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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  $  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====
<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></table>

====Comments====

====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 {{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 <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) {{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>