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

This work contains a list of all known results on the quotient filtration on the Milnor K-groups of a complete discrete valuation field in terms of differential modules over the residue field . Author's recent study of the case of a tamely ramified field of characteristic 0 with residue field of characteristic p by using an exponential map and a syntomic complex is explained.

Proposed is a program for what we call Geometric Arithmetic, based on our works on non-abelian zeta functions and non-abelian class field theory. Key words are stability and adelic intersection-cohomology theory.

This is is a survey of applications of Fontaine's theory of p-adic representations of local fields (Phi-Gamma-modules) to Galois cohomology of local fields and explicit formulas for the Hilbert symbol in relation with two-dimensional local objects.

These are the notes accompanying three lectures given by the second author at the Motivic Geometry program at CAS, which aim to give an introduction and an overview of some recent developments in the field of reciprocity sheaves.

This work sketches the author classification of complete discrete valuation fields K of characteristic 0 with residue field of characteristic p into two classes depending on the behaviour of the torsion part of a differential module. For each of these classes, the quotient filtration of the Milnor K-groups of K is characterized for all sufficiently large members of the filtration, as a quotient of differential modules. For a higher local field the previous result and higher l...

We give a new approach for the local class field theory of Serre and Hazewinkel. We also discuss two-dimensional local class field theory in this framework.

This is a presentation of explicit methods to construct higher local class field theory by using topological K-groups, explicit symbols and a generalization of Neukirch-Hazewinkel's axiomatic approaches. The existence theorem is discussed as well.

This is an introduction to noncommutative local reciprocity maps for totally ramified Galois extensions with arithmetically profinite group. These maps in general are not homomorphisms but Galois cycles; a description of their image and kernel is included.

Tate gave a famous construction of the residue symbol on curves by using some non-commutative operator algebra in the context of algebraic geometry. We explain Beilinson's multidimensional generalization, which is not so well-documented in the literature. We provide a new approach using Hochschild homology.

In this paper we show the existence of an action of Chow correspondences on the cohomology of reciprocity sheaves. In order to do so, we prove a number of structural results, such as a projective bundle formula, a blow-up formula, a Gysin sequence, and the existence of proper pushforward. In this way we recover and generalize analogous statements for the cohomology of Hodge sheaves and Hodge-Witt sheaves. We give several applications of the general theory to problems which ...