Talk:Milnor K-theory

"Milnor ring" (an alternative name) redirects here. BTotaro (talk) 17:40, 4 November 2015 (UTC)

Article improvements

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

• graded commutative ring structure
• 1.2 and 1.3

computations

• example 1.5 and implications
• example 1.6 with generators, also include $\mathbb {R} (t)$ from section 2
• $K_{n}\mathbb {Q} \cong \mathbb {Z} /2$ for $n\geq 3$ can be deduced from later methods

structure results

• 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

• 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

• theorem 1.4 for arithmetic

Other articles

• 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

Wundzer (talk) 17:15, 22 January 2021 (UTC)

Higher local class field theory

This article should mention the main theorem of local class field theory. The statement can be found in

• https://arxiv.org/abs/math/0012136

and

• https://msp.org/gtm/2000/03/

contains other useful pdfs. Also,

• https://www.maths.nottingham.ac.uk/plp/pmzibf/book/vol.pdf

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

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)