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$\mathbb{Z}$-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, 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, 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 $R_k$;

such that we 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?

$\mathbb{Z}$-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, 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, 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?

$\mathbb{Z}$-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 Ulrich–Nikolaus's Twisted differential cohomology, 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, 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 $R_k$;

such that we 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?

$\mathbb{Z}$-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, 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, 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 $R_k$;

such that we 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?

$\mathbb{Z}$-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 Ulrich–Nikolaus's Twisted differential cohomology, 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. Since $\mathsf{Ho}(\mathbb{S})\cong\tau_{\leq1}(\mathbb{S})$, 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 $R_k$;

such that the component $$\mu_{n,m}\colon R_n\otimes R_m \to R_{n+m}$$ of the multiplication map of $R_\bullet$ at $(n,m)$ is commutative if $nm$ is even, and otherwise is so up to the automorphism $\sigma_{n+m}$ in that we have $$ ba = \begin{cases} ab &\text{if $nm$ is even,}\\ \sigma_{n+m}(ab) &\text{if $nm$ is odd} \end{cases} $$ for each $a\in R_n$ and each $b\in R_m$.

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

$\mathbb{Z}$-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 Ulrich–Nikolaus's Twisted differential cohomology, 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, 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 $R_k$;

such that we 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?