Naive question in Cech cohomology

Question

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.

Answer 1

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 $U_i\cap V \hookrightarrow U_i\cap U_{i_0}$