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Motivic cohomology is a cohomological theory in mathematics, the existence of which was first conjectured by Alexander Grothendieck during the 1960s. At that time, it was conceived as a theory constructed on the basis of the so-called standard conjectures on algebraic cycles, in algebraic geometry. It had a basis in category theory for drawing consequences from those conjectures; Grothendieck and Enrico Bombieri showed the depth of this approach by deriving a conditional proof of the Weil conjectures by this route. The standard conjectures, however, resisted proof.

This left the motive (motif in French) theory as having heuristic status. Serre, for example, preferred to work more concretely with a compatible system of ℓ-adic representations, which at least conjecturally should be as good as having a motive, but instead listed the data obtainable from a motive by means of its 'realisations' in the étale cohomology theories with l-adic coefficients, as l varied over prime numbers. From the Grothendieck point of view, motives should further contain the information provided by algebraic de Rham cohomology, and crystalline cohomology. In some sense motivic cohomology would be the mother of all cohomology theories in algebraic geometry; the other cohomology theories would be specializations.

Grothendieck gave a solution for Weil cohomology theories over a field in 1967. This involved extending the category of smooth projective varieties to the category of Chow motives. This is an additive category, but not an abelian category unless one takes rational coefficients and passes to numerical equivalence. The study of motives for arbitrary varieties (mixed motives) began in the early 1970s with Deligne's notion of 1-motives. The hope is that there is something like an abelian category of mixed motives, containing all varieties over the field, and a universal cohomology theory on mixed motives in the sense of Homological Algebra. However, progress towards realizing this picture was slow; Deligne's absolute Hodge cycles provided one technical fix. Beilinson's absolute Hodge cohomology provided a universal cohomology theory with rational coefficients (and without any category of motives) using algebraic K-theory.

Recent progress

In the mid-1990s, several people proposed candidates for the derived category of the conjectural category of motives. The most successful has been Vladimir Voevodsky's $DM$ construction. By applying techniques from homotopy theory and K-theory to algebraic geometry, Voevodsky constructed a bigraded motivic cohomology theory

$H^{p,q}(X)$

for algebraic varieties. It is not known whether these groups vanish for negative $p$; this property is known as the vanishing conjecture. Otherwise, this theory is known to satisfy all of the properties suggested by Grothendieck. Voevodsky provided two constructions of motivic cohomology for algebraic varieties, via:

1. a homotopy theory for algebraic varieties, in the form of a model category, and
2. a triangulated category $DM$ of motives.

If the vanishing conjecture holds, there is an abelian category of motives, and $DM$ is its derived category.

See also

• Motive (algebraic geometry)

References

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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.
  
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  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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  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