I'd like to understand the concept of spectral scheme in two basic examples.

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 $S^\infty$, or equivalently, its 0-truncation.

Let $\Sigma^\infty(S)$ be the collection of all spectral deformations 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 $R$. Special cases of arithmetic $R$, e.g., $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{F}_p$, $\mathbb{\bar{Q}}$ would be specially interesting.

(2) What is $\Sigma^\infty(\mathbb{P}^n_R)$, or just $\Sigma^\infty(\mathbb{P}^1_\mathbb{C})$? Here $\mathbb{P}^n_R$ is the $n$-dimensional projective space over a commutative ring $R$.

Also,

(3) What is the structure of $\Sigma^\infty(S)$? Is $\Sigma^\infty$ functorial for morphisms 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 $\Sigma^\infty(S)$.

I'd like to understand the concept of spectral scheme in two basic examples.

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 of $S$ if $\pi_0(\mathcal{O}^\infty_X) = \mathcal{O}_X$, i.e., if $S$ is the scheme underlying the spectral scheme $S^\infty$, or equivalently, its 0-truncation.

Let $\Sigma^\infty(S)$ be the collection of all spectral deformations 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 $R$. Special cases of arithmetic $R$, e.g., $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{F}_p$, $\mathbb{\bar{Q}}$ would be specially interesting.

(2) What is $\Sigma^\infty(\mathbb{P}^n_R)$, or just $\Sigma^\infty(\mathbb{P}^1_\mathbb{C})$? Here $\mathbb{P}^n_R$ is the $n$-dimensional projective space over a commutative ring $R$.

Also,

(3) What is the structure of $\Sigma^\infty(S)$?

I'd like to understand the concept of spectral scheme in two basic examples.

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 $S^\infty$, or equivalently, its 0-truncation.

Let $\Sigma^\infty(S)$ be the collection of all spectral deformations 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 $R$. Special cases of arithmetic $R$, e.g., $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{F}_p$, $\mathbb{\bar{Q}}$ would be specially interesting.

(2) What is $\Sigma^\infty(\mathbb{P}^n_R)$, or just $\Sigma^\infty(\mathbb{P}^1_\mathbb{C})$? Here $\mathbb{P}^n_R$ is the $n$-dimensional projective space over a commutative ring $R$.

Also,

(3) What is the structure of $\Sigma^\infty(S)$? Is $\Sigma^\infty$ functorial for morphisms of schemes?