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====References====
 
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<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>
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<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>
  
  
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<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>
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<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>

Revision as of 21:57, 30 March 2012

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 -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 be a projective smooth connected scheme over a fixed algebraically closed field and let be a field of characteristic zero. Then Weil cohomology with coefficient field is a contravariant functor from the category of varieties into the category of finite-dimensional graded anti-commutative -algebras, which satisfies the following conditions:

1) If , then is isomorphic to , and the mapping

defined by the multiplication in , is non-degenerate for all ;

2) (Künneth formula);

3) Mapping of cycles. There exists a functorial homomorphism from the group of algebraic cycles in of codimension into which maps the direct product of cycles to the tensor product and is non-trivial in the sense that, for a point , becomes the canonical imbedding of into . The number

is known as the -th Betti number of the variety .

Examples. If , classical cohomology of complex manifolds with coefficients in is a Weil cohomology. If is a prime number distinct from the characteristic of the field , then étale -adic cohomology

is a Weil cohomology with coefficients in the field .

The Lefschetz formula

is valid for Weil cohomology. In the above formula, is the intersection index in of the graph of the morphism with the diagonal , which may also be interpreted as the number of fixed points of the endomorphism , while is the trace of the endomorphism which is induced by in . Moreover, this formula is also valid for correspondences, i.e. elements .

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


Comments

References

[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 -adique et fonctions . 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=24012
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article