let P be a closed point of X , C ⊂ X be an irreducible curve such that P ∈ C . If X andC are smooth atP , then we lett…

### Eight papers translated from the Russian

- Mathematics
- 1989

On tetragonal curves by S. G. Dalayan On pointwise estimates of some capacity potentials by I. V. Skrypnik Some singular boundary value problems for partial differential equations by V. P. Palamodov…