Given a ring spectrum $R$ and an $R$-module $E$, we have the [spectral symmetric algebra](https://ncatlab.org/nlab/show/spectral+symmetric+algebra) $\mathrm{Sym}\_R(E)$ of $E$ over $R$, defined by
$$
\begin{align\*}
\mathrm{Sym}\_R(E) &\overset{\mathrm{def}}{=} \mathrm{colim}\_{\mathbb{F}}(\Delta\_{E})\\
 &\cong \bigoplus\_{n\in\mathbb{N}}E^{\otimes\_\mathbb{S}n}\_{\mathsf{h}\Sigma\_{n}},
\end{align\*}
$$
where $\mathbb{F}\overset{\mathrm{def}}{=}\mathsf{FinSets}^\simeq$ is the groupoid of finite sets and permutations. As [A Rock and a Hard Place showed here](https://mathoverflow.net/a/403254), the $\mathbb{E}\_\infty$-ring $\mathrm{Sym}\_R(E)$ comes with a natural grading by the sphere spectrum, inducing a $\mathbb{Z}$-grading on $\pi\_0(\mathrm{Sym}\_R(E))\cong\mathrm{Sym}\_{\pi\_0(R)}(\pi\_0(E))$. So e.g. picking $R=E=\mathbb{S}$, gives
$$
\begin{align\*}
\pi\_0(\mathrm{Sym}\_{\mathbb{S}}(\mathbb{S})) &\overset{\mathrm{def}}{=} \pi\_0(\mathbb{S}\{t\})\\
&\cong \mathbb{Z}[t],
\end{align\*}
$$
which carries the natural $\mathbb{Z}$-grading.


However, the $\pi\_0$ of an $\mathbb{S}$-graded ring can be [more complicated](https://mathoverflow.net/q/403217) than just a commutative $\mathbb{Z}$-grading, and for instance allows for the multiplication on the $\pi\_0$ to be supercommutative, satisfying $ab=(-1)^{\deg(a)\deg(b)}ba$. This led me to the following pair of questions:


1. Is there an "spectral exterior algebra" construction $\bigwedge\_RE$, whose $\pi\_0$ is the $\mathbb{Z}$-graded supercommutative exterior algebra $\bigwedge\_{\pi\_0(R)}\pi\_0(E)$? If so, does it come with an $\mathbb{S}$-grading?
2. [One of the more homotopy-theoretic points of view](https://arxiv.org/abs/1110.4753) on symmetric and exterior algebras is that the passage from the former to the latter corresponds to considering a larger portion of the sphere spectrum. More generally, do we have an $\mathbb{N}$-indexed sequence of "higher exterior algebra" constructions $\mathrm{Sym}\_R(E)$, $\bigwedge\_R(E)$, $\bigwedge^{\mathbf{2}}\_R(E)$, $\ldots$?



Is there a "spectral exterior algebra" construction in higher algebra?
Interesting question! I can't give a real answer, but here are some idle musings:


Note that one way to encode exterior powers is as $\Lambda^i\_R(E) = \Sigma^{-i}(\mathrm{Sym}^i\_R(\Sigma(E)))$ (since placing the generators in degree one switches, by virtue of the Koszul sign rule from the usual $\Sigma\_n$-action to the alternating one). So the symmetric algebra in SAG or DAG on a $1$-shifted module looks like a sheared version of the exterior algebra. That is, if $R$ is an ordinary commutative ring and $M$ an $R$-module, then
$$
\mathrm{Sym}^\*\_R(\Sigma (M)) \simeq \Sigma^{\*}(\Lambda\_R^\*(M))
$$
(though here both symmetric and exterior algebras are understood in the spectral or derived sense, and may not agree with their usual algebraic counterparts unless $M$ is flat). In the DAG context, we similarly have
$$
\mathrm{Sym}^\*\_R(\Sigma^2(M)) \simeq \Sigma^{2\*}(\Gamma\_R^\*(M)),
$$
where $\Gamma^\*\_R(M)$ is the free divided power $R$-algebra generated by $M$. See Section 25.2 of [the SAG book](https://www.math.ias.edu/%7Elurie/papers/SAG-rootfile.pdf) for the proofs.


Therefore, perhaps the candidates for your degree $n$ higher exterior algebras might be some kind of $n\*$-fold desuspensions of the symmetric algebra $\mathrm{Sym}^\*\_R(\Sigma^n(R))$. For $n=0$, that would recover (flatness issues notwithstanding) the usual symmetric algebra, for $n=1$ the exterior algebra, for $n=2$ the free divided power algebra, and for $n\ge 3$, who knows? :)


Operating on this provisional understanding of what the higher exterior algebras are, let us discuss the $S$-grading. We see from the construction that the "unshearing" of it is the symmetric algebra $\mathrm{Sym}^\*\_R(\Sigma^n(R))$, which does indeed carry an $S$-grading. The question about whether the higher exterior algebras themselves carry a canonical $S$-grading now becomes a question about how, if at all, some kind of a "shearing" functor $(M\_n)\mapsto (\Sigma^n M\_n)$ interacts with $S$-gradings. I believe this going to be a very non-trivial question.


Even if we were not talking about the much-more-complicated $S$-gradings, but rather $\mathbf Z$-gradings, the shearing functor $\Sigma^\*$ is indeed an automorphism of the $\infty$-category of $\mathbf Z$-graded spectra $\mathrm{Fun}(\mathbf Z,\mathrm{Sp})$, but I don't think it is symmetric monoidal. Indeed, consider rather its inverse squared $\Sigma^{-2\*}$. If it were symmetric monoidal, it would send the polynomial $\mathbb E\_\infty$-algebra $S[t]\simeq \bigoplus\_{n\ge 0}S$, with its usual $\mathbf Z$-grading, into the "$2$-shifted polynomial $\mathbb E\_\infty$-algebra" $S[\beta] \simeq \bigoplus\_{n\ge 0}\Sigma^{-2n}S$. The latter thing does indeed exist as an $\mathbb E\_2$-ring, studied in Section 2.8 of [this paper by Lurie](https://people.math.harvard.edu/%7Elurie/papers/Waldhaus.pdf). But I have been told before that, due to some obstructions one can calculate, it can be shown that said $\mathbb E\_2$-ring does not support an $\mathbb E\_\infty$-structure. That would seem to imply that the shearing functor $\Sigma^{-2\*}$ (and consequently $\Sigma^\*$) can not be symmetric monoidal.


And that was just in the $\mathbf Z$-graded world, let alone all the additional complications that the $S$-graded setting might bring!


