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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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    $\begingroup$ 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. $\endgroup$ – Laurent Moret-Bailly Oct 15 '10 at 8:55
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    $\begingroup$ In the case of Voevodsky's h-topology the local rings are the valuation rings having algebraically closed residue field. $\endgroup$ – Tom Goodwillie Oct 15 '10 at 12:25
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    $\begingroup$ I meant to say "algebraically closed fraction field". $\endgroup$ – Tom Goodwillie Oct 16 '10 at 13:38
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    $\begingroup$ Is the problem for $\tau=fppf$ open? $\endgroup$ – Harry Gindi Dec 1 '10 at 3:09
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    $\begingroup$ 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 $R\rightarrow \mathcal{O}/p$? $\endgroup$ – Peter Scholze Mar 16 '12 at 21:58
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I was just moments ago told about a paper of Ofer Gabber and Shane Kelly where they answer this question in a whole host of cases (not fppf or fpqc, but ostensibly all the others, including some I've never heard of).

Here is a screenshot of their table: Table of topologies and points

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It looks like there was a paper published a few years ago by Stefan Schröer that answers this question:

Points in the fppf topology, Ann. Sc. Norm. Super. Pisa Cl. Sci Vol. XVII, issue 2 (2017) 419–447, doi:10.2422/2036-2145.201408_001, arXiv:1407.5446.

From the abstract:

"Using methods from commutative algebra and topos-theory, we construct topos-theoretical points for the fppf topology of a scheme. These points are indexed by both a geometric point and a limit ordinal. The resulting stalks of the structure sheaf are what we call fppf-local rings. We show that for such rings all localizations at primes are henselian with algebraically closed residue field, and relate them to AIC and TIC rings. Furthermore, we give an abstract criterion ensuring that two sites have point spaces with identical sobrification. This applies in particular to some standard Grothendieck topologies considered in algebraic geometry: Zariski, etale, syntomic, and fppf."

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    $\begingroup$ This is a cool paper, but I'm not sure they give a classification of all fppf points, rather it seems they give a lot of examples of fppf points without showing they are all the examples $\endgroup$ – Denis Nardin Jan 15 at 8:58
  • $\begingroup$ @DenisNardin Not sure. Maybe an expert can weigh in or I'll email the author. $\endgroup$ – Harry Gindi Jan 15 at 9:19

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