On the K-theory of local fields

@article{Hesselholt1999OnTK,
  title={On the K-theory of local fields},
  author={Lars Hesselholt and Ib Henning Madsen},
  journal={Annals of Mathematics},
  year={1999},
  volume={158},
  pages={1-113}
}
In this paper we establish a connection between the Quillen K-theory of certain local fields and the de Rham-Witt complex of their rings of integers with logarithmic poles at the maximal ideal. The fieldsK we consider are complete discrete valuation fields of characteristic zero with perfect residue field k of characteristic p > 2. When K contains the pth roots of unity, the relationship between the K-theory with Z/p-coefficients and the de Rham-Witt complex can be described by a sequence 
View PDF on arXiv
174 Citations
Highly Influential Citations
8
Background Citations
26
Methods Citations
17
Results Citations
2

174 Citations

On the K-theory of complete regular local Fp-algebras
The de Rham-Witt complex and p-adic vanishing cycles
We determine the structure modulo p of the de Rham-Witt complex of a smooth scheme X over a discrete valuation ring of mixed characteristic with log-poles along the special fiber Y and show that the…
K-THEORY AND MOTIVIC COHOMOLOGY OF SCHEMES
We examine some of the basic properties satisfied by Bloch’s cycle complexes for quasi-projective varieties over a field, and extend some of them to the cycle complex of a scheme of finite type over…
On the K-theory of finite algebras over witt vectors of perfect fields
On the K-theory of truncated polynomial algebras, revisited
  • M. Speirs
  • Mathematics
    Advances in Mathematics
  • 2020
On relative and bi-relative algebraic $K$-theory of rings of finite characteristic
We prove that the relative K-groups associated with a nilpotent extension of Z/p^N Z-algebras and the bi-relative K-groups associated with a Milnor square of Z/p^N Z-algebras are p-primary torsion…
Derived representation theory and the algebraic K‐theory of fields
In this paper, we prove a conjecture on the relationship of the algebraic K‐theory of a field F, with abelian separable Galois group GF, containing an algebraically closed subfield with the K‐theory…
On the topological cyclic homology of the algebraic closure of a local field
The cyclotomic trace provides a comparison of the algebraic K-theory spectrum and a pro-spectrum TR that is built from the cyclic fixed points of topological Hochschild homology. In a previous paper…
Rational algebraic K-theory of topological K-theory
We show that after rationalization there is a homotopy fiber sequence BBU -> K(ku) -> K(Z). We interpret this as a correspondence between the virtual 2-vector bundles over a space X and their…
Slices of hermitian $K$–theory and Milnor's conjecture on quadratic forms
We advance the understanding of K-theory of quadratic forms by computing the slices of the motivic spectra representing hermitian K-groups and Witt-groups. By an explicit computation of the slice…
...
...

References

SHOWING 1-10 OF 89 REFERENCES
On the K-theory of finite algebras over witt vectors of perfect fields
On the K-theory of local fields
On the K-Theory Spectrum of a Ring of Algebraic Integers⋆
Suppose that F is a number field (i.e. a finite algebraic extension of the field Q of rational numbers) and that OF is the ring of algebraic integers in F . One of the most fascinating and apparently…
Higher Algebraic K-Theory of Schemes and of Derived Categories
In this paper we prove a localization theorem for the K-theory of commutative rings and of schemes, Theorem 7.4, relating the K-groups of a scheme, of an open subscheme, and of the category of those…
ALGEBRAIC K-THEORY OF SPACES I
The algebraic K–theory of spaces is a variant, invented by F. Waldhausen in the late 1970’s, of the standard algebraic K–theory of rings. Until that time, applications of algebraic K–theory to…
Cyclic polytopes and the $K$-theory of truncated polynomial algebras
This paper calculates the relative algebraic K -theory K∗(k [x ]/(x n ), (x )) of a truncated polynomial algebra over a perfect field k of positive characteristic p. Since the ideal generated by x is…
The cyclotomic trace and algebraic K-theory of spaces
The cyclotomic trace is a map from algebraic K-theory of a group ring to a certain topological refinement of cyclic homology. The target is naturally mapped to topological Hochschild homology, and…
Formal Groups and Applications
TLDR
This book is a comprehensive treatment of the theory of formal groups and its numerous applications in several areas of mathematics, including very important applications in algebraic topology, number theory, and algebraic geometry.
On the topological cyclic homology of the integers
<abstract abstract-type="TeX"><p>This article provides a computation of the mod <i>p</i> homotopy groups of the fixed points of the Topological Hochschild Homology of the ring of integers under the…
Lectures on curves on an algebraic surface
These lectures, delivered by Professor Mumford at Harvard in 1963-1964, are devoted to a study of properties of families of algebraic curves, on a non-singular projective algebraic curve defined over…
...
...