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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 22nd 2014
   • (edited Aug 22nd 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexYou may remember that a good while back we had had lively discussion here on [sphere-spectrum augmented infinity-groups of units](http://ncatlab.org/nlab/show/infinity-group+of+units#AugmentedDefinition), partly due to to the remarks at [superalgebra -- Abstract idea](http://ncatlab.org/nlab/show/super+algebra#AbstractIdea). 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](http://ncatlab.org/nlab/show/nonunital+Ek-algebra#RelationToAugmentedEkAlgebras)) 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](http://ncatlab.org/nlab/show/infinity-group+of+units#FiberSequenceForExtendedGl1). $GL_1(\mathbb{S})$ 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.

   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.