##### Actions

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.