$\mathbb{Z}$-graded rings play an important role in algebra and algebraic geometry, so when moving to derived algebra and spectral algebraic geometry, it's natural to ask about ring spectra graded in the sphere spectrum $\mathbb{S}$.


As discussed in Section 2 of [Bunke–Nikolaus's *Twisted differential cohomology*](https://arxiv.org/abs/1406.3231), such an object may be defined as a symmetric lax monoidal functor $\mathbb{S}\to\mathsf{Sp}$, where $\mathbb{S}$ is considered as a monoidal $\infty$-groupoid.


In particular, given an $\mathbb{S}$-graded ring spectrum $R$, it follows that the ring $\pi\_0(R)$ is $\mathsf{Ho}(\mathbb{S})$-graded. [By the description here](https://mathoverflow.net/a/403221), such a $\tau\_{\leq1}\mathbb{S}$-graded ring consists of a pair $(R\_\bullet,\{\sigma\_k\}\_{k\in\mathbb{Z}})$ with


* $R\_\bullet$ a $\mathbb{Z}$-graded ring;
* $\{\sigma\_k\colon R\_k\to R\_k\}\_{k\in\mathbb{Z}}$ a family of order $2$ automorphisms, one for each $R\_k$.


Moreover, if $R$ is an $\mathbb{E}\_{\infty}$-ring, then $\pi\_0(R)$ is "$\mathsf{Ho}(\mathbb{S})$-graded commutative", in that we additionally have
$$
ab
=
\begin{cases}
ba &\text{if $\deg(a)\deg(b)$ is even,}\\
\sigma\_{\deg(a)+\deg(b)}(ab) &\text{if $\deg(a)\deg(b)$ is odd}
\end{cases}
$$
for each $a,b\in R\_\bullet$.


This includes in particular $\mathbb{Z}$-graded commutative algebras by picking $\sigma\_k(a)\overset{\mathrm{def}}{=}-a$ for each $k\in\mathbb{Z}$, as in that case the above condition becomes
$$ab=(-1)^{\deg(a)\deg(b)}ba.$$




---


**Question.** So, are there any interesting/non-trivial "in nature" examples of $\mathbb{S}$-graded ring spectra?



Grading ring spectra over the sphere spectrum
One of the default examples of ordinary graded commutative rings is the polynomial ring $\mathbf Z[t]$. Let us first examine the analogue of that, and then see where else that leads!


**1. $S$-grading on $S\{t\}$**


For the sake of clarity, allow me to denote the underlying infinite loop space (equivalently: grouplike $\mathbb E\_\infty$-space) of the sphere spectrum $S$ by $\Omega^\infty(S)$.
Recall that, by the Barrat-Priddy-Quillen Theorem, the ininite loop space $\Omega^\infty(S)$ can be described as the group completion of the groupoid of finite sets and isomorphisms $\mathcal F\mathrm{in}^\simeq$. Consider the constant functor $\mathcal{F}\mathrm{in}^\simeq \to\mathrm{Sp}$ with value $S$ - since the latter is the monoidal unit, this is a symmetric monoidal functor. Left Kan extension along the canonical map $\mathcal F\mathrm{in}^\simeq \to (\mathcal F\mathrm{in}^\simeq)^\mathrm{gp}\simeq \Omega^\infty(S)$ produces a symmetric monoidal functor $\Omega^\infty(S)\to \mathrm{Sp}$, and as such exhibits an $S$-graded $\mathbb E\_\infty$-ring.


But which one? To figure out which one, note that the passage to the "underlying $\mathbb E\_\infty$-ring" of an $S$-graded $\mathbb E\_\infty$-ring is given by passage to the colimit. Thus the underlying $\mathbb E\_\infty$-ring, which we have just adorned with an $S$-grading, is
$$
\varinjlim\_{\Omega^\infty(S)}\mathrm{LKan}^{\Omega^\infty(S)}\_{\mathcal{F}\mathrm{in}^\simeq}(S)\simeq \varinjlim\_{\mathcal F\mathrm{in}^\simeq}S,
$$
where we used that the left Kan extension of a functor does not change its colimit. Now we can use the explicit description of the groupoid of finite sets $\mathcal F\mathrm{in}^\simeq \simeq \coprod\_{n\ge 0}\mathrm B\Sigma\_n$ to obtain
$$
\varinjlim\_{\mathcal F\mathrm{in}^\simeq} S\simeq \bigoplus\_{n\ge 0}S\_{h\Sigma\_n}.
$$
We may recognize this as the free $\mathbb E\_\infty$-ring on a single generator $S\{t\}$, corresponding in terms of spectral algebraic geometry to the smooth affine line $\mathbf A^1$ (as opposed to the flat affine line $\mathbf A^1\_\flat = \mathrm{Spec}(S[t])$ for the polynomial $\mathbb E\_\infty$-ring $S[t]\simeq \bigoplus\_{n\ge 0}S$).


**2. $S$-grading on symmetric algebras**


The previous example can be easily generalized by observing that $\mathcal F\mathrm{in}^\simeq$ is the free $\mathbb E\_\infty$-space ( = symmetric monoidal $\infty$-groupoid) on a single generator. That means that a symmetric monoidal functor $\mathcal F\mathrm{in}^\simeq \to \mathrm{Sp}$ (which always factors through the maximal subgroupoid $\mathrm{Sp}^\simeq\subseteq\mathrm{Sp}$) is equivalent to the data of a spectrum $M\in \mathrm{Sp}$ (the image of the singleton set). The functor is given by sending a finite set $I$ to the smash product $M^{\otimes I}$, and is evidently symmetric monoidal. The same Kan extension game as before now gives rise to an $S$-graded $\mathbb E\_\infty$-ring spectrum, this time with underlying $\mathbb E\_\infty$-ring
$$
\mathrm{Sym}^\*(M)\simeq \bigoplus\_{n\ge 0}M^{\otimes n}\_{h\Sigma\_n},
$$
the free $\mathbb E\_\infty$-ring generated by $M$. We recover the prior situation by setting $M=S$. For $M = S^{\oplus n}$, we obtain an $S$-grading on $S\{t\_1, \ldots, t\_n\}$, corresponding to the spectral-algebro-geometric smooth affine $n$-space $\mathbf A^n$.


**3. $S$-grading on $S\{t^{\pm 1}\}$**


Another way to generalize the example of the $S$-grading on $S\{t\}$ is to take directly the constant functor $\Omega^\infty(S)\to \mathrm{Sp}$ with value $S$, instead of starting with a constant functor on $\mathcal F\mathrm{in}^\simeq$ and Kan-extending it. This produces a perfectly good $S$-graded $\mathbb E\_\infty$-ring, which let us denote $S\{t^{\pm 1}\}$.


From the group-completion relationship between $\Omega^\infty(S)$ and $\mathcal F\mathrm{in}^\simeq$, it may be deduced that $S\{t^{\pm 1}\}$ and $S\{t\}$ are related in terms of $\mathbb E\_\infty$-ring localization as $S\{t^{\pm 1}\}\simeq S\{t\}[t^{-1}]$, justifying our notation. Here we are localizing $S\{t\}$ at the element $t\in \mathbf Z[t] = \pi\_0(S\{t\})$. In terms of spectral algebraic geometry, this is encoding the spectral scheme $\mathrm{GL}\_1$, the smooth punctured line.


**4. Remark on non-negative grading**


In algebraic geometry, we often prefer to think about non-negatively graded commutative rings than graded commutative rings. Just as the latter are equivalent to lax symmetric monoidal functor $\mathbf Z\to\mathrm{Ab}$, so are the former equivalent to lax symmetric monoidal functors $\mathbf Z\_{\ge 0}\to \mathrm{Ab}$.


By analogy, the "non-negatively $S$-graded $\mathbb E\_\infty$-rings" are lax symmetric monoidal functors $\mathcal{F}\mathrm{in}^\simeq \to\mathrm{Sp}$. Indeed, just as $\mathbf Z\_{\ge 0}$ is the free commutative monoid on one generator, so is $\mathcal F\mathrm{in}^\simeq$ the free $\mathbb E\_\infty$-space on one generator. That is the reason why we were encountering such functors above (and Kan extending them along group completion, as we would to view as $\mathbf Z\_{\ge 0}$-grading as a special case of a $\mathbf Z$-grading).


**5. Some actual "non-tautological" examples though**


So far, a not-completely-unreasonable complaint might be that all the examples of $S$-graded $\mathbb E\_\infty$-rings were sort of tautological.


For a very non-tautological example, see the main result of [this paper of Hadrian Heine](https://arxiv.org/abs/1712.00521). It shows that there exists an $S$-graded $\mathbb E\_\infty$-ring spectrum such that its $S$-graded modules are equivalent to the $\infty$-category of cellular motivic spectra. In fact, much more is proved: this situation is very common, and under some not-too-harsh compact-dualizable-generation assumptions, a symmetric monoidal stable $\infty$-category will be equivalent to $S$-graded modules over some $S$-graded $\mathbb E\_\infty$-ring. So you may just as well view this result as a wellspring of potentially interesting examples of $S$-gradings "occurring in nature"! :)


