Timeline for Universal homeomorphisms and the étale topology
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 3, 2010 at 10:37 | vote | accept | Laurent Moret-Bailly | ||
| Nov 24, 2010 at 22:21 | answer | added | Angelo | timeline score: 6 | |
| Nov 24, 2010 at 18:44 | comment | added | Bhargav | Brian, great! So in these two cases all etale locally trivial G-torsors on X are actually Zariski locally trivial. Is that generally true? It seems mildly plausible that some variant of the restriction of scalars argument gives an affirmative answer when the inclusion of reduced subscheme admits a retraction, and so should be okay in equicharacteristic when the reduction is smooth... | |
| Nov 24, 2010 at 18:29 | comment | added | BCnrd | Bhargav, great. OK for $\mu_p$, not sure for $\alpha_p$. For $G = \mu_p$, finiteness of $f:X = G \times E \rightarrow E$ implies ${\rm{H}}^1(X,F) = {\rm{H}}^1(E,f_{\ast}F)$ for abelian etale $F$ on $X$. Take $F$ to be functor of pts of $G$ on etale $X$-schemes, so $f_{\ast}F$ is functor of pts of ${\rm{R}}_{X/E}(G) = {\rm{R}}_{G/k}(G) \times E$ on etale $E$-schemes (all reduced). By left-exactness, this is $E$-gp obtained from $k$-gp ${\rm{R}}_{G/k}(\mathbf{G}_m)[p] = \mu_p \times (1 + (t))/(1 + (t^p)) = \mu_p \times \mathbf{G}_a^{p-1}$ (use truncated log), underlying reduced is $O_E^{p-1}$. | |
| Nov 24, 2010 at 17:20 | comment | added | Bhargav | I'm probably being dense, but here's a thought: work over a perfect field k of char p, let X = E x alpha_p, where E is an elliptic curve over k, and let G = alpha_p viewed as a sheaf on X_et. Any etale V -> X is of the form U x alpha_p -> X for some etale U -> E by top. invariance. So if I'm not mistaken, this means the following: under the identification of the small etale topoi of E and X, the sheaf defined by G on X is identified by G_a^{p-1} on E (compute maps k[x]/x^p -> R[x]/x^p with R reduced). In particular, there exist lots of non-trivial etale locally trivial G-torsors on X. | |
| Nov 24, 2010 at 16:27 | comment | added | BCnrd | Dear Angelo: Ah whoops, I misread the formulation of the question! OK, will need to think about it some more. | |
| Nov 24, 2010 at 15:47 | comment | added | Angelo | Dear Brian, does your comment answer Laurent's question? I may be confused (this happens a lot), but I don't see it. | |
| Nov 24, 2010 at 14:06 | comment | added | André Henriques | This looks like an interesting question that I'd like to think about. Unfortunately, you're using some terminology that I'm not familiar with. What is a universal homeomorphism of schemes? What is a locally free group scheme? | |
| Nov 24, 2010 at 13:44 | comment | added | BCnrd | Better: $f^{\ast}$ is equivalence of etale sites (apply to $S$-etale $S'$ and $S$); a footnote w/o details in 1970 ed. of SGA1. Proof: By EGA IV$_4$, 18.12.11, univ. homeo. = integral, radiciel, & surj. So WLOG $S$ and $X$ affine and (by nilimmersion case: 18.1.2!) reduced. Then ring map is injective, integral, radiciel, so WLOG (by limits) finite and can focus on septd etale objects (good for ZMT). Now $X$ is str. hens. whenever $S$ is, so full faithfulness is seen via ZMT. Essential surj. follows for finite etale schemes by etale descent, then in general via stratification (tricky). | |
| Nov 24, 2010 at 9:11 | history | asked | Laurent Moret-Bailly | CC BY-SA 2.5 |