Let $X$ be a scheme and $\mathfrak{U} = \{U_i\}$ be an open cover of $X$. Let $\mathscr{F}$ be a sheaf on $X$. Denote $C^1$ for a first component of Cech complex. What I observe is :
$C^1 = \Pi_{i < j} \mathscr{F}(U_i \cap U_j) = \Pi_{i < j} \Delta_*\mathscr{F}(U_i \times U_j)$
Since $C^1$ is related to the first sheaf cohomology of $\mathscr{F}$ and $\Delta_*\mathscr{F}(U_i \times U_j)$ is related to the global sections of the diagonal, I think there might be good relation between them. Is there any relation? (I did not assume $X$ is separated or quasi separated.)