Let $(C, O)$ be a ringed site -- i.e., $C$ is a small category with a grothendieck topology $\tau$ and $O$ a sheaf of rings on the site $(C,\tau)$. In this context, for any object $U$ of $C$ one can define the sheaf cohomology groups $\mathrm{H}^{\bullet}(U,O)$ and the cech cohomology groups $\check{\mathrm{H}}^{\bullet}(U,O)$, and there will be natural group homomorphisms

$$e_n: \check{\mathrm{H}}^{n}(U,O) \to \mathrm{H}^{n}(U,O) $$

which is an isomorphism of groups for $n= 0$. This material is essentially contained in "cohomology on sites," Stacks Project.

I am assuming that in general bijectivity for $e_1$ fails as the stacks project does not mention it. I would like to figure out if $e_1$ will be bijective for any sheaf of rings $O$ on the site $(\mathbf{Sch}/S, \tau)$ where $\tau$ is any grothendieck topology on $\mathbf{Sch}/S$, category of schemes of finite type over $S$, with $S$ any scheme. Perhaps, one has to consider a reasonable grothendieck topology to get the bijectivity of $e_1$ in this context? Where could I look to find out why things fail in the more general context of any ringed site? I hope this question is not overly naive.