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First let me note that your definition of coherent sheaf is misleading: it implies that $\mathcal O_X$ is coherent by definition, whereas in reality coherence of $\mathcal O_X$ is a very deep theorem due to Oka .

The correct definition is that a sheaf $\mathcal F$ of $\mathcal O_X$-Modules is coherent if :

1. $\mathcal F$ is locally finitely generated : $X$ can be covered by open subsets $U$ on which there exist surjections $\mathcal O_U^N\to \mathcal F\mid U$ .
   and
2. For any open subset $V\subset X$ and any morphism $f:\mathcal O_V^s\to\mathcal F\mid V $ the sheaf $Ker(f)$ on $V$ is locally finitely generated .

As for the consequences of coherence, here is a typical application: if a sequence of coherent sheaves $ \mathcal F' \to \mathcal F\to\mathcal {F''} $ is exact at $x\in X$ (stalkwise) then it is exact on an open neighbourhood of $x$.

The real power of coherence however comes through Cartan's Theorems A and B for coherent sheaves on a Stein manifold.
These theorems have innumerable consequences: on a Stein manifold
(i) Every meromorphic function is the quotient of two global holomorphic functions.
(ii) Every topological line bundle has one and only one holomorphic structure,
(iii) Every closed analytic subset is the zero set of a family of globally defined holomorphic functions.
(iv) Every holomorphic function on a closed analytic subset can be extended to a holomorphic function on the whole Stein manifold.
(v) Given global holomorphic functions $f_1,...,f_r$ without a common zero, there exist global holomorphic functions $g_1,...,g_r$ with $f_1g_1+\cdots +f_rg_r=1$.

Just to illustrate how easy theorems become with those powerful tools let me prove (ii).
Start from the exponential sequence of sheaves $ 0\to \mathbb Z \to \mathcal O\to \mathcal O^*\to 0 $ on the Stein manifold $X$ .
The associated long exact sequence has as a fragment $$ \cdots \to H^1(X,\mathcal O) \to H^1(X,\mathcal O^*) \to H^2(X,\mathbb Z) \to H^2 (X,\mathcal O) \to \cdots $$ Since $H^1(X,\mathcal O) = H^2 (X,\mathcal O)=0 $ by theorem B, we get the isomorphism $H^1(X,\mathcal O^*) \cong H^2(X,\mathbb Z)$.
It factorizes as $H^1(X,\mathcal O^*)\to H^1(X,\mathcal C^*) \to H^2(X,\mathbb Z)$ and since $ H^1(X,\mathcal C^*) \to H^2(X,\mathbb Z)$ is well known to be the isomorphism given by the first chern class we get the result (ii).

First let me note that your definition of coherent sheaf is misleading: it implies that $\mathcal O_X$ is coherent by definition, whereas in reality coherence of $\mathcal O_X$ is a very deep theorem due to Oka .

The correct definition is that a sheaf $\mathcal F$ of $\mathcal O_X$-Modules is coherent if :

1. $\mathcal F$ is locally finitely generated : $X$ can be covered by open subsets $U$ on which there exist surjections $\mathcal O_U^N\to \mathcal F\mid U$ .
   and
2. For any open subset $V\subset X$ and any morphism $f:\mathcal O_V^s\to\mathcal F\mid V $ the sheaf $Ker(f)$ on $V$ is locally finitely generated .

As for the consequences of coherence, here is a typical application: if a sequence of coherent sheaves $ \mathcal F' \to \mathcal F\to\mathcal {F''} $ is exact at $x\in X$ (stalkwise) then it is exact on an open neighbourhood of $x$.

The real power of coherence however comes through Cartan's Theorems A and B for coherent sheaves on a Stein manifold.
These theorems have innumerable consequences: on a Stein manifold
(i) Every meromorphic function is the quotient of two global holomorphic functions.
(ii) Every topological line bundle has one and only one holomorphic structure,
(iii) Every closed analytic subset is the zero set of a family of globally defined holomorphic functions.
(iv) Every holomorphic function on a closed analytic subset can be extended to a holomorphic function on the whole Stein manifold.
(v) Given global holomorphic functions $f_1,...,f_r$ without a common zero, there exist global holomorphic functions $g_1,...,g_r$ with $f_1g_1+\cdots +f_rg_r=1$.

Just to illustrate how easy theorems become with those powerful tools in hand, let me prove (ii).
Start from the exponential sequence of sheaves $ 0\to \mathbb Z \to \mathcal O\to \mathcal O^\ast \to 0 $ on the Stein manifold $X$ .
The associated long exact sequence has as a fragment $$ \cdots \to H^1(X,\mathcal O) \to H^1(X,\mathcal O^\ast) \to H^2(X,\mathbb Z) \to H^2 (X,\mathcal O) \to \cdots $$ Since $H^1(X,\mathcal O) = H^2 (X,\mathcal O)=0 $ by theorem B, we get the isomorphism $$H^1(X,\mathcal O^\ast ) \cong H^2(X,\mathbb Z)$$
It factorizes as $H^1(X,\mathcal O^\ast )\to H^1(X,\mathcal C^\ast) \to H^2(X,\mathbb Z)$ and since $ H^1(X,\mathcal C^\ast) \stackrel {\cong}{\to }H^2(X,\mathbb Z)$ is well known to be the isomorphism given by the first chern class we get the result (ii) in the more precise form:

On a Stein manifold $X$ the first Chern class induces an isomorphism of abelian groups $$\text {Pic}(X)=H^1(X,\mathcal O^\ast )\stackrel {c_1}{\cong}H^2(X,\mathbb Z)$$

First let me note that your definition of coherent sheaf is misleading: it implies that $\mathcal O_X$ is coherent by definition, whereas in reality coherence of $\mathcal O_X$ is a very deep theorem due to Oka .

The correct definition is that a sheaf $\mathcal F$ of $\mathcal O_X$-Modules is coherent if :

1. $\mathcal F$ is locally finitely generated : $X$ can be covered by open subsets $U$ on which there exist surjections $\mathcal O_U^N\to \mathcal F\mid U$ .
   and
2. For any open subset $V\subset X$ and any morphism $f:\mathcal O_V^s\to\mathcal F\mid V $ the sheaf $Ker(f)$ on $V$ is locally finitely generated .

As for the consequences of coherence, here is a typical application: if a sequence of coherent sheaves $ \mathcal F' \to \mathcal F\to\mathcal {F''} $ is exact at $x\in X$ (stalkwise) then it is exact on an open neighbourhood of $x$.

The real power of coherence however comes through Cartan's Theorems A and B for coherent sheaves on a Stein manifold.
These theorems have innumerable consequences: on a Stein manifold
(i) Every meromorphic function is the quotient of two global holomorphic functions.
(ii) Every topological line bundle has one and only one holomorphic structure,
(iii) Every closed analytic subset is the zero set of a family of globally defined holomorphic functions.
(iv) Every holomorphic function on a closed analytic subset can be extended to a holomorphic function on the whole Stein manifold.
(v) Given global holomorphic functions $f_1,...,f_r$ without a common zero, there exist global holomorphic functions $g_1,...,g_r$ with $f_1g_1+\cdots +f_rg_r=1$.

Just to illustrate how easy theorems become with those powerful tools let me prove (ii).
Start from the exponential sequence of sheaves $ 0\to \mathbb Z \to \mathcal O\to \mathcal O^*\to 0 $ on the Stein manifold $X$ .

The associated long exact sequence has as a fragment $$ \cdots \to H^1(X,\mathcal O) \to H^1(X,\mathcal O^*) \to H^2(X,\mathbb Z) \to H^2 (X,\mathcal O) \to \cdots $$ Since $H^1(X,\mathcal O) = H^2 (X,\mathcal O)=0 $ by theorem B, we get the isomorphism $H^1(X,\mathcal O^*) \cong H^2(X,\mathbb Z)$.
It factorizes as $H^1(X,\mathcal O^*)\to H^1(X,\mathcal C^*) \to H^2(X,\mathbb Z)$ and since $ H^1(X,\mathcal C^*) \to H^2(X,\mathbb Z)$ is well known to be the isomorphism given by the first chern class we get the result (ii).

First let me note that your definition of coherent sheaf is misleading: it implies that $\mathcal O_X$ is coherent by definition, whereas in reality coherence of $\mathcal O_X$ is a very deep theorem due to Oka .

The correct definition is that a sheaf $\mathcal F$ of $\mathcal O_X$-Modules is coherent if :

1. $\mathcal F$ is locally finitely generated : $X$ can be covered by open subsets $U$ on which there exist surjections $\mathcal O_U^N\to \mathcal F\mid U$ .
   and
2. For any open subset $V\subset X$ and any morphism $f:\mathcal O_V^s\to\mathcal F\mid V $ the sheaf $Ker(f)$ on $V$ is locally finitely generated .

As for the consequences of coherence, here is a typical application: if a sequence of coherent sheaves $ \mathcal F' \to \mathcal F\to\mathcal {F''} $ is exact at $x\in X$ (stalkwise) then it is exact on an open neighbourhood of $x$.

The real power of coherence however comes through Cartan's Theorems A and B for coherent sheaves on a Stein manifold.
These theorems have innumerable consequences: on a Stein manifold
(i) Every meromorphic function is the quotient of two global holomorphic functions.
(ii) Every topological line bundle has one and only one holomorphic structure,
(iii) Every closed analytic subset is the zero set of a family of globally defined holomorphic functions.
(iv) Every holomorphic function on a closed analytic subset can be extended to a holomorphic function on the whole Stein manifold.
(v) Given global holomorphic functions $f_1,...,f_r$ without a common zero, there exist global holomorphic functions $g_1,...,g_r$ with $f_1g_1+\cdots +f_rg_r=1$.

Just to illustrate how easy theorems become with those powerful tools let me prove (ii).
Start from the exponential sequence of sheaves $ 0\to \mathbb Z \to \mathcal O\to \mathcal O^*\to 0 $ on the Stein manifold $X$ .
The associated long exact sequence has as a fragment   $$ \cdots \to H^1(X,\mathcal O) \to H^1(X,\mathcal O^*) \to H^2(X,\mathbb Z) \to H^2 (X,\mathcal O) \to \cdots $$ Since $H^1(X,\mathcal O) = H^2 (X,\mathcal O)=0 $ by theorem B, we get the isomorphism $H^1(X,\mathcal O^*) \cong H^2(X,\mathbb Z)$.
It factorizes as $H^1(X,\mathcal O^*)\to H^1(X,\mathcal C^*) \to H^2(X,\mathbb Z)$ and since $ H^1(X,\mathcal C^*) \to H^2(X,\mathbb Z)$ is well known to be the isomorphism given by the first chern class we get the result (ii).

First let me note that your definition of coherent sheaf is misleading: it implies that $\mathcal O_X$ is coherent by definition, whereas in reality coherence of $\mathcal O_X$ is a very deep theorem due to Oka .

The correct definition is that a sheaf $\mathcal F$ of $\mathcal O_X$-Modules is coherent if :

1. $\mathcal F$ is locally finitely generated : $X$ can be covered by open subsets $U$ on which there exist surjections $\mathcal O_U^N\to \mathcal F\mid U$ .
   and
2. For any open subset $V\subset X$ and any morphism $f:\mathcal O_V^s\to\mathcal F\mid V $ the sheaf $Ker(f)$ on $V$ is locally finitely generated .

As for the consequences of coherence, here is a typical application: if a sequence of coherent sheaves $ \mathcal F' \to \mathcal F\to\mathcal {F''} $ is exact at $x\in X$ (stalkwise) then it is exact on an open neighbourhood of $x$.

The real power of coherence however comes through Cartan's Theorems A and B for coherent sheaves on a Stein manifold.
These theorems have innumerable consequences: on a Stein manifold
(i) Every meromorphic function is the quotient of two global holomorphic functions.
(ii) Every topological line bundle has one and only one holomorphic structure,
(iii) Every closed analytic subset is the zero set of a family of globally defined holomorphic functions.
(iv) Every holomorphic function on a closed analytic subset can be extended to a holomorphic function on the whole Stein manifold.
(v) Given global holomorphic functions $f_1,...,f_r$ without a common zero, there exist global holomorphic functions $g_1,...,g_r$ with $f_1g_1+\cdots +f_rg_r=1$.
 (vi,vii,viii,...)

First let me note that your definition of coherent sheaf is misleading: it implies that $\mathcal O_X$ is coherent by definition, whereas in reality coherence of $\mathcal O_X$ is a very deep theorem due to Oka .

The correct definition is that a sheaf $\mathcal F$ of $\mathcal O_X$-Modules is coherent if :

1. $\mathcal F$ is locally finitely generated : $X$ can be covered by open subsets $U$ on which there exist surjections $\mathcal O_U^N\to \mathcal F\mid U$ .
   and
2. For any open subset $V\subset X$ and any morphism $f:\mathcal O_V^s\to\mathcal F\mid V $ the sheaf $Ker(f)$ on $V$ is locally finitely generated .

As for the consequences of coherence, here is a typical application: if a sequence of coherent sheaves $ \mathcal F' \to \mathcal F\to\mathcal {F''} $ is exact at $x\in X$ (stalkwise) then it is exact on an open neighbourhood of $x$.

The real power of coherence however comes through Cartan's Theorems A and B for coherent sheaves on a Stein manifold.
These theorems have innumerable consequences: on a Stein manifold
(i) Every meromorphic function is the quotient of two global holomorphic functions.
(ii) Every topological line bundle has one and only one holomorphic structure,
(iii) Every closed analytic subset is the zero set of a family of globally defined holomorphic functions.
(iv) Every holomorphic function on a closed analytic subset can be extended to a holomorphic function on the whole Stein manifold.
(v) Given global holomorphic functions $f_1,...,f_r$ without a common zero, there exist global holomorphic functions $g_1,...,g_r$ with $f_1g_1+\cdots +f_rg_r=1$.

Just to illustrate how easy theorems become with those powerful tools let me prove (ii).
Start from the exponential sequence of sheaves $ 0\to \mathbb Z \to \mathcal O\to \mathcal O^*\to 0 $ on the Stein manifold $X$ .

The associated long exact sequence has as a fragment $$ \cdots \to H^1(X,\mathcal O) \to H^1(X,\mathcal O^*) \to H^2(X,\mathbb Z) \to H^2 (X,\mathcal O) \to \cdots $$ Since $H^1(X,\mathcal O) = H^2 (X,\mathcal O)=0 $ by theorem B, we get the isomorphism $H^1(X,\mathcal O^*) \cong H^2(X,\mathbb Z)$.
It factorizes as $H^1(X,\mathcal O^*)\to H^1(X,\mathcal C^*) \to H^2(X,\mathbb Z)$ and since $ H^1(X,\mathcal C^*) \to H^2(X,\mathbb Z)$ is well known to be the isomorphism given by the first chern class we get the result (ii).