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Local questions:

1) Given a commutative ring $A,$ is $Sh_\infty\left(Spec(A)\right)$ hypercomplete?

2) Given a commutative ring $A,$ is $Sh_\infty\left(Et\left(A\right)\right)$ hypercomplete, where $Et\left(A\right)$ is the small etale site?

3) and 4) The same questions but for $A$ a simplicial commutative ring

5) and 6) The same questions but for $A$ and $E_\infty$-ring spectrum

Or perhaps something general is known about when $Spec^{\mathcal{G}}\left(X\right)$ is hypercomplete, where $\mathcal{G}$ is a geometry in the sense of Lurie's DAG V?

"Large" Global questions:

Ignoring size issues (e.g. by using universes) are infinity sheaves on any of the following hypercomplete?

A) Affine schemes with the Zariski topology

B) Affine schemes with the Etale topology (or flat, etc.)

C) and D) same question for simplicial affine schemes

E) and F) same question for spectral schemes

(I have a feeling that the answer might depend on things being Noetherian)

Any references would be great also, thanks!

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Zariski/Nisnevich sheaves on a Noetherian scheme of finite Krull dimension (and hence on any small category of such schemes which is "closed under coverings") are hypercomplete (even limit of their Postnikov tower): see HTT §7.2.4 for Zariski and Morel-Voevodsky or DAG XI for Nisnevich. I'm guessing that the étale topos is almost never hypercomplete, because (1) it seems well-known that Postnikov towers do not converge and (2) HTT shows that the classifying topos of the pro-group $\mathbb{Z}_{p}$, which is also the étale topos of some field, is not hypercomplete. –  Marc Hoyois Mar 2 '13 at 1:02
Oh and DAG XI works with $E_\infty$-rings as well (whose $\pi_0$ are Noetherian and of finite dimension). –  Marc Hoyois Mar 2 '13 at 1:08
Thansk Marc. So, I take it that "well behaved schemes" when considered in either the Zariski/Nisnevich topology are hypercomplete. How big is the class of rings for which their small etale sites are hypercomplete? Does it contain most of the rings people care about, or are there important examples that fall outside of this? –  David Carchedi Mar 2 '13 at 16:39
I don't know the answer, but fields are certainly important examples, and already there's an example of a non-hypercomplete étale topos there (one whose absolute Galois group is $\mathbb{Z}_p$). For a positive example, the étale topos of a real closed field is the topos of $\mathbb{Z}/2$-equivariant homotopy types which is hypercomplete. If I had to make a conjecture it would be that real closed and separably closed fields (the only fields with finite Galois groups) are the only examples of fields whose étale topoi are hypercomplete. –  Marc Hoyois Mar 2 '13 at 17:57
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