nLab arithmetic jet space

Idea

An analog of jet spaces in arithmetic geometry.

Definition

Notice that the p-adic integers $\mathbb{Z}_p$ are (by the discussion at p-adic integer – as formal power series) the analog in arithmetic geometry of a formal power series ring (around the point $p \in$ Spec(Z)), hence their formal spectrum $Spf(\mathbb{Z}_p)$ is an incarnation in arithmetic geometry of an abstract formal disk.

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

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

Moreover the map that sends an commutative ring, hence an arithmetic variety, to its $p$-formal power series in this sense is the construction of the ring of Witt vectors ($p$-typical Witt vectors if one fixes one prime, and “big Witt vectors” if one considers all at once) - see e.g. Hartl 06, section 1.1.

The following definition says essentially this, but further sends the resulting space to F1-geometry in the sense of Borger's absolute geometry:

For $X= Spec(R)$ an affine scheme over Spec(Z) (hence the formal dual of a ring), then the arithmetic jet space of $X$ at prime $p$ is $(W_n)_\ast$ applied to the $p$-adic completion of $X$, where $(W_n)_\ast$ is the ring of Witt vectors-construction, the direct image of Borger's absolute geometry $Et(Spec(\mathbb{Z})) \to Et(Spec(\mathbb{F}_1))$.

The definition is originally due to (Buium 96, section 2, Buium 05, section 3.1), reviewed in (Buium 13, 1.2.3) as part of his arithmetic differential geometry program. The above formulation is in (Borger 10, (12.8.2)). Buium and Borger have also defined the notion of an arithmetic jet space for a finite set of primes in (BB09).

Related concepts

• arithmetic differential geometry

References

The original articles are

• Alexandru Buium, Geometry of $p$-jets. Duke Math. J., 82(2):349–367, 1996. (Euclid)

• Alexandru Buium, Arithmetic differential equations, volume 118 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005. (pdf)

Introduction and survey is in

• Alexandru Buium, Differential calculus with integers (arXiv:1308.5194, slightly differing pdf)

• Alexandru Buium, Lectures on arithmetic differential equations (pdf)

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

Discussion in the context of the function field analogy is in

• Urs Hartl, A Dictionary between Fontaine-Theory and its Analogue in Equal Characteristic (arXiv:math/0607182)

Discussion in the context of Borger's absolute geometry over F1 is in

• James Borger, The basic geometry of Witt vectors, II: Spaces (arXiv:1006.0092)

See also

• Alexandru Buium, Taylor Dupuy, Arithmetic differential equations on $GL_n$, I: differential cocycles (arXiv:1308.0748)

• James Borger, Alexandru Buium, Differential forms on arithmetic jet spaces, (arXiv:0908.2512)