# ag.algebraic geometry – Computing \$H^1\$ with coefficients in a torsion-free abelian group

Let $$k$$ be a number field and denote by $$H^i(k,-)$$ the Galois cohomology functor $$H^i(mathrm{Gal}(bar{k}/k),-)$$. Let $$X$$ be a smooth geometrically integral curve over $$k$$. One can easily show that the map $$H^1(k,mathrm{Pic}^0(X_{bar{k}})) rightarrow H^1(k,mathrm{Pic}(X_{bar{k}}))$$ is surjective. Indeed, applying the functor $$H^1(k,-)$$ to the exact sequence $$0 rightarrow mathrm{Pic}^0(X_{bar{k}}) rightarrow mathrm{Pic}(X_{bar{k}}) rightarrow mathrm{NS}(X_{bar{k}}) rightarrow 0,$$ we get the exact sequence $$H^1(k,mathrm{Pic}^0(X_{bar{k}})) rightarrow H^1(k,mathrm{Pic}(X_{bar{k}})) rightarrow H^1(k,mathbb{Z}).$$ The last term is zero because $$mathbb{Z}$$ has trivial Galois action and so any 1-cocycle $$f in H^1(k,mathbb{Z})$$ is simply an element of $$mathrm{Hom}(mathrm{Gal}(bar{k}/k),mathbb{Z})$$. But $$mathbb{Z}$$ is torsion-free and thus all such maps are zero maps.

Question 0. Here I did not assume that $$X$$ is projective, but would all of the above still hold without this assumption? For example, I’ve never seen the Neron-Severi group of an affine curve discussed in any literature.

Now moving on, we have an exact sequence of Galois modules $$0 rightarrow mathrm{Pic}(X_{bar{k}})_{mathrm{tor}} rightarrow mathrm{Pic}(X_{bar{k}}) rightarrow mathrm{Pic}(X_{bar{k}})_{mathrm{free}} rightarrow 0$$ where $$mathrm{Pic}(X_{bar{k}})_mathrm{tor}$$ denotes the maximal torsion subgroup of $$mathrm{Pic}(X_{bar{k}})$$ and $$mathrm{Pic}(X_{bar{k}})_mathrm{free} = mathrm{Pic}(X_{bar{k}})/mathrm{Pic}(X_{bar{k}})_mathrm{tor}$$, the maximal free quotient.

Question 1. In this case $$mathrm{Pic}(X_{bar{k}})_mathrm{free}$$ certainly does not have trivial Galois action, so we cannot reduce 1-cocycles to group homomorphisms $$mathrm{Gal}(bar{k}/k) rightarrow mathrm{Pic}(X_{bar{k}})_mathrm{free}$$. Therefore how do we go about computing $$H^1(k,mathrm{Pic}(X_{bar{k}})_mathrm{free})$$?

I have thought about first studying $$H^1(k,mathrm{Pic}(X_{bar{k}}))$$ using a wild idea as follows:

We apply the (étale) cohomology functor $$H^i(X_{bar{k}},-)$$ to the Kummer sequence $$0 rightarrow mu_n rightarrow mathbb{G_m} rightarrow mathbb{G}_m rightarrow 0$$ to obtain the long cohomology sequence $$0 rightarrow mu_n(bar{k}) rightarrow bar{k}^* rightarrow bar{k}^* rightarrow H^1(X_bar{k},mu_n) rightarrow mathrm{Pic}(X_{bar{k}}) rightarrow mathrm{Pic}(X_{bar{k}}) rightarrow H^2(X_{bar{k}},mu_n) rightarrow H^2(X_{bar{k}},mathbb{G}_m)=0.$$

Then we apply $$H^1(k,-)$$ to $$H^1(X_bar{k},mu_n) rightarrow mathrm{Pic}(X_{bar{k}}) rightarrow mathrm{Pic}(X_{bar{k}}) rightarrow H^2(X_{bar{k}},mu_n)$$ and study the result. But I’m not sure if there are any spectral sequences we can use, or if this approach is even feasible.