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'''Motivic cohomology''' is a [[Cohomology|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.
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This left the motive (''motif'' in French) theory as having heuristic status. [[Jean-Pierre Serre|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, [[Motive (mathematics)|motive]]s 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.  
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Grothendieck gave a solution for [[Weil cohomology theory|Weil cohomology theories]] over
a field in 1967. This involved extending the category of smooth [[projective variety|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 cycle]]s 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 <math>DM</math> construction. By applying techniques from [[homotopy theory]] and [[K-theory]] to algebraic geometry, Voevodsky constructed a [[bigraded]] motivic cohomology theory
 
:<math>H^{p,q}(X)</math>
 
for algebraic varieties. It is not known whether these groups vanish for negative <math>p</math>; 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:
 
# a homotopy theory for algebraic varieties, in the form of a [[model category]], and
# a [[triangulated category]] <math>DM</math> of motives.
 
If the vanishing conjecture holds, there is an [[abelian category]] of motives, and <math>DM</math> is its derived category.
 
==See also==
* [[Motive (algebraic geometry)]]
 
==References==
* {{Citation | last1=Kleiman | first1=Steven L. | editor1-last=Oort | editor1-first=F. | title=Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer-School in Math., Oslo, 1970) | publisher=Wolters-Noordhoff | location=Groningen | mr=0382267  | year=1972 | chapter=Motives | pages=53–82}}
* {{Citation | editor1-last=Serre | editor1-first=Jean-Pierre | editor1-link=Jean-Pierre Serre | editor2-last=Jannsen | editor2-first=Uwe | editor3-last=Kleiman | editor3-first=Steven L. | title=Motives | publisher=[[American Mathematical Society]] | location=Providence, R.I. | isbn=978-0-8218-1637-0 | year=1994 | author=Uwe Jannsen ... eds.}}
* {{Citation | author=C. Mazza, V. Voevodsky and C. Weibel | title= Lectures in Motivic Cohomology  | url= http://math.rutgers.edu/~weibel/motiviclectures.html | series=Clay Math Monographs vol.2| publisher= [[American Mathematical Society]]| isbn=978-0-8218-3847-1 |  year=2006 | author=Carlo Mazza, Vladimir Voevodsky, Charles Weibel.}}
 
{{DEFAULTSORT:Motivic Cohomology}}
[[Category:Cohomology theories]]
[[Category:Topological methods of algebraic geometry]]

Latest revision as of 16:43, 20 April 2014

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If you would like more information on any of the home business ideas listed above or you would like to get more ideas make sure to visit HomeBizScout.

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A home run business opportunity can free you from worries over the national economy and losing your employed job. You are able to position your financial safety on something more substantial than a changeable corporation that could let you go the next day. Earn supplemental income by choosing among the best work from home ideas and give up your normal line of work in a few months time. This will mind the subject of your contempt becoming noticeable to the buyers and furnish you with an online revenue.

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