Difference between revisions of "Weil cohomology"
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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); | ||
− | is | + | 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 $. | ||
− | is a Weil cohomology with coefficients in the field | + | 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 | 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, | + | 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">[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 & Masson (1968) pp. 359–386 {{MR|0292838}} {{ZBL|0198.25902}} </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 & Masson (1968) pp. 359–386 {{MR|0292838}} {{ZBL|0198.25902}} </TD></TR></table> | ||
− | |||
− | |||
====Comments==== | ====Comments==== | ||
− | |||
====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 {{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> | <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 08:29, 6 June 2020
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 |
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 ![]() ![]() |
[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 |
Weil cohomology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weil_cohomology&oldid=24012