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In algebraic geometry, a motive (or sometimes motif) refers to 'some essential part of an algebraic variety'. Mathematically, the theory of motives is then the conjectural "universal" cohomology theory for such objects. 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 1960-1970, that hope for it remains unfulfilled. On the other hand, by a different route, motivic cohomology now has a technically-adequate definition.

There is therefore no well-established theory of motives yet. Instead, we know some facts and relationships between them that (as generally accepted among mathematicians) point to the existence of general underlying framework. Some mathematicians prefer the word motif to motive for the singular, following French usage.

Contents 1 Introduction 2 Pure motives 3 Mixed motives 4 Standard conjectures 5 Tannakian category approach 6 Remarks 7 References

[edit] Introduction

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 over the complex numbers
• for étale cohomology (l-adic cohomology) over fields of characteristic \neq l
• for the number of points over any finite field, and in multiplicative notation for local zeta-functions

???* for the conductor (which is of course trivial for above)

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 many things one may be interested in for an algebraic variety, such as computing the number of rational points in some finite field. This information is already given by the Weil conjectures (which are now proven), ???? and the standard conjectures are part of the effort to extend these results to characteristic 0.

There are different Weil cohomology theories, they apply in different situations and have values in different categories.

• 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 ℂ) comes with a mixed Hodge structure
• [[l-adic cohomology|étale cohomology]] (over any field) has a canonical Galois group action, i.e. has values in representations of the (absolute) Galois group
• crystalline cohomology

The above invariants reflect the structure of a variety in question. All these cohomology theories share common properties, e.g. existence of Mayer-Vietoris-sequences, homotopy invariance (H^*(X)≅H^*(X × A¹), the product of X with the affine line) and others. 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].

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

[edit] Pure motives

The category of pure motives is constructed in the following steps:

• Consider the category of smooth projective varieties over some field k. Instead of the usual morphisms of varieties, take the more general correspondences as morphisms, i.e. Hom(X, Y) consists of formal finite linear combinations ∑ n_i⋅V_i where the V_i are so-called algebraic cycles (irreducible, closed subsets) of the product X × Y. Morphisms f : X → Y of varieties are enclosed into this by taking the graph Γ_f ⊂ X × Y.
• The composition of such cycles is given by means of intersection theory. As usual, in order to get a well-defined intersection, the cycles have to intersect properly, i.e. the dimension of the intersection is as small as possible (depending on the dimensions of the intersecting cycles). The idea is to move the cycles appropriately, such that they do intersect properly. Technically, one chooes a so-called adequate equivalence relation on cycles and considers cycles up to this equivalence relation. Possible adequate equivalences include the following:

	definition	remarks
rational equivalence	Z ∼rat Z' if there is a cycle V on X × ℙ1, such that V ∩ X × {0} = Z and V ∩ X × {∞} = Z' .	the finest adequate equivalence relation. "∩" denotes intersection in the cycle-theoretic sense (i.e. with multiplicities)
algebraic equivalence	Z ∼alg Z' if there is a curve C and a cycle V on X × C, such that V ∩ X × {c} = Z and V ∩ X × {d} = Z' for two points c and d on the curve.
homological	for a given Weil cohomology H, Z ∼hom Z' if the image of the cycles under the cycle class map agrees	depends a priori of the choice of H
numerical equivalence	Z ∼num Z' if Z ∩ T = Z' ∩ T, where T is any cycle such that dim T = codim Z (so that the intersection is a linear combination of points)	the coarsest equivalence relation

• The resulting category has direct sums (given by disjoint unions of varieties) and tensor products (given by products), but is not abelian. Taking the Karoubi envelope (formally adding all kernels of projectors, i.e. morphisms p such that p ○ p = p) of this category yields a pseudoabelian category; this is the category of effective motives.
• Using the (formally adjoined) kernel of the projector p: ℙ¹ → point → ℙ¹, the projective line ℙ¹ canonically decomposes as a direct sum [point]⊕L, where L is the so-called Lefschetz motive. Morally, it corresponds to chopping off the degree zero part of the cohomology H^*(ℙ¹). The tensor inverse of L, the Tate motive, is then formally adjoined to yield the category of pure motives.

If the equivalence relation is chosen to be rational equivalence, it is also called category of Chow motives, because morphisms in this category are given by Chow groups.

[edit] Mixed motives

For a fixed base field k, the category of mixed motives is a conjectural abelian tensor category MM(k), together with a contravariant functor Var(k) → MM(X) (all varieties, not just smooth projective ones as it was the case with pure motives) such that motivic cohomology defined by Ext^*_MM(1, ?) 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 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 D^b(MM(k)). Getting MM back from DM would then be acomplished by a (conjectural) motivic t-structure.

The current state of the theory is, that we do have a category DM. Already this category is very useful. 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.

• Start with the category Sm of smooth varieties over a perfect field. Similarly to the construction of pure motives above, instead of usual morphisms smooth correspondences are allowed. Compared to the (quite general) cycles used aboe, the definition of these correspondences is more restrictive; in particular they always intersect properly, so no moving of cycles and hence no equivalence relation is needed to get a well-defined composition of correspondences. This category is denoted SmCor, it is additive.
• As a technical intermediate step, take the bounded homotopy category K^b(SmCor) of complexes of smooth schemes and correspondences.
• Apply localization of categories to force any variety X to be isomorphic to X × A¹ (product with the affine line) and also, that a Mayer-Vietoris-sequence holds, i.e. X = U ∪ V (union of two open subvarieties) shall be isomorphic to U ∩ V → U ⊔ V.
• Finally, as above, take the pseudo-abelian envelope.

The resulting category is called category of effective geometric motives. Again, formally inverting the Tate object, one gets the category DM of geometric motives.

[edit] Standard conjectures

The standard conjectures are conjectures about the category of pure motives. So, in this subsection motive will mean pure motive. All of the conjectures deal with "algebraic" cohomology classes, which means a morphism on cohomology H*(X) → H*(X) induced by an algebraic cycle on the product X × X via the cycle class map (which is part of the structure of a Weil cohomology theory).

Lefschetz type standard conjecture, also called conjecture B: To formulate the conjecture, consider a fixed Weil cohomology theory H. One of the axioms of a Weil theory is the so-called strong Lefschetz theorem (or axiom): for a fixed smooth hyperplane section W = H ∩ X (H some hyperplane in the ambient projective space ℙ^N containing the given smooth projective variety X), the Lefschetz operator L : Hⁱ(X) → Hⁱ⁺², which is defined by intersecting cohomology classes with W gives an isomorphism

L^n-i: Hⁱ(X) → H^2n-i(X) (i ≦ n = dim X). Define Λ : Hⁱ(X) → H^i-2(X) for 'i ≦ n be the composition (L^n-i+2)^-1 ○ L ○ (L^n-i)^-1 and Λ : H^2n-i+2(X) → H^2n-i(X) by (L^n-i) ○ L ○ (L^n-i+2)^{-1 -1}.

The Lefschetz conjecture states that the operator Λ is induced by an algebraic cycle.

The Lefschetz conjecture implies the Künneth type standard conjecture, also called conjecture C: Again, fix a Weil cohomology theory H. It is conjectured that the projectors H^*(X) ↠ Hⁱ(X) ↣ H^*(X) are algebraic, i.e. induced by a cycle πⁱ ⊂ X × X with rational coefficients. This implies that every motive M decomposes in graded pieces of weight n: M = ⊕Gr_nM. The terminology weights comes from a similar decomposition of, say, de-Rham cohomology of smooth projective varieties, see Hodge theory. The conjecture is known to hold for curves, surfaces and abelian varieties.

Conjecture D states that numerical equivalence and homological equivalence agree. (In particular the latter does not depend on the choice of the Weil cohomology theory). This conjecture implies the Lefschetz conjecture. The converse holds if the Hodge conjecture holds:

it states the positive definiteness of the cup product pairing on primitive algebraic cohomology classes or, equivalently, that every Hodge class is algebraic.

The above conjectures are commonly considered to be very hard. 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 above conjectures to hold.

[edit] Tannakian category approach

Through the technical machinery of Tannakian category theory (going back to Tannaka-Krein duality, but a purely algebraic theory), categories of motives are or should be equivalent to the category of linear representations of an algebraic group (or pro-algebraic group, if there is not a finite set of generating objects). An important application is to define a motivic Galois group; it is to the theory of motives what the Mumford-Tate group is to Hodge theory. Its purpose is to shed light on both the Hodge conjecture and the Tate conjecture, the outstanding questions in algebraic cycle 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.)

[edit] Remarks

The word 'motivic' occurring in the phrase motivic Galois group and elsewhere signifies a conceptual connection to the theory, but it must be accepted that the theory may not yet be in final form. See also motivic polylogarithm.

[edit] References

• Y. André: Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Société Mathématique de France, 2004
• U. Jannsen, S. Kleiman, J.-P. Serre (eds.): Motives, Proc. of Symposia in Pure Math., Volume 55.
  • L. Breen: Tannakian categories.
  • S. Kleiman: The standard conjectures.
  • A. Scholl: Classical motives. (detailed exposition of Chow motives)
• S. Kleiman: Motives, Algebraic Geometry, Oslo 1970, (F. Oort, ed.). (adequate equivalence relations on cycles).
• B. Mazur: What is... a Motive? AMS Notices Vol. 51, No. 10. (motives-for-dummies text).
• J.-P. Serre: Motifs, Astérisque No. 198-200 (1991), 11, 333--349 (1992). (non-technical introduction to motives).
• V. Voevodsky. Lectures in Motivic Cohomology. Notes by Carlo Mazza and Charles Weibel.
• V. Voevodsky, A. Suslin, E. Friedlander: Cycles, Transfers and Motivic Homology Theories, Princeton University Press, 2000, available at [1]. (Voevodsky's definition of mixed motives. Highly technical).