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In [[algebraic K-theory]], a field of [[mathematics]], the '''Steinberg group''' St(''A'') of a ring ''A'', is the [[universal central extension]] of the [[commutator subgroup]] of the stable [[general linear group]]. | |||
It is named after [[Robert Steinberg]], and is connected with [[Algebraic K-theory#Lower K-groups|lower K-groups]], notably ''K''<sub>2</sub> and ''K''<sub>3</sub>. | |||
==Definition== | |||
Abstractly, given a ring ''A'', the Steinberg group St(''A'') is the [[universal central extension]] of the [[commutator subgroup]] of the [[stable general linear group]] (the commutator subgroup is perfect, hence has a universal central extension). | |||
Concretely, it can also be described by [[Presentation of a group|generators and relations]]. | |||
===Steinberg relations=== | |||
[[Elementary matrices]]—meaning matrices of the form ''e<sub>pq</sub>''(λ) := '''1''' + ''a<sub>pq</sub>''(λ), where '''1''' is the identity matrix, ''a<sub>pq</sub>''(λ) is the matrix with λ in the (''p'', ''q'') entry and zeros elsewhere, and ''p'' ≠ ''q''—satisfy the following relations, called the '''Steinberg relations''': | |||
:<math>\begin{align} | |||
e_{ij}(\lambda) e_{ij}(\mu) &= e_{ij}(\lambda+\mu) \\ | |||
\left[ e_{ij}(\lambda),e_{jk}(\mu) \right] &= e_{ik}(\lambda \mu) && \mbox{for } i \neq k\\ | |||
\left[ e_{ij}(\lambda),e_{kl}(\mu) \right] &= \mathbf{1} && \mbox{for } i \neq l, j \neq k | |||
\end{align}</math> | |||
The '''unstable Steinberg group''' of order ''r'' over ''A'', St<sub>''r''</sub>(''A''), is defined by the generators ''x<sub>ij</sub>''(λ), 1 ≤ ''i'' ≠ ''j'' ≤ ''r'', λ ∈ ''A'', subject to the Steinberg relations. The '''stable Steinberg group''', St(''A''), is the [[direct limit]] of the system St<sub>''r''</sub>(''A'') → St<sub>''r''+1</sub>(''A''). It can also be thought of as the Steinberg group of infinite order. | |||
Mapping ''x<sub>ij</sub>''(λ) ↦ ''e<sub>ij</sub>''(λ) yields a [[group homomorphism]] φ : St(''A'') → GL(''A''). As the elementary matrices generate the [[commutator subgroup]], this map is onto the commutator subgroup. | |||
==Relation to K-theory== | |||
===K<sub>1</sub>=== | |||
[[Algebraic_K-theory#K1|''K''<sub>1</sub>(''A'')]] is the [[cokernel]] of the map φ : St(''A'') → GL(''A''), as ''K''<sub>1</sub> is the abelianization of GL(''A'') and φ is onto the commutator subgroup. | |||
===K<sub>2</sub>=== | |||
[[Algebraic_K-theory#K2|''K''<sub>2</sub>(''A'')]] is the [[centre of a group|center]] of the Steinberg group; this was Milnor's definition, and also follows from more general definitions of higher K-groups. | |||
It is also the kernel of the map φ : St(''A'') → GL(''A''), and indeed there is an [[exact sequence]] | |||
:1 → ''K''<sub>2</sub>(''A'') → St(''A'') → GL(''A'') → ''K''<sub>1</sub>(''A'') → 1. | |||
Equivalently, it is the [[Schur multiplier]] of the group of [[elementary matrices]], and thus is also a homology group: ''K''<sub>2</sub>(''A'') = ''H''<sub>2</sub>(E(''A''), '''Z'''). | |||
===K<sub>3</sub>=== | |||
{{harvtxt|Gersten|1973}} showed that ''K''<sub>3</sub>(''A'') = ''H''<sub>3</sub>(St(''A''), '''Z'''). | |||
==References== | |||
* {{citation | |||
|title=''K''<sub>3</sub> of a Ring is ''H''<sub>3</sub> of the Steinberg Group | |||
|first=S. M.|last= Gersten | |||
|journal=Proceedings of the American Mathematical Society | |||
|volume=37 | |||
|issue=2 | |||
|pages=366–368 | |||
|doi=10.2307/2039440 | |||
|year=1973 | |||
|jstor=2039440 | |||
|publisher=American Mathematical Society | |||
}} | |||
*{{Citation | last1=Milnor | first1=John Willard | author1-link=John Milnor | title=Introduction to algebraic ''K''-theory | publisher=[[Princeton University Press]] | series=Annals of Mathematics Studies | mr=0349811 | year=1971 | volume=72}} | |||
*{{Citation | last1=Steinberg | first1=Robert | title=Lectures on Chevalley groups | url=http://www.math.ucla.edu/~rst/ | publisher=Yale University, New Haven, Conn. | mr=0466335 | year=1968}} | |||
[[Category:K-theory]] | |||
Latest revision as of 09:14, 17 July 2013
In algebraic K-theory, a field of mathematics, the Steinberg group St(A) of a ring A, is the universal central extension of the commutator subgroup of the stable general linear group.
It is named after Robert Steinberg, and is connected with lower K-groups, notably K2 and K3.
Definition
Abstractly, given a ring A, the Steinberg group St(A) is the universal central extension of the commutator subgroup of the stable general linear group (the commutator subgroup is perfect, hence has a universal central extension).
Concretely, it can also be described by generators and relations.
Steinberg relations
Elementary matrices—meaning matrices of the form epq(λ) := 1 + apq(λ), where 1 is the identity matrix, apq(λ) is the matrix with λ in the (p, q) entry and zeros elsewhere, and p ≠ q—satisfy the following relations, called the Steinberg relations:
![{\displaystyle {\begin{aligned}e_{ij}(\lambda )e_{ij}(\mu )&=e_{ij}(\lambda +\mu )\\\left[e_{ij}(\lambda ),e_{jk}(\mu )\right]&=e_{ik}(\lambda \mu )&&{\mbox{for }}i\neq k\\\left[e_{ij}(\lambda ),e_{kl}(\mu )\right]&=\mathbf {1} &&{\mbox{for }}i\neq l,j\neq k\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35636bb6b75e9e78249ff1344a958afdd3faea62)
The unstable Steinberg group of order r over A, Str(A), is defined by the generators xij(λ), 1 ≤ i ≠ j ≤ r, λ ∈ A, subject to the Steinberg relations. The stable Steinberg group, St(A), is the direct limit of the system Str(A) → Str+1(A). It can also be thought of as the Steinberg group of infinite order.
Mapping xij(λ) ↦ eij(λ) yields a group homomorphism φ : St(A) → GL(A). As the elementary matrices generate the commutator subgroup, this map is onto the commutator subgroup.
Relation to K-theory
K1
K1(A) is the cokernel of the map φ : St(A) → GL(A), as K1 is the abelianization of GL(A) and φ is onto the commutator subgroup.
K2
K2(A) is the center of the Steinberg group; this was Milnor's definition, and also follows from more general definitions of higher K-groups.
It is also the kernel of the map φ : St(A) → GL(A), and indeed there is an exact sequence
- 1 → K2(A) → St(A) → GL(A) → K1(A) → 1.
Equivalently, it is the Schur multiplier of the group of elementary matrices, and thus is also a homology group: K2(A) = H2(E(A), Z).
K3
Template:Harvtxt showed that K3(A) = H3(St(A), Z).
References
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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
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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
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
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