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Let $\tau$ be a subcanonical topology on the category of affine schemes of finite type over $Spec(\mathbf{Z})$. Call this site $(S,\tau)$ or just $S$, and call its associated topos $\mathcal{S}$. Recall that given a topos $T$, we have an equivalence of categories $Hom_{Topos}(T,\mathcal{S})\cong Hom_{Sites}(S,T)$, where $T$ is given the canonical topology. It is a theorem of M. Hakim that $Hom_{Sites}(S,T)$ gives the category of commutative ring objects in $T$ when $\tau$ is the chaotic topology, the category of local rings in $T$ when $\tau$ is the Zariski topology, and the category of "strict local rings" in $T$ when $\tau$ is the étale topology.

In particular, when $T$ is the category of sets, it means that the points of $\mathcal{S}$ are precisely the commutative rings, local rings, and Henselian rings with separably closed residue fields (strict Henselian) respectively. It is also well-known that when $\tau$ is the Nisnevich topology, the local rings are precisely the Henselian rings.

There are other subcanonical Grothendieck topologies on the category of affine schemes of finite type. What are the local rings, for example, when we look at the fppf and fpqc topologies? (Just a guess, but fppf-local is going to be complete local rings? (Wrong! See Laurent Moret-Bailly's comment)).

How about for more obscure subcanonical topologies?

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In the fppf case, I think we are looking for local rings $A$ such that every fppf $A$-algebra $B$ has an $A$-morphism $B\to A$. These have to be strictly henselian but also in some sense "infinitely ramified". "Complete with alg. closed residue field" is neither sufficient (e.g. $\overline{\mathbb{Q}}[[t]]$ doesn't work) nor necessary: I think the ring of integers of $\overline{\mathbb{Q}}_p$ is an example. –  Laurent Moret-Bailly Oct 15 '10 at 8:55
In the case of Voevodsky's h-topology the local rings are the valuation rings having algebraically closed residue field. –  Tom Goodwillie Oct 15 '10 at 12:25
I meant to say "algebraically closed fraction field". –  Tom Goodwillie Oct 16 '10 at 13:38
Is the problem for $\tau=fppf$ open? –  Harry Gindi Dec 1 '10 at 3:09
As a concrete example where I do not know the answer: If $\mathcal{O}$ is the ring of integers of $\bar{\mathbb{Q}}_p$, then is $\mathcal{O}/p$ local in the fppf-topology? In other words, does every fppf $\mathcal{O}/p$-algebra $R$ admit a map $R\rightarrow \mathcal{O}/p$? –  Peter Scholze Mar 16 '12 at 21:58

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