Timeline for How far can Spectral Algebraic Geometry be developed over $\mathbb{E}_2$-rings (instead of $\mathbb{E}_\infty$-rings)?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 29, 2019 at 16:35 | comment | added | Emily | @skd Wow! Thank you for the references (and also for taking notes in the 2017 Talbot with Eva Belmont, these look incredible)! | |
| Jun 27, 2019 at 16:17 | comment | added | skd | ... which you can find in Behrens' notes on the construction of tmf. Regarding obstruction theory in SAG: part of the point of SAG is to obtain these highly structured "designer" ring spectra (like Lubin-Tate theory, KO, TMF) "without obstruction theory", but rather via spectrifying the moduli problems in classical algebraic geometry used to define the algebraic analogues of these ring spectra. | |
| Jun 27, 2019 at 16:14 | comment | added | skd | For the obstruction theory of structured ring spectra, you should check out Goerss-Hopkins (see Goerss' webpage), as well as the Talbot 2017 notes (math.mit.edu/conferences/talbot/2017/talbot-notes-2017.pdf). For E_oo-power operations in the chromatic setting, check out work of Rezk (in particular, his talk slides titled "Power operations in Morava E-theory: a survey"). As for this stuff in SAG: afaik, there's no source talking about a purely SAG-construction of power operations, but it is implicit in the height 2 case in the construction of tmf... | |
| Jun 19, 2019 at 22:45 | comment | added | Dylan Wilson | For certain things it might make more sense to use E_k-modules instead of left modules. Two reasons for this: (1) the category of E_k-modules over an E_k ring is still E_k monoidal, so you don’t get this loss of monoidality that others were pointing out; (2) deformations of E_k rings are controlled by the E_k-cotangent complex which is an E_k-module not a left module. (When k=infty you get this nice fact that left modules agree with E_infty-modules). | |
| Jun 19, 2019 at 19:29 | comment | added | Emily | @skd and thank you for answering my $E_k$-modules question! | |
| Jun 19, 2019 at 19:27 | comment | added | Emily | @skd Those are nice to know too! Would you recommend some particular references for these? (That is, obstruction theory for $E_\infty$-rings and $E_\infty$-power operations in chromatic homotopy theory, both classically and via SAG?) | |
| Jun 19, 2019 at 19:26 | history | edited | Emily | CC BY-SA 4.0 |
Added @crystalline's questions
|
| Jun 19, 2019 at 19:22 | comment | added | Emily | @crystalline These are nice questions! I mentioned them (now, in an edit) in the question too. | |
| Jun 19, 2019 at 19:18 | comment | added | skd | The (∞-)category of modules over an E_k-ring admits an E_{k-1}-monoidal structure. | |
| Jun 19, 2019 at 19:14 | comment | added | Emily | @DenisNardin That's nice to know! What happens for modules over $E_k$-rings for other $k$? Do they get braided monoidal structures or something similar? | |
| Jun 19, 2019 at 18:43 | comment | added | skd | Haha, didn't think my blog would show up on MathOverflow! Two comments: first, the obstruction theory for E_∞-rings is a lot better than that of E_k-rings for 1<k<∞, because for E_k-rings, one enters "bracket hell" (a term I learnt from Tyler Lawson); and second (relatedly), as I wrote at the end of the linked post, E_∞-power operations in chromatic homotopy theory have a nice algebro-geometric interpretation, but I don't know of anything similar for E_2-power operations. Note that the pure braid group is infinite. | |
| Jun 19, 2019 at 8:13 | comment | added | crystalline | a lot of the results in Lurie's book just go through word for word, the only thing you really have to be careful about is the tensor product-- it's not the same as the coproduct in the category of E_2-algebras! | |
| Jun 19, 2019 at 8:12 | comment | added | crystalline | let me ask a different question, what results of SAG does one actually want in the E_2-setting? and what are some concrete things those results would buy you? | |
| Jun 19, 2019 at 5:29 | comment | added | Denis Nardin | Not quite what you are asking for, but if $E$ is only a $E_2$-ring, then $E$-modules have only a monoidal structure (and not a symmetric monoidal structure) | |
| Jun 18, 2019 at 22:40 | review | First posts | |||
| Jun 18, 2019 at 22:42 | |||||
| Jun 18, 2019 at 22:39 | history | asked | Emily | CC BY-SA 4.0 |