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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 23rd 2014
   • (edited Jul 23rd 2014)
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   Author: Urs
   Format: MarkdownItexcreated _[[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.)

   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.)

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 24th 2014
   • (edited Jul 24th 2014)
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   Author: Urs
   Format: MarkdownItexI 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](http://ncatlab.org/nlab/show/infinitesimal+cohesive+%28infinity%2C1%29-topos#FormalModuliProblems).

   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.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 24th 2014
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   Author: David_Corfield
   Format: MarkdownItexThe arrows in the diagram there aren't right, are they?

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

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJul 24th 2014
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   Author: Urs
   Format: MarkdownItexMaybe I am too rushed to see what you mean (and my battery is about to quit again). Do you mean [this diagram](http://ncatlab.org/nlab/show/infinitesimal+cohesive+%28infinity%2C1%29-topos#FormalModuliProblems)?

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

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 24th 2014
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   Author: David_Corfield
   Format: MarkdownItexI 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?

   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?

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJul 24th 2014
   • (edited Jul 24th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexNow 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.

   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.

7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeAug 3rd 2014
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   Author: zskoda
   Format: MarkdownItex6: 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.

   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.

8. • CommentRowNumber8.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 21st 2017
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   Author: David_Corfield
   Format: MarkdownItexI added a reference to Buium's new book: * [[Alexandru Buium]], _Foundations of arithmetic differential geometry_, 2017, AMS, Mathematical Surveys and Monographs Vol. 222, ([AMS](http://bookstore.ams.org/surv-222/), [Preface and Introduction](http://www.math.unm.edu/~buium/foundationsintro.pdf)). 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 $\mathbb{F}_1$.

   I added a reference to Buium’s new book:

   • Alexandru Buium, Foundations of arithmetic differential geometry, 2017, AMS, Mathematical Surveys and Monographs Vol. 222, (AMS, Preface and Introduction).

   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 $\mathbb{F}_1$.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJun 21st 2017
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   Author: Urs
   Format: MarkdownItexThanks! I have added more cross-links from other relevant pages.

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

10. • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 22nd 2017
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    Author: David_Corfield
    Format: MarkdownItexI added a reference for the generalisation of jet spaces to a finite set of primes.

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