Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 22nd 2014
    • (edited Aug 22nd 2014)

    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 E_\infty-arithmetic geometry under Spec(𝕊)Spec(\mathbb{S}) at the end of differential cohesion and idelic structure. There one runs into “nonunital affine” E E_\infty-varieties

    (E Ring nu) op (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 Ring nuE Ring /𝕊 E_\infty Ring^{nu} \simeq E_\infty Ring_{/\mathbb{S}}

    between nonunital E 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 E_\infty-ring EE, write E +𝕊E^+\to \mathbb{S} for its image under

    E RingforgetE Ring nuE Ring /𝕊. 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. in Higher Algebra one finds that indeed there is a homotopy fiber sequence of underlying spectra

    EE +𝕊 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 1GL_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)GL 1(E +)GL 1(𝕊). GL_1(E) \longrightarrow GL_1(E^+) \longrightarrow GL_1(\mathbb{S}) \,.

    This is at least similar to Sagave’s augmentation here. GL 1(𝕊)GL_1(\mathbb{S}) is just 𝕊\mathbb{S} restricted to the 2\mathbb{Z}_2-subgroup of π 0(𝕊)=(,×)\pi_0(\mathbb{S}) = (\mathbb{Z},\times).

    In conclusion, I am beginning to wonder if the appearance of nonunital E E_\infty-geometry in “arithmetic cohesion” is more of a virtue than a bug.