##### Actions

Let be a prime number; an -adic sheaf on a scheme is a projective system of étale Abelian sheaves such that, for all , the transfer homomorphisms are equivalent to the canonical morphism . An -adic sheaf is said to be constructible (respectively, locally constant) if all sheaves are constructible (locally constant) étale sheaves. There exists a natural equivalence of the category of locally constant constructible sheaves on a connected scheme and the category of modules of finite type over the ring of integral -adic numbers which are continuously acted upon from the left by the fundamental group of the scheme . This proves that locally constant constructible sheaves are abstract analogues of systems of local coefficients in topology. Examples of constructible -adic sheaves include the sheaf , and the Tate sheaves (where is the constant sheaf on associated with the group , while is the sheaf of -th power roots of unity on ). If is an Abelian scheme over , then (where is the kernel of multiplication by in ) forms a locally constant constructible -adic sheaf on , called the Tate module of .
Let be a scheme over a field , let be the scheme obtained from by changing the base from to the separable closure of the field , and let be an -adic sheaf on ; the étale cohomology then defines a projective system of -modules. The projective limit is naturally equipped with the structure of a -module on which acts continuously with respect to the -adic topology. It is called the -th -adic cohomology of the sheaf on . If , the usual notation is . The fundamental theorems in étale cohomology apply to -adic cohomology of constructible -adic sheaves. If is the field of rational -adic numbers, then the -spaces are called the rational -adic cohomology of the scheme . Their dimensions (if defined) are called the -th Betti numbers of . For complete -schemes the numbers are defined and are independent of ( ). If is an algebraically closed field of characteristic and if , then the assignment of the spaces to a smooth complete -variety defines a Weil cohomology. If is the field of complex numbers, the comparison theorem is valid.