Good evening everyone, The exponential exact sequence is defined by: $ 0 \to \mathbb{Z} \to^{i} \mathcal{O} \to^{g} \mathcal{O}^{*} \to 0 $ with : $ g(f) = e^{ 2 \pi i f } $ and $ i $ is the sheaf embedding map. Could you explain me please, how we define its exact long exact sequence: $$ 0 \to H^0 ( X , \mathbb{Z} ) \to^{i_{0}} H^0 ( X , \mathcal{O} ) \to^{g_{0}} H^0 ( X , \mathcal{O}^{*} ) \to^{c_{0}} H^1 ( X , \mathbb{Z} ) \\ \to^{i_{1}} H^1 ( X , \mathcal{O} ) \to^{g_{1}} H^1 ( X , \mathcal{O}^{*} ) \to^{c_{1}} H^2 ( X , \mathbb{Z} ) \\ \to^{i_{2}} H^2 ( X , \mathcal{O} ) \to^{g_{2}} H^2 ( X , \mathcal{O}^{*} ) \to^{c_{2}} H^3 ( X , \mathbb{Z} ) \\ \to^{i_{3}} H^3 ( X , \mathcal{O} ) \to^{g_{3}} H^3 ( X , \mathcal{O}^{*} ) \to^{c_{3}} H^4 ( X , \mathbb{Z} )$$
I mean, how are defined explicitly : $H^{k-1} ( X , \mathcal{O}^* ) \to^{c_{k-1}} H^k ( X , \mathbb{Z} )$ and $H^k ( X , \mathbb{Z} ) \to^{i_{k}} H^k ( X , \mathcal{O} )$ and $ H^k ( X , \mathcal{O} ) \to^{g_{k}} H^k ( X , \mathcal{O}^{*} ) $ with : $ k \geq 0 $ and how do we obtain them explicitly ?.
Thank you very much.
