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In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, motives (or sometimes motifs, following French usage) is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved
cohomology theories 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 viewed ...
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 viewed ...
,
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 adapte ...
, etale cohomology, and
crystalline cohomology In mathematics, crystalline cohomology is a Weil cohomology theory for schemes ''X'' over a base field ''k''. Its values ''H'n''(''X''/''W'') are modules over the ring ''W'' of Witt vectors over ''k''. It was introduced by and developed by . ...
. Philosophically, a "motif" is the "cohomology essence" of a
variety Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film), ...
. 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
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) * ...
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, morphisms a ...
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 Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
:M_B, M_, M_, M_ over the
rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
:\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 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 adapte ...
, ''l''-adic cohomology, and
crystalline cohomology In mathematics, crystalline cohomology is a Weil cohomology theory for schemes ''X'' over a base field ''k''. Its values ''H'n''(''X''/''W'') are modules over the ring ''W'' of Witt vectors over ''k''. It was introduced by and developed by . ...
. The general hope is that equations like * rojective line=
ine INE, Ine or ine may refer to: Institutions * Institut für Nukleare Entsorgung, a German nuclear research center * Instituto Nacional de Estadística (disambiguation) * Instituto Nacional de Estatística (disambiguation) * Instituto Nacional Elec ...
+ 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 each ...
+
ine INE, Ine or ine may refer to: Institutions * Institut für Nukleare Entsorgung, a German nuclear research center * Instituto Nacional de Estadística (disambiguation) * Instituto Nacional de Estatística (disambiguation) * Instituto Nacional Elec ...
+ 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 A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...
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 i ...
.


Definition of pure motives

The
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) * ...
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 Correspondence may refer to: *In general usage, non-concurrent, remote communication between people, including letters, email, newsgroups, Internet forums, blogs. Science * Correspondence principle (physics): quantum physics theories must agree ...
. 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 In mathematics, the tensor product V \otimes W of two vector spaces and (over the same 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 element of V \otimes W ...
(). It is a
preadditive category In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-category C is a category such that every hom ...
. 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 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. Recall that an idempotent morphism p is an endomorphism of an object with the property tha ...
. 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) 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 element of V \otimes W ...
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 Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear o ...
. 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 In number theory, a cyclotomic character is a character of a Galois group giving the Galois action on a group of roots of unity. As a one-dimensional representation over a ring , its representation space is generally denoted by (that is, it is ...
'', ''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 proce ...
:D^b(MM(k)). Getting ''MM'' back from ''DM'' would then be accomplished by a (conjectural) ''motivic
t-structure In the branch of mathematics called homological algebra, a ''t''-structure is a way to axiomatize the properties of an abelian subcategory of a derived category. A ''t''-structure on \mathcal consists of two subcategories (\mathcal^, \mathcal^) ...
''. 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 The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...
-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'' wi ...
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 smoo ...
and a
variety Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film), ...
call an
integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
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 be ...
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 categ ...
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 due to ...
. 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 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, 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 numbers. Mo ...
, 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 rational fu ...
. 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 matrices. ...
, 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 can ...
. 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 extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
of a smooth projective
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
''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 viewe ...
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 INE, Ine or ine may refer to: Institutions * Institut für Nukleare Entsorgung, a German nuclear research center * Instituto Nacional de Estadística (disambiguation) * Instituto Nacional de Estatística (disambiguation) * Instituto Nacional Elec ...
* 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 each ...
+
ine INE, Ine or ine may refer to: Institutions * Institut für Nukleare Entsorgung, a German nuclear research center * Instituto Nacional de Estadística (disambiguation) * Instituto Nacional de Estatística (disambiguation) * Instituto Nacional Elec ...
+ 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 adapte ...
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 viewe ...
, ''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, subtr ...
, and in
multiplicative notation In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referred t ...
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 algebr ...
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 form ...
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 o ...
and is a topological invariant * de Rham cohomology (for varieties over \Complex) comes with a
mixed Hodge structure In algebraic geometry, a mixed Hodge structure is an algebraic structure containing information about the cohomology of general algebraic varieties. It is a generalization of a Hodge structure, which is used to study smooth projective varieties. ...
, 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 pol ...
action, i.e. has values in
representations ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...
of the (absolute) Galois group *
crystalline cohomology In mathematics, crystalline cohomology is a Weil cohomology theory for schemes ''X'' over a base field ''k''. Its values ''H'n''(''X''/''W'') are modules over the ring ''W'' of Witt vectors over ''k''. It was introduced by and developed by . ...
All these cohomology theories share common properties, e.g. existence of
Mayer-Vietoris sequence Mayer-Vietoris may refer to: * Mayer–Vietoris axiom * 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 topol ...
s, homotopy invariance H^*(X) \cong H^*(X\times \mathbb^1), the product of ''X'' with the
affine line In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
) 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 INE, Ine or ine may refer to: Institutions * Institut für Nukleare Entsorgung, a German nuclear research center * Instituto Nacional de Estadística (disambiguation) * Instituto Nacional de Estatística (disambiguation) * Instituto Nacional Elec ...
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 Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geo ...
'' 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 o ...
. 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 in the triangulated category.


Conjectures related to motives

The standard conjectures were first formulated in terms of the interplay of algebraic cycles and Weil cohomology theories. The category of pure motives provides a categorical framework for these conjectures. The standard conjectures are commonly considered to be very hard and are open in the general case. Grothendieck, with Bombieri, showed the depth of the motivic approach by producing a conditional (very short and elegant) proof of the
Weil conjectures 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. Th ...
(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 cohom ...
. ''Conjecture D'', stating the concordance of numerical and
homological equivalence Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor *Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chromo ...
, 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 conjectu ...
, 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 bicon ...
the ''Hodge realization'' mapping any pure motive with rational coefficients (over a subfield k of \Complex) to its Hodge structure is a
full functor In category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties is called a full and faithful functor. Formal definitions Explicitly, let ''C'' an ...
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 In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The c ...
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 ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...
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 pol ...
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 to ...
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 re ...
theory (going back to
Tannaka–Krein duality In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topologic ...
, 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 conjectu ...
and the
Tate conjecture In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The c ...
, 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 ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...
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. Man ...
''G'', known as the motivic Galois group. The motivic Galois group is to the theory of motives what the
Mumford–Tate group In algebraic geometry, the Mumford–Tate group (or Hodge group) ''MT''(''F'') constructed from a Hodge structure ''F'' is a certain algebraic group ''G''. When ''F'' is given by a rational representation of an algebraic torus, the definition of ' ...
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 cohom ...
. 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 pol ...
; however in terms of the
Tate conjecture In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The c ...
and
Galois representation In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring ...
s 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 conjecture ...
, 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 Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
.)


See also

*
Ring of periods In algebraic geometry, a period is a number that can be expressed as an integral of an algebraic function over an algebraic domain. Sums and products of periods Closure (mathematics), remain periods, so the periods form a ring (mathematics), r ...
*
Motivic cohomology Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geo ...
*
Presheaf with transfers In algebraic geometry, a presheaf with transfers is, roughly, a presheaf that, like cohomology theory, comes with pushforwards, “transfer” maps. Precisely, it is, by definition, a contravariant additive functor from the category of finite corre ...
*
Mixed Hodge module In mathematics, mixed Hodge modules are the culmination of Hodge theory, mixed Hodge structures, intersection cohomology, and the decomposition theorem yielding a coherent framework for discussing variations of degenerating mixed Hodge structures ...
* L-functions of motives


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