to a definition of $\widehat{\mathcal{D}}_{W(X)}$ for any smooth $X$ over $k$.
We need to show:

######  Proposition 2.32.

1) Let $\psi^{\\#}:B\to A$ be an etale morphism of smooth $k$-algebras, where
$B$ admits local coordinates. Then there is an isomorphism

$W(A)\widehat{\otimes}_{W(B)}\widehat{\mathcal{D}}_{W(B)}\tilde{\to}\widehat{\mathcal{D}}_{W(A)}$

where on the left hand side the $\widehat{\otimes}$ denotes completion with
respect to the filtration $V^{i}(W(A))$. In particular, we have for each
$r\geq 0$

$W_{r+1}(A)\otimes_{W_{r+1}(B)}\mathcal{D}_{W_{r+1}(B)}\tilde{\to}\mathcal{D}_{W_{r+1}(A)}$

2) Let $\varphi^{\\#}:C\to B$ any morphism, and $\psi^{\\#}:B\to A$ as above.
Let $X=\text{Spec}(B)$, $Y=\text{Spec}(C)$, and $U=\text{Spec}(A)$, we obtain
morphisms of affine formal schemes $W\varphi:W(X)\to W(Y)$ and $W\psi:W(U)\to
W(X)$, as well as the composition $W(U)\to W(Y)$. Then there is an isomorphism

$W(A)\widehat{\otimes}_{W(B)}\widehat{\mathcal{D}}_{W(X)\to
W(Y)}\tilde{\to}\widehat{\mathcal{D}}_{W(U)\to W(Y)}$

###### Proof.

1) As HS derivations extend canonically over etale morphisms, one sees easily
that there is a morphism $\mathcal{E}W_{B}\to\mathcal{E}W_{A}$ which extends
to a morphism $\widehat{\mathcal{D}}_{W(B)}\to\widehat{\mathcal{D}}_{W(A)}$;
extending this by linearity and completing gives the morphism of the
statement. The fact that it is an isomorphism follows directly from Theorem
2.17 for both $\widehat{\mathcal{D}}_{W(A)}$ and
$\widehat{\mathcal{D}}_{W(B)}$; noting the fact that

$W(A)\otimes_{W(B)}F^{-r}(W(B))\tilde{\to}F^{-r}(W(A))$

where $F^{-r}(W(B))$ and $F^{-r}(W(A))$ are regarded as sub-modules of
$W(B)[p^{-1}]$ and $W(A)[p^{-1}]$, respectively; this fact, in turn, can be
easily checked using local coordinates for $B$ and the fact that $A$ is etale
over $B$. Now the last sentence follows from the identification of
$\widehat{I^{(r+1)}}$ with $\widehat{D}_{W(A),b,r+1}$.

2) This follows exactly as in $1)$ but using 2.31 instead of Theorem 2.17. ∎

This leads to the

######  Definition 2.33.

1) Let $X$ be a smooth scheme over $k$. We define
$\widehat{\mathcal{D}}_{W(X)}$ as the unique sheaf of rings (in the Zariski
topology of $X$) which assigns to each open affine $\text{Spec}(A)\subset X$
the ring $\widehat{\mathcal{D}}_{W(A)}$. By the previous lemma this is an
inverse limit of quasicoherent sheaves on $\\{W_{r+1}(X)\\}$.

2) Let $\varphi:X\to Y$ a morphism of smooth schemes. We define
$\widehat{\mathcal{D}}_{W(X)\to W(Y)}$ as the unique sheaf of
$(\widehat{\mathcal{D}}_{W(X)},W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}))$
bimodules which assigns, to each open affine $\text{Spec}(A)\subset X$ such
that $\varphi(\text{Spec}(A))\subset\text{Spec}(B)\subset Y$, the bimodule
$\widehat{\mathcal{D}}_{W(\text{Spec}(A))\to W(\text{Spec}(B))}$. This is an
inverse limit of quasicoherent sheaves on $\\{W_{r+1}(X)\\}$.

###  2.2. Compliment: The algebra
$\widehat{\mathcal{D}}_{W(X),\text{crys}}^{(0)}$

In this subsection we wish to discuss the analogue of the above results and
constructions for a slightly different algebra, which will turn out to be a
completion of $\widehat{\mathcal{D}}_{W(X)}^{(0)}$. As above we start in the
case where $X=\text{Spec}(A)$.

######  Definition 2.34.

The operator filtration on $\widehat{\mathcal{D}}_{W(A)}^{(0)}$ is the
filtration defined by

$F^{i}(\widehat{\mathcal{D}}_{W(A)}^{(0)})=\\{P\in\widehat{\mathcal{D}}_{W(A)}^{(0)}|P(W(A)\subset
V^{i}(W(A))\\}$

This is a filtration by two sided ideals, and we denote the completion along
it by $\widehat{\mathcal{D}}_{W(X),\text{crys}}^{(0)}$.

There is a description of $\widehat{\mathcal{D}}_{W(X),\text{crys}}^{(0)}$ in
local coordinates:

######  Proposition 2.35.

Suppose $A$ admits local coordinates, and let the notation be as in Theorem
2.17. Then an element

$\sum_{I}(\sum_{r=0}^{\infty}\sum_{J_{I}}F^{-r}(\alpha_{J_{I}})\\{\partial\\}_{J_{I}/p^{r}})\\{\partial\\}^{I}\in\widehat{\mathcal{D}}_{W(A)}^{(0)}$

is contained in $F^{i}(\widehat{\mathcal{D}}_{W(A)}^{(0)})$ iff, whenever
$f_{I}<i$ and $r<i$, we have $\alpha_{J_{I}}\in V^{i-(r+f_{I})}(W(A))$ (where
we set $V^{i-(r+f_{I})}(W(A)=W(A)$ for $i-(r+f_{I})<0$).

Therefore, every element of $\widehat{\mathcal{D}}_{W(A),\text{crys}}^{(0)}$
has a unique expression

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

where $I=(i_{1},\dots,i_{n})\in\mathbb{N}^{n}$,
$\\{\partial\\}^{I}:=\\{\partial_{1}\\}_{1}^{i_{1}}\cdots\\{\partial_{n}\\}_{1}^{i_{n}}$,
and $J_{I}=(j_{1},\dots,j_{n})$ satisfies $0\leq j_{l}<p^{r}$ for each $j_{l}$
and $(p,j_{l})=1$ for at least one $l$; unless $r=0$ and $J_{I}=0$.

###### Proof.

This follows from Lemma 2.26 above. ∎

Exactly as above, this implies that the sheafification
$\widehat{\mathcal{D}}_{W(X),\text{crys}}^{(0)}$ has the property that
$\Gamma(U,\widehat{\mathcal{D}}_{W(X),\text{crys}}^{(0)})=\widehat{\mathcal{D}}_{W(A),\text{crys}}^{(0)}$
whenever $U=\text{Spec}(A)$ has local coordinates. We also see that

######  Corollary 2.36.

The completion map
$\widehat{\mathcal{D}}_{W(X)}^{(0)}\to\widehat{\mathcal{D}}_{W(X),\text{crys}}^{(0)}$
is injective.

##  3\. Accessibility

In this chapter we will define the main category of interest in this paper,
the category of accessible modules over $\widehat{\mathcal{D}}_{W(X)}^{(m)}$
(here, $m\geq 0$). In particular, we shall prove Theorem 1.5, Theorem 1.6,
1.7, as well as several other related results. To get things off the ground,
we need to defined and study the fundamental bimodule
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$ where $\mathcal{A}$ is, as
usual, a smooth $W(k)$ algebra lifting $A$ which admits local coordinates. We
let $F:\mathcal{A}\to\mathcal{A}$ be some coordinatized lift of Frobenius and
$\Phi:\mathcal{A}\to W(A)$ the associated map to Witt vectors.

######  Lemma 3.1.

1) The action map
$\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}\to\mathcal{E}nd_{W(k)}(\mathcal{A})$
is injective.

2) Let $\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$ denote the
completion of
$W(A)\otimes_{\mathcal{A}}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$ along the
filtration
$V^{i}(W(A))\otimes_{\mathcal{A}}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$.
Then there is an embedding

$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}\to\mathcal{H}om_{W(k)}(\mathcal{A},W(A))$

which takes $1\otimes 1$ to $\Phi$. This embedding preserves the natural right
$\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$-module structures on both sides.

###### Proof.

1) This is a well-known fact, whose proof boils down to a (vastly) simplified
version of Lemma 2.26.

2) There is a map
$\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}\to\mathcal{H}om_{W(k)}(\mathcal{A},W(A))$
given by taking an element $Q\in\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$ to
$\Phi\circ Q:\mathcal{A}\to W(A)$ (where we regard $Q$ as acting on
$\mathcal{A}$). Since $\Phi$ is injective this map is injective also; in
addition, the right action of $\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$ on
itself clearly corresponds to the right action of
$\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$ on
$\mathcal{H}om_{W(k)}(\mathcal{A},W(A))$ through the action on $\mathcal{A}$.
Further, the (left) action of $W(A)$ on itself gives
$\mathcal{H}om_{W(k)}(\mathcal{A},W(A))$ the structure of a $W(A)$-module, so
that there is an induced map

$\iota:W(A)\otimes_{\mathcal{A}}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}\to\mathcal{H}om_{W(k)}(\mathcal{A},W(A))$

and, for elements $a_{i}\in
V^{i}(W(A))\otimes_{\mathcal{A}}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$,
the map $\iota(a_{i})$ has image in $V^{i}(W(A))$, so that the sum
${\displaystyle\sum_{i}\iota(a_{i})}$ is well-defined in
$\mathcal{H}om_{W(k)}(\mathcal{A},W(A))$. Thus there is an induced map, which
we also call $\iota$,
$\iota:\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}\to\mathcal{H}om_{W(k)}(\mathcal{A},W(A))$,
which clearly preserves the right
$\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$-module structures.

To show that this map is injective, let $\\{T_{1},\dots,T_{n}\\}$ be local
coordinates on $\mathcal{A}$. Then any element of $W(A)$ can be written
uniquely as
${\displaystyle\sum_{r=0}^{\infty}\sum_{I}V^{r}(T^{I})\cdot\Phi(a_{I})}$
where, for any $r$, $I=(i_{1},\dots,i_{n})$ ranges over elements of
$\mathbb{N}^{n}$ such that $0\leq i_{j}<p^{r}$ and $(p,i_{j})=1$ for at least
one index $j$; and $a_{I}\in\mathcal{A}$. Thus any element
$Q\in\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$ can be written
uniquely as ${\displaystyle\sum_{r=0}^{\infty}\sum_{I}V^{r}(T^{I})\cdot
Q_{I}}$ where $Q_{I}\in\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$ and
$I$ satisfies the same conditions as above. But such an element acts as $0$ on
$\mathcal{A}$ iff each $Q_{I}$ acts as zero on $\mathcal{A}$ which implies
that $Q=0$; therefore $\iota$ is injective as claimed. ∎

The natural action of $\widehat{\mathcal{D}}_{W(A)}^{(m)}$ on $W(A)$ endows
the module $\mathcal{H}om_{W(k)}(\mathcal{A},W(A))$ with a left action by
$\widehat{\mathcal{D}}_{W(A)}^{(m)}$.

######  Proposition 3.2.

The submodule $\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$ of
$\mathcal{H}om_{W(k)}(\mathcal{A},W(A))$ is preserved under the action of
$\widehat{\mathcal{D}}_{W(A)}^{(m)}$. Thus
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$ has the structure of a
$(\widehat{\mathcal{D}}_{W(A)}^{(m)},\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)})$-bimodule.
If $m\geq 0$ the map
$\mathcal{D}_{W(\mathcal{A})}^{(0)}\to\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}$
which takes $1\to\Phi$ is onto.

###### Proof.

Let us consider the first statement. We shall suppose that $m=0$, the other
cases being extremely similar. We begin by considering an element of the form
$\\{\partial\\}_{J/p^{r}}$, where $r\geq 0$ and $0\leq j_{i}\leq p^{r}$ for
all $i\in\\{1,\dots n\\}$). We shall show that
$\\{\partial\\}_{J/p^{r}}\circ\Phi\in\text{Hom}_{W(k)}(\mathcal{A},W(A))$
agrees with an element of $\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}$.

The map $\\{\partial\\}_{J/p^{r}}\circ\Phi$ induces, for each $r^{\prime}\geq
r$, a map $\mathcal{A}/p^{r^{\prime}+1}\to W_{r^{\prime}+1}(A)$. Further,
$\mathcal{A}/p^{r^{\prime}+1}$ is etale over a polynomial ring
$W_{r^{\prime}+1}(k)[T_{1},\dots,T_{n}]$. Therefore we must compute the action
of $\\{\partial\\}_{J/p^{r}}$ on a monomial $T_{1}^{k_{1}}\cdots
T_{n}^{k_{n}}\in W_{r^{\prime}+1}(k)[T_{1},\dots,T_{n}]$. We have

$\\{\partial\\}_{J/p^{r}}(T_{1}^{k_{1}}\cdots
T_{n}^{k_{n}})=\prod_{i=1}^{n}\binom{k_{i}p^{r^{\prime}}}{j_{i}p^{r^{\prime}-r}}\cdot
T^{k_{i}-j_{i}/p^{r}}\in W_{r^{\prime}+1}(k[T_{1},\dots,T_{n}])$

When $j_{i}\neq 0$ we have the identity

$\binom{k_{i}p^{r^{\prime}}}{j_{i}p^{r^{\prime}-r}}=\prod_{l=1}^{j_{i}p^{r^{\prime}-r}}\frac{k_{i}p^{r^{\prime}}-j_{i}p^{r^{\prime}-r}+l}{l}=\prod_{l=1}^{j^{\prime}_{i}p^{r^{\prime}-r+v_{i}}}\frac{k_{i}p^{r^{\prime}}-j^{\prime}_{i}p^{r^{\prime}-r+v_{i}}+l}{l}$

where we have written $j_{i}=j_{i}^{\prime}p^{v_{i}}$ where
$v_{i}=\text{val}(j_{i})$. Our aim is to show that this expression is a
polynomial in $k_{i}$, with coefficients in $\mathbb{Z}/p^{r^{\prime}+1}$. If
$j_{i}=p^{r}$, then we have

$\binom{k_{i}p^{r^{\prime}}}{j_{i}p^{r^{\prime}-r}}=\binom{k_{i}p^{r^{\prime}}}{p^{r^{\prime}}}=\prod_{l=1}^{p^{r^{\prime}}}\frac{(k_{i}-1)p^{r^{\prime}}+l}{l}$
$=k_{i}\cdot\prod_{l=1}^{p^{r^{\prime}}-1}((k_{i}-1)\frac{p^{r^{\prime}}}{l}+1)$

which is evidently a polynomial in $k_{i}$, with coefficients in
$\mathbb{Z}/p^{r^{\prime}+1}$. We remark that if $q_{i,r^{\prime}}(t)$ denotes
this polynomial, then $t|q_{i,r^{\prime}}(t)$.

Next, if $j_{i}<p^{r}$ then $v_{i}<r$ and we have

(3.1)
$\frac{k_{i}p^{r^{\prime}}-j^{\prime}_{i}p^{r^{\prime}-r+v_{i}}-l}{l}=\frac{p^{r^{\prime}-r+v_{i}}(k_{i}p^{r-v_{i}}-j^{\prime}_{i})-l}{l}$

and since $\text{val}(j_{i}^{\prime})=0$ we have
$\text{val}(k_{i}p^{r-v_{i}}-j^{\prime}_{i})=0$ and so, if $\text{val}(l)\leq
r^{\prime}-r+v_{i}$ we have

$\frac{p^{r^{\prime}-r+v_{i}}(k_{i}p^{r-v_{i}}-j^{\prime}_{i})-l}{l}=\frac{p^{r^{\prime}-r+v_{i}}}{l}(k_{i}p^{r-v_{i}}-j^{\prime}_{i})-1\in\mathbb{Z}/p^{r^{\prime}+1}$

However, if $\text{val}(l)>r^{\prime}-r+v_{i}$ then the term (3.1), considered
as a element of $\mathbb{Q}$, is not in $\mathbb{Z}_{p}$. For each such $l$ we
define $l_{j^{\prime}}:=j^{\prime}p^{r^{\prime}-r+v_{i}}-l$; since $1\leq
l\leq j^{\prime}p^{r^{\prime}-r+v_{i}}$ the same is true of $l_{j^{\prime}}$,
and we have $\text{val}(l_{j^{\prime}})=r^{\prime}-r+v_{i}$. We have that

$\frac{k_{i}p^{r^{\prime}}-j^{\prime}_{i}p^{r^{\prime}-r+v_{i}}-l}{l}\cdot\frac{k_{i}p^{r^{\prime}}-j^{\prime}_{i}p^{r^{\prime}-r+v_{i}}-l_{j^{\prime}}}{l_{j^{\prime}}}$
$=\frac{k_{i}p^{r^{\prime}}-l_{j^{\prime}}}{l_{j^{\prime}}}\cdot\frac{k_{i}p^{r^{\prime}}-l}{l}=(k_{i}\frac{p^{r^{\prime}}}{l_{j^{\prime}}}-1)(k_{i}\frac{p^{r^{\prime}}}{l}-1)$

So, in sum, we have

$\binom{k_{i}p^{r^{\prime}}}{j_{i}p^{r^{\prime}-r}}=\prod(\frac{p^{r^{\prime}-r+v_{i}}}{l}(k_{i}p^{r-v_{i}}-j^{\prime}_{i})-1)\cdot\prod_{\\{l|\text{val}(l)>r^{\prime}-r+v_{i}\\}}(k_{i}\frac{p^{r^{\prime}}}{l_{j^{\prime}}}-1)(k_{i}\frac{p^{r^{\prime}}}{l}-1)$

where the first product ranges over
$l\in\\{1,\dots,j^{\prime}_{i}p^{r^{\prime}-r+v_{i}}\\}$ such that
$\text{val}(l)\leq r^{\prime}-r+v_{i}$ and $l$ is not of the form
$l_{j^{\prime}}$. This expression is evidently a polynomial in $k_{i}$, with
coefficients in $\mathbb{Z}/p^{r^{\prime}+1}$; as above we denote this
polynomial by $q_{i,r^{\prime}}(t)$ . When
$l=j^{\prime}_{i}p^{r^{\prime}-r+v_{i}}$ we get the term ${\displaystyle
k_{i}\frac{p^{r-v_{i}}}{j_{i}^{\prime}}}$, so that again
$t|q_{i,r^{\prime}}(t)$. By inspection one sees that the image of
$q_{i,r^{\prime}}(t)$ in $\mathbb{Z}/p^{r^{\prime}}[t]$ is
$q_{i,r^{\prime}-1}(t)$.

Now we note that, for any polynomial
$q^{\prime}(t)\in\mathbb{Z}/p^{r^{\prime}+1}[t]$, and for a fixed index
$i\in\\{1,\dots n\\}$, the endomorphism of
$W_{r^{\prime}+1}(k)[T_{1},\dots,T_{n}]$ given by

$\prod_{j=1}^{n}T_{j}^{k_{j}}\to k_{i}q^{\prime}(k_{i})\prod_{j\neq
i}T_{j}^{k_{j}}\cdot T_{i}^{k_{i}-1}$

is given by the action of an operator in
$D_{\mathcal{A}/p^{r^{\prime}+1}}^{(0)}$; indeed we have that
$\frac{\partial}{\partial T_{i}}\cdot
q^{\prime}{\displaystyle(T_{i}\frac{\partial}{\partial
T_{i}})(\prod_{j=1}^{n}T_{j}^{k_{j}})=k_{i}q^{\prime}(k_{i})\cdot\prod_{j\neq
i}T_{j}^{k_{j}}\cdot T_{i}^{k_{i}-1}}$. So we have

$\\{\partial\\}_{J/p^{r}}(T_{1}^{k_{1}}\cdots
T_{n}^{k_{n}})=(\prod_{\\{i|j_{i}\neq
0\\}}p^{r-v_{i}}T_{i}^{1-j_{i}/p^{r}}\cdot
q_{i,r^{\prime}}(k_{i}))(T_{1}^{k_{1}-\epsilon_{1}}\cdots
T_{n}^{k_{n}-\epsilon_{n}})$

(where $\epsilon_{i}=1$ if $j_{i}\neq 0$ and $\epsilon_{i}=0$ if $j_{i}=0$),
which, since $t|q_{i,r^{\prime}}(t)$, is given by the application of an
operator

$\prod_{\\{i|j_{i}\neq 0\\}}p^{r-v_{i}}T_{i}^{1-j_{i}/p^{r}}\cdot
Q_{i,r^{\prime}}$

for appropriate $Q_{i,r^{\prime}}\in D_{\mathcal{A}/p^{r^{\prime}+1}}^{(0)}$;
since $\mathcal{A}/p^{r^{\prime}+1}$ is etale over
$W_{r^{\prime}+1}(k)[T_{1},\dots,T_{n}]$ and both $\\{\partial\\}_{J/p^{r}}$
and ${\displaystyle\prod_{\\{i|j_{i}\neq
0\\}}p^{r-v_{i}}T_{i}^{1-j_{i}/p^{r}}\cdot Q_{i,r^{\prime}}}$ are differential
operators, we deduce that they are equal on all of
$\mathcal{A}/p^{r^{\prime}+1}$. Furthermore the image of $Q_{i,r^{\prime}}$ in
$D_{\mathcal{A}/p^{r^{\prime}}}^{(0)}$ is $Q_{i,r^{\prime}-1}$ since this is
true for the polynomials $q_{i,r^{\prime}}$. Therefore, we have that the
operator

$\prod_{\\{i|j_{i}\neq 0\\}}p^{r-v_{i}}T_{i}^{1-j_{i}/p^{r}}\cdot
Q_{i}=\lim_{r^{\prime}\to\infty}\prod_{\\{i|j_{i}\neq
0\\}}p^{r-v_{i}}T_{i}^{1-j_{i}/p^{r}}\cdot Q_{i,r^{\prime}}$

acts as $\\{\partial\\}_{J/p^{r}}$ on $\mathcal{A}$, which shows that
$\\{\partial\\}_{J/p^{r}}\circ\Phi$ agrees with an element of
$\Phi^{*}\mathcal{D}_{\mathcal{A}}^{(0)}$.

Now suppose, in addition, that $J=(j_{1},\dots,j_{n})$ has the property that
at least one $j_{i}$, say $j_{i^{\prime}}$, satisfies
$\text{val}(j_{i^{\prime}})=0$. Then we may consider the operator
$F^{-r}(\alpha)\cdot\\{\partial\\}_{J/p^{r}}$, and the previous discussion
shows that this operator acts as

$V^{r}(\alpha T^{p^{r}-j_{i^{\prime}}})Q_{i^{\prime}}\cdot\prod_{i\neq
i^{\prime}}p^{r-v_{i}}T_{i}^{1-j_{i}/p^{r}}\cdot
Q_{i}\in\Phi^{*}\mathcal{D}_{\mathcal{A}}^{(0)}$

This implies further that, for any $Q\in\mathcal{D}_{\mathcal{A}}^{(0)}$, the
operator $F^{-r}(\alpha)\\{\partial\\}_{J/p^{r}}\circ\Phi\circ Q$ agrees with
$V^{r}(\alpha T^{p^{r}-j_{i^{\prime}}})Q_{i^{\prime}}\cdot\prod_{i\neq
i^{\prime}}p^{r-v_{i}}T_{i}^{1-j_{i}/p^{r}}\cdot Q_{i}\cdot Q$, as an operator
on $\mathcal{A}$, and is therefore also contained in
$\Phi^{*}\mathcal{D}_{\mathcal{A}}^{(0)}$. Thus we see that, for any sum
${\displaystyle\sum_{r=1}^{\infty}F^{-r}(\alpha)\cdot\\{\partial\\}_{J/p^{r}}}$,
we have

$\sum_{r=1}^{\infty}F^{-r}(\alpha)\cdot\\{\partial\\}_{J/p^{r}}\circ\Phi\circ
Q={\displaystyle\sum_{r=1}^{\infty}}V^{r}(\alpha
T^{p^{r}-j_{i^{\prime}}})Q_{i^{\prime}}\cdot\prod_{i\neq
i^{\prime}}p^{r-v_{i}}T_{i}^{1-j_{i}/p^{r}}\cdot Q_{i}\cdot Q$

which is a convergent sum in $\Phi^{*}\mathcal{D}_{\mathcal{A}}^{(0)}$.

Next consider any element of the form
$\\{\partial\\}^{I}=\\{\partial_{1}\\}^{i_{1}}\cdots\\{\partial_{n}\\}^{i_{n}}$.
Suppose, inductively, that
$\\{\partial_{1}\\}^{i_{1}-1}\cdots\\{\partial_{n}\\}^{i_{n}}\circ\Phi\in\Phi^{*}\mathcal{D}_{\mathcal{A}}^{(0)}$,
and write

$\\{\partial_{1}\\}^{i_{1}-1}\cdots\\{\partial_{n}\\}^{i_{n}}\circ\Phi=\sum_{r=0}^{\infty}V^{r}(\alpha_{r})\cdot\Phi\circ
Q_{r}$

for $Q_{r}\in\mathcal{D}_{\mathcal{A}}^{(0)}$. Then

$\\{\partial_{1}\\}\sum_{r=0}^{\infty}V^{r}(\alpha_{r})\cdot\Phi\circ
Q_{r}=\sum_{r=0}^{\infty}[\\{\partial_{1}\\},V^{r}(\alpha_{r})]\cdot\Phi\circ
Q_{r}+\sum_{r=0}^{\infty}V^{r}(\alpha_{r})\\{\partial\\}_{1}\circ\Phi\circ
Q_{r}$

and since
${\displaystyle[\\{\partial_{1}\\},V^{r}(\alpha_{r})]=\sum_{l=1}^{\infty}\sum_{j\leq
p^{l}}F^{-l}(\beta_{jl})\\{\partial\\}_{j/p^{l}}}$ (as in the proof of (2.5)),
the previous remarks imply that
$\\{\partial\\}^{I}\circ\Phi\in\Phi^{*}\mathcal{D}_{\mathcal{A}}^{(0)}$. As
above this implies directly that $\\{\partial\\}^{I}\circ\Phi\circ
Q\in\Phi^{*}\mathcal{D}_{\mathcal{A}}^{(0)}$ for all
$Q\in\mathcal{D}_{\mathcal{A}}^{(0)}$, and then, using the representation of
an element of $\widehat{D}_{W(A)}^{(0)}$ as in Theorem 2.17, the first result
follows.

Now we consider the second statement, again when $m=0$ ($m>0$ is essentially
identical). We may, by the fact that
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}$ is $p$-adically complete,
prove the analogous fact for the reduction
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}/p$; let us call this object
$\Phi^{*}\mathcal{D}_{A}^{(0)}$. We have, for each $1\leq i\leq n$, that
$\\{\partial_{i}\\}\cdot(1\otimes 1)=1\otimes\partial_{i}$ inside
$\Phi^{*}\mathcal{D}_{A}^{(0)}$ (this follows directly from the above
description of the action of $\\{\partial_{i}\\}$). So we see that the image
of $\widehat{\mathcal{D}}_{W(A)}^{(0)}$ acting on $1\otimes
1\in\Phi^{*}\mathcal{D}_{A}^{(0)}$ contains every element of the form
$\alpha\otimes P$ for $P\in\mathcal{D}_{A}^{(0)}$. As every element of
$\Phi^{*}\mathcal{D}_{A}^{(0)}$ is a (convergent) sum of such elements, the
result follows. ∎

Let us note here that this bimodule sheafifies well:

######  Lemma 3.3.

Let $\mathcal{A}$ and $\Phi:\mathcal{A}\to W(A)$ be as above, and let
$X=\text{Spec}(A)$, $\mathfrak{X}=\text{Specf}(\mathcal{A})$. For each affine
open $U\subset X$ we obtain, by complete localization of the map $\Phi$, a map
$\Phi:\mathcal{O}_{\mathfrak{X}}(U)\to W(\mathcal{O}_{X}(U))$, and, therefore,
a bimodule $\Phi^{*}\mathcal{O}_{\mathfrak{X}}(U)$. Thus, regarding $\Phi$ as
a morphism $\mathcal{O}_{\mathfrak{X}}\to W(\mathcal{O}_{X})$, this sheaf is
the sheaf of bi-sub-modules of
$\mathcal{H}om_{W(k)}(\mathcal{O}_{\mathfrak{X}},\mathcal{O}_{W(X)})$ locally
generated by $\Phi$. It is also the inverse limit of the quasicoherent sheaves
$(W_{r}(A)\otimes_{\mathcal{A}}\mathcal{D}_{\mathcal{A}_{r}}^{(0)})^{\sim}$ on
$\text{Spec}(W_{r}(A))$, and hence quasicoherent on $W(X)$.

###### Proof.

Everything is clear from the above except, perhaps, the quasicoherence
statement. For that, recall that, for a $\\{\mathcal{F}_{n}\\}$ a surjective
system of quasicoherent sheaves and $U\subset X$ any open affine, we have
${\displaystyle\lim_{n}\mathcal{F}_{n}(U)\tilde{=}(\lim_{n}\mathcal{F}_{n})(U)}$
(by, e.g., [28], lemma 1.1.6). But for each such $U$ the theorem above implies

$\lim_{r}(W_{r}(A)\otimes_{\mathcal{A}}\mathcal{D}_{\mathcal{A}_{r}}^{(0)})^{\sim}(U)\tilde{=}\Phi^{*}\mathcal{O}_{\mathfrak{X}}(U)$

and the result follows. ∎

Before proceeding, we would like to give the crystalline version of these
results. Fix $r\geq 1$, and consider $\mathfrak{X}_{r}$ a smooth formal scheme
over $W_{r}(k)$ with special fibre $X$. We recall that the differential
operators $\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}$ posses a two sided ideal
$\mathcal{I}_{r}$ consisting of sections $P$ which annihilate
$\mathcal{O}_{\mathfrak{X}_{r}}$. If $\mathfrak{X}_{r}$ is affine and possess
local coordinates $\\{T_{1},\dots,T_{n}\\}$, then this ideal is generated by
sections of the form $\\{p^{j}\partial_{i}^{p^{r-j}}\\}$ for $1\leq i\leq n$
and $0\leq j\leq r-1$. The completion of
$\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}$ along the powers of $\mathcal{I}_{r}$
is denoted $\mathcal{D}_{\mathfrak{X}_{r},\text{crys}}^{(0)}$; if
$\mathfrak{X}_{r}=\text{Spec}(\mathcal{A}_{r})$ then we denote
$\mathcal{D}_{\mathcal{A}_{r},\text{crys}}^{(0)}$ the global sections of
$\mathcal{D}_{\mathfrak{X}_{r},\text{crys}}^{(0)}$. Taking the inverse limit
over $r$, we obtain the algebra
$\mathcal{\widehat{D}}_{\mathcal{A},\text{crys}}^{(0)}$, which is a further
completion of $\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}$; similarly we have
the sheaf $\mathcal{\widehat{D}}_{\mathfrak{X},\text{crys}}^{(0)}$. One can
show without much difficulty that elements of this ring have unique
expressions of the form

$\sum_{I}a_{I}\partial^{I}$

where $a_{I}\in\mathcal{A}$. Since $\partial^{I}\to 0$ as $I\to\infty$ as
operators on $\mathcal{A}$, this algebra acts on $\mathcal{A}$, and the action
is faithful.

Now, repeating the proof of the above theorems essentially verbatim, we obtain

######  Theorem 3.4.

1) Let $\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A},\text{crys}}^{(0)}$ denote
the completion of
$W(A)\otimes_{\mathcal{A}}\widehat{\mathcal{D}}_{\mathcal{A},\text{crys}}^{(0)}$
along the filtration
$V^{i}(W(A))\otimes_{\mathcal{A}}\widehat{\mathcal{D}}_{\mathcal{A},\text{crys}}^{(0)}$.
Then there is an embedding

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

which takes $1\otimes 1$ to $\Phi$. This embedding preserves the natural right
$\widehat{\mathcal{D}}_{\mathcal{A},\text{crys}}^{(0)}$-module structures on
both sides.

2) The submodule
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A},\text{crys}}^{(0)}$ of
$\mathcal{H}om_{W(k)}(\mathcal{A},W(A))$ is preserved under the action of
$\widehat{\mathcal{D}}_{W(A)}^{(0)}$. Thus
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A},\text{crys}}^{(0)}$ has the
structure of a
$(\widehat{\mathcal{D}}_{W(A)}^{(0)},\widehat{\mathcal{D}}_{\mathcal{A},\text{crys}}^{(0)})$-bimodule.

3) Let $X=\text{Spec}(A)$, $\mathfrak{X}=\text{Specf}(\mathcal{A})$. Then
there is a
$(\widehat{\mathcal{D}}_{W(X)}^{(0)},\widehat{\mathcal{D}}_{\mathfrak{X},\text{crys}}^{(0)})$,
which is an inverse limit of quasicoherent sheaves on $W(X)$.

Now we want to compare two bimodules
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$ and
$\Psi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$ for two morphisms
$\Phi,\Psi:\mathcal{A}\to W(A)$ coming from two lifts of Frobenius on
$\mathcal{A}$. In fact, assuming $p>2$ the situation is as nice as possible.
We begin by showing

######  Lemma 3.5.

Let $\Phi$ and $\Psi$ be two morphisms $\mathcal{A}\to W(A)$, coming from
coordinatized lifts of Frobenius on $\mathcal{A}$. Make $W(A)$ into an
$\mathcal{A}$-module via $\Phi$. Then, for any $n\geq 1$, the map
$\delta_{n}:\mathcal{A}/p^{n}\to W(A)/p^{n}$, given by the reduction mod
$p^{n}$ of $\delta=\Phi-\Psi$, is a differential operator of degree $\leq n$
over $\mathcal{A}$.

###### Proof.

We start by noting the formula

$\delta(ab)=a\delta(b)+b\delta(a)-\delta(a)\delta(b)$

for any $a,b\in\mathcal{A}$. Indeed, we have

$\delta(ab)=\Phi(ab)-\Psi(ab)=\Phi(a)\Phi(b)-\Psi(a)\Psi(b)$
$=\Phi(a)\Phi(b)-\Phi(a)\Psi(b)+\Phi(a)\Psi(b)-\Psi(a)\Psi(b)$
$=\Phi(a)\delta(b)+\delta(a)\Psi(b)=\Phi(a)\delta(b)+\delta(a)(-\delta(b)+\Phi(b))$
$=a\delta(b)+b\delta(a)-\delta(a)\delta(b)$

as claimed. Now, the map $\delta$ takes values in the ideal $V(W(A))$, as the
reduction mod $V(W(A))$ of both $\Phi$ and $\Psi$ is, by construction, the
natural quotient map $\mathcal{A}\to A$. So, after reduction to $W(A)/p$ we
have $\delta(a)\delta(b)=0$; in other words, $\delta_{1}:A\to W(A)/p$ is a
derivation.

Now we suppose $n\geq 2$. For any $a\in\mathcal{A}/p^{n}$ we have the operator
$\delta_{n,a}:\mathcal{A}/p^{n}\to W(A)/p^{n}$ defined by
$\delta_{n,a}(b)=\delta_{n}(ab)-a\cdot\delta_{n}(b)$. Iterating, we obtain for
any sequence $(a_{1},\dots a_{r})$ in $\mathcal{A}/p^{n}$ an operator
$\delta_{n,a_{1},\dots,a_{r}}:\mathcal{A}/p^{n}\to W(A)/p^{n}$ defined
inductively by

$\delta_{n,a_{1},\dots,a_{r}}(b)=\delta_{n,a_{1},\dots,a_{r-1}}(a_{r}\cdot
b)-a_{r}\cdot\delta_{n,a_{1},\dots,a_{r-1}}(b)$

To show that $\delta_{n}$ is a differential operator of order $\leq n$ we must
show that $\delta_{n,a_{1},\dots,a_{n+1}}=0$ for any sequence $(a_{1},\dots
a_{n+1})$ of length $n+1$.

On the other hand, for the sequence $(a_{1},\dots,a_{r})$ we can define the
operator $\epsilon_{a_{1},\dots a_{r}}:\mathcal{A}/p^{n}\to W(A)/p^{n}$

$\epsilon_{a_{1},\dots
a_{r}}(b)=\delta_{n}(a_{1})\cdots\delta_{n}(a_{r})\delta_{n}(b)$

I claim that for each $r\geq 1$, the map

$\delta_{a_{1},\dots,a_{r}}+(-1)^{r+1}\epsilon_{a_{1},\dots
a_{r}}:\mathcal{A}/p^{n}\to W(A)/p^{n}$

is an $\mathcal{A}/p^{n}$-linear map. We proceed by induction on $r$; when
$r=1$ this follows from

$\delta_{n,a}(b):=\delta_{n}(ab)-a\cdot\delta_{n}(b)=b\cdot\delta_{n}(a)-\delta_{n}(a)\delta_{n}(b)$

Supposing the result holds for $r$, we have

$\delta_{n,a_{1},\dots,a_{r+1}}(b):=\delta_{n,a_{1},\dots,a_{r}}(a_{r+1}\cdot
b)-a_{r+1}\cdot\delta_{n,a_{1},\dots,a_{r}}(b)$

By induction, we have that

$\delta_{n,a_{1},\dots,a_{r}}=(-1)^{r+1}\epsilon_{a_{1},\dots,a_{r}}+\varphi$

where $\varphi:\mathcal{A}/p^{n}\to W(A)/p^{n}$ is an
$\mathcal{A}/p^{n}$-linear map. So

$\delta_{n,a_{1},\dots,a_{r}}(a_{r+1}\cdot
b)=(-1)^{r+1}\epsilon_{a_{1},\dots,a_{r}}(a_{r+1}\cdot b)+a_{r+1}\varphi(b)$

and

$a_{r+1}\cdot\delta_{n,a_{1},\dots,a_{r}}(b)=(-1)^{r+1}a_{r+1}\cdot\epsilon_{a_{1},\dots,a_{r}}(b)+a_{r+1}\varphi(b)$

So we see that

$\delta_{n,a_{1},\dots,a_{r+1}}(b)=(-1)^{r+1}(\epsilon_{a_{1},\dots,a_{r}}(a_{r+1}\cdot
b)-a_{r+1}\cdot\epsilon_{a_{1},\dots,a_{r}}(b))$

But

$\epsilon_{a_{1},\dots,a_{r}}(a_{r+1}\cdot
b)=\delta_{n}(a_{1})\cdots\delta_{n}(a_{r})\delta_{n}(a_{r+1}b)$
$=b\cdot\delta_{n}(a_{1})\cdots\delta_{n}(a_{r})\delta_{n}(a_{r+1})+a_{r+1}\cdot\delta_{n}(a_{1})\cdots\delta_{n}(a_{r})\delta_{n}(b)-\delta_{n}(a_{1})\cdots\delta_{n}(a_{r})\delta_{n}(a_{r+1})\delta_{n}(b)$
$=b\cdot\delta_{n}(a_{1})\cdots\delta_{n}(a_{r})\delta_{n}(a_{r+1})+a_{r+1}\epsilon_{a_{1},\dots,a_{r}}(b)-\epsilon_{a_{1},\dots,a_{r+1}}(b)$

so that

$\epsilon_{a_{1},\dots,a_{r}}(a_{r+1}\cdot
b)-a_{r+1}\cdot\epsilon_{a_{1},\dots,a_{r}}(b)=b\cdot\delta_{n}(a_{1})\cdots\delta_{n}(a_{r})\delta_{n}(a_{r+1})-\epsilon_{a_{1},\dots,a_{r+1}}(b)$

Therefore

$\delta_{n,a_{1},\dots,a_{r+1}}(b)=(-1)^{r+1}(\epsilon_{a_{1},\dots,a_{r}}(a_{r+1}\cdot
b)-a_{r+1}\cdot\epsilon_{a_{1},\dots,a_{r}}(b))$
$=(-1)^{r+1}(b\cdot\delta_{n}(a_{1})\cdots\delta_{n}(a_{r})\delta_{n}(a_{r+1})-\epsilon_{a_{1},\dots,a_{r+1}}(b))$
$=(-1)^{r+2}\epsilon_{a_{1},\dots,a_{r+1}}(b)+(-1)^{r+1}b\cdot\delta_{n}(a_{1})\cdots\delta_{n}(a_{r})\delta_{n}(a_{r+1})$

as required.

Finally, to finish the proof, we note that

$\epsilon_{a_{1},\dots
a_{n}}(b)=\delta_{n}(a_{1})\cdots\delta_{n}(a_{n})\delta_{n}(b)=0$

for all $b$, as the product of $n+1$ elements of the form $V(\alpha)$ (in
$W(A)$) is contained in the ideal generated by $p^{n}$. Thus
$\delta_{n,a_{1},\dots,a_{n}}$ is a linear operator, and the result follows. ∎

Using this, we can prove

######  Theorem 3.6.

Let $\Phi$ and $\Psi$ be two morphisms $\mathcal{A}\to W(A)$, coming from
coordinatized lifts of Frobenius on $\mathcal{A}$.

1) Suppose $p>2$. Then there is a canonical isomorphism of
$(\widehat{\mathcal{D}}_{W(A)}^{(0)},\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)})$-bimodules
$\epsilon_{\Phi,\Psi}:\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}\tilde{\to}\Psi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}$.
If $\Xi$ is a third such lift then we have the cocycle condition
$\epsilon_{\Xi,\Phi}\circ\epsilon_{\Phi,\Psi}=\epsilon_{\Xi,\Psi}$.

2) For any $p$, there is a canonical isomorphism of
$(\widehat{\mathcal{D}}_{W(A),\text{crys}}^{(0)},\widehat{\mathcal{D}}_{\mathcal{A},\text{crys}}^{(0)})$-bimodules
$\epsilon_{\Phi,\Psi}:\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A},\text{crys}}^{(0)}\tilde{\to}\Psi^{*}\widehat{\mathcal{D}}_{\mathcal{A},\text{crys}}^{(0)}$.
If $\Xi$ is a third such lift then we have the cocycle condition
$\epsilon_{\Xi,\Phi}\circ\epsilon_{\Phi,\Psi}=\epsilon_{\Xi,\Psi}$. When $p>2$
this isomorphism is a completion of the one constructed in part $1)$ above.

3) Let $m>0$. Then for all $p$ there is a canonical isomorphism of
$(\widehat{\mathcal{D}}_{W(A)}^{(m)},\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)})$-bimodules
$\epsilon_{\Phi,\Psi}:\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}\tilde{\to}\Psi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$;
we again have the cocycle condition for a third such lift.

###### Proof.

1) By construction (c.f. 3.2), we have canonical inclusions
$\Psi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)},\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}\subset\text{Hom}_{W(k)}(\mathcal{A},W(A))$.
We shall show that, when $p>2$, these inclusions have the same image; this
immediately proves part $1)$.

To proceed, we shall show that $\Psi:\mathcal{A}\to W(A)$ is contained in the
image of $\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}$ in
$\text{Hom}_{W(k)}(\mathcal{A},W(A))$. This implies directly that the image of
$\Psi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}$ is contained in image of
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}$. By symmetry, we also have
the reverse inclusion, and hence the required equality.

Since $\Phi:\mathcal{A}\to W(A)$ is contained in the image of
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}$ by definition, it suffices
to show that $\delta=\Phi-\Psi$ is contained in the image of
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}$. By the previous lemma,
the reduction $\delta_{n}:\mathcal{A}/p^{n}\to W(A)/p^{n}$ is a differential
operator of order $\leq n$ (as in that lemma, we regard $W(A)$ as being a
$\mathcal{A}$-module via $\Phi$). Therefore it suffices to construct a
sequence of operators $(\gamma_{n})$ in the image of
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}$ in
$\text{Hom}_{W(k)}(\mathcal{A},W(A))$, such that $\gamma_{n}-\delta_{n}=0$ on
$\mathcal{A}/p^{n}$ and such that $(\gamma_{n})$ is convergent in
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}$. Then
$\gamma:=\lim\gamma_{n}$ will agree with $\delta$ and the result will follow.

We construct $\gamma_{n}$ by induction on $n$. We know that $\delta_{1}:A\to
W(A)/p$ is a derivation. So, we can write

$\delta_{1}=\sum_{i=1}^{n}\sum_{(I,r)}\bar{a}_{I,i}(\overline{p^{r}T^{I/p^{r}}})\cdot\partial_{i}$

where the second sum is over pairs $(I,r)$ where $I$ is a multi-index, each of
whose entries is $<p^{r}$, and at least one entry of which is coprime to $p$;
and $\bar{a}_{I,i}\in A$ (as usual we write $p^{r}T^{I/p^{r}}\in W(A)$ for
$V^{r}(T^{I})$). Choosing lifts of the $\bar{a}_{I,i}$ to
$a_{I,i}\in\mathcal{A}$, we obtain a derivation $\gamma_{1}:\mathcal{A}\to
W(A)$

$\gamma_{1}=\sum_{i=1}^{n}\sum_{(I,r)}a_{I,i}p^{r}T^{I/p^{r}}\partial_{i}$

so $\gamma_{1}-\delta_{1}=0$ on $\mathcal{A}/p$, and
$\gamma_{1}\in\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}$ by
construction.

Suppose we have constructed $(\gamma_{1},\gamma_{2},\dots,\gamma_{n})$. Then
the operator $\gamma_{n}-\delta:\mathcal{A}\to W(A)$ takes values in
$p^{n}W(A)$. Looking at the reduction mod $p^{n+1}$ of this operator, we
obtain a map

$\gamma_{n}-\delta:\mathcal{A}/p^{n+1}\to W(A)/p^{n+1}W(A)$

Since $\gamma_{n}\in\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}$ has
order $\leq n$ by induction, and $\delta_{n+1}$ is a differential operator of
order $\leq n+1$, we see that this map takes the form

$\gamma_{n}-\delta=\sum_{|J|\leq
n+1}\sum_{(I,r)}p^{n}\bar{a}_{I,J}(\overline{p^{r}T^{I/p^{r}}})\partial^{[J]}$

where the notation is as above. Now,

$\partial^{[J]}=\frac{\partial_{1}^{j_{1}}}{j_{1}!}\cdots\frac{\partial_{d}^{j_{d}}}{j_{d}!}$

with $j_{1}+\dots+j_{d}\leq n+1$. Recalling the formula

$\text{val}(j!)=\frac{j-s_{p}(j)}{p-1}$

where $s_{p}(j)$ is the sum of the digits in the $p$-adic expansion of $j$, we
see that

$\text{val}_{p}(j_{1}!\cdots
j_{d}!)=\sum_{i=1}^{d}\text{val}_{p}(j_{i}!)=\sum_{i=1}^{d}\frac{j_{i}-s_{p}(j_{i})}{p-1}\leq\frac{n}{p-1}$

where in the last inequality we used that $j_{1}+\dots+j_{d}\leq n+1$.
Therefore we have

$p^{n}\partial^{[J]}=\alpha\cdot p^{n-\text{val}_{p}(j_{1}!\cdots
j_{d}!)}\partial^{J}$

for some $\alpha\in\mathbb{Z}_{p}$. So we have

$\gamma_{n}-\delta=\sum_{|J|\leq n+1}\sum_{I}p^{n-\text{val}_{p}(j_{1}!\cdots
j_{d}!)}\alpha a_{I,J}(p^{r}T^{I/p^{r}})\partial^{J}$

and we can thus define $\gamma_{n+1}$ as

$\gamma_{n}-\sum_{|J|\leq n+1}\sum_{I}p^{n-\text{val}_{p}(j_{1}!\cdots
j_{d}!)}\alpha a_{I,J}(p^{r}T^{I/p^{r}})\partial^{J}$

which makes sense as a map $\mathcal{A}\to W(A)$ (and is clearly an element of
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}$).

Further, since $p>2$, we see that

$n-\text{val}_{p}(j_{1}!\cdots j_{d}!)\to\infty$

as $n\to\infty$. Thus we see that, in this case, the sequence $(\gamma_{n})$
is convergent inside $\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}$ as
claimed.

2) The proof proceeds identically to that of part $1)$, until we need to
evaluate $n-\text{val}_{p}(j_{1}!\cdots j_{d}!)$. At that point, we simply
have the estimate

$\text{val}_{p}(j_{1}!\cdots j_{d}!)\leq n$

(indeed, they can be equal in characteristic $2$). Thus the sequence of
operators $(\gamma_{n})$ may not converge in
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}$, but only in
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A},\text{crys}}^{(0)}$. Thus we
obtain the isomorphism
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A},\text{crys}}^{(0)}\tilde{\to}\Psi^{*}\widehat{\mathcal{D}}_{\mathcal{A},\text{crys}}^{(0)}$
as desired.

3) We again repeat the proof, but note that we can now use operators of the
form

$\sum_{|J|\leq n+1}\sum_{I}p^{n-\text{val}_{p}(j_{1}!\cdots j_{d}!)}\alpha
a_{I,J}(p^{r}T^{I/p^{r}})(\frac{\partial^{p^{m}}}{p^{m}!})^{J}$

and when $m>1$ obtain the required convergence even when $p=2$. ∎

Later on, we shall use this result to explain the relationship between
$\mathcal{D}_{\mathfrak{X}_{r}}^{(m)}$-modules on a smooth scheme
$\mathfrak{X}_{r}$ over $W(k)/p^{r}$ and modules over
$\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r}$. To that end, we shall record the
following result, which follows directly from the construction:

######  Corollary 3.7.

Let $m\geq 0$, and let $p>2$ if $m=0$. Let $\Phi$ and $\Psi$ be two morphisms
$\mathcal{A}\to W(A)$, coming from coordinatized lifts of Frobenius on
$\mathcal{A}$. Suppose that these maps agree after reduction mod $p^{r}$. Then
the reduction mod $p^{r}$ of the canonical isomorphism
$\epsilon_{\Phi,\Psi}:\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}/p^{r}\tilde{\to}\Psi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}/p^{r}$
constructed above agrees with the obvious identification
$\Phi^{*}(\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}/p^{r})\tilde{\to}\Psi^{*}(\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}/p^{r})$
coming from the fact that $\Phi=\Psi$ on $\mathcal{A}/p^{n}$.

We should also remark here on the relation between these constructions and
some facts from the usual crystalline cohomology theory. Recall that the
algebra $\mathcal{D}_{\mathcal{A}_{r},\text{crys}}^{(0)}$ is a major player in
that story, under the name of the HPD differential operators. They are
constructed as the dual of the divided power envelope of the diagonal of
$\mathfrak{X}_{r}$ (c.f. [7], chapter 4 for details).

We consider the two maps

$\Phi,\Psi:\mathcal{A}_{r}\to W(A)/p^{r}$

Via the standard divided power structure on $W(A)$, both of these maps may be
considered as (inverse limits of) divided power thickenings. Therefore the
theory of the crystalline site (or, in this case, the description of
$\mathcal{D}_{\mathcal{A}_{r},\text{crys}}^{(0)}$ as the HPD differential
operators) yields a canonical isomorphism

$\eta_{r}:\Phi^{*}\mathcal{D}_{\mathcal{A}_{r},\text{crys}}^{(0)}\tilde{\to}\Psi^{*}\mathcal{D}_{\mathcal{A}_{r},\text{crys}}^{(0)}$

Taking the inverse limit over $r$yields an isomorphism

$\eta:\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A},\text{crys}}^{(0)}\tilde{\to}\Psi^{*}\widehat{\mathcal{D}}_{\mathcal{A},\text{crys}}^{(0)}$

and we have

######  Corollary 3.8.

The isomorphism $\eta$ agrees with the isomorphism $\epsilon$ constructed
above in Theorem 3.6.

###### Proof.

This will follow if we know that the maps
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A},\text{crys}}^{(0)}\to\text{Hom}_{W(k)}(\mathcal{A},W(A))$
and
$\Psi^{*}\widehat{\mathcal{D}}_{\mathcal{A},\text{crys}}^{(0)}\to\text{Hom}_{W(k)}(\mathcal{A},W(A))$
coming from the action of differential operators on $\mathcal{A}$ are
compatible with $\eta$. But the analogous fact is true for $\eta_{r}$, simply
from the construction of $\eta_{r}$ which is given by factoring
$\Phi\times\Psi$ through a divided power neighborhood of the diagonal (c.f.
[7], chapters 3 and 4). Taking the inverse limit over $r$ we see that this is
true for $\eta$ as well. ∎

To finish off this section, we shall develop another fundamental property of
the bimodules $\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$, namely

######  Theorem 3.9.

1) As a right $\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$-module,
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$ is faithfully flat.

2) Let $m\geq 0$. As a left $\widehat{\mathcal{D}}_{W(A)}$-module,
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$ is projective and cyclic;
in particular, it is a summand of $\widehat{\mathcal{D}}_{W(A)}^{(m)}$ itself.

The analogous statements hold for
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A},\text{crys}}^{(m)}$ as a
$(\widehat{\mathcal{D}}_{W(A),\text{crys}}^{(m)},\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A},\text{crys}}^{(m)})$
bimodule.

This result will allow us, in the next section, to define categories of
$\widehat{\mathcal{D}}_{W(X)}^{(m)}$-modules which are surprisingly well
behaved.

###### Proof.

(of part $1)$ As $\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$ is
$p$-torsion free and $p$-adically complete, by [28], corollary 1.6.7 it
suffices to show that $\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}/p$ is
faithfully flat over $\mathcal{D}_{A}^{(m)}$. However,
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}/p$ is an inverse limit of
free modules over $\mathcal{D}_{A}^{(m)}$, namely modules of the form

$\bigoplus_{I<p^{r}}V^{r}(T^{I})\cdot\mathcal{D}_{A}^{(m)}$

where $I=(i_{1},\dots,i_{n})$ is a multi-index and $I<p^{r}$ means that each
$i_{j}<p^{r}$. As the maps in this inverse system are surjective, one deduces
immediately that $\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}/p$ is
faithfully flat over $\mathcal{D}_{A}^{(m)}$ as needed. An identical proof
works for the crystalline version of the theory. ∎

To prove part $2)$, we will have to work a bit harder. We start with the
special case in which $A=k[T]$ ; the lift of this algebra is given by
$W(k)<<T>>$ and we shall use the standard Frobenius lift $T\to T^{p}$. In this
context we write ${\displaystyle d=\frac{d}{dt}}$. For
$i\in\\{0,\dots,p^{r}-1\\}$, we write $\pi_{(i,p^{r})}:k[T]\to k[T]$ for the
$k$-linear projection operator which takes $T^{i+ap^{r}}$ to itself (for all
$a$) and which takes all other monomials to zero. Then

######  Lemma 3.10.

The operator $\pi_{(i,p^{r})}$ is a differential operator of order $<p^{r}$,
namely

$\pi_{(i,p^{r})}=T^{i}d^{[p^{r}-1]}T^{p^{r}-1-i}=\sum_{l=i}^{p^{r}-1}{p^{r}-1-i\choose
p^{r}-1-l}T^{l}\cdot d^{[l]}$

###### Proof.

We begin with the case $i=0$. Then, for any $a\geq 0$ we have

$d^{[p^{r}-1]}T^{p^{r}-1}(T^{p^{r}a})={p^{r}a+p^{r}-1\choose
p^{r}-1}T^{p^{r}a}$

But

${p^{r}a+p^{r}-1\choose
p^{r}-1}=\prod_{m=1}^{p^{r}-1}\frac{p^{r}a+m}{m}=\prod_{m=1}^{p^{r}-1}(1+a\frac{p^{r}}{m})=1$

since $\text{val}_{p}(m)<r$ for all terms in the product; so we see
$d^{[p^{r}-1]}T^{p^{r}-1}(T^{p^{r}a})=T^{p^{r}a}$. On the other hand, for any
natural number of the form $p^{r}a+b$ where $1\leq b\leq p^{r}-1$, we have

$d^{[p^{r}-1]}T^{p^{r}-1}(T^{p^{r}a+b})=T^{p^{r}(a+1)}\cdot
d^{[p^{r}-1]}(T^{b-1})=0$

where we have used the well known identity (over $k$)
$[d^{[p^{r}-1]},T^{p^{r}(a+1)}]=0$. This proves the result when $i=0$.

Now we consider the general case. Working for a moment over the ring
$k[T,T^{-1}]$, it follows from the case $i=0$ that the operator
$\pi_{(i,p^{r})}$ has the form
${\displaystyle T^{i-p^{r}}(d^{(p^{r}-1)}T^{p^{r}-1})T^{p^{r}-i}}$. Now write
$2p^{r}-1-i=p^{r}+b$ for $b\in\\{0,\dots,p^{r}-2\\}$. Then

$T^{i-p^{r}}(d^{[p^{r}-1]}T^{p^{r}-1})T^{p^{r}-i}=T^{i-p^{r}}d^{[p^{r}-1]}T^{p^{r}+b}=T^{i}d^{[p^{r}-1]}T^{b}$

is the operator of the required form, where in the last equality we have used
$[d^{[p^{r}-1]},T^{p^{r}}]=0$.

Finally to prove the equality

$T^{i}d^{[p^{r}-1]}T^{p^{r}-1-i}=\sum_{l=i}^{p^{r}-1}{p^{r}-1-i\choose
p^{r}-1-l}T^{l}\cdot d^{[l]}$

we note that by the Hasse-Schmidt property we have

$[d^{[p^{r}-1]},T^{p^{r}-1-i}]=\sum_{l=0}^{p^{r}-2}d^{[p^{r}-1-l]}(T^{p^{r}-1-i})\cdot
d^{[l]}$ $=\sum_{l=0}^{p^{r}-2}{p^{r}-1-i\choose p^{r}-1-l}T^{l-i}\cdot
d^{[l]}$

from which the equality follows directly. ∎

Our goal is to study the actions of these operators when lifted to
$W_{r+1}(A)$; we start with the

######  Definition 3.11.

Let $f:\mathbb{Z}_{\geq 0}\to\mathbb{Z}/p^{r+1}\mathbb{Z}$ be a function, and
let $j>0$ be some natural number. We say that $f$ is polynomial in arithmetic
progressions (of length $j$) if, for each $1\leq a\leq j$, there is a
polynomial $p_{a}\in(\mathbb{Z}/p^{r+1}\mathbb{Z})[x]$ such that
$f(a+bj)=p_{a}(b)$ for all $b\in\mathbb{Z}_{\geq 0}$.

We note that if $f$ is polynomial in arithmetic progressions (of length $j$),
then $f$ is also polynomial in arithmetic progressions of length $j^{\prime}$
for all $j^{\prime}$ which are divisible by $j$.

Then the result we need is

######  Proposition 3.12.

Let $1\leq i<p^{n}$. Fix $r\geq 0$. Then each operator of the form
$T^{i}d^{[i]}$ on $W_{r+1}(k)[T]$ takes the form

$T^{i}d^{[i]}(T^{a})=f(a)T^{a}$

where $f:\mathbb{Z}_{\geq 0}\to\mathbb{Z}/p^{r+1}\mathbb{Z}$ is polynomial in
arithmetic progressions of length $p^{n-1}$.

###### Proof.

We begin with the case of the operator $T^{p^{m}}d^{[p^{m}]}$ for $m<n$. Fix
$a\in\\{0,\dots,p^{m}-1\\}$. In this case, we have

$T^{p^{m}}d^{[p^{m}]}(T^{a+bp^{m}})={a+bp^{m}\choose p^{m}}T^{a+bp^{m}}$

We shall show that ${\displaystyle{a+bp^{m}\choose p^{m}}}$ is polynomial in
$b$ with coefficients in $\mathbb{Z}_{p}$, and therefore $f$ is arithmetic
progressions of length $p^{m}$ (as remarked above, this implies that it is
also polynomial in arithmetic progressions of length $p^{n-1}$). We have

${a+bp^{m}\choose p^{m}}=\prod_{l=1}^{p^{m}}\frac{a+(b-1)p^{m}+l}{l}$

For each $l\in\\{1,\dots,p^{m}\\}$ set

$a+l=t+\epsilon_{l}p^{m}$

where $t\in\\{1,\dots,p^{m}\\}$ and $\epsilon_{t}\in\\{0,1\\}$. Note that, as
$l$ ranges over $\\{1,\dots,p^{m}\\}$, $t$ also ranges over
$\\{1,\dots,p^{m}\\}$. Therefore we have

$\prod_{l=1}^{p^{m}}\frac{a+(b-1)p^{m}+l}{l}=\prod_{t=1}^{p^{m}}\frac{(b-1)p^{m}+t+\epsilon_{l}p^{m}}{t}$
$=\prod_{t=1}^{p^{m}}\frac{(b-1+\epsilon_{t})p^{m}+t}{t}=b\prod_{t=1}^{p^{m}-1}\frac{(b-1+\epsilon_{t})p^{m}+t}{t}$

Now, in the last expression, we have
$\text{val}_{p}(t)=\text{val}_{p}((b-1+\epsilon_{t})p^{m}+t)$. It follows that
this product is a polynomial in $b$ with coefficients in $\mathbb{Z}_{p}$ as
desired.

In order to handle the general case, we recall that, if $i<p^{n}$ has $p$-adic
expansion ${\displaystyle i=\sum_{j=0}^{m}a_{j}p^{j}}$, then we have

$d^{[i]}=u\cdot(d^{[p^{j}]})^{a_{j}}\cdots(d)^{a_{0}}$

where $u$ is a unit in $\mathbb{Z}_{p}$. From this, and the standard relation

$d^{[m]}T^{m}\cdot d^{[n]}T^{n}=$ $\sum_{l=0}^{m}{n\choose m-l}{l+n\choose
l}T^{(l+n)}d^{[l+n]}$

we see that $T^{i}d^{[i]}$ is itself a polynomial (with coefficients in
$\mathbb{Z}_{p}$) is the operators
$\\{T^{p^{m}}d^{[p^{m}]}\\}_{m=0}^{p^{n-1}}$, and the result follows. ∎

######  Corollary 3.13.

Let $\Xi:k[T]\to k[T[$ be an operator which satisfies $\Xi(T^{a})=f(a)T^{a}$,
where $f:\mathbb{Z}_{\geq 0}\to\mathbb{Z}/p\mathbb{Z}$ is polynomial in
arithmetic progressions of length $p^{n-1}$. Then $\Xi$ is given by a
differential operator of the form

$\Xi=\sum_{i=0}^{p^{n}-1}g_{i}(T^{i}d^{[i]})$

where $g_{i}$ are polynomials in $\mathbb{Z}/p\mathbb{Z}[T]$.

###### Proof.

By (the proof of) the previous proposition, we have that

$T^{p^{m}}d^{[p^{m}]}T^{(a+bp^{m})}=b\cdot T^{(a+bp^{m})}$

for each $0\leq a<p^{m}-1$. Thus, if $g$ is any polynomial in
$\mathbb{Z}/p\mathbb{Z}$, the operator $g(T^{p^{m}}d^{[p^{m}]})$ takes
$T^{(a+bp^{m})}$ to $g(b)T^{a+bp^{m}}$. Furthermore, the projector
$\pi_{(a,p^{m})}$ is an operator of the form
${\displaystyle\sum_{i=0}^{p^{m}-1}c_{i}T^{i}d^{[i]}}$ for
$c_{i}\in\mathbb{Z}$ (by Lemma 3.10). So the composition
$g(T^{p^{m}}d^{[p^{m}]})\cdot\pi_{a/p^{m}}$ acts with eigenvalue $g(b)$ on
each element of the form $T^{a+bp^{m}}$ and is zero on all other monomials; as
above we see that this product is a polynomial in the $\\{T^{i}d^{[i]}\\}$.
Adding up such operators over $a\in\\{0,\dots,p^{m}-1\\}$ yields the result. ∎

Now, for any $r\geq 0$, consider for any $i\in\\{0,\dots,p^{r}-1\\}$ the
projection operator $\pi_{i/p^{r}}$ on $W_{r+1}(A)$ which takes
$p^{m}T^{a/p^{m}}$ to itself whenever $p^{r-m}a\equiv
i\phantom{i}\text{mod}\phantom{i}p^{r}$ and $0$ otherwise. Then we have

######  Proposition 3.14.

Let $r\geq 0$. For each $i\in\\{1,\dots,p^{r}-1\\}$ the projector
$\pi_{i/p^{r}}$ (on $W_{r+1}(A)$) is given by the action of an element in
$\mathcal{D}_{W(A)}^{(0)}$, of the form

$\psi_{i/p^{r}}=\sum_{i=1}^{p^{r}-1}g_{i}(T^{i/p^{r}}\\{d\\}_{i/p^{r}})$

where $g_{i}\in\mathbb{Z}_{p}[x]$. We have that $\psi_{i/p^{r}}\in
V^{r-\text{val}_{p}(i)}(\mathcal{D}_{W(A)})$. Therefore, the element

$\pi_{r}=1-\sum_{i=1}^{p^{r}-1}\psi_{i/p^{r}}$

acts as a projector from $W_{r+1}(A)$ to $\mathcal{A}_{r}$. Further, one has
$\pi_{r}\equiv\pi_{r-1}\phantom{i}\text{mod}\phantom{i}V^{r}(\mathcal{\widehat{D}}_{W(A)}^{(0)})$.
Therefore the sequence $\\{\pi_{r}\\}$, approaches a limit,
$\pi\in\mathcal{\widehat{D}}_{W(A)}^{(0)}$, which acts as a projector from
$W(A)$ to $\mathcal{A}$.

###### Proof.

We proceed by induction on $r$, the case $r=0$ being trivial. Suppose the
theorem holds at level $r-1$. Consider any $i\in\\{1,\dots,p^{r}-1\\}$ with
$\text{val}(i)=0$. We will work now with the copy of $W_{r+1}(A^{(r)})$
contained in $\mathcal{A}_{r+1}=W_{r+1}(k)[T]$ and construct the projector
there.

The projector $\pi_{(i,p^{r})}$ on $k[T]$ is given by an expression of the
form ${\displaystyle\sum_{j=0}^{p^{r}-1}c_{j}T^{j}d^{[j]}}$ (for
$c_{j}\in\mathbb{Z}$); where, since $\text{val}(i)=0$, we have $c_{j}=0$
whenever $\text{val}(j)\neq 0$ (to see this, note that the restriction of the
operator to the subalgebra $k[T^{p}]$ is $0$). Now, consider the action of the
term ${\displaystyle\sum_{j=0}^{p^{r}-1}c_{j}T^{j}d^{[j]}}$ on
$W_{r+1}(A^{(r)})\subset\mathcal{A}_{r+1}$. As $\text{val}(j)=0$ for all
nonzero $c_{j}$, this operator takes every term of the form
$p^{m}T^{ap^{r-m}}$ to a term of the form $f_{m}(a)\cdot p^{r}T^{ap^{r-m}}$
for some $f_{m}(a)\in\mathbb{Z}/p\mathbb{Z}$. To analyze the functions
$f_{m}$, we shall apply 3.12. In particular, we see that $a\to f_{m}(a)$ is
polynomial on arithmetic progressions of length $p^{m}$, and is equal to zero
on all terms of the form $a=pa^{\prime}$ (when $m>0$).

Applying the previous result, we see that there is an operator of the form

$\sum_{i=0}^{p^{r+1}-1}g_{i,.m}(p^{r-\text{val}(i)}T^{i}d^{[i]})$

where $g_{i,m}\in\mathbb{Z}_{p}[X]$, which takes $p^{m}T^{ap^{r-m}}\to
f_{m}(a)\cdot p^{r}T^{ap^{r-m}}$ for all $a$. By construction this operator is
contained in $V_{r}(\mathcal{D}_{W(A)}^{(0)})$, and we have

$\psi_{i/p^{r}}=\sum_{j=0}^{p^{r}-1}c_{j}T^{j}d^{[j]}-\sum_{m=0}^{r-1}\sum_{i=0}^{p^{r+1}-1}g_{i}(p^{r-\text{val}(i)}T^{i}d^{[i]})$

Now suppose $\text{val}(i)>0$. Write $i=pi^{\prime}$. By induction we have
already constructed the operator $\psi_{i^{\prime}/p^{r-1}}$ on $W_{r}(A)$,
and it is an operator of the form
${\displaystyle\sum_{i=0}^{p^{r}-1}h_{i}(T^{i/p^{r-1}}\\{d\\}_{i/p^{r-1}})}$
for polynomials $h_{i}\in\mathbb{Z}_{p}[X]$. Translating to
$W_{r}(A^{(r-1)})\subset W_{r}(k)[T]$ it is an operator of the form
${\displaystyle{\displaystyle\sum_{i=0}^{p^{r}-1}h_{i}(T^{i}d^{[i]})}}$.
Lifting to $W_{r+1}(A^{(r))})\subset W_{r+1}(k)[T]$ yields the operator
${\displaystyle\sum_{i=0}^{p^{r}-1}h_{i}(T^{pi}d^{[pi]})}$. By construction,
the operator

$\psi_{i/p^{r}}-\sum_{i=0}^{p^{r}-1}h_{i}(T^{pi}d^{[pi]})$

takes each term of the form $p^{m}T^{ap^{r-m}}$ to a term of the form
$f_{m}(a)\cdot p^{r}T^{ap^{r-m}}$ for some $f_{m}(a)\in\mathbb{Z}/p\mathbb{Z}$
which is polynomial on arithmetic progressions of length $p^{m}$ (again by
3.12). Arguing as above, we find a term in $V_{r}(\mathcal{D}_{W(A)}^{(0)})$
which acts as
${\displaystyle\psi_{i/p^{r}}-\sum_{i=0}^{p^{r}-1}h_{i}(T^{pi}d^{[pi]})}$ on
$W_{r+1}(A^{(r)})$, and thus construct $\psi_{i/p^{r}}$ as required. ∎

Now we can give the

######  Corollary 3.15.

Let $A$ be any smooth affine algebra which is equipped with local coordinates,
and let $\Phi:\mathcal{A}\to W(A)$ be the map coming from a coordinatized
Frobenius lift. Then there is an element
$\pi\in\mathcal{\widehat{D}}_{W(A)}^{(m)}$ which acts as a projector onto
$\Phi(\mathcal{A})$, i.e., $\pi^{2}=\pi$ and $\pi(W(A))=\Phi(\mathcal{A})$.

Further, the map
$\mathcal{\widehat{D}}_{W(A)}^{(m)}\to\Phi^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$
which takes $1\to\Phi$ induces an isomorphism
$\mathcal{\widehat{D}}_{W(A)}^{(m)}\cdot\pi\tilde{\to}\Phi^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$;
therefore, $\widehat{\Phi}^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$ is a
summand of $\mathcal{D}_{W(A)}^{(m)}$. The analogous facts are true for
$\mathcal{\widehat{D}}_{W(A),\text{crys}}^{(0)}$ and
$\Phi^{*}\mathcal{\widehat{D}}_{\mathcal{A},\text{crys}}^{(0)}$.

Note that this immediately proves part $2)$ of Theorem 3.9

###### Proof.

If $A=k[T_{1},\dots,T_{d}]$, then we have inclusions
$\mathcal{\widehat{D}}_{W(k[T_{i}])}^{(0)}\to\mathcal{\widehat{D}}_{W(A)}^{(0)}$
coming from sending $p^{r}T_{i}^{a/p^{r}}\to p^{r}T_{i}^{a/p^{r}}$ and
$\\{d\\}_{a/p^{r}}\to\\{\partial_{i}\\}_{a/p^{r}}$; and we can set
${\displaystyle\pi=\prod_{i=1}^{d}\pi_{i}}$ where
$\pi_{i}\in\mathcal{D}_{W(k[T_{i}])}^{(0)}$ is the element constructed
directly above. In this case we see that
$\pi:W(A)\to\mathcal{A}=W(k)<<T_{1},\dots,T_{d}>>$ is the standard coordinate
projection; i.e., the continuous map which takes $p^{r}T^{I/p^{r}}$to $0$
whenever $r\geq 1$ and $T^{I}\to T^{I}$ for $I\in(\mathbb{Z}^{\geq 0})^{d}$.

For the general case, the choice of local coordinates yields an etale map
$k[T_{1},\dots,T_{d}]\to A$. By the etale local nature of Witt-differential
operators (2.32), we see that $\pi$ extends uniquely to
$\mathcal{D}_{W(A)}^{(0)}$; this is the required element.

To obtain the last statement, note that the element
$\Phi\in\widehat{\Phi}^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}$ has
annihilator consisting of
$\\{\varphi\in\mathcal{\widehat{D}}_{W(A)}^{(0)}|\varphi\cdot\Phi(\mathcal{A})=0\\}$;
this is also the left annihilator of $\pi$; this yields the isomorphism
$\mathcal{D}_{W(A)}^{(0)}\cdot\pi\tilde{\to}\widehat{\Phi}^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}$.
Since $\pi$ is an idempotent the rest of the statement follows directly. We
deduce the statement for higher $m$ by looking at the image of $\pi$ in
$\mathcal{\widehat{D}}_{W(A)}^{(m)}$.

Similarly, to handle the crystalline version, we recall that there is an
injective map
$\mathcal{\widehat{D}}_{W(A)}^{(0)}\to\mathcal{\widehat{D}}_{W(A),\text{crys}}^{(0)}$,
and so the image of $\pi$ in $\mathcal{\widehat{D}}_{W(A),\text{crys}}^{(0)}$
does the job. ∎

###  3.1. Accessible modules

In this section we use the fundamental isomorphism of Theorem 3.6 to define
and study our categories of interest.

Let us begin in positive characteristic. Translating 3.7 to this situation
(c.f. 5.15 for a proof that works in all characteristics) yields

######  Proposition 3.16.

Let $m\geq 0$. There is a well-defined
$(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p,\mathcal{D}_{X}^{(m)})$ bimodule,
denoted $\mathcal{B}_{X}^{(m)}$, which is locally projective over
$\mathcal{\widehat{D}}_{W(X)}^{(m)}/p$, and locally faithfully flat as a right
$\mathcal{D}_{X}^{(0)}$-module. The associated functor

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

is exact and fully faithful, and admits an exact right adjoint
$\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}/p}(\mathcal{B}_{X}^{(m)},?)$.
We have

$\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}/p}(\mathcal{B}_{X}^{(m)},\mathcal{B}_{X}^{(m)}\otimes_{\mathcal{D}_{X}^{(m)}}\mathcal{M})\tilde{\to}\mathcal{M}$

for all $\mathcal{M}\in\mathcal{D}_{X}^{(m)}-\text{mod}$. Therefore, the
functor

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

is fully faithful and we have

$R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}/p}(\mathcal{B}_{X}^{(m)},\mathcal{B}_{X}^{(m)}\otimes_{\mathcal{D}_{X}^{(m)}}\mathcal{M}^{\cdot})\tilde{\to}\mathcal{M}^{\cdot}$

for all $\mathcal{M}^{\cdot}\in D(\mathcal{D}_{X}^{(m)}-\text{mod})$.

###### Proof.

The existence of the bimodule follows immediately from Theorem 3.6 and 3.7 (or
5.15). By Theorem 3.9, we see that it is locally faithfully flat as a right
$\mathcal{D}_{X}^{(m)}$-module and locally projective over
$\mathcal{\widehat{D}}_{W(X)}^{(m)}/p$. The rest follows directly once we
prove that the adjunction map

(3.2)
$\mathcal{M}\to\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}/p}(\mathcal{B}_{X}^{(m)},\mathcal{B}_{X}^{(m)}\otimes_{\mathcal{D}_{X}^{(m)}}\mathcal{M})$

is an isomorphism for all $\mathcal{M}\in\mathcal{D}_{X}^{(m)}-\text{mod}$. To
see this, we may work locally and assume $X=\text{Spec}(A)$. Let
$\Phi:\mathcal{A}\to W(A)$ be the morphism coming from a coordinatized lift of
Frobenius on $\mathcal{A}$, and $\pi:W(A)\to\mathcal{A}$ the associated
projector. Now note that, as a right $\mathcal{D}_{X}^{(m)}$-module, we have

$\mathcal{B}_{X}^{(m)}=(1-\pi)\mathcal{B}_{X}^{(m)}\oplus\pi\mathcal{B}_{X}^{(m)}=(1-\pi)\mathcal{B}_{X}^{(m)}\oplus\mathcal{D}_{X}^{(m)}$

so that

$\mathcal{B}_{X}^{(m)}\otimes_{\mathcal{D}_{X}^{(m)}}\mathcal{M}=((1-\pi)\mathcal{B}_{X}^{(m)}\otimes_{\mathcal{D}_{X}^{(m)}}\mathcal{M})\oplus\mathcal{M}$

As the functor
$\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}/p}(\mathcal{B}_{X}^{(m)},\mathcal{N})$
agrees with $\\{n\in\mathcal{N}|(1-\pi)n=0\\}$, we see that (3.2) is an
isomorphism as required. ∎

This yields the

######  Definition 3.17.

1) A module $\mathcal{N}\in\mathcal{\widehat{D}}_{W(X)}^{(m)}/p-\text{mod}$ is
accessible if is of the form
$\mathcal{B}_{X}^{(m)}\otimes_{\mathcal{D}_{X}^{(m)}}\mathcal{M}$ for some
$\mathcal{M}\in\mathcal{D}_{X}^{(m)}-\text{mod}$.

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

3) Let $r\geq 1$. A complex $\mathcal{N}^{\cdot}\in
D(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod})$ is accessible if
$\mathcal{N}^{\cdot}\otimes_{W_{r}(k)}^{L}k$ is accessible in
$D(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p-\text{mod})$. Similarly, a complex
$\mathcal{N}^{\cdot}\in D_{cc}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$
is accessible if $\mathcal{N}^{\cdot}\otimes_{W(k)}^{L}k$ is accessible in
$D(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p-\text{mod})$.

As the full subcategory of accessible complexes in
$D(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p-\text{mod})$ is a thick triangulated
subcategory, the same is true for the accessible complexes in
$D(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{n}-\text{mod})$ and
$D(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$. We denote these categories
by
$D_{\text{acc}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{n}-\text{mod})$ and
$D_{\text{acc}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$, respectively.

In the presence of a lift of Frobenius, we have the following characterization
of accessibility. Before giving it, let us set some notation: Let $A$ be
smooth affine which possesses local coordinates and let $\Phi:\mathcal{A}\to
W(A)$ be the morphism coming from a coordinatized lift of Frobenius. For
$\mathcal{M}^{\cdot}\in D(\mathcal{D}_{X_{n}}^{(m)}-\text{mod})$, let
$\Phi^{*}\mathcal{M}^{\cdot}:=\Phi^{*}\mathcal{D}_{X_{n}}^{(m)}\otimes_{\mathcal{D}_{X_{n}}^{(m)}}^{L}\mathcal{M}^{\cdot}$.
For $\mathcal{M}^{\cdot}\in
D_{cc}(\mathcal{D}_{\mathfrak{X}}^{(m)}-\text{mod})$, let
$\Phi^{*}\mathcal{M}^{\cdot}:=\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{D}_{\mathfrak{X}}^{(m)}}^{L}\mathcal{M}^{\cdot}$,
the cohomological completion of
$\Phi^{*}\mathcal{M}^{\cdot}:=\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}\otimes_{\mathcal{D}_{\mathfrak{X}}^{(m)}}^{L}\mathcal{M}^{\cdot}$.
Then we have

######  Theorem 3.18.

Let $A$, $\mathcal{A}$ and $\Phi:\mathcal{A}\to W(A)$ be the morphism coming
from a coordinatized lift of Frobenius. Let $X=\text{Spec}(A)$. Then
$\mathcal{N}^{\cdot}\in
D(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod})$ is accessible iff
$\mathcal{N}^{\cdot}\tilde{=}\Phi^{*}\mathcal{M}^{\cdot}$ for some
$\mathcal{M}^{\cdot}\in D(\mathcal{D}_{\mathfrak{X}_{r}}^{(m)}-\text{mod})$
(where $\mathfrak{X}_{r}=\text{Spec}(\mathcal{A}/p^{r})$).

Similarly, $\mathcal{N}^{\cdot}\in
D_{cc}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$ is accessible iff
$\mathcal{N}^{\cdot}\tilde{=}\Phi^{*}\mathcal{M}^{\cdot}$ for some
$\mathcal{M}^{\cdot}\in D_{cc}(\mathcal{D}_{\mathfrak{X}}^{(m)}-\text{mod})$.

###### Proof.

Let $\mathcal{N}^{\cdot}\in
D(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p-\text{mod})$. It follows from 3.16 that
$\mathcal{N}^{\cdot}$ is accessible iff the adjunction

$\mathcal{N}^{\cdot}\to\mathcal{B}_{W(X)}^{(m)}\otimes_{\mathcal{D}_{X}^{(m)}}^{L}R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}/p}(\mathcal{B}_{W(X)}^{(m)},\mathcal{N}^{\cdot})$

is an isomorphism.

Now let $\mathcal{N}^{\cdot}\in
D(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod})$. Suppose
$\mathcal{N}^{\cdot}$ is accessible. We have the bimodule
$\mathcal{B}_{\mathfrak{X}_{r}}^{(m)}=\Phi^{*}\mathcal{D}_{\mathfrak{X}_{r}}^{(m)}$,
which, by Theorem 3.9, is locally projective as a left
$\widehat{\mathcal{D}}_{W(X)}^{(m)}/p^{n}$-module and is locally faithfully
flat as a right $\mathcal{D}_{\mathfrak{X}_{r}}^{(m)}$-module. Therefore

$R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}}(\Phi^{*}\mathcal{D}_{X_{n}}^{(m)},\mathcal{N}^{\cdot})\otimes_{W_{r}(k)}^{L}k\tilde{\to}R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}/p}(\mathcal{B}_{W(X)}^{(m)},\mathcal{N}^{\cdot}\otimes_{W_{r}(k)}^{L}k)$

Therefore,

$(\Phi^{*}\mathcal{D}_{\mathfrak{X}_{r}}^{(m)}\otimes_{\mathcal{D}_{\mathfrak{X}_{r}}^{(m)}}^{L}R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}}(\Phi^{*}\mathcal{D}_{\mathfrak{X}_{r}}^{(m)},\mathcal{N}^{\cdot}))\otimes_{W_{r}(k)}^{L}k$
$\tilde{\to}\mathcal{B}_{X}^{(m)}\otimes_{\mathcal{D}_{X}^{(m)}}^{L}R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}/p}(\mathcal{B}_{W(X)}^{(m)},\mathcal{N}^{\cdot}\otimes_{W_{r}(k)}^{L}k)$

so that the adjunction map

$\mathcal{N}^{\cdot}\to\Phi^{*}\mathcal{D}_{\mathfrak{X}_{r}}^{(m)}\otimes_{\mathcal{D}_{\mathfrak{X}_{r}}^{(m)}}^{L}R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}}(\Phi^{*}\mathcal{D}_{\mathfrak{X}_{r}}^{(m)},\mathcal{N}^{\cdot})$

becomes an isomorphism after applying $\otimes_{W_{r}(k)}^{L}k$, and therefore
is already an isomorphism by Nakayama. Thus we have
$\mathcal{N}^{\cdot}\tilde{=}\Phi^{*}\mathcal{M}^{\cdot}$ for some
$\mathcal{M}^{\cdot}\in\text{Qcoh}(\mathcal{D}_{\mathfrak{X}_{r}}^{(m)})$; the
converse direction is clear.

The statement for $\mathcal{N}^{\cdot}\in
D_{cc}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$ is essentially
identical, using the Nakayama lemma for cohomologically complete complexes and
the fact that

$R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}(\Phi^{*}\mathcal{D}_{\mathfrak{X}_{n}}^{(m)},\mathcal{N}^{\cdot})$

is cohomologically complete if $\mathcal{N}^{\cdot}$ is (since this sheaf is a
summand of $\mathcal{N}^{\cdot}$). ∎

From this, we obtain

######  Corollary 3.19.

Let
$D_{\text{acc},\text{qcoh}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod})$
denote the full subcategory of
$D_{\text{acc}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod})$
consisting of complexes $\mathcal{N}^{\cdot}$ such that

$\mathcal{N}^{\cdot}\otimes_{W_{r}(k)}^{L}k\tilde{\to}\mathcal{B}_{X}^{(m)}\otimes_{\mathcal{D}_{X}^{(m)}}^{L}\mathcal{M}^{\cdot}$

where
$\mathcal{H}^{i}(\mathcal{M}^{\cdot})\in\text{Qcoh}(\mathcal{D}_{X}^{(m)})$
for all $i$. Then, if $X=\text{Spec}(A)$ as above, we have
$\mathcal{N}^{\cdot}\in
D_{\text{acc},\text{qcoh}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod})$
iff $\mathcal{N}^{\cdot}\tilde{=}\Phi^{*}\mathcal{M}^{\cdot}$ for some
$\mathcal{M}^{\cdot}\in
D_{\text{qcoh}}(\mathcal{D}_{\mathfrak{X}_{r}}^{(m)}-\text{mod})$.

###### Proof.

By the previous theorem, it suffices to show the following: let
$\mathcal{M}^{\cdot}\in D(\mathcal{O}_{\mathfrak{X}_{r}}-\text{mod})$. Then
$\mathcal{M}^{\cdot}\in D(\text{QCoh}(\mathfrak{X}_{r}))$ iff
$\mathcal{M}^{\cdot}\otimes_{W_{r}(k)}^{L}k\in
D(\text{QCoh}(\mathfrak{X}_{r}))$. The forward direction is obvious. For the
reverse, consider the short exact sequence

$0\to W_{r-1}(k)\to W_{r}(k)\to k\to 0$

where on the left we identify $pW_{n}(k)\tilde{=}W_{n-1}(k)$. This yields a
distinguished triangle

$\mathcal{M}^{\cdot}\otimes_{W_{r}(k)}^{L}W_{r-1}(k)\to\mathcal{M}^{\cdot}\to\mathcal{M}^{\cdot}\otimes_{W_{r}(k)}^{L}k$

As
$(\mathcal{M}^{\cdot}\otimes_{W_{r}(k)}^{L}W_{r-1}(k))\otimes_{W_{r-1}(k)}^{L}k\tilde{\to}\mathcal{M}^{\cdot}\otimes_{W_{r}(k)}^{L}k$,
by induction we may assume
$\mathcal{M}^{\cdot}\otimes_{W_{r}(k)}^{L}W_{r-1}(k)\in
D(\text{Qcoh}(\mathfrak{X}_{r-1}))$; hence both edges of the triangle can be
regarded as members of $D(\text{Qcoh}(\mathfrak{X}_{r}))$, therefore
$\mathcal{M}^{\cdot}\in D(\text{Qcoh}(\mathfrak{X}_{r}))$ as well. ∎

We shall also find it useful to consider the category
$D_{\text{acc},\text{qcoh}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$
consisting of complexes $\mathcal{N}^{\cdot}$ such that

$\mathcal{N}^{\cdot}\otimes_{W_{r}(k)}^{L}k\tilde{\to}\mathcal{B}_{X}^{(m)}\otimes_{\mathcal{D}_{X}^{(m)}}^{L}\mathcal{M}^{\cdot}$

where
$\mathcal{H}^{i}(\mathcal{M}^{\cdot})\in\text{Qcoh}(\mathcal{D}_{X}^{(m)})$
for all $i$. I don’t know a local characterization of it in terms of the
functor $\Phi^{*}$ as above. However, we do have:

######  Corollary 3.20.

Let
$D_{\text{acc},\text{coh}}^{b}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$
denote the full subcategory of
$D_{\text{acc}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$ consisting of
bounded complexes $\mathcal{N}^{\cdot}$ such that

$\mathcal{N}^{\cdot}\otimes_{W(k)}^{L}k\tilde{\to}\mathcal{B}_{X}^{(m)}\otimes_{\mathcal{D}_{X}^{(m)}}^{L}\mathcal{M}^{\cdot}$

where
$\mathcal{H}^{i}(\mathcal{M}^{\cdot})\in\text{Coh}(\mathcal{D}_{X}^{(m)})$ for
all $i$. Then, if $X=\text{Spec}(A)$ as above, we have $\mathcal{N}^{\cdot}\in
D_{\text{acc},\text{coh}}^{b}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$
iff $\mathcal{N}^{\cdot}\tilde{=}\Phi^{*}\mathcal{P}^{\cdot}$ for some
$\mathcal{P}^{\cdot}\in
D_{\text{coh}}^{b}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}-\text{mod})$.
Furthermore, a bounded complex $\mathcal{N}^{\cdot}$ is contained in
$D_{\text{acc},\text{coh}}^{b}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$
iff each $\mathcal{H}^{i}(\mathcal{N}^{\cdot})$, considered as a complex in
degree $0$, is contained in
$D_{\text{acc},\text{coh}}^{b}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$.
In particular, the elements of
$D_{\text{acc},\text{coh}}^{b}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$
in homological degree zero form an abelian subcategory of
$D_{\text{acc}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$.

###### Proof.

Note that, as the functor $\otimes_{W(k)}^{L}k$ has finite homological
dimension, we have that the complex $\mathcal{M}^{\cdot}$ in the statement is
bounded. Further, $\mathcal{N}^{\cdot}\tilde{=}\Phi^{*}\mathcal{P}^{\cdot}$
for some $\mathcal{P}^{\cdot}\in
D_{\text{coh}}^{b}(\mathcal{D}_{\mathfrak{X}}^{(m)}-\text{mod})$ clearly
implies

$\mathcal{N}^{\cdot}\otimes_{W(k)}^{L}k\tilde{\to}\mathcal{B}_{X}^{(m)}\otimes_{\mathcal{D}_{X}^{(m)}}^{L}(\mathcal{P}^{\cdot}\otimes_{W(k)}^{L}k)$

which gives the converse direction of the if and only if. For the forward
direction, note that $\mathcal{N}^{\cdot}\tilde{=}\Phi^{*}\mathcal{P}^{\cdot}$
for some cohomologically complete complex $\mathcal{P}^{\cdot}$ which
satisfies $\mathcal{P}^{\cdot}\otimes_{W(k)}^{L}k\in
D_{\text{coh}}^{b}(\mathcal{D}_{X}^{(m)}-\text{mod})$; therefore , applying
[28], theorem 1.6.4, we have $\mathcal{P}^{\cdot}\in
D_{\text{coh}}^{b}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}-\text{mod})$.

Now, we have that
$\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{D}_{\mathfrak{X}}^{(m)}}^{L}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}$
is the cohomological completion of

$\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}\otimes_{\mathcal{D}_{\mathfrak{X}}^{(m)}}^{L}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}=\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}$

which, being $p$-torsion-free and $p$-adically complete, is already
cohomologically complete. Thus, for any locally free coherent
$\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}$-module $\mathcal{P}$, we have

$\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{D}_{\mathfrak{X}}^{(m)}}^{L}\mathcal{P}\tilde{=}\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}\otimes_{\mathcal{D}_{\mathfrak{X}}^{(m)}}^{L}\mathcal{P}=\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}\otimes_{\mathcal{D}_{\mathfrak{X}}^{(m)}}\mathcal{P}$

As $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}$ has finite homological
dimension, we have that any coherent
$\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}$-module $\mathcal{M}$ is quasi-
isomorphic to a finite complex of locally free coherent
$\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}$-modules, and so, as
$\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}$ is flat over
$\mathcal{D}_{\mathfrak{X}}^{(m)}$, we see

$\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{D}_{\mathfrak{X}}^{(m)}}^{L}\mathcal{M}\tilde{=}\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}\otimes_{\mathcal{D}_{\mathfrak{X}}^{(m)}}\mathcal{M}$

for all such $\mathcal{M}$; by induction on the cohomological length we see
that

$\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{D}_{\mathfrak{X}}^{(m)}}^{L}\mathcal{M}^{\cdot}\tilde{=}\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}\otimes_{\mathcal{D}_{\mathfrak{X}}^{(m)}}^{L}\mathcal{M}^{\cdot}$

for all $\mathcal{M}^{\cdot}\in
D_{\text{coh}}^{b}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}-\text{mod})$.
Therefore, a bounded complex $\mathcal{N}^{\cdot}$ is contained in
$D_{\text{acc},\text{coh}}^{b}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$
iff we have
$\mathcal{H}^{i}(\mathcal{N}^{\cdot})=\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}\otimes_{\mathcal{D}_{\mathfrak{X}}^{(m)}}\mathcal{M}$
for some coherent $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}$-module
$\mathcal{M}$; indeed the forward direction of this follows immediately from
the above discussion, and the converse follows by induction on the
cohomological length; the rest of the corollary follows immediately. ∎

######  Remark 3.21.

In the course of the above proof we showed that

$\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{D}_{\mathfrak{X}}^{(m)}}^{L}\mathcal{M}\tilde{=}\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}\otimes_{\mathcal{D}_{\mathfrak{X}}^{(m)}}\mathcal{M}$

for any coherent $\mathcal{D}_{\mathfrak{X}}^{(m)}$-module $\mathcal{M}$. In
fact this isomorphism holds whenever $\mathcal{M}$ is a cohomologically
complete $\mathcal{D}_{\mathfrak{X}}^{(m)}$-module with bounded $p$-torsion;
i.e., there is some $N\in\mathbb{N}$ such that, if a section $m$ is killed by
a power of $p$, then it is killed by $p^{N}$. Let $\mathcal{M}_{\text{tors}}$
denote this subsheaf. Then, since modules of bounded torsion are
cohomologically complete, we see that
$\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{D}_{\mathfrak{X}}^{(m)}}^{L}\mathcal{M}_{\text{tors}}=\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}\otimes_{\mathcal{D}_{\mathfrak{X}}^{(m)}}^{L}\mathcal{M}_{\text{tors}}=\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}\otimes_{\mathcal{D}_{\mathfrak{X}}^{(m)}}\mathcal{M}_{\text{tors}}$.
Further, since $\mathcal{M}/\mathcal{M}_{\text{tors}}$ is $p$-torsion-free, so
is
$\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}\otimes_{\mathcal{D}_{\mathfrak{X}}^{(m)}}\mathcal{M}/\mathcal{M}_{\text{tors}}$
(by the flatness of $\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}$) and we have
that
$\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{D}_{\mathfrak{X}}^{(m)}}^{L}\mathcal{M}/\mathcal{M}_{\text{tors}}$
is simply the $p$-adic completion of
$\Phi^{*}\mathcal{D}_{\mathfrak{X}}^{(m)}\otimes_{\mathcal{D}_{\mathfrak{X}}^{(m)}}\mathcal{M}/\mathcal{M}_{\text{tors}}$,
which lives in homological degree $0$. Therefore the result follows for
$\mathcal{M}$ from the short exact sequence

$0\to\mathcal{M}_{\text{tors}}\to\mathcal{M}\to\mathcal{M}/\mathcal{M}_{\text{tors}}\to
0$

When working mod $p^{r}$ we can do even better:

######  Corollary 3.22.

1) Let $\mathcal{N}^{\cdot}\in
D(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod})$. Then
$\mathcal{N}^{\cdot}$ is accessible iff $\mathcal{H}^{i}(\mathcal{N}^{\cdot})$
(considered as a complex concentrated in degree $0$) is accessible for all
$i$. Thus the full subcategory of
$\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod}$ consisting of accessible
modules (when considered as complexes in degree $0$) is abelian. The same
holds for accessible quasicoherent and coherent modules.

###### Proof.

The statement is local, so we can apply 3.19 using the exactness of $\Phi^{*}$
and the functor
$\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}}(\Phi^{*}\mathcal{D}_{\mathfrak{X}_{r}}^{(m)},)$.
∎

We denote by
$\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod}_{\text{acc}}$,
$\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod}_{\text{acc},\text{qcoh}}$,
and
$\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod}_{\text{acc},\text{coh}}$
the abelian category of accessible modules, accessible quasicoherent modules,
and accessible coherent modules, respectively.

Now let us note what happens in the presence of a global smooth lift
$\mathfrak{X}_{r}$ of $X$ (over $W_{r}(k)$; we allow $r=\infty$ here to cover
the case of a smooth formal $\mathfrak{X}$ over $W(k)$).

######  Corollary 3.23.

Suppose that $p>2$ if $m=0$. There is a
$(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r},\mathcal{D}_{\mathfrak{X}_{r}}^{(m)})$-bimodule
$\mathcal{B}_{\mathfrak{X}_{r}}^{(m)}$ which is locally isomorphic to
$\Phi^{*}\mathcal{D}_{\mathfrak{X}_{r}}^{(m)}$ whenever we have
$X=\text{Spec}(A)$ as above. This bimodule induces an equivalence of
categories

$\mathcal{D}_{\mathfrak{X}_{r}}^{(m)}-\text{mod}\to\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod}_{\text{acc}}$

as well as a derived equivalence

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

and when $r=\infty$ and equivalence

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

This follows immediately from the above discussions and 3.7 and Theorem 3.6.

To finish off this section we make some remarks about the completions in this
theory, which will be useful when discussing right $\mathcal{D}$-modules and
the de Rham-Witt resolution. Let us begin by noting the

######  Lemma 3.24.

Let $A$ be smooth affine which possesses local coordinates, and let
$\Phi:\mathcal{A}\to W(A)$ be the morphism coming from a coordinatized lift of
Frobenius. Let $X=\text{Spec}(A)$. For $r\geq 1$ consider the filtrations
$\\{V^{m}(W(A)/p^{r})\\}_{m\geq 1}$ and

$F^{m}(W(A)/p^{r})=\\{\sum_{(I,p)=1}a_{I}p^{s}T^{I/p^{s}}|a_{I}=0\phantom{i}\text{for}\phantom{i}s<m\\}$

(for $m\geq 1$). These filtrations satisfy $F^{m}(W(A)/p^{r})\subset
V^{m}(W(A)/p^{r})$ (for all $m$) and $V^{m+r-1}(W(A)/p^{r})\subset
F^{m}(W(A)/p^{r})$ (for all $m$).

###### Proof.

The first inclusion is obvious since $p^{s}T^{I/p^{s}}\in V^{m}$ whenever
$s\geq m$. For the second, set $s=r+m-1$ and note that for
$I\in\mathbb{Z}_{\geq 0}^{d}$

$V^{s}(T^{I})=p^{s}T^{I/p^{s}}$

Write $T^{I}=T^{I_{1}}T^{I_{2}}$ where the valuation of each entry in $I_{1}$
is $\geq s$ and the valuation of each entry in $I_{2}$ is $<s$ (if there are
no entries in the second category we take $I_{2}=\emptyset$). Let
$J_{1}=I_{1}/p^{s}$. Then

$p^{s}T^{I/p^{s}}=T^{J_{1}}\cdot p^{s}T^{I_{2}/p^{s}}$

If $I_{2}=\emptyset$ then $p^{s}T^{I/p^{s}}=p^{s}T^{J_{1}}=0$ (as $s\geq r$).
If $I_{2}\neq\emptyset$ then let $t=\text{min}\\{\text{val}_{p}(i)|i\in
I_{2}\\}$, and set $J_{2}=I_{2}/p^{t}$. Then

$p^{s}T^{I_{2}/p^{s}}=p^{t}p^{s-t}T^{J_{2}/p^{s-t}}$

and if $t<r$ then $s-t\geq m$. So we have either $p^{s}T^{I_{2}/p^{s}}=0$
(because $t\geq r$) or $p^{s}T^{I_{2}/p^{s}}\in F^{m}(W(A)/p^{r})$; so we see
that $V^{r+m-1}(T^{I})\in F^{m}(W(A)/p^{r})$ for all $I$. As $W(A)/p^{r}$ is
the completion of the span over $\mathcal{A}/p^{r}$ of terms of the form
$T^{I}$ and $V^{j}(T^{I})$ for some $j>0$, the result follows. ∎

This allows us to consider limits in the theory. Recall that if
$\mathcal{F}_{i}$ is a sequence of objects in a triangulated category with
maps $\mathcal{F}_{i}\to\mathcal{F}_{i-1}$ an object $\mathcal{L}$ is said to
be a homotopy limit of the $\mathcal{F}_{i}$ if there is a distinguished
triangle

$\mathcal{L}\to\prod_{i}\mathcal{F}_{i}\to\prod_{i}\mathcal{F}_{i}$

where the second map is $\text{Id }-\text{Shift}$. Even if they exist,
homotopy limits are not functorial in a general triangulated category.
However, in this paper all of our triangulated cateogries will be the derived
category of modules over some sheaf of rings on $X$, or a full subcategory
theoreof. In this setting, functorial cones and limits do exists; indeed if
$\mathcal{R}$ is a sheaf of rings on $X$ then, by a fundamental result of
Spaltenstein, we have

$D(\mathcal{R}-\text{mod})\tilde{\to}K(\text{k-inj}(\mathcal{R}))$

where $\text{k-inj}(\mathcal{R})$ is the category of k-injective complexes in
$\mathcal{R}-\text{mod}$, and $K()$ is the homotopy category thereof. In other
words, we may replace any element $\mathcal{M}^{\cdot}\in
D(\mathcal{R}-\text{mod})$ with a k-injective complex, and this replacement is
unique up to homotopy. As the category $K(\text{k-inj}(\mathcal{R}))$ admits
functorial cones, so too does the category $D(\mathcal{R}-\text{mod})$. So in
this setting we will write $\text{holim}(\mathcal{F}_{i})$ for the
functorially defined homotopy limit of the $\mathcal{F}_{i}$.

Specializing further, suppose
$\mathcal{M}^{\cdot}\in\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod}$.
Then we define the derived completion of $\mathcal{M}^{\cdot}$ along
$V^{i}(\mathcal{O}_{W(X)}/p^{r})$ to be

$\widehat{\mathcal{M}}^{\cdot}:=\text{holim}(\mathcal{M}^{\cdot}\otimes_{\mathcal{O}_{W(X)/p^{r}}}^{L}(\mathcal{O}_{W(X)}/p^{r})/V^{i}(\mathcal{O}_{W(X)}/p^{r}))$

In the accessible, quasicoherent case, this obect admits a more direct
description:

######  Proposition 3.25.

Let $r\geq 1$. Let $(\mathcal{M}^{j},d)$ be a complex of
$\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}$ modules, and suppose its image
$\mathcal{M}^{\cdot}$ in
$D(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod})$ is accessible and
quasicoherent. Form the complex $\mathcal{\mathcal{N}}^{j}$ whose terms are
the completions

$\mathcal{\widehat{M}}^{j}:=\lim_{i}\mathcal{M}^{j}/V^{i}(\mathcal{O}_{W(X)}/p^{r})$

Then there is an isomorphism

$\mathcal{N}^{\cdot}\tilde{\to}\widehat{\mathcal{M}}^{\cdot}$

In particular, if $X=\text{Spec}(A)$ possesses local coordinates and
$\Phi:\mathcal{A}_{r}\to W(A)/p^{r}$ is as above, then for any
$\mathcal{M}^{\cdot}\in
D_{\text{acc}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod})$, if
$\mathcal{M}^{\cdot}=\Phi^{*}\mathcal{N}^{\cdot}$ then

$R\lim_{i}(\mathcal{M}^{\cdot}\otimes_{\mathcal{O}_{W(X)/p^{r}}}^{L}(\mathcal{O}_{W(X)}/p^{r})/V^{i})\tilde{=}\widehat{\Phi}^{*}\mathcal{N}^{\cdot}$

where $\widehat{\Phi}^{*}$ denotes pullback followed by completion. For each
$\mathcal{N}^{i}$, $\widehat{\Phi}^{*}\mathcal{N}^{i}$ is an inverse limit of
a direct sum of copies of $\mathcal{N}^{i}$.

###### Proof.

Let $\mathcal{M}^{\cdot}\to\mathcal{F}^{\cdot}$ be a K-flat resolution in
$\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod}$. Then
$\mathcal{M}^{\cdot}\otimes_{\mathcal{O}_{W(X)/p^{r}}}^{L}(\mathcal{O}_{W(X)}/p^{r})/V^{i}$
is represented by the complex
$\mathcal{F}^{\cdot}\otimes_{\mathcal{O}_{W(X)/p^{r}}}(\mathcal{O}_{W(X)}/p^{r})/V^{i})$
which is the complex whose terms are
$\mathcal{F}^{j}/V^{i}(\mathcal{O}_{W(X)}/p^{r})$. Thus we obtain a map of
complexes whose terms are
$\mathcal{M}^{j}/V^{i}(\mathcal{O}_{W(X)}/p^{r})\to\mathcal{F}^{j}/V^{i}(\mathcal{O}_{W(X)}/p^{r})$.
Taking the inverse limit we obtain a map of complexes

$\mathcal{N}{}^{\cdot}\to(\mathcal{\widehat{F}})^{\cdot}$

and the latter is a complex representing $\widehat{\mathcal{M}}$. To show this
is an isomorphism, we may work locally and assume $X=\text{Spec}(A)$ possesses
local coordinates. Choose a quasi-isomorphism
$\mathcal{F}^{\cdot}\to\mathcal{H}^{\cdot}$, where $\mathcal{H}^{\cdot}$ is a
k-flat complex whose terms are direct sums of
$\Phi^{*}\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}$ (this is possible as
$\mathcal{M}^{\cdot}$ is accessible and quasicoherent). It suffices to show
that the induced map
$(\widehat{\mathcal{M}})^{\cdot}\to(\mathcal{\widehat{H}})^{\cdot}$ is an
isomorphism.

To see it, apply $\otimes_{W_{r}(k)}^{L}k$. We can evaluate it using the fact
that each $\mathcal{M}^{i}\tilde{=}\Phi^{*}\mathcal{N}^{i}$, and that the
completion along $V^{i}$ is equal to the completion along the filtration
$F^{i}$ (by the lemma above); this makes $\widehat{\mathcal{M}}^{i}$ an
inverse limit of a surjective system of modules, each of which is a direct sum
of copies of $\mathcal{M}^{i}$. This shows that

$(\widehat{\mathcal{M}})^{\cdot}\otimes_{W_{r}(k)}^{L}k\tilde{=}\widehat{\Phi}^{*}(\mathcal{N}^{\cdot}\otimes_{W(k)}^{L}k)$

By the same token, if
$\mathcal{H}^{i}={\displaystyle\bigoplus_{J_{i}}}\Phi^{*}\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}$,
then

$(\mathcal{\widehat{H}})^{i}\otimes_{W_{r}(k)}^{L}k\tilde{=}\widehat{\Phi}^{*}({\displaystyle\bigoplus_{J_{i}}}\mathcal{D}_{X}^{(0)})$

and since $\widehat{\Phi}^{*}$ is exact and conservative we obtain the result.
∎

###  3.2. The crystalline version

To relate the theory considered here to crystals, we run the above program
with $\mathcal{\widehat{D}}_{W(X),crys}^{(m)}$ in place of
$\mathcal{\widehat{D}}_{W(X)}^{(m)}$. All of the theorems above go through
without any change, and we obtain

######  Theorem 3.26.

1) There is a well-defined
$(\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p,\mathcal{D}_{X,\text{crys}}^{(0)})$
bimodule, denoted $\mathcal{B}_{X,\text{crys}}^{(0)}$ which is locally
projective over $\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p$, and
locally faithfully flat as a right $\mathcal{D}_{X,\text{crys}}^{(0)}$-module.
The associated functor

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

is exact and fully faithful, and admits an exact right adjoint
$\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p}(\mathcal{B}_{X,\text{crys}}^{(0)},?)$.
We have

$\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p}(\mathcal{B}_{X,\text{crys}}^{(0)},\mathcal{B}_{X,\text{crys}}^{(0)}\otimes_{\mathcal{D}_{X,\text{crys}}^{(0)}}\mathcal{M})\tilde{\to}\mathcal{M}$

for all $\mathcal{M}\in\mathcal{D}_{X,\text{crys}}^{(0)}-\text{mod}$.
Therefore, the functor

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

is fully faithful and we have

$R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p}(\mathcal{B}_{X,\text{crys}}^{(0)},\mathcal{B}_{X,\text{crys}}^{(0)}\otimes_{\mathcal{D}_{X,\text{crys}}^{(0)}}\mathcal{M}^{\cdot})\tilde{\to}\mathcal{M}^{\cdot}$

for all $\mathcal{M}^{\cdot}\in
D(\mathcal{D}_{X,\text{crys}}^{(0)}-\text{mod})$.

2) A module $\mathcal{M}^{\cdot}\in
D(\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p-\text{mod})$ is called
accessible if it is of the form
$\mathcal{B}_{X,\text{crys}}^{(0)}\otimes_{\mathcal{D}_{X,\text{crys}}^{(0)}}^{L}\mathcal{N}^{\cdot}$
for some $\mathcal{N}^{\cdot}\in
D(\mathcal{D}_{X,\text{crys}}^{(0)}-\text{mod})$. Let $\mathcal{M}^{\cdot}\in
D_{cc}(\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}-\text{mod})$. Then
$\mathcal{M}^{\cdot}$ is said to be accessible if
$\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k$ is accessible inside
$D(\mathcal{\widehat{D}}_{W(X)}^{(0)}/p-\text{mod})$; similarly
$\mathcal{M}^{\cdot}\in
D(\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p^{r}-\text{mod})$ is said to
accessible if $\mathcal{M}^{\cdot}\otimes_{W_{r}(k)}^{L}k$ 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

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

for a complex $\mathcal{N}^{\cdot}\in
D_{cc}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)}-\text{mod})$ In particular,
the latter condition is independent of the choice of $\Phi$; one has the
analogous statement for
$D(\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p^{r}-\text{mod})$, in which
case $\mathcal{M}^{\cdot}$ is accessible iff
$\mathcal{H}^{i}(\mathcal{M}^{\cdot})$ is for all $i$.

3) 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

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

and the analogous fact holds for schemes $\mathfrak{X}_{r}$ which are smooth
over $W_{r}(k)$.

In addition, we have a new subcategory to consider:

######  Definition 3.27.

An object $\mathcal{M}$ in
$\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p^{r}-\text{mod}_{\text{acc}}$
is said to be locally nilpotent if each local section is annihilated by
$F^{m}(\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p^{r})$ for all $m>>0$
(here, $F^{m}$ is the image of the operator filtration on
$\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}$). If $X=\text{Spec}(A)$
possesses local coordinates this is equivalent to
$\mathcal{M}\tilde{=}\Phi^{*}\mathcal{N}$ where $\mathcal{N}$ is locally
nilpotent over $\mathcal{D}_{\mathfrak{X}_{r},\text{crys}}^{(0)}$ in the usual
sense (each section is killed by a power of the ideal $\mathcal{I}_{r}$). One
makes the same definition over $\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}$, and
the category of locally nilpotent accessible
$\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}$-modules is equivalent to the
category of locally nilpotent accessible
$\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p^{r}$-modules; indeed, the
local nilpotence condition ensures that the
$\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}$-module structure extends uniquely
to a $\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p^{r}$-module.

We use the subscript ln to denote locally nilpotent objects in a given
category. Then we have

######  Proposition 3.28.

The map
$\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}\to\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p^{r}$
yields a functor

$\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p^{r}-\text{mod}_{\text{acc},\text{qcoh,ln}}\to\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}-\text{mod}_{\text{acc},\text{qcoh}}$

This functor is fully faithful, and its image consists of all sheaves
$\mathcal{M}$ such which are accessible, quasicoherent, and locally nilpotent.

###### Proof.

The map of algebras
$\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}\to\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p^{r}$
yields a forgetful functor
$D(\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p^{r}-\text{mod})\to
D(\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}-\text{mod})$. Suppose $\mathcal{M}$
is an accessible, quasicoherent, and locally nilpotent module over
$\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p^{r}$. Restricting to
$X=\text{Spec}(A)$ for some $A$ which possesses local coordinates, we may
write

$\mathcal{M}\tilde{\to}\Phi^{*}\mathcal{D}_{\mathfrak{X}_{r},\text{crys}}^{(0)}\otimes_{\mathcal{D}_{\mathfrak{X}_{r},\text{crys}}^{(0)}}^{L}\mathcal{N}$

As $\mathcal{N}$ is quasicoherent and locally nilpotent over
$\mathcal{D}_{\mathfrak{X}_{r},\text{crys}}^{(0)}$, it is a union of its
coherent $\mathcal{D}_{\mathfrak{X}_{r},\text{crys}}^{(0)}$-submodules which
are nilpotent (i.e. the entire module is annihilated by some power of
$\mathcal{I}_{r}$). Let $\mathcal{N}^{\prime}$ be such a coherent submodule.
Then the natural map

$\Phi^{*}\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}\otimes_{\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}}^{L}\mathcal{N}^{\prime}\to\Phi^{*}\mathcal{D}_{\mathfrak{X}_{r},\text{crys}}^{(0)}\otimes_{\mathcal{D}_{\mathfrak{X}_{r},\text{crys}}^{(0)}}^{L}\mathcal{N}^{\prime}$

is an isomorphism. To see this, note that $\mathcal{N}^{\prime}$ is also
coherent over $\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}$. Therefore both sides are
isomorphic to
${\displaystyle\lim_{\leftarrow}(\mathcal{O}_{W(X)}/p^{r})/V^{m}\otimes_{\mathcal{O}_{\mathfrak{X}_{r}}}\mathcal{N}^{\prime}}$
(use a presentation of $\mathcal{N}^{\prime}$ over
$\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}$ on the left hand side and over
$\mathcal{D}_{\mathfrak{X}_{r},\text{crys}}^{(0)}$ on the right hand side). As
the tensor product commutes with inductive limits, we see that

$\Phi^{*}\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}\otimes_{\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}}^{L}\mathcal{N}\tilde{\to}\Phi^{*}\mathcal{D}_{\mathfrak{X}_{r},\text{crys}}^{(0)}\otimes_{\mathcal{D}_{\mathfrak{X}_{r},\text{crys}}^{(0)}}^{L}\mathcal{N}$

In other words, $\mathcal{M}$ is also accessible and quasicoherent when
regarded as a module over $\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}$. Thus the
natural functor

$\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p^{r}-\text{mod}_{\text{acc},\text{qcoh,ln}}\to\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}-\text{mod}_{\text{acc},\text{qcoh}}$

has image in
$\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}-\text{mod}_{\text{acc},\text{qcoh}}$
as claimed, and is clearly onto the category of accessible, quasicoherent, and
locally nilpotent modules. For the full faithfulness, for any two objects
$\mathcal{M}_{1},\mathcal{M}_{2}$ in
$\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p^{r}-\text{mod}_{\text{acc},\text{qcoh,ln}}$,
we consider the morphism of sheaves

$\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p^{r}}(\mathcal{M}_{1},\mathcal{M}_{2})\to\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}}(\mathcal{M}_{1},\mathcal{M}_{2})$

we will be done if we can show it is an isomorphism; to do so, we can work
locally and suppose $X=\text{Spec}(A)$ and
$\mathcal{M}_{i}=\Phi^{*}\mathcal{N}_{i}$ (for $i=1,2$). Then we need to show
that

$\mathcal{H}om_{\mathcal{\mathcal{D}}_{\mathfrak{X}_{r},\text{crys}}^{(0)}}(\mathcal{N}_{1},\mathcal{N}_{2})\to\mathcal{H}om_{\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}}(\mathcal{N}_{1},\mathcal{N}_{2})$

is an isomorphism; but this is clear. ∎

Now we give the relation with crystals in the usual sense. It reads

######  Theorem 3.29.

Consider the category $\text{Crys}_{W_{r}(k)}(X)$ of crystals on $X$ over
$W_{r}(k)$. There is an exact functor

$\eta:\text{Crys}_{W_{r}(k)}(X))\to\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p^{r}-\text{mod}_{\text{acc},}$

which is fully faithful. Let

$\epsilon:\text{Crys}_{W_{r}(k)}(X))\to\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}-\text{mod}$

denote the composition of the restriction of $\eta$ with the forgetful functor
to $\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}-\text{mod}$. Then, upon
restriction to $\text{Qcoh}(\text{Crys}_{W_{r}(k)}(X)))$ $\epsilon$ lands in
$\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}-\text{mod}_{\text{acc,qcoh}}$. The
functor

$\epsilon:D_{\text{qcoh}}(\text{Crys}_{W_{r}(k)}(X))\to
D_{\text{acc},\text{qcoh}}(\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}-\text{mod})$

is also fully faithful. The image of $\epsilon$ consists of all complexes
$\mathcal{M}^{\cdot}$ such that, for each $i$,
$\mathcal{H}^{i}(\mathcal{M}^{\cdot})$ is accessible, quasicoherent, and
locally nilpotent.

###### Proof.

Let $\tilde{\mathcal{M}}$ be an element of $\text{Crys}_{W_{r}(k)}(X)$. For
every open subset of the form $U=\text{Spec}(A)$ (which possesses local
coordinates), and every flat lift $\mathcal{A}_{r}$ to $W_{r}(k)$,
$\tilde{\mathcal{M}}$ produces a canonically defined sheaf with locally
nilpotent flat connection on $\mathcal{A}_{r}$, denoted
$(\mathcal{N},\nabla)$. For any map $\Phi:\mathcal{A}_{r}\to W(A)/p^{r}$
coming from a coordinatized lift of Frobenius, we have
$\widehat{\Phi}^{*}\mathcal{N}$, the completion (along
$V^{i}(\mathcal{O}_{W(X)}/p^{r})$) of $\Phi^{*}\mathcal{N}$; this is exactly
the sheaf over $W(X)_{p^{r}=0}$ that the theory of the crystalline site
attaches to $\mathcal{N}$ (coming from the fact that $\Phi:\mathcal{A}_{r}\to
W(A)/p^{r}$ is an inverse limit of pd thickenings).

For any other such map $\Psi$, the crystalline theory yields a canonical
isomorphism
$\widehat{\Phi}^{*}\mathcal{N}\tilde{\to}\widehat{\Psi}^{*}\mathcal{N}$, and,
using 3.8, we have that this map agrees with the canonical isomorphism coming
from theory of accessible
$\mathcal{\widehat{D}}_{W(A),\text{crys}}^{(0)}/p^{r}$-modules (this is
because the morphism is realized via
$\Phi^{*}\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}\widehat{\otimes}_{\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}}\mathcal{N}\tilde{\to}\Psi^{*}\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}\widehat{\otimes}_{\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}}\mathcal{N}$).
So the sheaf $\tilde{\mathcal{M}}_{W(X)_{p^{r}=0}}$ on $W(X)_{p^{r}=0}$ which
the theory of the crystalline site attaches to $\tilde{\mathcal{M}}$ carries a
canonical $\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p^{r}$ action. This
sheaf is not accessible, however, there is an exact functor

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

which is right adjoint to the inclusion functor (c.f. the discussion right
below 4.5 below); we denote it $\mathcal{M}\to\mathcal{M}_{\text{acc}}$; if
$X=\text{Spec}(A)$ then
$(\widehat{\Phi}^{*}\mathcal{N})_{\text{acc}}=\Phi^{*}\mathcal{N}$. Therefore
we define the functor

$\eta(\tilde{\mathcal{M}})=(\tilde{\mathcal{M}}_{W(X)_{p^{r}=0}})_{\text{acc}}$

Clearly its restriction to quasicoherent crystals lands in quasicoherent
accessible modules (this is the argument of the previous proposition). There
is a canonical map
$\mathcal{H}om_{\text{crys}}(\tilde{\mathcal{M}}_{1},\tilde{\mathcal{M}}_{2})\to\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(0)}/p^{r}}(\eta(\tilde{\mathcal{M}}_{1}),\eta(\tilde{\mathcal{M}}_{2}))$
of sheaves in the Zariski topology of $X$; to show it is an isomorphism we may
work locally, but there the claim reduces to the full-faithfulness of
$\Phi^{*}$.

Now, after restricting $\eta$ to quasicoherent sheaves, we obtain $\epsilon$,
which is also fully faithful. Let us consider the derived version. Let
$\mathcal{M}_{i}^{\cdot}$ ($i=1,2$) be elements of
$D_{\text{qcoh}}(\text{Crys}_{W_{r}(k)}(X)))$; and replace $\mathcal{M}_{2}$
with a K-injective resolution, $\mathcal{K}^{\cdot}$ Then we have

$R\mathcal{H}om_{\text{crys}}(\mathcal{M}_{1}^{\cdot},\mathcal{M}_{2}^{\cdot})=\mathcal{H}om_{\text{crys}}(\mathcal{M}_{1},\mathcal{K}^{\cdot})\to\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}}(\epsilon(\mathcal{M}_{1}),\epsilon(\mathcal{K}^{\cdot}))$

where the latter two $\mathcal{H}om$ indicate Hom in the homotopy category of
chain complexes. Now we have a map

$R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}}(\epsilon(\mathcal{M}_{1}),\epsilon(\mathcal{K}^{\cdot}))\to\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}}(\epsilon(\mathcal{M}_{1}),\epsilon(\mathcal{K}^{\cdot}))$

and we will show that both of these maps are isomorphisms. This is a local
question, which boils down to the following: if $\mathcal{N}_{i}^{\cdot}$
($i=1,2$) are elements of the derived category
$\text{Qcoh}_{\text{ln}}(\mathcal{D}_{\mathfrak{X}_{r}}^{(0)})$ (the category
of quasicoherent $\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}$-modules which are
locally nilpotent), then the map

$R\mathcal{H}om_{\text{Qcoh}_{\text{ln}}(\mathcal{D}_{\mathfrak{X}_{r}}^{(0)})}(\mathcal{N}_{1}^{\cdot},\mathcal{N}_{2}^{\cdot})\to
R\mathcal{H}om_{\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}-\text{mod}}(\mathcal{N}_{1}^{\cdot},\mathcal{N}_{2}^{\cdot})$

is an isomorphism. This follows from by regarding a quasicoherent
$\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}$-module as a quasicoherent sheaf on the
center $\mathcal{Z}(\mathcal{D}_{\mathfrak{X}_{r}}^{(0)})$, and applying the
functor of local cohomology with support along the zero section (compare, e.g.
[8], lemma 3.1.7). ∎

It is worth noting that we also have equivalences

$D(\text{Qcoh}(\text{Crys}_{W_{r}(k)}(X)))\tilde{\to}D_{\text{qcoh}}(\text{Crys}_{W_{r}(k)}(X))$

and

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

These can be proved in a very similar way to the classical statement

$D(\text{QCoh}(X))\tilde{\to}D_{\text{qcoh}}(\mathcal{O}_{X}-\text{mod})$

c.f. [12], corollary 5.5.

###  3.3. Frobenius descent

In this subsection we explain how Berthelot’s fundamental theorem on the
independence of the Frobenius in arithmetic $\mathcal{D}$-module theory also
follows from our results. To set things up, let us recall the main results of
that theory:

######  Theorem 3.30.

1) Let $A$ be a smooth algebra which possesses local coordinates; let
$\mathcal{A}$ be a lift of $A$ and let $F:\mathcal{A}\to\mathcal{A}$ be a
coordinatized lift of Frobenius. Let consider the
$(\widehat{\mathcal{D}}_{\mathcal{A}}^{(m+1)},\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)})$
bisubmodule of $\text{End}_{W(k)}(\mathcal{A})$ generated by $F$. Then the
natural map
$F^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}\to\text{End}_{W(k)}(\mathcal{A})$
which takes $a\otimes P\to a\cdot F(P)$ is an isomorphism onto this
bisubmodule. Furthermore, 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
$\mathcal{\widehat{D}}_{\mathcal{A}}^{(m+1)}-\text{mod}$.

2) If $F_{1},F_{2}$ are two lifts of Frobenius as a above, then (assuming
$p>2$ when $m=0$), there is a canonical isomorphism of bimodules

$F_{1}^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}\tilde{\to}F_{2}^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$

In particular, if $X$ is an arbitrary smooth scheme over $k$ and
$\mathfrak{X}$ is a lift, then there is a globally defined equivalence of
categories
$F^{*}:\mathcal{D}_{\mathfrak{X}}^{(m)}-\text{mod}\to\mathcal{D}_{\mathfrak{X}}^{(m+1)}-\text{mod}$.

This theorem is proved in [5], section 2.3. (for part $1)$) and [5], theorem
2.2.5 for part $2)$; c.f. also [19], corollary 13.3.8.

Part $1)$ can be proved rather rapidly from the methods of this paper. Our
construction of $\Phi^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$ is a more
elaborate version of the construction of
$F^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$, and, the proof of Theorem
3.9 is a more elaborate version of the proof that
$F^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$ is projective over
$\mathcal{D}_{\mathcal{A}}^{(m+1)}$. Once this is known, the crucial
isomorphism

$\mathcal{\widehat{D}}_{\mathcal{A}}^{(m+1)}\tilde{\to}\text{End}_{\mathcal{\widehat{D}}_{\mathcal{A}}^{(m),\text{opp}}}(F^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)})$

can be proved after reduction mod $p$, where it is an elementary computation
(c.f. [17], proposition 4.18 for the case $m=0$).

Now let us address part $2)$. We in fact have the following for Witt
differential operators:

######  Lemma 3.31.

There is a morphism of sheaves of algebras
$F:\widehat{\mathcal{D}}_{W(X)}^{(m)}\to\widehat{\mathcal{D}}_{W(X)}^{(m+1)}$
whose restriction to $\mathcal{O}_{W(X)}$ is the Witt-vector Frobenius map.

###### Proof.

It suffices to show this locally, so without loss of generality we assume
$X=\text{Spec}(A)$ possesses local coordinates. Then for an operator
$P\in\widehat{\mathcal{D}}_{W(A)}^{(m)}$ we consider the operator $F\circ
P\circ F^{-1}$. When $P$ is multiplication by an element $a\in W(A)$, this
operator is $F(a)$. Using 2.10, we see that
$F\\{\partial_{i}\\}_{\lambda}F^{-1}=\\{\partial_{i}\\}_{p\lambda}$ for all
$\lambda$; so that Theorem 2.17 ensures us that $F\circ P\circ
F^{-1}\in\widehat{\mathcal{D}}_{W(A)}^{(m+1)}$ as required. ∎

Now, we can prove

######  Theorem 3.32.

Let $(\mathcal{A},F)$ be as above, and $X=\text{Spec(A)}$; let
$\Phi:\mathcal{A}\to W(A)$ the associated map. Then for
$\mathcal{M}^{\cdot}\in
D_{\text{acc}}(\widehat{\mathcal{D}}_{W(X)}^{(m)}-\text{mod})$, if
$\mathcal{M}^{\cdot}=\Phi^{*}\mathcal{N}^{\cdot}$ then

$F^{*}\mathcal{M}^{\cdot}\tilde{\to}\Phi^{*}(F^{*}\mathcal{N}^{\cdot})$

where on the left we have the pullback with respect to the algebra morphism
$\widehat{\mathcal{D}}_{W(X)}^{(m)}\to\widehat{\mathcal{D}}_{W(X)}^{(m+1)}$,
and on the right we have Berthelot’s Frobenius pullback. In particular for two
such morphisms $F_{1},F_{2}$ we obtain a canonical isomorphism
$F_{1}^{*}\mathcal{N}^{\cdot}\tilde{\to}F_{2}^{*}\mathcal{N}^{\cdot}$.

###### Proof.

By definition we have

$F^{*}\mathcal{M}^{\cdot}=\widehat{\mathcal{D}}_{W(X)}^{(m+1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot}$

where the action of $\widehat{\mathcal{D}}_{W(X)}^{(m)}$ on
$\widehat{\mathcal{D}}_{W(X)}^{(m+1)}$ is via the map $F$. As
$\mathcal{M}^{\cdot}=\Phi^{*}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}}^{L}\mathcal{N}^{\cdot}$
we are left to ponder

$\widehat{\mathcal{D}}_{W(X)}^{(m+1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(m)}}^{L}\Phi^{*}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}$

As $\Phi^{*}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}$ is locally projective
over $\widehat{\mathcal{D}}_{W(X)}^{(m)}$ this is concentrated in a single
degree, and is in fact a summand of $\widehat{\mathcal{D}}_{W(X)}^{(m+1)}$. We
are considering $\widehat{\mathcal{D}}_{W(X)}^{(m+1)}$ as a
$(\widehat{\mathcal{D}}_{W(X)}^{(m+1)},\widehat{\mathcal{D}}_{W(X)}^{(m)})$
bimodule; and it is exactly the bisubmodule of
$\mathcal{E}nd_{W(k)}(\mathcal{O}_{W(X)})$ which is locally generated by $F$.
As $\Phi^{*}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}$ is the
$(\widehat{\mathcal{D}}_{W(X)}^{(m)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)})$
bisubmodule of
$\mathcal{H}om_{W(k)}(\mathcal{O}_{\mathfrak{X}},\mathcal{O}_{W(X)})$ locally
generated by $\Phi$, we conclude that

$\widehat{\mathcal{D}}_{W(X)}^{(m+1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(m)}}^{L}\Phi^{*}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}$

is the
$(\widehat{\mathcal{D}}_{W(X)}^{(m+1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)})$
bisubmodule of
$\mathcal{H}om_{W(k)}(\mathcal{O}_{\mathfrak{X}},\mathcal{O}_{W(X)})$ locally
generated by $F\circ\Phi$. By the same token we have that

$\Phi^{*}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m+1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m+1)}}^{L}F^{*}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}$

is the
$(\widehat{\mathcal{D}}_{W(X)}^{(m+1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)})$
bisubmodule of
$\mathcal{H}om_{W(k)}(\mathcal{O}_{\mathfrak{X}},\mathcal{O}_{W(X)})$ locally
generated by $\Phi\circ F$. As $F\circ\Phi=\Phi\circ F$ we obtain that these
two bimodules are isomorphic and so

$\widehat{\mathcal{D}}_{W(X)}^{(m+1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot}=\widehat{\mathcal{D}}_{W(X)}^{(m+1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(m)}}^{L}\Phi^{*}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}}^{L}\mathcal{N}^{\cdot}$
$\tilde{\to}\Phi^{*}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m+1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m+1)}}^{L}F^{*}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}}^{L}\mathcal{N}^{\cdot}$
$=\Phi^{*}(F^{*}\mathcal{N}^{\cdot})$

as required. ∎

Restricting oneself to the subcategory of coherent accessible modules, we
obtain an isomorphism of functors $F_{1}^{*}\tilde{\to}F_{2}^{*}$ on that
category. This recovers part $2)$ of Berthelot’s theorem above. In addition,
one may use this technique to get a rather quick proof that the category of
coherent modules in Berthelot’s theory of overconvergent
$\mathcal{D}$-modules, admits a Frobenius action. We will return to this topic
elsewhere.

##  4\. Operations on Accessible Modules

In this chapter we develop the basic operations on the accessible
$\widehat{\mathcal{D}}_{W(X)}^{(m)}$-modules; we assume throughout that the
level $m\geq 0$.

###  4.1. Operations on modules: Right Modules and the left-right interchange

In this section, we’ll construct the category of accessible _right_
$\mathcal{D}_{W(X)}^{(m)}$-modules and explain the left-right interchange in
this context. Before doing so, we give a quick review of the left-right
interchange in the classical situation: on $\mathfrak{X}$, the line bundle
$\omega_{\mathfrak{X}}$ carries the natural structure of a right
$\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}$ module. One can define the
structure of a sheaf of algebras on

$\omega_{\mathfrak{X}}\otimes_{\mathcal{O}_{\mathfrak{X}}}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\otimes_{\mathcal{O}_{\mathfrak{X}}}\omega_{\mathfrak{X}}^{-1}$

via

$(s_{1}\otimes\Phi_{1}\otimes t_{1})\cdot(s_{2}\otimes\Phi_{2}\otimes
t_{2})=s_{1}\otimes\Phi_{1}<t_{1},s_{1}>\Phi_{2}\otimes t_{2}$

snd we have an there is an isomorphism of sheaves of algebras

$\omega_{\mathfrak{X}}\otimes_{\mathcal{O}_{\mathfrak{X}}}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\otimes_{\mathcal{O}_{\mathfrak{X}}}\omega_{\mathfrak{X}}^{-1}\tilde{=}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m),\text{op}}$

(this uses the right action of $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}$ on
$\omega_{\mathfrak{X}}$). One obtains from this the fact that
$\mathcal{M}\to\omega_{\mathfrak{X}}\otimes_{\mathcal{O}_{\mathfrak{X}}}\mathcal{M}$
is an equivalence of categories from
$\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}-\text{mod}$ to
$\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m),\text{op}}-\text{mod}$. This
extends to an equivalence on the derived categories, which preserves
cohomologically complete objects on each side. We wish to extend this to the
accessible $\widehat{\mathcal{D}}_{W(X)}^{(m)}$-modules.

Let’s begin with the local situation:

######  Proposition 4.1.

Let $X=\text{Spec}(A)$ where $A$ possesses local coordinates; let
$F:\mathcal{A}\to\mathcal{A}$ a coordinatized lift of Frobenius, and let
$\Phi:\mathcal{A}\to W(A)$; let $\pi:W(A)\to\mathcal{A}$ be the associated
projection as in 3.15. Then

$\Phi^{!}(\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}):=\text{Hom}_{\mathcal{\widehat{D}}_{W(A)}^{(m)}}(\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)},\mathcal{\widehat{D}}_{W(A)}^{(m)})\tilde{=}\pi\cdot\mathcal{\widehat{D}}_{W(A)}^{(m)}$

is a
$(\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)},\mathcal{\widehat{D}}_{W(A)}^{(m)})$
bimodule, which is a projective as a right
$\mathcal{\widehat{D}}_{W(A)}^{(m)}$-module. Further, there is an isomorphism
of right $\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$-modules

$\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)})\tilde{\to}\Phi^{!}(\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)})$

where
$\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)})\subset\text{Hom}_{\mathcal{A}}(W(A),\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)})$
consists of those $\mathcal{A}$-linear morphisms
$\epsilon:W(A)\to\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$ which satisfy
$\epsilon(V^{r}(W(A)))\subset p^{r}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$.

###### Proof.

The first line follows immediately from
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}=\mathcal{\widehat{D}}_{W(A)}^{(m)}\cdot\pi$;
the
$(\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)},\mathcal{\widehat{D}}_{W(A)}^{(m)})$-bimodule
structure is deduced from the right action of
$\mathcal{\widehat{D}}_{W(A)}^{(m)}$ on
$\text{Hom}_{\mathcal{\widehat{D}}_{W(A)}^{(m)}}(\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)},\mathcal{\widehat{D}}_{W(A)}^{(m)})$
and the right action of $\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$ on
$\Phi^{*}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$. So the only thing that
needs to be done is to construct the isomorphism

$\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)})\tilde{\to}\Phi^{!}(\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)})$

First, define $\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{A})$ to be the
set of $\mathcal{A}$ linear maps $\epsilon:W(A)\to\mathcal{A}$ satisfying
$\epsilon(V^{r}(W(A)))\subset p^{r}\mathcal{A}$. By the lemma below we have
that if $\epsilon\in\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{A})$, then
$\Phi\circ\epsilon\in\pi\cdot\mathcal{\widehat{D}}_{W(A)}^{(m)}$; thus we
obtain an inclusion
$\iota:\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{A})\subset\Phi^{!}(\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)})$.
Now, I claim that there is an isomorphism

(4.1)
$\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)})\tilde{=}\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{A})\widehat{\otimes}_{\mathcal{A}}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$

where $\widehat{\otimes}$ here denotes the completion with respect to the
filtration

$F^{m}(\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{A})):=\\{\epsilon\in\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{A})|\epsilon(W(A))\subseteq
p^{m}\mathcal{A}\\}$

To prove this note that, via the inclusion
$\mathcal{A}\subset\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$ we have an
inclusion
$\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{A})\subset\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)})$,
which induces a morphism

$\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{A})\otimes_{\mathcal{A}}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}\to\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)})$

whose completion induces (4.1) (that it is an isomorphism is easily checked in
local coordinates).

So, via the right action of $\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$ on
$\Phi^{!}(\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)})$ we obtain a map

$\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{A})\otimes_{\mathcal{A}}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}\to\Phi^{!}(\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)})$

given by $(\epsilon,P)\to\iota(\epsilon)\cdot P$. As elements of
$F^{m}(\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{A}))$ are contained in
$V^{r}(\mathcal{\widehat{D}}_{W(A)}^{(0)})$ (as shown in the lemma directly
below), we may complete to obtain a map

$a:\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{A})\widehat{\otimes}_{\mathcal{A}}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}\to\Phi^{!}(\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)})$

Identifying
$\Phi^{!}(\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)})\tilde{=}\pi\cdot\mathcal{\widehat{D}}_{W(A)}^{(m)}$
with a set of $W(k)$-linear morphisms from $W(A)$ to $\mathcal{A}$, this map
is given by the composition

$(\epsilon,P)\to P\circ\epsilon:W(A)\to\mathcal{A}$

It follows directly that the map $a$ is injective. To see the surjectivity, it
suffices to show that the image of $a$ is preserved under the right action of
$\mathcal{\widehat{D}}_{W(A)}^{(m)}$ (since $\pi$ is clearly contained in the
image). The proof of this is given in the lemma directly below ∎

In the proof above, we used the

######  Lemma 4.2.

1) Let $\phi_{I/p^{r}}:W(A)\to\mathcal{A}$ be the $\mathcal{A}$-linear map
which takes $p^{r}T^{I/p^{r}}$ to $p^{r}$ and which takes
$p^{r^{\prime}}T^{J/p^{r^{\prime}}}$ to $0$ for all $J\neq I$. Then
$\Phi\circ\phi_{I/p^{r}}\in\pi\cdot\mathcal{\widehat{D}}_{W(A)}^{(0)}$.

2) Let $P\in\mathcal{\widehat{D}}_{W(A)}^{(0)}$. Then
$\phi_{I/p^{r}}(P\cdot):W(A)\to\mathcal{A}$ is contained in the image of
$\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{A})\widehat{\otimes}_{\mathcal{A}}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$
in $\text{Hom}_{W(k)}(W(A),\mathcal{A})$.

###### Proof.

These will be variants of the proofs of 3.15 and 3.2.

1) Start with case $A=k[T]$, $\mathcal{A}=W(k)<<T>>$, and the Frobenius lift
is the standard one. Let $i\in\\{1,\dots p^{r}-1\\}$ so that
$\text{val}_{p}(i)=0$. Let $r^{\prime}\geq r$. As in 3.14, let
$\pi_{i/p^{r}}:W_{r^{\prime}+1}(A)\to W_{r^{\prime}+1}(A)$ denote the
projection operator which takes any monomial of the form $p^{r}T^{j+i/p^{r}}$
to itself (for each $j\in\mathbb{Z}_{\geq 0}$), and all other monomials to
$0$. By induction on $r^{\prime}$ we shall construct an operator satisfying

(4.2)
$\Phi_{r^{\prime}}\circ\phi_{i/p^{r}}^{r^{\prime}}:=\\{d\\}_{i/p^{r}}\circ\pi_{i/p^{r}}-\sum_{i=1}^{p^{r^{\prime}+1}-1}f_{i}(T^{i/p^{r^{\prime}}}\\{d\\}_{i/p^{r^{\prime}}})\circ\\{d\\}_{i/p^{r}}\circ\pi_{i/p^{r}}$

for some $f_{i}\in\mathbb{Z}_{p}[X]$, so that
$\phi_{i/p^{r}}^{r^{\prime}}\equiv\phi_{i/p^{r}}^{r^{\prime}-1}\phantom{i}\text{mod}\phantom{i}V^{r}(\mathcal{\widehat{D}}_{W(A)}^{(0)})$.
The inverse limit of these operators is then the required $\phi_{i/p^{r}}$.

We will work now with the copy of $W_{r^{\prime}+1}(A^{(r^{\prime})})$
contained in $\mathcal{A}_{r^{\prime}+1}=W_{r^{\prime}+1}(k)[T]$ and construct
the operator there. We will proceed by induction; when $r^{\prime}=r$ we have
that

$d^{[i]}\circ\pi_{i/p^{r}}$

is already equal to $\phi_{i/p^{r}}^{r}$. Assuming by induction that
$\phi_{i/p^{r}}^{r^{\prime}-1}$ has been constructed which satisfies (4.2),
consider the operator

$d^{[p^{r^{\prime}-r}i]}\circ\pi_{i/p^{r}}-\sum_{i=1}^{p^{r^{\prime}}-1}f_{i}(T^{i}d^{[i]})\circ
d^{[p^{r^{\prime}-r}i]}\circ\pi_{i/p^{r}}$

on $\mathcal{A}_{r^{\prime}+1}$. Arguing exactly as in the proof of 3.12, and
employing the induction hypothesis, we see that the evaluation of this
operator on a term of the form $p^{r}T^{p^{r^{\prime}-r}i}T^{ap^{r^{\prime}}}$
yields $p^{r^{\prime}}h(a)T^{ap^{r^{\prime}}}$ where $h$ is a polynomial in
$\mathbb{Z}_{p}[X]$. So we can add a term of the form

$-p^{r^{\prime}}h(T^{p^{r^{\prime}+1}}d^{[p^{r^{\prime}+1}]})\circ
d^{[p^{r^{\prime}-r}i]}\circ\pi_{i/p^{r}}$

to construct $\phi^{r^{\prime}}$, as desired.

To construct $\phi_{I/p^{r}}$ in general, use the inclusions
$\mathcal{\widehat{D}}_{W(k[T_{i}])}^{(0)}\to\mathcal{\widehat{D}}_{W(A)}^{(0)}$
(as noted in 3.15) and take products of operators in
$\mathcal{\widehat{D}}_{W(k[T_{i}])}^{(0)}$.

$2)$ As
$\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{A})\widehat{\otimes}_{\mathcal{A}}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}\to\text{Hom}_{W(k)}(W(A),\mathcal{A})$
is injective, we identify
$\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{A})\widehat{\otimes}_{\mathcal{A}}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$
with its image. By an $F$-conjugation argument, we can assume $m=0$ from now
on. As $A$ is etale over $k[T_{1},\dots,T_{n}]$, we can assume without loss of
generality that $A=k[T_{1},\dots,T_{n}]$ with the coordinate lift of
Frobenius.

We begin by considering $\phi_{I/p^{m}}(\\{\partial_{j}\\}_{1/p^{r}}\cdot)$;
without loss of generality suppose $j=1$. Choose $l\geq r,m$ and consider
$\phi_{I/p^{m}}$ acting on $W_{l+1}(A)$; as usual we work with the copy of
$W_{l+1}(A^{(l)})\subset\mathcal{A}_{l+1}=W(k)[T_{1},\dots,T_{n}]$. Then
$\phi_{I/p^{m}}$ is nonzero on monomials of the form $T^{K}$ where
$K=(p^{l-m}i_{1}+p^{l}j_{1},\dots,p^{l-m}i_{n}+p^{l}j_{n})$ where
$j_{i}\in\mathbb{Z}_{\geq 0}$ for all $i$, and zero on all other monomials. In
this context, $\\{\partial_{1}\\}_{1/p^{r}}$ becomes the operator
$\partial_{1}^{[p^{l-r}]}$. The operator
$\phi_{I/p^{m}}(\partial_{1}^{[p^{l-r}]}\cdot)$ is nonzero on monomials of the
form $T^{K^{\prime}}$where
$K^{\prime}=(p^{l-m}i_{1}+p^{l}j_{1}+p^{l-r},\dots,p^{l-m}i_{n}+p^{l}j_{n})$,
and zero on all the others.

We have

$d_{1}^{[p^{l-r}]}T_{1}^{p^{l-m}i_{1}+p^{l}j_{1}+p^{l-r}}T_{2}^{p^{l-m}i_{2}+p^{l}j_{2}}\cdots
T_{n}^{p^{l-m}i_{n}+p^{l}j_{n}}$ $={p^{l-m}i_{1}+p^{l}j_{1}+p^{l-r}\choose
p^{l-r}}T_{1}^{p^{l-m}i_{1}+p^{l}j_{1}}T_{2}^{p^{l-m}i_{2}+p^{l}j_{2}}\cdots
T_{n}^{p^{l-m}i_{n}+p^{l}j_{n}}$

However,

${p^{l-m}i_{1}+p^{l-r}(p^{r}j_{1}+1)\choose p^{l-r}}=f_{l}(p^{r}j_{1}+1)$

is a polynomial in $j_{1}$ (by 3.12). So we deduce that

$\phi_{I/p^{m}}(\partial_{1}^{[p^{l-r}]}\cdot)=p^{a}\lim_{l}f_{l}(1+p^{r}T_{1}\partial_{1})\circ\phi_{I^{\prime}/p^{b}}$

where ${\displaystyle\lim_{l}f_{l}(1+p^{r}T_{1}\partial_{1})}$ is understood
as an element of $\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}$,
$I^{\prime}/p^{b}$ is the term $I/p^{m}+(1/p^{r},0,\dots,0)$, and
$a=|b-\text{max}\\{r,m\\}|$.; thus we see that
$\phi_{I/p^{m}}(\partial_{1}^{[p^{l-r}]}\cdot)\in\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{A})\widehat{\otimes}_{\mathcal{A}}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}$.
As any element of
$\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{A})\widehat{\otimes}_{\mathcal{A}}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}$
can be written as a sum of the form

$\sum_{I,r}\phi_{I/p^{r}}\cdot P_{I}$

we see that
$\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\mathcal{A})\widehat{\otimes}_{\mathcal{A}}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0)}$
is closed under the action of $\\{\partial_{j}\\}_{1/p^{r}}$, and therefore
under the action of every term of the form $\\{\partial\\}_{J/p^{r}}$. By
Theorem 2.17, it suffices to show that it is closed under the action of terms
of the form $T^{K/p^{r}}\\{\partial\\}_{J/p^{r}}$. That follows from the above
by noting that the operator

$\phi_{I/p^{m}}(T^{K/p^{r}}\cdot):V^{r}(W(A))\to\mathcal{A}$

is the restriction to $V^{r}(W(A))$ of a sum of operators of the form
$\phi_{J/p^{l}}$ (and this is straightforward to verify). ∎

From the above we obtain

######  Definition 4.3.

Set
$\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}/p}(\mathcal{B}_{X}^{(m)},\mathcal{\widehat{D}}_{W(X)}^{(m)}/p):=\mathcal{B}_{X}^{(m),r}$.
This is a sheaf of
$(\widehat{\mathcal{D}}_{X}^{(m)},\mathcal{\widehat{D}}_{W(X)}^{(m)}/p)$
bimodules, which is locally isomorphic to
$\Phi^{!}(\mathcal{\widehat{D}}_{X}^{(m)})$ for a coordinatized lift of
Frobenius $\Phi$.

In particular, this bimodule always exists when $n=1$. We can thus copy over
3.17 and obtain

######  Definition 4.4.

1) A module
$\mathcal{N}\in\mathcal{\widehat{D}}_{W(X)}^{(m),\text{op}}/p-\text{mod}$ is
accessible if is of the form
$\mathcal{M}\otimes_{\mathcal{D}_{X}^{(m)}}\mathcal{B}_{X}^{(m),r}$ for some
$\mathcal{M}\in\mathcal{D}_{X}^{(m),\text{op}}-\text{mod}$.

2) A complex $\mathcal{N}^{\cdot}\in
D(\mathcal{\widehat{D}}_{W(X)}^{(m),\text{op}}/p-\text{mod})$ is accessible if
is of the form
$\mathcal{M}^{\cdot}\otimes_{\mathcal{D}_{X}^{(m)}}^{L}\mathcal{B}_{X}^{(m),r}$
for some $\mathcal{M}^{\cdot}\in
D(\mathcal{D}_{X}^{(m),\text{op}}-\text{mod})$.

3) Let $r\geq 1$. A complex $\mathcal{N}^{\cdot}\in
D(\mathcal{\widehat{D}}_{W(X)}^{(m),\text{op}}/p^{r}-\text{mod})$ is
accessible if $\mathcal{N}^{\cdot}\otimes_{W_{r}(k)}^{L}k$ is accessible in
$D(\mathcal{\widehat{D}}_{W(X)}^{(m),\text{op}}/p-\text{mod})$. Similarly, a
complex $\mathcal{N}^{\cdot}\in
D_{cc}(\mathcal{\widehat{D}}_{W(X)}^{(m),\text{op}}-\text{mod})$ is accessible
if $\mathcal{N}^{\cdot}\otimes_{W(k)}^{L}k$ is accessible in
$D(\mathcal{\widehat{D}}_{W(X)}^{(m),\text{op}}/p-\text{mod})$.

The analogues of all the basic results on accessibility (Theorem 3.18 through
3.23) all hold without any change, and with identical proofs, for right
modules.

With this in place, we turn to defining the fundamental bimodule
$\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)}$ and then describing the left-
right swap for $\mathcal{\widehat{D}}_{W(X)}^{(m)}$-modules.

First suppose we are working locally, i.e., $X=\text{Spec}(A)$ possesses local
coordinates. Then the functor

$\mathcal{N}^{\cdot}\to\Phi^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}^{L}R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}(\Phi^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)},\mathcal{N}^{\cdot})$
$\tilde{\to}\Phi^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}^{L}(\Phi^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}^{L}\mathcal{N}^{\cdot})$

is the right adjoint to the inclusion
$D_{acc}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})\to
D_{cc}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$. Applying this functor
to $\widehat{\mathcal{D}}_{W(X)}^{(m)}$ itself, we obtain
$\Phi^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}^{L}\Phi^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}$.
As $\Phi^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}$ is $p$-torsion free,
we see from 3.21 that
$\Phi^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}^{L}\Phi^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}$
is concentrated in degree $0$; i.e., it is a sheaf. By the uniqueness of
adjoints, we obtain for any pair of lifts $\Phi_{1},\Phi_{2}$, a canonical
isomorphism

(4.3)
$\Phi_{1}^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}^{L}\Phi_{1}^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\tilde{\to}\Phi_{2}^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}^{L}\Phi_{2}^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}$

of sheaves; for a third lift $\Phi_{3}$ we necessarily have the cocycle
condition. Therefore, for arbitrary $X$, the object
$\Phi^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}^{L}\Phi^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}$
glues to form a sheaf on $X$.

######  Definition 4.5.

The sheaf of bimodules constructed just above is called
$\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)}$.

As the natural map

$\Phi^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}^{L}\Phi^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\to\mathcal{\widehat{D}}_{W(X)}^{(m)}$

coming from the adjunction is compatible with the isomorphism (4.3), we obtain
a morphism of bimodules
$\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)}\to\mathcal{\widehat{D}}_{W(X)}^{(m)}$.

This, in turn, implies that the right adjoint to the inclusion
$D_{acc}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})\to
D_{cc}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$ is defined on any $X$,
namely, it is given by the functor

$\mathcal{N}^{\cdot}\to\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}^{L}\mathcal{N}^{\cdot}$

Note that

$\mathcal{N}^{\cdot}\to\mathcal{N}^{\cdot}\widehat{\otimes}_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}^{L}\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)}$

is the right adjoint to the inclusion
$D_{acc}(\text{mod}-\mathcal{\widehat{D}}_{W(X)}^{(m)})\to
D_{cc}(\text{mod}-\mathcal{\widehat{D}}_{W(X)}^{(m)})$ of the category of
right accessible $\mathcal{\widehat{D}}_{W(X)}^{(m)}$ modules (this discussion
completes the proof of 1.7, part $2$). Note the the same argument, applied to
$\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)}/p^{r}$, shows that the
inclusion $D_{acc}(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod})\to
D(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod})$ admits a right adjoint
as well; in this case the functor preserves the abelian subcategories of
accessible modules as well. We note here also that replacing everywhere
$\mathcal{\widehat{D}}_{W(X)}^{(m)}$ with
$\mathcal{\widehat{D}}_{W(X),\text{crys}}^{(m)}$ yields the analogous results
for those categories of modules.

Now, let’s give an application of the construction of $\Phi^{!}$:

######  Proposition 4.6.

For each $m\geq 0$ the sheaf $W\omega_{X}$ admits the structure of an
accessible right $\mathcal{\widehat{D}}_{W(X)}^{(m)}$-module. If
$X=\text{Spec}(A)$ then choosing a coordinatized lift of Frobenius, and the
corresponding map $\Phi:\mathcal{A}\to W(A)$, we have
$W\omega_{X}\tilde{=}\Phi^{!}(\omega_{\mathfrak{X}})$. For each $m\geq 0$ we
have $F^{!}W\omega_{X}\tilde{\to}W\omega_{X}$.

The proof of this takes a few steps. Echoing the construction of the functor
$F^{*}$ for left $\mathcal{D}$-modules, we have:

######  Corollary 4.7.

Let $\mathfrak{X}=\text{Specf}(\mathcal{A})$ be equipped with a coordinatized
lift of Frobenius $F$, and let $\Phi:\mathcal{A}\to W(A)$ be the associated
morphism. Let $\mathcal{A}^{F}$ denote $\mathcal{A}$ as an
$\mathcal{A}$-module via $F$ and let $\pi:\mathcal{A}\to F(\mathcal{A})$ be an
$F(\mathcal{A})$-linear projection.

1) Let $F^{!}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$ denote the
$(\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)},\widehat{\mathcal{D}}_{\mathcal{A}}^{(m+1)})$
bi-sub-module of $\text{Hom}_{W(k)}(\mathcal{A},\mathcal{A})$ generated by
$\pi$. Then the natural map

$\text{Hom}_{\mathcal{A}}(\mathcal{A}^{F},\mathcal{A})\otimes_{\mathcal{A}}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}\to\text{Hom}_{W(k)}(\mathcal{A},\mathcal{A})$

which takes $(\phi,P)$ to $P\circ\phi$ induces isomorphisms

$\text{Hom}_{\mathcal{A}}(\mathcal{A}^{F},\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)})\tilde{=}\text{Hom}_{\mathcal{A}}(\mathcal{A}^{F},\mathcal{A})\otimes_{\mathcal{A}}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}\tilde{\to}F^{!}\widehat{\mathcal{D}}_{\mathcal{A}}^{(m)}$

Sheafifying to $\mathfrak{X}=\text{Specf}(\mathcal{A})$, we obtain a sheaf of
bimodules $F^{!}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}$. Thus there is a
functor $\mathcal{M}\to
F^{!}\mathcal{M}:=\mathcal{M}\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}}F^{!}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)})$
which is in fact an equivalence of categories
$\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}-\text{mod}\to\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m+1)}-\text{mod}$.
When $\mathcal{M}\in\text{Coh}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)})$ we
have

$F^{!}\mathcal{M}\tilde{\to}\text{Hom}_{\mathcal{O}_{\mathfrak{X}}}(F_{*}\mathcal{O}_{\mathfrak{X}},\mathcal{M})$

2) Let $F^{!}\widehat{\mathcal{D}}_{W(A)}^{(m)}$ denote the
$(\widehat{\mathcal{D}}_{W(A)}^{(m)},\widehat{\mathcal{D}}_{W(A)}^{(m+1)})$
bi-sub-module of $\text{Hom}_{W(k)}(W(A),W(A))$ generated by the projection
$\pi:W(A)\to\mathcal{A}^{(1)}$. Sheafifying, we obtain a sheaf of bimodules
$F^{!}\widehat{\mathcal{D}}_{W(X)}^{(m)}$, and the induced functor
$\mathcal{M}\to
F^{!}\mathcal{M}:=\mathcal{M}\otimes_{\widehat{\mathcal{D}}_{W(X)}^{(m)}}\widehat{\mathcal{D}}_{W(X)}^{(m+1)}$
is an equivalence from right-accessible
$\widehat{\mathcal{D}}_{W(X)}^{(m)}$-modules to right-accessible
$\widehat{\mathcal{D}}_{W(X)}^{(m+1)}$-modules. In fact, if
$\mathcal{M}\in\text{Qcoh}(\widehat{\mathcal{D}}_{W(X)}^{(m)})$ we have
$F^{!}\Phi_{m}^{!}\mathcal{M}\tilde{\to}\Phi_{m+1}^{!}(F^{!}\mathcal{M})$
where on the left $F$ is the Witt-vector Frobenius and on the right it is the
lift of Frobenius on $\mathfrak{X}$.

The proof of this result is extremely similar to that of Theorem 3.32 and 4.1,
and will be left to the reader.

It will be necessary to also verify an additional compatibility. To state it,
we recall a few basic facts from Grothendieck duality theory (c.f. [23], as
well as [19], appendix for a useful discussion of the Cartier isomorphism in
this context). Recall that if $f:X\to Y$ is a proper morphism of locally
noetherian separated schemes then we have a functor
$f^{!}:D^{b}\text{Coh}(Y)\to D^{b}\text{Coh}(X)$ which satisfies

$R\mathcal{H}om_{\mathcal{O}_{Y}}(Rf_{*}\mathcal{N},\mathcal{M})\tilde{\to}Rf_{*}R\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{N},f^{!}\mathcal{M})$

for any $\mathcal{M}\in D^{b}\text{Coh}(Y)$ and $\mathcal{N}\in
D^{b}\text{Coh}(X)$. If $f$ is a finite morphism finite then we can in fact
define

$f^{!}(\mathcal{M}):=f^{-1}\mathcal{H}om_{\mathcal{O}_{Y}}(f_{*}\mathcal{O}_{X},\mathcal{M})$

from $\text{Coh}(Y)$ to $\text{Coh}(X)$ (the $\mathcal{O}_{X}$-module
structure is given by the action of $\mathcal{O}_{X}$ on
$f^{-1}(f_{*}\mathcal{O}_{X})$). In most cases of interest in this paper $f$
will also be a topological isomorphism, in which case we will suppress the
$f^{-1}$ in the formula.

Let $\mathfrak{X}_{r}$ be a flat lift of $X$ over $W_{r}(k)$; denote the
morphism to $W_{r}(k)$ by $p_{r}$. Then there is a canonical isomorphism
$p_{r}^{!}(W_{r}(k))\tilde{\to}\omega_{\mathfrak{X}_{r}}$. It follows that, if
$F:\mathfrak{X}_{r}\to\mathfrak{X}_{r}$ is a lift of Frobenius, we have