In algebraic geometry, a motive (or sometimes motif, following French usage) is an abstract object introduced by Alexander Grothendieck in the 1960s as part of a proposed universal cohomology theory for algebraic varieties. The idea is that theories such as Betti cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology should arise as different realizations of the same underlying object. Thus, a motive of a variety is meant to capture the part of its geometry common to these cohomology theories.
For smooth varieties, pure motives can be constructed from algebraic correspondences and idempotent projections. The Chow motives give one example. For more general varieties, the (partially conjectural) theory of mixed motives extends this picture to relate motives to motivic cohomology and algebraic K-theory. Vladimir Voevodsky constructed triangulated categories of motives that provide much of the formalism expected of mixed motives.
The theory of motives is connected to algebraic cycles, Weil cohomology, and the study of motivic Galois groups. It also provides a unifying framework for open problems such as the Hodge conjecture and Tate conjecture.
Introduction The theory of motives was originally conjectured as an attempt to unify a rapidly multiplying array of cohomology theories, including Betti cohomology, de Rham cohomology, l-adic cohomology, and crystalline cohomology. The general hope is that equations like [projective line] = [line] + [point] [projective plane] = [plane] + [line] + [point] can be put on increasingly solid mathematical footing with a deep meaning. Of course, the above equations are already known to be true in many senses, such as in the sense of CW-complex where "+" corresponds to attaching cells, and in the sense of various cohomology theories, where "+" corresponds to the direct sum.
From another viewpoint, motives continue the sequence of generalizations from rational functions on varieties to divisors on varieties to Chow groups of varieties. The generalization happens in more than one direction, since motives can be considered with respect to more types of equivalence than rational equivalence. The admissible equivalences are given by the definition of an adequate equivalence relation.
Grothendieck and Deligne formulations In the formulation of Grothendieck for smooth projective varieties, a motive is a triple , where is a smooth projective variety, is an idempotent correspondence, and m an integer; however, such a triple contains almost no information outside the context of Grothendieck's category of pure motives, where a morphism from to is given by a correspondence of degree . A more object-focused approach is taken by Pierre Deligne in Le Groupe Fondamental de la Droite Projective Moins Trois Points. In that article, a motive is a "system of realisations" – that is, a tuple
consisting of modules
over the rings
respectively, various comparison isomorphisms
between the obvious base changes of these modules, filtrations , an action of the absolute Galois group on and a "Frobenius" automorphism of . This data is modeled on the cohomologies of a smooth projective -variety and the structures and compatibilities they admit, and gives an idea about what kind of information is contained in a motive.
Definition of pure motives The category of pure motives often proceeds in three steps. Below we describe the case of Chow motives , where k is any field.
First step: category of (degree 0) correspondences, Corr(k) The objects of are simply smooth projective varieties over k. The morphisms are correspondences. They generalize morphisms of varieties , which can be associated with their graphs in , to fixed dimensional Chow cycles on .
It will be useful to describe correspondences of arbitrary degree, although morphisms in are correspondences of degree 0. In detail, let X and Y be smooth projective varieties and consider a decomposition of X into connected components:
If , then the correspondences of degree r from X to Y are
where denotes the Chow-cycles of codimension k. Correspondences are often denoted using the "⊢"-notation, e.g., . For any and their composition is defined by
where the dot denotes the product in the Chow ring (i.e., intersection).
Returning to constructing the category notice that the composition of degree 0 correspondences is degree 0. Hence we define morphisms of to be degree 0 correspondences.
The following association is a functor (here denotes the graph of ):
Just like the category has direct sums () and tensor products (). It is a preadditive category. The sum of morphisms is defined by
Second step: category of pure effective Chow motives, Choweff(k)
The transition to motives is made by taking the pseudo-abelian envelope of :
.
In other words, effective Chow motives are pairs of smooth projective varieties X and idempotent correspondences α: X ⊢ X, and morphisms are of a certain type of correspondence:
Composition is the above defined composition of correspondences, and the identity morphism of (X, α) is defined to be α : X ⊢ X.
The association,
,
where ΔX := [idX] denotes the diagonal of X × X, is a functor. The motive [X] is often called the motive associated to the variety X.
As intended, Choweff(k) is a pseudo-abelian category. The direct sum of effective motives is given by
The tensor product of effective motives is defined by
where
The tensor product of morphisms may also be defined. Let f1 : (X1, α1) → (Y1, β1) and f2 : (X2, α2) → (Y2, β2) be morphisms of motives. Then let γ1 ∈ A(X1 × Y1) and γ2 ∈ A(X2 × Y2) be representatives of f1 and f2. Then
,
where πi : X1 × X2 × Y1 × Y2 → Xi × Yi are the projections.
Third step: category of pure Chow motives, Chow(k) To proceed to motives, we adjoin to Choweff(k) a formal inverse (with respect to the tensor product) of a motive called the Lefschetz motive. The effect is that motives become triples instead of pairs. The Lefschetz motive L is
.
If we define the motive 1, called the trivial Tate motive, by 1 := h(Spec(k)), then the elegant equation
holds, since
The tensor inverse of the Lefschetz motive is known as the Tate motive, T := L−1. Then we define the category of pure Chow motives by
.
A motive is then a triple
such that morphisms are given by correspondences
and the composition of morphisms comes from composition of correspondences.
As intended, is a rigid pseudo-abelian category.
Other types of motives In order to define an intersection product, cycles must be "movable" so we can intersect them in general position. Choosing a suitable equivalence relation on cycles will guarantee that every pair of cycles has an equivalent pair in general position that we can intersect. The Chow groups are defined using rational equivalence, but other equivalences are possible, and each defines a different sort of motive. Examples of equivalences, from strongest to weakest, are Rational equivalence Algebraic equivalence Smash-nilpotence equivalence (sometimes called Voevodsky equivalence) Homological equivalence (in the sense of Weil cohomology) Numerical equivalence The literature occasionally calls every type of pure motive a Chow motive, in which case a motive with respect to algebraic equivalence would be called a Chow motive modulo algebraic equivalence.
Mixed motives For a fixed base field k, the category of mixed motives is a conjectural abelian tensor category , together with a contravariant functor
taking values on all varieties (not just smooth projective ones as it was the case with pure motives). This should be such that motivic cohomology defined by
coincides with the one predicted by algebraic K-theory, and contains the category of Chow motives in a suitable sense (and other properties). The existence of such a category was conjectured by Alexander Beilinson.
Instead of constructing such a category, it was proposed by Deligne to first construct a category DM having the properties one expects for the derived category
.
Getting MM back from DM would then be accomplished by a (conjectural) motivic t-structure.
The current state of the theory is that we do have a suitable category DM. Already this category is useful in applications. Vladimir Voevodsky's Fields Medal-winning proof of the Milnor conjecture uses these motives as a key ingredient.
There are different definitions due to Hanamura, Levine and Voevodsky. They are known to be equivalent in most cases and we will give Voevodsky's definition below. The category contains Chow motives as a full subcategory and gives the "right" motivic cohomology. However, Voevodsky also shows that (with integral coefficients) it does not admit a motivic t-structure.
Geometric mixed motives
We will fix a field of characteristic and let be our coefficient ring.
Smooth varieties with correspondences Given a smooth variety and a variety call an integral closed subscheme which is finite over and surjective over a component of a prime correspondence from to . Then, we can take the set of prime correspondences from to and construct a free -module . Its elements are called finite correspondences. Then, we can form an additive category whose objects are smooth varieties and morphisms are given by smooth correspondences. The only non-trivial part of this "definition" is the fact that we need to describe compositions. These are given by a push-pull formula from the theory of Chow rings.
Typical examples of prime correspondences come from the graph of a morphism of varieties .
Localizing the homotopy category From here we can form the homotopy category of bounded complexes of smooth correspondences. Here smooth varieties will be denoted . If we localize this category with respect to the smallest thick subcategory (meaning it is closed under extensions) containing morphisms
and
then we can form the triangulated category of effective geometric motives Note that the first class of morphisms are localizing -homotopies of varieties while the second will give the category of geometric mixed motives the Mayer–Vietoris sequence.
Also, note that this category has a tensor structure given by the product of varieties, so .
Inverting the Tate motive Using the triangulated structure we can construct a triangle
from the canonical map . We will set and call it the Tate motive. Taking the iterative tensor product lets us construct . If we have an effective geometric motive we let denote Moreover, this behaves functorially and forms a triangulated functor. Finally, we can define the category of geometric mixed motives as the category of pairs for an effective geometric mixed motive and an integer representing the twist by the Tate motive. The hom-groups are then the colimit
Examples of motives
Tate motives There are several elementary examples of motives which are readily accessible. One of them being the Tate motives, denoted , , or , depending on the coefficients used in the construction of the category of motives. These are fundamental building blocks in the category of motives because they form the "other part" besides Abelian varieties.
Motives of curves
The motive of a curve can be explicitly understood with relative ease: their Chow ring is justfor any smooth projective curve , hence Jacobians embed into the category of motives.
Explanation for non-specialists