View source for L-adic-cohomology

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One of the constructions of cohomology of abstract algebraic varieties and schemes. Etale cohomologies (cf. [[Etale cohomology|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|Lefschetz formula]] and its application to zeta-functions. It is obtained from étale cohomology by passing to the projective limit.

Let  $  \ell $
be a prime number; an  $  \ell $-adic sheaf on a scheme  $  X $
is a projective system  $  {( F _ {n} ) } _ {n \in \mathbf N }  $
of étale Abelian sheaves  $  F _ {n} $
such that, for all  $  n $, 
the transfer homomorphisms  $  F _ {n+ 1} \rightarrow F _ {n} $
are equivalent to the canonical morphism  $  F _ {n+ 1} \rightarrow F _ {n+ 1} / \ell  ^ {n} F _ {n+ 1} $. 
An  $  \ell $-adic sheaf  $  F $
is said to be constructible (respectively, locally constant) if all sheaves  $  F _ {n} $
are constructible (locally constant) étale sheaves. There exists a natural equivalence of the category of locally constant constructible sheaves on a connected scheme  $  X $
and the category of modules of finite type over the ring  $  \mathbf Z _ {\ell} $
of integral  $  \ell $-adic numbers which are continuously acted upon from the left by the fundamental group of the scheme  $  X $. 
This proves that locally constant constructible sheaves are abstract analogues of systems of local coefficients in topology. Examples of constructible  $  \ell $-adic sheaves include the sheaf  $  \mathbf Z _ {l,X} = {( ( \mathbf Z / \ell  ^ {n} \mathbf Z ) _ {X} ) } _ {n \in \mathbf N }  $, 
and the Tate sheaves  $  \mathbf Z _ {\ell} ( m) _ {X} = ( \mu _ {\ell  ^ {n}  , X } ^ {\otimes  ^ {m} } ) _ {n \in \mathbf N }  $ (where  $  ( \mathbf Z / \ell  ^ {n} \mathbf Z ) _ {X} $
is the constant sheaf on  $  X $
associated with the group  $  \mathbf Z / \ell  ^ {n} \mathbf Z $, 
while  $  \mu _ {\ell  ^ {n}  , X } $
is the sheaf of  $  \ell  ^ {n} $-th power roots of unity on  $  X $). 
If  $  A $
is an [[Abelian scheme|Abelian scheme]] over  $  X $, 
then  $  T _ {\ell} ( A) = {( A _ {\ell  ^ {n}  } ) } _ {n \in \mathbf N }  $ (where  $  A _ {\ell  ^ {n}  } $
is the kernel of multiplication by  $  \ell  ^ {n} $
in  $  A $) 
forms a locally constant constructible  $  \ell $-adic sheaf on  $  X $, 
called the Tate module of  $  A $.

Let  $  X $
be a scheme over a field  $  k $, 
let  $  \overline{X} = X \otimes _ {k} \overline{k} _ {s} $
be the scheme obtained from  $  X $
by changing the base from  $  k $
to the separable closure  $  \overline{k} _ {s} $
of the field  $  k $, 
and let  $  F = ( F _ {n} ) $
be an  $  \ell $-adic sheaf on  $  X $; 
the étale cohomology  $  H  ^ {i} ( \overline{X} , \overline{F} _ {n} ) $
then defines a projective system  $  ( H  ^ {i} ( \overline{X} , \overline{F} _ {n} )) _ {n \in \mathbf N }  $
of  $   \mathop{\rm Gal} ( \overline{k} _ {s} / k ) $-modules. The projective limit  $  H  ^ {i} ( \overline{X} , F  ) = \lim\limits _ {\leftarrow n }   H  ^ {i} ( \overline{X} , \overline{F} _ {n} ) $
is naturally equipped with the structure of a  $  \mathbf Z _ {\ell} $-module on which  $   \mathop{\rm Gal} ( \overline{k} _ {s} / k ) $
acts continuously with respect to the  $  \ell $-adic topology. It is called the  $  i $-th  $  \ell $-adic cohomology of the sheaf  $  F $
on  $  X $. 
If  $  k = \overline{k} _ {s} $, 
the usual notation is  $  H  ^ {i} ( \overline{X} , F  ) = H  ^ {i} ( X, F  ) $. 
The fundamental theorems in étale cohomology apply to  $  \ell $-adic cohomology of constructible  $  \ell $-adic sheaves. If  $  \mathbf Q _ {\ell} $
is the field of rational  $  \ell $-adic numbers, then the  $  \mathbf Q _ {\ell} $-spaces  $  H _ {\ell}  ^ {i} ( \overline{X} ) = H  ^ {i} ( \overline{X} , \mathbf Z _ {\ell} ) \otimes \mathbf Q _ {\ell} $
are called the rational  $  \ell $-adic cohomology of the scheme  $  X $. 
Their dimensions  $  b _ {i} ( X;  \ell) $ (if defined) are called the  $  i $-th Betti numbers of  $  X $. 
For complete  $  k $-schemes the numbers  $  b _ {i} ( X;  \ell) $
are defined and are independent of  $  \ell $ ($  \ell \neq  \mathop{\rm char}  k $). 
If  $  k $
is an algebraically closed field of characteristic  $  p $
and if  $  \ell \neq p $, 
then the assignment of the spaces  $  H _ {\ell}  ^ {i} ( X) $
to a smooth complete  $  k $-variety defines a [[Weil cohomology|Weil cohomology]]. If  $  k = \mathbf C $
is the field of complex numbers, the comparison theorem  $  H _ {\ell}  ^ {i} = H  ^ {i} ( X, \mathbf Q ) \otimes \mathbf Q _ {\ell} $
is valid.

====Comments====
The fact (mentioned above) that for complete $k$-schemes the Betti numbers are independent of $\ell$ follows from Deligne's proof of the Weil conjectures (cf. also [[Zeta-function|Zeta-function]]).

====References====
<table>
<tr><td valign="top">[1]</td> <td valign="top"> A. Grothendieck, "Formule de Lefschetz et rationalité des fonctions $L$", ''Sém. Bourbaki'', '''17''' :  279  (1964–1965)</td></tr>
<tr><td valign="top">[a1]</td> <td valign="top"> A. Grothendieck, "Cohomologie $\ell$-adique et fonctions $L$", ''SGA 5'' , ''Lect. notes in math.'' , '''589''' , Springer  (1977). {{ISBN|0-387-08248-4}} {{ZBL|0345.00011}}</td></tr>
<tr><td valign="top">[a2]</td> <td valign="top"> J.S. Milne, "Etale cohomology", Princeton Univ. Press  (1980)</td></tr>
<tr><td valign="top">[a3]</td> <td valign="top"> E. Freitag, R. Kiehl, "Etale cohomology and the Weil conjectures", Springer  (1988)</td></tr>
<tr><td valign="top">[a4]</td> <td valign="top"> P. Deligne, "La conjecture de Weil I" ''Publ. Math. IHES'', '''43'''  (1974)  pp. 273–307</td></tr>
<tr><td valign="top">[a5]</td> <td valign="top"> P. Deligne, "La conjecture de Weil II" ''Publ. Math. IHES'', '''52'''  (1980)  pp. 137–252</td></tr>
</table>