nLab spectral symmetric algebra

Idea

Given a ring spectrum $R$ and module spectrum $E$ over $R$, there is the free $R$-algebra spectrum induced by $E$, this is the symmetric algebra spectrum $Sym_R E$, in direct analogy to the construction of symmetric algebras on vector spaces (e.g. Khan 16).

The underlying spectrum of the symmetric algebra is simply the direct sum (wedge sum of spectra) of all the $n$-fold smash products of $E$ over $R$ homotopy quotiented by the canonical ∞-action of the symmetric group $\Sigma(n)$ (by permutation of factors):

$$Sym_R E \;\coloneqq\; \underset{n \in \mathbb{N}}{\vee} (E^{\wedge^n})/\Sigma(n) \,.$$

In contrast to ordinary symmetric algebras on ordinary modules over ordinary rings, this means that the symmetric algebra on $R$ itself, regarded as a module spectrum itself has interesting structure:

$$\begin{aligned} Sym_R R & \simeq \underset{n \in \mathbb{N}}{\vee} R / \Sigma(n) \\ & \simeq \underset{n \in \mathbb{N}}{\vee} R \wedge (B \Sigma(n))_+ \\ & \simeq R \wedge \left( \underset{n \in \mathbb{N}}{\coprod} B \Sigma(n) \right)_+ \end{aligned} \,,$$

where $B \Sigma_n$ denotes the homotopy type of the classifying space of the symmetric group on $n$ elements (given for instance by the topological space $Emb(\{1, \cdots, n\}, \mathbb{R}^{\infty})/\Sigma(n)$).

In particular for $R = \mathbb{S}$ the sphere spectrum, then the absolute spectral affine line $Sym_{\mathbb{S}}\mathbb{S}$ is nontrivial. This, and its structure of a Hopf ring spectrum, is discussed in Strickland-Turner 97.

More generally,

$$R\{x_1, \cdots, x_n\} \coloneqq Sym_R (R^{\vee^n})$$

is like a polynomial algebra spectrum over $R$. Beware that there is a similar construction that is in general different, namely

$$R[x_1, \cdots, x_n] \coloneqq R \wedge \Sigma^\infty(\mathbb{N}_+)$$

where on the right we have the “monoid $\infty$-algebra” of the natural numbers, directly analogous to the ∞-group ∞-ring construction. There is a canonical comparison homomorphisms

$$R\{x_1, \cdots, x_n\} \longrightarrow R[x_1, \cdots, x_n] \,.$$

This is an equivalence if $R$ is of characteristic zero (Khan 16, prop. 2.7.4).

Similarly, if $E = \Sigma^n R \simeq R \wedge S^n$ is the $n$-fold suspension of $R$, regarded as an $R$-module spectrum, then

$$\begin{aligned} Sym_R(\Sigma^n R) & \simeq R \wedge \left( \underset{k \in \mathbb{N}}{\coprod} S^{n k}/\Sigma(k) \right)_+ \\ & \simeq R \wedge \left( \underset{k \in \mathbb{N}}{\coprod} (B \Sigma(k))^{n \tau_k} \right) \end{aligned} \,,$$

where on the right we have the Thom space of the vector bundle $\tau_n$ associated to the $\Sigma(n)$-universal principal bundle via the canonical action of $\Sigma(n)$ on $\mathbb{R}^n$ (see also at symmetric group – Classifying space and Thom space).

The operation $Sym_R$ is of course functorial, and hence any choice of $R$-linear map $f_x \colon R \to E$ induces morphisms

$$R \wedge B \Sigma(n)_+ \simeq Sym^n_R(R) \longrightarrow Sym_R^n(E) \,.$$

These correspond to power operation in generalized (Eilenberg-Steenrod) cohomology (Rezk 10, slide 4).

Examples

Spectral affine lines

For $R$ a (connective) E-∞ ring, the

$$\mathbb{A}^1_R \coloneqq Spec( Sym_R(R) )$$

is the spectral affine line over $R$.

Spectral superpoints

By the discussion at spectral super scheme, it makes good sense to regard the concept of a spectral scheme over an even periodic ring spectrum $R$ as the lift to spectral geometry of the concept of ordinary super schemes. Under this identification the ordinary superpoint, which is the spectrum of a commutative ring of the graded symmetric algebra on a single odd generator (“ring of dual numbers”)

$$\mathbb{A}_k^{0 \vert 1} \;\simeq\; Spec( Sym_k (k[1]) )$$

is given by the corresponding spectral scheme given by the spectral symmetric algebra on the suspension spectrum of $R$:

$$\begin{aligned} R^{0 \vert 1} &\coloneqq Spec \left( Sym_R (\Sigma R) \right) \\ & \simeq Spec\left( R \wedge Sym_{\mathbb{S}}(\Sigma \mathbb{S}) \right) \end{aligned}$$

Free $\mathbb{E}_\infty$-algebras

The free $\mathbb{E}_\infty$-algebra over $R$ on $n$ generators is the spectrum $Sym_R(R^{\vee n})$. This is typically denoted $R\{x_1, \ldots, x_n\}$. If $R$ is connective, $\pi_0(R\{x_1, \ldots, x_n\})$ can be identified with the polynomial algebra $(\pi_0 R)[x_1, \ldots, x_n]$. The spectrum $R\{x_1, \ldots, x_n\}$ satisfies the following universal property: for any other $\mathbb{E}_\infty$-$R$-algebra $T$,

$$Map_{Alg_R^{\mathbb{E}_\infty}}(R\{x_1, \ldots, x_n\}, T) \simeq (\Omega^\infty T)^{\times n}.$$

See (Lurie 2018, Notation B.1.1.2)

References

The symmetric algebra spectrum of the sphere spectrum, and its structure as a Hopf ring spectrum is discussed in

• Neil Strickland, Paul Turner, Rational Morava $E$-theory and $D S^0$, Topology Volume 36, Issue 1, January 1997, Pages 137-151 (pdf)

Symmetric algebras in the context of power operation on generalized (Eilenberg-Steenrod) cohomology are discussed in

• Charles Rezk, Power operations in Morava E-theory – a survey (2009) (pdf)

• Charles RezkPower operations in Morava $E$-Theory (2010) (pdf)

• Charles Rezk, Isogenies, power operations, and homotopy theory, article (pdf) and talk at ICM 2014 (pdf)

See also

• Jacob Lurie, examples 3.5.4 in Elliptic Cohomology I (pdf)

• Jacob Lurie, notation B.1.1.2 in Spectral Algebraic Geometry (pdf)

• Adeel Khan, sections 2.6 and 2.7 of Brave new motivic homotopy theory I (arXiv:1610.06871)