The classical $exponential sequence$ on a complex manifold $X$ is the short exact sequence $$0\longrightarrow\mathbb Z\longrightarrow\mathcal O_{X}\longrightarrow\mathcal O_{X}^{*}\longrightarrow 0.$$
Here, $\mathbb Z$ is the locally constant sheaf, $\mathcal O_{X}$ is the sheaf of holomorphic functions, $\mathcal O_{X}^{*}$ is the sheaf of non-vanishing holomorphic functions. Then,we can get a long exact cohomology sequence $$H^{1}(X,\mathbb Z)\longrightarrow H^{1}(X,\mathcal O_{X})\longrightarrow H^{1}(X,\mathcal O_{X}^{*})\longrightarrow H^{2}(X,\mathbb Z).$$
My question is,when $X$ is compact ,then we can get a claim: $$H^{1}(X,\mathbb Z)\longrightarrow H^{1}(X,\mathcal O_{X})$$is injective.Maybe this is a very trivial problem,can anyone give me some advice?Thanks a lot!
You have to look at the $H^0$ level of the exact sequence. Since it gives a short exact sequence (on a compact complex manifold, the only global holomorphic functions are constants), the final map is surjective, and this means that the exact sequence you wrote down in fact starts with a $0\longrightarrow$.