In algebraic geometry, motives (or sometimes motifs, following French usage) is a theory proposed by Alexander Grothendieck in the 1960's to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically, a 'motif' is the 'cohomology essence' of a variety.

In the formulation of Grothendieck for smooth projective varieties, a motive is a triple, where X is a smooth projective variety,is an idempotentcorrespondence, and m an integer, however, such a triple contains almost no information outside the context of Grothendieck's category of pure motives, where amorphismfromtois given by a correspondence of degree. A more object focussed approach is taken byPierre Delignein 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, a-actiononand a"Frobenius" automorphismof. 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 a motive.

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

[point]
[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 admissiable equivalences are given by the definition of an adequate equivalence relation.

Definition of pure motives

Thecategoryof 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,

The objects ofare simply smooth projective varieties over k. The morphisms arecorrespondences. They generalize morphisms of varieties, which can be associated with their graphs in, to fixed dimensionalChow cycleson.

It will be useful to describe correspondences of arbitrary degree, although morphisms inare 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

wheredenotes the Chow-cycles of codimension k. Correspondences are often denoted using the "⊢"-notation, e.g.,. For anyandtheir composition is defined by

where the dot denotes the product in the Chow ring (i.e., intersection).

Returning to constructing the categorynotice that the composition of degree 0 correspondences is degree 0. Hence we define morphisms ofto be degree 0 correspondences.

The following association is a functor (heredenotes the graph of):

Just likethe categoryhas direct sums (X ⊕ Y := X ∐ Y) andtensor products(X ⊗ Y := X × Y). It is apreadditive category. The sum of morphisms is defined by

Second step: category of pure effective Chow motives,

The transition to motives is made by taking thepseudo-abelian envelopeof:

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 arigidpseudo-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 abeliantensor 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

Notation

Here we will fix a fieldkof characteristic0and letbe our coefficient ring. Setas the category of quasi-projective varieties overkare separated schemes of finite type. We will also letbe the subcategory of smooth varieties.

Smooth varieties with correspondences

Given asmooth varietyXand avarietyYcall anintegralclosed subschemewhich is finite overXand surjective over a component ofYa prime correspondence fromXtoY. Then, we can take the set of prime correspondences fromXtoYand construct a freeA-module. Its elements are called finite correspondences. Then, we can form an additive categorywhose 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.

Examples

Typical examples of prime correspondences come from the graphof a morphism of varieties.

Localizing the homotopy category

From here we can form thehomotopy categoryof bounded complexes of smooth correspondences. Here smooth varieties will be denoted. If welocalizethis category with respect to the smallest thick subcategory (meaning it is closed under extensions) containing morphisms

and

then we can form thetriangulated categoryof effective geometric motivesNote that the first class of morphisms are localizing-homotopies of varieties while the second will give the category of geometric mixed motives theMayer–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 setand call it the Tate motive. Taking the iterative tensor product lets us construct. If we have an effective geometric motiveMwe letdenoteMoreover, this behaves functorially and forms a triangulated functor. Finally, we can define the category of geometric mixed motivesas the category of pairsforMan effective geometric mixed motive andnan integer representing the twist by the Tate motive. The hom-groups are then the colimit

Explanation for non-specialists

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 non-linear 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 that 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.

The search for a universal cohomology

Each algebraic variety X has a corresponding motive [X], so the simplest examples of motives are:

[point]
[projective line] = [point] + [line]
[projective plane] = [plane] + [line] + [point]

These 'equations' hold in many situations, namely for de Rham cohomology and Betti cohomology, l-adic cohomology, the number of points over any finite field, and in multiplicative notation for local zeta-functions.

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:

Betti cohomology is defined for varieties over (subfields of) the complex numbers, it has the advantage of being defined over the integers and is a topological invariant
de Rham cohomology (for varieties over ) comes with a mixed Hodge structure, it is a differential-geometric invariant
l-adic cohomology (over any field of characteristic ≠ l) has a canonical Galois group action, i.e. has values in representations of the (absolute) Galois group
crystalline cohomology

All these cohomology theories share common properties, e.g. existence ofMayer-Vietoris sequences, homotopy invariancethe product of X with theaffine line) and others. Moreover, they are linked by comparison isomorphisms, for example Betti cohomologyof a smooth variety X overwith finite coefficients is isomorphic to l-adic 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

[projective line] = [line]+[point].

In particular, calculating the motive of any variety X directly gives all the information about the several Weil cohomology theories *HBetti(X), *HDR(X) etc.

Beginning with Grothendieck, people have tried to precisely define this theory for many years.

Motivic cohomology

Motivic cohomology itself had been invented before the creation of mixed motives by means of algebraic K-theory. The above category provides a neat way to (re)define it by

where n and m are integers andis the m-th tensor power of the Tate objectwhich in Voevodsky's setting is the complexshifted by –2, and [n] means the usualshiftin 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 πi ⊂ X × X inducing the canonical projectors H*(X) → Hi(X) ↣ H*(X) (for any Weil cohomology H) implies that every pure motive M decomposes in graded pieces of weight n: M = ⊕GrnM. The terminology weights comes from a similar decomposition of, say, de-Rham 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.

TheHodge conjecture, may be neatly reformulated using motives: it holdsiffthe Hodge realization mapping any pure motive with rational coefficients (over a subfieldof) to its Hodge structure is afull functor(rationalHodge structures). Here pure motive means pure motive with respect to homological equivalence.

Similarly, theTate conjectureis equivalent to: the so-called Tate realization, i.e. ℓ-adic cohomology is a faithful functor(pure motives up to homological equivalence, continuousrepresentationsof the absoluteGalois groupof the base field k), which takes values in semi-simple representations. (The latter part is automatic in the case of the Hodge analogue).

Tannakian formalism and motivic Galois group

To motivate the (conjectural) motivic Galois group, fix a field k and consider the functor

finite separable extensions K of k → non-empty finite sets with a (continuous) transitive action of the absolute Galois group of k

which maps K to the (finite) set of embeddings of K into an algebraic closure of k. InGalois theorythis functor is shown to be an equivalence of categories. Notice that fields are 0-dimensional. Motives of this kind are called Artin motives. By-linearizing the above objects, another way of expressing the above is to say that Artin motives are equivalent to finite-vector spaces together with an action of the Galois group.

The objective of the motivic Galois group is to extend the above equivalence to higher-dimensional varieties. In order to do this, the technical machinery ofTannakian categorytheory (going back toTannaka–Krein duality, but a purely algebraic theory) is used. Its purpose is to shed light on both theHodge conjectureand theTate conjecture, the outstanding questions inalgebraic cycletheory. Fix a Weil cohomology theory H. It gives a functor from *Mnum (pure motives using numerical equivalence) to finite-dimensional -vector 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 tensor-functor. Applying the Tannakian formalism, one concludes that *Mnum is equivalent to the category of representationsof analgebraic groupG, known as the motivic Galois group.

The motivic Galois group is to the theory of motives what the Mumford–Tate 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.)