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

    (ERingnu)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

    ERingnuERing/𝕊

    between 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

    ERingforgetERingnuERing/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

    EE+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 2-subgroup of π0(𝕊)=(,×).

    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.