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created arithmetic jet space, so far only highlighting the statement that at prime $p$ these are $X \underset{Spec(\mathbb{Z})}{\times}Spec(\mathbb{Z}_p)$ (regarded so in Borger’s absolute geometry by applying the Witt ring construction $(W_n)_\ast$ 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 $Spec(\mathbb{Z}_p)$ regarded as the $p$th abstract formal disk.
Well, or at least this is what Buium defines. Borger instead calls $(W_n)_\ast$ 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 $[\Pi_{dR}(-),-]$ 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 $Et(Spec(\mathbb{F}_1))$ that makes the cohesive fracture square reproduce the arithmetic one.
If Borger’s absolute direct image were base change to $Spec(\mathbb{Z}_p)$ 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 $Et(Spec(Z))$, fine, but why this one? Need to further think about it.)
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
$\Gamma \colon Et(Spec(\mathbb{Z})) \to Et(Spec(\mathbb{F}_1))$in Borger’s absolute geometry is analogous to the direct image
$\Gamma^i \colon SynthDiff\infty Grpd \longrightarrow L_\infty Alg^{op}_{gen}$here.
The arrows in the diagram there aren’t right, are they?
Maybe I am too rushed to see what you mean (and my battery is about to quit again). Do you mean this diagram?
I meant in arithmetic jet space, there is
Therefore in the sense of synthetic differential geometry the $p$-formal neighbourhood of any arithmetic scheme $X$ around a global point $x \colon Spec(\mathbb{Z}) \to X$ is the space of lifts
$\array{ Spf(\mathbb{Z}_p) && \stackrel{\hat x}{\longrightarrow}&& X \\ & \searrow && \swarrow \\ && Spec(\mathbb{Z}) } \,.$So the two lower arrows should go the other way?
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.
6: I am totally confused – if you label $x \colon Spec(\mathbb{Z}) \to X$ 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.
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 $\lambda$-ring framework [13] for $\mathbb{F}_1$; indeed, roughly speaking, $\lambda$-ring structure leads to zero curvature. For each individual prime, though, our theory is consistent with Borger’s philosophy of $\mathbb{F}_1$.
Thanks! I have added more cross-links from other relevant pages.
I added a reference for the generalisation of jet spaces to a finite set of primes.
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