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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m0650401.png" /> as a replacement for the cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m0650402.png" /> in the classical theory of correspondences, and the use of this theory in the study of the [[Zeta-function|zeta-function]] of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m0650403.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m0650404.png" /> with constant coefficients, every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m0650405.png" />-adic cohomology theory for various prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m0650406.png" /> 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.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m0650407.png" /> be the category of smooth projective varieties over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m0650408.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m0650409.png" /> be a contravariant functor of global [[Intersection theory|intersection theory]] from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504010.png" /> into the category of commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504011.png" />-algebras, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504012.png" /> is a fixed ring. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504013.png" /> is the [[Chow ring|Chow ring]] of classes of algebraic cycles (cf. [[Algebraic cycle|Algebraic cycle]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504014.png" /> modulo a suitable (rational, algebraic, numerical, etc.) equivalence relation, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504015.png" /> is the Grothendieck ring, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504016.png" /> is the ring of cohomology classes of even dimension, etc. The category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504017.png" /> and the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504018.png" /> enable one to define a new category, the category of correspondences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504019.png" />, whose objects are varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504020.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504021.png" />, and whose morphisms are defined by the formula
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504022.png" /></td> </tr></table>
+
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.
  
with the usual composition law for correspondences (see [[#References|[1]]]). Let the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504023.png" /> take values in the category of commutative graded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504024.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504025.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504026.png" /> will be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504027.png" />-additive category of graded correspondences. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504028.png" /> will have direct sums and tensor products.
+
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
  
The category whose objects are the varieties from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504029.png" /> and whose morphisms are correspondences of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504030.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504031.png" />. A natural functor from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504032.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504033.png" /> has been defined, and the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504034.png" /> extends to a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504035.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504036.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504037.png" />. The category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504038.png" />, like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504039.png" />, is not Abelian. Its pseudo-Abelian completion, the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504040.png" />, has been considered. It is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504041.png" /> by the formal addition of the images of all projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504042.png" />. More precisely, the objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504043.png" /> are pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504044.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504047.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504048.png" /> is the set of correspondences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504049.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504050.png" /> modulo a correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504051.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504052.png" />. The category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504053.png" /> is imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504054.png" /> by means of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504055.png" />. The natural functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504056.png" /> is called the functor of motive cohomology spaces and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504057.png" /> is called the category of effective motives.
+
$$
 +
\mathop{\rm Hom} ( \overline{X} , \overline{Y} )  = C ( X \times Y )
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504058.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504059.png" /> is the class of any rational point on the projective line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504060.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504061.png" />. Then
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504062.png" /></td> </tr></table>
+
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.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504063.png" /> is the projectivization of a locally free sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504064.png" /> of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504065.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504066.png" />, then
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504067.png" /></td> </tr></table>
+
$$
 +
h ( P  ^ {n} )  = 1 \oplus L \oplus
 +
L ^ {\otimes 2 } \oplus \dots \oplus
 +
L ^ {\otimes n } .
 +
$$
 +
 
 +
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 [[#References|[1]]]), motives of Abelian manifolds (see [[#References|[2]]]), and motives of Weil hypersurfaces have also been calculated.
 
Motives of a monodial 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504068.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504069.png" /> by the formal addition of negative powers of the motives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504070.png" />. By analogy with [[L-adic-cohomology|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504071.png" />-adic cohomology]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504072.png" /> is called the Tate motive. Tensor multiplication with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504073.png" /> is called twisting by the Tate motive. Twisting enables one to define the level of a motive as in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504074.png" />-adic cohomology theory. Any functor of the Weil cohomology factors through the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504075.png" />. There is the conjecture that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504076.png" /> does not, in some sense, depend on the intersection theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504077.png" />, and that the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504078.png" /> 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).
+
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====
 
====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,  "Motives des variétés algébrique" , ''Sem. 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>
 
<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,  "Motives des variétés algébrique" , ''Sem. 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====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504079.png" />-functions (see [[#References|[a4]]]).
+
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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Deligne,  "Theory 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504080.png" />-theory 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065040/m06504081.png" />-functions'' , Acad. Press  (1988)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Deligne,  "Theory 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 $K$-theory 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>

Latest revision as of 09:02, 8 May 2022


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

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, "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)
How to Cite This Entry:
Motives, theory of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Motives,_theory_of&oldid=28019
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article