In the formulation of Grothendieck for smooth projective varieties, a motive is a triple
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
A more object-focused approach is taken by Pierre Deligne in Le Groupe Fondamental de la Droite Projective Moins Trois Points.
-variety and the structures and compatibilities they admit, and gives an idea about what kind of information is contained in a motive.
The general hope is that equations like can be put on increasingly solid mathematical footing with a deep meaning.
In detail, let X and Y be smooth projective varieties and consider a decomposition of X into connected components: If
their composition is defined by where the dot denotes the product in the Chow ring (i.e., intersection).
The sum of morphisms is defined by The transition to motives is made by taking the pseudo-abelian envelope of
, together with a contravariant functor taking values on all varieties (not just smooth projective ones as it was the case with pure motives).
Vladimir Voevodsky's Fields Medal-winning proof of the Milnor conjecture uses these motives as a key ingredient.
as the category of quasi-projective varieties over k are separated schemes of finite type.
-homotopies of varieties while the second will give the category of geometric mixed motives the Mayer–Vietoris sequence.
The hom-groups are then the colimit There are several elementary examples of motives which are readily accessible.
These are fundamental building blocks in the category of motives because they form the "other part" besides Abelian varieties.
The motive of a curve can be explicitly understood with relative ease: their Chow ring is just
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.
There are different Weil cohomology theories, they apply in different situations and have values in different categories, and reflect different structural aspects of the variety in question: All these cohomology theories share common properties, e.g. existence of Mayer-Vietoris sequences, homotopy invariance
Beginning with Grothendieck, people have tried to precisely define this theory for many years.
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 (which are proven by different means by Deligne), assuming the standard conjectures to hold.
For example, the Künneth standard conjecture, which states the existence of algebraic cycles πi ⊂ X × X inducing the canonical projectors H*(X) → Hi(X) ↣ H*(X) (for any Weil cohomology H) implies that every pure motive M decomposes in graded pieces of weight n: M = ⨁GrnM.
The terminology weights comes from a similar decomposition of, say, de-Rham cohomology of smooth projective varieties, see Hodge theory.
(In particular the former category of motives would not depend on the choice of the Weil cohomology theory).
To motivate the (conjectural) motivic Galois group, fix a field k and consider the functor which maps K to the (finite) set of embeddings of K into an algebraic closure of k. In Galois theory this functor is shown to be an equivalence of categories.
The objective of the motivic Galois group is to extend the above equivalence to higher-dimensional varieties.
Fix a Weil cohomology theory H. It gives a functor from Mnum (pure motives using numerical equivalence) to finite-dimensional
Again speaking in rough terms, the Hodge and Tate conjectures are types of invariant theory (the spaces that are morally the algebraic cycles are picked out by invariance under a group, if one sets up the correct definitions).