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Cohomology of algebraic varieties with coefficients in a field of characteristic zero, with formal properties required to obtain the [[Lefschetz formula|Lefschetz formula]] for the number of fixed points. The necessity for such a theory was pointed out by A. Weil [[#References|[1]]], who showed that the rationality of the [[Zeta-function|zeta-function]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w0976001.png" />-function of a variety over a finite field follow from the Lefschetz formula, whereas the remaining hypotheses about the zeta-function can naturally be formulated in cohomological terms. Let the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w0976002.png" /> be a projective smooth connected scheme over a fixed algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w0976003.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w0976004.png" /> be a field of characteristic zero. Then Weil cohomology with coefficient field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w0976005.png" /> is a contravariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w0976006.png" /> from the category of varieties into the category of finite-dimensional graded anti-commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w0976007.png" />-algebras, which satisfies the following conditions:
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1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w0976008.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w0976009.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760010.png" />, and the mapping
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/w/w097/w097600/w09760011.png" /></td> </tr></table>
+
Cohomology of algebraic varieties with coefficients in a field of characteristic zero, with formal properties required to obtain the [[Lefschetz formula|Lefschetz formula]] for the number of fixed points. The necessity for such a theory was pointed out by A. Weil [[#References|[1]]], who showed that the rationality of the [[zeta-function]] and  $  L $-
 +
function of a variety over a finite field follow from the Lefschetz formula, whereas the remaining hypotheses about the zeta-function can naturally be formulated in cohomological terms. Let the variety  $  X $
 +
be a projective smooth connected scheme over a fixed algebraically closed field  $  k $
 +
and let  $  K $
 +
be a field of characteristic zero. Then Weil cohomology with coefficient field  $  K $
 +
is a contravariant functor  $  X \rightarrow H  ^ {*} ( X) $
 +
from the category of varieties into the category of finite-dimensional graded anti-commutative  $  K $-
 +
algebras, which satisfies the following conditions:
  
defined by the multiplication in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760012.png" />, is non-degenerate for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760013.png" />;
+
1) If  $  n= { \mathop{\rm dim} }  ( X) $,  
 +
then  $  H  ^ {2n} ( X) $
 +
is isomorphic to  $  K $,
 +
and the mapping
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760014.png" /> (Künneth formula);
+
$$
 +
H  ^ {i} ( X) \times H  ^ {2n-i} ( X) \rightarrow  H  ^ {2n} ( X) ,
 +
$$
  
3) Mapping of cycles. There exists a functorial homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760015.png" /> from the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760016.png" /> of algebraic cycles in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760017.png" /> of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760018.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760019.png" /> which maps the direct product of cycles to the tensor product and is non-trivial in the sense that, for a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760021.png" /> becomes the canonical imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760022.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760023.png" />. The number
+
defined by the multiplication in $  H  ^ {*} ( X) $,
 +
is non-degenerate for all  $  i $;
  
<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/w/w097/w097600/w09760024.png" /></td> </tr></table>
+
2)  $  H  ^ {*} ( X) \otimes _ {K} H  ^ {*} ( Y)  \widetilde \rightarrow    H  ^ {*} ( X \times Y) $(
 +
Künneth formula);
  
is known as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760025.png" />-th Betti number of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760026.png" />.
+
3) Mapping of cycles. There exists a functorial homomorphism  $  \gamma _ {X} $
 +
from the group  $  C  ^ {p} ( X) $
 +
of algebraic cycles in  $  X $
 +
of codimension  $  p $
 +
into  $  H  ^ {2p} ( X) $
 +
which maps the direct product of cycles to the tensor product and is non-trivial in the sense that, for a point  $  P $,
 +
$  \gamma _ {P} $
 +
becomes the canonical imbedding of  $  \mathbf Z $
 +
into  $  K $.  
 +
The number
  
Examples. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760027.png" />, classical cohomology of complex manifolds with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760028.png" /> is a Weil cohomology. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760029.png" /> is a prime number distinct from the characteristic of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760030.png" />, then étale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760031.png" />-adic cohomology
+
$$
 +
b _ {i} ( X)  =   \mathop{\rm dim} _ {K}  H  ^ {i} ( X)
 +
$$
  
<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/w/w097/w097600/w09760032.png" /></td> </tr></table>
+
is known as the  $  i $-
 +
th Betti number of the variety  $  X $.
  
is a Weil cohomology with coefficients in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760033.png" />.
+
Examples. If  $  k = \mathbf C $,
 +
classical cohomology of complex manifolds with coefficients in  $  \mathbf C $
 +
is a Weil cohomology. If  $  l $
 +
is a prime number distinct from the characteristic of the field $  k $,
 +
then étale  $  l $-
 +
adic cohomology
  
The Lefschetz formula
+
$$
 
+
X  \mapsto  \left [ \lim\limits _ {\\vec{nu} } H _ {et} ^ {*}
<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/w/w097/w097600/w09760034.png" /></td> </tr></table>
+
( X, \mathbf Z / l ^ \nu \mathbf Z ) \right ] \otimes _ {\mathbf Z _ {l} }
 
+
\mathbf Q _ {l} $$
is valid for Weil cohomology. In the above formula, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760035.png" /> is the intersection index in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760036.png" /> of the graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760037.png" /> of the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760038.png" /> with the diagonal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760039.png" />, which may also be interpreted as the number of fixed points of the endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760040.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760041.png" /> is the trace of the endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760042.png" /> which is induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760043.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760044.png" />. Moreover, this formula is also valid for correspondences, i.e. elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760045.png" />.
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Weil,  "Numbers of solutions of equations in finite fields" ''Bull. Amer. Math. Soc.'' , '''55'''  (1949)  pp. 497–508</TD></TR><TR><TD valign="top">[2]</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>
 
  
 +
is a Weil cohomology with coefficients in the field  $  \mathbf Q _ {l} $.
  
 +
The Lefschetz formula
  
====Comments====
+
$$
 +
\langle  u \cdot \Delta \rangle  = \sum_{i=0}^ { 2n }  (- 1)  ^ {i}  \mathop{\rm Tr} ( u _ {i} )
 +
$$
  
 +
is valid for Weil cohomology. In the above formula,  $  \langle  u \cdot \Delta \rangle $
 +
is the intersection index in  $  X \times X $
 +
of the graph  $  \Gamma $
 +
of the morphism  $  u :  X \rightarrow X $
 +
with the diagonal  $  \Delta \subset  X \times X $,
 +
which may also be interpreted as the number of fixed points of the endomorphism  $  u $,
 +
while  $  { \mathop{\rm Tr} } ( u _ {i} ) $
 +
is the trace of the endomorphism  $  u _ {i} $
 +
which is induced by  $  u $
 +
in  $  H  ^ {i} ( X) $.
 +
Moreover, this formula is also valid for correspondences, i.e. elements  $  u \in H  ^ {2n} ( X \times X) $.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Grothendieck,   "The cohomology theory of abstract algebraic varieties" J.A. Todd (ed.) , ''Proc. Internat. Congress Mathematicians (Edinburgh, 1958)'' , Cambridge Univ. Press (1960) pp. 103–118</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Grothendieck,   I. Bucur,   C. Honzel,   L. Illusie,   J.-P. Jouanolou,   J.-P. Serre,   "Cohomologie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760046.png" />-adique et fonctions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097600/w09760047.png" />. SGA 5" , ''Lect. notes in math.'' , '''589''' , Springer (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.S. Milne,   "Etale cohomology" , Princeton Univ. Press (1980)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E. Freitag,   R. Kiehl,   "Étale cohomology and the Weil conjecture" , Springer (1988)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> R. Hartshorne,   "Algebraic geometry" , Springer (1977) pp. 272</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top"> A. Weil, "Numbers of solutions of equations in finite fields" ''Bull. Amer. Math. Soc.'' , '''55''' (1949) pp. 497–508 {{MR|0029393}} {{ZBL|0032.39402}} </TD></TR>
 +
<TR><TD valign="top">[2]</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 {{MR|0292838}} {{ZBL|0198.25902}} </TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Grothendieck, "The cohomology theory of abstract algebraic varieties" J.A. Todd (ed.) , ''Proc. Internat. Congress Mathematicians (Edinburgh, 1958)'' , Cambridge Univ. Press (1960) pp. 103–118 {{MR|0130879}} {{ZBL|0119.36902}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Grothendieck, I. Bucur, C. Honzel, L. Illusie, J.-P. Jouanolou, J.-P. Serre, "Cohomologie $\ell$-adique et fonctions $L$. SGA 5" , ''Lect. notes in math.'' , '''589''' , Springer (1977) {{MR|491704}} {{ZBL|}} </TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top"> J.S. Milne, "Etale cohomology" , Princeton Univ. Press (1980) {{MR|0559531}} {{ZBL|0433.14012}} </TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top"> E. Freitag, R. Kiehl, "Étale cohomology and the Weil conjecture" , Springer (1988) {{MR|0926276}} {{ZBL|0643.14012}} </TD></TR>
 +
<TR><TD valign="top">[a5]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 272 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>

Latest revision as of 08:12, 21 January 2024


Cohomology of algebraic varieties with coefficients in a field of characteristic zero, with formal properties required to obtain the Lefschetz formula for the number of fixed points. The necessity for such a theory was pointed out by A. Weil [1], who showed that the rationality of the zeta-function and $ L $- function of a variety over a finite field follow from the Lefschetz formula, whereas the remaining hypotheses about the zeta-function can naturally be formulated in cohomological terms. Let the variety $ X $ be a projective smooth connected scheme over a fixed algebraically closed field $ k $ and let $ K $ be a field of characteristic zero. Then Weil cohomology with coefficient field $ K $ is a contravariant functor $ X \rightarrow H ^ {*} ( X) $ from the category of varieties into the category of finite-dimensional graded anti-commutative $ K $- algebras, which satisfies the following conditions:

1) If $ n= { \mathop{\rm dim} } ( X) $, then $ H ^ {2n} ( X) $ is isomorphic to $ K $, and the mapping

$$ H ^ {i} ( X) \times H ^ {2n-i} ( X) \rightarrow H ^ {2n} ( X) , $$

defined by the multiplication in $ H ^ {*} ( X) $, is non-degenerate for all $ i $;

2) $ H ^ {*} ( X) \otimes _ {K} H ^ {*} ( Y) \widetilde \rightarrow H ^ {*} ( X \times Y) $( Künneth formula);

3) Mapping of cycles. There exists a functorial homomorphism $ \gamma _ {X} $ from the group $ C ^ {p} ( X) $ of algebraic cycles in $ X $ of codimension $ p $ into $ H ^ {2p} ( X) $ which maps the direct product of cycles to the tensor product and is non-trivial in the sense that, for a point $ P $, $ \gamma _ {P} $ becomes the canonical imbedding of $ \mathbf Z $ into $ K $. The number

$$ b _ {i} ( X) = \mathop{\rm dim} _ {K} H ^ {i} ( X) $$

is known as the $ i $- th Betti number of the variety $ X $.

Examples. If $ k = \mathbf C $, classical cohomology of complex manifolds with coefficients in $ \mathbf C $ is a Weil cohomology. If $ l $ is a prime number distinct from the characteristic of the field $ k $, then étale $ l $- adic cohomology

$$ X \mapsto \left [ \lim\limits _ {\\vec{nu} } H _ {et} ^ {*} ( X, \mathbf Z / l ^ \nu \mathbf Z ) \right ] \otimes _ {\mathbf Z _ {l} } \mathbf Q _ {l} $$

is a Weil cohomology with coefficients in the field $ \mathbf Q _ {l} $.

The Lefschetz formula

$$ \langle u \cdot \Delta \rangle = \sum_{i=0}^ { 2n } (- 1) ^ {i} \mathop{\rm Tr} ( u _ {i} ) $$

is valid for Weil cohomology. In the above formula, $ \langle u \cdot \Delta \rangle $ is the intersection index in $ X \times X $ of the graph $ \Gamma $ of the morphism $ u : X \rightarrow X $ with the diagonal $ \Delta \subset X \times X $, which may also be interpreted as the number of fixed points of the endomorphism $ u $, while $ { \mathop{\rm Tr} } ( u _ {i} ) $ is the trace of the endomorphism $ u _ {i} $ which is induced by $ u $ in $ H ^ {i} ( X) $. Moreover, this formula is also valid for correspondences, i.e. elements $ u \in H ^ {2n} ( X \times X) $.

References

[1] A. Weil, "Numbers of solutions of equations in finite fields" Bull. Amer. Math. Soc. , 55 (1949) pp. 497–508 MR0029393 Zbl 0032.39402
[2] 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 MR0292838 Zbl 0198.25902
[a1] A. Grothendieck, "The cohomology theory of abstract algebraic varieties" J.A. Todd (ed.) , Proc. Internat. Congress Mathematicians (Edinburgh, 1958) , Cambridge Univ. Press (1960) pp. 103–118 MR0130879 Zbl 0119.36902
[a2] A. Grothendieck, I. Bucur, C. Honzel, L. Illusie, J.-P. Jouanolou, J.-P. Serre, "Cohomologie $\ell$-adique et fonctions $L$. SGA 5" , Lect. notes in math. , 589 , Springer (1977) MR491704
[a3] J.S. Milne, "Etale cohomology" , Princeton Univ. Press (1980) MR0559531 Zbl 0433.14012
[a4] E. Freitag, R. Kiehl, "Étale cohomology and the Weil conjecture" , Springer (1988) MR0926276 Zbl 0643.14012
[a5] R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 272 MR0463157 Zbl 0367.14001
How to Cite This Entry:
Weil cohomology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weil_cohomology&oldid=15121
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article