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.
This Long exact sequence arises from the Snake lemma. Other link.
You obtain the long exact sequence from the short exact sequence by constructing connecting homomorphisms.