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Split off p-adic integer from p-adic number.
Thanks!
To the section As formal neighbourhood of a prime (which I had started earlier) I have
added a pointer to Buium 13, where this perspective turns out to be highlighted;
included the infinitesimal and local - table, into which in turn I have added a row mentioning the $p$-adic integers.
Now, the natural thing to do would be to proceed as in synthetic differential geometry and define arithmetic jet spaces as (certain subspaces of) mapping spaces out of $Spec(\mathbb{Z}_p)$.
It seems that what Buium does is at least roughly like this (according to theorem 2.1 1) in the above). I am wondering why it’s not exactly like this, or if it secretly is.
or if it secretly is.
I hadn’t realised Joyal had worked on anything like this:
the work of Joyal [49] and Borger [4, 5] on the Witt functor; the Witt functor is a right adjoint to the forgetful functor from “δ-rings” to rings as opposed to the arithmetic jet functor which is a left adjoint to the same forgetful functor. As it is usually the case the left and right stories turn out to be rather diﬀerent. (Differential calculus with integers, p. 17, slight different version from one on Arxiv).
Interesting contrast on same page
…the work of Soul´e, Deitmar, Connes, Berkovich, and many others on the “geometry over the ﬁeld F1 with one element”. In their work passing from the geometry over Z to the geometry over F1 amounts to removing part of the structure deﬁning commutative rings, e.g. removing addition and hence considering multiplicative monoids instead of rings. On the contrary our theory can be seen as a tentative approach to F1 (cf. the Introduction to [16]) that passes from Z to F1 by adding structure to the commutative rings, speciﬁcally adding the operator(s) δp. This point of view was independently proposed (in a much more systematic form) by Borger [6]. Borger’s philosophy is global in the sense that it involves all the primes (instead of just one prime as in our work) and it also proposes to see “positivity” as the corresponding story at the “inﬁnite” prime; making our theory ﬁt into Borger’s larger picture is an intriguing challenge.
Presumably that last comment is about Borger’s Witt vectors, semirings, and total positivity
…because of this adelic ﬂavor, it is natural to ask whether the inﬁnite prime plays a role. The answer is yes, and the purpose of this chapter is to explore how it does, at least as related to positivity, which we will regard as p-adic integrality for the inﬁnite prime.
Thanks. Yes, as Borger nicely highlights sort of in between the lines of his introductions, his proposal for $\mathbb{F}_1$ stands out from the others in that it produces a rich realm of mathematics, instead just reformulating simple statements about finite sets in terms of $\mathbb{F}_1$-newspeak.
Also, it seems striking to me that his theory is all about abstracting the structure of Frobenius endomorphisms. That certainly sounds promising for a global picture of geometry that is to stand a chance to provide a unified perspective on zeta functions.
But I am still not sure if I understand why the exact definition Borger gives is “the right” one. I am all sympathetic to it, but I need to better understand what it achieves.
For instance why is the direct image $Et(Spec(\mathbb{Z})) \to Et(Spec(\mathbb{F}_1))$ given by just $(W_n)_\ast$ and not $(W_n)_\ast ( Spec(\mathbb{A}_{\mathbb{Z}})\underset{Spec(\mathbb{Z})}{\times} (-)) = J^n(-)$. The latter would fit better into my world-view.
I have added to p-adic integers the short exact sequence
$0 \to \mathbb{Z}_{(p)} \overset{p \cdot (-)}{\longrightarrow} \mathbb{Z}_{(p)} \longrightarrow \mathbb{Z}/p\mathbb{Z} \to 0 \,.$(here)
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