3
$\begingroup$

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.

$\endgroup$
3
  • $\begingroup$ By $\check{H}$, do you mean with respect to a particular cover, or the filtered colimit over all covers? In the latter case, $e_1$ is also a natural isomorphism. $\endgroup$
    – Zhen Lin
    Oct 12, 2013 at 12:45
  • $\begingroup$ I mean the filtered colimit. Perhaps, writing symbols $U$ for an object in $C$ is confusing as "the classical notation" reserves the symbol $U$ for a cover. $\endgroup$ Oct 12, 2013 at 12:51
  • $\begingroup$ It is always so? That is great. I would like to find a reference, but perhaps, it is easy enough to prove without one. $\endgroup$ Oct 12, 2013 at 12:53

2 Answers 2

3
$\begingroup$

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 $(\mathcal{C}_{/ U}, \tau_{/ U})$.

$\endgroup$
1
  • $\begingroup$ maybe this argument should be added to this particular chapter of the stacks project. thank you for the reference as well. $\endgroup$ Oct 12, 2013 at 14:31
1
$\begingroup$

Another reference is Cor. 3.4.7 in Tamme's Introduction to étale cohomology.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.