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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. 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).

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), 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 $\pi_*(\mathbb{S})$?

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    $\begingroup$ I am not sure whether I am mistaken, but the first localization seems to be simply $\mathbb Q$, since higher homotopy groups of the sphere are finite. $\endgroup$
    – Z. M
    Commented Mar 31, 2023 at 6:28
  • $\begingroup$ @Z.M I'm not sure about the precise form of $(\pi_0(\mathbb{S})\setminus\{0\})^{-1}\mathbb{S}$, but I think we should be fine since we are inverting only the nonzero elements in $\pi_0(\mathbb{S})$, which are all non-torsion. HA 7.2.3.19 and 7.2.3.20 seem relevant here; e.g. I wonder if we have elements like $[\nu]/k$ in $\pi_1((\pi_0(\mathbb{S})\setminus)\mathbb{S})$. $\endgroup$
    – Emily
    Commented Mar 31, 2023 at 16:24
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    $\begingroup$ It is worth pointing out that this localization far predates Lurie's Higher Algebra. You can find it (that is, localizing a spectrum at a subset of homotopy elements) in writings of Bousfield from the 1970s, and Adams even earlier. $\endgroup$
    – David White
    Commented Apr 5, 2023 at 1:35

1 Answer 1

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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.

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  • $\begingroup$ Hi Tyler, thank you so much for your answer! I now see why the positive degree elements end up all vanishing. (That said it's such a shame though! I was really hoping there'd be a more interesting ring spectrum counterpart to $\mathbb{Q}$ than just $H\mathbb{Q}$ :/) $\endgroup$
    – Emily
    Commented Mar 31, 2023 at 22:16
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    $\begingroup$ @Emily There are other well-known localizations that still "invert things" but those things are no longer strictly represented in the coefficient ring; these are things like K(n) and T(n) localization in chromatic homotopy theory. $\endgroup$
    – Tyler Lawson
    Commented Apr 13, 2023 at 14:13
  • $\begingroup$ Oh I had never though about localization of spectra in that way, thanks! Also, this is probably a silly question, but is $L_{H\mathbb{Z}}$ (and in particular $L_{H\mathbb{Z}}\mathbb{S}$) anything interesting? (Also sorry if case the question ends up not making sense; it's been ages since I read about localization, and I haven't had a chance to properly learn it yet) $\endgroup$
    – Emily
    Commented Apr 13, 2023 at 21:31
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    $\begingroup$ @Emily Sorry for the late reply. The localization $L_{H\Bbb Z}$ doesn't do anything interesting to connective spectra like the sphere spectrum - they are already homotopy limits of their Postnikov towers, and you can use this to prove that they are local. But it has rather drastic effects on some important nonconnective spectra (the Morava K-theories and E-theories). $\endgroup$
    – Tyler Lawson
    Commented May 25, 2023 at 17:27
  • $\begingroup$ Oh, I see! Thank you for explaining it to me :) $\endgroup$
    – Emily
    Commented May 25, 2023 at 21:12

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