Motivic cohomology
Encyclopedia
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
.
of the
conjectural category of motives. The most successful has been Vladimir Voevodsky
's
construction. By applying techniques from homotopy theory and K-theory
to algebraic geometry, Voevodsky constructed a bigraded motivic cohomology theory
for algebraic varieties. It is not known whether these groups vanish for negative
; 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:
If the vanishing conjecture holds, there is an abelian category
of motives, and
is its derived category.
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
theory in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the existence of which was first conjectured by Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
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
Standard conjectures on algebraic cycles
In mathematics, the standard conjectures about algebraic cycles is a package of several conjectures describing the relationship of algebraic cycles and Weil cohomology theories. One of the original applications of these conjectures, envisaged by Alexander Grothendieck, was to prove that his...
, in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
. It had a basis in category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
for drawing consequences from those conjectures; Grothendieck and Enrico Bombieri
Enrico Bombieri
Enrico Bombieri is a mathematician who has been working at the Institute for Advanced Study in Princeton, New Jersey. Bombieri's research in number theory, algebraic geometry, and mathematical analysis have earned him many international prizes --- a Fields Medal in 1974 and the Balzan Prize in 1980...
showed the depth of this approach by deriving a conditional proof
Conditional proof
A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent....
of the Weil conjectures
Weil conjectures
In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields....
by this route. The standard conjectures, however, resisted proof.
This left the motive (motif in French) theory as having heuristic status. Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
, 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
Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
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
De Rham cohomology
In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes...
, and crystalline cohomology
Crystalline cohomology
In mathematics, crystalline cohomology is a Weil cohomology theory for schemes introduced by and developed by . Its values are modules over rings of Witt vectors over the base field....
. 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
Weil cohomology theory
In algebraic geometry, a Weil cohomology or Weil cohomology theory is a cohomology satisfying certain axioms concerning the interplay of algebraic cycles and cohomology groups. The name is in honor of André Weil...
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
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
. 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
Algebraic K-theory
In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....
.
Recent progress
In the mid-1990s, several people proposed candidates for the derived categoryDerived category
In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C...
of the
conjectural category of motives. The most successful has been Vladimir Voevodsky
Vladimir Voevodsky
Vladimir Voevodsky is a Russian American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal in 2002.- Biography :...
's
construction. By applying techniques from homotopy theory and K-theoryK-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
to algebraic geometry, Voevodsky constructed a bigraded motivic cohomology theory
for algebraic varieties. It is not known whether these groups vanish for negative
; this property is known as thevanishing 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 categoryModel categoryIn mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes...
, and - a triangulated categoryTriangulated categoryA triangulated category is a mathematical category satisfying some axioms that are based on the properties of the homotopy category of spectra, and the derived category of an abelian category. A t-category is a triangulated category with a t-structure.- History :The notion of a derived category...
of motives.
If the vanishing conjecture holds, there is an abelian category
Abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
of motives, and
is its derived category.