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Let $R$ be a commutative ring with one and $S$ commutative faithfully flat $R$-algebra (that is there is a faithfully flat ring map let $\phi: R \to S$). Then the so called Amitsur complex $R \to S^{\otimes \bullet +1}$ is exact and determines a faithfully flat resolution of $R$.

Applying the multiplicative group scheme functor $\mathbb{G}_m$ to this complex and taking cohomology gives exactly the fppf Cech cohomology $H^i_{Cech, fppf}(S^{\otimes \bullet +1}, \mathbb{G}_m)$ with respect the fppf cover $Spec(S) \to Spec(R)$.

We want to compare the fppf Cech cohomology with fppf cohomology of $H^i_{fppf}(R, \mathbb{G}_m)$. In general there exist only a morphism

$$ H^i_{Cech, fppf}(S^{\otimes \bullet +1}, \mathbb{G}_m) \to H^i_{fppf}(R, \mathbb{G}_m) $$

since often Cech cannot resolve 'fine' enough. For which class of rings $R$ there exist a fppf resolution $R \to S \to ...$ of $R$ which induces an isomorphism between associated Cech and fppf cohomology? What about the same problem with etale cohomology instead of fppf? Where (if that's true) a complete proof can be found? For which rings do moreover the fppf cohomology $H^i_{fppf}(R, \mathbb{G}_m)$ and etale cohomology $H^i_{etale}(R, \mathbb{G}_m)$ coinside?

Note that this question is identical to the question I recently asked in mse.

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    $\begingroup$ For the last question, by a theorem of Grothendieck, for a smooth group scheme $G$ the fppf cohomology and the etale cohomology coincide. $\endgroup$
    – Piotr Achinger
    Commented May 4, 2021 at 20:51
  • $\begingroup$ Note that for the Zariski topology, the Cech complex often computes the right cohomology because the cohomology of quasi-coherent sheaves on open affine subschemes is trivial. I think the issue for the fppf and etale topology is that for many significant sheaves, one cannot find covers on which the sheaf cohomology is trivial. On the other hand, if ones takes the direct limit of the Cech cohomology of all the covers, then one does get the right cohomology, at least in the etale situation and for any noetherian ring or any scheme quasi-projective over a noetherian ring. $\endgroup$
    – Damian Rössler
    Commented May 5, 2021 at 14:44
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    $\begingroup$ ... see p. 22 of the book of Freitag-Kiehl. Note that in general, there is a 'Cech spectral sequence' from Cech to usual cohomology (for a given cover). Cech cohomology corresponds to the situation where this sequence degenerates. $\endgroup$
    – Damian Rössler
    Commented May 5, 2021 at 14:47
  • $\begingroup$ For proofs of Grothendieck's theorem that etale cohomology suffices for smooth groups, of Artin's theorem that Cech suffices for etale cohomology, and for the spectral sequence relating Cech and derived cohomology, see III, 3.9; III, 2.17; and III, 2.7, of Milne's book on Etale Cohomology. $\endgroup$
    – user166831
    Commented May 5, 2021 at 20:12
  • $\begingroup$ Alright, thank you all! $\endgroup$
    – user267839
    Commented May 29, 2021 at 18:12

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