I am reading my teacher's lecture notes on a course about modular forms and I am stuck at the following problem:
Assume $\Gamma\subseteq\mathbb{C}^{n}$ is a lattice and $A:=\mathbb{C}^{n}/\Gamma$ is an $n$-dimensional torus.
Then there exists a diagram of short exact sequences
$\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c} 0 & \ras{} & \mathbb{Z} & \ras{} & \mathbb{C} & \ras{e^{2\pi i\cdot}} & \mathbb{C}^{*} & \ras{} & 0 \\ & & \da{=} & & \da{} & & \da{} & & \\ 0 & \ra{} & \mathbb{Z} & \ra{} & \mathcal{O}_{A} & \ra{} & \mathcal{O}^{*}_{A} & \ra{} & 0\text{.} \end{array}$
This induces a diagram of long exact sequences
$\begin{array}{c} H^{1}(A,\mathbb{Z}) & \ras{} & H^{1}(A,\mathbb{C}) & \ras{} & H^{1}(A,\mathbb{C}^{*}) & \ras{} & H^{2}(A,\mathbb{Z})\\ \da{=} & & \da{} & & \da{} & & \da{=}\\ H^{1}(A,\mathbb{Z}) & \ra{} & H^{1}(A,\mathcal{O}_{A}) & \ra{} & H^{1}(A,\mathcal{O}_{A}^{*}) & \ra{} & H^{2}(A,\mathbb{Z})\text{.} \end{array}$
Questions:
(1.) How do the two maps $\mathbb{C}\longrightarrow\mathcal{O}_{A}$ and $\mathbb{C}^{*}\longrightarrow\mathcal{O}^{*}_{A}$ come about? Are they induced from $\begin{array}\mathbb{C}& \ras{a\mapsto(a,a,a,...)} & \mathbb{C}^{n} & \ras{} & A\end{array}$?
(2.) My teacher makes the claim that $H^{1}(A,\mathbb{C})\longrightarrow H^{1}(A,\mathcal{O}_{A})$ is a surjection. Why is that the case?
Thanks in advance.