The two examples I have in mind are curves and abelian varieties. To be precise, if $C$ is a smooth projective algebraic curve over a number field $K$, then, for a prime $l$

$H^1_{et}(C_{\bar{K}},\mathbb{Q}_l)\cong T_l(J(C))\otimes\mathbb{Q}_l,$

where $J(C)$ is the Jacobian of $C$. The isomorphism is as $G_K$-modules, where $G_K$ is the absolute Galois group of $K$. A similar statement is true for abelian varieties.

What about smooth projective varieties of dimension $d$? Is there some generalization of the Jacobian with Tate module dual to the first etale cohomology group? When applied to an abelian variety, does this construction return the original abelian variety?

  • 14
    $\begingroup$ Yes, the generalized Jacobian is the Picard scheme and the result is basic Kummer theory, i.e. long cohomology exact sequences associated to the short exact sequences $0\to \mu_{\ell^n} \to \mathbb{G}_m \to \mathbb{G}_m \to 0$ of etale sheaves on $X$, together with the isomorphism $H^1_{et}(X, \mathbb{G}_m) = H^1(X, \mathcal{O}_X^\ast) = Pic(X)$. $\endgroup$ Jul 3, 2013 at 14:09
  • 4
    $\begingroup$ Ah, I should have been able to guess this might be the case. It would be very useful to me to know a reference covering the details, do you know one? $\endgroup$
    – Tom163
    Jul 4, 2013 at 11:15
  • $\begingroup$ We need also a duality between $H^1_{et}(X_{\overline{K}},\mathbb{Z}/l^{n})$ and $H^1_{et}(X_{\overline{K}},\mu_{l^{n}})$. $\endgroup$ Sep 8, 2016 at 22:51

1 Answer 1


You have an exact sequence of sheaves for the étale topology:

$0\rightarrow \mu_{l^{n}} \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m \rightarrow 0$.

If you take an injective resolution of this complex, you can see immediately that the p^power commute the diagramme in the first arrow of this complex.

By applying global section and the fact that we are in characteristic 0:

we have a short exact sequence of etale group cohomology: $0\rightarrow H^1_{et}(X_{\overline{K}},\mu_{l^{n}})\rightarrow H^{1}_{et}(X_{\overline{K}}, \mathbb{G}_m) \rightarrow H^{1}_{et}(X_{\overline{K}},\mathbb{G}_m) \rightarrow 0$. Wa have also a duality between $H^1_{et}(X_{\overline{K}},\mu_{l^{n}})$ and $H^1_{et}(X_{\overline{K}},\mathbb{Z}/l^{n})$, and it is easy to see using complex Cech that $H^{1}_{et}(X_{\overline{K}}, \mathbb{G}_m)$ is the picard group and $H^1_{et}(X_{\overline{K}},\mu_{l^{n}})$ classifies the invertible sheaf with trivialisation of its $l^{n}$ tensor power. Thus, by taking the projective limits over $n$ you find that $H^1_{et}(X_{\overline{K}},\mathbb{Z}_p)$ is the dual of the tate module of the Picard scheme of $X$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.