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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∞-arithmetic geometry under Spec(𝕊) at the end of differential cohesion and idelic structure. There one runs into “nonunital affine” E∞-varieties
(E∞Ringnu)opbeing 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∞Ringnu≃E∞Ring/𝕊between nonunital E∞-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∞-ring E, write E+→𝕊 for its image under
E∞Ringforget⟶E∞Ringnu≃⟶E∞Ring/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⟶E+⟶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 GL1 preserves ∞-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
GL1(E)⟶GL1(E+)⟶GL1(𝕊).This is at least similar to Sagave’s augmentation here. GL1(𝕊) is just 𝕊 restricted to the ℤ2-subgroup of π0(𝕊)=(ℤ,×).
In conclusion, I am beginning to wonder if the appearance of nonunital E∞-geometry in “arithmetic cohesion” is more of a virtue than a bug.
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