# Witt Differential Operators

Christopher Dodd

###### Abstract.

For a smooth scheme $X$ over a perfect field $k$ of positive characteristic,
we define (for each $m\in\mathbb{Z}$) a sheaf of rings
$\mathcal{\widehat{D}}_{W(X)}^{(m)}$ of differential operators (of level $m$)
over the Witt vectors of $X$. If $\mathfrak{X}$ is a lift of $X$ to a smooth
formal scheme over $W(k)$, then for $m\geq 0$ modules over
$\mathcal{\widehat{D}}_{W(X)}^{(m)}$ are closely related to modules over
Berthelot’s ring $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}$ of differential
operators of level $m$ on $\mathfrak{X}$. Our construction therefore gives an
description of suitable categories of modules over these algebras, which
depends only on the special fibre $X$. There is an embedding of the category
of crystals on $X$ (over $W_{r}(k)$) into modules over
$\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}$; and so we obtain an alternate
description of this category as well. For a map $\varphi:X\to Y$ we develop
the formalism of pullback and pushforward of
$\mathcal{\widehat{D}}_{W(X)}^{(m)}$-modules and show all of the expected
properties. When working mod $p^{r}$, this includes compatibility with the
corresponding formalism for crystals, assuming $\varphi$ is smooth. In this
case we also show that there is a “relative de Rham Witt resolution”
(analogous to the usual relative de Rham resolution in $\mathcal{D}$-module
theory) and therefore that the pushforward of (a quite general subcategory of)
modules over $\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}$ can be computed via
the reduction mod $p^{r}$ of Langer-Zink’s relative de Rham Witt complex.
Finally we explain a generalization of Bloch’s theorem relating integrable de
Rham-Witt connections to crystals.

###### Contents

  1. 1 Introduction
    1. 1.1 Notations and conventions
    2. 1.2 Acknowledgements
  2. 2 Higher Derivations on Witt Vectors
    1. 2.1 Local Coordinates
    2. 2.2 Compliment: The algebra $\widehat{\mathcal{D}}_{W(X),\text{crys}}^{(0)}$
  3. 3 Accessibility
    1. 3.1 Accessible modules
    2. 3.2 The crystalline version
    3. 3.3 Frobenius descent
  4. 4 Operations on Accessible Modules
    1. 4.1 Operations on modules: Right Modules and the left-right interchange
    2. 4.2 Operations on modules: Pull-Back
    3. 4.3 Operations on modules: Pushforward
    4. 4.4 The de Rham-Witt resolution
  5. 5 The algebra $\widehat{\mathcal{U}}$, and applications
    1. 5.1 The algebra $\widehat{\mathcal{U}}$.
    2. 5.2 De Rham-Witt connections mod $p^{r}$

##  1\. Introduction

Let $X$ be a smooth algebraic variety over $\mathbb{C}$. Then one may attach
to $X$ its (algebraic) de Rham cohomology,
$\mathbb{H}_{\text{dR}}^{\cdot}(X)$, which is defined as the hypercohomology
of the algebraic de Rham complex of $X$. It is a fundamental theorem of
Grothendieck that this cohomology is isomorphic to the singular cohomology of
the analytic space attached to $X$, $X^{\text{an}}$. Guided by the philosophy
that any cohomology theory should come with a good theory of local
coefficients, Grothendieck defined a site, the infinitesimal site of $X$,
which has the property that the de Rham cohomology of $X$ is isomorphic to the
cohomology of the structure sheaf of $X$ in the infinitesimal site. He further
showed that the data of a sheaf on the infinitesimal site is equivalent to the
data of an $\mathcal{O}_{X}$-module $\mathcal{M}$ (in the usual Zariski
topology of $X$), equipped with a flat connection. Let us recall that this is
a morphism

$\nabla:\mathcal{M}\to\mathcal{M}\otimes\Omega_{X}^{1}$

which satisfies the Leibniz rule $\nabla(f\cdot m)=f\cdot\nabla(m)+m\otimes
df$, as well as the following: we can extend $\nabla$ to a morphism

$\nabla:\mathcal{M}\otimes\Omega_{X}^{i}\to\mathcal{M}\otimes\Omega_{X}^{i+1}$

via the rule $\nabla(m\otimes\phi)=\nabla(m)\cdot\phi+m\otimes d\phi$, where
the multiplication in the first term denotes the map
$(\mathcal{M}\otimes\Omega_{X}^{1})\otimes\Omega_{X}^{i}\to\mathcal{M}\otimes\Omega_{X}^{i+1}$
coming from multiplication in the de Rham complex. Then we demand that
$\nabla\circ\nabla=0$ (this is what it means for a connection to be flat).

Therefore, to any such $(\mathcal{M},\nabla)$ we can associated a complex of
sheaves
$(\mathcal{M}\otimes\Omega_{X}^{\cdot},\nabla)$, whose hypercohomology
$\mathbb{H}_{\text{dR}}^{\cdot}(\mathcal{M})$ is the de Rham cohomology of
$\mathcal{M}$. If $\tilde{\mathcal{M}}$ is the sheaf in the infinitesimal site
associated to $(\mathcal{M},\nabla)$, then there is an isomorphism

$\mathbb{H}_{\text{dR}}^{\cdot}(\mathcal{M})\tilde{\to}\mathbb{H}_{\text{inf}}^{\cdot}(\tilde{\mathcal{M}})$

where the cohomology group on the right is the cohomology of
$\tilde{\mathcal{M}}$ in the infinitesimal site (c.f. [22]).

At around the same time, the theory of $\mathcal{O}_{X}$-modules with flat
connection, in a rather different guise, was beginning to be developed, by
both M. Sato and his collaborators in Kyoto and by J. Bernstein and his
collaborators in Moscow. This different guise goes under the name of
$\mathcal{D}$-modules111To be completely precise, Sato and his school worked
with complex analytic spaces, while Bernstein worked with algebraic varieties.
The two theories are similar enough, and had enough influence on each other,
that we will mention them both in this introduction, where $\mathcal{D}$
stands for the sheaf of (finite order) differential operators on $X$. In fact,
a sheaf of modules over $\mathcal{D}$ is equivalent to an
$\mathcal{O}_{X}$-module with flat connection. Thus $\mathcal{D}$-modules
(i.e. sheaves of modules over $\mathcal{D}$) provide another (equivalent)
answer to the question of how to find a good theory of local coefficients for
the de Rham cohomology. Furthermore, the fact that $\mathcal{D}$ is itself a
locally noetherian sheaf of algebras with a nice presentation allows one to
develop the theory in new directions. For instance, there is a well behaved
notion of coherence for $\mathcal{D}$-modules, and to a coherent
$\mathcal{D}$-module one may attach an invariant called the singular support,
which is a closed subvariety of $T^{*}X$, the cotangent bundle on $X$.

Furthermore, if $\varphi:X\to Y$ is a morphism of smooth algebraic varieties
over $\mathbb{C}$, then there is formalism of push-forward and pull-back. As
there is, in general, no morphism of sheaves of rings between
$\varphi^{-1}(\mathcal{D}_{Y})$ and $\mathcal{D}_{X}$, this formalism is a bit
non-trivial. Let us take a moment to review it. First, the pullback: as a
$\mathcal{D}$-module is equivalent to a module with flat connection; if one
has a flat connection on some $\mathcal{O}_{Y}$-module $\mathcal{M}$, then the
composed map

$\varphi^{*}\mathcal{M}\xrightarrow{\varphi^{*}\nabla}\varphi^{*}\mathcal{M}\otimes\varphi^{*}\Omega_{Y}^{1}\to\varphi^{*}\mathcal{M}\otimes\Omega_{X}^{1}$

is a flat connection on $\varphi^{*}\mathcal{M}$. Applying this to
$\mathcal{D}_{Y}$ itself, one endows $\varphi^{*}\mathcal{D}_{Y}$ with the
structure of a left $\mathcal{D}_{X}$-module. As it is also a right
$\varphi^{-1}(\mathcal{D}_{Y})$-module via the right action of
$\mathcal{D}_{Y}$ on itself, we see that $\varphi^{*}\mathcal{D}_{Y}$ acquires
the structure of a $(\mathcal{D}_{X},\varphi^{-1}(\mathcal{D}_{Y}))$ bimodule;
this object is denoted $\mathcal{D}_{X\to Y}$. Thus we may define a functor on
derived categories

$L\varphi^{*}:D(\mathcal{D}_{Y}-\text{mod})\to D(\mathcal{D}_{X}-\text{mod})$

via

$L\varphi^{*}\mathcal{M}^{\cdot}:=\mathcal{D}_{X\to
Y}\otimes_{\varphi^{-1}(\mathcal{D}_{Y})}^{L}\varphi^{-1}(\mathcal{M}^{\cdot})$

If $\mathcal{M}$ is an object in $\mathcal{D}_{Y}-\text{mod}$, then
$\mathcal{H}^{0}(L\varphi^{*}\mathcal{M})=\varphi^{*}\mathcal{M}$, equipped
with the flat connection defined above.

Now let us consider the push-forward. The existence of the bimodule
$\mathcal{D}_{X\to Y}$ allows us to construct a push-forward on _right_
$\mathcal{D}$-modules as follows: if $\mathcal{N}^{\cdot}$ is a complex of
right $\mathcal{D}$-modules on $X$, then

$\mathcal{N}^{\cdot}\otimes_{\mathcal{D}_{X}}^{L}\mathcal{D}_{X\to Y}$

is a complex of _right $\varphi^{-1}(\mathcal{D}_{Y})$-_modules. Therefore we
obtain a functor

$\int_{\varphi}:D(\text{mod}-\mathcal{D}_{X})\to
D(\text{mod}-\mathcal{D}_{Y})$

(where $\text{mod}-\mathcal{D}$ stands for the category of right
$\mathcal{D}$-modules) defined as

$\int_{\varphi}\mathcal{N}^{\cdot}:=R\varphi_{*}(\mathcal{N}^{\cdot}\otimes_{\mathcal{D}_{X}}^{L}\mathcal{D}_{X\to
Y})$

Of course, as our aim is to study left $\mathcal{D}$-modules, and as
$\mathcal{D}$ is a non-commutative sheaf of rings, this does not satisfy our
original goal. However, the situation is salvaged by the following:

######  Proposition 1.1.

(c.f. [24], proposition 1.2.12) There is an equivalence of categories
$\mathcal{D}_{X}-\text{mod}\to\text{mod}-\mathcal{D}_{X}$. On the underlying
$\mathcal{O}_{X}$-modules, this functor is given by
$\mathcal{M}\to\mathcal{M}\otimes_{\mathcal{O}_{X}}\omega_{X}$. This is
referred to as the left-right interchange.

Therefore, one may obtain a push-forward functor on left $\mathcal{D}$-modules
by applying the equivalence to right modules, applying the above functor
${\displaystyle\int_{\varphi}}$, and then applying the equivalence from right
to left $\mathcal{D}_{Y}$-modules. More efficiently, one may use the left-
right interchange to directly define from $\mathcal{D}_{X\to Y}$ a
$(\varphi^{-1}(\mathcal{D}_{Y}),\mathcal{D}_{X})$-bimodule, denoted
$\mathcal{D}_{Y\leftarrow X}$ (c.f. the beginning of LABEL:subsec:Operations-
on-modulesPush!: below for details) and then define

$\int_{\varphi}:D(\mathcal{D}_{X}-\text{mod})\to
D(\mathcal{D}_{Y}-\text{mod})$

via

$\int_{\varphi}\mathcal{M}^{\cdot}:=R\varphi_{*}(\mathcal{D}_{Y\leftarrow
X}\otimes_{\mathcal{D}_{X}}^{L}\mathcal{M}^{\cdot})$

Though one may easily prove many basic properties of the pushforward from this
formula, it does not seem to lend itself to any obvious evaluation. For
instance, if $Y=\text{Spec}(\mathbb{C})$ is a point, and
$\mathcal{M}^{\cdot}=\mathcal{M}$ is a left $\mathcal{D}_{X}$-module, we would
hope that the pushforward of $\mathcal{N}$ agrees with the de Rham cohomology.
That this is indeed the case (up to a harmless homological shift) is contained
in the following fundamental

######  Theorem 1.2.

(c.f. [24], section 1.5) Suppose $Y$ is a point. Then
$\mathcal{D}_{Y\leftarrow X}=\omega_{X}$ (with its natural right
$\mathcal{D}_{X}$-module structure). Consider the de Rham complex of
$\mathcal{D}_{X}$,
$(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\Omega_{X}^{\cdot},\nabla)$. Giving
each term $\mathcal{D}_{X}\otimes\Omega_{X}^{i}$ the structure of a right
$\mathcal{D}_{X}$-module via the right multiplication of $\mathcal{D}_{X}$ on
itself makes this complex into a complex of right $\mathcal{D}_{X}$-modules.
This complex is exact except at the rightmost term (where $i=\text{dim}(X)$),
and we have that the action map

$\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X}\to\omega_{X}$

induces an isomorphism
$\mathcal{H}^{\text{dim}(X)}(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\Omega_{X}^{\cdot})\tilde{\to}\omega_{X}$.

In other words,
$(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\Omega_{X}^{\cdot},\nabla)$ is a
locally free resolution of $\omega_{X}$ in the category of right
$\mathcal{D}_{X}$-modules.

Plugging this resolution into the definition of the pushforward yields

$\int_{\varphi}\mathcal{M}\tilde{\to}\mathbb{H}_{\text{dR}}^{\cdot}(\mathcal{M})[\text{dim}(X)]$

Along with the resolution of $\omega_{X}$ as a right $\mathcal{D}_{X}$-module,
there is (via the left-right-interchange) a corresponding resolution of
$\mathcal{O}_{X}$ as a left $\mathcal{D}_{X}$-module, called the Spencer
resolution. It allows one to prove the other basic interpretation of the de
Rham cohomology in terms of the category of modules with flat connection:

######  Corollary 1.3.

There is a locally free resolution of $\mathcal{O}_{X}$, in the category of
left $\mathcal{D}_{X}$-modules, whose terms are
$\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{T}_{X}^{i}$ (here,
$\mathcal{T}_{X}^{i}$ is the $i$th exterior power of the tangent sheaf); and
the differential is obtained from the differential in the de Rham complex of
$\mathcal{D}_{X}$ via the left-right interchange. Therefore, we have for any
$\mathcal{M}\in\mathcal{D}_{X}-\text{mod}$ an isomorphism

$\text{Ext}{}_{\mathcal{D}_{X}}^{i}(\mathcal{O}_{X},\mathcal{M})\tilde{\to}\mathbb{H}_{dR}^{i}(\mathcal{M})$

After identifying $\mathcal{D}$-modules with sheaves on the infinitesimal
site, this yields another proof of the fact that de Rham cohomology of a
connection agrees with its infinitesimal cohomology.

In fact there is a generalization of this to any smooth morphism $\varphi:X\to
Y$. In that case, one may consider the relative connection
$\nabla:\mathcal{D}_{X}\to\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\Omega_{X/Y}^{1}$
and the associated relative de Rham complex

$(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\Omega_{X/Y}^{\cdot},\nabla)$

This is again a locally free complex of right $\mathcal{D}_{X}$-modules, which
is a resolution of $\mathcal{D}_{Y\leftarrow X}$. Using this, one may go on to
show

######  Theorem 1.4.

Let $\varphi:X\to Y$ be smooth of relative dimension $d$. Let
$\mathcal{M}\in\mathcal{D}_{X}-\text{mod}$, and
$(\mathcal{M}\otimes_{\mathcal{O}_{X}}\Omega_{X/Y}^{\cdot},\nabla)$ the
corresponding relative de Rham complex. Then

1) There is an isomorphism

$\int_{\varphi}\mathcal{M}[-d]\tilde{=}R\varphi_{*}(\mathcal{M}\otimes_{\mathcal{O}_{X}}\Omega_{X/Y}^{\cdot},\nabla)$

in the category $D(\mathcal{O}_{Y})-\text{mod}$. The resulting
$\mathcal{D}_{Y}$-module structure on the sheaf
$R^{i}\varphi_{*}(\mathcal{M}\otimes_{\mathcal{O}_{X}}\Omega_{X/Y}^{\cdot},\nabla)$
corresponds to the Gauss-Manin connection.

2) Let $\varphi^{!}=L\varphi^{*}[d]$. Then there is an isomorphism

$R\varphi_{*}R\mathcal{H}om_{\mathcal{D}_{X}}(\varphi^{!}\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})\tilde{\to}R\mathcal{H}om{}_{\mathcal{D}_{Y}}(\mathcal{N}^{\cdot},\int_{\varphi}\mathcal{M}^{\cdot})$

for any $\mathcal{N}^{\cdot}\in D(\mathcal{D}_{Y}-\text{mod})$ and any
$\mathcal{M}^{\cdot}\in D(\mathcal{D}_{X}-\text{mod})$. In particular,
${\displaystyle\int_{\varphi}}$ is the right adjoint to $\varphi^{!}$.

For a nice exposition of this, we refer the reader to [27], chapter 4. Without
going into any more detail, let us mention that the theory of
$\mathcal{D}$-modules allows one to study many other interesting phenomena
related to the topology of algebraic varieties. For instance, there is a very
rich theory of nearby and vanishing cycles for $\mathcal{D}$-modules, there is
a $\mathcal{D}$-module duality which is generalization of Poincare duality,
and that there are deep connections with Hodge theory.

Now let us turn to the main interest of this paper, the theory of varieties in
characteristic $p>0$. Fix now a perfect field $k$ of characteristic $p>0$;
letters $X$ and $Y$ will now denote smooth varieties over $k$. Let $W(k)$
denote the $p$-typical Witt vectors of $k$, and set $W_{r}(k)=W(k)/p^{r}$. In
this case it was predicted by Grothendieck that there should be a cohomology
theory, the crystalline cohomology, denoted
$\mathbb{H}_{\text{crys}}^{\cdot}(X;W_{r}(k))$, with coefficients in modules
over $W_{r}(k)$ (for each $r$). The crystalline cohomology is supposed to have
the following property: if $\mathfrak{X}_{r}$ is a flat lift of $X$ to a
scheme over $W_{r}(k)$, then there is an isomorphism

(1.1)
$\mathbb{H}_{\text{dR}}^{\cdot}(\mathfrak{X}_{r})\tilde{\to}\mathbb{H}_{\text{crys}}^{\cdot}(X;W_{r}(k))$

Furthermore, the crystalline cohomology should be the cohomology of the
structure sheaf in a suitable site, the crystalline site of $X$. One should
then be able to take the limit over $r$ and obtain a cohomology
$\mathbb{H}_{\text{crys}}^{\cdot}(X)$ with values in $W(k)$.

This prediction was borne out in Berthelot’s monumental work [3]. In that
work, he constructs the crystalline site of $X$ (it is modeled after the
infinitesimal site) and defines crystalline cohomology as the cohomology of
the structure sheaf in the crystalline site; and proves, among many other
things, the isomorphism 1.1. As the crystalline cohomology is defined for any
$X$, without using a lift, the theory shows that the de Rham cohomology of
such a lift depends only on special fibre $X$.

There is, in some cases, an interpretation of sheaves of $\mathcal{O}$-modules
on the crystalline site over $W_{r}(k)$ (we shall refer to them as _$W_{r}(k)$
-crystals _from now on) in terms of flat connections. Namely, if
$\mathfrak{X}_{r}$ is a lift of $X$ as above, then there is an equivalence
from $W_{r}(k)$-crystals on $X$ to $\mathcal{O}_{\mathfrak{X}_{r}}$-modules
with flat connection (satisfying an additional condition called nilpotence).
This means that crystals always have such a description locally on $X$. In
general, however, $X$ admits no such lift, and so there is no “concrete”
description of the category of crystals analogous to the theory of
$\mathcal{D}_{X}$-modules sketched above for complex varieties.

In order to find such a description, the first step is to identify the
analogue of the de Rham complex for crystals. This task was accomplished in
the monumental work [25] of Illusie (following ideas of Bloch [10] and
Deligne). The required complex is called the de Rham-Witt complex, denoted
$W\Omega_{X}^{i}$. This is a complex of sheaves on the formal scheme of Witt
vectors of $X$, $W(X)$. As $W(X)$ is a functorially defined lift of $X$ to
mixed characteristic, it had been suspected (since at least [36]) that it
might be a good place to look for cohomology theories which take values in
$W(k)$, and the de Rham-Witt complex realizes this vision. The theory of this
complex is extensively developed in [25], but for now let us only mention the
fact that there is a canonical isomorphism

$\mathbb{H}_{\text{crys}}^{\cdot}(X)\tilde{\to}\mathbb{H}^{\cdot}(W\Omega_{X}^{i})$

Furthermore, Etesse (in [20]) has constructed a functor from crystals to
modules equipped with a de Rham-Witt connection, and has shown that, when the
underlying $\mathcal{O}$-module $\mathcal{M}$ is a vector bundle, there is a
comparison

$\mathbb{H}_{\text{crys}}^{\cdot}(\mathcal{M})\tilde{\to}\mathbb{H}_{dRW}^{\cdot}(\mathcal{M})$

where on the right we have the hypercohomology of the de Rham-Witt complex
with values in $\mathcal{M}$.

It therefore seems natural to look for the analogue of the above theory of
$\mathcal{D}_{X}$-modules and try to interpret crystals in terms of some kind
of $\mathcal{D}$-modules on $W(X)$; one would hope also for the analogues of
the basic functorialities discussed above, as well as suitable comparisons
between crystalline cohomology, the $\mathcal{D}$-module pushforward, and the
de Rham-Witt cohomology.

This paper accomplishes exactly such a construction. In the remainder of this
introduction, we will explain the idea behind our theory, as well as stating
the main theorems of the paper and giving an outline of the contents.

Let us begin by supposing that $A$ admits local coordinates $\\{T_{1},\dots
T_{n}\\}$, i.e., there is an etale map $\text{Spec}(A)\to\mathbb{A}_{k}^{n}$ .
Then one has a concrete description of the ring of ($k$-linear) differential
operators $\mathcal{D}_{A}$ as the subring of $\text{End}_{k}(A)$ generated by
multiplication by elements of $A$ and operators of the form
$\partial_{i}^{[j]}$ where $\partial_{i}^{[j]}$ is the unique differential
operator on $A$ satisfying

$\partial_{i}^{[j]}(T_{i}^{m})={m\choose j}T_{i}^{m-j}$

and $\partial_{i}^{[j]}(T_{l}^{m})=0$ for $l\neq i$. The collection of
operators $\partial_{i}^{[j]}$ is very well behaved; for instance, we have for
all $j$ the relation

$\partial_{i}^{[pj]}\circ F=F\circ\partial_{i}^{[j]}$

where $F:A\to A$ is the Frobenius. The subring of $\mathcal{D}_{A}$ generated
by $A$ and the operators $\\{\partial_{i}^{[j]}\\}_{j\leq p^{m}}$ is
denoted222The notation in this area is not entirely standardized; see directly
below for our conventions on rings of differential operators
$\mathcal{\overline{D}}_{A}^{(m)}$, and the above relation is the key to
showing that the functor of pullback by Frobenius can be upgraded to a functor

$F^{*}:\mathcal{\overline{D}}_{A}^{(m)}-\text{mod}\to\mathcal{\overline{D}}_{A}^{(m+1)}-\text{mod}$

In fact, this functor is an equivalence of categories.

Now consider the ring $W(A)$. It is a $W(k)$ algebra, equipped with a lift of
Frobenius $F$ and an operator $V$ which acts as $p\cdot F^{-1}$. The
filtration $V^{i}(W(A))$ is a decreasing filtration by ideals, with respect to
which $W(A)$ is complete. One has that $p^{i}\in V^{i}(W(A))$, and $W(A)$ is
also $p$-adically complete. There is a canonical isomorphism

(1.2) $W(A)/V(W(A))\tilde{\to}A$

On this algebra we construct a family of operators, denoted
$\\{\partial_{i}\\}_{\lambda}$, where $\lambda\in\mathbb{Q}^{+}$ is a positive
rational number of the form $j/p^{r}$ with $j$ a positive integer and $r\geq
0$. For all $\lambda$ the operators $\\{\partial_{i}\\}_{\lambda}$ preserve
the filtration $\\{V^{i}(W(A))\\}$. If $\lambda\in\mathbb{Z}$ then
$\\{\partial_{i}\\}_{\lambda}$ is a lift of $\partial_{i}^{[\lambda]}$, via
the isomorphism (1.2). If $\text{val}_{p}(\lambda)=r<0$ then
$\\{\partial_{i}\\}_{\lambda}(W(A))\subset V^{r}(W(A))$; this means that for
any such $\lambda$ and any $\alpha\in W(A)$ we can define the operator
$F^{-r}(\alpha)\cdot\\{\partial_{i}\\}_{\lambda}:W(A)\to W(A)$ (this is
because, while $F^{-r}(\alpha)\notin W(A)$ in general, its action on
$V^{r}(W(A))$ is well-defined).

Most importantly, the operators satisfy the relations

$\\{\partial_{i}\\}_{\lambda}F=F\\{\partial_{i}\\}_{\lambda/p}$

and therefore

$\\{\partial_{i}\\}_{\lambda}V=V\\{\partial_{i}\\}_{p\lambda}$

As in the case of $\partial_{i}^{[j]}$, there is a formula for the action of
$\\{\partial_{i}\\}_{\lambda}$ on $W(A)$ in terms of coordinates; more
important than this is the fact there is an intrinsic construction of these
operators in terms of Hasse-Schmidt derivations on $W(A)$ (this is detailed in
Theorem 2.5 below). In particular, the algebra of $W(k)$-linear endomorphisms
of $W(A)$ generated333strictly speaking, we need to work with a kind of
completion of this algebra by $W(A)$ itself and terms of the form
$F^{-v}(\alpha)\cdot\\{\partial_{i}\\}_{\lambda}$ (where
$v=\text{max}\\{0,-\text{val}_{p}(\lambda)\\}$ can be defined independently of
any choice of coordinates. We term this algebra the Witt-differential
operators of $W(A)$, denoted $\widehat{\mathcal{D}}_{W(A)}$. For each integer
$m$ (positive or negative), we have also the differential operators of level
$\leq m$; this is the subalgebra of $\widehat{\mathcal{D}}_{W(A)}$
generated444as above, generated in the topological sense by operators
$F^{-v}(\alpha)\cdot\\{\partial_{i}\\}_{\lambda}$ for which
$\text{val}_{p}(\lambda)\leq m$. This algebra, denoted
$\widehat{\mathcal{D}}_{W(A)}^{(m)}$ can be defined independently of the
choice of coordinates (all of this is detailed in 2.1 below).

At this point the reader might (quite reasonably) ask: what is the
justification for this definition? And what is the relationship between
modules over this ring and crystals (or flat connections)? Both of these
questions are answered by the following construction: as $A$ is affine, it
admits a lift to a $p$-adically complete, $W(k)$-flat algebra $\mathcal{A}$.
We may choose an endomorphism $F:\mathcal{A}\to\mathcal{A}$ which lifts the
Frobenius map (in fact, as $A$ admits local coordinates, we can even demand
that $\mathcal{A}$ admits local coordinates $\\{T_{1},\dots T_{n}\\}$ for
which $F(T_{i})=T_{i}^{p}$; we call such lifts _coordinatized_). By a basic
property of the Witt vectors, such a lift induces a morphism

$\Phi:\mathcal{A}\to W(A)$

which intertwines the Frobenius lift $F$ with the Witt vector Frobenius on
$W(A)$. On $\mathcal{A}$ we have, for each $m\geq 0$, Berthelot’s ring of
arithmetic differential operators $\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$.
For the purpose of this introduction we focus on the case $m=0$, in which case
$\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}$ is simply the $p$-adic completion
of subalgebra of $\text{End}_{W(k)}(\mathcal{A})$ generated by $\mathcal{A}$
and the continuous, $W(k)$-linear derivations of $\mathcal{A}$. This algebra
acts faithfully on $\mathcal{A}$ itself.

Then we have

######  Theorem 1.5.

Consider the
$(\widehat{\mathcal{D}}_{W(A)}^{(0)},\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)})$-bimodule
$\text{Hom}_{W(k)}(\mathcal{A},W(A))$-this is a bimodule via the obvious
actions of $\widehat{\mathcal{D}}_{W(A)}^{(0)}$ on $W(A)$ and
$\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}$ on $\mathcal{A}$. Let
$\Phi^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}$ denote the completion of
$W(A)\otimes_{\mathcal{A}}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}$ along the
filtration
$V^{i}(W(A))\otimes_{\mathcal{A}}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}$.
Then there is an embedding

$\Phi^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}\to\text{Hom}_{W(k)}(\mathcal{A},W(A))$

which takes a tensor $\alpha\otimes P$ to the map $a\to\alpha\cdot\Phi(P(a))$.
The image of this embedding is exactly the
$(\widehat{\mathcal{D}}_{W(A)}^{(0)},\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)})$-bisubmodule
generated by $\Phi$. The object
$\Phi^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}$ is faithfully flat as a
right $\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}$-module, and projective as a
left $\widehat{\mathcal{D}}_{W(A)}^{(0)}$-module; in fact it is a summand of
$\widehat{\mathcal{D}}_{W(A)}^{(0)}$ itself.

This is a combination of 3.2 and Theorem 3.9 below. This bimodule allows us to
closely relate modules over $\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}$ (which
are, essentially, $\mathcal{A}$-modules with continuous flat connection over
$\mathcal{A}$) to modules over $\widehat{\mathcal{D}}_{W(A)}^{(0)}$. Before
giving the exact statement, let us mention that these results are directly
inspired by Berthelot’s Frobenius descent. To state it, recall that, for each
$m\geq 0$, Berthelot puts the structure of a
$(\mathcal{\widehat{D}}_{\mathcal{A}}^{(m+1)},\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)})$
bimodule on $F^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$. This bimodule
induces an equivalence of categories

$\mathcal{M}\to
F^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}\otimes_{\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}}\mathcal{M}:=F^{*}\mathcal{M}$

from $\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}-\text{mod}$ to
$\mathcal{\widehat{D}}_{\mathcal{A}}^{(m+1)}-\text{mod}$. The bimodule
$F^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$ can be identified with the
$(\mathcal{\widehat{D}}_{\mathcal{A}}^{(m+1)},\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)})$
bi-submodule of $\text{End}_{W(k)}(\mathcal{A})$ generated by $F$. So this
theory shows that taking an $m$th power Frobenius map corresponds to
increasing the order of the differential operators by $m$. Roughly speaking,
the map $\Phi:\mathcal{A}\to W(A)$ is something like a perfection map, or, an
infinite-power Frobenius map. The Witt differential operators, then, are
exactly the (infinite order) differential operators that correspond to
Berthelot’s arithmetic differential operators after taking the pullback by
$\Phi$. This is, in fact, how the definition of these operators was originally
arrived at.

Now let us explain some important consequences of Theorem 1.5. Reducing
everything mod $p$, we obtain a
$(\widehat{\mathcal{D}}_{W(A)}^{(0)}/p,\mathcal{D}_{A}^{(0)})$-bimodule
$\Phi^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}/p$ (here
$\mathcal{D}_{A}^{(0)}$ are the pd-differential operators on $A$, which by
definition are the reduction mod $p$ of the arithmetic differential operators
of level $0$). We have

######  Theorem 1.6.

Let $\Phi_{1},\Phi_{2}$ be two coordinatized lifts of Frobenius on
$\mathcal{A}$.

1) There is a canonical isomorphism of
$(\widehat{\mathcal{D}}_{W(A)}^{(0)}/p,\mathcal{D}_{A}^{(0)})$-bimodules

$\epsilon_{12}:\Phi_{1}^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}/p\tilde{\to}\Phi_{2}^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}/p$

If $\Phi_{3}$ is a third such lift we have the cocycle condition
$\epsilon_{23}\circ\epsilon_{12}=\epsilon_{13}$.

2) Suppose $p>2$. Then there in fact a canonical isomorphism of
$(\widehat{\mathcal{D}}_{W(A)}^{(0)},\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)})$-bimodules
$\epsilon_{12}:\Phi_{1}^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}\tilde{\to}\Phi_{2}^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}$.
For a third lift $\Phi_{3}$ we have the cocycle condition
$\epsilon_{23}\circ\epsilon_{12}=\epsilon_{13}$.

The second statement of this theorem is proved in Theorem 3.6, by showing that
$\Phi_{1}-\Phi_{2}$ is itself a differential operator of the required type;
this requires $p>2$. To prove the first statement (for all $p$) we use a
totally different technique, which relies on the fact that $W(A)/p\to A$ is a
square zero infinitesimal extension. Such extensions are ubiquitous in the
theory of flat connections, because, as observed by Grothendieck, giving a
flat connection on a sheaf is essentially the same as giving a canonical
extension of that sheaf to every square zero extension (c.f. [7], intro, for a
detailed discussion of this). Our proof of part $1)$ of the above theorem is
essentially a rephrasing of this argument (see 5.12 and 5.15 for details).

Now let us suppose that $X$ is an arbitrary smooth scheme over $k$. We have
the formal scheme $W(X)$, as well as the formal scheme $W(X)_{p=0}$, whose
structure sheaf is given by $\mathcal{O}_{W(X)}/p$ (in all cases the
topological space is $X$ itself). From the construction, there is a sheaf of
$\widehat{\mathcal{D}}_{W(X)}^{(0)}$, whose sections over an open affine
$\text{Spec}(A)$ which admits local coordinates agree with
$\widehat{\mathcal{D}}_{W(A)}^{(0)}$. In that case where $X=\text{Spec}(A)$ is
as above, we let $\mathfrak{X}=\text{Specf}(\mathcal{A})$. Then
$\Phi^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}$ sheafifies to
$\Phi^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)}$, a sheaf of
$(\widehat{\mathcal{D}}_{W(X)}^{(0)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)})$
bimodules.

So, globalizing all of the above yields

######  Corollary 1.7.

1) There is a sheaf of
$(\widehat{\mathcal{D}}_{W(X)}^{(0)}/p,\mathcal{D}_{X}^{(0)})$ bimodules on
$X$, denoted $\mathcal{B}_{X}^{(0)}$, whose sections over an open affine
$\text{Spec}(A)$ which admits local coordinates are equal to
$\Phi^{*}\mathcal{D}_{A}^{(0)}$. The functor
$\mathcal{M}\to\mathcal{B}_{X}^{(0)}\otimes_{\mathcal{D}_{X}^{(0)}}\mathcal{M}$
from $\mathcal{D}_{X}^{(0)}-\text{mod}$ to
$\mathcal{\widehat{D}}_{W(X)}^{(0)}/p-\text{mod}$ is exact and fully faithful,
and the derived functor

$\mathcal{B}_{X}^{(0)}\otimes_{\mathcal{D}_{X}^{(0)}}^{L}:D(\mathcal{D}_{X}^{(0)}-\text{mod})\to
D(\mathcal{\widehat{D}}_{W(X)}^{(0)}/p-\text{mod})$

is fully faithful as well. A complex $\mathcal{M}^{\cdot}\in
D(\mathcal{\widehat{D}}_{W(X)}^{(0)}/p-\text{mod})$ is called accessible if it
is of the form
$\mathcal{B}_{X}^{(0)}\otimes_{\mathcal{D}_{X}^{(0)}}^{L}\mathcal{N}^{\cdot}$
for some $\mathcal{N}^{\cdot}\in D(\mathcal{D}_{X}^{(0)}-\text{mod})$.

2) Let $\mathcal{M}^{\cdot}\in
D(\mathcal{\widehat{D}}_{W(X)}^{(0)}-\text{mod})$ be a cohomologically
complete555This is a technical condition, which can be considered the analogue
for complexes of being $p$-adically complete. It ensures that the functor
$\otimes_{W(k)}^{L}k$ is well behaved  complex. Then $\mathcal{M}^{\cdot}$ is
said to be accessible if $\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k$ is
accessible inside $D(\mathcal{\widehat{D}}_{W(X)}^{(0)}/p-\text{mod})$. The
complex $\mathcal{M}^{\cdot}$ is accessible iff, for any open affine
$\text{Spec}(A)\subset X$, which admits local coordinates, and any
coordinatized lift of Frobenius $\Phi$, we have

$\mathcal{M}^{\cdot}\tilde{\to}\Phi^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}}^{L}\mathcal{N}^{\cdot}$

for a cohomologically complete complex $\mathcal{N}^{\cdot}\in
D(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)}-\text{mod})$ (here, the symbol
$\Phi^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}}^{L}$
denotes the derived tensor product, followed by the cohomological completion).
In particular, the latter condition is independent of the choice of $\Phi$.
The inclusion functor
$D_{acc}(\mathcal{\widehat{D}}_{W(X)}^{(0)}-\text{mod})\to
D_{cc}(\mathcal{\widehat{D}}_{W(X)}^{(0)}-\text{mod})$ admits a right adjoint.

3) Suppose $p>2$, and suppose that $\mathfrak{X}$ is a smooth formal scheme
over $W(k)$ whose special fibre is $X$ (it might not exist in general). Then
there is an equivalence of categories

$D_{cc}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)}-\text{mod})\to
D_{\text{acc}}(\mathcal{\widehat{D}}_{W(X)}^{(0)}-\text{mod})$

where on the left, $D_{cc}$ stands for cohomologically complete complexes, and
on the right we have the accessible complexes inside
$D(\mathcal{\widehat{D}}_{W(X)}^{(0)}-\text{mod})$.

These facts are proved below in 3.1 (and for the existence of the right
adjoint see 4.5). There, one will also find analogues of the above for abelian
categories, at least under certain circumstances. For instance, one may work
with $\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}-\text{mod}$ (for some $r\geq
1$) and in this situation there is an abelian category of accessible modules;
denoted $\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}-\text{mod}_{\text{acc}}$,
and inside that there are categories of quasicoherent666We warn the reader
that this is not quite the usual notion of quasicoherence over a sheaf of
rings. The same goes for coherent. and coherent modules. There is also an
abelian category of coherent accessible modules over
$\mathcal{\widehat{D}}_{W(X)}^{(0)}$ (c.f. 3.20). These results have some
other natural variants as well. Most importantly, one can replace
$\mathcal{\widehat{D}}_{W(X)}^{(0)}$ with a certain completion,
$\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}$. We have also the algebras
$\mathcal{D}_{X,\text{crys}}^{(0)}$ and
$\mathcal{\widehat{D}}_{\mathfrak{X},\text{crys}}^{(0)}$, modules over which
correspond to sheaves with topologically nilpotent flat connection. The
analogue of 1.7 holds in this context, and in fact the restriction $p>2$ is
unnecessary in part $3)$ (c.f. Theorem 3.6 below). In particular, we obtain an
embedding of the category of quasi-coherent crystals on $X$ (over $W_{r}(k)$)
into the category of accessible
$\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}$-modules, and the same holds at the
derived level (this is detailed in 3.2 below).

Another useful variant is the category of complete accessible modules over
$\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}$; these are modules which are given
as the completion (along the topology defined by $V^{i}(\mathcal{O}_{W(X)})$)
of an accessible module; we show (c.f. Lemma 3.24) that this filtration is
extremely well-behaved on accessible modules, and therefore that the resulting
completion functor is exact and conservative; there is therefore a version in
the derived category as well.

With the basic categories defined, the rest of the paper is devoted to the
construction of the basic operations (the left-right interchange, pullback and
pushforward), the analogue of the de Rham resolution (which we call the de
Rham-Witt resolution), and the functor from accessible modules to de Rham-Witt
connections. We give an overview of these sections now.

The first basic operation we discuss is the left-right interchange on
accessible modules. To do so, we need to define the notion of an accessible
_right_ module over $\widehat{\mathcal{D}}_{W(X)}^{(0)}$. Starting in the
local case, we need the right handed version of the bimodule
$\Phi^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}$; one may swiftly define it
as
$\text{Hom}_{\widehat{\mathcal{D}}_{W(X)}^{(0)}}(\Phi^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)},\widehat{\mathcal{D}}_{W(A)}^{(0)}):=\Phi^{!}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}$.
There is also a characterization of it as a certain submodule of
$\text{Hom}_{\mathcal{A}}(W(A),\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)})$ (in
line with one would predict from Grothendieck duality; c.f. 4.1 below). It is
a
$(\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)},\widehat{\mathcal{D}}_{W(A)}^{(0)})$
bimodule and the analogues of Theorem 1.6 and 1.7 hold without any essential
change. This already shows (via the left-right interchange on
$\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}$-modules) that there is _local_
left-right interchange; to obtain the global interchange we need a little
more.

First of all, we have

$\omega_{\mathcal{A}}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}}^{L}\Phi^{!}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}\tilde{=}\omega_{\mathcal{A}}\otimes{}_{\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}}\Phi^{!}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}\tilde{\to}W\omega_{A}$

where $W\omega_{A}$ is the highest nonzero entry in the de Rham-Witt complex.
In particular $W\omega_{A}$ inherits the structure of a right
$\widehat{\mathcal{D}}_{W(A)}^{(0)}$-module, which can be shown to be
independent of all the choices involved.

Next, we make the important

######  Definition 1.8.

4.5 Let $\widehat{\mathcal{D}}_{W(X),\text{acc}}^{(0)}$ be the image of
$\mathcal{\widehat{D}}_{W(X)}^{(0)}$ under the right adjoint to the inclusion
functor $D_{acc}(\mathcal{\widehat{D}}_{W(X)}^{(0)}-\text{mod})\to
D_{cc}(\mathcal{\widehat{D}}_{W(X)}^{(0)}-\text{mod})$. This sheaf admits the
structure of a
$(\mathcal{\widehat{D}}_{W(X)}^{(0)},\mathcal{\widehat{D}}_{W(X)}^{(0)})$
bimodule.

Then, when $X=\text{Spec}(A)$ admits local coordinates, we have

$\widehat{\mathcal{D}}_{W(X),\text{acc}}^{(0)}=\Phi^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)}}\Phi^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)}$

(here $\widehat{\otimes}$ denotes the $p$-adic completion). This formula was
inspired by Berthelot’s isomorphism
$\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m+1)}\tilde{=}F^{!}F^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\tilde{=}F^{*}F^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}$
in his Frobenius descent theory. So, one may say that, this bimodule, rather
than $\mathcal{\widehat{D}}_{W(X)}^{(0)}$ itself, can be considered the
fundamental object of the accessible theory777There are interesting aspects to
Witt-differential operator theory beyond the accessible realm; however, we
don’t discuss them much in this paper, as they relate mostly to the case where
the level of differential operator is negative. Now, using this fact, one may
put two commuting structures of a right
$\mathcal{\widehat{D}}_{W(X)}^{(0)}$-module on the object

$W\omega_{X}\widehat{\otimes}_{\mathcal{O}_{W(X)}}\widehat{\mathcal{D}}_{W(X),\text{acc}}^{(0)}$

(where the hat denotes suitable completion, c.f. 4.10 for details). Then we
have

######  Corollary 1.9.

The functor
$\mathcal{M}^{\cdot}\to(W\omega_{X}\widehat{\otimes}_{\mathcal{O}_{W(X)}}\widehat{\mathcal{D}}_{W(X),\text{acc}}^{(0)})\otimes_{\widehat{\mathcal{D}}_{W(X)}^{(0)}}^{L}\mathcal{M}^{\cdot}$
yields an equivalence from left accessible to right accessible modules over
$\widehat{\mathcal{D}}_{W(X)}^{(0)}$. If $X=\text{Spec}(A)$ admits local
coordinates, then this equivalence is compatible with the classical
equivalence from left to right
$\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$-modules.

This is proved in 4.11 below.

Now we turn to pullback and pushforward. Let $\varphi:X\to Y$ be a morphism of
smooth varieties over $k$. There is, by the functoriality of the Witt vectors,
a canonical morphism $W\varphi:W(X)\to W(Y)$, which lifts $\varphi$. As in the
classical case, the operations are defined by suitable bimodules; we in fact
have both a
$(\widehat{\mathcal{D}}_{W(X)}^{(0)},W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(0)}))$-bimodule
$\widehat{\mathcal{D}}_{W(X)\to W(Y),\text{acc}}^{(0)}$ and a
$(W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(0)}),\widehat{\mathcal{D}}_{W(X)}^{(0)})$
bimodule, $\widehat{\mathcal{D}}_{W(Y)\leftarrow W(X),\text{acc}}^{(0)}$
(which is obtained via the left right swap). The definitions are slightly
involved, however, one can say that they are closely modeled on the analogous
definitions in the classical case (c.f. 3.2 below). We have the following

######  Definition 1.10.

The functor $LW\varphi^{*}:D(\widehat{\mathcal{D}}_{W(Y)}^{(0)}-\text{mod})\to
D(\widehat{\mathcal{D}}_{W(X)}^{(0)}-\text{mod})$ is defined as

$\mathcal{M}^{\cdot}\to\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(0)}\otimes_{W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(0)})}^{L}W\varphi^{-1}(\mathcal{M}^{\cdot})$

The basic properties of this functor are given in the

######  Theorem 1.11.

1) $LW\varphi^{*}$ takes
$D_{\text{acc}}(\widehat{\mathcal{D}}_{W(Y)}^{(0)}-\text{mod})$ to
$D_{\text{acc}}(\widehat{\mathcal{D}}_{W(X)}^{(0)}-\text{mod})$.

2) Let $LW\varphi^{*}:D(\widehat{\mathcal{D}}_{W(Y)}^{(0)}/p-\text{mod})\to
D(\widehat{\mathcal{D}}_{W(X)}^{(0)}/p-\text{mod})$ denote the functor defined
by the bimodule $\widehat{\mathcal{D}}_{W(X)\to W(Y),\text{acc}}^{(0)}/p$.
Then, under the equivalence of categories

$D(\mathcal{D}_{X}^{(0)}-\text{mod})\tilde{\to}D_{\text{acc}}(\widehat{\mathcal{D}}_{W(X)}^{(0)}/p-\text{mod})$

(and the analogous one for $Y$, c.f. 1.7, part $1)$ above), the functor
$LW\varphi^{*}$ corresponds to $L\varphi^{*}$, the usual
$\mathcal{D}^{(0)}$-module pullback.

3) Let $p>2$, and suppose that $\mathfrak{X}$ and $\mathfrak{Y}$ are smooth
formal schemes (over $W(k)$), lifting $X$ and $Y$, respectively, and let
$\varphi:\mathfrak{X}\to\mathfrak{Y}$ be a morphism lifting $\varphi$. Then,
under the equivalence of categories

$D_{cc}(\mathcal{D}_{\mathfrak{X}}^{(0)}-\text{mod})\tilde{\to}D_{\text{acc}}(\widehat{\mathcal{D}}_{W(X)}^{(0)}-\text{mod})$

(and the analogous one for $Y$, c.f. 1.7, part $3)$ above), the functor
$LW\varphi_{\text{}}^{*}$ corresponds to $L\varphi^{*}$, the usual
$\mathcal{\widehat{D}}^{(0)}$-module pullback.

These facts are proved in 4.2 below. Various standard consequences, such as
the fact that pullback commutes with composition, are also derived there;
there is also a version of part $3)$ for lifts of $X$ and $Y$ to $W_{r}(k)$
for some $r>1$. If $X$ and $Y$ are both affine, and we have coordinatized
lifts of Frobenius on both of them, there is an “explicit” description of
$\widehat{\mathcal{D}}_{W(X)\to W(Y),\text{acc}}^{(0)}$, which is 4.15 (this
is the key to proving the above theorem).

Now let us turn to pushforward, where the situation is (mostly) similar.

######  Definition 1.12.

The functor
${\displaystyle\int_{W\varphi}:}D_{\text{acc}}(\widehat{\mathcal{D}}_{W(X)}^{(0)}-\text{mod})\to
D_{\text{acc}}(\widehat{\mathcal{D}}_{W(Y)}^{(0)}-\text{mod})$ is defined by

$\int_{W\varphi}\mathcal{M}^{\cdot}:=R(W\varphi)_{*}(\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(0)}\otimes_{\widehat{\mathcal{D}}_{W(X)}^{(0)}}^{L}\mathcal{M}^{\cdot})_{\text{acc}}$

where $()_{\text{acc}}$ denotes the right adjoint to the inclusion from
accessible $\widehat{\mathcal{D}}_{W(Y)}^{(0)}$-modules to all
$\widehat{\mathcal{D}}_{W(Y)}^{(0)}$-modules. There is also, for $r\geq 1$ the
analogous functor

$\int_{W\varphi}\mathcal{M}^{\cdot}:=R(W\varphi)_{*}(\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(m)}/p^{r}\otimes_{\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r}}^{L}\mathcal{M}^{\cdot})_{\text{acc}}$

Unlike in the case of pullback, I don’t have a proof that the object

$R(W\varphi)_{*}(\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(m)}\otimes_{\widehat{\mathcal{D}}_{W(X)}^{(0)}}^{L}\mathcal{M}^{\cdot})$

is accessible when $\mathcal{M}^{\cdot}$ is- this question involves tricky
issues related to when the projection formula holds for some extremely large
objects. At any rate, with this definition we have the following analogue of
Theorem 1.11:

######  Theorem 1.13.

1) Under the equivalence of categories

$D(\mathcal{D}_{X}^{(0)}-\text{mod})\tilde{\to}D_{\text{acc}}(\widehat{\mathcal{D}}_{W(X)}^{(0)}/p-\text{mod})$

(and the analogous one for $Y$, c.f. 1.7, part $1)$ above), the functor
${\displaystyle\int_{W\varphi}}$ corresponds to
${\displaystyle\int_{\varphi}}$, the usual $\mathcal{D}^{(0)}$-module
pushforward.

2) Let $p>2$, and suppose that $\mathfrak{X}$ and $\mathfrak{Y}$ are smooth
formal schemes (over $W(k)$), lifting $X$ and $Y$, respectively, and let
$\varphi:\mathfrak{X}\to\mathfrak{Y}$ be a morphism lifting $\varphi$. Then,
under the equivalence of categories

$D_{cc}(\mathcal{D}_{\mathfrak{X}}^{(0)}-\text{mod})\tilde{\to}D_{\text{acc}}(\widehat{\mathcal{D}}_{W(X)}^{(0)}-\text{mod})$

(and the analogous one for $Y$, c.f. 1.7, part $3)$ above), the functor
${\displaystyle\int_{W\varphi}}$ corresponds to
${\displaystyle\int_{\varphi}}$, the usual
$\mathcal{\widehat{D}}^{(0)}$-module pushforward.

3) Fix $r\geq 1$ and consider the functor

$\widehat{\int}_{W\varphi}\mathcal{M}^{\cdot}:=R(W\varphi)_{*}(\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{$c-$acc}}^{(0)}/p^{r}\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r}}^{L}\mathcal{\widehat{M}}^{\cdot})$

where on the right we have the (derived) completion of $\mathcal{M}^{\cdot}$
with respect to $V^{i}(\mathcal{O}_{W(X)}/p^{r})$, and
$\widehat{\mathcal{D}}_{W(Y)\leftarrow W(X),\text{$c-$acc}}^{(0)}/p^{r}$ is a
suitable completion of $\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(0)}/p^{r}$. Then
${\displaystyle\widehat{\int}_{W\varphi}\mathcal{M}^{\cdot}}$ is isomorphic to
the (derived) completion of
${\displaystyle\int_{W\varphi}\mathcal{M}^{\cdot}}$ with respect to
$V^{i}(\mathcal{O}_{W(Y)}/p^{r})$. In other words, the completed pushforward
is given by the “naive” formula.

As in the case of pullback, we can deduce many of the basic properties of
push-forward from this theorem (this is all done in LABEL:subsec:Operations-
on-modulesPush!: below).

Now, let us discuss how these functors are related. The main result is

######  Theorem 1.14.

1) Let $\varphi:X\to Y$ be smooth of relative dimension $d$, and let
$W\varphi^{!}=LW\varphi^{*}[d]$. Then there is an isomorphism

$R\varphi_{*}R\mathcal{H}om_{\widehat{\mathcal{D}}_{W(X)}^{(0)}}(W\varphi^{!}\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})\tilde{\to}R\mathcal{H}om{}_{\widehat{\mathcal{D}}_{W(Y)}^{(0)}}(\mathcal{N}^{\cdot},\int_{W\varphi}\mathcal{M}^{\cdot})$

for any $\mathcal{N}^{\cdot}\in
D_{\text{acc}}(\widehat{\mathcal{D}}_{W(Y)}^{(0)}-\text{mod})$ and any
$\mathcal{M}^{\cdot}\in
D_{\text{acc}}(\widehat{\mathcal{D}}_{W(X)}^{(0)}-\text{mod})$. In particular,
${\displaystyle\int_{W\varphi}}$ is the right adjoint to $\varphi^{!}$.

2) Fix $r\geq 1$, and suppose
$\mathcal{M}\in\widehat{\mathcal{D}}_{W(X)}^{(0)}-\text{mod}_{\text{qcoh}}$ is
an accessible quasicoherent $\widehat{\mathcal{D}}_{W(X)}^{(0)}$-module which
is nilpotent. Let $\mathcal{\tilde{M}}$ denote the associated crystal on $X$
over $W_{r}(k)$. Then, for each $i$
${\displaystyle\mathcal{H}^{i}(\int_{W\varphi}\mathcal{M})}$ is an accessible
quasicoherent $\widehat{\mathcal{D}}_{W(Y)}^{(0)}$-module which is nilpotent,
and there is a functorial isomorphism

$\mathcal{H}^{i-d}\tilde{(\int_{W\varphi}\mathcal{M})}\tilde{\to}R^{i}\varphi_{*,\text{crys}}(\tilde{\mathcal{M}})$

This is theorems 4.32 and 4.34 below.

To finish things off, let us discuss the de Rham-Witt theory. By a theorem of
Etesse, there is a natural functor from crystals over $W_{r}(k)$ to de Rham-
Witt connections over $W_{r}\Omega_{X}^{1}$; and one may easily show that it
extends to a functor to continuous de Rham-Witt connections over
$W\Omega_{X}^{1}/p^{r}$ (this object, being flat over $W_{r}(k)$, is better
behaved). Using this theory we put a canonical continuous de Rham-Witt
connection on the object
$\widehat{\mathcal{D}}_{W(X),c-\text{acc}}^{(0)}/p^{r}$ (this is the
completion of $\widehat{\mathcal{D}}_{W(X),\text{acc}}^{(0)}/p^{r}$ along
$V^{i}(\mathcal{O}_{W(X)}/p^{i})$). From this we deduce

######  Theorem 1.15.

There is an equivalence of categories from the (abelian) category of
accessible quasicoherent $\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r}$ modules to
the category of quasicoherent modules over $W(X)_{p^{r}=0}$ with continuous
flat connection over $W\Omega_{X}^{1}/p^{r}$.

The fact that this functor is fully faithful is actually quite
straightforward; the essential surjectivity is trickier. In fact, the case of
vector bundles is handled by the paper of Bloch [11], whose central lemma also
forms the core of our proof (c.f. Lemma 5.18 below).

Finally, let us present the de Rham-Witt resolution. If $\varphi:X\to Y$ is a
smooth morphism of relative dimension $d$, we have the relative de Rham-Witt
complex of Langer-Zink, as developed in [32], denoted $W\Omega_{X/Y}^{\cdot}$.
The de Rham-Witt connection on
$\widehat{\mathcal{D}}_{W(X),c-\text{acc}}^{(0)}/p^{r}$ yields also a
continuous connection map

$\nabla:\widehat{\mathcal{D}}_{W(X),c-\text{acc}}^{(0)}/p^{r}\to\widehat{\mathcal{D}}_{W(X),c-\text{acc}}^{(0)}/p^{r}\widehat{\otimes}_{\mathcal{O}_{W(X)}/p^{r}}W\Omega_{X/Y}^{1}/p^{r}$

(as above the hat denotes completion with respect to
$V^{i}(\mathcal{O}_{W(X)}/p^{r})$), and therefore an associated de Rham-Witt
complex
$(\widehat{\mathcal{D}}_{W(X),c-\text{acc}}^{(0)}/p^{r}\widehat{\otimes}_{\mathcal{O}_{W(X)}/p^{r}}W\Omega_{X/Y}^{\cdot}/p^{r},\nabla)$.
We have

######  Theorem 1.16.

The complex
$(\widehat{\mathcal{D}}_{W(X),c-\text{acc}}^{(0)}/p^{r}\widehat{\otimes}_{\mathcal{O}_{W(X)}/p^{r}}W\Omega_{X/Y}^{\cdot}/p^{r},\nabla)$
is exact except at the right-most term (which is the $d$th) and we have

$\mathcal{H}^{d}((\widehat{\mathcal{D}}_{W(X),c-\text{acc}}^{(0)}/p^{r}\widehat{\otimes}_{\mathcal{O}_{W(X)}/p^{r}}W\Omega_{X/Y}^{\cdot}/p^{r},\nabla))\tilde{\to}\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{c-acc}}^{(0)}/p^{r}$

where $\widehat{\mathcal{D}}_{W(Y)\leftarrow W(X),\text{c-acc}}^{(0)}/p^{r}$
is the completion of $\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(0)}/p^{r}$ along $V^{i}(\mathcal{O}_{W(X)}/p^{r})$; when
$Y$ is a point it is simply $W\omega_{X}/p^{r}$. It follows that, for any
$\mathcal{M}\in\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r}-\text{mod}_{\text{acc}}$
there is an isomorphism

$\widehat{\int}_{W\varphi_{p^{r}=0}}\mathcal{M}\tilde{\to}R\varphi_{*}(\widehat{\mathcal{M}}\widehat{\otimes}_{\mathcal{O}_{W(X)}/p^{r}}W\Omega_{X/Y}^{\cdot}/p^{r})$

of sheaves over $\mathcal{O}_{W(Y)}/p^{r}$; as above all of the completions
are along $V^{i}(\mathcal{O}_{W(X)}/p^{r})$.

This is Theorem 4.39 below. The second part of the theorem therefore implies
that crystalline cohomology for an arbitrary quasicoherent crystal can be
computed via the de Rham-Witt complex mod $p^{r}$. This result has antecedents
in the work of Etesse [20] and Berthelot [6], who worked instead with the de
Rham-Witt complex $W_{r}\Omega_{X}^{\cdot}$. To get a comparison theorem when
working with that complex forces one to consider crystals which are flat over
$W_{r}(k)$ (Etesse considered only vector bundles). Despite these differences,
the proofs (including the one in this paper) all rely essentially on the basic
fact that the de-Rham-Witt complex mod $p$ is quasi-isomorphic to the de Rham
complex of $X$. In our strategy, though, evaluating the complex on a single
object (namely $\widehat{\mathcal{D}}_{W(X),c-\text{acc}}^{(0)}/p^{r}$) allows
one to deduce the result for the entire category, including for those
accessible modules which are not nilpotent. This appears to be new.

###  1.1. Notations and conventions

Throughout the entire paper a perfect field $k$ of characteristic $p>0$ is
fixed; as indicated in the introduction, some of the results depend on weather
$p=2$, we will indicate this when it occurs. We let $W(k)$ denote the
$p$-typical Witt vectors of $k$.

Letters $X,Y,Z$ denote a smooth varieties over $k$. Letters $A,B,C$ denote
smooth $k$-algebras, i.e., finite type $k$-algebras such that the induced
morphism $\text{Spec}(A)\to\text{Spec}(k)$ is smooth. Gothic letters
$\mathfrak{X}$, $\mathfrak{Y}$,$\mathfrak{Z}$ denote smooth formal schemes
over $W(k)$, whose special fibres are denoted by the corresponding standard
letters. We use $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ to denote smooth
algebra over $W(k)$; we use $\mathfrak{X}_{r}$, $\mathfrak{Y}_{r}$,
$\mathfrak{Z}_{r}$ to denote smooth schemes over $W_{r}(k)$ and
$\mathcal{A}_{r}$, $\mathcal{B}_{r}$, $\mathcal{C}_{r}$ smooth algebras over
$W_{r}(k)$.

We shall make essential use of Berthelot’s arithmetic differential operators;
c.f. [4]. In particular, for $m\geq 0$,
$\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}$ denotes the arithmetic
differential operators of level $m$, a $W(k)$-flat and $p$-adically complete
sheaf of algebras whose reduction mod $p$ is denoted $\mathcal{D}_{X}^{(m)}$.
When $m=0$ this algebra is the usual PD differential operators of [7]; for a
quick definition and exposition of the basic properties of
$\mathcal{D}_{X}^{(0)}$ we recommend [8], chapter $2$. In addition, the
construction has been extended by Shiho to define algebras
$\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}$ for $m<0$; c.f. [38] for details.

For all such $X$ over $k$ we have the formal scheme of $p$-typical Witt
vectors $W(X)$, whose underlying topological space is $X$ and whose topology
is defined by a collection of ideals $V^{i}(\mathcal{O}_{W(X)})$ (c.f.
[13],[14] for a detailed account). None of these are locally finitely
generated, but we have the finite type scheme $W_{m}(X)$ whose structure sheaf
is $\mathcal{O}_{W(X)}/V^{m}(\mathcal{O}_{W(X)})$. We also have the formal
schemes $W(X)_{p^{r}=0}$ for any $r\geq 1$, whose underlying sheaf of rings
are given by $\mathcal{O}_{W(X)}/p^{r}$. By a quasicoherent sheaf on this
scheme we mean an inverse limit of quasicoherent sheaves on the finite type
schemes $W_{m}(X)_{p^{r}=0}$. The morphisms in this category are continuous
maps of $\mathcal{O}_{W(X)}/p^{r}$-modules. This category is additive, though
not abelian and otherwise terribly behaved.

Finally, let us take a moment to discuss the notion of cohomological
completeness. This notion, which also goes under the name derived
completeness, has been developed extensively in, e.g., [28] [39], Tag 091N,
[18]. We will use as a reference [28], chapter $1$ as it develops the notion
in the context we need- a noncommutative sheaf of rings $\mathcal{R}$ over a
topological space $X$, so that $\mathcal{R}$ possesses a global central
element $p$ (called $h$ in [28]) and such that $\mathcal{R}$ has no nonzero
$p$-torsion. For us $\mathcal{R}$ will in fact be a flat $W(k)$-module. In
this case a complex $\mathcal{M}^{\cdot}\in D(\mathcal{R}-\text{mod})$ is
called cohomologically complete if

$R\mathcal{H}om_{\mathcal{R}}(\mathcal{R}[p^{-1}],\mathcal{M}^{\cdot})=R\mathcal{H}om_{W(k)}(W(k)[p^{-1}],\mathcal{M}^{\cdot})=0$

A $p$-torsion-free sheaf of modules is cohomologically complete iff it is
$p$-adically complete in the usual sense (c.f. [28], lemma 1.5.4). Further,
any complex in the image of the natural functor
$D(\mathcal{R}/p^{n}-\text{mod})\to D(\mathcal{R}-\text{mod})$ is
cohomologically complete (for any $n\geq 1$). The collection of such complexes
forms a thick triangulated subcategory, which we denote
$D_{cc}(\mathcal{R}-\text{mod})$. The main facts about
$D_{cc}(\mathcal{R}-\text{mod})$ that we need are, firstly, the Nakayama
lemma: if $\mathcal{M}^{\cdot}\in D_{cc}(\mathcal{R}-\text{mod})$, then
$\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k=0$ iff $\mathcal{M}^{\cdot}=0$ (this
is [28], proposition 1.5.8.). Secondly, we shall use the fact that the
inclusion functor $D_{cc}(\mathcal{R}-\text{mod})\to
D(\mathcal{R}-\text{mod})$ admits a right adjoint, the derived completion
functor (c.f. [28], proposition 1.5.6), concretely, this functor is given by

$\mathcal{M}^{\cdot}\to
R\mathcal{H}om_{\mathcal{R}}(\mathcal{R}[p^{-1}]/\mathcal{R}[-1],\mathcal{M}^{\cdot}):=\widehat{\mathcal{M}^{\cdot}}$

For complexes over the various $W(k)$-torsion-free rings appearing in this
paper, the symbol $\widehat{?}$ will denote cohomological completion. We
should mention that this symbol is also sometimes used for complexes which are
annihilated by $p^{r}$, where it stands for the (derived) completion with
respect to $V^{i}(\mathcal{O}_{W(X)}/p^{r})$ (c.f. 3.25 and the discussion
directly above). As complexes annihilated by $p^{r}$ are automatically
cohomologically complete (in the sense used in this paper), the two notions
are essentially disjoint, and hopefully this will not cause undo confusion.

###  1.2. Acknowledgements

I would like to thank Michel Gros for inviting me to speak about this work at
Rennes, and for helpful conversations afterwards. I would also like to that
Bernard Le Stum for friendly conversations about related topics. I would like
to gratefully acknowledge the support of the NSF.

##  2\. Higher Derivations on Witt Vectors

In this section we define, for each $m\in\mathbb{Z}$, the sheaf
$\mathcal{E}_{W(X)}^{(m)}$ of higher derivations on the formal scheme $W(X)$
(of level $m$), as well as the algebra $\mathcal{\widehat{D}}_{W(X)}^{(m)}$ of
Witt-differential operators, which is (the completion of) the sheaf of
algebras generated by $\mathcal{O}_{W(X)}$ and $\mathcal{E}_{W(X)}^{(m)}$. The
key to defining $\mathcal{E}_{W(X)}^{(m)}$ is the observation that Hasse-
Schmidt derivations on $X$ can be lifted canonically to the Witt-vectors.

In order to make all this precise, we recall the

######  Definition 2.1.

Let $R$ be a commutative ring, and let $D_{0}:B\to A$ be a morphism of
commutative $R$-algebras. A Hasse-Schmidt derivation (or higher derivation) of
length $n$ from $B$ to $A$ over $R$ is a sequence of $R$-linear operators
$D=(D_{0},\dots,D_{n})$ (we allow $n=\infty$ for such a sequence indexed by
$\mathbb{N}$) such that

$D_{l}(xy)=\sum_{i+j=l}D_{i}(x)D_{j}(y)$

If $A=B$, we shall suppose that $D_{0}=Id$ unless otherwise specified. In that
case we simply refer to a Hasse-Schmidt derivation of $A$ (over $R$). The
operator $D_{i}$ is referred to as the $i$th component of $D$.

We refer the reader to [40] and [33] for more details and some interesting
applications (e..g to jet spaces in algebraic geometry). At any rate it
follows immediately from the definition that $D_{1}\in\text{Der}_{R}(A,B)$,
and then by a quick induction it follows that each $D_{i}$ is an $R$-linear
differential operator from $B$ to $A$ of order $\leq i$.

In the sequel, it will be convenient to use the elementary fact that a Hasse-
Schmidt derivation is equivalent to a map of $R$-algebras

$\varphi_{D}:B\to A[t]/t^{n+1}$

where the sequence $D=(D_{0},\dots,D_{n})$ corresponds to

$\varphi_{D}(b)=\sum_{i=0}^{n}D_{i}(b)t^{i}$

(here, if $n=\infty$, then the target is taken to be the power series ring
$A[[t]]$). For instance, from this characterization and the infinitesimal
lifting property (c.f. [15], ch. 2, proposition 6), we see immediately that if
$B\to B^{\prime}$ is an etale morphism of $R$-algebras such that the structure
morphism $D_{0}:B\to A$ extends to a morphism $D^{\prime}_{0}:B^{\prime}\to
A$, then any Hasse-Schmidt derivation $D=(D_{0},\dots,D_{n})$ from $B$ to $A$
over $R$ extends uniquely to
$D^{\prime}=(D^{\prime}_{0},\dots,D^{\prime}_{n})$ from $B^{\prime}$ to $A$
over $R$. In particular, a Hasse-Schmidt derivation of $A$ over $R$ extends
uniquely to any localization $A_{g}$ (for $g\in A$).

It turns out that, after applying appropriate Frobenius twists, higher
derivations are very well-behaved on Witt vectors. To explain this, we start
with the following basic result:

######  Lemma 2.2.

Let $A$ be a smooth $k$-algebra. For any $r\geq 0$, let $\mathcal{A}_{r+1}$ be
a flat $W_{r+1}(k)$-algebra such that $\mathcal{A}_{r+1}/p\tilde{=}A$. Then
there is an embedding

$W_{r+1}(A)\to\mathcal{A}_{r+1}$

which is $F^{r}$-semilinear over $W_{r+1}(k)$. The map is given as follows: if
$(f_{1},f_{2}\dots,f_{r+1})\in W_{r+1}(A)$ we choose any lifts $\tilde{f}_{i}$
of $f_{i}$ in $\mathcal{A}_{r+1}$ and send

$(f_{1},f_{2}\dots,f_{r+1})\to\tilde{f}_{1}^{p^{r}}+p\tilde{f}_{2}^{p^{r-1}}+\dots+p^{r}\tilde{f}_{r+1}$

Further, if $F$ is any lift of the absolute Frobenius map to
$\mathcal{A}_{r+1}$, then the restriction of $F$ to the image of $W_{r+1}(A)$
is isomorphic to the Witt-vector Frobenius; in particular, we have

$F(\tilde{f}_{1}^{p^{r}}+p\tilde{f}_{2}^{p^{r-1}}+\dots+p^{r}\tilde{f}_{r+1})=\tilde{f}_{1}^{p^{r+1}}+p\tilde{f}_{2}^{p^{r}}+\dots+p^{r}\tilde{f}_{r+1}^{p}$

For each $i\geq 0$, the ideal $V^{i}(W_{r+1}(A))$ is given by the intersection
of $p^{i}\mathcal{A}_{r+1}$ with $W_{r+1}(A)$.

If $\varphi^{\\#}:B\to A$ is a morphism of smooth $k$-algebras, and we
consider any lift to
$\varphi_{r+1}^{\\#}:\mathcal{B}_{r+1}\to\mathcal{A}_{r+1}$, then
$\varphi_{r+1}^{\\#}(W_{r+1}(B))\subset W_{r+1}(A)$, and the induced map

$\varphi_{r+1}^{\\#}:W_{r+1}(B)\to W_{r+1}(A)$

agrees with the functorial map $W\varphi^{\\#}$ coming from Witt vector
theory.

###### Proof.

By definition, the ghost map

$w_{r}:W_{r+1}(\mathcal{A}_{r+1})\to\mathcal{A}_{r+1}$

given by

$(\tilde{f}_{1},\tilde{f}_{2}\dots,\tilde{f}_{r+1})\to\tilde{f}_{1}^{p^{r}}+p\tilde{f}_{2}^{p^{r-1}}+\dots+p^{r}\tilde{f}_{r+1}$

is a ring homomorphism, and since $p^{r+1}$ annihilates $\mathcal{A}_{r+1}$,
we see that $w_{r}$ factors through $W_{r+1}(\mathcal{A}_{r+1}/p)=W_{r+1}(A)$.
Call the image of this map $W^{\prime}$. I claim that the surjection
$W_{r+1}(A)\to W^{\prime}$ is also injective. For, if

$\tilde{f}_{1}^{p^{r}}+p\tilde{f}_{2}^{p^{r-1}}+\dots+p^{r}\tilde{f}_{r+1}=0$

then, taking the image in $A$, we see that $f_{1}^{p^{r}}=0$ which implies
$f_{1}=0$ and so $\tilde{f}_{1}=p\tilde{g}_{1}$; but then
$\tilde{f}_{1}^{p^{r}}=0$. Therefore
$p\tilde{f}_{2}^{p^{r-1}}+\dots+p^{r}\tilde{f}_{r+1}=0$ in $pA_{r+1}$ but
since $A_{r+1}$ is flat over $W_{r+1}(k)$ this implies
$\tilde{f}_{2}^{p^{r-1}}+\dots+p^{r-1}\tilde{f}_{r+1}=0$ in
$A_{r}=A_{r+1}/p^{r}$. Thus an induction on $r$ shows that each $f_{i}=0$
(where as above $f_{i}$ is the image of $\tilde{f}_{i}$ in $A$) as required.

Now let us show that the map $w_{r}$ is $F^{r}$-semilinear. Choose a lift of
Frobenius on $\mathcal{A}_{r+1}$ whose restriction to $W_{r+1}(k)$ is the
Witt-vector Frobenius. By a universal property of Witt-vectors (c.f., e.g.,
[37] proposition 1.1.23) there is a unique map

$s:\mathcal{A}_{r+1}\to W_{r+1}(\mathcal{A}_{r+1})$

so that $w_{r}\circ S=F^{r}$. By our choice of $F$ the map $s$ is a morphism
of $W_{r+1}(k)$-algebras; therefore $w_{r}$ is $F^{r}$-semilinear over
$W_{r+1}(k)$ as claimed.

Next let $F$ be any lift of Frobenius on $A_{r+1}$. As $F$ is a map of
algebras we have

$F(\tilde{f}_{1}^{p^{r}}+p\tilde{f}_{2}^{p^{r-1}}+\dots+p^{r}\tilde{f}_{r+1})=F(\tilde{f}_{1})^{p^{r}}+pF(\tilde{f}_{2})^{p^{r-1}}+\dots+p^{r}F(\tilde{f}_{r+1})$

so that $F(W^{\prime})\subset W^{\prime}$; further, by what we have just
proved the term
$F(\tilde{f}_{1})^{p^{r}}+pF(\tilde{f}_{2})^{p^{r-1}}+\dots+p^{r}F(\tilde{f}_{r+1})$
depends only on the images $F(f_{i})=f_{i}^{p}$ in $A$; whence the statement.
The fact that

$V^{i}(W_{r+1}(A))=p^{i}\mathcal{A}_{r+1}\cap W_{r+1}(A)$

follows immediately from the definition of $V^{i}(W_{r+1}(A))$.

Finally, consider a map $\varphi^{\\#}:B\to A$, which, by the infinitesimal
lifting property, can be lifted to a map
$\varphi_{r+1}^{\\#}:\mathcal{B}_{r+1}\to\mathcal{A}_{r+1}$. Then

$\varphi^{\\#}(\tilde{f}_{1}^{p^{r}}+p\tilde{f}_{2}^{p^{r-1}}+\dots+p^{r}\tilde{f}_{r+1})=\varphi^{\\#}(\tilde{f}_{1})^{p^{r}}+\dots+p^{r}\varphi^{\\#}(\tilde{f}_{r+1})$

from which the result follows immediately. ∎

In the sequel, we shall make extensive use of this map, writing
$W_{r+1}(A^{(r)})\subset\mathcal{A}_{r+1}$ for the image.

An important special case of this lemma is given by the

######  Example 2.3.

Suppose $A=k[T_{1},\dots,T_{n}]$, and
$\mathcal{A}_{r+1}=W_{r+1}(k)[T_{1},\dots,T_{n}]$. Then the image of
${\displaystyle W_{r+1}(A^{(r)})\to\mathcal{A}_{r+1}}$ is the
$W_{r+1}(k)$-submdule spanned by

$\\{p^{j}\prod_{i=1}^{n}T_{i}^{a_{i}p^{r-j}}\\}$

where the $\\{a_{i}\\}$ are any natural numbers. Indeed, it is clear that all
of the displayed monomials are contained in $W_{r+1}(A^{(r)})$. To see the
converse, we use induction on $r$, the case $r=0$ being trivial. Consider a
term of the form $p^{j}f^{p^{r-j}}\in W_{r+1}(A^{(r)})$, for $j<r$ (when $j=r$
the result is obvious). By induction we have an expression

$p^{j}f^{p^{r-1-j}}=\sum_{j=0}^{r-1}\sum_{i}b_{i}p^{j}T_{i}^{a_{i}p^{r-1-j}}\text{mod}(p^{r})$

We may consider the lift of Frobenius which takes $T_{i}\to T_{i}^{p}$.
Applying this map to the displayed equality yields

$p^{j}f^{p^{r-j}}=\sum_{j=0}^{r-1}\sum_{i}b_{i}p^{j}T_{i}^{a_{i}p^{r-j}}\text{mod}(p^{r})$

which implies

$p^{j}f^{p^{r-j}}=\sum_{j=0}^{r-1}\sum_{i}b_{i}p^{j}T_{i}^{a_{i}p^{r-j}}+\sum_{i}b_{i}p^{r}T^{a_{i}}$

as claimed.

We will construct our canonical lifts of Hasse-Schmidt derivations using the
above construction of $W_{r+1}(A^{(r)})$; to apply it, we make use of the
following straightforward application of the infinitesimal lifting property:

######  Lemma 2.4.

For any $r\geq 0$, let $\mathcal{A}_{r+1},\mathcal{B}_{r+1}$ be flat
$W_{r+1}(k)$-algebras such that $\mathcal{A}_{r+1}/p\tilde{=}A$ and
$\mathcal{B}_{r+1}/p\tilde{=}B$. Let $D=(D_{0},\dots,D_{n})$ be any Hasse-
Schmidt derivation from $B$ to $A$, over $k$. Then there is a lift to a Hasse-
Schmidt derivation $\tilde{D}=(\tilde{D}_{0},\dots,\tilde{D}_{n})$, over
$W_{r+1}(k)$, from $\mathcal{B}_{r+1}$ to $\mathcal{A}_{r+1}$,

In general, there will be many lifts of a given Hasse-Schmidt derivation.
However, upon restricting to Witt vectors we have the following essential
result:

######  Theorem 2.5.

Let $B,A$ be smooth $k$-algebras, and let $D=(D_{0},D_{1},\dots):B\to A$ be a
Hasse-Schmidt derivation (of any length). Let
$\tilde{D}=(\tilde{D}_{0},\tilde{D}_{1},\dots)$ be a lift of $D$ to a Hasse-
Schmidt derivation from $\mathcal{B}_{r+1}$ to $\mathcal{A}_{r+1}$ (where, as
above, these are flat lifts of $B$ and $A$, respectively). Then $\tilde{D}$
takes the subalgebra $W_{r+1}(B^{(r)})$ into $W_{r+1}(A^{(r)})$. Further, the
induced map $\tilde{D}:W_{r+1}(B^{(r)})\to W_{r+1}(A^{(r)})$ is independent of
the choice of lift $\tilde{D}$.

###### Proof.

We begin by showing that $\tilde{D}$ takes $W_{r+1}(B^{(r)})$ into
$W_{r+1}(A^{(r)})$. We recall the following formula for the action of Hasse-
Schmidt derivations on powers, which is easily checked by induction:

$\tilde{D}_{j}(f^{n})=\sum_{i_{1}+\dots+i_{n}=j}\tilde{D}_{i_{1}}(f)\cdots\tilde{D}_{i_{n}}(f)$

so that we have

(2.1)
$\tilde{D}_{j}(p^{l}f^{p^{r-l}})=p^{l}\sum_{i_{1}+\dots+i_{p^{r-l}}=j}\tilde{D}_{i_{1}}(f)\cdots\tilde{D}_{i_{p^{r-l}}}(f)$

Now, the set

$\\{(i_{1},\dots,i_{p^{r-l}})\in\mathbb{N}^{p^{r-l}}|i_{1}+\dots+i_{p^{r-l}}=j\\}$

is acted upon by the symmetric group $S_{p^{r-l}}$, and, after grouping like
terms together, the coefficient of a term
$\tilde{D}_{i_{1}}(f)\cdots\tilde{D}_{i_{p^{r-l}}}(f)$ in (2.1) is precisely
the size of the $S_{p^{r-l}}$ orbit of $(i_{1},\dots,i_{p^{r-l}})$.

To calculate this size, suppose that there are $m$ distinct numbers occurring
in $(i_{1},\dots,i_{p^{r-l}})$, call them $\\{j_{1},\dots,j_{m}\\}$. Let
$\\{C_{n}\\}_{n=1}^{m}$ denote the corresponding $m$ subsets of
$\\{1,\dots,p^{r-l}\\}$, so that

$C_{n}=\\{t|i_{t}=j_{n}\\}$

The stabilizer of the action of $S_{p^{r-l}}$ on $(i_{1},\dots,i_{p^{r-l}})$
is then the group

$S_{C_{1}}\times\cdots\times S_{C_{m}}$

of permutations which preserve each $C_{i}$. If we let $c_{i}=|C_{i}|$ then we
deduce that

$|\mathcal{O}_{(i_{1},\dots,i_{p^{r-l}})}|=\frac{(p^{r-l})!}{c_{1}!\cdots
c_{m}!}$

where the left hand side denotes the size of the orbit. Summing up, we deduce
that

(2.2)
$\tilde{D}_{j}(p^{l}f^{p^{r-l}})=\sum_{\mathcal{O}}p^{l}\frac{(p^{r-l})!}{c_{1}!\cdots
c_{m}!}(\tilde{D}_{j_{1}}(f))^{c_{1}}\cdots(\tilde{D}_{j_{m}}(f))^{c_{m}}$

where $\mathcal{O}$ ranges over the set of orbits of $S_{p^{r-l}}$ on
$\\{(i_{1},\dots,i_{p^{r-l}})\in\mathbb{N}^{p^{r-l}}|i_{1}+\dots+i_{p^{r-l}}=j\\}$.

So, we must show that

$p^{l}\frac{(p^{r-l})!}{c_{1}!\cdots
c_{m}!}(\tilde{D}_{j_{1}}(f))^{c_{1}}\cdots(\tilde{D}_{j_{m}}(f))^{c_{m}}\in
W_{r+1}(A^{(r)})$

Consider the map $W_{r+1}(k)[T_{1},\dots,T_{m}]\to A_{r+1}$ given by sending
$T_{i}\to\tilde{D}_{j_{i}}(f)$. This map takes
$W_{r+1}(k[T_{1},\dots,T_{m}]^{(r)})$ to $W_{r+1}(A^{(r)})$, and so it
suffices to prove that

$p^{l}\frac{(p^{r-l})!}{c_{1}!\cdots
c_{m}!}(T_{1})^{c_{1}}\cdots(T_{m})^{c_{m}}\in
W_{r+1}(k[T_{1},\dots,T_{m}]^{(r)})$

On the other hand, the multinomial formula tells us that

$p^{l}(T_{1}+\dots+T_{m})^{p^{r-l}}=\sum_{k_{1}+\dots
k_{m}=p^{r-l}}p^{l}\frac{(p^{r-l})!}{k_{1}!\cdots k_{m}!}T_{1}^{k_{1}}\cdots
T_{m}^{k_{m}}$

and by 2.3 all of the terms of this sum are in
$W_{r+1}(k[T_{1},\dots,T_{m}]^{(r)})$, since the left hand side is. So putting
$k_{i}=c_{i}$ implies the result.

To see the uniqueness, we note that we have just proved an equality

$p^{l}\frac{(p^{r-l})!}{c_{1}!\cdots
c_{m}!}(\tilde{D}_{j_{1}}(f))^{c_{1}}\cdots(\tilde{D}_{j_{m}}(f))^{c_{m}}=b_{c_{1},\dots,c_{m}}p^{\alpha}(\tilde{D}_{j_{1}}(f))^{c^{\prime}_{1}p^{r-\alpha}}\cdots(\tilde{D}_{j_{m}}(f))^{c^{\prime}_{m}p^{r-\alpha}}$

where

$\alpha=\text{val}(p^{l}\frac{(p^{r-l})!}{c_{1}!\cdots c_{m}!})$

and $b_{c_{1},\dots,c_{m}}$ is the image of ${\displaystyle
p^{l}\frac{(p^{r-l})!}{c_{1}!\cdots c_{m}!}p^{-\alpha}}$ in
$\mathbb{Z}/p^{r+1}$, and $c_{i}=c^{\prime}_{i}p^{r-\alpha}$ for all $i$. But
by Lemma 2.2, such a term depends only on the image of
$\\{\tilde{D}_{j_{1}}(f),\dots,\tilde{D}_{j_{m}}(f)\\}$ in $A$, i.e., it
depends only on the Hasse-Schmidt derivation $D$. Since
$\tilde{D}_{j}(p^{l}f^{p^{r-l}})$ is a sum of such terms, we see that it is
independent of the choice of lift $\tilde{D}$ as claimed. ∎

If we unpack the proof of the theorem, we in fact obtain a formula for the
Hasse-Schmidt derivation as a sequence of maps $W_{r+1}(B^{(r)})\to
W_{r+1}(A^{(r)})$; if we then apply the isomorphisms $A\tilde{=}A^{(r)}$ and
$B\tilde{=}B^{(r)}$ we obtain a Hasse-Schmidt derivation $W_{r+1}(B)\to
W_{r+1}(A)$. It will be useful to record this explicitly:

######  Corollary 2.6.

Let $A,B$ be smooth $k$-algebras and $D$ a Hasse-Schmidt derivation from $B$
to $A$ as above. Let $j\geq 1$. For $l\in\\{0,\dots,r\\}$, we have the set
$\mathcal{S}_{l}=\\{(i_{1},\dots,i_{p^{r-l}})\in\mathbb{N}^{p^{r-l}}|i_{1}+\dots+i_{p^{r-l}}=j\\}$.
To each element $v\in\mathcal{S}_{l}$, we may associate a partition
$\\{C_{i}\\}_{i=1}^{m}$ of $\\{1,\dots,p^{r-l}\\}$. Let $c_{i}=|C_{i}|$. The
number $c_{i}$ depends only on the associated orbit $\mathcal{O}$ of the
action of the symmetric group $S_{p^{r-l}}$; as does the set
$\\{j_{1},\dots,j_{m}\\}$ of distinct numbers888We index so that each element
of $C_{i}$ is $j_{i}$ occurring in $v$. Then we have

1)
$\alpha_{\mathcal{O}}:={\displaystyle\text{val}(p^{l}\frac{(p^{r-l})!}{c_{1}!\cdots
c_{m}!})}\geq l$ and if $\alpha_{\mathcal{O}}\leq r$ we have
$p^{r-\alpha_{\mathcal{O}}}|c_{i}$ for each $i\in\\{1,\dots m\\}$.

2) Let $c_{i}=c^{\prime}_{i}p^{r-\alpha_{\mathcal{O}}}$, and let
$b_{\mathcal{O}}$ be the image of ${\displaystyle
p^{l}\frac{(p^{r-l})!}{c_{1}!\cdots c_{m}!}p^{-\alpha}}$ in $\mathbb{F}_{p}$.
Define

$\tilde{D}_{j}(0,\dots f_{l+1},\dots
0)=\sum_{\mathcal{O}}(0,\dots,b_{\mathcal{O}}\prod_{i=1}^{m}D_{j_{i}}(f_{l+1})^{c^{\prime}_{i}},\dots,0)$

where on the left hand side the term $f_{l+1}$ is in the $l+1$ position, and
on the right the term $b_{\mathcal{O}}\prod_{i=1}^{m}D_{j_{i}}(f_{l+1})$
occurs in the $\alpha_{\mathcal{O}}+1$ position (the term is taken to be $0$
if $\alpha_{\mathcal{O}}>r$). If we extend this map to all of $W_{r+1}(A)$ by
addition of components, then $\tilde{D}$ is a $W_{r+1}(k)$-linear Hasse-
Schmidt derivation from $W_{r+1}(B)$ to $W_{r+1}(A)$.

###### Proof.

Everything is immediate from the proof of the previous theorem except
(perhaps) the $W_{r+1}(k)$-linearity; for a Hasse-Schmidt derivation this is
equivalent to demanding that $\tilde{D}_{j}(W(k))=0$ for all $j\geq 1$. But
this is immediate from the formula; by definition we have
${\displaystyle\sum_{i=1}^{m}c_{i}j_{i}=j}$, so that if $j\geq 1$ then at
least one $j_{i}$ is $\geq 1$ as well; and we have $D_{j_{i}}(\lambda)=0$ for
$\lambda\in k$ and $j_{i}\geq 1$. ∎

######  Remark 2.7.

I an unaware of a _direct_ proof that the output of the formula recorded above
actually is a Hasse-Schmidt derivation; one deduces this implicitly in the
theorem from the fact that the lift $\tilde{D}$ is a Hasse-Schmidt derivation.
In fact, it seems non-obvious that the maps written above are even additive.

######  Definition 2.8.

The Hasse-Schmidt derivations occurring in 2.6 are called _canonical_ Hasse-
Schmidt derivations.

######  Remark 2.9.

In fact, the construction can be extended to the case where $A$ and $B$ are
finite type over a noetherian $k$-algebra $R$; in this case, one obtains (by
the same formula), $W_{r+1}(R)$-linear HS derivations $W_{r+1}(B)\to
W_{r+1}(A)$. We won’t use this generalization in this work.

Until further notice we fix a morphism $B\to A$ of algebras, which are each
smooth over $k$. Let us show that the canonical Hasse-Schmidt derivations
enjoy many favorable properties.

######  Proposition 2.10.

Let $(D_{0},\dots D_{n})$ be a Hasse-Schmidt derivation from $B$ to $A$ (over
$R$). Then for any $j\leq n$ we have

$\tilde{D}_{j}F=F\tilde{D}_{j/p}$

where $\tilde{D}$ denotes the canonical lift of $D$, $F$ is the Witt-vector
Frobenius (on $W_{r+1}(B)$ and $W_{r+1}(A)$, respectively), and the right hand
side is defined to be $0$ if $p$ does not divide $j$.

###### Proof.

Let us first show this result when $r=0$. In this case we have

$D_{j}(F(f))=D_{j}(f^{p})=\sum_{i_{1}+\dots i_{p}=j}D_{i_{1}}(f)\cdots
D_{i_{p}}(f)$

For each term in the sum, either $i_{1}=i_{2}=\cdots=i_{p}$, or there are at
least two distinct indices. In the latter case, grouping like terms together
and arguing as in the proof of Theorem 2.5 gives a coefficient of the form

$\frac{p!}{c_{1}!\cdots c_{m}!}D_{j_{1}}(f)^{c_{1}}\dots
D_{j_{m}}(f)^{c_{m}}=0$

since each $c_{i}<p$ and $p=0$ in $A$. Thus we obtain

$D_{j}(f^{p})=D_{j/p}(f){}^{p}$

as required (where the right hand side is taken to be $0$ if $p$ does not
divide $j$).

Now let us consider the case of the action of $\tilde{D}_{j}$ from
$W_{r+1}(B)$ to $W_{r+1}(A)$. From 2.6, we obtain the formula

$\tilde{D}_{j}(0,\dots F(f_{l+1}),\dots
0)=\sum_{\mathcal{O}}(0,\dots,b_{\mathcal{O}}\prod_{i=1}^{m}D_{j_{i}}(F(f_{l+1}))^{c^{\prime}_{i}},\dots,0)$

where the notation is as above. In particular, with
$c_{i}=c^{\prime}_{i}p^{r-\alpha_{\mathcal{O}}}$ we have
${\displaystyle\sum_{i=1}^{m}c_{i}j_{i}=j}$. From the case $r=0$ of the
proposition we therefore obtain an equality

$\sum_{\mathcal{O}}(0,\dots,b_{\mathcal{O}}\prod_{i=1}^{m}D_{j_{i}}(F(f_{l+1}))^{c^{\prime}_{i}},\dots,0)=\sum_{\mathcal{O}}(0,\dots,b_{\mathcal{O}}\prod_{i=1}^{m}F(D_{j_{i}/p}(f_{l+1}))^{c^{\prime}_{i}},\dots,0)$
$=F(\sum_{\mathcal{O}}(0,\dots,b_{\mathcal{O}}\prod_{i=1}^{m}D_{j_{i}/p}(f_{l+1})^{c^{\prime}_{i}},\dots,0))$

By definition any term in which $p$ does not divide all the $j_{i}$ is zero.
So, if $p$ does not divide $j$, then from the equality
${\displaystyle\sum_{i=1}^{m}c_{i}j_{i}=j}$ we see that $p$ cannot divide all
the $j_{i}$, so that the sum is zero.

If $p$ does divide $j$, then multiplication by $p$ induces a bijection between
$\\{(i_{1},\dots,i_{p^{r-l}})\in\mathbb{N}^{p^{r-l}}|i_{1}+\dots+i_{p^{r-l}}=j/p\\}$
and the subset of
$\\{(i_{1},\dots,i_{p^{r-l}})\in\mathbb{N}^{p^{r-l}}|i_{1}+\dots+i_{p^{r-l}}=j\\}$
in which $p$ divides every term; and this bijection preserves the numbers
$\\{c_{1},\dots,c_{m}\\}$ and $\\{c^{\prime}_{1},\dots,c^{\prime}_{m}\\}$
associated to each orbit. Thus from the displayed formula we see that
$\tilde{D}_{j}F=F\tilde{D}_{j/p}$ as desired. ∎

We would like to consider the interaction of the canonical Hasse-Schmidt
derivations with the natural quotient maps on Witt vectors. To that end we let
$R:W_{r+1}(A)\to W_{r}(A)$ denote the natural projection.

Then we have

######  Proposition 2.11.

Let $D=(D_{0},\dots,D_{n})$ be a Hasse-Schmidt derivation from $B$ to $A$, and
let $\tilde{D}$ be the associated canonical Hasse-Schmidt derivation from
$W_{r+1}(B)$ to $W_{r+1}(A)$. Then $\tilde{D}$ takes $V^{i}(W_{r+1}(B))$ into
$V^{i}(W_{r+1}(A))$. Furthermore,

a) Under the isomorphisms

$R:W_{r+1}(A)/V^{r}(W_{r+1}(A))\tilde{\to}W_{r}(A)$

and $R:W_{r+1}(B)/V^{r}(W_{r+1}(B))\tilde{\to}W_{r}(B)$ we have
$R\circ\tilde{D}_{j}=\tilde{D}_{j/p}\circ R$, where the right hand side is
zero if $p$ does not divide $j$.

b) Now suppose $\text{val}(j)=a\leq r$. Then

$\tilde{D}_{j}(W_{r+1}(B))\subset V^{r-a}(W_{r+1}(A))$

###### Proof.

The fact that $\tilde{D}$ takes $V^{i}(W_{r+1}(B))$ into $V^{i}(W_{r+1}(A))$
is immediate from 2.6.

Next, to prove $a)$, we first prove that if $p$ does not divide $j$, then
$R\circ\tilde{D}_{j}=0$. To see this, we use the formula

$\tilde{D}_{j}(0,\dots f_{l+1},\dots
0)=\sum_{\mathcal{O}}(0,\dots,b_{\mathcal{O}}\prod_{i=1}^{m}D_{j_{i}}(f_{l+1})^{c^{\prime}_{i}},\dots,0)$

Since, in this formula, ${\displaystyle\sum c_{i}j_{i}=j}$, the assumption
that $p$ does not divide $j$ implies that, for each term in the sum, at least
one $c_{i}$ is not divisible by $p$. Therefore the fact that
$p^{r-\alpha_{\mathcal{O}}}|c_{i}$ implies $\alpha_{\mathcal{O}}\geq r$ for
each orbit $\mathcal{O}$, which shows that on the right hand side each nonzero
term is in the $r+1$ position. Therefore $R$ annihilates this term.

Now suppose $p$ divides $j$. Let $r>l$. For an $S_{p^{r-l}}$ orbit inside
$\mathcal{S}_{l}=\\{(i_{1},\dots,i_{p^{r-l}})\in\mathbb{N}^{p^{r-l}}|i_{1}+\dots+i_{p^{r-l}}=j\\}$,
we write $p|\mathcal{O}$ if each of the numbers $c_{i}$ is divisible by $p$.
If we choose a member of $\mathcal{O}$ for which each of the sets $C_{i}$ is
an interval in $\\{1,\dots,p^{r-l}\\}$ then $p|\mathcal{O}$ implies that each
$C_{i}$ is an interval of the form $\\{pm_{i}+1,\dots,pm_{i+1}\\}$. Thus to
the orbit $\mathcal{O}$ we may associate the partition of
$\\{1,\dots,p^{r-l-1}\\}$ where $C^{\prime}_{i}=\\{m_{i}+1,\dots,m_{i+1}\\}$.
Further, to each $C_{i}$ is associated the number $j_{i}$ such that
${\displaystyle\sum_{i=1}^{m}c_{i}j_{i}=j}$; this implies
${\displaystyle\sum_{i=1}^{m}(c_{i}/p)j_{i}=j/p}$, so if we associate $j_{i}$
to each $C_{i}^{\prime}$ we obtain an element of
$\mathcal{\tilde{S}}_{l}=\\{(i_{1},\dots,i_{p^{r-l-1}})\in\mathbb{N}^{p^{r-l-1}}|i_{1}+\dots+i_{p^{r-l-1}}=j/p\\}$.
Clearly, if we choose a different member of $\mathcal{O}$ the resulting
elements of $\mathcal{\tilde{S}}_{l}$ will be conjugate. Thus we obtain a map

$\\{S_{p^{r-l}}\phantom{i}\text{orbits
on}\phantom{i}\mathcal{S}_{l}\phantom{i}\text{with}\phantom{i}p|\mathcal{O}\\}\to\\{S_{p^{r-l-1}}\phantom{i}\text{orbits
on}\phantom{i}\mathcal{\tilde{S}}_{l}\\}$

and this map is a bijection. Indeed, given an orbit $\mathcal{O}^{\prime}$ in
$\tilde{S}_{l}$ we may choose an element for which the members of the
partition $C_{i}^{\prime}$ consists of intervals
$\\{m_{i}+1,\dots,m_{i+1}\\}$; and to this partition we attach the partition
$\\{pm_{i}+1,\dots,pm_{i+1}\\}$ of $\\{1,\dots,p^{r-l}\\}$, with the same
associated number $j_{i}$ as before.

Further, by the claim directly below, we have

$p^{l}\frac{(p^{r-l})!}{c_{1}!\cdots c_{m}!}\equiv
p^{l}\frac{(p^{r-l-1})!}{(c_{1}/p)!\cdots(c_{m}/p)!}\phantom{i}\text{mod}\phantom{i}p^{r}$

which tells us that $\alpha_{\mathcal{O}}=\alpha_{\mathcal{O}^{\prime}}$ if
$\alpha_{O}<r$, and in this case, $b_{\mathcal{O}}=b_{\mathcal{O}^{\prime}}$.
Recall that the numbers $\\{c_{i}\\}$ satisfy
$c_{i}=c^{\prime}_{i}p^{r-\alpha_{\mathcal{O}}}$; thus in this case we also
have $c_{i}/p=c_{i}^{\prime}p^{r-1-\alpha_{\mathcal{O}^{\prime}}}$.

Now consider $\tilde{D}_{j}$ acting on $(0,\dots f_{l+1},\dots 0)$; if $l=r$
then $R$ annihilates this term so suppose $r>l$. As above, if $\mathcal{O}$ is
an orbit such that $p$ does not divide $c_{i}$ for some $i$, then

$R\circ(0,\dots,b_{\mathcal{O}}\prod_{i=1}^{m}D_{j_{i}}(f_{l+1})^{c^{\prime}_{i}},\dots,0)=0$

and the same is true for any orbit $\mathcal{O}$ in which
$\alpha_{\mathcal{O}}\geq r$. Thus we have

$R\circ\tilde{D}_{j}(0,\dots f_{l+1},\dots
0)=\sum_{p|\mathcal{O}}R\circ(0,\dots,b_{\mathcal{O}}\prod_{i=1}^{m}D_{j_{i}}(f_{l+1})^{c^{\prime}_{i}},\dots,0)$
$=\sum_{\\{p|\mathcal{O},\alpha_{\mathcal{O}}<r\\}}R\circ(0,\dots,b_{\mathcal{O}}\prod_{i=1}^{m}D_{j_{i}}(f_{l+1})^{c^{\prime}_{i}},\dots,0)$

Now, we have

$\sum_{\\{p|\mathcal{O},\alpha_{\mathcal{O}}<r\\}}R\circ(0,\dots,b_{\mathcal{O}}\prod_{i=1}^{m}D_{j_{i}}(f_{l+1})^{c^{\prime}_{i}},\dots,0)=\sum_{\\{\mathcal{O}^{\prime},\alpha_{\mathcal{O}^{\prime}}<r\\}}(0,\dots,b_{\mathcal{O}^{\prime}}\prod_{i=1}^{m}D_{j_{i}}(f_{l+1})^{c^{\prime}_{i}},\dots,0)$
$=\sum_{\mathcal{O}^{\prime}}(0,\dots,b_{\mathcal{O}^{\prime}}\prod_{i=1}^{m}D_{j_{i}}(f_{l+1})^{c^{\prime}_{i}},\dots,0)$

where the last equality follows because $\alpha_{\mathcal{O}^{\prime}}\geq r$
implies that the term
$(0,\dots,b_{\mathcal{O}^{\prime}}\prod_{i=1}^{m}D_{j_{i}}(f_{l+1})^{c^{\prime}_{i}},\dots,0)$
is zero in $W_{r}(A)$. But the above discussion implies that this last sum is
$\tilde{D}_{j/p}\circ R(0,\dots f_{l+1},\dots 0)$ as claimed.

Finally, part $b)$ follows immediately from $a)$ and induction on $r$. ∎

To finish the proof we need the

######  Claim 2.12.

Let $r>l$ and let $\\{c_{1},\dots,c_{m}\\}$ be positive integers such that
$\sum c_{i}=p^{r-l}$. Suppose $c_{i}$ is divisible by $p$ for each $i$. We
have

$p^{l}\frac{(p^{r-l})!}{c_{1}!\cdots c_{m}!}\equiv
p^{l}\frac{(p^{r-l-1})!}{(c_{1}/p)!\cdots(c_{m}/p)!}\phantom{i}\text{mod}\phantom{i}p^{r}$

###### Proof.

Consider the polynomial ring $\mathbb{Z}/p^{r+1}[T_{1},\dots,T_{m}]$. The
multinomial formula yields

(2.3) $p^{l}(T_{1}+\dots+T_{m})^{p^{r-l}}=\sum_{k_{1}+\dots
k_{m}=p^{r-l}}p^{l}\frac{(p^{r-l})!}{k_{1}!\cdots k_{m}!}T_{1}^{k_{1}}\cdots
T_{m}^{k_{m}}$

If we take the image of $p^{l}(T_{1}+\dots+T_{m})^{p^{r-l}}$ in
$\mathbb{Z}/p^{r}[T_{1},\dots,T_{m}]$, then we have

$p^{l}(T_{1}+\dots+T_{m})^{p^{r-l}}=p^{l}((T_{1}+\dots+T_{m})^{p})^{p^{r-1-l}}=F_{1}(p^{l}(T_{1}+\dots+T_{m})^{p^{r-1-l}})$

where $F_{1}$ is an appropriately chosen lift of Frobenius. Since
$p^{l}(T_{1}+\dots+T_{m})^{p^{r-1-l}}\in W_{r}(\mathbb{F}_{p}[T_{1},\dots
T_{m}]^{(r-1)})\subset\mathbb{Z}/p^{r}[T_{1},\dots,T_{m}]$, the same equation
holds if we replace $F_{1}$ with any lift of Frobenius, by Lemma 2.2. Now,
applying the multinomial formula again yields

$p^{l}(T_{1}+\dots+T_{m})^{p^{r-1-l}}=\sum_{j_{1}+\dots
j_{m}=p^{r-l-1}}p^{l}\frac{(p^{r-l-1})!}{j_{1}!\cdots
j_{m}!}T_{1}^{j_{1}}\cdots T_{m}^{j_{m}}$

and so if we apply the map $F$ for which $F(T_{i})=T_{i}^{p}$ we obtain

$F(p^{l}(T_{1}+\dots+T_{m})^{p^{r-1-l}})=\sum_{j_{1}+\dots
j_{m}=p^{r-l-1}}p^{l}\frac{(p^{r-l-1})!}{j_{1}!\cdots
j_{m}!}T_{1}^{pj_{1}}\cdots T_{m}^{pj_{m}}$

and if we consider the term where $j_{i}=c_{i}/p$ and compare with (2.3) we
obtain the result. ∎

With these preliminaries in hand, we can give the definition of Witt-
differential operators over the algebra $A$. We shall be working with the
space
$\text{Hom}_{W(k)}^{V}(W(B),W(A))$ of continuous, $W(k)$-linear endomorphisms
of $W(A)$ which preserve $V^{i}(W(A))$ for each $i\geq 0$. We note that
$W(A)\to\text{Hom}_{W(k)}^{V}(W(B),W(A))$ via the map which takes $a\in W(A)$
to $a\cdot\varphi^{\\#}$, and $W(B)\to\text{Hom}_{W(k)}^{V}(W(B),W(A))$ by the
map which takes $b$ to $\varphi^{\\#}(b\cdot)$.

For each $r\geq 0$, we define

$\text{Hom}_{W(k)}^{V}(W_{r+1}(B),W_{r+1}(A))\subset\text{Hom}_{W(k)}(W_{r+1}(B),W_{r+1}(A))$

to be the subspace of $W(k)$-linear endomorphisms of $W_{r+1}(A)$ which
preserve the filtration $V^{i}(W_{r+1}(A))$.

We have the natural restriction map

$R:\text{Hom}_{W(k)}^{V}(W_{r+1}(B),W_{r+1}(A))\to\text{Hom}_{W(k)}^{V}(W_{r}(B),W_{r}(A))$

and from the definitions it follows directly that

$\text{Hom}_{W(k)}^{V}(W(B),W(A))\tilde{=}\lim_{r}\text{Hom}_{W(k)}^{V}(W_{r+1}(B),W_{r+1}(A))$

We have the following elementary observation:

######  Lemma 2.13.

For each $j\geq 0$, and each $r\geq 0$ (including $r=\infty$) there is a well-
defined operation

$W_{r+1}(A)\times V^{j}(W_{r+1}(A))\to V^{j}(W_{r+1}(A))$

given by

$(a,V^{j}(b))\to V^{j}(a\cdot b):=F^{-j}(a)\cdot b$

and this operation is $F^{-j}$-semilinear over $W_{r+1}(k)$, and preserves the
filtration $V^{i}(W_{r+1}(A))$.

To justify the notation, note that when $r=\infty$, the product
$F^{-j}(x)\cdot y$ actually makes sense inside $W(A)[p^{-1}]$, and agrees with
the operation as defined above.

######  Definition 2.14.

Let $m\in\mathbb{Z}$, and $r\geq-m$. An element

$\varphi\in\text{Hom}_{W(k)}^{V}(W_{r+1}(B),W_{r+1}(A))$

is an elementary Witt-differential operator of level $\leq m$ if there is a
canonical Hasse-Schmidt derivation $D=(\tilde{D}_{0},\dots,\tilde{D}_{i})$ of
length $0<i\leq p^{r+m}$ on $W_{r+1}(A)$ so that
$\varphi=F^{\text{val}(i)-r}(a)\cdot\tilde{D}_{i}$ for some $a\in W_{r+1}(A)$
(this is a well-defined operator by 2.11).

We define $\mathcal{E}W_{r+1}^{(m)}(B;A)$ to be the $W_{r+1}(A)$-submodule of
$\text{Hom}_{W(k)}^{V}(W_{r+1}(B),W_{r+1}(A))$ generated by the elementary
Witt-differential operators of level $\leq m$. By 2.11, we have that the map

$R:\mathcal{E}W_{r+1}^{(m)}(B;A)\to\text{Hom}_{W(k)}^{V}(W_{r}(B),W_{r}(A))$

has image contained in $\mathcal{E}W_{r}(A)$; thus we may define

$\mathcal{E}W^{(m)}(B;A):=\lim_{r}\mathcal{E}W_{r+1}^{(m)}(B;A)\subset\text{Hom}_{W(k)}^{V}(W(B),W(A))$

When $A=B$ and $\varphi^{\\#}$ is the identity map we write
$\mathcal{E}W_{r+1}^{(m)}(A)$ and $\mathcal{E}W^{(m)}(A)$ for
$\mathcal{E}W_{r+1}^{(m)}(B;A)$ and $\mathcal{E}W^{(m)}(B;A)$, respectively.
In this case, The $W(A)$-module $\mathcal{E}W^{(m)}(A)$ is naturally filtered
by

$\mathcal{E}W^{(m),r}(A)=\\{\varphi\in\mathcal{E}W^{(m)}(A)|\varphi(W(A))\subset
V^{r}(W(A))\\}$

Now we give the

######  Definition 2.15.

The ring $\mathcal{D}_{W(A)}^{(m)}$ of (uncompleted) Witt-differential
operators of level $\leq m$ on $A$ is defined to be the subalgebra of
continuous $W(k)$-linear endomorphisms of $W(A)$ generated by multiplication
by elements of $W(A)$ and the elements of $\mathcal{E}W_{A}^{(m)}$. This ring
possesses a filtration by two sided ideals

$I^{(r)}:=(V^{r}(W(A)),\mathcal{E}W^{(m),r}(A))$

and we define
$\mathcal{D}_{W_{r}(A)}^{(m)}:=\mathcal{D}_{W(A)}^{(m)}/I^{(r)}$.

We define $\widehat{\mathcal{D}}_{W(A)}^{(m)}$, the ring of Witt-differential
operators on $A$, of level $\leq m$, to be the completion of
$\mathcal{D}_{W(A)}^{(m)}$ along $\\{I^{(r)}\\}$; i.e., the inverse limit of
the $D_{W_{r}(A)}^{(m)}$. Since each element of $I^{(r)}$ takes $W(A)$ to
$V^{r}(W(A))$, this ring also acts on $W(A)$.

Finally, we let $\widehat{\mathcal{D}}_{W(A)}$ be the inductive limit of
$\widehat{\mathcal{D}}_{W(A)}^{(m)}$ under the obvious maps
$\widehat{\mathcal{D}}_{W(A)}^{(m)}\to\widehat{\mathcal{D}}_{W(A)}^{(m+1)}$;
similarly $\mathcal{D}_{W(A)}$ is the inductive limit of
$\mathcal{D}_{W(A)}^{(m)}$.

This definition is inspired by Berthelot’s definition of arithmetic
differential operators of level $\leq m$ (c.f. [4],[5], as well as [17] for a
more operator-theoretic point of view) and, as we shall see below, yields
closely related categories of modules, at least when $m\geq 0$.

###  2.1. Local Coordinates

In this subsection we shall give a presentation of the modules
$\mathcal{E}W^{(m)}(B;A)$ and the algebra $\widehat{\mathcal{D}}_{W(A)}^{(m)}$
for $m\in\mathbb{Z}$ in terms of local coordinates on $B$ (or $A$ in the
latter case) and deduce some basic structural results. Therefore, throughout
this subsection, we shall suppose that that there is an etale map
$k[T_{1},\dots,T_{n}]\to B$. This map yields Hasse-Schmidt derivations $B\to
A$, which we shall denote by $\partial_{l}^{[i]}$; these are defined on
$k[T_{1},\dots T_{n}]$ as

$\partial_{l}^{[i]}(T_{l}^{j})={j\choose i}\varphi^{\\#}(T_{l}^{j-i})$

and

$\partial_{l}^{[i]}(T_{l^{\prime}}^{j})=0$

for $l\neq l^{\prime}$ and $i>0$. By [40] these operators extend uniquely to
Hasse-Schmidt derivations on $B$. When $B=A$ and $\varphi^{\\#}=Id$ these
Hasse-Schmidt derivations are iterative; i.e, we have

$\partial_{l}^{[i]}\partial_{l}^{[j]}={i+j\choose i}\partial_{l}^{[i+j]}$

Further, one sees immediately from the definition of these operators that they
all mutually commute; i.e.,

$\partial_{l}^{[i]}\partial_{l^{\prime}}^{[j]}=\partial_{l^{\prime}}^{[j]}\partial_{l}^{[i]}$

for all $l,l^{\prime},i,j$.

If $I=(i_{1},\dots,i_{n})$ is a multi-index, we shall also use the notation
$\partial^{[I]}=\partial_{1}^{[i_{1}]}\cdots\partial_{n}^{[i_{n}]}$. If $J$ is
another multi-index, then we also have
$\partial^{J}=\partial_{1}^{j_{1}}\cdots\partial_{n}^{j_{n}}$ and
$(\partial^{[I]})^{J}=(\partial_{1}^{[i_{1}]})^{j_{1}}\cdots(\partial_{n}^{[i_{n}]})^{j_{n}}$.

Now we give the

######  Definition 2.16.

Let $\alpha\in W(A)$ and $r,j\in\mathbb{Z}$ such that $j>0$. We define the
operator $\\{\partial_{l}\\}_{jp^{r}}$ as follows: it is equal to
$\varphi^{\\#}(\partial_{l}^{[jp^{r^{\prime}+r}]}):W_{r^{\prime}+1}(B)\to
W_{r^{\prime}+1}(A)$ where $\partial_{l}^{[jp^{r^{\prime}+r}]}$ is the
canonical lift of the HS derivation $\partial_{l}^{[jp^{r^{\prime}+r}]}$ to
$W_{r^{\prime}+1}(B)$. The inverse limit of these operators is well defined on
$W(B)$ by 2.11.

Now suppose $r>0$ and let $v_{j}:=\text{val}(j)$. Then we define the operator

$F^{v_{j}-r}(\alpha)\cdot\\{\partial_{l}\\}_{j/p^{r}}$

as the operator
$F^{v_{j}-r}(\alpha)\cdot\varphi^{\\#}(\partial_{l}^{[jp^{r^{\prime}-r}]})$
from $W_{r^{\prime}+1}(B)\to W_{r^{\prime}+1}(A)$; here, we denote by
$\partial_{l}^{[jp^{r^{\prime}-r}]}$ the canonical lift of
$\partial_{l}^{[jp^{r^{\prime}-r}]}$ to $W_{r^{\prime}+1}(B)$ for any
$r^{\prime}$ such that $r^{\prime}\geq r-\text{val}(j)$, for
$r^{\prime}<r-\text{val}(j)$ this operator is taken to be $0$. By 2.11, we
have

$\partial_{l}^{[jp^{r^{\prime}-r}]}(W_{r^{\prime}+1}(B))\subset
V^{r-v_{j}}(W_{r^{\prime}+1}(A))$

so that this operator is well-defined on $W_{r^{\prime}+1}(B)$.

Further, we note that by 2.11, $a)$, we have; for $R:W_{r^{\prime}+1}(A)\to
W_{r^{\prime}}(A)$

$R(F^{v_{j}}(\alpha)\partial_{l}^{[jp^{r^{\prime}-r}]})=F^{v_{j}}(\alpha)\partial_{l}^{[jp^{r^{\prime}-1-r}]}R$

so that the inverse limit of these operators is in fact well-defined on
$W(B)$. We note that when $1\leq j\leq p^{r}$, we have
$F^{v_{j}-r}(\alpha)\cdot\\{\partial_{l}\\}_{j/p^{r}}\in\mathcal{E}W(A)$. We
may extend this definition to the case $j=0$ by defining $v_{0}=r$; then we
have

$F^{v_{j}-r}(\alpha)\cdot\\{\partial_{l}\\}_{j/p^{r}}=\alpha$

for $j=0$ and any $r\geq 0$.

If $J=(j_{1},\dots,j_{n})$ denotes a multi-index in $\mathbb{N}$ for each
$j_{l}$, let $v_{J}=\text{min}_{1\leq l\leq n}\text{val}\\{j_{l}\\}$. Suppose
this minimum is realized at index $l^{\prime}$. Then we define

$F^{v_{J}-r}(\alpha)\cdot\\{\partial\\}_{J/p^{r}}:=F^{v_{J}-r}(\alpha)\prod_{l=1}^{n}\\{\partial_{l}\\}_{j_{l}/p^{r}}=F^{v_{l^{\prime}}-r}(\alpha)\\{\partial_{l^{\prime}}\\}_{j_{l^{\prime}}/p^{r}}\cdot\prod_{l=1,l\neq
l^{\prime}}^{n}\\{\partial_{l}\\}_{j_{l}/p^{r}}$

and we note that this product is in $\mathcal{D}_{W(A)}$ whenever each
$j_{i}\leq p^{r}$. If $J=(0,\dots,0)$ then we set $v_{J}=r$, this gives
${\displaystyle F^{v_{J}-r}(\alpha)\cdot\\{\partial\\}_{J/p^{r}}=\alpha}$ as
above.

We shall show that (products of) such operators form a basis
$\widehat{\mathcal{D}}_{W(A)}^{(0)}$, in a suitable sense. Namely,

######  Theorem 2.17.

Let $m\in\mathbb{Z}$. Let $Q\in\widehat{\mathcal{D}}_{W(A)}^{(m)}$. Then we
may write

$Q=\sum_{I}(\sum_{r=1}^{\infty}\sum_{J_{I}}F^{-r}(\alpha_{J_{I}})\\{\partial\\}_{J_{I}/p^{r}}+\sum_{K_{I}}\alpha_{K_{I}}\\{\partial\\}_{K_{I}})\\{\partial\\}_{p^{m}}^{I}$

where $\alpha_{J_{I}},\alpha_{K_{I}}\in W(A)$,
$I=(i_{1},\dots,i_{n})\in\mathbb{N}^{n}$,
$\\{\partial\\}_{p^{m}}^{I}:=\\{\partial_{1}\\}_{p^{m}}^{i_{1}}\cdots\\{\partial_{n}\\}_{p^{m}}^{i_{n}}$,
$J_{I}=(j_{1},\dots,j_{n})$ satisfies $0\leq j_{l}<p^{r}$ for each $j_{l}$ and
$(p,j_{l})=1$ for at least one $l$; unless $r=1$ where we allow $J_{I}=0$, and
$K_{I}=(k_{1},\dots,k_{n})$ satisfies $0\leq k_{i}<p^{m}$ with at least one
$k_{i}\neq 0$ (if $m\leq 0$ this sum is taken to be empty). We demand that the
terms

$\sum_{r=1}^{\infty}\sum_{J_{I}}F^{-r}(\alpha_{J_{I}})\\{\partial\\}_{J_{I}/p^{r}}+\sum_{K_{I}}\alpha_{K_{I}}\\{\partial\\}_{K_{I}}$

approach $0$ (in the topology of $\widehat{\mathcal{D}}_{W(A)}^{(m)}$) as
$I\to\infty$. Furthermore, the elements $\alpha_{I_{J}}$ are unique.

We shall break the proof of this up into several steps, beginning by showing
that we can write the elementary Witt-differential operators (of level $0$) in
the required form.

######  Lemma 2.18.

Let $\tilde{D}$ be a canonical Hasse-Schmidt derivation of length $\leq p^{r}$
from $W_{r+1}(B)$ to $W_{r+1}(A)$. Then, for any $0<j\leq p^{r}$ the
associated canonical Hasse-Schmidt derivation $\tilde{D}_{j}$ on $W_{r+1}(A)$
is equal to

$\sum_{I}F^{v_{I}-r}(\beta_{I})\cdot\\{\partial\\}_{I/p^{r}}$

where $I=(i_{1},\dots,i_{n})\backslash\\{(0,\cdots,0)\in\mathbb{N}^{n}$
satisfies ${\displaystyle\sum_{l=1}^{n}i_{l}\leq p^{r}}$, and $\beta_{I}\in
W_{r+1}(A)$. Furthermore, if $v_{I}\geq\text{val}(j)$, then we have
$\beta_{I}\in V^{v_{I}-\text{val}(j)}(W_{r+1}(A))$.

###### Proof.

We recall that the operator $\tilde{D}_{j}$ is obtained by choosing any lift
of the Hasse-Schmidt derivation $D$ to a Hasse-Schmidt derivation
$\mathcal{B}_{r+1}\to\mathcal{A}_{r+1}$ and restricting to $W_{r+1}(B^{(r)})$.
If we write

$D_{j}=\sum_{I}\alpha_{I}\partial^{[I]}$

for $\alpha_{I}\in A$ (and
$\partial^{[I]}=\varphi^{\\#}(\partial_{1}^{[i_{1}]}\cdots\partial_{n}^{[i_{n}]})$)
then since $D_{j}$ is a differential operator of order $j\leq p^{r}$ we obtain
$|I|\leq p^{r}$ for all $I$ such that $\alpha_{I}\neq 0$.

Now, write

$\tilde{D}_{j}=\sum_{I}\tilde{\alpha}_{I}\partial^{[I]}$

for $\tilde{\alpha}_{I}\in A_{r+1}$ lifting $\alpha_{I}$. To prove the lemma,
we must show that, after possibly altering $\tilde{\alpha}_{I}$ in a way that
does not change the action on $W_{r+1}(A^{(r)})$, we have
$F^{r-v_{I}}(\tilde{\alpha}_{I})\in W_{r+1}(A^{(r)})$ (where
$F:\mathcal{A}_{r+1}\to\mathcal{A}_{r+1}$ is a lift of Frobenius); and that
$\tilde{\alpha}_{I}\in p^{v_{I}-\text{val(j)}}\mathcal{A}_{r+1}$ if
$v_{I}>\text{val}(j)$.

Let $r\geq 0$. We’ll first show that $\tilde{\alpha}_{I}\in
p^{v_{I}-\text{val(j)}}\mathcal{A}_{r+1}$ if $v_{I}>\text{val}(j)$. If this
does not hold, let $I_{0}$ be the least element (in the lexicographic
ordering) amongst $I$ such that $\tilde{\alpha}_{I}\notin
p^{v_{I}-\text{val(j)}}\mathcal{A}_{r+1}$ and $v_{I}>\text{val}(j)$.

Then we have $\partial^{[I_{0}]}(p^{r-v_{I_{0}}}T^{I_{0}})=p^{r-v_{I_{0}}}$,
while $\partial^{[J]}(p^{r-v_{I}}T^{I_{0}})=0$ for all other $J$ with
$\tilde{\alpha}_{J}\notin p^{v_{J}-\text{val(j)}}\mathcal{A}_{r+1}$ and
$v_{J}>\text{val}(j)$. For $K$ such that $v_{K}\leq\text{val}(j)$ we have
$\partial^{[K]}(p^{r-v_{I}}T^{I_{0}})\subset
V^{r-v_{K}}(W_{r+1}(A^{(r)}))\subset V^{r-\text{val}(j)}(W_{r+1}(A^{(r)}))$
(by 2.11 applied to the HS derivations $\partial_{m}^{[i]}$). So, in order for
$\tilde{D}_{j}(p^{r-v_{I_{0}}}T^{I_{0}})\subset
V^{r-\text{val}(j)}(W_{r+1}(A^{(r)}))$ we must have

$p^{r-v_{I_{0}}}\tilde{\alpha}_{I_{0}}\in
p^{r-\text{val}(j)}\mathcal{A}_{r+1}$

which yields $\tilde{\alpha}_{I_{0}}\in
p^{v_{I_{0}}-\text{val}(j)}\mathcal{A}_{r+1}$ after all.

Now we check that, after possibly altering $\tilde{\alpha}_{I}$ in a way that
does not change the action on $W_{r+1}(A^{(r)})$, we have
$F^{r-v_{I}}(\tilde{\alpha}_{I})\in W_{r+1}(A^{(r)})$ for each $I$ such that
$v_{I}\leq r$. Note that every term of the form
$\tilde{\alpha}_{I}\partial^{[I]}$, for which
$F^{r-v_{I}}(\tilde{\alpha}_{I})\in W_{r+1}(A^{(r)})$, takes
$W_{r+1}(B^{(r)})$ to $W_{r+1}(A^{(r)})$ (as explained above in 2.16). So let
$I_{0}$ be the least element (in the lexicographic ordering) amongst $I$ such
that $F^{r-v_{I}}(\tilde{\alpha}_{I})\notin W_{r+1}(A^{(r)})$. Then

$\tilde{D}_{j}(p^{r-v_{I_{0}}}T^{I_{0}})=\sum_{I}\tilde{\alpha}_{I}\partial^{[I]}(p^{r-v_{I_{0}}}T^{I_{0}})$
$=\sum_{\\{I|F^{r-v_{I}}(\tilde{\alpha}_{I})\in
W_{r+1}(A^{(r)})\\}}\tilde{\alpha}_{I}\partial^{[I]}(p^{r-v_{I_{0}}}T^{I_{0}})+\sum_{\\{I|F^{r-v_{I}}(\tilde{\alpha}_{I})\notin
W_{r+1}(A^{(r)})\\}}\tilde{\alpha}_{I}\partial^{[I]}(p^{r-v_{I_{0}}}T^{I_{0}})$
$=\beta+p^{r-v_{I_{0}}}\tilde{\alpha}_{I_{0}}$

where $\beta\in W_{r+1}(A^{(r)})$; the last equality comes from noting that
each $\tilde{\alpha}_{I}\partial^{[I]}(p^{r-v_{I_{0}}}T^{I_{0}})$ in the first
sum is contained in $W_{r+1}(A^{(r)})$, and in the second sum each term is
zero except the one coming from $I_{0}$. Since $\tilde{D}_{j}$ itself takes
$W_{r+1}(B^{(r)})$ to $W_{r+1}(A^{(r)})$, we see that
$\beta+p^{r-v_{I_{0}}}\tilde{\alpha}_{I_{0}}\in W_{r+1}(A^{(r)})$, which
forces $p^{r-v_{I_{0}}}\tilde{\alpha}_{I_{0}}\in W_{r+1}(A^{(r)})$. Writing

$p^{r-v_{I_{0}}}\tilde{\alpha}_{I_{0}}=\sum_{i=r-v_{I_{0}}}^{r}p^{i}a^{p^{r-i}}$

we obtain

$\tilde{\alpha}_{I_{0}}=\sum_{i=0}^{v_{I_{0}}}p^{i}a^{p^{v_{I_{0}}-i}}+p^{v_{I_{0}}+1}\gamma$

for some $\gamma\in\mathcal{A}_{r+1}$. Therefore

$F^{r-v_{I_{0}}}(\tilde{\alpha}_{I_{0}})=\sum_{i=0}^{v_{I_{0}}}p^{i}a^{p^{r-i}}+p^{v_{I_{0}}+1}F^{r-v_{I_{0}}}(\gamma)$

And since $\partial^{[I_{0}]}(W_{r+1}(B^{(r)}))\subset
V^{r-v_{I_{0}}}(W_{r+1}(A^{(r)}))$, we see that
$p^{v_{I_{0}}+1}\gamma\partial^{[I_{0}]}$ annihilates $W_{r+1}(B^{(r)})$.
Therefore, altering $\tilde{\alpha}_{I_{0}}$ to
$\tilde{\alpha}_{I_{0}}-p^{v_{I_{0}}+1}\gamma$, we have that
$F^{r-v_{I_{0}}}(\tilde{\alpha}_{I_{0}})\in W_{r+1}(A^{(r)})$. Continuing up
the lexicographic ordering in this way, we obtain the result. ∎

Next we have

######  Lemma 2.19.

Every element $\phi\in\mathcal{E}W_{r+1}^{(0)}(B;A)$ admits a representation

$\phi=\sum_{I}F^{v_{I}-r}(\alpha_{I})\\{\partial\\}_{I/p^{r}}$

where $\alpha_{I}\in W_{r+1}(A)$,
$I=(i_{1},\dots,i_{n})\in\mathbb{N}^{n}\backslash\\{(0,\dots,0)\\}$, with
${\displaystyle\sum_{l=1}^{n}i_{l}\leq p^{r}}$; and, as above,
$v_{I}=\text{min}\\{\text{val}(i_{l})\\}$. Furthermore,

$\sum_{I}F^{v_{I}-r}(\alpha_{I})\\{\partial\\}_{I/p^{r}}=\sum_{I}F^{v_{I}-r}(\alpha^{\prime}_{I})\\{\partial\\}_{I/p^{r}}$

iff $\alpha^{\prime}_{I}-\alpha_{I}\in V^{v_{I}+1}(W_{r+1}(A))$ for all $I$.

###### Proof.

In the previous lemma we showed that such a representation exists for $\phi$
if $\phi$ is itself a canonical Hasse-Schmidt derivation; more precisely, if
$\phi=\tilde{D}_{j}$ for some $j\leq p^{r}$ then

$\phi=\sum_{I}F^{v_{I}-r}(\beta_{I})\\{\partial\\}_{I/p^{r}}$

where the sum ranges over $I$ such that ${\displaystyle\sum_{l=1}^{n}i_{l}\leq
j}$, and we have $\beta_{I}\in V^{v_{I}-\text{val}(j)}(W_{r+1}(A^{(r)}))$
whenever $v_{I}>\text{val}(j)$. We must show that a similar representation
exists for operators of the form $F^{\text{val}(j)-r}(\alpha)\cdot\phi$ for
some $\alpha\in W_{r+1}(A)$. We write

$F^{\text{val}(j)-r}(\alpha)\cdot\phi$
$=\sum_{\\{I|v_{I}\leq\text{val}(j)\\}}F^{\text{val}(j)-r}(\alpha)F^{v_{I}-r}(\beta_{I})\\{\partial\\}_{I/p^{r}}+\sum_{\\{I|v_{I}>\text{val}(j)\\}}F^{\text{val}(j)-r}(\alpha)F^{v_{I}-r}(\beta_{I})\\{\partial\\}_{I/p^{r}}$
$=\sum_{\\{I|v_{I}\leq\text{val}(j)\\}}F^{\text{val}(j)-v_{I}}(\alpha)\cdot
F^{v_{I}-r}(\alpha\beta_{I})\\{\partial\\}_{I/p^{r}}+\sum_{\\{I|v_{I}>\text{val}(j)\\}}F^{\text{val}(j)-r}(\alpha)F^{v_{I}-r}(\beta_{I})\\{\partial\\}_{I/p^{r}}$

Now, the first sum is already in the required form since we may write

$F^{\text{val}(j)-v_{I}}(\alpha)\cdot
F^{v_{I}-r}(\alpha\beta_{I})=F^{v_{I}-r}(F^{r-\text{val}(j)-2v_{I}}(\alpha)\cdot\beta_{I})$

and $r-\text{val}(j)-2v_{I}=(r-v_{I})+(\text{val}(j)-v_{I})\geq 0$.

On the other hand, for $I$ such that $v_{I}>\text{val}(j)$, we may, by the
previous lemma, write $\beta_{I}=V^{v_{I}-\text{val}(j)}(\beta^{\prime}_{I})$
so that

$\sum_{\\{I|v_{I}>\text{val}(j)\\}}F^{\text{val}(j)-r}(\alpha)F^{v_{I}-r}(\beta_{I})\\{\partial\\}_{I/p^{r}}$
$=\sum_{\\{I|v_{I}>\text{val}(j)\\}}F^{\text{val}(j)-r}(\alpha)F^{v_{I}-r}(V^{v_{I}-\text{val}(j)}(\beta^{\prime}_{I}))\\{\partial\\}_{I/p^{r}}$
$=\sum_{\\{I|v_{I}>\text{val}(j)\\}}F^{v_{I}-r}(V^{v_{I}-\text{val}(j)}(\alpha\beta^{\prime}_{I}))\\{\partial\\}_{I/p^{r}}$

which is a sum of the required form; therefore, the claimed representation
exists.

Now we deal with the uniqueness. First note that, if
$\alpha_{I}^{\prime}-\alpha_{I}\in V^{v_{I}+1}(W_{r+1}(A))$, then, since the
operator $\\{\partial\\}_{I/p^{r}}$ takes $W_{r+1}(B)$ into
$V^{r-v_{I}}(W_{r+1}(A))$ (by 2.11), we have that
$F^{v_{I}-r}(\alpha^{\prime}_{I}-\alpha_{I})\cdot\\{\partial\\}_{I/p^{r}}$
acts as $0$. So we must show the other implication; namely, that if an
operator

$\sum_{I}F^{v_{I}-r}(\beta_{I})\\{\partial\\}_{I/p^{r}}$

acts as $0$, then $\beta_{I}\in V^{v_{I}+1}(W_{r+1}(A))$. This is equivalent
to the assertion that, if a differential operator

$\sum_{I}\tilde{\beta}_{I}\partial^{[I]}$

(with $F^{r-v_{I}}(\tilde{\beta}_{I})\in W_{r+1}(A^{(r)})$) is zero on
$W_{r+1}(B^{(r)})$, then

$F^{r-v_{I}}(\tilde{\beta}_{I})\in V^{v_{I}+1}(W_{r+1}(A^{(r)}))$

for all $I$; we shall in fact show that $\tilde{\beta}_{I}\in
p^{v_{I}+1}A_{r+1}$, which immediately implies the above statement.

Now, for any $I$ we may consider $p^{r-v_{I}}T^{I}\in W_{r+1}(A^{(r)})$. If
$I_{1}$ is the least (in the lexicographic ordering) index such that
$\tilde{\beta}_{I_{1}}\neq 0$, we have

$0=(\sum_{I}\tilde{\beta}_{I}\partial^{[I]})(p^{r-v_{I_{1}}}T^{I_{1}})=p^{r-v_{I_{1}}}\tilde{\beta}_{I_{1}}$

which implies $\tilde{\beta}_{I_{1}}\in p^{v_{I}+1}A_{r+1}$ as claimed.
Further, by the forward implication, this implies that the term
$\tilde{\beta}_{I_{1}}\partial^{[I_{1}]}$ acts as zero on $W_{r+1}(B^{(r)})$ ,
we see that the operator
${\displaystyle\sum_{I>I_{1}}\tilde{\beta}_{I}\partial^{(I)}}$ acts as zero on
$W_{r+1}(B^{(r)})$ as well. Thus we may replace
${\displaystyle\sum_{I}\tilde{\beta}_{I}\partial^{(I)}}$ by
${\displaystyle\sum_{I>I_{1}}\tilde{\beta}_{I}\partial^{(I)}}$ and run the
argument again; continuing in this way implies the result. ∎

From this, we deduce

######  Corollary 2.20.

Every element $\phi\in\mathcal{E}W^{(0)}(B;A)$ may be written as

$\phi=\sum_{r=0}^{\infty}\sum_{I}F^{-r}(\alpha_{I})\\{\partial\\}_{I/p^{r}}$

where $\alpha_{I}\in W(A)$,
$I=(i_{1},\dots,i_{n})\in\mathbb{N}^{n}\backslash\\{(0,\dots,0)\\}$, with
${\displaystyle\sum_{l=1}^{n}i_{l}\leq p^{r}}$, and $v_{I}=0$ for all $I$.
Furthermore, this representation is unique.

###### Proof.

By definition

$\mathcal{E}W^{(0)}(B;A)=\lim_{r}\mathcal{E}W_{r+1}^{(0)}(B;A)$

and, if $\phi_{r}$ is the image of $\phi$ in $\mathcal{E}W_{r+1}^{(0)}(B;A)$,
then, by slightly rewriting the previous lemma, we have an expression

$\phi_{r}=\sum_{r^{\prime}=0}^{r}\sum_{I}F^{-r^{\prime}}(\alpha_{I,r})\\{\partial\\}_{I/p^{r^{\prime}}}$

with $v_{I}=0$ for all $I$ (in this expression, terms of the form
$F^{-r^{\prime}}(\alpha_{I,r})\\{\partial\\}_{I/p^{r^{\prime}}}$ correspond to
terms with $v_{I}=r-r^{\prime}$ in the statement of the previous lemma).

Thus, by the uniqueness statement of the previous lemma, the image

$\overline{\alpha_{I,r}}\in W_{r+1}(A)/V^{r-r^{\prime}+1}(W_{r+1}(A))$

is well-defined. So, fixing an index $I/p^{r^{\prime}}$, we obtain an element

$\alpha_{I}:=\lim_{r}\overline{\alpha}_{I,r}\in\lim_{r}W_{r+1}(A)/V^{r-r^{\prime}+1}(W_{r+1}(A))\tilde{=}W(A)$

so that the action of
${\displaystyle\sum_{r=0}^{\infty}\sum_{I}F^{v_{I}-r}(\alpha_{I})\\{\partial\\}_{I/p^{r}}}$
agrees with $\phi$ on $W(A)$.

To see that this representation is unique, suppose that a term of the form
${\displaystyle\sum_{r=0}^{\infty}\sum_{I}F^{v_{I}-r}(\alpha_{I})\\{\partial\\}_{I/p^{r}}}$
acts as zero on $W(A)$. Consider a term
$F^{v_{I}-r^{\prime}}(\alpha_{I})\\{\partial\\}_{I/p^{r^{\prime}}}$ for some
index $r^{\prime}$. Then, for each $r\geq r^{\prime}$, the uniqueness
statement of the previous lemma implies that $\alpha_{I}\in
V^{r-r^{\prime}+1}(W_{r+1}(A))$. Since this is true for all $r$, we see
$\alpha_{I}=0$ as claimed. ∎

Now we can deduce from this the analogous presentation for
$\mathcal{E}W^{(m)}(B;A)$:

######  Corollary 2.21.

Let $m\in\mathbb{Z}$. Every element $\phi\in\mathcal{E}W^{(m)}(B;A)$ may be
written as

$\phi=\sum_{r=0}^{\infty}\sum_{I}F^{-r}(\alpha_{I})\\{\partial\\}_{I/p^{r}}+\sum_{J}\alpha_{J}\\{\partial\\}_{J}$

where, in the first sum, $\alpha_{I}\in W(A)$,
$I=(i_{1},\dots,i_{n})\in\mathbb{N}^{n}\backslash\\{(0,\dots,0)\\}$, with
${\displaystyle\sum_{l=1}^{n}i_{l}\leq p^{r}}$, and $v_{I}=0$ for all $I$,
and, in the second sum, we have $\alpha_{J}\in W(A)$ and
${\displaystyle\sum_{l=1}^{n}j_{l}\leq p^{m}}$. Furthermore, this
representation is unique.

###### Proof.

By 2.10 we have $\tilde{D}_{p^{m}j}F^{m}=F^{m}\tilde{D}_{j}$ for any canonical
Hasse-Schmidt derivation from $W_{r+1}(B)$ to $W_{r+1}(A)$. Therefore if
$\phi$ is a canonical Hasse-Schmidt derivation of length $\leq p^{m}$, we have
$\phi F^{m}=F^{m}\psi$ where $\psi\in\mathcal{E}W_{r+1}^{(0)}(B;A)$. So, by
Lemma 2.19 we have

$\phi\circ
F^{m}=F^{m}(\sum_{I}F^{v_{I}-r}(\alpha_{I})\\{\partial\\}_{I/p^{r}})=\sum_{I}F^{m+v_{I}-r}(\alpha_{I})\\{\partial\\}_{p^{m}I/p^{r}}\circ
F^{m}$

where the notation is as in loc cit. So for an arbitrary element of the form
${\displaystyle\sum_{i}F^{m_{i}}(\alpha_{i})\phi_{i}\in}\mathcal{E}W_{r+1}^{(m)}(B;A)$
(where $\phi_{i}$ is the $i$th component of a canonical Hasse-Schmidt
derivation, and $m_{i}=\text{min}\\{0,\text{val}(i)-r\\}$) we have

$\sum_{i}F^{m_{i}}(\alpha_{i})\phi_{i}\circ
F^{m}=\sum_{i}\sum_{I}F^{m_{i}}(\alpha_{i})\cdot
F^{m+v_{I}-r}(\alpha_{I,i})\\{\partial\\}_{p^{m}I/p^{r}}\circ F^{m}$

As in the proof of Lemma 2.19, taking the inverse limit and re-indexing yields
an expression

$\phi\circ
F^{m}=(\sum_{r=0}^{\infty}\sum_{I}F^{-r}(\alpha_{I})\\{\partial\\}_{I/p^{r}}+\sum_{J}\alpha_{J}\\{\partial\\}_{J})\circ
F^{m}$

for $\phi\in\mathcal{E}W^{(m)}(B;A)$; this easily implies the result (as
$F^{m}$ is an isomorphism on $W(A)[p^{-1}]$). ∎

Now we turn to proving the full statement of Theorem 2.17. To do this, we need
to also analyze products. We start with the elementary

######  Lemma 2.22.

Inside $\mathcal{D}_{W(A)}$ for any $i\geq 1$, $r\geq 0$, we have

$\\{\partial_{l}\\}_{1/p^{r}}^{i}=u\cdot i!\\{\partial_{l}\\}_{i/p^{r}}$

where $u\in\mathbb{Z}_{p}$ is a unit (which depends on $r$ and $i$). In
particular, for $I=(i_{1},\dots,i_{n})$ with each $i_{j}<p^{r}$ we have

$\\{\partial\\}_{I/p^{r}}=u^{\prime}\prod_{l,r^{\prime}}\\{\partial_{l}\\}_{1/p^{r^{\prime}}}^{i_{r^{\prime}}}$

where $r^{\prime}\in\\{1\dots,r-1\\}$, $0\leq i_{r^{\prime}}<p$, and
$u^{\prime}$ is a unit in $\mathbb{Z}_{p}$.

###### Proof.

Let $r^{\prime}\geq r$. Then $\\{\partial_{l}\\}_{i/p^{r}}$ acts on
$W_{r^{\prime}+1}(A)$ via the action of $\partial_{l}^{[ip^{r^{\prime}-r}]}$
on $W_{r+1}(A^{(r)})\subset\mathcal{A}_{r^{\prime}+1}$. But we have

$(\partial_{l}^{[p^{r^{\prime}-r}]})^{i}=\prod_{j=1}^{i}{jp^{r^{\prime}-r}\choose
p^{r^{\prime}-r}}\partial_{l}^{[ip^{r^{\prime}-r}]}$

(this follows easily from the fact that $\\{\partial_{l}^{[i]}\\}$ form an
iterative HS derivation). Now, we have that

${jp^{r^{\prime}-r}\choose
p^{r^{\prime}-r}}=\prod_{m=0}^{p^{r^{\prime}-r}-1}\frac{(jp^{r^{\prime}-r}-m)}{(p^{r^{\prime}-r}-m)}=j\cdot\prod_{m=1}^{p^{r^{\prime}-r}-1}\frac{(jp^{r^{\prime}-r}-m)}{(p^{r^{\prime}-r}-m)}$

For each $m\in\\{1,\dots,p^{r^{\prime}-r}-1\\}$, we have
$\text{val}(p^{r^{\prime}-r}-m)=\text{val}(m)=\text{val}(jp^{r^{\prime}-r}-m)$.
Therefore

$\text{val}{jp^{r^{\prime}-r}\choose p^{r^{\prime}-r}}=\text{val}(j)$

and we see that

$(\partial_{l}^{[p^{r^{\prime}-r}]})^{i}=u_{r^{\prime}}\cdot
i!\partial_{l}^{[ip^{r^{\prime}-r}]}$

where $u_{r^{\prime}}$ is a unit in $\mathbb{Z}/p^{r+1}$. The fact that
$u_{r^{\prime}+1}\equiv u_{r^{\prime}}$ mod $p^{r^{\prime}}$ follows from the
claim directly below, therefore we may set ${\displaystyle
u:=\lim_{r^{\prime}}u_{r^{\prime}}\in\mathbb{Z}_{p}}$ to prove the result. ∎

To finish the proof, we need to show

######  Claim 2.23.

We have the identity

(2.4) ${lp^{r}\choose p^{j}}\equiv{lp^{r-1}\choose
p^{j-1}}\phantom{q}\text{mod}\phantom{q}p^{r}$

for all natural numbers $l$, $r$, and $j$.

###### Proof.

This is not difficult to check directly. However, since ${\displaystyle
d^{([p]}(T^{lp^{r}})={lp^{r}\choose p^{j}}}T^{lp^{r}-p^{j}}$ inside
$\mathbb{Z}/p^{r+1}[T]$, and ${\displaystyle
d^{[p^{j-1}]}(T^{lp^{r-1}})={lp^{r-1}\choose p^{j-1}}}T^{lp^{r-1}-p^{j-1}}$
inside $\mathbb{Z}/p^{r}[T]$, this equality also follows from $R\circ
d^{[p^{j}]}=d^{[p^{j-1}]}\circ R$, which is 2.11. ∎

For the course of the next few lemmas, we write
$\widehat{\mathcal{D}}_{W(A),b}^{(m)}$ for the set of elements in
$\widehat{\mathcal{D}}_{W(A)}^{(m)}$ which have a representation as in Theorem
2.17. If

$Q=\sum_{I}(\sum_{r=1}^{\infty}\sum_{J_{I}}F^{-r}(\alpha_{J_{I}})\\{\partial\\}_{J_{I}/p^{r}}+\sum_{K_{I}}\alpha_{K_{I}}\\{\partial\\}_{K_{I}})\\{\partial\\}_{p^{m}}^{I}$

is an element of $\widehat{\mathcal{D}}_{W(A),b}^{(m)}$, we say that
$Q\in\widehat{\mathcal{D}}_{W(A),b,i}^{(m)}$ if $Q$ admits a representation as
above so that for each $r<i$ and each associated $J_{I}$, we have
$\alpha_{J_{I}}\in V^{i-r}(W(A))$; we demand also that $\alpha_{K_{I}}\in
V^{i}(W(A))$ for all $K_{I}$. Note that if we have for each $s$ an element
$\phi_{s}\in\widehat{D}_{W(A),b,s}$ then

$\sum_{s=i}^{\infty}\phi_{s}\in\widehat{\mathcal{D}}_{W(A),b,i}^{(m)}$

for all $i\geq 0$.

######  Lemma 2.24.

Fix $m\in\mathbb{Z}$. Suppose $\phi\in\mathcal{E}W^{(m)}(A)$, and
$Q\in\widehat{\mathcal{D}}_{W(A),b}^{(m)}$. Then $\phi\cdot
Q\in\widehat{\mathcal{D}}_{W(A),b}^{(m)}$. Further, if
$\phi\in\mathcal{E}W^{m,(r)}(A)$, then $\phi\cdot
Q\in\widehat{\mathcal{D}}_{W(A),b.r}^{(m)}$. If
$Q\in\widehat{\mathcal{D}}_{W(A),b,r}^{(m)}$, then $\phi\cdot
Q\in\widehat{\mathcal{D}}_{W(A),b.r}^{(m)}$ for any $\phi$. Similarly, $a\cdot
Q\in\widehat{\mathcal{D}}_{W(A),b,r}^{(m)}$ for any $Q$ if $a\in V^{r}(W(A))$.

###### Proof.

By 2.21, if $\phi\in\mathcal{E}W^{(m)}(A)$ we can write

$\phi=\sum_{r=0}^{\infty}\sum_{I}F^{-r}(\alpha_{I})\\{\partial\\}_{I/p^{r}}+\sum_{J}\alpha_{J}\\{\partial\\}_{J}\in\widehat{\mathcal{D}}_{W(A),b}^{(m)}$

Therefore we must compute

$\phi\cdot Q=$

(2.5)
$(\sum_{r=0}^{\infty}\sum_{I}F^{-r}(\alpha_{I})\\{\partial\\}_{I/p^{r}}+\sum_{J}\alpha_{J}\\{\partial\\}_{J})(\sum_{I}(\sum_{r=1}^{\infty}\sum_{J_{I}}F^{-r}(\alpha_{J_{I}})\\{\partial\\}_{J_{I}/p^{r}}+\sum_{K_{I}}\alpha_{K_{I}}\\{\partial\\}_{K_{I}})\\{\partial\\}_{p^{m}}^{I})$

In order to compute this term, we first note that if $D=(D_{0},\dots,D_{j})$
is any Hasse-Schmidt derivation (on an arbitrary ring $R$), we have an
equality of operators

$[D_{j},f]=\sum_{i=1}^{j}D_{i}(f)D_{j-i}=\sum_{i=0}^{j-1}D_{j-i}(f)D_{i}$

(where $f\in R$ denotes the operator $g\to fg$). Now choose any $j$ so that
$j\leq p^{r+m}$. We have, for any $r^{\prime}\geq r-\text{val}(j)$ and
$\alpha\in W_{r^{\prime}+1}(A)$,

$[\partial_{l}^{[jp^{r^{\prime}-r}]},\alpha]=\sum_{i=0}^{jp^{r^{\prime}-r}-1}\partial_{l}^{[jp^{r^{\prime}-r}-i]}(\alpha)\partial_{l}^{[i]}$

which may be rewritten as

$[\\{\partial_{l}\\}_{j/p^{r}},\alpha]=\sum_{i=0}^{jp^{r^{\prime}-r}-1}\\{\partial_{l}\\}_{(jp^{r^{\prime}-r}-i)/p^{r^{\prime}}}(\alpha)\cdot\\{\partial_{l}\\}_{i/p^{r^{\prime}}}$
$=\sum_{i=0}^{jp^{r^{\prime}-r}-1}\\{\partial_{l}\\}_{(jp^{r^{\prime}-r}-i)/p^{r^{\prime}}}(\alpha)\cdot\\{\partial_{l}\\}_{i/p^{r^{\prime}}}$

So that we obtain the equality

$[\\{\partial_{l}\\}_{j/p^{r}},\alpha]=\\{\partial_{l}\\}_{j/p^{r}}(\alpha)+\sum_{r^{\prime}\geq
r-\text{val}(j)}^{\infty}\sum_{i=1}^{jp^{r^{\prime}-r}-1}\\{\partial_{l}\\}_{(jp^{r^{\prime}-r}-i)/p^{r^{\prime}}}(\alpha)\cdot\\{\partial_{l}\\}_{i/p^{r^{\prime}}}$

where on the right hand side we sum over $i$ with $(i,p)=1$. This is an
equality of operators on $W(A)$; with both sides in $\mathcal{D}_{W(A)}^{(m)}$
(in fact the right hand side is in $\mathcal{E}W_{A}$). Thus we see from this
expression that
$[\\{\partial_{l}\\}_{j/p^{r}},\alpha]\in\widehat{\mathcal{D}}_{W(A),b,r}^{(m)}$,
and consequently that
$F^{-r}(\alpha)\\{\partial\\}_{I/p^{r}}\cdot\alpha_{I_{0}}\in\widehat{\mathcal{D}}_{W(A),b,r}^{(m)}$
for any $\alpha,\alpha_{I_{0}}\in W(A)$.

Next, we consider a product of the form
$F^{-r}(\alpha)\\{\partial_{l}\\}_{j/p^{r}}\cdot
F^{-s}(\beta)\\{\partial_{t}\\}_{j^{\prime}/p^{s}}$, where $j^{\prime}$ is not
divisible by $p$, and $s\geq r$ (the case $s<r$ is essentially identical; as
is the case where we need to multiply $\alpha_{J}\\{\partial\\}_{J}\cdot
F^{-s}(\beta)\\{\partial_{t}\\}_{j^{\prime}/p^{s}}$). To compute this term, we
may, by extending linearly, regard both $\\{\partial_{l}\\}_{j/p^{r}}$ and
$F^{-s}(\beta)\\{\partial_{t}\\}_{j^{\prime}/p^{s}}$ as endomorphisms of
$W(A)[p^{-1}]$. Since $F^{-s}(\beta)\in W(A)[p^{-1}]$, the previous equality
implies an equality

$[\\{\partial_{l}\\}_{j/p^{r}},F^{-s}(\beta)]$
$=\\{\partial_{l}\\}_{j/p^{r}}(F^{-s}(\beta))+\sum_{r^{\prime}\geq
r-\text{val}(j)}^{\infty}\sum_{i=1}^{jp^{r^{\prime}-r}-1}\\{\partial_{l}\\}_{(jp^{r^{\prime}-r}-i)/p^{r^{\prime}}}(F^{-s}(\beta))\cdot\\{\partial_{l}\\}_{i/p^{r^{\prime}}}$

of operators on $W(A)[p^{-1}]$. Therefore

(2.6)
$[\\{\partial_{l}\\}_{j/p^{r}},F^{-s}(\beta)\\{\partial_{m}\\}_{j^{\prime}/p^{s}}]$
$=\\{\partial_{l}\\}_{j/p^{r}}(F^{-s}(\beta))\\{\partial_{t}\\}_{j^{\prime}/p^{s}}$
$+\sum_{r^{\prime}\geq
r-\text{val}(j)}^{\infty}\sum_{i=1}^{jp^{r^{\prime}-r}-1}\\{\partial_{l}\\}_{(jp^{r^{\prime}-r}-i)/p^{r^{\prime}}}(F^{-s}(\beta))\cdot\\{\partial_{l}\\}_{i/p^{r^{\prime}}}\\{\partial_{t}\\}_{j^{\prime}/p^{s}}$

(using that $\\{\partial_{l}\\}_{j/p^{r}}$ and $\\{\partial_{t}\\}_{i/p^{s}}$
commute); as above we suppose $(i,p)=1$ in the right hand sum.

Now, 2.10, applied to the operators $\\{\partial_{l}\\}_{j/p^{r}}$, yields a
relation

$\\{\partial_{l}\\}_{j/p^{r}}F=F\\{\partial_{l}\\}_{j/p^{r+1}}$

which implies

$F^{-1}\\{\partial_{l}\\}_{j/p^{r}}=\\{\partial_{l}\\}_{j/p^{r+1}}F^{-1}$

in $W(A)[p^{-1}]$. We note that this makes sense for all $j,r\in\mathbb{Z}$
such that $j>0$ and $r\geq 0$; in particular we have

$\\{\partial_{l}\\}_{(j/p^{r}-i/p^{r^{\prime}})}(F^{-s}(\beta))=F^{-s}(\\{\partial_{l}\\}_{(jp^{r^{\prime}-r}-i)p^{s}/p^{r^{\prime}}}(\beta))$

So that

$F^{-r}(\alpha)\\{\partial_{l}\\}_{j/p^{r}}\cdot
F^{-s}(\beta)\\{\partial_{t}\\}_{j^{\prime}/p^{s}}=F^{-s}(\beta
F^{s-r}(\alpha))\\{\partial_{t}\\}_{j^{\prime}/p^{s}}\\{\partial_{l}\\}_{j/p^{r}}$
$+F^{-s}(\\{\partial_{l}\\}_{p^{s}j/p^{r}}(\beta)\cdot
F^{s-r}(\alpha))\\{\partial_{t}\\}_{j^{\prime}/p^{s}}$ $+\sum_{r^{\prime}\geq
r-\text{val}(j)}^{\infty}\sum_{i=1}^{jp^{r^{\prime}-r}-1}F^{-s}(\\{\partial_{l}\\}_{(jp^{r^{\prime}-r}-i)p^{s}/p^{r^{\prime}}}(\beta)F^{s-r}(\alpha))\cdot\\{\partial_{l}\\}_{i/p^{r^{\prime}}}\\{\partial_{t}\\}_{j^{\prime}/p^{s}}$

as operators on $W(A)$. We note that all of the terms involved both sides of
this equality are in $\mathcal{D}_{W(A)}^{(m)}$; indeed, since
$\mathcal{D}_{W(A)}^{(m)}$ is an algebra, this is clear for the product
$F^{-r}(\alpha)\\{\partial_{l}\\}_{j/p^{r}}\cdot
F^{-s}(\beta)\\{\partial_{t}\\}_{i/p^{s}}$, and it is also clear for the terms
$F^{-s}(\beta
F^{s-r}(\alpha))\\{\partial_{t}\\}_{j^{\prime}/p^{s}}\\{\partial_{l}\\}_{j/p^{r}}$
and $F^{-s}(\\{\partial_{l}\\}_{p^{s}j/p^{r}}(\beta)\cdot
F^{s-r}(\alpha))\\{\partial_{t}\\}_{j^{\prime}/p^{s}}$, therefore it is true
for the last term as well.

So, since $\mathcal{D}_{W(A)}^{(m)}\subset\text{End}_{W(k)}(W(A))$, the
equality of (2.6) is in fact an equality of elements of
$\mathcal{D}_{W(A)}^{(m)}$, and so we obtain that the image of
$F^{-r}(\alpha)\\{\partial_{l}\\}_{j/p^{r}}\cdot
F^{-s}(\beta)\\{\partial_{t}\\}_{i/p^{s}}$ in
$\widehat{\mathcal{D}}_{W(A)}^{(m)}$ is contained in
$\widehat{\mathcal{D}}_{W(A),b}^{(m)}$.

Now, we claim that $F^{-r}(\alpha)\\{\partial_{l}\\}_{j/p^{r}}\cdot
F^{-s}(\beta)\\{\partial_{t}\\}_{i/p^{s}}$ is actually contained in
$\widehat{\mathcal{D}}_{W(A),b,s}^{(m)}$. This is obvious in the above sum
unless $r^{\prime}=s$ and $l=t$. In that case, we have

$\\{\partial_{l}\\}_{i/p^{s}}\\{\partial_{l}\\}_{j^{\prime}/p^{s}}=u\cdot{i+j^{\prime}\choose
i}\\{\partial_{l}\\}_{(i+j^{\prime})/p^{s}}$

by Lemma 2.22. Let $\nu=\text{val}(i+j^{\prime})$. Then we need to show that

$\text{val}{i+j^{\prime}\choose i}\geq\nu$

This only has content if $v>0$, so we assume this from now on; let
$\alpha=(i+j^{\prime})/p$. To see this, we consider the element
$(T_{1}+T_{2})^{\alpha}$ inside $\mathbb{Z}/p^{v}[T_{1},T_{2}]$. Since
$\text{val}(\alpha)=\nu-1$, we have $(T_{1}+T_{2})^{\alpha}\in
W_{v}(\mathbb{F}_{p}[T_{1},T_{2}]^{(\nu-1)})$. Therefore
$F(T_{1}+T_{2})^{\alpha}=(T_{1}+T_{2})^{i+j^{\prime}}$ for any lift of
Frobenius on $\mathbb{Z}/p^{v}[T_{1},T_{2}]$; choosing $F$ to be the lift
which takes $T_{i}\to T_{i}^{p}$, we see that the coefficient of
$T_{1}^{a}T_{2}^{b}$ in $(T_{1}+T_{2})^{i+j^{\prime}}$ is zero if either $a$
or $b$ is coprime to $p$. As the coefficient of $T_{1}^{i}T_{2}^{j^{\prime}}$
is ${\displaystyle{i+j^{\prime}\choose i}}$ , and as $i$ is coprime to $p$, we
deduce that ${\displaystyle{i+j^{\prime}\choose i}}$ is zero in
$\mathbb{Z}/p^{v}$ as required.

Now the result follows, via additivity, from the remark above that
${\displaystyle\sum_{s=i}^{\infty}\phi_{s}\in\widehat{\mathcal{D}}_{W(A),b,i}^{(m)}}$
whenever $\phi_{s}\in\widehat{\mathcal{D}}_{W(A),b,s}^{(m)}$. ∎

From this we deduce

######  Corollary 2.25.

Every element of $\widehat{\mathcal{D}}_{W(A)}^{(m)}$ is contained in
$\widehat{\mathcal{D}}_{W(A),b}^{(m)}$. Further, the ideal $\widehat{I^{(i)}}$
is equal to $\widehat{\mathcal{D}}_{W(A),b.i}^{(m)}$.

###### Proof.

Since $W(A)\subset\widehat{\mathcal{D}}_{W(A),b}^{(m)}$ and
$\mathcal{E}W^{(m)}(A)\subset\widehat{\mathcal{D}}_{W(A),b}^{(0)}$, we see
that the image of
$\mathcal{D}_{W(A)}^{(m)}\to\widehat{\mathcal{D}}_{W(A)}^{(m)}$ is contained
in $\widehat{\mathcal{D}}_{W(A),b}^{(m)}$. By the last statement of Lemma
2.24, the image of $I^{(i)}$ is contained in
$\widehat{\mathcal{D}}_{W(A),b,i}^{(m)}$. Since an arbitrary sum
${\displaystyle\sum_{s=0}^{\infty}Q_{s}}$ converges if
$Q_{s}\in\widehat{D}_{W(A),b,s}$, we deduce that
$\widehat{\mathcal{D}}_{W(A)}^{(m)}=\widehat{\mathcal{D}}_{W(A),b}^{(m)}$,
with $\widehat{I^{(i)}}\subset\widehat{\mathcal{D}}_{W(A),b,i}^{(m)}$. But the
containment $\widehat{\mathcal{D}}_{W(A),b,i}^{(m)}\subset\widehat{I^{(i)}}$
is clear from the definitions. ∎

Finally, we need to address uniqueness. It will follow from the a
generalization of Lemma 2.19; to state this, set, for $i\in\mathbb{N}$,
${\displaystyle f_{i}:=\text{val}(i!)}$, and, for a multi-index $I$, set
${\displaystyle f_{I}:=\sum_{j=1}^{n}f_{i_{j}}}$.

######  Lemma 2.26.

An element

$Q=\sum_{I}(\sum_{r=1}^{\infty}\sum_{J_{I}}F^{-r}(\alpha_{J_{I}})\\{\partial\\}_{J_{I}/p^{r}}+\sum_{K_{I}}\alpha_{K_{I}}\\{\partial\\}_{K_{I}})\\{\partial\\}_{p^{m}}^{I}$

(where the notation is as in Theorem 2.17) acts trivially on
$W_{r^{\prime}+1}(A)$ iff we have $\alpha_{J_{I}}\in
V^{r^{\prime}+1-r-f_{I}}(W(A))$ for each $I$ and $J_{I}$ (where we take the
superscript to be $0$ if $r^{\prime}+1-r-f_{I}<0$), and $\alpha_{K_{I}}\in
V^{r^{\prime}+1}(W(A))$ for all $K_{I}$.

###### Proof.

We consider the operator

$\sum_{I}(\sum_{r=-m}^{r^{\prime}}\sum_{J_{I}}F^{-r}(\alpha_{J_{I}})\partial^{[p^{r^{\prime}-r}J_{I}]}+\sum_{K_{I}}\alpha_{K_{I}}\partial^{[p^{r^{\prime}}K_{I}]})(\partial^{[p^{r^{\prime}+m}]})^{I}$

we note that by our conventions on sums in Theorem 2.17 we have
$\alpha_{J_{I}},\alpha_{K_{I}}\to 0$ as $I\to\infty$, so for the purposes of
considering the action on $W_{r^{\prime}+1}(A)$ we may regard this as being a
finite sum. We claim that

$((\partial^{[p^{r^{\prime}+m}]})^{I})(W_{r^{\prime}+1}(A^{(r^{\prime})}))\subset
V^{f_{I}}(W_{r^{\prime}+1}(A^{(r^{\prime})}))$

To show this it suffices to show
$(\partial_{l}^{[p^{r^{\prime}+m}]})^{i}(W_{r^{\prime}+1}(A^{(r^{\prime})}))\subset
p^{f_{i}}(W_{r^{\prime}+1}(A^{(r^{\prime})}))$ for each $l$ and $i$. We have,
by Lemma 2.22 the relation

$(\partial_{l}^{[p^{r^{\prime}+m}]})^{i}=up^{f_{i}}\partial_{l}^{[ip^{r^{\prime}+m}]}$

for an appropriate unit $u\in\mathbb{Z}_{p}$; the operator
$\partial_{l}^{[ip^{r^{\prime}}]}$ is the canonical lift of the Hasse-Schmidt
derivation $\partial_{l}^{[ip^{r^{\prime}}]}$ and hence preserves
$W_{r^{\prime}+1}(A^{(r^{\prime})})$, so that
$(\partial_{l}^{[p^{r^{\prime}}]})^{i}$ must take
$W_{r^{\prime}+1}(A^{(r^{\prime})})$ to
$p^{f_{i}}(W_{r^{\prime}+1}(A^{(r^{\prime})}))$ as required.

Next, we claim that
$F^{-r}(\alpha_{J_{I}})\partial^{[p^{r^{\prime}-r}J_{I}]}(V^{f_{I}}(W_{r^{\prime}+1}(A^{(r^{\prime})})))=0$
if
$\alpha_{J_{I}}\in V^{r^{\prime}-r+1-f_{I}}(W(A))$; this follows directly from
the definitions as in Lemma 2.19. This, along with the fact that
$\alpha_{K_{I}}\partial^{[p^{r^{\prime}}K_{I}]}(W_{r^{\prime}+1}(A^{(r^{\prime})}))=0$
if
$\alpha_{K_{I}}\in V^{r^{\prime}+1}(W_{r^{\prime}+1}(A^{(r^{\prime})}))$,
gives us the reverse direction of the lemma.

For the forward direction, consider the (finite) set of indices
$\\{p^{r^{\prime}-r}J_{I}+p^{r^{\prime}+m}I|\alpha_{J_{I}}\neq 0\\}$, union
with $\\{p^{r^{\prime}K_{I}}+p^{r^{\prime}+m}I|\alpha_{K_{I}}\neq 0\\}$.
Suppose the least (in the lexicographic ordering) index in this set has the
form $p^{r^{\prime}-r}J_{I_{1}}+p^{r^{\prime}}I_{1}$ (the case where the least
element has the form $p^{r^{\prime}K_{I}}+p^{r^{\prime}+m}I$ is similar, but
simpler). Then we have

$0=\sum_{I}(\sum_{r=-m}^{r^{\prime}}\sum_{J_{I}}F^{-r}(\alpha_{J_{I}})\partial^{[p^{r^{\prime}-r}J_{I}]}+\sum_{K_{I}}\alpha_{K_{I}}\partial^{[p^{r^{\prime}}K_{I}]})(\partial^{[p^{r^{\prime}+m}]})^{I}(p^{r}T^{p^{r^{\prime}-r}J_{I_{1}}+p^{r^{\prime}}I_{1}})$
$=u\cdot p^{r+f_{I}}F^{-r}(\alpha_{J_{I}})$

for some $u$ such that $\text{val}(u)=0$; here we have used that
$\partial^{[p^{r^{\prime}-r}J_{I}]}(\partial^{[p^{r^{\prime}}]})^{I}(T^{I^{\prime}})=0$
whenever $I^{\prime}<p^{r^{\prime}-r}J_{I}+p^{r^{\prime}}I$, and that
$\partial^{[p^{r^{\prime}-r}J_{I_{1}}]}T^{p^{r^{\prime}-r}J_{I_{1}}}=1$ while
$(\partial^{[p^{r^{\prime}}]})^{I_{1}}T^{I_{1}}=up^{f_{I}}$. For this to hold
we must have $\alpha_{J_{I_{1}}}\in
V^{r^{\prime}+1-r-f_{I}}(W_{r^{\prime}+1}(A^{(r^{\prime})}))$; this shows,
using the direction of the lemma already proved, that
$F^{-r}(\alpha_{J_{I_{1}}})\partial^{[p^{r^{\prime}-r}J_{I_{1}}]}(\partial^{[p^{r^{\prime}}]})^{I_{1}}$
acts as zero on $W_{r^{\prime}+1}(A^{(r^{\prime})})$. Thus we may subtract
this element from

$\sum_{I}\sum_{r=0}^{r^{\prime}}\sum_{J_{I}}F^{-r}(\alpha_{J_{I}})\partial^{[p^{r^{\prime}-r}J_{I}]}(\partial^{[p^{r^{\prime}}]})^{I}$

and obtain another operator which acts as zero; continuing in this way shows
that $\alpha_{J_{I}}\in
V^{r^{\prime}+1-r-f_{I}}(W_{r^{\prime}+1}(A^{(r^{\prime})}))$ for all
$\alpha_{J}$ as required. ∎

Now we put everything together for the

###### Proof.

(of Theorem 2.17) The existence, for each
$Q\in\widehat{\mathcal{D}}_{W(A)}^{(m)}$, of the claimed representation is
given by 2.25. As for the uniqueness, we note that if any element of the form

$\sum_{I}(\sum_{r=1}^{\infty}\sum_{J_{I}}F^{-r}(\alpha_{J_{I}})\\{\partial\\}_{J_{I}/p^{r}}+\sum_{K_{I}}\alpha_{K_{I}}\\{\partial\\}_{K_{I}})\\{\partial\\}_{p^{m}}^{I}$

acts as zero on $W(A)$, then by the previous lemma we have, for each index
$J_{I}$, $\alpha_{J_{I}}\in V^{r^{\prime}-r+1-f_{I}}(W(A))$ for all
$r^{\prime}\geq 0$; but this implies $\alpha_{J_{I}}=0$. ∎

The proof actually gave us a little more:

######  Corollary 2.27.

The natural map $\widehat{\mathcal{D}}_{W(A)}^{(m)}\to\text{End}_{W(k)}(W(A))$
is injective.

In addition, using the description of $I^{(1)}$ in 2.25, we obtain

######  Corollary 2.28.

For any $m\geq 0$ there is an isomorphism
$\widehat{\mathcal{D}}_{W(A)}^{(m)}/I^{(1)}\tilde{\to}\mathcal{D}_{A}^{(m)}$.

As well as

######  Corollary 2.29.

For each $m$ the obvious map
$\widehat{\mathcal{D}}_{W(A)}^{(m)}\to\widehat{\mathcal{D}}_{W(A)}^{(m+1)}$ is
injective. For each $m$ the natural map
$\widehat{\mathcal{D}}_{W(A)}^{(m)}\to\widehat{\mathcal{D}}_{W(A)}$ is
injective.

To close out this section, we would like to record for later use the analogue
of Theorem 2.17 when we have a morphism $\varphi^{\\#}:B\to A$ of smooth
$k$-algebras. By the functoriality of the Witt vectors there is a morphism
$W\varphi^{\\#}:W(B)\to W(A)$. Letting $X=\text{Spec}(A)$ and
$Y=\text{Spec}(B)$, we obtain a morphism of affine formal schemes
$W\varphi:W(X)\to W(Y)$.

######  Definition 2.30.

Let $\mathcal{D}_{W(X)\to W(Y)}^{(m)}$ be the
$(\mathcal{D}_{W(A)},\mathcal{D}_{W(B)})$ bi-submodule of
$\text{Hom}_{W(k)}(W(B),W(A))$ generated by $\mathcal{E}W^{(m)}(B;A)$. Let
$\widehat{\mathcal{D}}_{W(X)\to W(Y)}^{(m)}$ be the completion of this
bimodule along the filtration

$F^{l}(\mathcal{D}_{W(X)\to W(Y)}^{(m)})=\\{I^{(i)}\mathcal{D}_{W(X)\to
W(Y)}^{(m)}I^{(j)}\\}_{i+j\geq l}$

Suppose that $B$ possesses local coordinates $\\{T_{1},\dots,T_{d}\\}$. Then
we have

######  Corollary 2.31.

Every element of $\widehat{\mathcal{D}}_{W(X)\to W(Y)}^{(m)}$ can be uniquely
expressed as

$\sum_{I}(\sum_{r=0}^{\infty}\sum_{J_{I}}F^{-r}(\alpha_{J_{I}})\cdot
W\varphi^{\\#}(\\{\partial\\}_{J_{I}/p^{r}})\\{\partial\\}^{I})$

where $\alpha_{J_{I}}\in W(A)$, $\\{\partial\\}_{J_{I}/p^{r}}$ and
$\\{\partial\\}^{I}$ are in $\widehat{\mathcal{D}}_{W(B)}$, and the
convergence conditions are as in Theorem 2.17. In particular, for any $m\geq
0$ we have

$I^{(1)}\backslash\widehat{\mathcal{D}}_{W(X)\to
W(Y)}/I^{(1)}\tilde{\to}A\otimes_{B}\mathcal{D}_{B}^{(m)}$

The latter object is the usual transfer bimodule in the theory of
$\mathcal{D}^{(m)}$-modules.

###### Proof.

It follows directly from the description of $\mathcal{E}W^{(m)}(B;A)$ in 2.21
the set of such sums are contained in $\widehat{\mathcal{D}}_{W(X)\to W(Y)}$;
denote this set by $\widehat{\mathcal{D}}_{W(X)\to W(Y),b}$ and as above we
say that $Q\in\widehat{\mathcal{D}}_{W(X)\to W(Y),b,i}$ if $Q$ admits a
representation as above so that for each $r<i$ and each associated $J_{I}$, we
have $\alpha_{J_{I}}\in V^{i-r}(W(A))$. So, applying Theorem 2.17, we have to
show that for any element $\phi$ of $\mathcal{E}W^{(0),i}(A)$, we have that
$\phi\circ W\varphi^{\\#}$ is contained in $\widehat{D}_{W(A),b,i}$.

Using the local coordinates on $B$, we see that the argument(s) of Lemma 2.18,
Lemma 2.19, and 2.20 carry over to this situation with (essentially) no
change, and we conclude that $\widehat{\mathcal{D}}_{W(X)\to
W(Y)}=\widehat{\mathcal{D}}_{W(X)\to W(Y),b}$.

The uniqueness of the representation of an element of
$\widehat{\mathcal{D}}_{W(X)\to W(Y)}$ as a sum follows exactly as in Lemma
2.26, and the last sentence follows immediately from the description and the
definition of the ideal $I^{(1)}$. ∎

To finish out this section we show how to turn $\widehat{\mathcal{D}}_{W(A)}$
into a sheaf on the etale site of $X=\text{Spec}(A)$. This immediately leads