I'm reading the Gortz's Algebraic Geometry, proof of the Proposition 15.26 and stuck at understanding that some map appearing in the proposition is surjective. Let $X$ be an absolute curve ; i.e., a non-empty noetherian scheme with irreducible components $X_i$ ($1\le i \le n$) satisfying following equivalent condtions (Gortz, Prop.15.1)
(i) For every closed point $x \in X$, $\operatorname{dim}\mathcal{O}_{X,x} =1$ (ii) The closed irreducible subsets of $X$ are the $X_i$ and the closed points of $X$. (iii) $\operatorname{dim}X_i = 1$ for all $i$.
Let $C^{1}$ be the set of points $x\in X$ such that $\operatorname{dim}\mathcal{O}_{X,x}=1$ and let $j_x : \operatorname{Spec}\mathcal{O}_{X,x} \to X $ be the canonical morphism of $x\in C^{1}$. Note that $C^{1}$ is precisely the set of closed point of $X$ (If needed, I will upload detail).
Then Gortz claims that next homomorphism of abelian groups :
$$ \Psi : \operatorname{Div}(X) \to \bigoplus_{x\in C^{1}}\operatorname{Div}(\mathcal{O}_{X,x})$$ $$ D \mapsto \Sigma_{x\in C^{1}} j_{x}^{*}(D)$$
is isomorphism, where $\operatorname{Div}(X)$ is the Cartier divisors and $j_x^{*} : \operatorname{Div}(X) \to \operatorname{Div}(\operatorname{Spec} \mathcal{O}_{X,x})$ is the inverse image of cartier divisor (c.f. Gortz's book p.312)
Note that $\operatorname{Div}(\mathcal{O}_{X,x}) \cong (\operatorname{Frac}\mathcal{O}_{X,x})^{\times}/\mathcal{O}_{X,x}^{\times}$ since for $x \in C^{1}$, $\mathcal{O}_{X,x}$ is a local noetherian ring of dimension $1$. (His book Exercise 11.18) I don't know explicit isomorphism. I want to get explicit map, for later usage.
Now, let $\mathcal{K}_X$ be the sheaf of total fractions( or called sheaf of meromorphic functions) of $\mathcal{O}_{X}$.
For the surjectivity, he argues that
To show that surjectivity we first recall that $\operatorname{Frac}\mathcal{O}_{X,x}=\mathcal{K}_{X,x}$ and that $\Gamma(U, \mathcal{K}_X) = \operatorname{Frac}(\Gamma(U, \mathcal{O}_{X}))$ for every open affine subset $U$ of $X$ (Remark 11.23, $X$ is noetherian). Thus it suffices to show that for $f\in (\operatorname{Frac}\mathcal{O}_{X,x})^{\times}$ there exists an open affine neighborhood $U$ of $x$ and $g \in (\operatorname{Frac}\Gamma(U,\mathcal{O}_X))^{\times}$ such that $g_x = f$ and such that $g_y \in \mathcal{O}_{X,y}^{\times}$ for all $y\in U$, $y\neq x$.
My question is, why the condition is sufficient for the surjectivity of the $\Psi$?
My first attempt is, let $(\overline{f}_x)_{x\in C^{1}} \in \bigoplus_{x\in C^{1}}\operatorname{Div}(\mathcal{O}_{X,x})= \bigoplus_{x\in C^{1}}(\operatorname{Frac}\mathcal{O}_{X,x})^{\times}/\mathcal{O}_{X,x}^{\times}= \bigoplus_{x\in C^{1}}(\mathcal{K}_{X,x})^{\times}/\mathcal{O}_{X,x}^{\times} $ ($f_x \in (\operatorname{Frac}\mathcal{O}_{X,x})^{\times}$).
Then by the above condition, for each $x\in C^{1}$, there exists an open affine neighborhood $U_x$ of $x$ and $g_x \in (\operatorname{Frac}\Gamma(U_x,\mathcal{O}_X))^{\times} = \Gamma(U_x, \mathcal{K}_X)^{\times}$ such that $(g_x)_{,x} = f_x$ and such that $(g_x)_{,y} \in \mathcal{O}_{X,y}^{\times}$ for all $y \in U_x$ with $y\neq x$.
I could show that $\{ U_x\}_{x\in C^{1}}$ forms an open cover of $X$ (If needed, I will upload proof).
Now consider $D := (U_x , g_x)_{x\in C^{1}}$.
Then my question is,
Q.1) $D \in \operatorname{Div}(X)$ ; i.e., $D$ forms a Cartier divisor on $X$?
Q.2) $\Psi(D) = (j_x^{*}(D))_{x\in C^{1}} = (\overline{f}_x)_{x\in C^{1}} $?
To show Q.2), fix $x_0 \in C^{1}$. We need to show that $j_{x_0}^{*}(D) = \overline{f_{x_0}}$.
Note that $j_{x_0}^{*}(D) = j_{x_0}^{*}((U_x, g_x)_{x\in C^{1}}) := (j_{x_0}^{-1}(U_x), \overline{j_{x_0}^{\flat}}_{,U_x}(g_x))_{x\in C^{1}} = (j_{x_0}^{-1}(U_{x_0}), \overline{j_{x_0}^{\flat}}_{,U_{x_0}}(g_{x_0})) \in \operatorname{Div}(\mathcal{O}_{X,x_0}) $
, where $\overline{j_{x_0}^{\flat}} : \mathcal{K}_X \to j_{x_0,*}\mathcal{K}_{\operatorname{Spec}\mathcal{O}_{X,x_0}}$ is an extension of $j_{x_0}^{\flat} : \mathcal{O}_X \to j_{x_0,*}\mathcal{O}_{\operatorname{Spec}\mathcal{O}_{X,x_0}}$ and the final equality is true since $j_{x_0}$ is a homeomorphism onto the subspace which is the intersection of all open subsets of $X$ which contains $x_0$ (c.f.Gortz, p.69), so $j_{x_0}^{-1}(U_{x_0})$ forms an open cover of $\operatorname{Spec}\mathcal{O}_{X,x}$.
I asked that what is an explicit isomorphism $\operatorname{Div}(\mathcal{O}_{X,x_0}) \cong (\operatorname{Frac}\mathcal{O}_{X,x_0})^{\times}/\mathcal{O}_{X,x_0}^{\times}$.
My question about Q.2) is, upto this explicit isomorphism, $(j_{x_0}^{-1}(U_{x_0}), \overline{j_{x_0}^{\flat}}_{,U_{x_0}}(g_{x_0}))$ sends to $\overline{(g_{x_0})_{,x_0}}$ ? If so, then since $(g_{x_0})_{,x_0} = f_{x_0}$ as above, maybe we are done.
Furthur progress : Let $Y_0 : =\operatorname{Spec}\mathcal{O}_{X,x_0}$. Note that $\operatorname{DivCl}(Y_0) =0$ ( c.f. Gortz, p.318, Exercise 11.18-(a) ) ; i.e., every Cartier divisor on $Y_0$ is principal. So, there exists a homomorphism $ \eta : \operatorname{Div}(Y_0) \cong \Gamma(Y_0,\mathcal{K}_{Y_0}^{\times})/ \Gamma(Y_0,\mathcal{O}_{Y_0}^{\times})$ (c.f. Gortz, p.301, (11.11.2)) defined as follows : For every divisor $(Y_0, f) \in \operatorname{Div}(Y_0)$, $\eta ( (Y_0, f)) : = f + \Gamma(Y_0,\mathcal{O}_{Y_0}^{\times}) =f + \mathcal{O}_{X,x_0}^{\times} $.
Now let's go back to the above question. Note that $(j_{x_0}^{-1}(U_{x_0}), \overline{j_{x_0}^{\flat}}_{,U_{x_0}}(g_{x_0}))$ is a principal divisor on $Y_0$. So it sends via $\eta$ to $\overline{j_{x_0}^{\flat}}_{,U_{x_0}}(g_{x_0}) + \mathcal{O}_{X,x_0}^{\times} \in \Gamma(Y_0,\mathcal{K}_{Y_0}^{\times})/ \Gamma(Y_0,\mathcal{O}_{Y_0}^{\times})$.
So, it suffices to show that upto isomorphism ( since $X$ is noetherian ; c.f. Gortz, p.301, Remark 11.23)
$$\operatorname{Div}(Y_0) \cong \Gamma(Y_0,\mathcal{K}_{Y_0}^{\times})/ \Gamma(Y_0,\mathcal{O}_{Y_0}^{\times}) \cong (\operatorname{Frac}(\mathcal{O}_{X,x_0}))^{\times}/ \mathcal{O}_{X,x_0}^{\times} \cong (\mathcal{K}_{X,x_0})^{\times}/\mathcal{O}_{X,x_0}^{\times} $$
, $\overline{j_{x_0}^{\flat}}_{,U_{x_0}}(g_{x_0}) + \mathcal{O}_{X,x_0}^{\times}$ sends to $(g_{x_0})_{,x_0} + \mathcal{O}_{X,x_0}^{\times}$.
To sum it up, next question is true?
Q. For the canonical map $j_x : Y:= \operatorname{Spec}(\mathcal{O}_{X,x}) \to X$ and an open (affine) neighborhood $U \ni x$ and $g \in \mathcal{K}_{X}(U)^{\times}$, upto isomorphism :
$$ \Gamma(j_x^{-1}(U),\mathcal{K}_{Y}^{\times})/ \Gamma(Y,\mathcal{O}_{Y}^{\times}) =\Gamma(Y,\mathcal{K}_{Y}^{\times})/ \Gamma(Y,\mathcal{O}_{Y}^{\times}) \cong (\operatorname{Frac}(\mathcal{O}_{X,x}))^{\times}/ \mathcal{O}_{X,x}^{\times} \cong (\mathcal{K}_{X,x})^{\times}/\mathcal{O}_{X,x}^{\times} $$
, $\overline{j_{x}^{\flat}}_{,U}(g) + \mathcal{O}_{X,x}^{\times}$ sends to $g_{x} + \mathcal{O}_{X,x}^{\times}$, where $g_x$ is the germ in $\mathcal{K}_{X,x}^{\times}$ at $x$?
C.f. Next is full proof of the Proposition 15.26 of the Gortz's book :
Why the underlined statement is sufficient for the surjectivity of the $\Psi$?
Can anyone help?
