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In algebraic geometry, motives (or sometimes motifs, following
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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 In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
,
de Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) 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 adap ...
, 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 (X, p, m), where ''X'' is a smooth projective variety, p: X \vdash X is an idempotent correspondence, and ''m'' an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
, however, such a triple contains almost no information outside the context of Grothendieck's
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of pure motives, where a
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphis ...
from (X, p, m) to (Y, q, n) is given by a correspondence of degree n-m. A more object-focused approach is taken by
Pierre Deligne Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...
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 : \left (M_B, M_, M_, M_, \operatorname_, \operatorname_, \operatorname_, W, F_\infty, F, \phi, \phi_p \right ) consisting of modules :M_B, M_, M_, M_ over the rings :\Q, \Q, \mathbb^f, \Q_p, respectively, various comparison isomorphisms :\operatorname_, \operatorname_, \operatorname_ between the obvious base changes of these modules, filtrations W, F, a \operatorname(\overline, \Q)-action \phi on M_, and a "Frobenius" automorphism \phi_p of M_. This data is modeled on the cohomologies of a smooth projective \Q-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 In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be vie ...
,
de Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) 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 adap ...
, ''l''-adic cohomology, and crystalline cohomology. The general hope is that equations like * rojective line= ine+ oint* rojective plane=
lane In road transport, a lane is part of a roadway that is designated to be used by a single line of vehicles to control and guide drivers and reduce traffic conflicts. Most public roads ( highways) have at least two lanes, one for traffic in eac ...
+ ine+ ointcan 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 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 in ...
.


Definition of pure motives

The
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of pure motives often proceeds in three steps. Below we describe the case of Chow motives \operatorname(k), where ''k'' is any field.


First step: category of (degree 0) correspondences, Corr(''k'')

The objects of \operatorname(k) are simply smooth projective varieties over ''k''. The morphisms are correspondences. They generalize morphisms of varieties X \to Y, which can be associated with their graphs in X \times Y, to fixed dimensional Chow cycles on X \times Y. It will be useful to describe correspondences of arbitrary degree, although morphisms in \operatorname(k) are correspondences of degree 0. In detail, let ''X'' and ''Y'' be smooth projective varieties and consider a decomposition of ''X'' into connected components: :X = \coprod_i X_i, \qquad d_i := \dim X_i. If r\in \Z, then the correspondences of degree ''r'' from ''X'' to ''Y'' are :\operatorname^r(k)(X, Y) := \bigoplus_i A^(X_i \times Y), where A^k(X) denotes the Chow-cycles of codimension ''k''. Correspondences are often denoted using the "⊢"-notation, e.g., \alpha : X \vdash Y. For any \alpha\in \operatorname^r(X, Y) and \beta\in \operatorname^s(Y,Z), their composition is defined by :\beta \circ \alpha := \pi_ \left (\pi^_(\alpha) \cdot \pi^_(\beta) \right ) \in \operatorname^(X, Z), where the dot denotes the product in the Chow ring (i.e., intersection). Returning to constructing the category \operatorname(k), notice that the composition of degree 0 correspondences is degree 0. Hence we define morphisms of \operatorname(k) to be degree 0 correspondences. The following association is a functor (here \Gamma_f \subseteq X\times Y denotes the graph of f: X\to Y): :F : \begin \operatorname(k) \longrightarrow \operatorname(k) \\ X \longmapsto X \\ f \longmapsto \Gamma_f \end Just like \operatorname(k), the category \operatorname(k) has direct sums () and tensor products (). It is a preadditive category. The sum of morphisms is defined by :\alpha + \beta := (\alpha, \beta) \in A^(X \times X) \oplus A^(Y \times Y) \hookrightarrow A^ \left (\left (X \coprod Y \right ) \times \left (X \coprod Y \right ) \right ).


Second step: category of pure effective Chow motives, Choweff(''k'')

The transition to motives is made by taking the pseudo-abelian envelope of \operatorname(k): :\operatorname^\operatorname(k) := Split(\operatorname(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: :\operatorname \left (\operatorname^\operatorname(k) \right ) := \. :\operatorname((X, \alpha), (Y, \beta)) := \. Composition is the above defined composition of correspondences, and the identity morphism of (''X'', ''α'') is defined to be ''α'' : ''X'' ⊢ ''X''. The association, :h : \begin \operatorname(k) & \longrightarrow \operatorname(k) \\ X & \longmapsto := (X, \Delta_X) \\ f & \longmapsto := \Gamma_f \subset X \times Y \end, where Δ''X'' := '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 :( \alpha) \oplus ( \beta) := \left ( \left \coprod Y \right \alpha + \beta \right ), The
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
of effective motives is defined by :( \alpha) \otimes ( \beta) := (X \times Y, \pi_X^\alpha \cdot \pi_Y^\beta), where :\pi_X : (X \times Y) \times (X \times Y) \to X \times X, \quad \text \quad \pi_Y : (X \times Y) \times (X \times Y) \to Y \times Y. 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 :f_1 \otimes f_2 : (X_1, \alpha_1) \otimes (X_2, \alpha_2) \vdash (Y_1, \beta_1) \otimes (Y_2, \beta_2), \qquad f_1 \otimes f_2 := \pi^_1 \gamma_1 \cdot \pi^_2 \gamma_2, 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 :L := (\mathbb^1, \lambda), \qquad \lambda := pt \times \mathbb^1 \in A^1(\mathbb^1 \times \mathbb^1). If we define the motive 1, called the ''trivial Tate motive'', by 1 := h(Spec(''k'')), then the elegant equation : mathbb^1= \mathbf \oplus L holds, since :\mathbf \cong \left (\mathbb^1, \mathbb^1 \times \operatorname \right ). 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 :\operatorname(k) := \operatorname^\operatorname(k) /math>. A motive is then a triple :(X \in \operatorname(k), p: X \vdash X, n \in \Z ) such that morphisms are given by correspondences :f : (X, p, m) \to (Y, q, n), \quad f \in \operatorname^(X, Y) \mbox f \circ p = f = q \circ f, and the composition of morphisms comes from composition of correspondences. As intended, \operatorname(k) 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 * 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 In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left an ...
MM(k), together with a contravariant functor :\operatorname(k) \to MM(k) 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 :\operatorname^*_(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
Alexander Beilinson Alexander A. Beilinson (born 1957) 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. In 1 ...
. Instead of constructing such a category, it was proposed by
Deligne Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...
to first construct a category ''DM'' having the properties one expects for the
derived category In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pro ...
: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.
Vladimir Voevodsky Vladimir Alexandrovich Voevodsky (, russian: Влади́мир Алекса́ндрович Воево́дский; 4 June 1966 – 30 September 2017) was a Russian-American mathematician. His work in developing a homotopy theory for algebraic var ...
's Fields Medal-winning proof of the
Milnor conjecture In mathematics, the Milnor conjecture was a proposal by of a description of the Milnor K-theory (mod 2) of a general field ''F'' with characteristic different from 2, by means of the Galois (or equivalently étale) cohomology of ''F'' w ...
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 A =\Q,\Z be our coefficient ring. Set \mathcal/k as the category of quasi-projective varieties over are separated schemes of finite type. We will also let \mathcal/k be the subcategory of smooth varieties.


Smooth varieties with correspondences

Given a
smooth variety In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a s ...
and a variety call an
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
closed subscheme W \subset X \times Y 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 C_A(X,Y). Its elements are called finite correspondences. Then, we can form an additive category \mathcal 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 \Gamma_f \subset X\times Y of a morphism of varieties f:X \to Y.


Localizing the homotopy category

From here we can form the
homotopy category In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed b ...
K^b(\mathcal) of bounded complexes of smooth correspondences. Here smooth varieties will be denoted /math>. If we localize this category with respect to the smallest thick subcategory (meaning it is closed under extensions) containing morphisms : \times\mathbb^1\to /math> and : \cap V\xrightarrow oplus \xrightarrow /math> then we can form the
triangulated category In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy cate ...
of effective geometric motives \mathcal_\text^\text(k,A). Note that the first class of morphisms are localizing \mathbb^1-homotopies of varieties while the second will give the category of geometric mixed motives the
Mayer–Vietoris sequence In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is du ...
. Also, note that this category has a tensor structure given by the product of varieties, so otimes = \times Y/math>.


Inverting the Tate motive

Using the triangulated structure we can construct a triangle :\mathbb \to mathbb^1\to operatorname(k)\xrightarrow from the canonical map \mathbb^1 \to \operatorname(k). We will set A(1) = \mathbb 2/math> and call it the Tate motive. Taking the iterative tensor product lets us construct A(k). If we have an effective geometric motive we let M(k) denote M \otimes A(k). Moreover, this behaves functorially and forms a triangulated functor. Finally, we can define the category of geometric mixed motives \mathcal_ as the category of pairs (M,n) for an effective geometric mixed motive and an integer representing the twist by the Tate motive. The hom-groups are then the colimit :\operatorname_((A,n),(B,m))=\lim_ \operatorname_(A(k+n),B(k+m))


Examples of motives


Tate motives

There are several elementary examples of motives which are readily accessible. One of them being the Tate motives, denoted \mathbb(n), \mathbb(n), or A(n), 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 just\Z\oplus \text(C)for any smooth projective curve C, 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 Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
whose morphisms preserve this structure. Then one may ask when two given objects are 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 Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex number ...
, 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 In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rationa ...
. 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 is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matric ...
, for example a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
. 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 Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial n ...
of a smooth projective
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
''C'' which is an interesting invariant of the curve, is an integer, which can be read off the dimension of the first
Betti cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be vie ...
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: * oint* rojective line= oint+ ine* rojective plane=
lane In road transport, a lane is part of a roadway that is designated to be used by a single line of vehicles to control and guide drivers and reduce traffic conflicts. Most public roads ( highways) have at least two lanes, one for traffic in eac ...
+ ine+ oint These 'equations' hold in many situations, namely for
de Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) 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 adap ...
and
Betti cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be vie ...
, ''l''-adic cohomology, the number of points over any
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
, and in multiplicative notation for
local zeta-function In number theory, the local zeta function (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as :Z(V, s) = \exp\left(\sum_^\infty \frac (q^)^m\right) where is a non-singular -dimensional projective alge ...
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 number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s, it has the advantage of being defined over the
integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
and is a topological invariant * de Rham cohomology (for varieties over \Complex) 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 In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
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) \cong H^*(X\times \mathbb^1), the product of ''X'' with the affine line) and others. Moreover, they are linked by comparison isomorphisms, for example Betti cohomology H^*_(X, \Z/n) of a smooth variety ''X'' over \Complex 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 : rojective line= ine oint 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 Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
. The above category provides a neat way to (re)define it by :H^n(X,m) := H^n(X, \Z(m)) := \operatorname_(X, \Z(m) , 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 \mathbb^1 \to \operatorname shifted by –2, and '' ' means the usual
shift Shift may refer to: Art, entertainment, and media Gaming * ''Shift'' (series), a 2008 online video game series by Armor Games * '' Need for Speed: Shift'', a 2009 racing video game ** '' Shift 2: Unleashed'', its 2011 sequel Literature * ''Sh ...
in the triangulated category.


Conjectures related to motives

The
standard conjectures In mathematics, the standard conjectures about algebraic cycles are several conjectures describing the relationship of algebraic cycles and Weil cohomology theory, Weil cohomology theories. One of the original applications of these conjectures, envi ...
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 In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. ...
(which are proven by different means by
Deligne Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...
), 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 In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coh ...
. ''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 In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. In simple terms, the Hodge conjec ...
, may be neatly reformulated using motives: it holds
iff In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
the ''Hodge realization'' mapping any pure motive with rational coefficients (over a subfield k of \Complex) to its Hodge structure is a full functor H:M(k)_ \to HS_ (rational
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. 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 full functor H: M(k)_ \to \operatorname_ (\operatorname(k)) (pure motives up to homological equivalence, continuous representations of the absolute
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
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'' → non-empty finite sets with a (continuous) transitive action of the absolute Galois group of ''k'' which maps ''K'' to the (finite) set of embeddings of ''K'' into an algebraic closure of ''k''. In
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory t ...
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 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 ...
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 In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. In simple terms, the Hodge conjec ...
and the Tate conjecture, the outstanding questions in
algebraic cycle In mathematics, an algebraic cycle on an algebraic variety ''V'' is a formal linear combination of subvarieties of ''V''. These are the part of the algebraic topology of ''V'' that is directly accessible by algebraic methods. Understanding the al ...
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 In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. ...
''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 In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coh ...
. Again speaking in rough terms, the Hodge and Tate conjectures are types of
invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...
(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 In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
; however in terms of the Tate conjecture and Galois representations on
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 conject ...
, it predicts the image of the Galois group, or, more accurately, its
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
.)


See also

* Ring of periods * Motivic cohomology * Presheaf with transfers * Mixed Hodge module *
L-functions of motives In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may give ris ...


References


Survey Articles

* (technical introduction with comparatively short proofs)
Motives over Finite Fields
- J.S. Milne * (motives-for-dummies text). * (high-level introduction to motives in French). *


Books

* * ** L. Breen: ''Tannakian categories''. ** S. Kleiman: ''The standard conjectures''. ** A. Scholl: ''Classical motives''. (detailed exposition of Chow motives) * * * *


Reference Literature

* * (adequate equivalence relations on cycles). * Milne, James S
Motives — Grothendieck’s Dream
* (Voevodsky's definition of mixed motives. Highly technical). *


Future directions

* Musings on \mathbb(1/4): Arithmetic spin structures on elliptic curves
What are "Fractional Motives"?


External links

* {{wikiquote-inline Algebraic geometry Topological methods of algebraic geometry Homological algebra