In algebraic geometry, 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 such as singular cohomology, de Rham cohomology, 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 , where X is a smooth projective variety, is an idempotent correspondence, and m an integer, however, such a triple contains almost no information outside the context of Grothendieck's category of pure motives, where a morphism from to is given by a correspondence of degree . A more object-focused approach is taken by Pierre Deligne 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
over the rings
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 in 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, de Rham cohomology, l-adic cohomology, and crystalline cohomology. The general hope is that equations like [projective line] = [line] + [point] [projective plane] = [plane] + [line] + [point] can be put on increasingly solid mathematical footing with a deep meaning. Of course, the above equations are already known to be true in many senses, such as in the sense of CW-complex where "+" corresponds to attaching cells, and in the sense of various cohomology theories, where "+" corresponds to the direct sum.
From another viewpoint, motives continue the sequence of generalizations from rational functions on varieties to divisors on varieties to Chow groups of varieties. The generalization happens in more than one direction, since motives can be considered with respect to more types of equivalence than rational equivalence. The admissible equivalences are given by the definition of an adequate equivalence relation.
Definition of pure motives
The category 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. 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 (). It is a preadditive category. 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 ΔX := [idX] denotes the diagonal of X × X, is a functor. The motive [X] is often called the motive associated to the variety X.
As intended, Choweff(k) is a pseudo-abelian category. The direct sum of effective motives is given by