Perhaps the most common construction of the rational numbers is the one given by taking the field of fractions $\mathrm{Frac}(\mathbb{Z})\cong\mathbb{Q}$ of the ring $\mathbb{Z}$ of integers.


I'm wondering whether it's possible to do the same with the sphere spectrum $\mathbb{S}$, using the constructions given in [Lurie's *Higher Algebra*, Section 7.2.3](https://www.math.ias.edu/%7Elurie/papers/HA.pdf#subsection.7.2.3). Namely, in that section Lurie shows how to construct an $\mathbb{E}\_1$-ring spectrum $S^{-1}R$ starting from an $\mathbb{E}\_1$-ring spectrum $R$ and a subset $S$ of $\pi\_\*(R)$ consisting of homogeneous elements only and satisfying the left Ore condition ([HA 7.2.3.1](https://www.math.ias.edu/%7Elurie/papers/HA.pdf#theorem.7.2.3.1)).


**Question I.** What do we know about the ring spectrum $(\pi\_0(\mathbb{S})\setminus\{0\})^{-1}\mathbb{S}$? For instance, can we say anything useful about its homotopy groups¹? Is it also $\mathbb{E}\_\infty$? Does it arise naturally in other contexts in homotopy theory?


¹HA 7.2.3.19 and 7.2.3.20 seem relevant here: they seem to indicate (I don't understand the statements too well) that the elements $\pi\_\*((\pi\_0(\mathbb{S})\setminus\{0\})^{-1}\mathbb{S})$ might be of the form $[\nu]/k$ for $[\nu]$ in $\pi\_\*(\mathbb{S})$ and $k\in\mathbb{Z}\setminus\{0\}$.




---


Secondly, the subset $S$ of $\pi\_\*(\mathbb{S})$ given by all nonzero homogeneous elements does not satisfy the left Ore condition, so we cannot speak of the localisation of $\mathbb{S}$ at $S$. However, in the classical case it is possible to define the localisation of a noncommutative ring $R$ at an arbitrary subset $S$ of $R$ (see e.g. [MSE 177853](https://math.stackexchange.com/a/177853)), even if $S$ does not satisfy the left Ore condition or is not multiplicatively closed, although in this case $S^{-1}R$ usually behaves in a much worse way, with e.g. it being very hard to tell whether the canonical map $R\to S^{-1}R$ is injective or not.


**Question II.** Similarly to the classical case, do we have a notion of the localisation $S^{-1}R$ of an $\mathbb{E}\_1$-ring $R$ at an arbitrary subset $S$ of homogeneous elements of $\pi\_\*(R)$ (even if such a notion turns out to not be so well-behaved, again like in the classical case)? If so, what do we get when $R=\mathbb{S}$ and $S$ is the set of nonzero homogeneous elements of $\pi\_\*(\mathbb{S})$?



The “field of fractions” of the sphere spectrum (localization at $\pi_0(\mathbb{S})\setminus\{0\}$, the non-zero integers)
Z.M.'s first comment is correct. If $S \subset \Bbb Z \cong \pi\_0(\Bbb S)$ is a multiplicatively closed subset, there is a localization $S^{-1} \Bbb S$ whose homotopy groups lift the algebraic localization:
$$
\pi\_\*(S^{-1} \Bbb S) \cong S^{-1} \pi\_\*(\Bbb S)
$$
This localization (a "smashing localization" of the sphere) is a lift of Serre's algebraic localization techniques. These are all $E\_\infty$ rings.


In particular, if you localize the sphere spectrum at the set of nonzero elements in degree zero, you get the Eilenberg-Mac Lane spectrum $H\Bbb Q$ due to finiteness of the stable homotopy groups of spheres.


For elements in nonzero degree (where I do believe that there is a left Ore condition available due to graded-commutativity), you run into the Nishida nilpotence theorem:



> 
> Any positive-degree element in $\pi\_\* \Bbb S$ is nilpotent.
> 
> 
> 


This forces any ring spectrum that inverts a positive-degree element to be the zero ring.


