Timeline for Construction of the petit Zariski topos out of the gros topos of a scheme
Current License: CC BY-SA 4.0
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| Dec 26, 2021 at 19:48 | comment | added | Mendieta | @IngoBlechschmidt , I think it would be useful to cast part of your answer in terms of MacLane-Moerdijk's presentation. Let $\mathbf{fp}$ be the category of finitely presented rings. Let $\mathcal{Z}$ be the Zariski topos. For $A$ in $\mathbf{fp}$: is the petit topos of $A$ a subtopos ${\mathcal{Z}/\mathbf{fp}(A,-)}$ ? In that case, is it the largest subtopos satisfying certain property? | |
| May 2, 2019 at 11:15 | comment | added | Ingo Blechschmidt | The fpqc and fppf topologies coincide in the case that the site only contains schemes which are (locally) of finite presentation. Without this finiteness condition, not even the big Zariski topos will classify any reasonable theory. That said, the big fppf topos classifies fppf-local rings (Definition 21.4 in the linked notes). Conjecturally, these are the same as algebraically closed local rings. Hence the big fppf topos is the largest subtopos of the big Zariski topos where $\mathbf{A}^1$ is fppf-local. The answer for the Nisnevich topology is to the best of my knowledge unknown. | |
| May 1, 2019 at 19:36 | comment | added | მამუკა ჯიბლაძე | That's nice! Do you know characterizations of some other sheaves in a similar way? Fpqc, fppf, Nisnevich...? | |
| May 1, 2019 at 17:56 | history | answered | Ingo Blechschmidt | CC BY-SA 4.0 |