Invitation to higher local fields, Part II, section 2: Adelic constructions for direct images of differentials and symbols

For a projective morphism of an smooth algebraic surface $X$ onto a smooth algebraic curve $S$, both given over a perfect field $k$, we construct the direct image morphism in two cases: from $H^i(X,\Omega^2_X)$ to $H^{i-1}(S,\Omega^1_S)$ and when $char k =0$ from $H^i(X,K_2(X))$ to $H^{i-1}(S,K_1(S))$. (If i=2, then the last map is the Gysin map from $CH^2(X)$ to $CH^1(S)$.) To do this in the first case we use the known adelic resolution for sheafs $\Omega^2_X$ and $\Omega^...

It is a survey paper on n-dimensional local fields and adeles on n-dimensional schemes.

This is a review of the vast area of explicit formulas for the (wild) Hilbert symbol (not only in the one-dimensional case but in the higher dimensional case as well). An extensive bilbiography is included.

We construct a theory of higher local symbols along Parsin chains for reciprocity sheaves. Applying this formalism to differential forms, gives a new construction of the Parsin-Lomadze residue maps, and applying it to the torsion characters of the fundamental group gives back the reciprocity map from Kato's higher local class field theory in the geometric case. The higher local symbols satisfy various reciprocity laws. The main result of the paper is a characterization of the...

These notes are an introduction to higher dimensional local fields and higher dimensional adeles. As well as the foundational theory, we summarise the theory of topologies on higher dimensional local fields and higher dimensional local class field theory.

Building our previous work, we explore an analogy to the differential symbol map used in the Bloch-Gabber-Kato theorem. Within this framework, for an appropriate variety over a field, the higher Chow group corresponds to the 0-th Kato homology group. Using methods inspired by Akhtar's theorem on higher Chow groups, we investigate the structure of the 0-th Kato homology group for varieties over arithmetic fields, including finite fields, local fields, and global fields of posi...

We give a new construction of linear codes over finite fields on higher dimensional varieties using Grothendieck's theory of residues. This generalizes the construction of differential codes over curves to varieties of higher dimensions.

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, Poisson formula); representations of discrete Heisenberg groups; adelic Heisenberg groups and their representations arising from two-dimensional schemes.

We consider an algebraic surface. For an irreducible curve on this surface and for a point on this curve one can associate an artinian ring, which is a sum of two-dimensional local fields. An example of two-dimensional local field is iterated Laurent series. We consider an integer value symbol on a two-dimensional local field. This symbol appears for the description of non-ramified coverings of two-dimensional local fields and for the description of intersection index of divi...

This appendix discusses some basic definitions and properties of differential forms and Kato's cohomology groups in characteristic p and a sketch of the proof of Bloch-Kato-Gabber's theorem which describes the differential symbol from the Milnor K-group K_n(F)/p of a field F of positive characteristic p to the differential module \Omega_F^n.