# Motives, theory of

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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 monodial 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, "Motives des variétés algébrique" , Sem. 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

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]).
 [a1] P. Deligne, "Theory 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)