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In [[mathematics]], the '''Quillen–Lichtenbaum conjecture''' is a conjecture relating [[étale cohomology]] to [[algebraic K-theory]] introduced by {{harvtxt|Quillen|1975|p=175}}, who was inspired by earlier conjectures of {{harvtxt|Lichtenbaum|1973}}. {{harvtxt|Kahn|1997}} and {{harvtxt|Rognes|Weibel|2000}} proved the Quillen–Lichtenbaum conjecture at the prime 2 for some number fields. Rost and Voevodsky have announced proofs of the [[Bloch–Kato conjecture]], which implies the Quillen–Lichtenbaum conjecture for all primes.
 
==Statement==
 
The conjecture in Quillen's original form states that if ''A'' is a finitely-generated algebra over the integers and ''l'' is prime, then there is a spectral sequence analogous to the [[Atiyah-Hirzebruch spectral sequence]], starting at
 
:<math>E_2^{pq}=H^p_\text{etale}(\text{Spec }A[\ell^{-1}], Z_\ell(-q/2)),</math> (which is understood to be 0 if ''q'' is odd)
 
and abutting to
 
:<math>K_{-p-q}A\otimes Z_\ell</math>
 
for &minus;''p''&nbsp;&minus;&nbsp;''q''&nbsp;>&nbsp;1&nbsp;+&nbsp;dim&nbsp;''A''.
 
==K theory of the integers==
Assuming the Quillen–Lichtenbaum conjecture and the [[Vandiver conjecture]], the ''K''-groups of the integers, ''K''<sub>''n''</sub>('''Z'''), are given by:
 
*0 if ''n''&nbsp;=&nbsp;0 mod 8 and ''n''&nbsp;>&nbsp;0, '''Z''' if ''n''&nbsp;=&nbsp;0
*'''Z'''&nbsp;⊕&nbsp;'''Z'''/2 if ''n''&nbsp;=&nbsp;1 mod 8 and ''n''&nbsp;>&nbsp;1, '''Z'''/2 if ''n''&nbsp;=&nbsp;1.
*'''Z'''/''c''<sub>''k''</sub> ⊕ '''Z'''/2 if ''n''&nbsp;=&nbsp;2 mod&nbsp;8
*'''Z'''/8''d''<sub>''k''</sub> if ''n''&nbsp;=&nbsp;3 mod 8
*0 if ''n''&nbsp;=&nbsp;4 mod 8
*'''Z''' if ''n''&nbsp;=&nbsp;5 mod 8
*'''Z'''/''c''<sub>''k''</sub> if ''n''&nbsp;=&nbsp;6 mod 8
*'''Z'''/4''d''<sub>''k''</sub> if ''n''&nbsp;=&nbsp;7 mod 8
 
where ''c''<sub>''k''</sub>/''d''<sub>''k''</sub> is the [[Bernoulli number]] ''B''<sub>2''k''</sub>/''k'' in lowest terms and ''n'' is 4''k''&nbsp;&minus;&nbsp;1 or 4''k''&nbsp;&minus;&nbsp;2 {{harv|Weibel|2005}}.
 
==References==
 
*{{Citation | last1=Grayson | first1=Daniel R. | editor1-last=Jannsen | editor1-first=Uwe | editor2-last=Kleiman | editor2-first=Steven | editor3-last=Serre | editor3-first=Jean-Pierre | editor3-link=Jean-Pierre Serre | title=Motives (Seattle, WA, 1991) | url=http://books.google.com/books?id=v2CuklFFV5IC&lpg=PA232 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Proc. Sympos. Pure Math. | isbn=978-0-8218-1636-3  | mr=1265531 | year=1994 | volume=55 | chapter=Weight filtrations in algebraic K-theory | pages=207–237}}
*{{citation |first=Bruno |last=Kahn |title=The Quillen-Lichtenbaum conjecture at the prime 2|year=1997|url=http://www.math.uiuc.edu/K-theory/0208/QL2.pdf}}
*{{Citation | last1=Lichtenbaum | first1=Stephen |authorlink=Stephen Lichtenbaum| editor1-last=Bass | editor1-first=H. | title=Algebraic K-theory, II: Classical algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-06435-0 | doi=10.1007/BFb0073737 | mr=0406981 | year=1973 | volume=342 | chapter=Values of zeta-functions, étale cohomology, and algebraic K-theory | pages=489–501}}
*{{Citation | last1=Quillen | first1=Daniel | author1-link=Daniel Quillen | title=Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1 | url=http://mathunion.org/ICM/ICM1974.1/ | publisher=Canad. Math. Congress, Montreal, Que. | mr=0422392 | year=1975 | chapter=Higher algebraic K-theory | pages=171–176}}
*{{Citation | last1=Rognes | first1=J. | last2=Weibel | first2=Charles | title=Two-primary algebraic K-theory of rings of integers in number fields | url=http://www.math.uiuc.edu/K-theory/0220 | doi=10.1090/S0894-0347-99-00317-3 | mr=1697095 | year=2000 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | volume=13 | issue=1 | pages=1–54}}
*{{Citation | last1=Weibel | first1=Charles |authorlink1=Charles Weibel | editor1-last=Friedlander | editor1-first=Eric M. |editorlink1=Eric Friedlander | editor2-last=Grayson | editor2-first=Daniel R. | title=Handbook of K-theory. Vol. 1 | url=http://www.math.uiuc.edu/K-theory/0691/ | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-23019-9 | doi=10.1007/3-540-27855-9_5 | mr=2181823 | year=2005 | chapter=Algebraic K-theory of rings of integers in local and global fields | pages=139–190}}
 
{{DEFAULTSORT:Quillen-Lichtenbaum conjecture}}
[[Category:Algebraic K-theory]]

Latest revision as of 09:08, 2 August 2013

In mathematics, the Quillen–Lichtenbaum conjecture is a conjecture relating étale cohomology to algebraic K-theory introduced by Template:Harvtxt, who was inspired by earlier conjectures of Template:Harvtxt. Template:Harvtxt and Template:Harvtxt proved the Quillen–Lichtenbaum conjecture at the prime 2 for some number fields. Rost and Voevodsky have announced proofs of the Bloch–Kato conjecture, which implies the Quillen–Lichtenbaum conjecture for all primes.

Statement

The conjecture in Quillen's original form states that if A is a finitely-generated algebra over the integers and l is prime, then there is a spectral sequence analogous to the Atiyah-Hirzebruch spectral sequence, starting at

E2pq=Hetalep(Spec A[ℓ−1],Zℓ(−q/2)), (which is understood to be 0 if q is odd)

and abutting to

K−p−qA⊗Zℓ

for −p − q > 1 + dim A.

K theory of the integers

Assuming the Quillen–Lichtenbaum conjecture and the Vandiver conjecture, the K-groups of the integers, Kn(Z), are given by:

  • 0 if n = 0 mod 8 and n > 0, Z if n = 0
  • Z ⊕ Z/2 if n = 1 mod 8 and n > 1, Z/2 if n = 1.
  • Z/ck ⊕ Z/2 if n = 2 mod 8
  • Z/8dk if n = 3 mod 8
  • 0 if n = 4 mod 8
  • Z if n = 5 mod 8
  • Z/ck if n = 6 mod 8
  • Z/4dk if n = 7 mod 8

where ck/dk is the Bernoulli number B2k/k in lowest terms and n is 4k − 1 or 4k − 2 Template:Harv.

References

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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.

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    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.

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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.

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    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.

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  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    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
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    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.

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    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
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    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
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