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:
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* 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
, where ''X'' is a smooth projective variety,
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
to
is given by a correspondence of degree
. 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
:
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 ...
:
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 ...
:
respectively, various comparison isomorphisms
:
between the obvious base changes of these modules, filtrations
, a
-action
on
and a
"Frobenius" automorphism of
. This data is modeled on the cohomologies of a smooth projective
-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
, where ''k'' is any field.
First step: category of (degree 0) correspondences, Corr(''k'')
The objects of
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
, which can be associated with their graphs in
, to fixed dimensional
Chow cycles on
.
It will be useful to describe correspondences of arbitrary degree, although morphisms in
are correspondences of degree 0. In detail, let ''X'' and ''Y'' be smooth projective varieties and consider a decomposition of ''X'' into connected components:
:
If
, then the correspondences of degree ''r'' from ''X'' to ''Y'' are
:
where
denotes the Chow-cycles of codimension ''k''. Correspondences are often denoted using the "⊢"-notation, e.g.,
. For any
and
their composition is defined by
:
where the dot denotes the product in the Chow ring (i.e., intersection).
Returning to constructing the category
notice that the composition of degree 0 correspondences is degree 0. Hence we define morphisms of
to be degree 0 correspondences.
The following association is a functor (here
denotes the graph of
):
:
Just like
the category
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
:
Second step: category of pure effective Chow motives, Choweff(''k'')
The transition to motives is made by taking the
pseudo-abelian envelope of
:
:
.
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 Δ
''X'' :=
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, Chow
eff(''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
:
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
:
where
:
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 ''f
1'' and ''f
2''. Then
:
,
where ''π
i'' : ''X''
1 × ''X''
2 × ''Y''
1 × ''Y''
2 → ''X
i'' × ''Y
i'' are the projections.
Third step: category of pure Chow motives, Chow(''k'')
To proceed to motives, we
adjoin to Chow
eff(''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
:
.
If we define the motive 1, called the ''trivial Tate motive'', by 1 := h(Spec(''k'')), then the elegant equation
:
holds, since
:
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
: