I was looking at a proof of Abel's theorem saying that a degree zero divisor on a compact riemann surface is principal iff $A_0(D)=0$ using cohomology. I have the following diagram:
$\require{AMScd}$
\begin{CD}
@[email protected]\\
@.@VVV@VVV\\
@.PDiv(X)@.PDiv(X)\\
@.@VVV@VVV\\
0 @>>> Div_0(X) @>>> Div(X) @>deg>>\mathbb{Z} @>>> 0 \\
@. @VA_0 VV\# @V V V @VV V @. \\
0 @>>> {\frac{H^1(X,O_X)}{H^1(X,\mathbb{Z})}}=Jac(X) @>>{f}> {H^1(X,O_X^*)} @>>{}> {H^2(X,\mathbb{Z})} @>>> 0\\
\end{CD}
Where the last row comes from the long exact sequence in cohomology relative to the exponential sequence, the second row is obviously exact and the second column comes from the long exact sequence in cohomology relative to the short exact sequence $0\rightarrow O^*\rightarrow M^* \rightarrow M^*/O^*\rightarrow 0$ (it is the same as saying cartier divisor and weil divisor coincide in this case). Now I want to prove that the first column is exact implying the conclusion of the theorem, to do that I would use diagram chasing and to do that I need the diagram to commuted so it would easily follow that the first column is exact. So my question is how to prove that the square with the # commutes.
Addition to what I said above:
Considering the square # we see that taking $D=\sum p_i-\sum q_i\in Div_0X$ it is sent via $A_0$ to $(\sum_i\int_{p_i}^{q_i} \omega_j)_{j}$ modulo $\Lambda\cong H^1(X,\mathbb{Z})$, where $\omega_1,...,\omega_g$ is a basis of $H^1(X,O_X)$.The horizontal map is simply the inclusion sending $D$ to itself inside $DivX$ and the other vertical map is the map sending $D$ to its class $[D]$ inside $Pic(X)$. My problem is to understand what actually is the map $f$ in order to prove the diagrm is commutative.
I am really interested in this proof I found on Rick Miranda book on algebraic curves and riemann surfaces because it seems to me very beautiful, but it lacks in details about the commutativity of the diagram (it doesn't even mention it) and in this case the commutativity of the diagram is crucial (the rest follows with easy cohomological arguments and trivial diagram chasing). Hence any help will be much apprecited.