Timeline for Subsheaves of Spec K, K a field

9 events

when toggle format	what		by	license	comment
Dec 14, 2022 at 21:02	comment	added	Nico		Really nice! :)
Dec 10, 2022 at 23:08	history	edited	Zhen Lin	CC BY-SA 4.0	added 345 characters in body
Dec 9, 2022 at 7:52	history	edited	Zhen Lin	CC BY-SA 4.0	deleted 24 characters in body
Dec 8, 2022 at 23:05	history	edited	Zhen Lin	CC BY-SA 4.0	added 162 characters in body
Dec 8, 2022 at 22:06	comment	added	Zhen Lin		Oops. I was confused about what local rings in these topologies were. Let me fix that...
Dec 8, 2022 at 17:47	vote	accept	Nico
Dec 8, 2022 at 16:35	comment	added	R. van Dobben de Bruyn		Sorry, isn't $\phi\colon\operatorname{Spec}A\to\operatorname{Spec}K$ itself an fpqc cover for any nonzero $K$-algebra $A$? Then $h_{\operatorname{Spec}\phi}\colon h_{\operatorname{Spec}A}\to h_{\operatorname{Spec}K}$ is surjective as sheaves. In the Zariski, étale, or fppf cases I agree with you: take $K\to L$ a field extension that is not an isomorphism, not étale, or not finitely presented, respectively. Then $K\to B_i$ cannot factor through $K\to L$ since this forces an injection $L\to B_i$, so if $K\to B_i$ is an isomorphism, étale, or finitely presented, the same goes for $K\to L$.
Dec 8, 2022 at 16:03	history	edited	Zhen Lin	CC BY-SA 4.0	added 22 characters in body
Dec 8, 2022 at 15:51	history	answered	Zhen Lin	CC BY-SA 4.0