Timeline for $G$-invariant morphism and coarse moduli spaces
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 10, 2021 at 9:17 | vote | accept | Davide | ||
| Aug 18, 2021 at 15:19 | comment | added | Davide | You use $k$ algebraically closed only to show that $k$ points are dense. So, if we suppose $G$ reduced and $\mathop{char} k = 0$, we have that this result is true in general. Yes, probably the assumption that the author has in mind is $S$ reduced, but she doesn't mention it. I'm looking for a more general statement of this kind that does not require $S$ reduced. It's not clear to me even if it is a good purpose. | |
| Aug 18, 2021 at 15:18 | comment | added | Davide | Thanks very much for your edit! Now it's clear to me why we need $S$ separated. I think that in this case is automatic that schematic image commute with base change. According to prop 0.4 in these notes, we need that the schematic image of $\bigsqcup_{p \in G(k)} p \to G$ is quasi-compact (that's true because it is $G$) and that $G$ is quasi-separated (that's true because it is of finite presentation). | |
| Aug 17, 2021 at 3:24 | comment | added | afh | The other direction depends on your "moduli functor" $\mathcal{M}$. E.g. you can take $\mathcal{M}$ to be represented by a scheme and take $S$ to the trivial square-zero thickening $\mathcal{M}[\epsilon]$ with trivial action. Then the construction of the inverse does not go through. | |
| Aug 17, 2021 at 3:24 | comment | added | afh | Hello @merlino. I added a small explanation to my answer. For your other question, it seems to me that in Prop. 3.35 there are some implicit assumptions the author had in mind. Otherwise the proposition cannot be true as stated. The first problem is in forming the G-equivariant map as in your question (the forward direction). For that you probably want to impose some condition on $S$ (e.g. reducedness). | |
| Aug 17, 2021 at 2:38 | history | edited | afh | CC BY-SA 4.0 |
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| Aug 16, 2021 at 12:25 | comment | added | Davide | Thanks for your answer @afh. I'm happy to know that we can avoid assuming $X$ is reduced. Do you have any references for the last sentence you wrote? The notes I linked above still confuse me. Do you know if it's enough to assume that $G$ is reduced to prove proposition 3.35? | |
| Aug 16, 2021 at 0:57 | history | edited | afh | CC BY-SA 4.0 |
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| Aug 16, 2021 at 0:50 | history | edited | afh | CC BY-SA 4.0 |
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| Aug 16, 2021 at 0:41 | review | First posts | |||
| Aug 16, 2021 at 1:42 | |||||
| Aug 16, 2021 at 0:40 | history | answered | afh | CC BY-SA 4.0 |