Let $X$ be a smooth Riemann Surface (to be simple) and let $E\longrightarrow X$ be a coherent sheaf. The Serre Duality states that there exists a pairing $$ H^1(X,E) \otimes H^0(X, E^\vee \otimes K_X) \longrightarrow H^1(X,K_X)\simeq \mathbb C $$ This is usually (at least where I read) done in terms of a resolution of $E$ or using the Dolbeault cohomology when $E$ is a vector bundle.
My question is whether this pairing can be computed in terms of Cech cohomology.
It seems that for a acyclic cover $\{ U_i \}$ $$ (U_i\cap U_j , s_{ij}) \otimes (U_i, f_i \otimes \omega_i) \mapsto (U_i\cap U_j , f_i|_{U_i\cap U_j}(s_{ij})\omega_i|_{U_i\cap U_j}) $$ must work but I'm not sure.
The first thing to check is that your $1$-cochain is well-defined. So does $\sigma_{ij}=f_j|_{U_i\cap U_j}(s_{ij})\omega_j|_{U_i\cap U_j}$ equal $f_i|_{U_i\cap U_j}(s_{ij})\omega_i|_{U_i\cap U_j}$? (Well, recall that $\{f_i\otimes\omega_i\}$ is a $0$-cocycle.) Then it's a matter of checking that $\{\sigma_{ij}\}$ define a $1$-cocycle: You compute the coboundary $\sigma_{ij}-\sigma_{ik}+\sigma_{jk}$ on $U_i\cap U_j\cap U_k$. If you use the fact that $\{s_{ij}\}$ is a $1$-cocycle, this all turns to $0$.
(It's probably best to assume $i<j<k$ in these computations. But if not, we have, e.g., $s_{ij}=-s_{ji}$ for a $1$-cocycle. Then $\sigma_{ji}=f_j|_{U_j\cap U_i}(s_{ji})\omega_j|_{U_j\cap U_i} = -f_i|_{U_i\cap U_j}(s_{ij})\omega_i|_{U_i\cap U_j} = -\sigma_{ij}$, as is consistent.)