Arithmetic Jet Spaces

• Arithmetic jet spaces are p-adic analogues of classical jet spaces that represent maps into truncated Witt vector rings and encode arithmetic differential structures.
• They are constructed via adjunctions linking jet algebras with Witt vector functors, employing methods such as π-derivations and compatible Frobenius lifts.
• These spaces have practical applications in arithmetic differential geometry, including the construction of filtered F-isocrystals and connections to perfectoid spaces.

Arithmetic jet spaces are the $\pi$-adic or $p$-adic analogues of ordinary jet spaces in differential geometry. In the standard arithmetic-differential setup, if $X$ is a $\pi$-formal scheme over a complete discrete valuation ring $R$ with uniformizer $\pi$, then its $n$-th arithmetic jet space $J^nX$ represents the functor

$B \longmapsto X\bigl(W_n(B)\bigr),$

where $W_n(B)$ denotes the ring of truncated $p$0-typical Witt vectors of length $p$1. In Buium’s formulation, these spaces encode arithmetic differential data through $p$2-derivations and lifts of Frobenius; in Borger’s formulation, they arise as the functors left adjoint to Witt-space functors on schemes and algebraic spaces. Recent work has emphasized that the jet tower carries an internal structure closely analogous to the classical Witt-vector package $p$3, especially on the kernel towers attached to formal group schemes (Mishra et al., 30 Jan 2026).

1. Arithmetic-differential framework

Arithmetic jet spaces are built over bases that carry a canonical lift of Frobenius. One standard setup begins with a Dedekind domain $p$4, a fixed maximal ideal $p$5, residue field $p$6 of cardinality $p$7, and the $p$8-adic completion $p$9 with uniformizer $X$0. In this setting, a lift of Frobenius $X$1 relative to a structure map $X$2 is an $X$3-algebra map satisfying

$X$4

Equivalently, one may specify a $X$5-derivation $X$6, with

$X$7

$X$8

and

$X$9

This is the basic language of arithmetic differential geometry (Borger et al., 2017).

The relative theory is organized by prolongation sequences. A prolongation sequence is a chain

$\pi$0

equipped at each stage with compatible $\pi$1-derivations, or equivalently compatible lifts of Frobenius. In the related $\pi$2-typical notation over a Dedekind domain $\pi$3 with nonzero prime $\pi$4, the same structure is expressed by $\pi$5-derivations, and the resulting theory is formulated relative to an arbitrary base prolongation sequence $\pi$6 rather than only over the constant base (Bertapelle et al., 2020).

Witt vectors are the representing objects underlying the theory. For $\pi$7-typical Witt vectors, the ghost map is determined by

$\pi$8

and the standard operators restriction, Frobenius, Verschiebung, and Teichmüller govern the algebra. Arithmetic jet spaces may therefore be viewed as the geometric incarnation of Witt-vector-valued points endowed with arithmetic differential structure (Bertapelle et al., 2020).

2. Definition, adjunction, and representability

For a $\pi$9-scheme $R$0, the arithmetic jet spaces $R$1 form the canonical prolongation sequence

$R$2

characterized by the universal property that every prolongation sequence $R$3 with a map $R$4 admits a canonical morphism $R$5. In affine form, if $R$6 and $R$7, then

$R$8

so $R$9 represents maps from $\pi$0 into truncated Witt vectors (Borger et al., 2017).

Buium’s affine construction uses jet algebras. If

$\pi$1

then the $\pi$2-th $\pi$3-jet algebra is

$\pi$4

A central comparison theorem states that $\pi$5 is left adjoint to the Witt-vector functor $\pi$6: $\pi$7 For affine $\pi$8, this identifies Borger’s and Buium’s jet spaces canonically: $\pi$9 The same work shows that the formal $n$0-completion of the algebraic jet space agrees with Buium’s formal jet space, and their special fibers coincide (Bertapelle et al., 2020).

Borger generalized the theory from $n$1-typical formal schemes of finite type to arbitrary algebraic spaces over a separated base $n$2, allowing an arbitrary finite set $n$3 of pairwise coprime supramaximal ideals. In that language, the Witt-space functor $n$4 on sheaves has a left adjoint $n$5, called the $n$6-typical arithmetic jet-space functor of length $n$7. For affine $n$8,

$n$9

and if $J^nX$0 is an algebraic space, then $J^nX$1 is again an algebraic space; if $J^nX$2 is a scheme, then $J^nX$3 is a scheme (Borger, 2010).

3. Projections, Frobenius, and prolongation on fibers

The Witt-vector truncation maps induce natural projections

$J^nX$4

while Witt-vector Frobenius induces morphisms

$J^nX$5

These maps are functorial in $J^nX$6, and together they endow the jet tower with the structure of a prolongation sequence (Mishra et al., 30 Jan 2026).

A subtle point is that ordinary Frobenius on $J^nX$7 does not in general preserve fibers over a point or section. To address this, one studies inverse systems of the form

$J^nX$8

for a prolongation sequence $J^nX$9. The main theorem of this theory is that such an inverse system admits a canonical lift of Frobenius, called the lateral Frobenius,

$B \longmapsto X\bigl(W_n(B)\bigr),$0

When $B \longmapsto X\bigl(W_n(B)\bigr),$1, this map becomes transparent on ghost components: $B \longmapsto X\bigl(W_n(B)\bigr),$2 The general case is obtained by reduction to affine space via equalizers and then by descent from affine étale covers (Borger et al., 2017).

For a smooth group scheme $B \longmapsto X\bigl(W_n(B)\bigr),$3, taking $B \longmapsto X\bigl(W_n(B)\bigr),$4 with the identity section $B \longmapsto X\bigl(W_n(B)\bigr),$5 gives

$B \longmapsto X\bigl(W_n(B)\bigr),$6

The lateral Frobenius then yields canonical maps

$B \longmapsto X\bigl(W_n(B)\bigr),$7

so the inverse system $B \longmapsto X\bigl(W_n(B)\bigr),$8 becomes a prolongation sequence. This construction is specifically needed because the ordinary Frobenius lift on $B \longmapsto X\bigl(W_n(B)\bigr),$9 does not preserve $W_n(B)$0 in general; otherwise $W_n(B)$1 itself would inherit a Frobenius lift, which is false for many group schemes (Borger et al., 2017).

4. Kernels of projection maps and shifted Witt vectors

A more recent development isolates the successive kernels

$W_n(B)$2

for a $W_n(B)$3-formal group scheme $W_n(B)$4, rather than only the kernel of $W_n(B)$5. These $W_n(B)$6 are again $W_n(B)$7-formal group schemes, and the fundamental theorem is that Frobenius on arithmetic jet spaces restricts to this kernel tower: $W_n(B)$8 The significance is that the sequence

$W_n(B)$9

behaves as a shifted version of the Witt-vector Frobenius tower (Mishra et al., 30 Jan 2026).

The mechanism is a system of shifted $p$00-typical Witt vectors. Ordinary truncated $p$01-typical Witt vectors have coordinates $p$02 and ghost components of the form

$p$03

up to normalization. The kernel tower is represented not by the full Witt vectors $p$04, but by a shifted family adapted to truncation kernels. The induced map

$p$05

is then shown to come from a natural ring homomorphism between shifted Witt-vector rings, functorial both in the base algebra and in the formal group $p$06 (Mishra et al., 30 Jan 2026).

The additive formal group is the model case. For $p$07,

$p$08

and the group law is Witt-vector addition. The successive kernels are affine formal spaces, and under the identification with shifted Witt-vector schemes the restricted Frobenius is simply multiplication by $p$09: $p$10 In ghost coordinates this becomes

$p$11

The resulting system $p$12 is therefore presented as a geometric generalization of the classical Witt system $p$13 (Mishra et al., 30 Jan 2026).

A precursor, explicitly marked as superseded in its abstract, introduced $p$14-shifted $p$15-typical Witt vectors $p$16, a lateral Frobenius $p$17, and the isomorphism

$p$18

thereby anticipating the later kernel-tower formalism (Saha, 2022).

5. Geometric descriptions and comparison theorems

Borger’s global theory gives arithmetic jet spaces a concrete geometry. The adjunction

$p$19

shows that arithmetic jets are precisely the objects representing maps out of Witt spaces. Correspondingly, Witt ghost maps give rise by adjunction to co-ghost maps

$p$20

Away from the primes in $p$21, the jet space is product-like: $p$22 Over the special fibers, however, the geometry is controlled by Frobenius (Borger, 2010).

In the single-prime first-order case, if $p$23 is smooth over $p$24 locally at $p$25 and $p$26 is the ideal sheaf cutting out the graph of Frobenius on the special fiber inside $p$27, then

$p$28

where $p$29 is the subalgebra generated by $p$30. More generally, under a flatness hypothesis one has the recursive affine-modification formula

$p$31

This describes arithmetic jet spaces as affine modifications of $p$32 governed by Frobenius on the special fiber (Borger, 2010).

Comparison with Greenberg transforms clarifies the relation to special fibers. Over an arbitrary prolongation sequence, the special fiber of the algebraic jet space agrees with the special fiber of Buium’s formal jet space. For a finite totally ramified extension $p$33 of $p$34, there is a natural morphism from the Greenberg transform $p$35 to the special fiber of the arithmetic jet space $p$36, and this induces an isomorphism on inverse perfections. In the unramified $p$37-typical case, the comparison is an actual isomorphism (Bertapelle et al., 2020).

6. Arithmetic jet spaces in current research and scope

Arithmetic jet spaces now function as source objects for further $p$38-adic structures. For an abelian scheme $p$39 over a $p$40-adically complete discrete valuation ring, arithmetic differential characters

$p$41

and the kernel tower

$p$42

are used to construct a filtered $p$43-isocrystal $p$44. As a filtered vector space, $p$45 admits a natural map to the de Rham cohomology of $p$46, but its Frobenius is induced from arithmetic differential theory—more precisely from the lateral Frobenius on the kernels of jet projections—and is not the usual crystalline Frobenius (Borger et al., 2017).

A different development attaches perfectoid spaces to smooth schemes and to $p$47-morphisms by starting from the infinite arithmetic jet algebra $p$48 of a smooth $p$49-scheme and passing to a Frobenius-colimit construction. The resulting perfectoid space $p$50 satisfies

$p$51

so its reduction modulo $p$52 is the perfection of the Greenberg transform. In this way, arithmetic jet spaces provide a bridge between Buium’s $p$53-geometry and perfectoid geometry (Buium et al., 2019).

The term “jet space” also appears in several unrelated settings, and the distinction is essential. Model filiform Carnot groups $p$54, with their projection-like splittings and Hausdorff-dimension theory, are geometric jet spaces of smooth real functions and do not concern arithmetic jet spaces in the sense of Buium or Borger (Jung, 2018). Likewise, derived jet and arc spaces are enhancements of the classical algebro-geometric jet functor in derived algebraic geometry, designed to recover classical jets in the smooth case and to record higher homotopy data for singular schemes; they are a different theory from arithmetic jet spaces built from Witt vectors and $p$55-derivations (Docampo et al., 9 Apr 2026). This suggests that “jet space” denotes a common formal pattern of prolongation, but arithmetic jet spaces are distinguished by their dependence on Frobenius lifts, Witt vectors, and arithmetic differential structure.