[proofplan]
We first make explicit the transition functions associated to the Cousin II datum and verify that they form a multiplicative Čech $1$-cocycle with values in $\mathcal{O}^*$. This cocycle defines a holomorphic line bundle by the standard gluing construction for line bundles. We then prove the equivalence: a global meromorphic solution gives local nowhere-vanishing holomorphic correction factors, hence makes the cocycle a coboundary; conversely, a coboundary gives compatible local meromorphic functions that glue to a global meromorphic solution.
[/proofplan]
[step:Form the Čech cocycle from the local meromorphic data]
Let $\mathcal{U} = \{U_i\}_{i \in I}$ be the open cover on which the Cousin II datum is given, and let each $f_i \in \mathcal{M}^*(U_i)$ be the corresponding nonzero meromorphic function. For $i,j \in I$, define $U_{ij} := U_i \cap U_j$. The Cousin II condition says that the quotient
\begin{align*}
f_{ij}: U_{ij} &\to \mathbb{C}^* \\
x &\mapsto f_i(x) f_j(x)^{-1}
\end{align*}
is a nowhere-vanishing holomorphic function on $U_{ij}$, that is, $f_{ij} \in \mathcal{O}^*(U_{ij})$.
For $i,j,k \in I$, on $U_i \cap U_j \cap U_k$ we compute in the multiplicative group $\mathcal{M}^*$:
\begin{align*}
f_{ij} f_{jk}
=
(f_i f_j^{-1})(f_j f_k^{-1})
=
f_i f_k^{-1}
=
f_{ik}.
\end{align*}
Also $f_{ii} = 1$ on $U_i$. Hence $\{f_{ij}\}$ is a Čech $1$-cocycle in $Z^1(\mathcal{U}, \mathcal{O}^*)$.
[guided]
The first point is to check that the transition functions really are legal transition functions for a holomorphic line bundle. The Cousin II datum consists of nonzero meromorphic functions $f_i \in \mathcal{M}^*(U_i)$ such that their quotients are holomorphic and nowhere zero on overlaps. Thus for each pair $i,j \in I$ we define $U_{ij} := U_i \cap U_j$ and define
\begin{align*}
f_{ij}: U_{ij} &\to \mathbb{C}^* \\
x &\mapsto f_i(x) f_j(x)^{-1}.
\end{align*}
The target is $\mathbb{C}^*$, not merely $\mathbb{C}$, because the Cousin II condition requires $f_i f_j^{-1}$ to be nowhere vanishing and holomorphic.
Now we verify the Čech cocycle identity. On a triple overlap $U_i \cap U_j \cap U_k$, multiplication in the sheaf $\mathcal{M}^*$ gives
\begin{align*}
f_{ij} f_{jk}
=
(f_i f_j^{-1})(f_j f_k^{-1})
=
f_i f_k^{-1}
=
f_{ik}.
\end{align*}
The cancellation of $f_j^{-1} f_j$ is valid because all functions are considered inside the multiplicative group of nonzero meromorphic functions. Also $f_{ii} = f_i f_i^{-1} = 1$ on $U_i$. Therefore the family $\{f_{ij}\}$ is a multiplicative Čech $1$-cocycle with values in $\mathcal{O}^*$, namely $\{f_{ij}\} \in Z^1(\mathcal{U}, \mathcal{O}^*)$.
[/guided]
[/step]
[step:Build the holomorphic line bundle using the cocycle]
By the classification of holomorphic line bundles by $\check{H}^1(X,\mathcal{O}^*)$, the cocycle $\{f_{ij}\} \in Z^1(\mathcal{U},\mathcal{O}^*)$ defines a holomorphic line bundle $L \to X$. Explicitly, $L$ is obtained by gluing the product bundles $U_i \times \mathbb{C} \to U_i$ using the transition maps
\begin{align*}
\Phi_{ij}: U_{ij} \times \mathbb{C} &\to U_{ij} \times \mathbb{C} \\
(x,z) &\mapsto (x, f_{ij}(x)z).
\end{align*}
The cocycle identity verified above is exactly the compatibility condition $\Phi_{ik} = \Phi_{ij} \circ \Phi_{jk}$ on triple overlaps, so the gluing produces a well-defined holomorphic line bundle.
[/step]
[step:Derive a Čech coboundary from a global Cousin II solution]
Assume the Cousin II problem is solvable. Thus there exists a global nonzero meromorphic function $f \in \mathcal{M}^*(X)$ such that, for every $i \in I$,
\begin{align*}
g_i := f_i f^{-1} \in \mathcal{O}^*(U_i).
\end{align*}
Equivalently, define
\begin{align*}
g_i: U_i &\to \mathbb{C}^* \\
x &\mapsto f_i(x)f(x)^{-1}.
\end{align*}
Then $\{g_i\}_{i \in I}$ is a Čech $0$-cochain in $C^0(\mathcal{U},\mathcal{O}^*)$.
We use the multiplicative Čech coboundary convention
\begin{align*}
(\delta g)_{ij}: U_{ij} &\to \mathbb{C}^* \\
x &\mapsto g_i(x)g_j(x)^{-1}.
\end{align*}
On $U_{ij}$,
\begin{align*}
(\delta g)_{ij}
=
g_i g_j^{-1}
=
(f_i f^{-1})(f_j f^{-1})^{-1}
=
(f_i f^{-1})(f f_j^{-1})
=
f_i f_j^{-1}
=
f_{ij}.
\end{align*}
Hence $\{f_{ij}\} = \delta\{g_i\}$ is a Čech coboundary, so $[f_{ij}] = 0$ in $\check{H}^1(X,\mathcal{O}^*)$. By the same classification theorem, the corresponding line bundle $L$ is holomorphically straightforward.
[guided]
Suppose a global solution exists. This means there is a nonzero meromorphic function $f \in \mathcal{M}^*(X)$ whose divisor agrees locally with the divisor prescribed by each $f_i$. In sheaf-theoretic terms, this is exactly the condition that the quotient $f_i f^{-1}$ is holomorphic and nowhere zero on each $U_i$. Therefore we define, for every $i \in I$,
\begin{align*}
g_i: U_i &\to \mathbb{C}^* \\
x &\mapsto f_i(x)f(x)^{-1}.
\end{align*}
The solvability hypothesis gives $g_i \in \mathcal{O}^*(U_i)$, so $\{g_i\}_{i \in I}$ is a Čech $0$-cochain with values in $\mathcal{O}^*$.
Now we compare the original transition functions with the coboundary of this $0$-cochain. With the multiplicative Čech convention
\begin{align*}
(\delta g)_{ij}: U_{ij} &\to \mathbb{C}^* \\
x &\mapsto g_i(x)g_j(x)^{-1},
\end{align*}
we compute on every overlap $U_{ij}$:
\begin{align*}
(\delta g)_{ij}
=
g_i g_j^{-1}
=
(f_i f^{-1})(f_j f^{-1})^{-1}
=
(f_i f^{-1})(f f_j^{-1})
=
f_i f_j^{-1}
=
f_{ij}.
\end{align*}
Thus the cocycle $\{f_{ij}\}$ is the Čech coboundary $\delta\{g_i\}$. Therefore its cohomology class vanishes:
\begin{align*}
[f_{ij}] = 0 \quad \text{in } \check{H}^1(X,\mathcal{O}^*).
\end{align*}
By the classification of holomorphic line bundles by $\check{H}^1(X,\mathcal{O}^*)$, a line bundle represented by a coboundary cocycle is holomorphically straightforward. Hence the line bundle $L$ determined by the Cousin II datum admits a holomorphic product decomposition.
[/guided]
[/step]
[step:Glue local meromorphic functions from a straightforwardising cochain]
Conversely, assume $[f_{ij}] = 0$ in $\check{H}^1(X,\mathcal{O}^*)$. Then there exists a Čech $0$-cochain $\{g_i\}_{i \in I} \in C^0(\mathcal{U},\mathcal{O}^*)$ such that
\begin{align*}
f_{ij} = g_i g_j^{-1}
\end{align*}
on $U_{ij}$ for every $i,j \in I$.
For each $i \in I$, define a meromorphic function
\begin{align*}
h_i: U_i &\dashrightarrow \mathbb{C} \\
x &\mapsto f_i(x)g_i(x)^{-1}.
\end{align*}
Since $g_i \in \mathcal{O}^*(U_i)$, the inverse $g_i^{-1}$ is holomorphic and nowhere zero on $U_i$, so $h_i \in \mathcal{M}^*(U_i)$. On $U_{ij}$,
\begin{align*}
h_i
=
f_i g_i^{-1}
=
(f_{ij} f_j)g_i^{-1}
=
(g_i g_j^{-1} f_j)g_i^{-1}
=
f_j g_j^{-1}
=
h_j.
\end{align*}
Thus the family $\{h_i\}$ agrees on overlaps. By the gluing property for the sheaf of meromorphic functions, there exists a unique global meromorphic function $f \in \mathcal{M}^*(X)$ such that $f|_{U_i} = h_i$ for every $i \in I$.
[guided]
Now assume the cohomology class vanishes. The statement $[f_{ij}] = 0$ means that the cocycle $\{f_{ij}\}$ is a Čech coboundary. Therefore there is a family of nowhere-vanishing holomorphic functions $g_i \in \mathcal{O}^*(U_i)$ such that
\begin{align*}
f_{ij} = g_i g_j^{-1}
\end{align*}
on every overlap $U_{ij}$.
The goal is to construct one global meromorphic function from the local meromorphic functions $f_i$. The cochain $\{g_i\}$ tells us exactly how to correct the $f_i$ so that they match. For each $i \in I$, define
\begin{align*}
h_i: U_i &\dashrightarrow \mathbb{C} \\
x &\mapsto f_i(x)g_i(x)^{-1}.
\end{align*}
Because $g_i$ is holomorphic and nowhere zero, $g_i^{-1}$ is also holomorphic and nowhere zero. Multiplying the meromorphic function $f_i$ by $g_i^{-1}$ gives a nonzero meromorphic function $h_i \in \mathcal{M}^*(U_i)$.
We now check compatibility on overlaps. Since $f_{ij} = f_i f_j^{-1}$, we have $f_i = f_{ij} f_j$ on $U_{ij}$. Since $f_{ij} = g_i g_j^{-1}$, it follows that
\begin{align*}
h_i
=
f_i g_i^{-1}
=
(f_{ij} f_j)g_i^{-1}
=
(g_i g_j^{-1} f_j)g_i^{-1}
=
f_j g_j^{-1}
=
h_j.
\end{align*}
Thus the corrected meromorphic functions agree on every overlap. The gluing property for the sheaf of meromorphic functions applies because the $h_i$ are meromorphic sections on the open sets $U_i$ and agree on all pairwise overlaps. Hence there is a unique global meromorphic function $f \in \mathcal{M}^*(X)$ satisfying
\begin{align*}
f|_{U_i} = h_i = f_i g_i^{-1}
\end{align*}
for every $i \in I$.
[/guided]
[/step]
[step:Verify that the glued meromorphic function solves the Cousin II problem]
Let $f \in \mathcal{M}^*(X)$ be the global meromorphic function obtained above. For every $i \in I$, since $f|_{U_i} = f_i g_i^{-1}$, we have
\begin{align*}
f_i f^{-1}
=
f_i (f_i g_i^{-1})^{-1}
=
f_i(g_i f_i^{-1})
=
g_i
\end{align*}
on $U_i$. Since $g_i \in \mathcal{O}^*(U_i)$, the quotient $f_i f^{-1}$ is holomorphic and nowhere zero on $U_i$. Therefore $f$ has exactly the local divisor prescribed by the Cousin II datum, so the Cousin II problem is solvable.
Combining the two directions, the Cousin II datum is solvable if and only if $[f_{ij}] = 0$ in $\check{H}^1(X,\mathcal{O}^*)$. Equivalently, by the line-bundle classification theorem, the associated holomorphic line bundle $L \to X$ admits a holomorphic product decomposition.
[/step]
