spectral symmetric algebra



Stable Homotopy theory

Higher algebra



Given a ring spectrum RR and module spectrum EE over RR, there is the free RR-algebra spectrum induced by EE, this is the symmetric algebra spectrum Sym RESym_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 nn-fold smash products of EE over RR homotopy quotiented by the canonical ∞-action of the symmetric group Σ(n)\Sigma(n) (by permutation of factors):

Sym REn(E n)/Σ(n). 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 RR itself, regarded as a module spectrum itself has interesting structure:

Sym RR nR/Σ(n) nR(BΣ(n)) + R(nBΣ(n)) +, \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 BΣ nB \Sigma_n denotes the homotopy type of the classifying space of the symmetric group on nn elements (given for instance by the topological space Emb({1,,n}, )/Σ(n)Emb(\{1, \cdots, n\}, \mathbb{R}^{\infty})/\Sigma(n)).

In particular for R=𝕊R = \mathbb{S} the sphere spectrum, then the absolute spectral affine line Sym 𝕊𝕊Sym_{\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,,x n}Sym R(R n) R\{x_1, \cdots, x_n\} \coloneqq Sym_R (R^{\vee^n})

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

R[x 1,,x n]RΣ ( +) 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,,x n}R[x 1,,x n]. R\{x_1, \cdots, x_n\} \longrightarrow R[x_1, \cdots, x_n] \,.

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

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

Sym R(Σ nR) R(kS nk/Σ(k)) + R(k(BΣ(k)) nτ k), \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 τ n\tau_n associated to the Σ(n)\Sigma(n)-universal principal bundle via the canonical action of Σ(n)\Sigma(n) on n\mathbb{R}^n (see also at symmetric group – Classifying space and Thom space).

The operation Sym RSym_R is of course functorial, and hence any choice of RR-linear map f x:REf_x \colon R \to E induces morphisms

RBΣ(n) +Sym R n(R)Sym R n(E). 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).


Spectral affine lines

For RR a (connective) E-∞ ring, the

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

is the spectral affine line over RR.

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 RR 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”)

𝔸 k 0|1Spec(Sym k(k[1])) \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 RR:

R 0|1 Spec(Sym R(ΣR)) Spec(RSym 𝕊(Σ𝕊)) \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 𝔼 \mathbb{E}_\infty-algebras

The free 𝔼 \mathbb{E}_\infty-algebra over RR on nn generators is the spectrum Sym R(R n)Sym_R(R^{\vee n}). This is typically denoted R{x 1,,x n}R\{x_1, \ldots, x_n\}. If RR is connective, π 0(R{x 1,,x n})\pi_0(R\{x_1, \ldots, x_n\}) can be identified with the polynomial algebra (π 0R)[x 1,,x n](\pi_0 R)[x_1, \ldots, x_n]. The spectrum R{x 1,,x n}R\{x_1, \ldots, x_n\} satisfies the following universal property: for any other 𝔼 \mathbb{E}_\infty-RR-algebra TT,

Map Alg R 𝔼 (R{x 1,,x n},T)(Ω T) ×n. 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)


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

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 EE-Theory (2010) (pdf)

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

See also

Last revised on June 20, 2019 at 19:07:01. See the history of this page for a list of all contributions to it.