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.



1st cech cohomology groups on ringed sites
First things first: $\check{H}{}^n(U, \mathscr{F})$ (resp. $H^n(U, \mathscr{F})$) are same whether you regard $\mathscr{F}$ as an $\mathscr{O}$-module or as an abelian sheaf, so we may simplify things by considering only abelian sheaves.


Let $\mathscr{F}$ be an abelian sheaf and write $\check{H}{}^\* (U, -)$ for the right derived functors of $\Gamma(U, (-)^+) : \mathbf{AbPsh}(\mathcal{C}) \to \mathbf{Ab}$. (Recall, $\mathscr{F}^+$ is the *separated presheaf* associated with $\mathscr{F}$ and $\Gamma(U, \mathscr{F}^+)$ is the set of presheaf morphisms $\mathfrak{U} \to \mathscr{F}^+$ equipped with the componentwise abelian group structure.) One can check that $\check{H}{}^\* (U, -)$ defined in this way agrees with the "classical" construction via filtered colimits over covers of $U$. Let $\mathscr{H}^\* (-)$ be the right derived functors of the forgetful functor $\mathbf{AbSh}(\mathcal{C}, \mathcal{\tau}) \to \mathbf{AbPsh}(\mathcal{C})$. (It is not hard to check that $\mathscr{H}^\* (\mathscr{F})$ is the *presheaf* defined by $C \mapsto H^\* (C, \mathscr{F})$, where $H^\* (C, -)$ are the right derived functors of $\Gamma (C, -) : \mathbf{AbSh}(\mathcal{C}, \tau) \to \mathbf{Ab}$.)


Note that the sheaf condition implies $\check{H}{}^0 (U, \mathscr{H}^0 (\mathscr{F}))$ is naturally isomorphic to $\Gamma (U, \mathscr{F})$, while the forgetful functor $\mathbf{AbSh}(\mathcal{C}, \mathcal{\tau}) \to \mathbf{AbPsh}(\mathcal{C})$ preserves injective objects, so we may construct the Grothendieck spectral sequence:


* $E^{p,q}\_2 = \check{H}{}^p (U, \mathscr{H}^q (\mathscr{F}))$
* $E^{\bullet, \bullet}\_{\bullet}$ converges to $H^\* (U, \mathscr{F})$.


However,
$$\check{H}{}^0 (U, \mathscr{H}^n (\mathscr{F})) = 0 \text{ for all } n \ge 1$$
and so the comparison homomorphisms $\check{H}{}^\* (U, \mathscr{F}) \to H^\* (U, \mathscr{F})$ are isomorphisms in degrees 0 and 1 (and a monomorphism in degree 2).


You can find the above argument as Theorem 8.27 in [Johnstone, *Topos theory*]. An alternative argument goes via the classification of $\mathscr{F}$-torsors – this relies on the fact that there is an easy identification of $\check{H}{}^1 (U, \mathscr{F})$ and the set of isomorphism classes of $\mathscr{F}$-torsors over $(\mathcal{C}\_{/ U}, \tau\_{/ U})$.


