Motive (algebraic geometry)
Encyclopedia
In algebraic geometry
, a motive (or sometimes motif, following French
usage) denotes '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, following Alexander Grothendieck
, 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 correspondence
s. 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 quite different route, motivic cohomology
now has a technically-adequate definition.
, 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 admissiable equivalences are given by the definition of an adequate equivalence relation
.
of pure motives often proceeds in three steps. Below we describe the case of Chow motives Chow(k), where k is any field.
. 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 di := dim Xi. 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 ∈ Corrr(X, Y) and β ∈ Corrs(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 Γf ⊆ 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
.
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 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
.
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.
.
If we define the motive 1, called the trivial Tate motive, by 1 := h(Spec(k)), then the pleasant equation
holds, since 1 ≅ (P1, P1 × 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
pseudo-abelian category.
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.
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 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 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.
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.
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 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.
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-function
s.
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 (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 HBetti∗(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
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.
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 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.
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 k of ℂ) to its Hodge structure is a full functor H : M(k)ℚ → HSℚ (rational Hodge structure
s). 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 → Repℓ(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).
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 Mnum (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 Mnum 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
.)
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, a motive (or sometimes motif, following French
French language
French is a Romance language spoken as a first language in France, the Romandy region in Switzerland, Wallonia and Brussels in Belgium, Monaco, the regions of Quebec and Acadia in Canada, and by various communities elsewhere. Second-language speakers of French are distributed throughout many parts...
usage) denotes 'some essential part of an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
'. 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, following Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
, mathematicians are working to find a suitable definition which will then provide a "universal" cohomology theory. In terms of category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, it was intended to have a definition via splitting idempotents in a category of algebraic correspondence
Correspondence (mathematics)
In mathematics and mathematical economics, correspondence is a term with several related but not identical meanings.* In general mathematics, correspondence is an alternative term for a relation between two sets...
s. The way ahead for that definition has been blocked for some decades, by the failure to prove the standard conjectures on algebraic cycles
Standard conjectures on algebraic cycles
In mathematics, the standard conjectures about algebraic cycles is a package of several conjectures describing the relationship of algebraic cycles and Weil cohomology theories. One of the original applications of these conjectures, envisaged by Alexander Grothendieck, was to prove that his...
. 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 quite different route, motivic cohomology
Motivic cohomology
Motivic cohomology is a cohomological theory in mathematics, the existence of which was first conjectured by Alexander Grothendieck during the 1960s. At that time, it was conceived as a theory constructed on the basis of the so-called standard conjectures on algebraic cycles, in algebraic geometry...
now has a technically-adequate definition.
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 cohomologyDe Rham cohomology
In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes...
, l-adic cohomology
Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
, and crystalline cohomology
Crystalline cohomology
In mathematics, crystalline cohomology is a Weil cohomology theory for schemes introduced by and developed by . Its values are modules over rings of Witt vectors over the base field....
. 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
Adequate equivalence relation
In algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on algebraic cycles of smooth projective varieties used to obtain a well-working theory of such cycles, and in particular, well-defined intersection products. Samuel formalized the concept of...
.
Definition of pure motives
The categoryCategory (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
of pure motives often proceeds in three steps. Below we describe the case of Chow motives Chow(k), where k is any field.
First step: category of (degree 0) correspondences, Corr(k)
The objects of Corr(k) are simply smooth projective varieties over k. The morphisms are correspondencesCorrespondence (mathematics)
In mathematics and mathematical economics, correspondence is a term with several related but not identical meanings.* In general mathematics, correspondence is an alternative term for a relation between two sets...
. They generalize morphisms of varieties X → Y, which can be associated with their graphs in X × Y, to fixed dimensional Chow cycles
Chow ring
In algebraic geometry, the Chow ring of an algebraic variety is an algebraic-geometric analogue of the cohomology ring of the variety considered as a topological space: its elements are formed out of actual subvarieties and its multiplicative structure is derived from the intersection of...
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 di := dim Xi. 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 ∈ Corrr(X, Y) and β ∈ Corrs(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 Γf ⊆ 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 productsMonoidal category
In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
(X ⊗ Y := X × Y). It is a preadditive category (see the convention for preadditive vs. additive in the preadditive category
Preadditive category
In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups...
article.) The sum of morphisms is defined by
.Second step: category of pure effective Chow motives, Choweff(k)
The transition to motives is made by taking the pseudo-abelian envelopeKaroubi envelope
In mathematics the Karoubi envelope of a category C is a classification of the idempotents of C, by means of an auxiliary category. Taking the Karoubi envelope of a preadditive category gives a pseudo-abelian category, hence the construction is sometimes called the pseudo-abelian completion...
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 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
Pseudo-abelian category
In mathematics, specifically in category theory, a pseudo-abelian category is a category that is preadditive and is such that every idempotent has a kernel...
. The direct sum of effective motives is given by
,The tensor product
Monoidal category
In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
of effective motives is defined by
.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 pleasant equation
holds, since 1 ≅ (P1, P1 × 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
Rigid category
In category theory, a branch of mathematics, a rigid category is a monoidal category where every object is rigid, that is, has a dual X* and a morphism 1 → X ⊗ X* satisfying natural conditions. The category is called right rigid or left rigid according to whether it has right duals or...
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- 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 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
Alexander Beilinson
Alexander A. Beilinson is the David and Mary Winton Green University Professor at the University of Chicago and works on mathematics. His research has spanned representation theory, algebraic geometry and mathematical physics...
. 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
Derived category
In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C...
- Db(MM(k)).
Getting MM back from DM would then be accomplished by a (conjectural) motivic t-structure
Triangulated category
A triangulated category is a mathematical category satisfying some axioms that are based on the properties of the homotopy category of spectra, and the derived category of an abelian category. A t-category is a triangulated category with a t-structure.- History :The notion of a derived category...
.
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
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
-winning proof of the Milnor conjecture
Milnor conjecture
In mathematics, the Milnor conjecture was a proposal by of a description of the Milnor K-theory of a general field F with characteristic different from 2, by means of the Galois cohomology of F with coefficients in Z/2Z. It was proved by .-Statement of the theorem:Let F be a field of...
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 categoryHomotopy category of chain complexesIn homological algebra in mathematics, the homotopy category K of chain complexes in an additive category A is a framework for working with chain homotopies and homotopy equivalences...
Kb(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.
Explanation for non-specialists
A commonly applied technique in mathematics is to study objects carrying a particular structure by introducing a categoryCategory (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
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
Birational geometry
In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension two, the birational geometry of algebraic surfaces was largely worked out by the Italian...
. 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
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, for example a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
. 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
Genus
In biology, a genus is a low-level taxonomic rank used in the biological classification of living and fossil organisms, which is an example of definition by genus and differentia...
of a smooth projective curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
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
De Rham cohomology
In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes...
and Betti cohomology, l-adic cohomology
Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
, the number of points over any finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
, and in multiplicative notation for local zeta-function
Local zeta-function
In number theory, a local zeta-functionis a function whose logarithmic derivative is a generating functionfor the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F.-Formulation:...
s.
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 numberComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s, 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 cohomologyÉtale cohomologyIn mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
(over any field of characteristic ≠ l) has a canonical Galois groupGalois groupIn mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
action, i.e. has values in representationsRepresentation (mathematics)In mathematics, representation is a very general relationship that expresses similarities between objects. Roughly speaking, a collection Y of mathematical objects may be said to represent another collection X of objects, provided that the properties and relationships existing among the...
of the (absolute) Galois group - crystalline cohomologyCrystalline cohomologyIn mathematics, crystalline cohomology is a Weil cohomology theory for schemes introduced by and developed by . Its values are modules over rings of Witt vectors over the base field....
All these cohomology theories share common properties, e.g. existence of Mayer-Vietoris
Mayer-Vietoris
Mayer-Vietoris may refer to:* Mayer–Vietoris axiom* Mayer–Vietoris sequence...
-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 HBetti∗(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 HBetti∗(X), HDR∗(X) etc.
Beginning with Grothendieck, people have tried to precisely define this theory for many years.
Motivic cohomology
Motivic cohomologyMotivic cohomology
Motivic cohomology is a cohomological theory in mathematics, the existence of which was first conjectured by Alexander Grothendieck during the 1960s. At that time, it was conceived as a theory constructed on the basis of the so-called standard conjectures on algebraic cycles, in algebraic geometry...
itself had been invented before the creation of mixed motives by means of algebraic K-theory
Algebraic K-theory
In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....
. The above category provides a neat way to (re)define it by
- Hn(X, m) := Hn(X, ℤ(m)) := HomDM(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
Triangulated category
A triangulated category is a mathematical category satisfying some axioms that are based on the properties of the homotopy category of spectra, and the derived category of an abelian category. A t-category is a triangulated category with a t-structure.- History :The notion of a derived category...
in the triangulated category.
Conjectures related to motives
The standard conjecturesStandard conjectures on algebraic cycles
In mathematics, the standard conjectures about algebraic cycles is a package of several conjectures describing the relationship of algebraic cycles and Weil cohomology theories. One of the original applications of these conjectures, envisaged by Alexander Grothendieck, was to prove that his...
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
Weil conjectures
In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields....
(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
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised...
.
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
Hodge conjecture
The Hodge conjecture is a major unsolved problem in algebraic geometry which relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they...
, may be neatly reformulated using motives: it holds iff
IFF
IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
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 → HSℚ (rational Hodge structure
Hodge structure
In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold...
s). Here pure motive means pure motive with respect to homological equivalence.
Similarly, the Tate conjecture
Tate conjecture
In mathematics, the Tate conjecture is a 1963 conjecture of John Tate linking algebraic geometry, and more specifically the identification of algebraic cycles, with Galois modules coming from étale cohomology...
is equivalent to: the so-called Tate realization, i.e. ℓ-adic cohomology is a faithful functor
H : M(k)ℚℓ → Repℓ(Gal(k)) (pure motives up to homological equivalence, continuous representations
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
of the absolute Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
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
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group 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
Tannakian category
In mathematics, a tannakian category is a particular kind of monoidal category C, equipped with some extra structure relative to a given field K. The role of such categories C is to approximate, in some sense, the category of linear representations of an algebraic group G defined over K...
theory (going back to Tannaka-Krein duality
Tannaka-Krein duality
In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. Its natural extension to the non-Abelian case is the Grothendieck duality theory....
, but a purely algebraic theory) is used. Its purpose is to shed light on both the Hodge conjecture
Hodge conjecture
The Hodge conjecture is a major unsolved problem in algebraic geometry which relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they...
and the Tate conjecture
Tate conjecture
In mathematics, the Tate conjecture is a 1963 conjecture of John Tate linking algebraic geometry, and more specifically the identification of algebraic cycles, with Galois modules coming from étale cohomology...
, the outstanding questions in algebraic cycle
Algebraic cycle
In mathematics, an algebraic cycle on an algebraic variety V is, roughly speaking, a homology class on V that is represented by a linear combination of subvarieties of V. Therefore the algebraic cycles on V are the part of the algebraic topology of V that is directly accessible in algebraic geometry...
theory.
Fix a Weil cohomology theory H. It gives a functor from Mnum (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 Mnum is equivalent to the category of representations
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
of an algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
G, which is called motivic Galois group.
It is to the theory of motives what the Mumford-Tate group is to Hodge theory
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised...
. Again speaking in rough terms, the Hodge and Tate conjectures are types of invariant theory
Invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...
(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
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
; however in terms of the Tate conjecture
Tate conjecture
In mathematics, the Tate conjecture is a 1963 conjecture of John Tate linking algebraic geometry, and more specifically the identification of algebraic cycles, with Galois modules coming from étale cohomology...
and Galois representations on étale cohomology
Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
, it predicts the image of the Galois group, or, more accurately, its Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
.)