# Canonical Witt formal scheme extensions and $p$-torsion groups

Alessandra Bertapelle1 1Università degli Studi di Padova, Dipartimento di
Matematica “Tullio Levi-Civita”, via Trieste 63, I-35121 Padova, Italy
<EMAIL_ADDRESS>,  Nicola Mazzari1<EMAIL_ADDRESS>and
Arnab Saha2 2Indian Institute of Technology Gandhinagar, Gujarat 382355
<EMAIL_ADDRESS>
###### Abstract.

We study the $n$-th arithmetic jet space of the $p$-torsion subgroup attached
to a smooth commutative formal group scheme. We show that the $n$-th jet space
above fits in the middle of a canonical short exact sequence between a power
of the formal scheme of Witt vectors of length $n$ and the $p$-torsion
subgroup we started with. This result generalizes a result of Buium on roots
of unity.

###### Key words and phrases:

Witt vectors, arithmetic jet spaces, $\pi$-derivation, group schemes

###### 2010 Mathematics Subject Classification:

Primary 13F35, 14L05, 14L15, 14B20, 14G20.

##  1\. Introduction

Buium in [8] introduced the theory of arithmetic jet spaces on (formal)
abelian schemes over $p$-adic rings and showed that the jet spaces of an
abelian scheme $A$ are naturally affine fibrations over $A$. Since then the
theory of arithmetic jet spaces has been developed in several articles such as
[1], [9], [12], [6], [7], [16], and has found remarkable applications in
diophantine geometry as in [8] and [13].

In this paper, we study the structure of the jet space functors associated to
the $p$-torsion subgroup $G[p^{\infty}]$ of a smooth commutative formal group
scheme $G$ over a fixed $p$-adic basis. Here we show that for any $n$, the
$n$-th jet space $J^{n}(G[p^{\infty}])$ is canonically an extension of
$G[p^{\infty}]$ by a power of the unipotent formal group scheme
$\widehat{\mathbb{W}}_{n-1}$, where $\widehat{\mathbb{W}}_{n-1}$ is
$\widehat{\mathbb{A}}^{n}$, the $n$-dimensional formal affine space endowed
with the group scheme structure of the additive Witt vectors of length $n$.
This generalizes results obtained by Buium in [11] for $G$ the multiplicative
group scheme.

Before stating our main result in detail, let us introduce some notation. Let
$K$ be a finite extension of $\mathbb{Q}_{p}$ with ramification index $e$,
uniformizer $\pi$ and ring of integers $\mathcal{O}$. We denote by $k$ the
residue field of $\mathcal{O}$ and let $q$ be its order. Then the identity map
of $\mathcal{O}$ is a lift of $q$-Frobenius. Fix a $\pi$-adically complete
$\pi$-torsion free $\mathcal{O}$-algebra $R$ with a lifting of Frobenius
$\phi$, i.e., an endomorphism of $R$ such that $\phi(r)-r^{q}\in\pi R$ for all
$r\in R$. As an example, consider the ring of restricted power series
$\mathcal{O}\langle x\rangle$ with $\phi$ the $\mathcal{O}$-algebra
endomorphism given by $\phi(x)=x^{q}$. Let $W_{n}$ be the functor of ramified
Witt vectors of length $n+1$ (following Borger’s convention, details in §
2.2).

Let $\mathbf{Nilp}_{R}$ be the category of $R$-algebras on which $\pi$ (or
equivalently, $p$) is nilpotent. Its opposite category is a site with respect
to the Zariski topology. Any adic $R$-algebra $A$ with ideal of definition $I$
containing $\pi$ gives rise to a sheaf (of sets)
$\operatorname{\mathrm{Spf}}(A)$ such that

$\operatorname{\mathrm{Spf}}(A)(B)=\varinjlim_{n}\mathrm{Hom}_{R}(A/I^{n},B)\
,$

for any $B$ in $\mathbf{Nilp}_{R}$. By a formal scheme over $R$ we mean a
sheaf on $\mathbf{Nilp}_{R}^{\rm op}$ admitting an open cover by open
subfunctors of the type $\operatorname{\mathrm{Spf}}(A)$ for $A$ as above.

Given a sheaf $X$ on $\mathbf{Nilp}_{R}^{\rm op}$ we define its $n$-th
$\pi$-jet by

(1.1) $J^{n}X(C)=X(W_{n}(C))$

for any $C\in\mathbf{Nilp}_{R}$. If $X$ is a formal scheme over $R$ the same
is $J^{n}X$ and it holds

(1.2)
$\mathrm{Hom}_{R}(\operatorname{\mathrm{Spf}}(C),J^{n}X)=\mathrm{Hom}_{R}(\operatorname{\mathrm{Spf}}(W_{n}(C)),X)$

[8], [4], [7], [2].

For a smooth commutative formal group scheme $G$ over $R$ there is a short
exact sequence of formal group schemes

(1.3) $0\rightarrow N^{n}G\rightarrow J^{n}G\rightarrow G\rightarrow 0,$

which we call the _canonical Witt formal scheme extension_ of $G$ because of
Theorem 3.18 below. Let $G[p^{\nu}]$ be the $p^{\nu}$-torsion formal subgroup
scheme of $G$ and let $G[p^{\infty}]$ denote the sheaf on
$\mathbf{Nilp}_{R}^{\rm op}$ such that
$G[p^{\infty}](C)=\varinjlim_{\nu}G[p^{\nu}](C)$, for any
$C\in\mathbf{Nilp}_{R}$. For each $\nu$ the closed immersions
$G[p^{\nu}]\hookrightarrow G[p^{\nu+1}]$ induce closed immersions of
$\pi$-jets $J^{n}(G[p^{\nu}])\hookrightarrow J^{n}(G[p^{\nu+1}])$ and

$J^{n}(G[p^{\infty}])=\varinjlim_{\nu}J^{n}(G[p^{\nu}])$

as sheaves on $\mathbf{Nilp}_{R}^{\rm op}$; see Lemma 4.2.

Consider the natural projection map $u\colon J^{n}(G[p^{\infty}])\rightarrow
G[p^{\infty}]$ and let $N^{n}(G[p^{\infty}])$ denote the kernel of $u$. Buium
in [11, Corollary 1.2] shows that if $e=1$, $p>2$ and $G$ is the formal
multiplicative group scheme over $R$, then the sheaf $N^{n}(G[p^{\infty}])$ is
representable by a formal $R$-scheme and is isomorphic to
$\widehat{\mathbb{A}}^{n}$, the $n$-dimensional affine space over
$\operatorname{\mathrm{Spf}}(R)$.

In this paper, we will enrich Buium’s result and extend it to any smooth
commutative formal $R$-group scheme $G$ of relative dimension $d\geq 1$. In
fact, in Theorem 3.18 we show that if $p\geq e+2$ the kernel
$N^{n}(G[p^{\infty}])$ is isomorphic to $(\widehat{\mathbb{W}}_{n-1})^{d}$
where $\widehat{\mathbb{W}}_{n-1}$ is $\widehat{\mathbb{A}}^{n}$ endowed with
the group structure of Witt vectors of length $n$. Further, we deduce the
following result (see Theorem 4.8).

######  Main Theorem.

Assume $p\geq e+2$. Given a smooth commutative formal group scheme $G$ of
relative dimension $d$ over $R$, for any positive integer $n$ the natural
morphism $J^{n}G\to G$ gives an exact sequence

$0\rightarrow(\widehat{\mathbb{W}}_{n-1})^{d}\rightarrow
J^{n}G[p^{\infty}]\rightarrow G[p^{\infty}]\rightarrow 0$

of sheaves on $\mathbf{Nilp}_{R}^{\rm op}$.

We remark that by Lemma 4.2 it is $J^{n}(G[p^{\infty}])=(J^{n}G)[p^{\infty}]$
as sheaves on $\mathbf{Nilp}_{R}^{\rm op}$; hence there is no possible
ambiguity in the above statement.

###  1.1. Plan of the paper

In Section 2 we recall the definition and properties of $\pi$-jets in the
setting of formal schemes, with particular attention to the adjunction between
jet algebras and Witt vectors (2.14).

In Section 3 we focus on the notion of shifted Witt vectors $W_{n}^{+}$,
introduced by [7], and show that $W_{n}^{+}$ induces an adjoint functor to
$N^{n}$ (Theorem 3.8). This is an important result, analogous to the
adjunction formula that involves $W_{n}$ and $J^{n}$ (1.2). Then we show that
given a smooth formal $R$-group scheme $G$ of dimension $d$, we have

  * i)

_For all $n>0$, $N^{n}G\simeq J^{n-1}(N^{1}G)$;_ see Theorem 3.15.

  * ii)

_Assume $p\geq e+2$. Then there is a natural isomorphism of formal group
schemes _ $N^{n}G\simeq(\widehat{\mathbb{W}}_{n-1})^{d}\ $; see Theorem 3.18.

The proof of the first fact reduces to a local computation in coordinates
which is detailed in the Appendix section. Another important ingredient is the
notion of _lateral Frobenius_ introduced in [7]. Both results i) and ii) are
generalized to the case of $m$-shifted Witt vectors in [17] by the third
author.

In Section 4 we apply the previous results to the study of the sheaves
$J^{n}(G[p^{\infty}])$ and $N^{n}(G[p^{\infty}])$ and deduce a statement
similar to Theorem 3.18 where $G$ is replaced by $G[p^{\infty}]$, see Theorem
4.8.

In this paper all rings are assumed to be commutative with unit and
${\mathbf{Alg}_{R}}$ denotes the category of $R$-algebras, i.e., of ring
homomorphisms $R\to B$.

##  2\. Arithmetic jets

###  2.1. Conventions

Let $R$ be the base ring fixed in the introduction. Given a formal scheme $X$
and a fixed point $x\colon\operatorname{\mathrm{Spf}}(R)\to X$, one can
consider the fibre of $x$ under the natural map $J^{n}X\to X$, which is the
closed formal subscheme
$N^{n}X=(J^{n}X)_{x}=J^{n}X\times_{X}\operatorname{\mathrm{Spf}}(R)$. Note
that if $G$ is a formal group scheme then $J^{n}G$ is naturally a formal group
scheme too and we set $N^{n}G=(J^{n}G)_{\varepsilon}=\ker(J^{n}G\to G)$ to be
the fibre along the unit section $\varepsilon$.

If $X$ is a functor on ${\mathbf{Alg}_{R}}$, we will usually denote by
$\widehat{X}$ the restriction of $X$ to $\mathbf{Nilp}_{R}$. Let $R\langle
x_{1},...,x_{n}\rangle$ be the $\pi$-adic completion of the $R$-polynomial
algebra in $n$ variables.

Let $\widehat{\mathbb{G}}_{\mathrm{a}}:=\operatorname{\mathrm{Spf}}(R\langle
x\rangle)$ be the additive formal group scheme over $R$. Note that this formal
group scheme should not be confused with the $(x)$-adic formal group
$\mathbb{G}_{a}^{\rm for}=\operatorname{\mathrm{Spf}}(R[[x]])$, the formal
completion of $\mathbb{G}_{a}$ along the zero section.

If $\mathcal{F}\in R[[\mathbf{x},{\bf y}]]$ is a commutative formal group law
of dimension $g$, let ${\mathcal{F}}\\{n\\}$ be the formal group law given by
$\pi^{-n}\mathcal{F}(\pi^{n}\mathbf{x},\pi^{n}{\bf y})$, for any $n\geq 1$.
Note that ${\mathcal{F}}\\{n\\}$ endows
$\operatorname{\mathrm{Spf}}(R\langle\mathbf{x}\rangle)$ with a structure of
formal group scheme over $R$.

If $B$ is an $R$-algebra, $\rho=\rho_{B}\colon R\to B$ always denote the
corresponding ring homomorphism. If the context is clear, we will write $r$ in
place of $\rho(r)\in B$.

###  2.2. Witt vectors over $R$

In the following pages $W_{n}$ denotes the functor of $\pi$-typical Witt
vectors of length $n+1$ on $R$-algebras. Hence, for any $R$-algebra $B$, the
ring $W_{n}(B)$ is always considered with its natural $R$-algebra structure,
which depends on $\phi$. We explain this briefly.

As functor on $\mathcal{O}$-algebras $W_{n}$ coincides with the so-called
functor of ramified Witt vectors of length $n+1$ (see [15], [3]). Let $w\colon
W_{n}(R)\rightarrow\prod_{i=0}^{n}R$ be the ghost map. Then for any Witt
vector ${\bf a}=(a_{0},\dots,a_{n})$, $w({\bf a})=(w_{0}({\bf
a}),\dots,w_{n}({\bf a}))$ where $w_{i}$ are the ghost polynomials

(2.1) $w_{i}=x_{0}^{q^{i}}+\pi x_{1}^{q^{i-1}}+\cdots+\pi^{i}x_{i}.$

Since $R$ has a lifting of Frobenius $\phi$, by the universal property of Witt
vectors there exists a ring homomorphism of $\mathcal{O}$-algebras
$\exp_{\delta}$ making the following diagram commute (see [2, (2.9)])

(2.2)
$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exp_{\delta}}$$\scriptstyle{(\phi^{0},\phi,\dots,\phi^{n})\
}$$\textstyle{W_{n}(R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w}$$\textstyle{\prod_{i=0}^{n}R}$

Let $B$ be an $R$-algebra. Then $W_{n}(B)$ is naturally endowed with the
$R$-algebra structure

$R\stackrel{{\scriptstyle\exp_{\delta}}}{{\longrightarrow}}W_{n}(R)\stackrel{{\scriptstyle
W_{n}(\rho_{B})}}{{\longrightarrow}}W_{n}(B).$

In [7, §3.2] the authors give an equivalent construction of the functor
$W_{n}$. The ghost map $w$ in (2.2) is $\mathcal{O}$-linear, but not
$R$-linear in general, if the ring $\prod_{i=0}^{n}R$ is endowed with the
direct product $R$-module structure. It is then preferable to change the
$R$-module structure on the product ring so that $w$ becomes $R$-linear. Let
${}^{\phi^{n}}\\!B$ denote the ring $B$ with the $R$-algebra structure induced
by $\rho_{B}\circ\phi^{n}\colon R\to B$, and let

(2.3) ${\prod}_{\phi}^{n}(B):=\prod_{i=0}^{n}({}^{\phi^{i}}\\!B)$

be the direct product algebra. Its underlying ring is $\prod_{i}B$ and there
is a commutative diagram of $R$-algebras

(2.4)
$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{({\rm
id},\phi,\dots,\phi^{n})\
}$$\textstyle{W_{n}(R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w}$$\scriptstyle{W_{n}(\rho_{B})}$$\textstyle{W_{n}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w}$$\textstyle{{\prod}_{\phi}^{n}(R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\prod_{i}(\rho_{B})}$$\textstyle{{\prod}_{\phi}^{n}(B)}$

Then Frobenius and Verschiebung maps can be described in terms of ghost
components as in the case of ramified Witt vectors, with caution when
considering the $R$-algebra structure. As for example, the Frobenius ring
homomorphism $F\colon W_{n}(B)\to W_{n-1}(B)$ described in terms of ghost
components as the left shift is $\phi$-semilinear. We prefer then to write it
as the homomorphism of $R$-algebras

(2.5) $F\colon W_{n}(B)\to W_{n-1}({}^{\phi}\\!B)$

corresponding to the homomorphism of $R$-algebras

(2.6)
$F_{w}\colon{\prod}_{\phi}^{n}\\!(B)\to{\prod}_{\phi}^{n{\text{--}}1}\\!({}^{\phi}\\!B),\quad(b_{0},\dots,b_{n})\mapsto(b_{1},\dots,b_{n}).$

Similarly, the Verschiebung map $V\colon W_{n}(B)\to W_{n+1}(B)$ is described
on ghost components as the right shift multiplied by $\pi$. Clearly it is
$\mathcal{O}$-linear but not $R$-linear in general. We prefer then to write it
as the homomorphism of $R$-modules

(2.7) $V\colon W_{n}({}^{\phi}\\!B)\to W_{n+1}(B)$

corresponding to the homomorphism of $R$-modules

(2.8)
$V_{w}\colon{\prod}_{\phi}^{n}\\!({}^{\phi}\\!B)\to{\prod}_{\phi}^{n{\text{+}}1}\\!(B),\quad(b_{0},\dots,b_{n})\mapsto(0,\pi
b_{0},\dots,\pi b_{n}).$

Then $FV$ is multiplication by $\pi$ on $W_{n}({}^{\phi}\\!B)$.

Since $\phi$ might not be invertible, one can not write $B$ in place of
${}^{\phi}\\!B$ in (2.5) and (2.7). However, since $\phi$ is the identity on
$\mathcal{O}$, the $\mathcal{O}$-module structure on $B$ and ${}^{\phi}\\!B$
are the same.

######  Remark 2.9.

If $B$ is a $\pi$-adic $R$-algebra (by this we mean $\pi$-adically complete
and separated) then the same is $W_{n}(B)$ for any $n$. The proof works as in
[19, Proposition 3].

###  2.3. Shifted Witt vectors

We recall the construction of $0$-shifted Witt vectors as introduced in [7]
and [17]. Here we simply refer to them as shifted Witt vectors. The general
theory of $m$-shifted Witt vectors is developed in [17].

Let $B$ be an $R$-algebra and set-theoretically define

(2.10) $\displaystyle W^{+}_{n}(B):=R\times_{B}W_{n}(B)\simeq R\times
W_{n-1}(B).$

Also define the product ring

${\prod}^{+}_{n}(B):=R\times\prod_{i=1}^{n}({}^{\phi^{i}}B)=R\times{\prod}_{\phi}^{n-1}({}^{\phi}\\!B),$

where ${\prod}_{\phi}^{n}(B)$ was introduced in (2.3). Note that there is an
isomorphism of $R$-algebras

${\prod}^{+}_{n}B:=R\times{\prod}_{\phi}^{n-1}({}^{\phi}\\!B)\simeq
R\times_{B}{\prod}_{\phi}^{n}(B)$

mapping $(r,b_{1},\dots,b_{n})$ to the element
$\left(r,(\rho(r),b_{1},\dots,b_{n})\right)$.

Define the _apriori_ set-theoretic map $w^{+}\colon
W^{+}_{n}(B)\rightarrow{\prod}^{+}_{n}(B)$ given by
$w^{+}(r,b_{1},\dots,b_{n})=(w_{0},\dots,w_{n})$ where

$w_{i}=r^{q^{i}}+\pi b_{1}^{q^{i-1}}+\dots+\pi^{i}b_{i},$

for all $i=0,\dots,n$. Then note that $W^{+}_{n}(B)$ naturally is endowed with
the Witt ring structure of addition and multiplication making $w^{+}$ a ring
homomorphism. Hence we have the following commutative diagram

(2.11)
$\textstyle{W^{+}_{n}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w^{+}}$$\scriptstyle{\square}$$\textstyle{{\prod}^{+}_{n}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
pr}}$$\scriptstyle{\square}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{W_{n}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w}$$\textstyle{{\prod}_{\phi}^{n}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
pr}_{0}}$$\textstyle{B}$

where ${\rm pr}_{0}$ is the projection onto the $0$-th component. The
$R$-algebra $W^{+}_{n}(B)$ was denoted by $\tilde{W}_{n}(B)$ in [7, §4] and by
$W_{0n}(B)$ in [17].

Since the lower horizontal arrows in (2.11) are homomorphisms of $R$-algebras,
the same are the upper horizontal arrows. Hence the left hand square in (2.11)
is a diagram of $R$-algebras and, up to the above identifications, it can be
illustrated as

(2.12) $\textstyle{R\times
W_{n-1}({}^{\phi}\\!B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w^{+}}$$\textstyle{R\times{\prod}_{\phi}^{n\text{--}1}({}^{\phi}\\!B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{n}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w}$$\textstyle{{\prod}_{\phi}^{n}(B)}$
$\textstyle{\big{(}r,(b_{1},\dots,b_{n})\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\big{(}r,(w_{1}(r,b_{.}),\dots,w_{n}(r,b_{.}))\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(r,b_{1},\dots,b_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(r,w_{1}(r,b_{.}),\dots,w_{n}(r,b_{.}))}$

where we have written $r$ in place of $\rho(r)$ in $B$ and $w_{i}$ are the
ghost polynomials in (2.1).

###  2.4. Prolongation sequences

For any formal schemes $Y$ and $Z$ over $\operatorname{\mathrm{Spf}}(R)$ we
say that $(u,\delta)\colon Z\rightarrow Y$ is a prolongation if $u\colon
Z\rightarrow Y$ is a morphism of formal schemes over
$\operatorname{\mathrm{Spf}}(R)$ and $\delta\colon\mathcal{O}_{Y}\rightarrow
u_{*}\mathcal{O}_{Z}$ is a $\pi$-derivation on the sheaves (cf. Appendix A).
Then a sequence of formal schemes $T^{*}=\\{T^{n}\\}_{n=0}^{\infty}$ is a
_prolongation sequence_ if for each $n$, $(u_{n},\delta_{n})\colon
T^{n+1}\rightarrow T^{n}$ is a prolongation of formal group schemes over
$\operatorname{\mathrm{Spf}}(R)$ satisfying
$u_{n-1}^{*}\circ\delta_{n}=\delta_{n-1}\circ u_{n}^{*}$ and making the
following diagram commute

$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{u_{*}\mathcal{O}_{Z}}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{\mathcal{O}_{Y}\
.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$

A morphism of prolongation sequences $T^{*}\to P^{*}$ is a system of morphisms
of formal schemes $f_{n}\colon T^{n}\to P^{n}$ that satisfy the expected
commutations: $f_{n}\circ u_{n}=u_{n}\circ f_{n+1}$ and
$f_{n}\circ\delta_{n}=\delta_{n}\circ f_{n+1}$. For each $n$, let
$S^{n}=\operatorname{\mathrm{Spf}}(R)$. Then the fixed $\pi$-derivation
$\delta$ on $R$ makes $S^{*}$ into a prolongation sequence. Let
$\mathcal{C}_{S^{*}}$ denote the category of prolongation sequences defined
over $S^{*}$.

###  2.5. Jet spaces

Given a formal scheme $X$ over $S^{0}=\operatorname{\mathrm{Spf}}(R)$, Buium
constructs the _canonical prolongation sequence_
$J^{*}X=\\{J^{n}X\\}_{n=0}^{\infty}$ where $J^{0}X=X$ and by [9, Proposition
1.1], $J^{*}X$ satisfies the following universal property: for any $T^{*}$ in
$\mathcal{C}_{S^{*}}$ it is

(2.13)
$\displaystyle\mathrm{Hom}_{S^{0}}(T^{0},X)=\mathrm{Hom}_{\mathcal{C}_{S^{*}}}(T^{*},J^{*}X)\
.$

Moreover, by [5] and [2] we have the following functorial description:

$J^{n}X(C)=X(W_{n}(C))$

for any $C\in\mathbf{Nilp}_{R}$. In particular in the affine case with
$X=\operatorname{\mathrm{Spf}}(A)$ and
$J^{n}X=\operatorname{\mathrm{Spf}}(J_{n}A)$, we have a natural adjunction

(2.14)
$\Theta\colon\mathrm{Hom}_{R}(J_{n}A,C)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{Hom}_{R}(A,W_{n}(C))$

such that $w_{0}\circ\Theta(g)=g\circ\iota$ with $\iota\colon A\to J_{n}A$ the
natural morphism.

Here we make the above adjunction explicit when
$X=\operatorname{\mathrm{Spf}}(R\langle x\rangle)$ is the formal affine line
over $\operatorname{\mathrm{Spf}}(R)$. Let $A=R\langle x\rangle$. Then
$J^{n}X=\operatorname{\mathrm{Spf}}(J_{n}A)$ and

(2.15) $J_{n}A=R\langle x,x^{\prime},\dots
x^{(n)}\rangle=R\langle\mathsf{p}_{0},\dots\mathsf{p}_{n}\rangle$

where $x,x^{\prime},\dots,x^{(n)}$ are the Buium-Joyal coordinates and
$\mathsf{p}_{0},\dots\mathsf{p}_{n}$ are the Witt coordinates and they satisfy
$\mathsf{p}_{0}=x$, $\mathsf{p}_{1}=x^{\prime}$ while the general relation
between the above two coordinate systems can be found in [2, Proposition
2.10].

If $g\in\mathrm{Hom}_{R}(J_{n}A,C)$, then
$\Theta(g)\in\mathrm{Hom}_{R}(A,W_{n}(C))$ is determined by by

$\Theta(g)(x)=(g(\mathsf{p}_{0}),\dots,g(\mathsf{p}_{n}))\ .$

Note that when $G=\widehat{\mathbb{G}}_{\mathrm{a}}$ we have the following
isomorphism of formal group schemes

(2.16) $J^{n}\widehat{\mathbb{G}}_{\mathrm{a}}=\widehat{\mathbb{W}}_{n}.$

where $\widehat{\mathbb{W}}_{n}$ is $\widehat{\mathbb{A}}^{n+1}$ endowed with
the additive group structure of Witt vectors of length $n+1$.

##  3\. The Kernel as a $\pi$-jet space

For any sheaf of groups $G$ on $\mathbf{Nilp}_{R}^{\rm op}$ one defines

$N^{n}G:=\ker(J^{n}G\stackrel{{\scriptstyle u}}{{\longrightarrow}}G)$

where $u$ is the natural morphism. Scope of this section is to take a closer
look at the kernel $N^{n}G$ in the case $G$ is representable by a smooth
formal scheme. Since the kernel is the fibre at the unit section, we will
first consider more general fibres.

Let $X$ be a smooth formal scheme over $R$ with a marked point $a$ and let
$u\colon
U\to\mathbb{A}=\operatorname{\mathrm{Spf}}(R\langle\mathbf{x}\rangle)$ be an
étale chart around $a$, where $\mathbf{x}$ denotes here a finite family of
indeterminates. Hence $U$ is an open affine formal subscheme of $X$, $u$ is
étale, $a$ factors through $u$ and $u\circ a$ is the zero section $0$ of the
affine space $\mathbb{A}$. By [10, Proposition 3.13 & Corollary 3.16] (see
also [2, Proposition 3.12]), we have $J^{n}U\simeq
J^{n}\mathbb{A}\times_{\mathbb{A}}U$ for all $n$ and hence

(3.1)
$N^{n}X=J^{n}X\times_{X,a}\operatorname{\mathrm{Spf}}(R)=J^{n}U\times_{U,a}\operatorname{\mathrm{Spf}}(R)=J^{n}\mathbb{A}\times_{\mathbb{A},0}\operatorname{\mathrm{Spf}}(R)=N^{n}\mathbb{A}\
;$

in particular, $N^{n}X$ is formal affine and isomorphic to $N^{n}U$. Up to
shrinking $U$, we may assume $U=\operatorname{\mathrm{Spf}}(A)$ with $A$ a
$\pi$-adically complete separated $R$-algebra. Then the point
$a\colon\operatorname{\mathrm{Spf}}(R)\to U$ induces an $R$-algebra morphism
$\varepsilon\colon A\rightarrow R$. Let
$J^{n}U=\operatorname{\mathrm{Spf}}(J_{n}A)$. It is immediate to check that

(3.2)
$N^{n}U=\operatorname{\mathrm{Spf}}(J_{n}A\hat{\otimes}_{A,\varepsilon}R).$

###  3.1. Adjunction

Let ${\mathbf{Alg}^{+}_{R}}$ denote the category of augmented (commutative)
$R$-algebras. Its objects are commutative $R$-algebras $A$ together with an
augmentation $\varepsilon$, i.e. an $R$-algebra morphism $\varepsilon\colon
A\to R$; morphisms in ${\mathbf{Alg}^{+}_{R}}$ are morphisms of $R$-algebras
$h\colon A_{1}\to A_{2}$ respecting augmentations, i.e. $\varepsilon_{2}\circ
h=\varepsilon_{1}$. For any $(A,\varepsilon)$ in ${\mathbf{Alg}^{+}_{R}}$ we
define the $R$-algebra

(3.3) $N_{n}A:=J_{n}A\otimes_{A,\varepsilon}R.$

Note that shifted Witt vectors yield objects in ${\mathbf{Alg}^{+}_{R}}$.
Indeed, let $B$ be an $R$-algebra and let $w_{0}^{+}\colon W_{n}^{+}(B)\to R$
denote the projection onto the first component, i.e., the composition of the
upper horizontal arrows in (2.11). Then $W_{n}^{+}(B)$ together with
$w_{0}^{+}$ is an augmented $R$-algebra. Hence the above construction defines
a functor

$W_{n}^{+}\colon{\mathbf{Alg}_{R}}\longrightarrow{\mathbf{Alg}^{+}_{R}}\
,\quad B\mapsto(W_{n}^{+}(B),w_{0}^{+})\ ,$

on the category of $R$-algebras.

We now prove a key result: $N_{n}$ and $W_{n}^{+}$ is a pair of adjoint
functors.

######  Theorem 3.4.

For any augmented $R$-algebra $(A,\varepsilon)$ and any $R$-algebra $B$ there
is a natural bijection

(3.5)
$\Theta^{+}\colon\mathrm{Hom}_{R}(N_{n}A,B)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{Hom}_{{\mathbf{Alg}^{+}_{R}}}(A,W^{+}_{n}(B))\
.$

###### Proof.

Let $\iota\colon A\to J_{n}A$ denote the natural morphism. Then

$\displaystyle\mathrm{Hom}_{R}(N_{n}A,B)=$
$\displaystyle\\{g\in\mathrm{Hom}_{R}(J_{n}A,B)\ |\
g\circ\iota=\rho_{B}\circ\varepsilon\\}$ $\displaystyle=$
$\displaystyle\\{f\in\mathrm{Hom}_{R}(A,W_{n}(B))\ |\ w_{0}\circ
f=\rho_{B}\circ\varepsilon\\}$ $\displaystyle=$
$\displaystyle\\{f^{+}\in\mathrm{Hom}_{R}(A,W^{+}_{n}(B))\ |\ w_{0}^{+}\circ
f^{+}=\varepsilon\\}$ $\displaystyle=$
$\displaystyle\mathrm{Hom}_{{\mathbf{Alg}^{+}_{R}}}(A,W^{+}_{n}(B)),$

where the first equality follows by (3.3), the second by (2.14) taking
$f=\Theta(g)$, the third by definition of $W_{n}^{+}(B)$ in (2.10). ∎

By Remark 2.9 an analogous adjunction holds when working with the category of
augmented formal $R$-algebras ${\mathbf{fAlg}^{+}_{R}}$. We make this fact
explicit.

######  Example 3.6.

Let $A=R\langle x\rangle$ and $\varepsilon(x)=0$. By (2.15)

$N_{n}A=J_{n}A\hat{\otimes}_{A,\varepsilon}R=R\langle
x^{\prime},\dots,x^{(n)}\rangle=R\langle\mathsf{p}_{1}^{+},\dots,\mathsf{p}_{n}^{+}\rangle,$

where $\mathsf{p}_{i}^{+}\in R[x^{\prime},\dots,x^{(n)}]$ denotes the
polynomial $\mathsf{p}_{i}\in R[x,x^{\prime},\dots,x^{(n)}]$ evaluated at
$x=0$. Then the formal counterpart of (3.5) works as follow: given
$g\in\mathrm{Hom}_{R}(N_{n}A,B)$, then $\Theta^{+}(g)$ maps $x$ to
$(0,g(\mathsf{p}_{1}^{+}),\dots,g(\mathsf{p}_{n}^{+}))$.

The higher dimensional case is analogous. Let $A=R\langle\mathbf{x}\rangle$
with $\mathbf{x}$ a collection of $r$ indeterminates $\\{x_{1},\dots,x_{r}\\}$
and let $\varepsilon$ be the zero section. Then $J_{n}A\simeq
R\langle\mathbf{p}_{0},\mathbf{p}_{1},\dots,\mathbf{p}_{n}\rangle$ where
$\mathbf{p}_{i}$ denotes a collection of polynomials
$\\{\mathsf{p}_{i,1},\dots,\mathsf{p}_{i,r}\\}$ and $\mathsf{p}_{i,j}\in
R[x_{j},x_{j}^{\prime},...,x_{j}^{(n)}]$ plays the role of $\mathsf{p}_{i}$ in
(2.15). Then

(3.7) $N_{n}A\simeq
R\langle\mathbf{p}_{0},\dots\mathbf{p}_{n}\rangle{\hat{\otimes}}_{R\langle\mathbf{x}\rangle,\varepsilon}\
R\simeq R\langle\mathbf{p}_{1}^{+},\dots,\mathbf{p}_{n}^{+}\rangle,$

where $\mathbf{p}_{i}^{+}$ denotes the collection of polynomials
$\\{\mathsf{p}_{i,1}^{+},\dots,\mathsf{p}_{i,r}^{+}\\}$ with
$\mathsf{p}_{i,j}^{+}$ obtained by evaluating $\mathsf{p}_{i,j}$ at $x_{j}=0$.
Finally for a homomorphism $g\colon N_{n}A\to B$, $\Theta^{+}(g)$ maps $x_{i}$
to $(0,g(\mathsf{p}_{1,i}^{+}),\dots,g(\mathsf{p}_{n,i}^{+}))$.

We can now describe the functor $N^{n}X$ on $R$-algebras as done in (1.2) for
$J^{n}X$.

######  Theorem 3.8.

Let $X$ be a smooth formal scheme over $R$ with a marked point $x$ and let $B$
be in $\mathbf{Nilp}_{R}$. Then

$N^{n}X(B)=\mathrm{Hom}_{R\text{\rm-
pt}}(\operatorname{\mathrm{Spf}}(W_{n}^{+}(B)),X),$

where on the right we are considering morphisms of $R$-pointed formal schemes.

###### Proof.

The result is clearly true if $X$ is affine by Theorem 3.4. For the general
case, assume first that $X$ is an $R$-scheme and consider the following
diagram

$\textstyle{\operatorname{\mathrm{Spec}}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{\rho}$$\scriptstyle{w_{0}}$$\textstyle{\operatorname{\mathrm{Spec}}(W_{n}(B))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\operatorname{\mathrm{Spec}}(R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x}$$\scriptstyle{w^{+}_{0}}$$\textstyle{\operatorname{\mathrm{Spec}}(W_{n}^{+}(B))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{+}}$$\textstyle{J^{n}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{X}$

where $w_{0}$ is induced by the projection on the first component on algebras,
$\rho$ is the structure morphism and $u$ is the natural map. Note that
$\operatorname{\mathrm{Spec}}(W_{n}^{+}(B))$ is the push-out of $w,\rho$ in
the category of _all_ schemes [18, 07RS]. Then $N^{n}X(B)=X(W_{n}^{+}(B))$.
Indeed,

$\displaystyle N^{n}X(B)=$
$\displaystyle\\{g\in\mathrm{Hom}_{R}(\operatorname{\mathrm{Spec}}(B),J^{n}X)\
|\ u\circ g=x\circ\rho\\}$ $\displaystyle=$
$\displaystyle\\{f\in\mathrm{Hom}_{R}(\operatorname{\mathrm{Spec}}(W_{n}(B)),X)\
|\ f\circ w_{0}=x\circ\rho\\}$ $\displaystyle=$
$\displaystyle\\{f^{+}\in\mathrm{Hom}_{R}(\operatorname{\mathrm{Spec}}(W_{n}^{+}(B)),X)\
|\ f^{+}\circ w^{+}_{0}=x\\}$ $\displaystyle=$
$\displaystyle\mathrm{Hom}_{R\text{\rm-
pt}}(\operatorname{\mathrm{Spec}}(W_{n}^{+}(B)),X).$

If $X$ is a formal scheme, then the above holds for all schemes
$X\times\operatorname{\mathrm{Spec}}R/(\pi^{m})$ and thus one concludes. ∎

###  3.2. A special case

Let $G$ be a formal group scheme over $R$ and denote by $G^{\mathrm{for}}$ the
formal completion of $G$ along the unit section. Let $\mathcal{F}\in
R[\\![x_{1},\dots,x_{r},y_{1},\dots,y_{r}]\\!]$ be the formal group law on
$G^{\mathrm{for}}$, ${\mathcal{F}}^{\phi}$ the one obtained by acting on the
coefficients of $\mathcal{F}$ by $\phi$ and
${\mathcal{F}}^{\phi}\\{1\\}:=\pi^{-1}\mathcal{F}^{\phi}(\pi\mathbf{x},\pi{\bf
y})$, where $\pi\mathbf{x}:=(\pi x_{1},\dots,\pi x_{r})$. By [8, Lemma 2.2] it
is $N^{1}G\simeq{\mathcal{F}}^{\phi}\\{1\\}$ as formal $R$-schemes. We give
below a direct computation of this fact.

######  Lemma 3.9.

Let the notation be as above. Then the formal group law on the formal
completion of $N^{1}G$ at the origin is isomorphic to
${\mathcal{F}}^{\phi}\\{1\\}$.

###### Proof.

As seen in Remark 3.6, we may write
$G^{\mathrm{for}}=\operatorname{\mathrm{Spf}}(R[\\![\mathbf{x}]\\!])$ and
$N^{1}G=\operatorname{\mathrm{Spf}}(R\langle\mathbf{p}_{1}^{+}\rangle)=\operatorname{\mathrm{Spf}}(R\langle\mathbf{x}^{\prime}\rangle)$.
Let $\delta\colon R[\\![\mathbf{x},{\bf y}]\\!]\to R[\\![\mathbf{x},{\bf
y},\mathbf{x}^{\prime},{\bf y}^{\prime}]\\!]$ be the $\pi$-derivation
compatible with that of $R$ and such that
$\delta(\mathbf{x})=\mathbf{x}^{\prime}$, $\delta({\bf y})={\bf y}^{\prime}$.
If $\mathcal{F}(\mathbf{x},{\bf y})$ is the formal group law of $G$, the
formal group law of $N^{1}G$ is $\delta(\mathcal{F}(\mathbf{x},{\bf y}))$
evaluated at $\mathbf{x}=\bm{0},{\bf y}=\bm{0}$. Write
$\mathcal{F}(\mathbf{x},{\bf
y})=\sum_{\alpha,\beta}a_{\alpha,\beta}\mathbf{x}^{\alpha}{\bf y}^{\beta}$
with $\alpha,\beta$ varying in $\mathbb{N}^{r}\smallsetminus\\{\bm{0}\\}$,
$\mathbf{x}^{\alpha}:=x_{1}^{\alpha_{1}}x_{1}^{\alpha_{2}}\dots
x_{r}^{\alpha_{r}}$, and the coefficients of the monomials of degree $1$ equal
to $1$. By induction, applying the usual rules of $\pi$-derivations, one
checks that

$\delta(\mathcal{F}(\mathbf{x},{\bf y}))_{|\mathbf{x}=\bm{0},{\bf
y}=\bm{0}}=\sum_{\alpha,\beta}\delta(a_{\alpha,\beta}\mathbf{x}^{\alpha}{\bf
y}^{\beta})_{|\mathbf{x}=\bm{0},{\bf y}=\bm{0}}=\\\
\sum_{\alpha,\beta}\pi^{-1}\phi(a_{\alpha,\beta})(\pi\mathbf{x}^{\prime})^{\alpha}(\pi{\bf
y}^{\prime})^{\beta}=\pi^{-1}\mathcal{F}^{\phi}(\pi\mathbf{x},\pi{\bf
y})=\mathcal{F}^{\phi}\\{1\\}.$

∎

###  3.3. Lateral Frobenius

Let $X$ be a formal $R$-scheme. As $n$ varies, the $\pi$-jet spaces $J^{n}X$
form an inverse system of formal schemes and, more precisely, a prolongation
sequence, whence a lifting of Frobenius $\phi_{J}$ exists on the limit.
Clearly, the transition maps $u=u^{n+1}_{n}\colon J^{n+1}X\to J^{n}X$ induce
homomorphisms $N^{n+1}X\to N^{n}X$, but the image of $\phi$ restricted to
$N^{n+1}X$ is not necessarily contained in $N^{n}X$ and hence $\phi$ does not
induce a lifting of Frobenius on the sequence of the kernels. For this reason,
the notion of _lateral Frobenius_ was introduced and studied in [7, 17].

On shifted Witt vectors, the lateral Frobenius $F^{+}$ is defined as the
homomorphism of $R$-algebras making the following diagram

(3.10)
$\textstyle{W^{+}_{n}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F^{+}}$$\textstyle{W^{+}_{n-1}({}^{\phi}\\!B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R\times
W_{n-1}({}^{\phi}\\!B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
id}\times F}$$\textstyle{R\times W_{n-2}({}^{\phi^{2}}\\!B),}$

commute, where vertical identifications are meant as sets, and $F$ is the
Frobenius on Witt vectors recalled in (2.5). The homomorphism $F^{+}$ then
corresponds to the homomorphism of $R$-algebras

(3.11) $F_{w}^{+}\colon
R\times{\prod}_{\phi}^{n{\text{--}}1}\\!({}^{\phi}\\!B)\to
R\times{\prod}_{\phi}^{n{\text{--}}2}\\!({}^{\phi^{2}}\\!B),\quad(r,b_{1},\dots,b_{n})\mapsto(r,b_{2},\dots,b_{n})$

via ghost map, i.e., it makes the following diagram

(3.12)
$\textstyle{W^{+}_{n}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w^{+}}$$\scriptstyle{F^{+}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\prod}^{+}_{n}(B)=R\times{\prod}_{\phi}^{n{\text{--}}1}\\!({}^{\phi}\\!B)}$$\scriptstyle{F^{+}_{w}}$$\textstyle{W^{+}_{n-1}({}^{\phi}\\!B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w^{+}}$$\textstyle{{\prod}^{+}_{n{\text{--}}1}\\!({}^{\phi}\\!B)=R\times{\prod}_{\phi}^{n{\text{--}}2}\\!({}^{\phi^{2}}\\!B)}$

commute. Then, the homomorphism $F^{+}$ induces via (2.14) a natural morphism

$\mathfrak{f}\colon N^{n}X\to N^{n-1}X$

called again _lateral Frobenius_. It is showed in [7, Theorem 4.3] that
$\mathfrak{f}$ is a lift of Frobenius and satisfies

$\phi\circ\phi\circ u=\phi\circ u\circ\mathfrak{f},$

where $u$ denotes the immersion $N^{m}X\to J^{m}X$ and $\phi$ denotes the
Frobenius morphism $J^{m}X\to J^{m-1}X$ for any $m$.

######  Remark 3.13.

For later use, note that the element $(0,b_{.})=(0,b_{1},\dots,b_{n})\in
W^{+}_{n}(B)$ traces in (3.12) the following images

$\textstyle{(0,b_{1},\dots,b_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(0,\pi
b_{1},\pi b_{1}^{q}+\pi^{2}b_{2},\dots,\pi
w_{n-1}(b_{.}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(0,c_{1},\dots,c_{n-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(0,\pi
b_{1}^{q}+\pi^{2}b_{2},\dots,\pi w_{n-1}(b_{.}))\ .}$

Hence $\pi c_{1}=\pi b_{1}^{q}+\pi^{2}b_{2}$ implies $c_{1}=b_{1}^{q}+\pi
b_{2}$ (for $B$ without $\pi$-torsion and hence for any $R$-algebra $B$ by
standard arguments). By recursion one sees

$(F^{+})^{i}(0,b_{1},\dots,b_{n})=(0,b_{1}^{q^{i}}+\pi
b_{2}^{q^{i-1}}+\dots+\pi^{i}b_{i},\dots)\in W^{+}_{n-i}(B)$

for any $i<n$.

If $G$ is smooth over $R$, the same are $J^{n}G$ and $N^{n}G$ for all $n$. As
seen in the previous section, $N^{n}G=\operatorname{\mathrm{Spf}}(N_{n}A)$ is
an affine space over $R$. In particular the $R$-algebras $N_{n}A$ are
$\pi$-torsion free and therefore the lateral Frobenius homomorphisms
$\mathfrak{f}^{*}\colon N_{n}A\rightarrow N_{n+1}A$ induce a unique
$\pi$-derivation $\Delta$ on the prolongation sequence
$N_{*}A:=\\{N_{n}A\\}_{n=1}^{\infty}$.

In order to describe $N^{n}$ as a jet functor we need a preparation lemma.

######  Lemma 3.14.

Let $\mathbb{A}=\operatorname{\mathrm{Spf}}(R\langle\mathbf{x}\rangle)$ with
$\mathbf{x}$ a collection of $r$ indeterminates, and choose the origin as
marked point. Let $\mathfrak{f}^{i}\colon N^{n}\mathbb{A}\to
N^{n-i}\mathbb{A}$ denote the $i$-th fold composition of lateral Frobenius for
any $i\leq n$. Then $\mathfrak{f}^{i}$ induces an homomorphism of $\pi$-adic
$R$-algebras

$(\mathfrak{f}^{i})^{*}\colon
R\langle\mathbf{p}^{+}_{1},\dots,\mathbf{p}^{+}_{n-i}\rangle\longrightarrow
R\langle\mathbf{p}^{+}_{1},\dots,\mathbf{p}^{+}_{n}\rangle$

such that

$(\mathfrak{f}^{i})^{*}(\mathbf{p}^{+}_{1})=(\mathbf{p}^{+}_{1})^{q^{i-1}}+\pi(\mathbf{p}^{+}_{2})^{q^{i-2}}+\cdots+\pi^{i-1}\mathbf{p}^{+}_{i}.$

###### Proof.

Recall from Example 3.6 that
$N^{n}\mathbb{A}=\operatorname{\mathrm{Spf}}(N_{n}R\langle\mathbf{x}\rangle)\simeq\operatorname{\mathrm{Spf}}(R\langle\mathbf{p}^{+}_{1},\dots,\mathbf{p}^{+}_{n}\rangle)$.
Then by definition of $\mathfrak{f}$ and (3.5) with
$B=N_{n}R\langle\mathbf{x}\rangle$ we have a commutative diagram of rings

$\textstyle{\mathrm{Hom}_{R}(R\langle\mathbf{p}^{+}_{1},\dots,\mathbf{p}^{+}_{n}\rangle,R\langle\mathbf{p}^{+}_{1},\dots,\mathbf{p}^{+}_{n}\rangle)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-\circ(\mathfrak{f}^{*})^{i}}$$\scriptstyle{\Theta^{+}}$$\scriptstyle{\sim}$$\textstyle{\mathrm{Hom}_{{\mathbf{fAlg}^{+}_{R}}}(R\langle\mathbf{x}\rangle,W^{+}_{n}(R\langle\mathbf{p}^{+}_{1},\dots,\mathbf{p}^{+}_{n}\rangle))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(F^{+})^{i}\circ-}$$\textstyle{\mathrm{Hom}_{R}(R\langle\mathbf{p}^{+}_{1},\dots,\mathbf{p}^{+}_{n-i}\rangle,R\langle\mathbf{p}^{+}_{1},\dots,\mathbf{p}^{+}_{n}\rangle)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta^{+}}$$\scriptstyle{\sim}$$\textstyle{\mathrm{Hom}_{{\mathbf{fAlg}^{+}_{R}}}(R\langle\mathbf{x}\rangle,W^{+}_{n-i}(R\langle\mathbf{p}^{+}_{1},\dots,\mathbf{p}^{+}_{n}\rangle))}$

By Remark 3.13 the identity map on
$R\langle\mathbf{p}^{+}_{1},\dots,\mathbf{p}^{+}_{n}\rangle$ traces the
following images

$\textstyle{{\rm
id}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{x}\mapsto(0,\mathbf{p}^{+}_{1},\dots,\mathbf{p}^{+}_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathfrak{f}^{*})^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{x}\mapsto(0,(\mathbf{p}^{+}_{1})^{q^{i-1}}+\pi(\mathbf{p}^{+}_{2})^{q^{i-2}}+\cdots+\pi^{i-1}\mathbf{p}^{+}_{i},\dots).}$

The conclusion follows by the explicit description of the map $\Theta^{+}$ as
in the lines below (3.7) ∎

We can now prove the main result of this section. This is a particular case of
[17, Theorem 1.3] for which we give a shorter proof.

######  Theorem 3.15.

Let $G$ be a smooth formal group scheme over $R$. Then for all $n$ we have

$N^{n}G\simeq J^{n-1}(N^{1}G).$

###### Proof.

Let $\mathbb{A}=\operatorname{\mathrm{Spf}}(R\langle\mathbf{x}\rangle)$ be an
étale coordinate system around the identity section of $G$. We have seen in
(3.1) and Example 3.16 that
$N^{n}G=N^{n}\mathbb{A}=\operatorname{\mathrm{Spec}}N_{n}A$ with $N_{n}A\simeq
R\langle\mathbf{p}^{+}_{1},\dots,\mathbf{p}^{+}_{n}\rangle$. Now by Lemma 3.14
the lateral Frobenius satisfies

$(\mathfrak{f}^{*})^{i}(\mathbf{p}^{+}_{1})=(\mathbf{p}^{+}_{1})^{q^{i-1}}+\pi(\mathbf{p}^{+}_{2})^{q^{i-2}}+\cdots+\pi^{i-1}\mathbf{p}^{+}_{i}$

for all $i=0,\dots,n$. Hence by Theorem A.8 we have $N_{n}A\simeq
J_{n-1}(N_{1}A)$ and we are done. ∎

Here we discuss examples for $G=\widehat{\mathbb{G}}_{\mathrm{a}}$ and
$\widehat{\mathbb{G}}_{\mathrm{m}}$ in the context of Theorem 3.15

######  Examples 3.16.

  1. (1)

Assume
$G=\widehat{\mathbb{G}}_{\mathrm{a}}=\operatorname{\mathrm{Spf}}(R\langle
x\rangle)$ with the comultiplication mapping $x$ to $x\otimes 1+1\otimes x$.
Then
$J^{1}\widehat{\mathbb{G}}_{\mathrm{a}}=\operatorname{\mathrm{Spf}}(R\langle
x,x^{\prime}\rangle)$ where the group law is described by

$\displaystyle x$ $\displaystyle\mapsto x\otimes 1+1\otimes x\ ,$
$\displaystyle x^{\prime}$ $\displaystyle\mapsto x^{\prime}\otimes 1+1\otimes
x^{\prime}+C_{\pi}(x\otimes 1,1\otimes x)$

with $C_{\pi}(X,Y)=\frac{X^{q}+Y^{q}-(X+Y)^{q}}{\pi}\in\mathcal{O}[X,Y]$.
Hence
$N^{1}\widehat{\mathbb{G}}_{\mathrm{a}}=\operatorname{\mathrm{Spf}}(\langle
x^{\prime}\rangle)=\widehat{\mathbb{G}}_{\mathrm{a}}$. By Theorem 3.15 and
equation (2.16) one concludes
$N^{n}\widehat{\mathbb{G}}_{\mathrm{a}}=J^{n-1}(\widehat{\mathbb{G}}_{\mathrm{a}})=\widehat{\mathbb{W}}_{n-1}$.

  2. (2)

Assume
$G=\widehat{\mathbb{G}}_{\mathrm{m}}=\operatorname{\mathrm{Spf}}(R\langle
x,y\rangle/(xy-1)$ with the comultiplication mapping $x$ to $x\otimes x$. Then
$J^{1}\widehat{\mathbb{G}}_{\mathrm{m}}=\operatorname{\mathrm{Spf}}(R\langle
x,y,x^{\prime},y^{\prime}\rangle/(xy-1,\delta(xy))=\operatorname{\mathrm{Spf}}(R\langle
x,x^{-1},x^{\prime}\rangle)$ with the group law described by

$\displaystyle x$ $\displaystyle\mapsto x\otimes x\ ,$ $\displaystyle
x^{\prime}$ $\displaystyle\mapsto x^{\prime}\otimes x^{q}+x^{q}\otimes
x^{\prime}+\pi x^{\prime}\otimes x^{\prime}\ .$

Hence
$N^{1}\widehat{\mathbb{G}}_{\mathrm{m}}=\operatorname{\mathrm{Spf}}\left(R\langle
x^{\prime},y^{\prime}\rangle/(x^{\prime}+y^{\prime}+\pi
x^{\prime}y^{\prime})\right)=\operatorname{\mathrm{Spf}}(R\langle
x^{\prime}\rangle)$ and the group law on the latter maps $x^{\prime}$ to
$x^{\prime}\otimes 1+1\otimes x^{\prime}+\pi x^{\prime}\otimes x^{\prime}$,
i.e.,
$N^{1}\widehat{\mathbb{G}}_{\mathrm{m}}=\mathbb{G}^{\mathrm{for}}_{m}\\{1\\}$
as formal group schemes. Now, $\mathbb{G}^{\mathrm{for}}_{m}\\{1\\}$, as
formal group law, has invariant differential $(1+\pi T)^{-1}dT$ and the
corresponding logarithm is

$\pi^{-1}\log(1+\pi T)=\sum_{j\geq 1}\frac{(-\pi)^{j-1}}{j}T^{j}=T+\sum_{j\geq
2}a_{j}T^{j}\in K[\\![T]\\!].$

Assume $p\geq e+1$. We prove that $\pi^{-1}\log(1+\pi
T)\in\mathcal{O}[\\![T]\\!]$ and indeed in $\mathcal{O}\langle T\rangle$. It
suffices to check that $v_{\pi}(a_{j})\geq 0$ tends to infinity as $j$ tends
to infinity. Let $r>0$ and note that $v_{\pi}(a_{p^{r}})=p^{r}-1-er\geq 0$
since

$p^{r}-1=(p-1)(p^{r-1}+\dots+1)\geq er.$

Further $v_{\pi}(a_{p^{r}})<v_{\pi}(a_{p^{r+1}})$ and for $p^{r}\leq
j<p^{r+1}$ we have

$v_{\pi}(a_{p^{r}})=p^{r}-1-er\leq j-1-v_{\pi}(j)=v_{\pi}(a_{j}).$

Hence $\pi^{-1}\log(1+\pi T)\in\mathcal{O}\langle T\rangle$ and it defines a
morphism of formal group schemes
$\mathbb{G}^{\mathrm{for}}_{m}\\{1\\}\to\widehat{\mathbb{G}}_{\mathrm{a}}$. It
is an isomorphism under the stronger hypothesis that $p\geq e+2$. Indeed the
inverse of $\pi^{-1}\log(1+\pi T)$ is

$\pi^{-1}(\exp(\pi T)-1)=\sum_{j\geq 1}\frac{(\pi T)^{j}}{j!}\in
K[\\![T]\\!],$

and the $\pi$-adic valuation of the $j$-th coefficient is

$v_{\pi}(\pi^{j}/j!)=j-v_{\pi}(j!)=j-e\cdot\frac{j-s_{p}(j)}{p-1}=\frac{j(p-1-e)+es_{p}(j)}{p-1},$

where $s_{p}(j)$ denotes the sum of the digits in the base-$p$ expansion of
$j$. Clearly if $p\geq e+2$ this valuation tends to infinity as $j$ tends to
infinity and hence $\pi^{-1}(\exp(\pi T)-1)\in\mathcal{O}\langle T\rangle$.
Then, if $p\geq e+2$, one concludes that
$N^{1}\widehat{\mathbb{G}}_{\mathrm{m}}\simeq\widehat{\mathbb{G}}_{\mathrm{a}}$
and, with arguments as in (1), that
$N^{n}\widehat{\mathbb{G}}_{\mathrm{m}}\simeq\widehat{\mathbb{W}}_{n-1}$.

The next result is an extension of [8, Lemma 2.3].

######  Lemma 3.17.

Let $\mathcal{F}$ be a commutative formal group law over $R$ of dimension $d$.
If $n(p-1)\geq e+1$ then
$\mathcal{F}\\{n\\}\simeq(\widehat{\mathbb{G}}_{\mathrm{a}})^{d}$ as formal
group schemes over $R$.

###### Proof.

In [8, Lemma 2.3] $R$ is a complete discrete valuation ring with algebraically
closed residue field. The proof in our hypothesis works the same. Indeed Buium
applies results in [15] that are valid for any $\mathbb{Z}_{(p)}$-algebra and
the key-point is showing that the coefficients of the logarithm and
exponential series of $\mathcal{F}\\{n\\}$ over $R[1/p]$ are indeed in $\pi
R$. This is done by explicit estimates for the $\pi$-valuation of those
coefficients. ∎

######  Theorem 3.18.

Let $G$ be a smooth commutative formal group scheme of relative dimension $d$
over $\operatorname{\mathrm{Spf}}(R)$. Assume $p\geq e+2$. Then there is a
natural isomorphism of formal group schemes

$N^{n}G\simeq(\widehat{\mathbb{W}}_{n-1})^{d}.$

###### Proof.

Let $G^{\mathrm{for}}$ be the formal completion of $G$ along the unit section
$\operatorname{\mathrm{Spf}}(R)\to G$. Let $\mathcal{F}\in
R[[x_{1},\dots,x_{d},y_{1},\dots,y_{d}]]$ be the formal group law on
$G^{\mathrm{for}}$, ${\mathcal{F}}^{\phi}$ the one obtained by acting the
coefficients of $\mathcal{F}$ by $\phi$ and
${\mathcal{F}}^{\phi}\\{1\\}\colon=\pi^{-1}\mathcal{F}^{\phi}(\pi x_{.},\pi
y_{.})$. By Lemma 3.9 (see also [8, Lemma 2.2]) it is
$N^{1}G\simeq{\mathcal{F}}^{\phi}\\{1\\}$ as formal group schemes. Note that
since $\phi(\pi)=\pi$ it is
${\mathcal{F}}^{\phi}\\{1\\}=({\mathcal{F}}\\{1\\})^{\phi}$. Now by hypothesis
and Lemma 3.17 we have
${\mathcal{F}}\\{1\\}\simeq(\widehat{\mathbb{G}}_{\mathrm{a}})^{d}$. Hence
${\mathcal{F}}\\{1\\}^{\phi}\simeq((\widehat{\mathbb{G}}_{\mathrm{a}})^{d})^{\phi}=(\widehat{\mathbb{G}}_{\mathrm{a}})^{d}$
and hence $N^{1}G\simeq(\widehat{\mathbb{G}}_{\mathrm{a}})^{d}$. By Theorem
3.15 and definition of $J^{n-1}$ it is

$N^{n}G\simeq
J^{n-1}(\widehat{\mathbb{G}}_{\mathrm{a}})^{d}\simeq(J^{n-1}\widehat{\mathbb{G}}_{\mathrm{a}})^{d}\simeq(\widehat{\mathbb{W}}_{n-1})^{d}.$

∎

######  Remark 3.19.

Assume $R=\mathcal{O}$, $p>2$ and let $G$ be as in the previous theorem. Then
passing to limit on $n$ we have a short exact sequence

$0\to N^{\infty}G(k)\to J^{\infty}G(k)=G(\mathcal{O})\to G(k)\to 0$

where $G(\mathcal{O})\to G(k)$ is the reduction map, and we recover the fact
that the kernel of the reduction map is isomorphic to
$\mathcal{O}^{d}=W(k)^{d}$ as groups.

##  4\. $p$-power torsion

For any formal commutative $R$-group scheme $G$ let $G[p^{\nu}]$ denote the
kernel of the multiplication by $p^{\nu}$ on $G$ and let $G[p^{\infty}]$ be
the sheaf on $\mathbf{Nilp}_{R}^{\rm op}$ such that

(4.1) $G[p^{\infty}](C)=\varinjlim_{\nu}G[p^{\nu}](C)$

for any $C$ in $\mathbf{Nilp}_{R}$. Note that $G[p^{\infty}]$ is a sheaf since
the above colimit commutes with equalizers [18, 04AX]. If $G$ is a formal
torus or a formal abelian scheme $G[p^{\infty}]$ is $p$-divisible, but not in
general.

For each $\nu>0$ the closed immersions $G[p^{\nu}]\hookrightarrow
G[p^{\nu+1}]$ induce closed immersions of $\pi$-jets
$J^{n}(G[p^{\nu}])\hookrightarrow J^{n}(G[p^{\nu+1}])$ [9, Proposition 1.7],
[2, Lemma 3.8, Theorem 3.9]. We can naturally pass to limit on $\nu$.

######  Lemma 4.2.

Let the notation be as above. Then

$J^{n}(G[p^{\infty}])=\varinjlim_{\nu}J^{n}(G[p^{\nu}])=J^{n}(G)[p^{\infty}]$

as sheaves on $\mathbf{Nilp}_{R}^{\rm op}$.

###### Proof.

Recall (1.1) and (4.1). For any $C$ in $\mathbf{Nilp}_{R}$ it is

$J^{n}(G[p^{\infty}])(C)=G[p^{\infty}](W_{n}(C))=\varinjlim_{\nu}G[p^{\nu}](W_{n}(C))=\varinjlim_{\nu}J^{n}(G[p^{\nu}])(C).$

Whence the first isomorphism. The second one follows by the fact that $J^{n}$
is left exact (being a right adjoint) and hence

$\varinjlim_{\nu}J^{n}(G[p^{\nu}])(C)=\varinjlim_{\nu}((J^{n}G)[p^{\nu}])(C)=(J^{n}(G)[p^{\infty}])(C).$

∎

Recall that $N^{n}F$ is defined as $\ker(J^{n}F\to F)$ for any sheaf of groups
on $\mathbf{Nilp}_{R}^{\rm op}$. We then deduce from Lemma 4.2 the following
result.

######  Lemma 4.3.

Let notation be as above. Then

$N^{n}(G[p^{\infty}])=\varinjlim_{\nu}N^{n}(G[p^{\nu}])=(N^{n}G)[p^{\infty}]$

as sheaves on $\mathbf{Nilp}_{R}^{\rm op}$.

###### Proof.

By Theorem 3.8 the functor $N^{n}$ is left exact; hence
$(N^{n}G)[p^{\nu}]=N^{n}(G[p^{\nu}])$ and

$(N^{n}G)[p^{\infty}]:=\varinjlim_{\nu}(N^{n}G)[p^{\nu}]=\varinjlim_{\nu}N^{n}(G[p^{\nu}]).$

Further, $(N^{n}G)[p^{\infty}]=\ker((J^{n}G)[p^{\infty}]\to
G[p^{\infty}])=\ker(J^{n}(G[p^{\infty}])\to
G[p^{\infty}])=N^{n}(G[p^{\infty}])$. ∎

We say that a commutative formal $R$-group scheme is _triangular_ if it admits
a finite filtration by formal subgroup schemes whose successive quotients are
isomorphic to $\widehat{\mathbb{G}}_{\mathrm{a}}$. It is called _triangular of
level $0$_ in [8, p. 322]. The formal $R$-group scheme
$\widehat{\mathbb{W}}_{n}$ is triangular.

######  Lemma 4.4.

If $H$ is a triangular formal $R$-group scheme then $H[p^{\infty}]=H$ as
sheaves on $\mathbf{Nilp}_{R}^{\rm op}$.

###### Proof.

We proceed by induction on the length $m$ of the filtration. The result is
clearly true for $H=\widehat{\mathbb{G}}_{\mathrm{a}}$ since $p$ is nilpotent
in any object $C$ of $\mathbf{Nilp}_{R}$. Assume $m>1$. Then $H$ is extension
of $\widehat{\mathbb{G}}_{\mathrm{a}}$ by $G$ where $G$ is a triangular formal
group scheme with a filtration of length $m-1$. One concludes by induction
hypothesis. ∎

Moreover, we have the following stronger result.

######  Theorem 4.5.

Assume $G$ is a smooth commutative formal $R$-group scheme and $p\geq e+2$.
Then $N^{n}(G[p^{\infty}])=N^{n}G$ as sheaves on $\mathbf{Nilp}_{R}^{\rm op}$.

###### Proof.

By Theorem 3.18, the formal $R$-group scheme $N^{n}G$ is triangular. Hence the
result follows by Lemmas 4.3 and 4.4. ∎

######  Corollary 4.6.

Assume $p\geq e+2$. The exact sequence (1.3) induces an exact sequence of
formal groups

(4.7) $0\rightarrow N^{n}G[p^{\infty}]\rightarrow
J^{n}G[p^{\infty}]\rightarrow G[p^{\infty}]\rightarrow 0,$

for any $n\geq 0$.

###### Proof.

Only the right exactness needs to be proved. Let $s$ be a section of $G$ such
that $p^{m}s=0$. It lifts to a section $s^{\prime}$ of $J^{n}G$. Now
$p^{m}s^{\prime}$ comes from a section of $N^{n}G$ and thus is $p$-power
torsion by the previous theorem. Hence $s^{\prime}$ is $p$-power torsion. ∎

We are now ready to proof the Main Theorem stated in the introduction.

######  Theorem 4.8.

Assume $p\geq e+2$. Given a smooth commutative formal group scheme $G$ of
relative dimension $d$ over $R$, for any positive integers $n$ the natural
morphism $J^{n}G\to G$ gives an exact sequence

$0\rightarrow(\widehat{\mathbb{W}}_{n-1})^{d}\rightarrow
J^{n}G[p^{\infty}]\rightarrow G[p^{\infty}]\rightarrow 0$

as sheaves on $\mathbf{Nilp}_{R}^{\rm op}$.

###### Proof.

This follows directly by applying Theorems 3.18, 4.5 and Corollary 4.6. ∎

We can now conclude the study of Examples 3.16.

######  Examples 4.9.

  1. (1)

Assume $G=\widehat{\mathbb{G}}_{\mathrm{a}}$. We have seen in Example 3.16(1)
that
$N^{1}\widehat{\mathbb{G}}_{\mathrm{a}}=\operatorname{\mathrm{Spf}}(R\langle
x^{\prime}\rangle)$ is isomorphic to
$\widehat{\mathbb{G}}_{\mathrm{a}}=\operatorname{\mathrm{Spf}}(R\langle
x\rangle)$ as formal group scheme mapping $x$ to $x^{\prime}$ on algebras. In
particular, for any $\nu\geq 1$, it is

$N^{1}\widehat{\mathbb{G}}_{\mathrm{a}}[p^{\nu}]\cong\widehat{\mathbb{G}}_{\mathrm{a}}[p^{\nu}].$

This result can be checked directly. Indeed
$\widehat{\mathbb{G}}_{\mathrm{a}}[p^{\nu}]=\operatorname{\mathrm{Spf}}(R\langle
x\rangle/(p^{\nu}x))$ and hence

$J^{1}\widehat{\mathbb{G}}_{\mathrm{a}}[p^{\nu}]=\operatorname{\mathrm{Spf}}\left(\frac{R\langle
x,x^{\prime}\rangle}{(p^{\nu}x,\delta(p^{\nu}x))}\right)\ ,\quad
N^{1}\widehat{\mathbb{G}}_{\mathrm{a}}[p^{\nu}]=\operatorname{\mathrm{Spf}}\left(\frac{R\langle
x^{\prime}\rangle}{(p^{\nu}x^{\prime})}\right)\\\ ,$

since
$(p^{\nu}x)^{q}+\pi\delta(p^{\nu}x)=\phi(p^{\nu}x)=p^{\nu}\phi(x)=p^{\nu}(x^{q}+\pi
x^{\prime})$. Passing to limit on $\nu$ we get an isomorphism
$N^{1}\widehat{\mathbb{G}}_{\mathrm{a}}[p^{\infty}]\cong\widehat{\mathbb{G}}_{\mathrm{a}}[p^{\infty}]$.

  2. (2)

Assume $G=\widehat{\mathbb{G}}_{\mathrm{m}}$. We have seen in Example 3.16(2)
that if $p\geq e+2$ it is
$N^{1}\widehat{\mathbb{G}}_{\mathrm{m}}\simeq\widehat{\mathbb{G}}_{\mathrm{a}}$.
Hence
$N^{1}\widehat{\mathbb{G}}_{\mathrm{m}}[p^{\infty}]\simeq\widehat{\mathbb{G}}_{\mathrm{a}}[p^{\infty}]$.

##  Appendix A Prolongation sequences of algebras

Let for this section $R$ be a flat $\mathcal{O}$-algebra with a fixed
$\pi$-derivation $\delta$. In particular, it has a lift of Frobenius. Let
$u\colon B\to C$ be a morphism of $R$-algebras. Recall that if $u\colon B\to
C$ is a morphism of $\mathcal{O}$-algebras, a $\pi$-derivation relative to $u$
is a map of sets $\partial\colon B\to C$ such that $\partial(0)=0=\partial(1)$
and for any $x,y\in B$

(A.1) $\displaystyle\partial(x+y)$
$\displaystyle=\partial(x)+\partial(y)+\frac{u(x)^{q}+u(y)^{q}-u(x+y)^{q}}{\pi}$
(A.2) $\displaystyle\partial(xy)$
$\displaystyle=u(x)^{q}\partial(y)+u(y)^{q}\partial(x)+\pi\partial(x)\partial(y)\
.$

Associated to $\partial$ there is a lift of Frobenius (relative to $u$)
$\Psi\colon B\rightarrow C$ given by $\Psi(x)=u(x)^{q}+\pi\partial(x)$ (see
[7, §3.1], [2, § 1]).

Fix a positive integer $r$. Let
$B_{n}=R[\mathbf{x}_{0},\dots,\mathbf{x}_{n}]$, where for each $i\geq 0$,
$\mathbf{x}_{i}$ denotes the $r$-tuple of variables $x_{i,1},\dots,x_{i,r}$
and denote by $u\colon B_{n}\to B_{n+1}$ the natural inclusions. Fix a
prolongation sequence

(A.3)
$B=B_{0}\xrightarrow{u,\partial}B_{1}\xrightarrow{u,\partial}B_{2}\to...$

i.e., for any $n$ we fix a $\pi$-derivation (relative to $u$) $\partial\colon
B_{n}\to B_{n+1}$ such that

  * •

$\partial$ is compatible with the $\pi$-derivation of $R$;

  * •

$\partial\circ u=u\circ\partial$ (and thus we avoided using subscripts).

The lift of Frobenius is then the homomorphism

$\Psi\colon B_{n}\to
B_{n+1},\qquad\mathbf{x}_{i}\mapsto\mathbf{x}_{i}^{q}+\pi\partial\mathbf{x}_{i},$

where we used a compact notation instead of writing $x_{i,j}\mapsto
x_{i,j}^{q}+\pi\partial x_{i,j}$ for all $i\leq n$ and $1\leq j\leq r$. Note
that we can handle all $B_{n}$ together by introducing
$B_{\infty}=R[\mathbf{x}_{0},\mathbf{x}_{1},\dots]=\bigcup_{n}B_{n}$ and again
denoting by $\partial$ the induced $\pi$-derivation associated with the
identity on $B_{\infty}$ and by $\Psi$ the associated lift of Frobenius.

On the other hand, starting with $B=R[\mathbf{x}_{0}]$ we have a prolongation
sequence

(A.4)
$B=B_{0}\xrightarrow{u,\delta}J_{1}B_{0}\xrightarrow{u,\delta}J_{2}B_{0}\to...$

with
$J_{n}B=R[\mathbf{x}_{0},\mathbf{x}_{0}^{\prime},\dots,\mathbf{x}_{0}^{(n)}]$
and $\mathbf{x}_{0}^{(i+1)}=\delta\mathbf{x}_{0}^{(i)}$. Let

$\Phi\colon J_{n}B\to
J_{n+1}B,\qquad\mathbf{x}_{0}\mapsto\mathbf{x}_{0}^{q}+\pi\mathbf{x}_{0}^{\prime}$

be the corresponding lift of Frobenius and define
$J_{\infty}B=\bigcup_{n}J_{n}B$.

By [2, (2.9)] the restriction on the first component $W(J_{\infty}B)\to
J_{\infty}B$ admits a homomorphic section $\exp$ such that the following
diagram

(A.5)
$\textstyle{W(J_{\infty}B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w}$$\textstyle{\prod_{i\in\mathbb{N}}B_{\infty}}$$\textstyle{J_{\infty}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exp}$$\scriptstyle{({\rm
id},\Phi,\Phi^{2},\dots)}$

commutes, with $w$ the ghost map of ramified Witt vectors. Let
$\exp(\mathbf{x}_{0})=({\bf z}_{0},{\bf z}_{1},{\bf z}_{2},\dots)$ so that
${\bf z}_{0}=\mathbf{x}_{0}$, ${\bf z}_{1}=\ \mathbf{x}_{0}^{\prime}$ and for
$n>1$

$\Phi^{n}(\mathbf{x}_{0})=\Phi^{n}({\bf z}_{0})={\bf z}_{0}^{q^{n}}+\pi{\bf
z}_{1}^{q^{n-1}}+\cdots+\pi^{n}{\bf z}_{n}.$

We will show in Theorem A.8 that if the indeterminates
$\mathbf{x}_{0},\mathbf{x}_{1},\dots$ satisfy the analogous property for
$\Psi$, i.e.,

(A.6)
$\Psi^{n}(\mathbf{x}_{0})=\mathbf{x}_{0}^{q^{n}}+\pi\mathbf{x}_{1}^{q^{n-1}}+\cdots+\pi^{n}\mathbf{x}_{n},$

then there is a unique isomorphism between the prolongation sequences (A.3)
and (A.4).

We start with a technical result. For brevity, let us define for any $n>1$ the
following polynomial in $2n-2$ indeterminates

$H_{n}(x_{0},\dots,x_{n-2};y_{0},\dots,y_{n-2}):=\\!\sum_{i=0}^{n-2}\left[\sum_{j=1}^{q^{n-1-i}}\\!\pi^{i+j}{\binom{q^{n-1-i}}{j}}x_{i}^{q(q^{n-1-i}-j)}y_{i}^{j}\right]$

######  Lemma A.7.

Let $B_{*}$ be the prolongation sequence in (A.3) and assume that it satisfies
(A.6) for any $n$. Then

$\mathbf{x}_{n}=\partial\mathbf{x}_{n-1}+H_{n}(\mathbf{x}_{0},\dots,\mathbf{x}_{n-2};\partial\mathbf{x}_{0},\dots,\partial\mathbf{x}_{n-2})$

and hence
$\mathbf{x}_{n}-\partial\mathbf{x}_{n-1}\in\mathcal{O}[\mathbf{x}_{0},\dots,\mathbf{x}_{n-2},\partial\mathbf{x}_{0},\dots,\partial\mathbf{x}_{n-2}]$
with trivial constant term. Similarly, in $J_{\infty}B$ it is

${\bf z}_{n}=\delta{\bf z}_{n-1}+H_{n}({\bf z}_{0},\dots,{\bf
z}_{n-2};\delta{\bf z}_{0},\dots,\delta{\bf z}_{n-2})$

###### Proof.

The second assertion is [2, Proposition 2.10]. The proof of the first one is
similar and we write below the main steps:

$\displaystyle\sum_{i=0}^{n}\pi^{i}\mathbf{x}_{i}^{q^{n-i}}$ $\displaystyle=$
$\displaystyle\Psi^{n}(\mathbf{x}_{0})$ $\displaystyle=$
$\displaystyle\Psi(\Psi^{n-1}(\mathbf{x}_{0}))$ $\displaystyle=$
$\displaystyle\Psi\left(\sum_{i=0}^{n-1}\pi^{i}\mathbf{x}_{i}^{q^{n-1-i}}\right)$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{n-1}\pi^{i}\Psi(\mathbf{x}_{i})^{q^{n_{i}}}$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{n-1}\pi^{i}\left(\mathbf{x}_{i}^{q}+\pi\partial\mathbf{x}_{i}\right)^{q^{n_{i}}}$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{n-1}\pi^{i}\left[\mathbf{x}_{i}^{q^{n-i}}+\sum_{j=1}^{q^{n_{i}}}{\binom{q^{n_{i}}}{j}}\mathbf{x}_{i}^{(q^{n_{i}}-j)q}\pi^{j}(\partial\mathbf{x}_{i})^{j}\right]$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{n-1}\pi^{i}\mathbf{x}_{i}^{q^{n-i}}+\sum_{i=0}^{n-1}\left[\sum_{j=1}^{q^{n_{i}}}\pi^{i+j}{\binom{q^{n_{i}}}{j}}\mathbf{x}_{i}^{(q^{n_{i}}-j)q}(\partial\mathbf{x}_{i})^{j}\right]$

with $n_{i}=n-1-i$. Hence cancelling the common terms on both sides of the
above equality we get

$\pi^{n}\mathbf{x}_{n}=\pi^{n}\partial\mathbf{x}_{n-1}+\sum_{i=0}^{n-2}\left[\sum_{j=1}^{q^{n-1-i}}\pi^{i+j}{\binom{q^{n-1-i}}{j}}\mathbf{x}_{i}^{q(q^{n-1-i}-j)}(\partial\mathbf{x}_{i})^{j}\right]$

and one can divide by $\pi^{n}$. ∎

We now prove that prolongation sequences $B_{*}$ as above satisfying condition
(A.6) for all $n$ are unique up to unique isomorphism.

######  Theorem A.8.

Let $B_{*}$ be the prolongation sequence in (A.3). Assume that the
indeterminates $\mathbf{x}_{i}$ satisfy (A.6) for all $n$ and let ${\bf
z}_{i}\in J_{\infty}R[\mathbf{x}_{0}]$ be the elements defined just below
(A.5). Then we have

  1. (i)

The inclusion $R[{\bf z}_{0},\dots,{\bf z}_{n}]\to J_{n}R[\mathbf{x}_{0}]$ is
an isomorphism for any $n$.

  2. (ii)

For any $n\geq 0$ the $R$-algebra homomorphism $h_{n}\colon
J_{n}R[\mathbf{x}_{0}]\to B_{n},\
\mathbf{x}_{0}^{(i)}\mapsto\partial^{i}\mathbf{x}_{0}$ is an isomorphism and
the following square

$\textstyle{R\left[\mathbf{x}_{0},\dots,\mathbf{x}_{0}^{(n)}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(u,\delta)}$$\scriptstyle{h_{n}}$$\textstyle{R\left[\mathbf{x}_{0},\dots,\mathbf{x}_{0}^{(n+1)}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{n+1}}$$\textstyle{R\left[\mathbf{x}_{0},\dots,\mathbf{x}_{n}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(u,\partial)}$$\textstyle{R\left[\mathbf{x}_{0},\dots,\mathbf{x}_{n+1}\right]}$

commutes, where $u$ denotes the inclusion map on both levels.

###### Proof.

The first assertion was proved in [2, Lemma 2.20] with ${\bf
z}_{i}=P_{i}(\mathbf{x})$. Commutativity of the squares is immediate by
definition of $h_{n}$. We are then left to prove that $h_{n}$ is an
isomorphism. Note hat $J_{n}R[\mathbf{x}_{0}]=R[{\bf z}_{0},\dots,{\bf
z}_{n}]$ by point (i) and $B_{n}=R[\mathbf{x}_{0},\dots,\mathbf{x}_{n}]$. If
we prove that $h_{n}({\bf z}_{i})=\mathbf{x}_{i}$, for all $0\leq i\leq n$,
the result is clear.

We proceed by strong induction on the subset $\\{(i,n),0\leq i\leq
n\\}\subset\mathbb{N}^{2}$ totally ordered as follows:

$(i_{1},n_{1})<(i_{2},n_{2})\quad\text{ if }\quad
n_{1}+i_{1}<n_{2}+i_{2}\quad\text{ or }\quad n_{1}+i_{1}=n_{2}+i_{2}\ \text{
and }\ i_{1}<i_{2}.$

The picture below illustrates the order.

$\textstyle{(0,4)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(0,3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(0,2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(2,2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(0,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(0,0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

It follows immediately by definition of $h_{n}$ that $h_{n}({\bf
z}_{0})=h_{n}(\mathbf{x}_{0})=\mathbf{x}_{0}$ and $h_{n}({\bf
z}_{1})=h_{n}(\mathbf{x}_{0}^{\prime})=\partial\mathbf{x}_{0}=\mathbf{x}_{1}$
for all $n\geq 0$. Hence the assertion $h_{n}({\bf z}_{i})=\mathbf{x}_{i}$ is
clear for $i\leq 1$ and any $n$, in particular for the base step $(0,0)$.
Assume then $i>1$ and that $h_{s}({\bf z}_{j})=\mathbf{x}_{j}$ for any
$(0,0)\leq(j,s)<(i,n)$. By Lemma A.7, the commutativity of the above square
and the induction step, we have

$\displaystyle h_{n}({\bf z}_{i})$ $\displaystyle=$ $\displaystyle
h_{n}\left(\delta{\bf z}_{i-1}+H_{i}({\bf z}_{0},\dots,{\bf
z}_{i-2},\delta{\bf z}_{0},\dots,\delta{\bf z}_{i-2})\right)$ $\displaystyle=$
$\displaystyle h_{n}(\delta{\bf z}_{i-1})+H_{i}(h_{n}({\bf
z}_{0}),\dots,h_{n}({\bf z}_{i-2}),h_{n}(\delta{\bf
z}_{0}),\dots,h_{n}(\delta{\bf z}_{i-2}))$ $\displaystyle=$
$\displaystyle\partial(h_{n-1}({\bf
z}_{i-1}))+H_{i}(\mathbf{x}_{0},\dots,\mathbf{x}_{i-2},\partial h_{n-1}({\bf
z}_{0}),\dots,\partial h_{n-1}({\bf z}_{i-2}))$ $\displaystyle=$
$\displaystyle\partial\mathbf{x}_{i-1}+H_{i}(\mathbf{x}_{0},\dots,\mathbf{x}_{i-2},\partial\mathbf{x}_{0},\dots,\partial\mathbf{x}_{i-2})$
$\displaystyle=$ $\displaystyle\mathbf{x}_{i}.$

∎

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