Motives, theory of

A generalization of the various cohomology theories of algebraic varieties. The theory of motives systematically generalizes the idea of using the Jacobi variety of an algebraic curve $ X $ as a replacement for the cohomology group $ H ^ {1} ( X , \mathbf Q ) $ in the classical theory of correspondences, and the use of this theory in the study of the zeta-function of a curve $ X $ over a finite field. The theory of motives is universal in the sense that every geometric cohomology theory, of the type of the classical singular cohomology for algebraic varieties over $ \mathbf C $ with constant coefficients, every $ \ell $-adic cohomology theory for various prime numbers $ \ell $ different from the characteristic of the ground field, every crystalline cohomology theory, etc. (see Weil cohomology) are functors on the category of motives.

Let $ V ( k) $ be the category of smooth projective varieties over a field $ k $ and let $ X \rightarrow C ( X) $ be a contravariant functor of global intersection theory from $ V ( k) $ into the category of commutative $ \Lambda $-algebras, where $ \Lambda $ is a fixed ring. For example, $ C ( X) $ is the Chow ring of classes of algebraic cycles (cf. Algebraic cycle) on $ X $ modulo a suitable (rational, algebraic, numerical, etc.) equivalence relation, or $ C ( X) = K ( X) $ is the Grothendieck ring, or $ C ( X) = H ^ {ev} ( X) $ is the ring of cohomology classes of even dimension, etc. The category $ V ( k) $ and the functor $ X \rightarrow C ( X) $ enable one to define a new category, the category of correspondences $ C V ( k) $, whose objects are varieties $ X \in V ( k) $, denoted by $ \overline{X} $, and whose morphisms are defined by the formula

$$ \mathop{\rm Hom} ( \overline{X} , \overline{Y} ) = C ( X \times Y ) $$

with the usual composition law for correspondences (see [1]). Let the functor $ C $ take values in the category of commutative graded $ \Lambda $-algebras $ A ( \Lambda ) $. Then $ C V ( k) $ will be the $ \Lambda $-additive category of graded correspondences. Moreover, $ C V ( k) $ will have direct sums and tensor products.

The category whose objects are the varieties from $ V ( k) $ and whose morphisms are correspondences of degree $ 0 $ is denoted by $ C V ^ {0} ( k) $. A natural functor from $ V ( k) $ into $ C V ^ {0} ( k) $ has been defined, and the functor $ C $ extends to a functor $ T $ from $ C V ^ {0} ( k) $ to $ A ( \Lambda ) $. The category $ C V ^ {0} ( k) $, like $ C V ( k) $, is not Abelian. Its pseudo-Abelian completion, the category $ M _ {C} ^ {+} ( k) $, has been considered. It is obtained from $ C V ^ {0} ( k) $ by the formal addition of the images of all projections $ p $. More precisely, the objects of $ M _ {C} ^ {+} ( k) $ are pairs $ ( \overline{X} , p ) $, where $ \overline{X} \in C V ^ {0} ( k) $ and $ p \in \mathop{\rm Hom} ( \overline{X} , \overline{X} ) $, $ p ^ {2} = p $, and $ H ( ( \overline{X} , p ) , ( \overline{Y} , q ) ) $ is the set of correspondences $ f : \overline{X} \rightarrow \overline{Y} $ such that $ f \circ p = = q \circ f $ modulo a correspondence $ g $ with $ g \circ p = p \circ g = 0 $. The category $ C V ^ {0} ( k) $ is imbedded in $ M _ {C} ^ {+} ( k) $ by means of the functor $ \overline{X} \rightarrow ( \overline{X} , \mathop{\rm id} ) $. The natural functor $ h : V ( k) \rightarrow M _ {C} ^ {+} ( k) $ is called the functor of motive cohomology spaces and $ M _ {C} ^ {+} ( k) $ is called the category of effective motives.

Let $ p = ( 1 \times e ) $, where $ e $ is the class of any rational point on the projective line $ P ^ {1} $, and let $ L = ( P ^ {1} , p ) $. Then

$$ h ( P ^ {n} ) = 1 \oplus L \oplus L ^ {\otimes 2 } \oplus \dots \oplus L ^ {\otimes n } . $$

If $ X = P ( E) $ is the projectivization of a locally free sheaf $ E $ of rank $ r $ on $ Y $, then

$$ h ( X) = \bigoplus _ {i = 0 } ^ { {r } - 1 } ( h ( Y) \otimes L ^ {\otimes i } ) . $$

Motives of a monoidal transformation with a non-singular centre, motives of curves (see [1]), motives of Abelian manifolds (see [2]), and motives of Weil hypersurfaces have also been calculated.

The category of motives $ M _ {C} ( k) $ is obtained from $ M _ {C} ^ {+} ( k) $ by the formal addition of negative powers of the motives $ L $. By analogy with $ \ell $-adic cohomology, $ T = L ^ {\otimes -1 } $ is called the Tate motive. Tensor multiplication with $ T $ is called twisting by the Tate motive. Twisting enables one to define the level of a motive as in an $ \ell $-adic cohomology theory. Any functor of the Weil cohomology factors through the functor $ h : V ( k) \rightarrow M _ {C} ( k) $. There is the conjecture that $ M _ {C} ( k) $ does not, in some sense, depend on the intersection theory of $ C $, and that the functor $ X \rightarrow h ( X) $ is itself a (universal) theory for the Weil cohomology. This conjecture is closely related to the standard Grothendieck conjectures (see [5]) on algebraic cycles (at present, 1982, not proved).

References

[1]	Yu.I. Manin, "Correspondences, motives and monoidal transformations" Math. USSR Sb. , 6 : 4 (1968) pp. 439–470 Mat. Sb. , 77 : 4 (1968) pp. 475–507
[2]	A.M. Shermenev, "The motif of an abelian variety" Uspekhi Mat. Nauk , 26 : 2 (1971) pp. 215–216 (In Russian)
[3]	M. Demazure, "Motifs des variétés algébrique" , Sém. Bourbaki Exp. 365 , Lect. notes in math. , 180 , Springer (1971) pp. 19–38
[4]	S.L. Kleiman, "Motives" P. Holm (ed.) , Algebraic Geom. Proc. 5-th Nordic Summer School Math. Oslo, 1970 , Wolters-Noordhoff (1972) pp. 53–96
[5]	S.L. Kleiman, "Algebraic cycles and the Weil conjectures" A. Grothendieck (ed.) J. Giraud (ed.) et al. (ed.) , Dix exposés sur la cohomologie des schémas , North-Holland & Masson (1968) pp. 359–386

Comments

The theory of motives has been created by A. Grothendieck in the 1960-s. Although the above-mentioned standard conjectures on algebraic cycles have not yet (1989) been proved, the theory of motives has played an important role in various recent developments, for instance: i) as a guide for the Deligne–Hodge theory ([a1]); ii) in the study of absolute Hodge cycles on Abelian varieties ([a2]), where a variant of the notion of a motive has been used; iii) in the study of Chow groups on certain varieties over a finite field ([a3]); and iv) in work on the Beilinson's conjectures on special values of $ L $-functions (see [a4]).

References

[a1]	P. Deligne, "Théorie de Hodge I" , Proc. Internat. Congress Mathematicians (Nice, 1970) , 1 , Gauthier-Villars (1971) pp. 425–430
[a2]	P. Deligne (ed.) J.S. Milne (ed.) A. Ogus (ed.) K. Shih (ed.) , Hodge cycles, motives and Shimura varieties , Lect. notes in math. , 900 , Springer (1980)
[a3]	C. Soulé, "Groupes de Chow et $K$-theory des variétés sur un corps fini" Math. Ann. , 268 (1984) pp. 317–345
[a4]	M. Rapoport (ed.) N. Schappacher (ed.) P. Schneider (ed.) , Beilinson's conjectures on special values of $L$-functions , Acad. Press (1988)