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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]]

Revision as of 20:28, 3 March 2014

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