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You may remember that a good while back we had had lively discussion here on sphere-spectrum augmented infinity-groups of units, partly due to to the remarks at superalgebra – Abstract idea.
I was reminded of this when discussing the $E_\infty$-arithmetic geometry under $Spec(\mathbb{S})$ at the end of differential cohesion and idelic structure. There one runs into “nonunital affine” $E_\infty$-varieties
$(E_\infty Ring^{nu})^{op}$being the formal duals of nonunital E-infinity rings.
Now by Lurie’s vast generalization (here) of the elementary fact that Zhen Lin had kindly highligted in recent discussion, there is an equivalence
$E_\infty Ring^{nu} \simeq E_\infty Ring_{/\mathbb{S}}$between nonunital $E_\infty$-rings and sphere-spectrum augmented E-∞ rings, given by unitalization and by forming augmentation ideals, respectively.
This message here is about noticing the following:
given a unital $E_\infty$-ring $E$, write $E^+\to \mathbb{S}$ for its image under
$E_\infty Ring \stackrel{forget}{\longrightarrow} E_\infty Ring^{nu} \stackrel{\simeq}{\longrightarrow} E_\infty Ring_{/\mathbb{S}} \,.$By inspection of the proof of prop. 5.2.3.15 in Higher Algebra one finds that indeed there is a homotopy fiber sequence of underlying spectra
$E \longrightarrow E^+ \longrightarrow \mathbb{S}$(which is just the statement that indeed the inverse equivalence to unitalization is is given by forming augmentation ideals).
But now the ∞-group of units-functor $GL_1$ preserves $\infty$-limits in any case (being given by homotopy pullback of values of right adjoint functors) and so from the above we canonically have a homotopy fiber (and hence cofiber) sequence of abelian ∞-groups
$GL_1(E) \longrightarrow GL_1(E^+) \longrightarrow GL_1(\mathbb{S}) \,.$This is at least similar to Sagave’s augmentation here. $GL_1(\mathbb{S})$ is just $\mathbb{S}$ restricted to the $\mathbb{Z}_2$-subgroup of $\pi_0(\mathbb{S}) = (\mathbb{Z},\times)$.
In conclusion, I am beginning to wonder if the appearance of nonunital $E_\infty$-geometry in “arithmetic cohesion” is more of a virtue than a bug.
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