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Jacob Lurie has extensively developed derived algebraic geometry in the setting of $\mathbb{E}_\infty$-ring spectra [SAG]. The resulting theory of Spectral Algebraic Geometry (SAG) gives (in particular) a way to view some of the spectra topologists care about algebro-geometrically, as spectral schemes. In particular, examples of such spectra would be

  1. The complex cobordism spectrum $\mathrm{MU}$;

  2. Complex $K$-theory $\mathrm{KU}$;

  3. The spectrum $\mathrm{TMF}$ of topological modular forms;

  4. The sphere spectrum $\mathbb{S}$.

On the other hand, there are many important spectra which don't admit $\mathbb{E}_\infty$-structures (and hence don't fit in Lurie's SAG) such as:

  1. The $p$-local Brown-Peterson spectrum $\mathrm{BP}$ and its connective covers $\rm{BP}\langle n\rangle$;

  2. The Morava $K$-theories $K(n)$;

  3. The Ravenel spectra $X(n)$.

There has been work on SAG over $\mathbb{E}_n$-rings, in particular by John Francis, in his thesis. Focusing on $n=2$, what results of SAG are expected to be troublesome to extend to the setting of $\mathbb{E}_2$-ring spectra?

I'm also tempted to ask here Sanath's question on this topic:

[...] what are some results in either the purely algebro-geometric or purely chromatic aspects of spectral algebraic geometry which rely upon using the entire $\mathbb{E}_\infty$-ring structure?

to be found in his A love letter to E_2-rings.

@crystalline raised two interesting questions in the comments:

[...] what results of SAG does one actually want in the E_2-setting? and what are some concrete things those results would buy you?

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    $\begingroup$ 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! $\endgroup$
    – crystalline
    Commented Jun 19, 2019 at 8:13
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    $\begingroup$ 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. $\endgroup$
    – skd
    Commented Jun 19, 2019 at 18:43
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    $\begingroup$ The (∞-)category of modules over an E_k-ring admits an E_{k-1}-monoidal structure. $\endgroup$
    – skd
    Commented Jun 19, 2019 at 19:18
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    $\begingroup$ @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?) $\endgroup$
    – Emily
    Commented Jun 19, 2019 at 19:27
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    $\begingroup$ 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... $\endgroup$
    – skd
    Commented Jun 27, 2019 at 16:14

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