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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 23rd 2014
    • (edited Jul 23rd 2014)

    created arithmetic jet space, so far only highlighting the statement that at prime pp these are X×Spec()Spec( p)X \underset{Spec(\mathbb{Z})}{\times}Spec(\mathbb{Z}_p) (regarded so in Borger’s absolute geometry by applying the Witt ring construction (W n) *(W_n)_\ast to it).

    This is what I had hoped that the definition/characterization would be, so I am relieved. Because this is of course just the definition of synthetic differential geometry with Spec( p)Spec(\mathbb{Z}_p) regarded as the ppth abstract formal disk.

    Well, or at least this is what Buium defines. Borger instead calls (W n) *(W_n)_\ast itself already the arithmetic jet space functor. I am not sure yet if I follow that.

    I am hoping to realize the following: in ordinary differential geometry then synthetic differential infinity-groupoids is cohesive over “formal moduli problems” and here the flat modality \flat is exactly the analog of the above “jet space” construction, in that it evaluates everything on formal disks. Moreover, \flat canonically sits in a fracture suare together with the “cohesive rationalization” operation [Π dR(),][\Pi_{dR}(-),-] and hence plays exactly the role of the arithmetic fracture square, but in smooth geometry. I am hoping that Borger’s absolute geometry may be massaged into a cohesive structure over the base Et(Spec(𝔽 1))Et(Spec(\mathbb{F}_1)) that makes the cohesive fracture square reproduce the arithmetic one.

    If Borger’s absolute direct image were base change to Spec( p)Spec(\mathbb{Z}_p) followed by the Witt vector construction, then this would come really close to being true. Not sure what to make of it being just that Witt vector construction. Presently I have no real idea of what good that actually is (apart from giving any base topos for Et(Spec(Z))Et(Spec(Z)), fine, but why this one? Need to further think about it.)

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 24th 2014
    • (edited Jul 24th 2014)

    I didn’t really say the above well yet. Have now included at arithmetic jet space at least a brief remark and a pointer on how the construction of rings of Witt vectors is an arithmetic analog of formal power series, namely of p-adics.

    If I had time I would now dig deeply into this, since this means that the direct image

    Γ:Et(Spec())Et(Spec(𝔽 1)) \Gamma \colon Et(Spec(\mathbb{Z})) \to Et(Spec(\mathbb{F}_1))

    in Borger’s absolute geometry is analogous to the direct image

    Γ i:SynthDiffGrpdL Alg gen op \Gamma^i \colon SynthDiff\infty Grpd \longrightarrow L_\infty Alg^{op}_{gen}


    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 24th 2014

    The arrows in the diagram there aren’t right, are they?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 24th 2014

    Maybe I am too rushed to see what you mean (and my battery is about to quit again). Do you mean this diagram?

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 24th 2014

    I meant in arithmetic jet space, there is

    Therefore in the sense of synthetic differential geometry the pp-formal neighbourhood of any arithmetic scheme XX around a global point x:Spec()Xx \colon Spec(\mathbb{Z}) \to X is the space of lifts

    Spf( p) x^ X Spec(). \array{ Spf(\mathbb{Z}_p) && \stackrel{\hat x}{\longrightarrow}&& X \\ & \searrow && \swarrow \\ && Spec(\mathbb{Z}) } \,.

    So the two lower arrows should go the other way?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 24th 2014
    • (edited Jul 24th 2014)

    Now I see what you mean. I did mean it the way it’s displayed, but you are right that the actual lft would be expressed by reversing the arrows, or maybe better by adding arrows on top. I should improve that whole entry, it’s a bit of a hasty remark.

    But now I first need to take care if some offline bureaucracy annoyance.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeAug 3rd 2014

    6: I am totally confused – if you label x:Spec()Xx \colon Spec(\mathbb{Z}) \to X why this is not straightforwardly good in triangle diagram but works only as “arrow on top” ? I suggest that at least Idea section has a unique convention and then additional things be put in main part if you feel so.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 21st 2017

    I added a reference to Buium’s new book:

    So then thought to start a new page arithmetic differential geometry. I hadn’t realised that his approach diverges from Borger’s:

    The non-vanishing curvature in our theory also prevents our arithmetic differential geometry from directly fitting into Borger’s λ\lambda-ring framework [13] for 𝔽 1\mathbb{F}_1; indeed, roughly speaking, λ\lambda-ring structure leads to zero curvature. For each individual prime, though, our theory is consistent with Borger’s philosophy of 𝔽 1\mathbb{F}_1.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2017

    Thanks! I have added more cross-links from other relevant pages.

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 22nd 2017

    I added a reference for the generalisation of jet spaces to a finite set of primes.