View source for Motives, theory of

<!--
m0650401.png
$#A+1 = 79 n = 2
$#C+1 = 79 : ~/encyclopedia/old_files/data/M065/M.0605040 Motives, theory of
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

A generalization of the various [[Cohomology|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|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|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|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|Chow ring]] of classes of algebraic cycles (cf. [[Algebraic cycle|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 [[#References|[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 [[#References|[1]]]), motives of Abelian manifolds (see [[#References|[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 [[L-adic-cohomology| $  \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 [[#References|[5]]]) on algebraic cycles (at present, 1982, not proved).

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  A.M. Shermenev, "The motif of an abelian variety"  ''Uspekhi Mat. Nauk'' , '''26''' :  2  (1971)  pp. 215–216  (In Russian)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  M. Demazure, "Motifs des variétés algébriques" , ''Sém. Bourbaki Exp. 365'' , ''Lect. notes in math.'' , '''180''' , Springer  (1971)  pp. 19–38</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  S.L. Kleiman,   "Motives"  P. Holm (ed.) , ''Algebraic Geom. Proc. 5-th Nordic Summer School Math. Oslo, 1970'' , Wolters-Noordhoff  (1972)  pp. 53–96</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top">  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 &amp; Masson  (1968)  pp. 359–386</TD></TR></table>

====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 ([[#References|[a1]]]); ii) in the study of absolute Hodge cycles on Abelian varieties ([[#References|[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 ([[#References|[a3]]]); and iv) in work on the Beilinson's conjectures on special values of  $  L $-functions (see [[#References|[a4]]]).

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Deligne,   "Théorie de Hodge I" , ''Proc. Internat. Congress Mathematicians (Nice, 1970)'' , '''1''' , Gauthier-Villars  (1971)  pp. 425–430</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C. Soulé,   "Groupes de Chow et $K$-théorie des variétés sur un corps fini"  ''Math. Ann.'' , '''268'''  (1984)  pp. 317–345</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Rapoport (ed.)  N. Schappacher (ed.)  P. Schneider (ed.) , ''Beilinson's conjectures on special values of $L$-functions'' , Acad. Press  (1988)</TD></TR>
</table>