L-adic-cohomology
One of the constructions of cohomology of abstract algebraic varieties and schemes. Etale cohomologies (cf. Etale cohomology) of schemes are torsion modules. Cohomology with coefficients in rings of characteristic zero is used for various purposes, mainly in the proof of the Lefschetz formula and its application to zeta-functions. It is obtained from étale cohomology by passing to the projective limit.
Let be a prime number; an
-adic sheaf on a scheme
is a projective system
of étale Abelian sheaves
such that, for all
, the transfer homomorphisms
are equivalent to the canonical morphism
. An
-adic sheaf
is said to be constructible (respectively, locally constant) if all sheaves
are constructible (locally constant) étale sheaves. There exists a natural equivalence of the category of locally constant constructible sheaves on a connected scheme
and the category of modules of finite type over the ring
of integral
-adic numbers which are continuously acted upon from the left by the fundamental group of the scheme
. This proves that locally constant constructible sheaves are abstract analogues of systems of local coefficients in topology. Examples of constructible
-adic sheaves include the sheaf
, and the Tate sheaves
(where
is the constant sheaf on
associated with the group
, while
is the sheaf of
-th power roots of unity on
). If
is an Abelian scheme over
, then
(where
is the kernel of multiplication by
in
) forms a locally constant constructible
-adic sheaf on
, called the Tate module of
.
Let be a scheme over a field
, let
be the scheme obtained from
by changing the base from
to the separable closure
of the field
, and let
be an
-adic sheaf on
; the étale cohomology
then defines a projective system
of
-modules. The projective limit
is naturally equipped with the structure of a
-module on which
acts continuously with respect to the
-adic topology. It is called the
-th
-adic cohomology of the sheaf
on
. If
, the usual notation is
. The fundamental theorems in étale cohomology apply to
-adic cohomology of constructible
-adic sheaves. If
is the field of rational
-adic numbers, then the
-spaces
are called the rational
-adic cohomology of the scheme
. Their dimensions
(if defined) are called the
-th Betti numbers of
. For complete
-schemes the numbers
are defined and are independent of
(
). If
is an algebraically closed field of characteristic
and if
, then the assignment of the spaces
to a smooth complete
-variety defines a Weil cohomology. If
is the field of complex numbers, the comparison theorem
is valid.
References
[1] | A. Grothendieck, "Formule de Lefschetz et rationalité des fonctions ![]() |
Comments
The fact (mentioned above) that for complete -schemes the Betti numbers are independent of
follows from Deligne's proof of the Weil conjectures (cf. also Zeta-function).
References
[a1] | A. Grothendieck, "Cohomologie ![]() ![]() |
[a2] | J.S. Milne, "Etale cohomology" , Princeton Univ. Press (1980) |
[a3] | E. Freitag, R. Kiehl, "Etale cohomology and the Weil conjectures" , Springer (1988) |
[a4] | P. Deligne, "La conjecture de Weil I" Publ. Math. IHES , 43 (1974) pp. 273–307 |
[a5] | P. Deligne, "La conjecture de Weil II" Publ. Math. IHES , 52 (1980) pp. 137–252 |
L-adic-cohomology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L-adic-cohomology&oldid=47543