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Let $C$ be a smooth geometrically integral curve over a number field $k$, we do not assume $C$ to be proper, i.e., $C$ is not projective.

Under the spectral sequence $H^p(k,H^q_{\mathrm{et}}(C_{\bar{k}},\mathbb{G}_m)) \implies H^{p+q}_{\mathrm{et}}(C,\mathbb{G}_m)$, we will obtain the cohomology groups $H^i(k,\bar{k}[C]^*)$.

Question 1: How does one understand the elements of such groups, say for $i=1,2,3$?

In the case where $C$ is proper, we have $\bar{k}^* = \bar{k}[C]^*$, in other words, the invertible regular functions on $C_{\bar{k}}$ are simply the nonzero constant functions. Therefore, for $i=1,2,3$, we have respectively, $0$ (by Hilbert 90), Br$(k)$, and $0$.

Now let $C_1$ be the smooth completion of $C$. $C_1$ is now a smooth projective geometrically integral curve. The inclusion $i : C \rightarrow C_1$ induces the map $i^*: H^1(k,\mathrm{Pic}(C_{1,\bar{k}})) \rightarrow H^1(k,\mathrm{Pic}(C_{\bar{k}}))$.

Question 2: Are there any ways to explicitly describe $i^*$?

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