In algebraic geometry, a motive (or sometimes motif, following French usage) denotes refers to 'some essential part of an algebraic variety'. To date, pure motives have been defined, while conjectural mixed motives have not. Pure motives are triples (X, p, m), where X is a smooth projective variety, p : X ⊢ X is an idempotent correspondence, and m an integer. A morphism from (X, p, m) to (Y, q, n) is given by a correspondence of degree n  m.
As far as mixed motives, mathematicians are working to find a suitable definition which will then provide a "universal" cohomology theory. In terms of category theory, it was intended to have a definition via splitting idempotents in a category of algebraic correspondences. The way ahead for that definition has been blocked for some decades, by the failure to prove the standard conjectures on algebraic cycles. This prevents the category from having 'enough' morphisms, as can currently be shown. While the category of motives was supposed to be the universal Weil cohomology much discussed in the years 19601970, that hope for it remains unfulfilled. On the other hand, by a quite different route, motivic cohomology now has a technicallyadequate definition.
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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, ladic cohomology, and crystalline cohomology. The general hope is that equations like
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 CWcomplex 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 admissiable equivalences are given by the definition of an adequate equivalence relation.
The category of pure motives often proceeds in three steps. Below we describe the case of Chow motives Chow(k), where k is any field.
The objects of Corr(k) are simply smooth projective varieties over k. The morphisms are correspondences. They generalize morphisms of varieties X → Y, which can be associated with their graphs in X × Y, to fixed dimensional Chow cycles on X × Y.
It will be useful to describe correspondences of arbitrary degree, although morphisms in Corr(k) are correspondences of degree 0. In detail, let X and Y be smooth projective varieties, let be the decomposition of X into connected components, and let d_{i} := dim X_{i}. If r ∈ Z, then the correspondences of degree r from X to Y are
Correspondences are often denoted using the "⊢"notation, e.g., α : X ⊢ Y. For any α ∈ Corr^{r}(X, Y) and β ∈ Corr^{s}(Y, Z), their composition is defined by
where the dot denotes the product in the Chow ring (i.e., intersection).
Returning to constructing the category Corr(k), notice that the composition of degree 0 correspondences is degree 0. Hence we define morphisms of Corr(k) to be degree 0 correspondences.
The association,
where Γ_{f} ⊆ X × Y is the graph of f : X → Y, is a functor.
Just like SmProj(k), the category Corr(k) has direct sums () and tensor products (X ⊗ Y := X × Y). It is a preadditive category (see the convention for preadditive vs. additive in the preadditive category article.) The sum of morphisms is defined by
The transition to motives is made by taking the pseudoabelian envelope of Corr(k):
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 Δ_{X} := [id_{X}] denotes the diagonal of X × X, is a functor. The motive [X] is often called the motive associated to the variety X.
As intended, Chow^{eff}(k) is a pseudoabelian category. The direct sum of effective motives is given by
The tensor product of effective motives is defined by
The tensor product of morphisms may also be defined. Let f_{1} : (X_{1}, α_{1}) → (Y_{1}, β_{1}) and f_{2} : (X_{2}, α_{2}) → (Y_{2}, β_{2}) be morphisms of motives. Then let γ_{1} ∈ A^{*}(X_{1} × Y_{1}) and γ_{1} ∈ A^{*}(X_{1} × Y_{1}) be representatives of f_{1} and f_{2}. Then
where π_{i} : X_{1} × X_{2} × Y_{1} × Y_{2} → X_{i} × Y_{i} are the projections.
To proceed to motives, we adjoin to Chow^{eff}(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 pleasant equation
holds, since 1 ≅ (P^{1}, P^{1} × pt). 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 (X ∈ SmProj(k), p : X ⊢ X, n ∈ Z) such that p ˆ p = p. Morphisms are given by correspondences
and the composition of morphisms comes from composition of correspondences.
As intended, Chow(k) is a rigid pseudoabelian category.
In order to define an intersection product, cycles must be "movable" so we can intersect them in general position. Choosing an 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
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.
For a fixed base field k, the category of mixed motives is a conjectural abelian tensor category MM(k), 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 Ktheory, and contains the category of Chow motives in a suitable sense (and other properties). The existence of such a category was conjectured by Beilinson. This category is yet to be constructed.
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 tstructure.
The current state of the theory is that we do have a suitable category DM. Already this category is useful in applications. Voevodsky's Fields medalwinning 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 tstructure.
The resulting category is called the category of effective geometric motives. Again, formally inverting the Tate object, one gets the category DM of geometric motives.
A commonly applied technique in mathematics is to study objects carrying a particular structure by introducing a category whose morphisms preserve this structure. Then one may ask, when are two given objects isomorphic and ask for a "particularly nice" representative in each isomorphism class. The classification of algebraic varieties, i.e. application of this idea in the case of algebraic varieties, is very difficult due to the highly nonlinear structure of the objects. The relaxed question of studying varieties up to birational isomorphism has led to the field of birational geometry. Another way to handle the question is to attach to a given variety X an object of more linear nature, i.e. an object amenable to the techniques of linear algebra, for example a vector space. This "linearization" goes usually under the name of cohomology.
There are several important cohomology theories which reflect different structural aspects of varieties. The (partly conjectural) theory of motives is an attempt to find a universal way to linearize algebraic varieties, i.e. motives are supposed to provide a cohomology theory which embodies all these particular cohomologies. For example, the genus of a smooth projective curve C which is an interesting invariant of the curve, is an integer, which can be read off the dimension of the first Betti cohomology group of C. So, the motive of the curve should contain the genus information. Of course, the genus is a rather coarse invariant, so the motive of C is more than just this number.
Each algebraic variety X has a corresponding motive [X], so the simplest examples of motives are:
These 'equations' hold in many situations, namely for de Rham cohomology and Betti cohomology, ladic cohomology, the number of points over any finite field, and in multiplicative notation for local zetafunctions.
The general idea is that one motive has the same structure in any reasonable cohomology theory with good formal properties; in particular, any Weil cohomology theory will have such properties. There are different Weil cohomology theories, they apply in different situations and have values in different categories, and reflect different structural aspects of the variety in question:
All these cohomology theories share common properties, e.g. existence of MayerVietorissequences, homotopy invariance (H^{∗}(X)≅H^{∗}(X × A^{1}), the product of X with the affine line) and others. Moreover, they are linked by comparison isomorphisms, for example Betti cohomology H_{Betti}^{∗}(X, ℤ/n) of a smooth variety X over ℂ with finite coefficients is isomorphic to ladic cohomology with finite coefficients.
The theory of motives is an attempt to find a universal theory which embodies all these particular cohomologies and their structures and provides a framework for "equations" like
In particular, calculating the motive of any variety X directly gives all the information about the several Weil cohomology theories H_{Betti}^{∗}(X), H_{DR}^{∗}(X) etc.
Beginning with Grothendieck, people have tried to precisely define this theory for many years.
Motivic cohomology itself had been invented before the creation of mixed motives by means of algebraic Ktheory. The above category provides a neat way to (re)define it by
where n and m are integers and ℤ(m) is the mth tensor power of the Tate object ℤ(1), which in Voevodsky's setting is the complex ℙ^{1} → point shifted by 2, and [n] means the usual shift in the triangulated category.
The standard conjectures were first formulated in terms of the interplay of algebraic cycles and Weil cohomology theories. The category of pure motives provides a categorical framework for these conjectures.
The standard conjectures are commonly considered to be very hard and are open in the general case. Grothendieck, with Bombieri, showed the depth of the motivic approach by producing a conditional (very short and elegant) proof of the Weil conjectures (which are proven by different means by Deligne), assuming the standard conjectures to hold.
For example, the Künneth standard conjecture, which states the existence of algebraic cycles π^{i} ⊂ X × X inducing the canonical projectors H^{∗}(X) ↠ H^{i}(X) ↣ H^{∗}(X) (for any Weil cohomology H) implies that every pure motive M decomposes in graded pieces of weight n: M = ⊕Gr_{n}M. The terminology weights comes from a similar decomposition of, say, deRham cohomology of smooth projective varieties, see Hodge theory.
Conjecture D, stating the concordance of numerical and homological equivalence, implies the equivalence of pure motives with respect to homological and numerical equivalence. (In particular the former category of motives would not depend on the choice of the Weil cohomology theory). Jannsen (1992) proved the following unconditional result: the category of (pure) motives over a field is abelian and semisimple if and only if the chosen equivalence relation is numerical equivalence.
The Hodge conjecture, may be neatly reformulated using motives: it holds iff the Hodge realization mapping any pure motive with rational coefficients (over a subfield k of ℂ) to its Hodge structure is a full functor H : M(k)_{ℚ} → HS_{ℚ} (rational Hodge structures). Here pure motive means pure motive with respect to homological equivalence.
Similarly, the Tate conjecture is equivalent to: the socalled Tate realization, i.e. ℓadic cohomology is a faithful functor H : M(k)_{ℚ}ℓ → Rep_{ℓ}(Gal(k)) (pure motives up to homological equivalence, continuous representations of the absolute Galois group of the base field k), which takes values in semisimple representations. (The latter part is automatic in the case of the Hodge analogue).
To motivate the (conjectural) motivic Galois group, fix a field k and consider the functor
which maps K to the (finite) set of embeddings of K into an algebraic closure of k. In Galois theory this functor is shown to be an equivalence of categories. Notice that fields are 0dimensional. Motives of this kind are called Artin motives. By ℚlinearizing the above objects, another way of expressing the above is to say that Artin motives are equivalent to finite ℚvector spaces together with an action of the Galois group.
The objective of the motivic Galois group is to extend the above equivalence to higherdimensional varieties. In order to do this, the technical machinery of Tannakian category theory (going back to TannakaKrein duality, but a purely algebraic theory) is used. Its purpose is to shed light on both the Hodge conjecture and the Tate conjecture, the outstanding questions in algebraic cycle theory. Fix a Weil cohomology theory H. It gives a functor from M_{num} (pure motives using numerical equivalence) to finitedimensional ℚvector spaces. It can be shown that the former category is a Tannakian category. Assuming the equivalence of homological and numerical equivalence, i.e. the above standard conjecture D, the functor H is an exact faithful tensorfunctor. Applying the Tannakian formalism, one concludes that M_{num} is equivalent to the category of representations of an algebraic group G, which is called motivic Galois group.
It is to the theory of motives what the MumfordTate group is to Hodge theory. Again speaking in rough terms, the Hodge and Tate conjectures are types of invariant theory (the spaces that are morally the algebraic cycles are picked out by invariance under a group, if one sets up the correct definitions). The motivic Galois group has the surrounding representation theory. (What it is not, is a Galois group; however in terms of the Tate conjecture and Galois representations on étale cohomology, it predicts the image of the Galois group, or, more accurately, its Lie algebra.)
