Ergodic sequence

In mathematics, Milnor K-theory was an early attempt to define higher algebraic K-theory, introduced by Template:Harvs.

Definition

The calculation of K₂ of a field F led Milnor to the following ad hoc definition of "higher" K-groups by

$K_{*}^M(F):=T^{*}{F}^{\times }/(a\otimes (1-a)),$

thus as graded parts of a quotient of the tensor algebra of the multiplicative group F^× by the two-sided ideal, generated by the

$a\otimes (1-a)$

for a ≠ 0, 1. For n = 0,1,2 these coincide with Quillen's K-groups of a field, but for n ≧ 3 they differ in general. We define the symbol $\{a_1,\dots ,a_n\}$ as the image of $a_1\otimes \cdots \otimes a_n$: the case n=2 is a Steinberg symbol.

The tensor product on the tensor algebra induces a product $K_m\times K_n\to K_{m+n}$ making $K_{*}^M(F)$ a graded ring which is graded-commutative.

Examples

For example, we have $K_n^M({\mathbb{𝔽}}_q)=0$ for n ≧ 2; $K_2^M(\mathbb{ℂ})$ is an uncountable uniquely divisible group; $K_2^M(\mathbb{ℝ})$ is the direct sum of a cyclic group of order 2 and an uncountable uniquely divisible group; $K_2^M({\mathbb{\mathbb{Q}}}_p)$ is the direct sum of the multiplicative group of ${\mathbb{𝔽}}_p$ and an uncountable uniquely divisible group; $K_2^M(\mathbb{\mathbb{Q}})$ is the direct sum of the cyclic group of order 2 and cyclic groups of order $p-1$ for all odd prime $p$.

Applications

Milnor K-theory plays a fundamental role in higher class field theory, replacing $K_1^M$ in the one dimensional class field theory.

Milnor K-theory modulo 2, denoted k_*(F) is related to étale (or Galois) cohomology of the field F by the Milnor conjecture, proven by Voevodsky. The analogous statement for odd primes is the Bloch–Kato conjecture, proved by Voevodsky, Rost, and others.

There are homomorphisms from k_n(F) to the Witt ring of F by taking the symbol

$\{a_1,\dots ,a_n\}\mapsto ⟨⟨a_1,a_2,...,a_n⟩⟩=⟨1,a_1⟩\otimes ⟨1,a_2⟩\otimes ...\otimes ⟨1,a_n⟩,$

where the image is a Pfister form of dimension 2ⁿ. The image can be taken as Iⁿ/Iⁿ⁺¹ and the map is surjective since the Pfister forms additively generate Iⁿ. The Milnor conjecture can be interpreted as stating that these maps are isomorphisms.

References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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Further reading

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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
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  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010