I want to calculate the Picard group of a Torus. For simplicity, a Torus of complex dimension 1: $X=\mathbb C/\Lambda$ So far, I looked at the exponential sequence
$$\to H^1(X,\mathbb Z) \to H^1(X,O_X) \to H^1(X,O^*_X) \to H^2(X,\mathbb Z) \to H^2(X,O_X) \to $$
and got the following:
$$\to \mathbb Z^2 \to \mathbb C \to \text{Pic}(X) \to \mathbb Z \to 0.$$
The first map is injective, as it is induced by the inclusion $\mathbb Z \subset O_X$.
Furthermore, under the isomorphism $H^1(X,O_X) = H^{0,1}(X)$, the first map should be given by $(1,1)\mapsto (1,\tau)$, where $\Lambda =\mathbb Z + \tau \mathbb Z$.
But how do I proceed from here? How can I figure out the other maps?
From what you have, you see that there is a short exact sequence of abelian groups $0\to\mathbb C/\Lambda\to Pic\to\mathbb Z\to 0$ and since $\mathbb Z$ is a free abelian group that sequence splits, so that $Pic$ is isomorphic to $\mathbb C/\Lambda\oplus \mathbb Z$. In terms fo this isomorphism, the maps in the sequence are the obvious ones.