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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.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l0570202.png" /> be a prime number; an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l0570204.png" />-adic sheaf on a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l0570205.png" /> is a projective system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l0570206.png" /> of étale Abelian sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l0570207.png" /> such that, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l0570208.png" />, the transfer homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l0570209.png" /> are equivalent to the canonical morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702010.png" />. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702011.png" />-adic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702012.png" /> is said to be constructible (respectively, locally constant) if all sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702015.png" /> are constructible (locally constant) étale sheaves. There exists a natural equivalence of the category of locally constant constructible sheaves on a connected scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702016.png" /> and the category of modules of finite type over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702017.png" /> of integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702018.png" />-adic numbers which are continuously acted upon from the left by the fundamental group of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702019.png" />. This proves that locally constant constructible sheaves are abstract analogues of systems of local coefficients in topology. Examples of constructible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702020.png" />-adic sheaves include the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702021.png" />, and the Tate sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702022.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702023.png" /> is the constant sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702024.png" /> associated with the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702025.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702026.png" /> is the sheaf of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702027.png" />-th power roots of unity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702028.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702029.png" /> is an [[Abelian scheme|Abelian scheme]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702030.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702031.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702032.png" /> is the kernel of multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702033.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702034.png" />) forms a locally constant constructible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702035.png" />-adic sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702036.png" />, called the Tate module of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702037.png" />.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702038.png" /> be a scheme over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702039.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702040.png" /> be the scheme obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702041.png" /> by changing the base from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702042.png" /> to the separable closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702043.png" /> of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702044.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702045.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702046.png" />-adic sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702047.png" />; the étale cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702048.png" /> then defines a projective system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702049.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702050.png" />-modules. The projective limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702051.png" /> is naturally equipped with the structure of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702052.png" />-module on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702053.png" /> acts continuously with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702054.png" />-adic topology. It is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702055.png" />-th <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702056.png" />-adic cohomology of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702057.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702058.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702059.png" />, the usual notation is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702060.png" />. The fundamental theorems in étale cohomology apply to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702061.png" />-adic cohomology of constructible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702062.png" />-adic sheaves. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702063.png" /> is the field of rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702064.png" />-adic numbers, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702065.png" />-spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702066.png" /> are called the rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702068.png" />-adic cohomology of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702069.png" />. Their dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702070.png" /> (if defined) are called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702072.png" />-th Betti numbers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702073.png" />. For complete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702074.png" />-schemes the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702075.png" /> are defined and are independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702076.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702077.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702078.png" /> is an algebraically closed field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702079.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702080.png" />, then the assignment of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702081.png" /> to a smooth complete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702082.png" />-variety defines a [[Weil cohomology|Weil cohomology]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702083.png" /> is the field of complex numbers, the comparison theorem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702085.png" /> is valid.
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====References====
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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.
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,   "Formule de Lefschetz et rationalité des fonctions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702086.png" />" , ''Sem. Bourbaki'' , '''17''' :  279  (1964–1965)</TD></TR></table>
 
  
 +
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====
 
====Comments====
The fact (mentioned above) that for complete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702087.png" />-schemes the Betti numbers are independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702088.png" /> follows from Deligne's proof of the Weil conjectures (cf. also [[Zeta-function|Zeta-function]]).
+
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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Grothendieck,   "Cohomologie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702089.png" />-adique et fonctions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057020/l05702090.png" />" , ''SGA 5'' , ''Lect. notes in math.'' , '''589''' , Springer  (1977)</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>
+
<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>

Latest revision as of 11:49, 8 April 2023


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 $ \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 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. 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).

References

[1] A. Grothendieck, "Formule de Lefschetz et rationalité des fonctions $L$", Sém. Bourbaki, 17 : 279 (1964–1965)
[a1] 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
[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
How to Cite This Entry:
L-adic-cohomology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L-adic-cohomology&oldid=15770
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article