## How far can Spectral Algebraic Geometry be developed over $\mathbb{E}_2$-rings (instead of $\mathbb{E}_\infty$-rings)?

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.