I just want to make sure I have the definition of etale cohomology groups right, since it is built up on a bunch of other definitions:

Let $X$ be a variety over a field $k$. Very concretely, an element of $H^i_{et}(X_{k^s}, \mathbb{Z}_l)$ is represented by an element of $\Gamma(X_{k^s}, I^i)$, where $\mathbb{Z}_l \to I^\bullet$ is an resolution of the constant sheaf $\mathbb{Z}_l$ by injective sheaves, all this taking place in the abelian category of sheaves of abelian groups on the etale topology on $X_{k^s}$.

Is that correct?

  • 1
    $\begingroup$ No. Etale cohomology does not behave well with respect to non torsion sheaves. But there is a trick for this : first define $H^i_{ét}(X_{k^s},\mathbb{Z}/l^n\mathbb{Z})$ by the definition you gave. When $n$ varies, you get a projective system and $H^i_{ét}(X_{k^s},\mathbb{Z}_l)$ is defined to be the projective limit. You can check that this is indeed a $\mathbb{Z}_l$-module. $\endgroup$ – Roland Jun 22 '17 at 20:52

To define $\mathbb{Z}_l$ cohomology, one first defines $\mathbb{Z}/l^n$ cohomology and then passes to projective limit. There are many ways to explain this.

$\text{}$1. For an abelian variety, the degree $1$ cohomology is the Tate module, and it is defined to be the projective limit of $\mathbb{Z}/l^n$ torsion points of the abelian variety.

$\text{}$2. To prove various deep properties of étale cohomology, one has to rely on geometric argument, and one has to reduce statements above general sheaves to statements about sheaves which are representable by étale schemes. Sheaves representable by étale schemes must have finite stalks, and hence are torison sheaves.

$\text{}$3. If one insists to define the cohomology of $\mathbb{Z}_\ell$ using injective resolution, one will get the wrong cohomology. For example, for a smooth curve $X$ of genus $g$, one always gets$$\text{H}^1(X, \mathbb{Z}_l) = 0.$$See SGA 4, IX, 3.7.


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