Motive (algebraic geometry)

In algebraic geometry, motives (or sometimes motifs, following French usage) is a theory proposed by Alexander Grothendieck in the 1960s 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 formula_1, where "X" is a smooth projective variety, formula_2 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 formula_1 to formula_4 is given by a correspondence of degree formula_5. A more object focussed 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 formula_10, a formula_11-action formula_12 on formula_13 and a "Frobenius" automorphism formula_14 of formula_15. This data is modeled on the cohomologies of a smooth projective formula_16-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
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.
Definition of pure motives.
The category of pure motives often proceeds in three steps. Below we describe the case of Chow motives formula_17, where "k" is any field.
First step: category of (degree 0) correspondences, formula_18.
The objects of formula_18 are simply smooth projective varieties over "k". The morphisms are correspondences. They generalize morphisms of varieties formula_20, which can be associated with their graphs in formula_21, to fixed dimensional Chow cycles on formula_21.
It will be useful to describe correspondences of arbitrary degree, although morphisms in formula_18 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 formula_25, then the correspondences of degree "r" from "X" to "Y" are
where formula_27 denotes the Chow-cycles of codimension "k". Correspondences are often denoted using the "⊢"-notation, e.g., formula_28. For any formula_29 and formula_30 their composition is defined by
where the dot denotes the product in the Chow ring (i.e., intersection).
Returning to constructing the category formula_32 notice that the composition of degree 0 correspondences is degree 0. Hence we define morphisms of formula_18 to be degree 0 correspondences.
The following association is a functor (here formula_34 denotes the graph of formula_35):
Just like formula_37 the category formula_18 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, formula_40.
The transition to motives is made by taking the pseudo-abelian envelope of formula_18:
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 "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"2 ∈ "A"("X"2 × "Y"2) be representatives of "f1" and "f2". Then
where "πi" : "X"1 × "X"2 × "Y"1 × "Y"2 → "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, formula_17 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
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 formula_57, 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 field of characteristic and let formula_61 be our coefficient ring. Set formula_62 as the category of quasi-projective varieties over are separated schemes of finite type. We will also let formula_63 be the subcategory of smooth varieties.
Smooth varieties with correspondences.
Given a smooth variety and a variety call an integral closed subscheme formula_64 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 formula_65. Its elements are called finite correspondences. Then, we can form an additive category formula_66 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.
Examples of correspondences.
Typical examples of prime correspondences come from the graph formula_67 of a morphism of varieties formula_68.
Localizing the homotopy category.
From here we can form the homotopy category formula_69 of bounded complexes of smooth correspondences. Here smooth varieties will be denoted formula_70. 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 formula_73 Note that the first class of morphisms are localizing formula_74-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 formula_75.
Inverting the Tate motive.
Using the triangulated structure we can construct a triangle
from the canonical map formula_77. We will set formula_78 and call it the Tate motive. Taking the iterative tensor product lets us construct formula_79. If we have an effective geometric motive we let formula_80 denote formula_81 Moreover, this behaves functorially and forms a triangulated functor. Finally, we can define the category of geometric mixed motives formula_82 as the category of pairs formula_83 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 formula_85, formula_86, or formula_87, 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 justformula_88for any smooth projective curve formula_89, hence Jacobians embed into the category of motives.
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:
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:
All these cohomology theories share common properties, e.g. existence of Mayer-Vietoris sequences, homotopy invariance formula_91 the product of "X" with the affine line) and others. Moreover, they are linked by comparison isomorphisms, for example Betti cohomology formula_92 of a smooth variety "X" over formula_90 with 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 
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.
"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 and formula_95 is the "m"-th tensor power of the Tate object formula_96 which in Voevodsky's setting is the complex formula_97 shifted by –2, and "[n]" means the usual shift in the triangulated category.
Conjectures related to motives.
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.
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 formula_98 of formula_90) to its Hodge structure is a full functor formula_100 (rational Hodge structures). Here pure motive means pure motive with respect to homological equivalence.
Similarly, the Tate conjecture is equivalent to: the so-called Tate realization, i.e. ℓ-adic cohomology, is a full functor formula_101 (pure motives up to homological equivalence, continuous representations of the absolute Galois group of 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
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 0-dimensional. Motives of this kind are called "Artin motives". By formula_16-linearizing the above objects, another way of expressing the above is to say that Artin motives are equivalent to finite formula_16-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 of Tannakian category theory (going back to Tannaka–Krein 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 "Mnum" (pure motives using numerical equivalence) to finite-dimensional formula_16-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 representations of an algebraic group "G", 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.)