Timeline for Affine scheme as algebraic space
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jul 16, 2020 at 9:05 | comment | added | user267839 | I understand why in your example $p: \overline{\Delta(U)} \to U$ is also etale but I'm not sure if this example covers all pathological phenomena can occure when passing from $\Delta(U)$ to $\overline{\Delta(U)}$ as subscheme of $U \times U$. Does your example prove a general argument why for arbitrary $U$ the projection $p: \overline{\Delta(U)} \to U$ should be etale? | |
| Jul 16, 2020 at 6:25 | comment | added | AlexIvanov | Look at the example where $U$ is the affine line with doubled origin. The closure of the diagonal is then an affine line with four points "at the origin". Each of the projection maps $\overline{\Delta{U}} \rightarrow U$ is an isomorphism outside origin and over each of the origins of $U$ there lie two of the four points of $\overline{\Delta{U}}$. This map is an isomorphism locally on the source, hence etale. See also math.stackexchange.com/questions/1438886/… | |
| Jul 15, 2020 at 23:17 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
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| Jul 15, 2020 at 23:04 | comment | added | user267839 | A curious nitpick beside: Do you know if, if we assume that $\overline{\Delta(U)} \not \cong \Delta(U)$ (eg $U$ not separated). Is the restricted projection map $p \vert _{\overline{\Delta(U)}}:\overline{\Delta(U)} \subset U \times U \to U$ nevertheless etale, or is it in general wrong? | |
| Jul 15, 2020 at 23:03 | comment | added | user267839 | @AlexI: yes I see. More precisely the restricted $p \vert _{\Delta(U)}: \Delta(U) \to U$ must be an isomorphism because it's nothing but the composition of isomorphisms $\Delta^{-1}: \Delta(U) \to U$ and $id_U= p \circ \Delta: U \xrightarrow{\Delta} U \times U \xrightarrow{\text{p}} U$. So indeed $R:=\Delta(U)$ is the right choice and not the schematic closure $\overline{\Delta(U)} \subset_c U \times U$, I think. | |
| Jul 15, 2020 at 21:57 | comment | added | AlexIvanov | The restriction of each of the projections $U \times U \rightarrow U$ to $\Delta(U)$ gives an isomrphism of $\Delta(U)$ to $U$, so it is in particular etale. | |
| Jul 15, 2020 at 21:06 | history | asked | user267839 | CC BY-SA 4.0 |