MOTIVE (ALGEBRAIC GEOMETRY)

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

Introduction

Pure motives

Mixed motives

Motivic cohomology

Conjectures related to motives

Tannakian formalism and motivic Galois group

References

Introduction

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, i.e. objects given by polynomial equations 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 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:

★ [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 ℂ) 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 of Mayer-Vietoris-sequences, homotopy invariance (''H
^★ (X)≅H
^★ (X × A¹)'', 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 ''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 ''H_Betti
^★ (X)'', ''H_DR
^★ (X)'' etc.
Beginning with Grothendieck, people have tried to precisely define this theory for many years.

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'' of cycles and considers cycles up to this equivalence relation. Possible adequate equivalences include (ranging from the finest to the coarsest possible relation) rational, algebraic, homological and numerical equivalence.

★ The resulting category has direct sums (coproducts geometrically given by disjoint unions of varieties) and tensor product (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.

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'')
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
:''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 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. 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 above, 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 the ''category of effective geometric motives''. Again, formally inverting the Tate object, one gets the category ''DM'' of 'geometric motives'.

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
:''Hⁿ(X, m) := Hⁿ(X, ℤ(m)) := Hom_DM(X, ℤ(m)[n])'',
where ''n'' and ''m'' are integers and ''ℤ(m)'' is the ''m''-th tensor power of the Tate object ''ℤ(1)'', which in Voevodsky's setting is the complex ℙ¹ → ''point'' 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, as of 2007 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 = ''⊕Gr_n''M''. 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).
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 so-called Tate realization, i.e. ℓ-adic cohomology is a faithful functor
''H : M(k)''_ℚℓ → ''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 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'' → ''finite sets with a (continuous) 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''.
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 ℚ-linearizing the above objects, another way of expressing the above is to say that Artin motives are equivalent to
finite ℚ-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 ''M_num'' (pure motives using numerical equivalence) to finite-dimensional ℚ-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 ''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 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.)

References

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★ (technical introduction with comparatively short proofs)

★

★
★ L. Breen: ''Tannakian categories''.

★
★ S. Kleiman: ''The standard conjectures''.

★
★ A. Scholl: ''Classical motives''. (detailed exposition of Chow motives)

★ (adequate equivalence relations on cycles).

★ (motives-for-dummies text).

★

★ (non-technical introduction to motives).

★ (Voevodsky's definition of mixed motives. Highly technical).