Spectral and derived deformations of schemes I'd like to understand how ordinary schemes deform or lift to spectral and derived schemes in two basic examples as well as what the structure of the space of deformations in general is.
Let $S = (X, \mathcal{O}_X)$ be a scheme, where $X$ is the underlying topological space, and $\mathcal{O}_X$, its structure sheaf of commutative rings.
Call a spectral scheme $S^\infty = (X, \mathcal{O}^\infty_X)$ a spectral deformation (or lift) of $S$ if $\pi_0(\mathcal{O}^\infty_X) = \mathcal{O}_X$, i.e., if $S$ is the scheme underlying the spectral scheme $S^\infty$.
Let $\Sigma^\infty(S)$ be the collection of all spectral deformations of $S$. Call it the spectral deformation space of $S$.
(1) What is $\Sigma^\infty(Spec\ R)$ for a commutative ring $R$? This is equivalent to asking for a description of the collection of all $\mathbb{E}_\infty$ rings whose $\pi_0$ is $R$. Examples for some elementary rings, e.g., $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{F}_p$, $\mathbb{Q}_p$,   $\mathbb{\bar{Q}}$ would be specially welcome.
(2) What is $\Sigma^\infty(\mathbb{P}^n_k)$? Here $\mathbb{P}^n_k$ is the $n$-dimensional projective space over a field $k$.
(3) What is the structure of $\Sigma^\infty(S)$? Is $\Sigma^\infty$ functorial for morphisms of schemes?
I'd also like to ask the same questions for the derived deformation space $\Sigma^{\triangledown}(S)$ of $S$ obtained by replacing spectral schemes by derived schemes in the discussion above, with a simple example in which it differs from $\Sigma^\infty(S)$.
 A: In general, these are incredibly hard questions. It seems to me that one natural question to ask (if you are interested in $\pi_0$ of ring spectra) would be about understanding even periodic $\mathbf{E}_\infty$-rings $A$ with $\pi_0 A = R$. Alternatively, you could attempt to understand those $\mathbf{E}_\infty$-rings $A$ with $\pi_\ast A = R_\ast$, where $R_\ast$ is some graded ring. In what follows, I'll attempt to address some of these questions (in particular, this is not a complete answer to your question). Let me begin with some generalities.
If you restrict to asking for $K(1)$-local $\mathbf{E}_\infty$-rings $A$ with $\pi_0 A = R$, then there are a few things you can say. For instance, if $p=0$ in $R$, then the collection of such $A$ is necessarily empty if $R$ is nonzero. If $R$ does not have a lift of Frobenius, then the collection of such $A$ is again empty. (Examples of such non-liftability results, in the $K(d)$-local $\mathbf{E}_n$-case for varying $d$ and $n$, are in the paper referred to above; there's an updated version on my webpage, which hasn't yet found its way onto the arXiv. Also see Schwanzl-Vogt-Waldhausen's "Adjoining roots of unity to E∞ ring spectra in good cases -- a remark".)
There are also some positive results: suppose $R_\ast$ is an even-periodic graded $p$-torsion free ring such that $R_0$ is the $p$-adic completion of a smooth $\mathbf{Z}_p$-algebra, which is equipped with:


*

*a formal group classified by a map $MUP_0 \to R_0$ for which the induced formal group over $R_0/p$ is of height $1$, and

*an action of $\mathbf{Z}_p^\times$ and a compatible $p$-derivation (a "$\theta$-algebra structure").


Then there is a(n even periodic, complex oriented) $K(1)$-local $\mathbf{E}_\infty$-ring $A$ such that $\pi_\ast A = R_\ast$. This can be deduced from the results in the following paper of Lawson's and Naumann's: https://arxiv.org/abs/1101.3897. Admittedly, the conditions specified above seem like a lot; however, I know of no way to get past this. I also do not know how one might generalize this to constructing $K(n)$-local $\mathbf{E}_\infty$-rings for $n\geq 2$. The obstruction stems from the fact that we just do not know as much about power operations at heights $\geq 2$ as we do at height $1$. (This is not to downplay the work of Rezk, Zhu, and others on height $2$ power operations.)
Let me now give some examples. Fix an $\mathbf{E}_\infty$-ring $A$.


*

*Suppose $\pi_0 A = \mathbf{Q}$. Then $A$ is an $\mathbf{E}_\infty$-$\mathbf{Q}$-algebra, and these are very well understood. Indeed, $\mathbf{E}_\infty$-$\mathbf{Q}$-algebras are the same as commutative dg-$\mathbf{Q}$-algebras. The same is true of any $\mathbf{E}_\infty$-$R$-algebras over any (ordinary) $\mathbf{Q}$-algebra $R$ (hence, in particular, $\mathbf{Q}_p$ and $\overline{\mathbf{Q}}$).

*Suppose $\pi_0 A = \mathbf{F}_p$. If $A$ is $p$-local, then a result of Hopkins and Mahowald implies that $A$ is an $\mathbf{E}_\infty$-$\mathbf{F}_p$-algebra. There can be many such $A$, because Steenrod operations exist. For example, take the $\mathbf{F}_2$-algebras $A_1 = \mathbf{F}_p \otimes^\mathbf{L}_{\mathbf{Z}} \mathbf{F}_p$ (the fiber product in "derived schemes") and $A_2 = \mathbf{F}_p \otimes_{\mathbb{S}} \mathbf{F}_p$ (the fiber product in "spectral schemes"). Then $\pi_\ast A_1 = \mathbf{F}_p[t]/t^2$ with $|t|=1$ (so $\pi_0 A_1 = \mathbf{F}_p$), but $\pi_\ast A_2$ is the mod $p$ dual Steenrod algebra (so $\pi_0 A_2 = \mathbf{F}_p$ as well), but one is clearly much larger than the other! This gives an example showing that (in your notation) $\Sigma^\infty(\mathbf{F}_p) \supsetneq \Sigma^\triangledown(\mathbf{F}_p)$.

*Suppose $\pi_0 A = \mathbf{Z}$. Then, there are a ton of possibilities for $A$. For instance, $A$ could be $\mathbb{S}, MU, KU, KO, TMF, Tmf, ...$ (and their connective covers). Again, even periodicity would be a reasonable condition to impose. Then, the number of examples is cut down significantly (in the list provided above, for instance, you're only left with $KU$).
Hopefully someone else more knowledgeable in these topics can provide a more comprehensive and satisfactory answer to your question.
