Talk:Milnor K-theory
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"Milnor ring" (an alternative name) redirects here. BTotaro (talk) 17:40, 4 November 2015 (UTC)
Article improvements edit
There are quite a few useful results from Milnor's original paper "Algebraic K-theory and Quadratic forms" which should be included in this article for a better understanding of Milnor K-theory.
structure of K-theory ring edit
- graded commutative ring structure
- 1.2 and 1.3
computations edit
- example 1.5 and implications
- example 1.6 with generators, also include from section 2
- for can be deduced from later methods
structure results edit
- example 1.7 gives partial computation of local fields : they are all divisible, moreover, using the same argument as 1.5 this gives all milnor K-groups
theorems edit
- theorem 2.3 (give exact sequence for Q(t), R(t), C(t) (or any algebraically closed field), showing the structure)
- C(t_1), C(t_1)(t_2), C(t_1)(t_2)(t_3), ...
- lemma 6.2 -> relation with Galois cohomology (refined further later on Bloch-Kato)
- A.2
Applications section edit
- theorem 1.4 for arithmetic
Other articles edit
- Also, mention Voevodsky's article on motivic cohomology (corollary 7.5 on page 97) https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/motivic_cohomology_with_Z2_coefficients_published.pdf
- Give motivic sheaves representing Milnor K-theory https://arxiv.org/abs/math/0107109
- Relate to motivic cohomology, higher chow groups, and higher algebraic K-theory, this shows Milnor K-theory is part of higher algebraic K-theory
Higher local class field theory edit
This article should mention the main theorem of local class field theory. The statement can be found in
and
contains other useful pdfs. Also,
contains useful stuff on Milnor K-theory, starting on page 292 of the pdf.
- Differential forms and Milnor K-theory
Kaptain-k-theory (talk) 20:40, 14 April 2021 (UTC)
Motivic steenrod algebra edit
Should discuss the relation between motivic cohomology and Milnor K-theory. In addition, discuss the various results about the motivic steenrod algebra and motivic eilenberg-maclane spaces. Some resources include
- http://www.mathematik.ur.de/hoyois/papers/motivicsteenrod.pdf
- https://people.math.rochester.edu/faculty/doug/otherpapers/powell.pdf
- https://conf.math.illinois.edu/K-theory/0170/milnor.pdf
Kaptain-k-theory (talk) 18:19, 28 April 2021 (UTC)