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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 $
 
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 $
 
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  $  l $-
+
with constant coefficients, every  $  \ell $-adic cohomology theory for various prime numbers  $  \ell $
adic cohomology theory for various prime numbers  $  l $
 
 
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.
 
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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and let  $  X \rightarrow C ( X) $
 
and let  $  X \rightarrow C ( X) $
 
be a contravariant functor of global [[Intersection theory|intersection theory]] from  $  V ( k) $
 
be a contravariant functor of global [[Intersection theory|intersection theory]] from  $  V ( k) $
into the category of commutative  $  \Lambda $-
+
into the category of commutative  $  \Lambda $-algebras, where  $  \Lambda $
algebras, where  $  \Lambda $
 
 
is a fixed ring. For example,  $  C ( X) $
 
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 $
 
is the [[Chow ring|Chow ring]] of classes of algebraic cycles (cf. [[Algebraic cycle|Algebraic cycle]]) on  $  X $
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enable one to define a new category, the category of correspondences  $  C V ( k) $,  
 
enable one to define a new category, the category of correspondences  $  C V ( k) $,  
 
whose objects are varieties  $  X \in V ( k) $,  
 
whose objects are varieties  $  X \in V ( k) $,  
denoted by  $  \overline{X}\; $,  
+
denoted by  $  \overline{X} $,  
 
and whose morphisms are defined by the formula
 
and whose morphisms are defined by the formula
  
 
$$  
 
$$  
  \mathop{\rm Hom} ( \overline{X}\; , \overline{Y}\; )  =  C ( X \times Y )
+
  \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 $
 
with the usual composition law for correspondences (see [[#References|[1]]]). Let the functor  $  C $
take values in the category of commutative graded  $  \Lambda $-
+
take values in the category of commutative graded  $  \Lambda $-algebras  $  A ( \Lambda ) $.  
algebras  $  A ( \Lambda ) $.  
 
 
Then  $  C V ( k) $
 
Then  $  C V ( k) $
will be the  $  \Lambda $-
+
will be the  $  \Lambda $-additive category of graded correspondences. Moreover,  $  C V ( k) $
additive category of graded correspondences. Moreover,  $  C V ( k) $
 
 
will have direct sums and tensor products.
 
will have direct sums and tensor products.
  
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by the formal addition of the images of all projections  $  p $.  
 
by the formal addition of the images of all projections  $  p $.  
 
More precisely, the objects of  $  M _ {C}  ^ {+} ( k) $
 
More precisely, the objects of  $  M _ {C}  ^ {+} ( k) $
are pairs  $  ( \overline{X}\; , p ) $,  
+
are pairs  $  ( \overline{X} , p ) $,  
where  $  \overline{X}\; \in C V  ^ {0} ( k) $
+
where  $  \overline{X} \in C V  ^ {0} ( k) $
and  $  p \in  \mathop{\rm Hom} ( \overline{X}\; , \overline{X}\; ) $,  
+
and  $  p \in  \mathop{\rm Hom} ( \overline{X} , \overline{X} ) $,  
 
$  p  ^ {2} = p $,  
 
$  p  ^ {2} = p $,  
and  $  H ( ( \overline{X}\; , p ) , ( \overline{Y}\; , q ) ) $
+
and  $  H ( ( \overline{X} , p ) , ( \overline{Y} , q ) ) $
is the set of correspondences  $  f :  \overline{X}\; \rightarrow \overline{Y}\; $
+
is the set of correspondences  $  f :  \overline{X} \rightarrow \overline{Y} $
 
such that  $  f \circ p = = q \circ f $
 
such that  $  f \circ p = = q \circ f $
 
modulo a correspondence  $  g $
 
modulo a correspondence  $  g $
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The category  $  C V  ^ {0} ( k) $
 
The category  $  C V  ^ {0} ( k) $
 
is imbedded in  $  M _ {C}  ^ {+} ( k) $
 
is imbedded in  $  M _ {C}  ^ {+} ( k) $
by means of the functor  $  \overline{X}\; \rightarrow ( \overline{X}\; ,  \mathop{\rm id} ) $.  
+
by means of the functor  $  \overline{X} \rightarrow ( \overline{X} ,  \mathop{\rm id} ) $.  
 
The natural functor  $  h :  V ( k) \rightarrow M _ {C}  ^ {+} ( k) $
 
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 functor of motive cohomology spaces and  $  M _ {C}  ^ {+} ( k) $
Line 87: Line 83:
 
$$  
 
$$  
 
h ( P  ^ {n} )  =  1 \oplus L \oplus
 
h ( P  ^ {n} )  =  1 \oplus L \oplus
L ^ {\otimes _ {2} } \oplus \dots \oplus
+
L ^ {\otimes 2 } \oplus \dots \oplus
L ^ {\otimes _ {n} } .
+
L ^ {\otimes n } .
 
$$
 
$$
  
Line 98: Line 94:
  
 
$$  
 
$$  
h ( X)  =  \oplus _ {i = 0 } ^ { {r }  - 1 }
+
h ( X)  =  \bigoplus _ {i = 0 } ^ { {r }  - 1 }
( h ( Y) \otimes L ^ {\otimes _ {i} } ) .
+
( h ( Y) \otimes L ^ {\otimes i } ) .
 
$$
 
$$
  
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is obtained from  $  M _ {C}  ^ {+} ( k) $
 
is obtained from  $  M _ {C}  ^ {+} ( k) $
 
by the formal addition of negative powers of the motives  $  L $.  
 
by the formal addition of negative powers of the motives  $  L $.  
By analogy with [[L-adic-cohomology| $  l $-
+
By analogy with [[L-adic-cohomology| $  \ell $-adic cohomology]],  $  T = L ^ {\otimes -1 } $
adic cohomology]],  $  T = L ^ {\otimes _ {-} 1 } $
 
 
is called the Tate motive. Tensor multiplication with  $  T $
 
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  $ l $-
+
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) $.  
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) $
 
There is the conjecture that  $  M _ {C} ( k) $
 
does not, in some sense, depend on the intersection theory of  $  C $,  
 
does not, in some sense, depend on the intersection theory of  $  C $,  
Line 121: Line 115:
  
 
====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  $  L $-
+
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]]]).
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=47909
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article