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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 13th 2014
    • (edited Aug 26th 2014)

    by the discussion here we have for each finite set of primes 𝔞\mathfrak{a} at least the top part of cohesion in affine E E_\infty-arithmetic geometry

    (Π 𝔞 𝔞):E Ring nu opE Ring nu op(\Pi_{\mathfrak{a}} \dashv \flat_{\mathfrak{a}}) \colon E_\infty Ring_{nu}^{op}\to E_\infty Ring_{nu}^{op}

    over formal duals of 𝔞\mathfrak{a}-torsion E E_\infty-rings.

    I have a decent geometric intuition of what 𝔞\flat_{\mathfrak{a}} does, namely the 𝔞\mathfrak{a}-adic completion that it encodes means picking in each E E_\infty-arithmetic space the collection of all formal neighbourhoods around all its points.

    On the other hand, I am presently lacking intuition as to what Π 𝔞\Pi_{\mathfrak{a}} is about. Of course the adjoint modality as such tells us that we are to think of this as forming fundamental \infty-groupoids/etale homotopy type relative not to points but to formal neighbourhoods. But what I am lacking intuition for presently is why that is given by forming 𝔞\mathfrak{a}-torsion approximation of nonunital E E_\infty-rings, as it is.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 27th 2014

    One thought:

    if we regard an elliptic curve CC as an abelian group hence as a \mathbb{Z}-module, then its pp-torsion approximation is the direct limit overs its groups C[p ν]C[p^\nu] of p νp^{\nu}-torsion points for ν\nu. A p νp^\nu-level structure on CC is an isomorphism /p ν×/p νC[p ν]\mathbb{Z}/{p^\nu} \times \mathbb{Z}/{p^\nu}\simeq C[p^\nu]. As ν\nu tends to infinity, this tends to the actual “geometric realization” in the sense of the fundamental group ×\mathbb{Z}\times\mathbb{Z} of a complex elliptic curve. Of course in a way this is rather the fundamental group of the dual elliptic curve.

    Might this be a hook into conceptually understanding how torsion approximation is analogous to geometric realization?