Let $X$ be a smooth, projective variety and $F$ a coherent sheaf on $X$. Let $\{U\_i\}\_{i \in I}$ be an open affine covering of $X$ and $\{f\_{ij}\}\_{i<j}$ with $f\_{ij} \in \Gamma(U\_{ij},F)$ satisfying the cocycle condition. Denote by $\alpha \in H^1(X,F)$ corresponding to the collection $\{f\_{ij}\}\_{i<j}$.


Choose $i\_0 \in I$ and a proper affine open subset $V \subset U\_{i\_0}$. Consider the new open affine covering $\mathcal{V}$ consisting of $V$ and $U\_i$ for $i \in I$. We know that $H^1(X,F)$ does not depend on the choice of the covering. What is the Cech cocyle corresponding to $\alpha$ under the new open covering $\mathcal{V}$? 


Any hint/reference will be most welcome.



Naive question in Cech cohomology
Let $U = \coprod\_i U\_i$ and $U'=V \amalg \coprod\_i U\_i$. Since there is $V \to U\_{i\_0}$, there is a map $U' \to U$ over $X$, and hence a map $U'\times\_X U'\to U\times\_X U$ (also over $X$). The original cocycle is a map $U\times\_X U\to F$, and then the new cocycle is the induced map $U'\times\_X U' \to U\times\_X U \to F$.


To unpack that, it means that on $V\cap U\_j$ the new cocycle is defined as the restriction of $f\_{i\_0j}$ along $V\cap U\_j \hookrightarrow U\_{i\_0}\cap U\_j$, and on $U\_i\cap V$ it is defined as the restriction of $f\_{ii\_0}$ along $U\_i\cap V \hookrightarrow U\_i\cap U\_{i\_0}$


