##### Actions

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 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. 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.