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  $  l $
be a prime number; an  $  l $-
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 / l  ^ {n} F _ {n+} 1 $. 
An  $  l $-
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 _ {l} $
of integral  $  l $-
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  $  l $-
adic sheaves include the sheaf  $  \mathbf Z _ {l,X} = {( ( \mathbf Z / l  ^ {n} \mathbf Z ) _ {X} ) } _ {n \in \mathbf N }  $, 
and the Tate sheaves  $  \mathbf Z _ {l} ( m) _ {X} = ( \mu _ {l  ^ {n}  , X } ^ {\otimes  ^ {m} } ) _ {n \in \mathbf N }  $(
where  $  ( \mathbf Z / l  ^ {n} \mathbf Z ) _ {X} $
is the constant sheaf on  $  X $
associated with the group  $  \mathbf Z / l  ^ {n} \mathbf Z $, 
while  $  \mu _ {l  ^ {n}  , X } $
is the sheaf of  $  l  ^ {n} $-
th power roots of unity on  $  X $). 
If  $  A $
is an [[Abelian scheme|Abelian scheme]] over  $  X $, 
then  $  T _ {l} ( A) = {( A _ {l  ^ {n}  } ) } _ {n \in \mathbf N }  $(
where  $  A _ {l  ^ {n}  } $
is the kernel of multiplication by  $  l  ^ {n} $
in  $  A $) 
forms a locally constant constructible  $  l $-
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  $  l $-
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 _ {l} $-
module on which  $   \mathop{\rm Gal} ( \overline{k}\; _ {s} / k ) $
acts continuously with respect to the  $  l $-
adic topology. It is called the  $  i $-
th  $  l $-
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  $  l $-
adic cohomology of constructible  $  l $-
adic sheaves. If  $  \mathbf Q _ {l} $
is the field of rational  $  l $-
adic numbers, then the  $  \mathbf Q _ {l} $-
spaces  $  H _ {l}  ^ {i} ( \overline{X}\; ) = H  ^ {i} ( \overline{X}\; , \mathbf Z _ {l} ) \otimes \mathbf Q _ {l} $
are called the rational  $  l $-
adic cohomology of the scheme  $  X $. 
Their dimensions  $  b _ {i} ( X;  l) $(
if defined) are called the  $  i $-
th Betti numbers of  $  X $. 
For complete  $  k $-
schemes the numbers  $  b _ {i} ( X;  l) $
are defined and are independent of  $  l $(
$  l \neq  \mathop{\rm char}  k $). 
If  $  k $
is an algebraically closed field of characteristic  $  p $
and if  $  l \neq p $, 
then the assignment of the spaces  $  H _ {l}  ^ {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 _ {l}  ^ {i} = H  ^ {i} ( X, \mathbf Q ) \otimes \mathbf Q _ {l} $
is valid.

====References====
<table><tr><td valign="top">[1]</td> <td valign="top">  A. Grothendieck,   "Formule de Lefschetz et rationalité des fonctions $L$" , ''Sem. Bourbaki'' , '''17''' :  279  (1964–1965)</td></tr></table>

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

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