Timeline for what is $\mathrm{Bun}(G)$?
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17 events
| when toggle format | what | by | license | comment | |
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| Apr 27, 2017 at 20:01 | comment | added | Qfwfq | (I suspected that flat analytic maps were already open, though I specified it cause I wasn't sure) | |
| Apr 27, 2017 at 19:09 | comment | added | nfdc23 | It is a non-trivial theorem that any flat map between complex-analytic spaces (by which I mean the induced maps between local rings are flat in the sense of commutative algebra) is necessarily open for the classical topology. Also, flat surjective maps between complex-analytic spaces admit a section "finite flat locally" on the base, which is pretty much the best one could hope for (motivated by a slightly weaker analogue for fppf maps of schemes in EGA, proved by the same "fibral slicing" method but exploiting that analytic maps become finite locally near points isolated in fibers). | |
| Apr 27, 2017 at 16:44 | comment | added | Will Sawin | @Qfwfq Sure, but the topology given by flat maps in the analytic cattegory is differnt from the fppf or fpqc topologies, which depend on flat maps in the algebraic category. | |
| Apr 27, 2017 at 16:26 | comment | added | Qfwfq | @WillSawin: sure, the example of a ramified cover settles the (classical, flat) comparison. About the thing you mention, that "there are classical open sets that are not covered by flat maps", I think it depends on the suitable definition of flat map in the analytic category: it's not unreasonable to define a flat map to be an open holomorphic map for which the local rings upstairs are flat modules over the local rings downstairs. So in this case a classical open would tautologically be a "flat open". | |
| Apr 27, 2017 at 16:23 | comment | added | Will Sawin | @Qfwfq There are sometimes classical torsors which are not etale-locally trivial, such as the Hopf surface, which is an elliptic curve torsor over $\mathbb P^1$. Perhaps for affine group schemes over proper varieties all analytic torsors are etale-locally trivial. | |
| Apr 27, 2017 at 16:21 | comment | added | Will Sawin | @Qfwfq As nfdc23 mentioned, ramified covers provide examples of flat maps without analytic slices. | |
| Apr 27, 2017 at 16:20 | comment | added | Will Sawin | @Qfwfq For classical and flat, neither direction is true - there are classical open sets that aren't covered by flat morphisms, and flat morphism that aren't covered by classical open sets. Torsors by smooth relative group schemes in the etale and flat topologies are equivalent - because every flat-local torsor is flat-locally trivial, hence flat-locally smooth, hence smooth, hence admits sections etale-locally, hence etale-locally trivial. In characteristic zero, all constant group schemes are smooth, so if $G$ is a constant group all flat torsors are etale (and hence classical) torsors. | |
| Apr 27, 2017 at 16:20 | comment | added | Qfwfq | And.. Does every flat map have (locally) an "analytic slice"? | |
| Apr 27, 2017 at 16:18 | comment | added | Qfwfq | @WillSawin: yes, of course, I was forgetting that not all topologies are comparable! | |
| Apr 27, 2017 at 16:16 | comment | added | Will Sawin | @Qfwfq Not all etale open sets are classical open sets, and vice versa. It's probably best to compare them in a joint topology where we define open sets to be complex analytic spaces with a map to $X$ that is a local analytic isomorphism. Doing this, we can state that every etale open set has a cover by classical open sets - which is exactly what nfdc23 said - which I think is an appropriate interpretation of "the classical topology is finer" for Grothendieck topologies. The reverse is not true, as some classical open sets are smaller. | |
| Apr 27, 2017 at 16:13 | comment | added | Qfwfq | I would also ask the same question about the pair (classical, flat). That may have already been asked on MO. Anyway, are there analytic $G$-torsors on $S$ that are flat-locally trivial but not analytically-(=classically)-locally trivial? What about families of $G$-torsors on $X$ parametrized by $S$? | |
| Apr 27, 2017 at 16:03 | comment | added | Qfwfq | So, technically speaking, can we say that classical and étale are equivalent? If not, which of the two is the finest? | |
| Apr 27, 2017 at 15:47 | comment | added | nfdc23 | The classical and etale topologies are very closely related, ultimately due to a combination of the Artin approximation theorem and various more basic features of etale morphisms (e.g., a map between finite type $\mathbf{C}$-schemes is etale if and only if the associated map of analytifications is a local analytic isomorphism, and more deeply via Artin approximation if $(X,x)$ and $(X',x')$ are such $\mathbf{C}$-schemes equipped with $\mathbf{C}$-points then there is a common etale neighborhood if and only if the analytifications have isomorphic actual neighborhoods of $x$ and $x'$). | |
| Apr 27, 2017 at 15:39 | comment | added | Qfwfq | @nfdc23: thanks for the comment. What about the comparison between the (Grothendieck topology induced by the) "classical" analytic topology (I mean, not the analytic-Zariski one) and the other topologies (at least from the point of view of principal $G$-bundles)? | |
| Apr 27, 2017 at 14:32 | comment | added | nfdc23 | In the complex-analytic setting the flat topology is not subsumed by the etale topology; the relation between the two involves the same issues as in the algebro-geometric context (and even there, the fppf topology has merits over the etale topology when one is studying some maps with "bad" fibers); admittedly the flat topology is less important in characteristic 0 than in characteristic $>0$ (for a variety of reasons), but it does have uses there too and as such cannot be replaced with the etale topology. The distinction already appears for branched covers of Riemann surfaces. | |
| Apr 27, 2017 at 14:24 | history | edited | Qfwfq | CC BY-SA 3.0 |
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| Apr 27, 2017 at 14:18 | history | answered | Qfwfq | CC BY-SA 3.0 |