$\mathcal{H}om_{\mathcal{O}_{\mathfrak{X}_{r}}}(F_{*}(\mathcal{O}_{\mathfrak{X}_{r}}),\omega_{\mathfrak{X}_{r}}):=F^{!}(\omega_{\mathfrak{X}_{r}})\tilde{\to}\omega_{\mathfrak{X}_{r}}$

######  Corollary 4.8.

Let notation be as in the previous corollary.

1) Let $\mathcal{M}\in\text{mod}-\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}$
be coherent and $p$-torsion free. Then there is a natural isomorphism

$\Phi^{!}(\mathcal{M})\tilde{\to}\tilde{\mathcal{H}om}_{\mathcal{O}_{\mathfrak{X}}}(\mathcal{O}_{W(X)},\mathcal{M})$

where
$\tilde{\mathcal{H}om}_{\mathcal{O}_{\mathfrak{X}}}(\mathcal{O}_{W(X)},\mathcal{M})\subset\mathcal{H}om_{\mathcal{O}_{\mathfrak{X}}}(\mathcal{O}_{W(X)},\mathcal{M})$
consists of those morphisms which satisfy
$\epsilon(V^{m}(\mathcal{O}_{W(X)}))\subset p^{m}\mathcal{N}$ for all $m$
(over every open set in $X$). In other words,

$\Phi^{!}(\mathcal{M})\tilde{\to}\lim_{m}\mathcal{H}om_{\mathcal{O}_{\mathfrak{X}_{m}}}(\mathcal{O}_{W_{m}(X)},\mathcal{M}/p^{m}\mathcal{M})$

2) Suppose $\mathcal{N}\in\text{mod}-\widehat{\mathcal{D}}_{W(X)}^{(m)}$ is a
right-accessible coherent module, which is $p$-torsion-free. Then

$F^{!}\mathcal{N}\tilde{=}\tilde{\mathcal{H}om}_{\mathcal{O}_{W(X)}}(F_{*}(\mathcal{O}_{W(X)}),\mathcal{N})$

where
$\tilde{\mathcal{H}om}_{\mathcal{O}_{W(X)}}(F_{*}(\mathcal{O}_{W(X)}),\mathcal{N})\subset\mathcal{H}om_{\mathcal{O}_{W(X)}}(F_{*}(\mathcal{O}_{W(X)}),\mathcal{N})$
consists of morphisms $\epsilon$ satisfying
$\epsilon(V^{l}(\mathcal{O}_{W(X)}))\subset p^{l}\mathcal{N}$ for all $m$
(over every open set in $X$). In other words,

$F^{!}\mathcal{N}\tilde{=}\lim_{l}\mathcal{H}om_{\mathcal{O}_{W_{l}(X)}}(F_{*}(\mathcal{O}_{W_{l}(X)}),\mathcal{N}/p^{l})=\lim_{l}F^{!}(\mathcal{N}/p^{l}\mathcal{N})$

###### Proof.

$1)$ There is a natural map

$\mathcal{M}\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}}\tilde{\mathcal{H}om}_{\mathcal{O}_{\mathfrak{X}}}(\mathcal{O}_{W(X)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)})\to\tilde{\mathcal{H}om}_{\mathcal{O}_{\mathfrak{X}}}(\mathcal{O}_{W(X)},\mathcal{M})$

taking $(m,\phi)$ to the morphism $f\to m\cdot\phi(f)$. As everything in sight
is $p$-torsion-free and $p$-adically complete, it suffices to show that the
reduction mod $p$

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

is an isomorphism. Further, we have

$\tilde{\mathcal{H}om}_{\mathcal{O}_{\mathfrak{X}}}(\mathcal{O}_{W(X)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)})/p=\prod_{(I,r)}p^{r}T^{I}\cdot\mathcal{D}_{X}^{(m)}$

is an infinite product of copies of $\mathcal{D}_{X}^{(m)}$, and similarly
$\tilde{\mathcal{H}om}_{\mathcal{O}_{\mathfrak{X}}}(\mathcal{O}_{W(X)},\mathcal{M})/p$
is an infinite product of copies of $\mathcal{M}/p$. Since $\mathcal{M}$ is
finitely generated and $\mathcal{D}_{X}^{(m)}$ is noetherian, the result
follows from this.

$2)$ Write $\mathcal{N}=\Phi_{m}^{!}\mathcal{M}$ for a coherent, $p$-torsion-
free $\mathcal{M}$. Then by the previous result we have

$F^{!}\Phi_{m}^{!}\mathcal{M}\tilde{\to}\Phi_{m+1}^{!}F^{!}\mathcal{M}$

as $\mathcal{M}$ is $p$-torsion-free and complete we have

$\Phi_{m+1}^{!}F^{!}\mathcal{M}=\lim_{l}\mathcal{H}om_{\mathcal{O}_{\mathfrak{X}_{l}}}(\mathcal{O}_{W_{l}(X)},\mathcal{H}om_{\mathcal{O}_{\mathcal{\mathfrak{X}}_{l}}}(F_{*}\mathcal{O}_{\mathfrak{X}_{l}},\mathcal{M}/p^{l}\mathcal{M}))$

using part $1)$ to evaluate $\Phi_{m+1}^{!}$. We wish to prove that

$F^{!}\mathcal{N}\tilde{=}\lim_{l}\mathcal{H}om_{\mathcal{O}_{W_{l}(X)}}(F_{*}(\mathcal{O}_{W_{l}(X)}),\mathcal{N}/p^{l})$

But again by part $1)$ we have

$\mathcal{H}om_{\mathcal{O}_{W_{l}(X)}}(F_{*}(\mathcal{O}_{W_{l}(X)}),\mathcal{N}/p^{l})\tilde{=}\mathcal{H}om_{\mathcal{O}_{W_{l}(X)}}(F_{*}(\mathcal{O}_{W_{l}(X)}),\mathcal{H}om_{\mathcal{O}_{\mathfrak{X}_{l}}}(\mathcal{O}_{W_{l}(X)},\mathcal{M}/p^{l}\mathcal{M}))$

So the result follows from the fact that $\Phi\circ F=F\circ\Phi$ and the
composition of morphisms in Grothendieck duality. ∎

Let us continue with $X=\text{Spec}(A)$ as above. Recalling that
$\omega_{\mathfrak{X}}$ has the structure of a right
$\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}$ module for every $m\geq 0$, and,
the above isomorphism in Grothendieck duality theory yields
$F^{!}\omega_{\mathfrak{X}}\tilde{=}\omega_{\mathfrak{X}}$ as
$\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m+1)}$ modules. Thus
$\omega_{\mathfrak{X}}$ acquires the structure of a
$\mathcal{\widehat{D}}_{\mathfrak{X}}:=\lim\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}$-module.
We have

######  Lemma 4.9.

For $X=\text{Spec}(A)$ with Frobenius lift $\Phi$ we have that there exists,
for each $m\geq 0$ a map of $\mathcal{\widehat{D}}_{W(X)}^{(m)}$-modules
$\eta_{m}:\Phi^{!}\omega_{\mathfrak{X}}\tilde{\to}W\omega_{X}$; the sequence
of maps $\eta_{m}$ are compatible with $F^{!}$ in the sense that
$\eta_{m+1}=F^{!}\eta_{m}$. Therefore $W\omega_{X}$ acquires the structure of
a right $\mathcal{\widehat{D}}_{W(X)}$-module and there is an isomorphism
$\eta_{\infty}:\Phi^{!}\omega_{\mathfrak{X}}\tilde{\to}W\omega_{X}$ satisfying
$\eta_{\infty}F^{!}=F^{!}\eta_{\infty}$.

###### Proof.

According to Illusie ([26], c.f. also [16], section 1.9) there is, for each
$l\geq 0$, a canonical isomorphism

$p_{l}^{!}(W_{l}(k))\tilde{\to}W_{l}\omega_{X}$

where $p_{l}$ maps $W_{l}(X)$ to a point. And therefore a canonical
isomorphism

$\Phi^{!}p_{l}^{!}(W_{l}(k))\tilde{\to}\Phi^{!}\omega_{\mathfrak{X}_{l}}\tilde{\to}W_{l}\omega_{X}$

taking the inverse limit over $l$ we obtain the required isomorphism at the
level of coherent sheaves. Endowing $\omega_{\mathfrak{X}}$ with its right
$\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}$-module structure as above gives
$W\omega_{X}$ the structure of a right
$\mathcal{\widehat{D}}_{W(X)}^{(m)}$-module and using 4.8 shows the
compatibility with $F^{!}$ as required. ∎

Now we can proceed to the

###### Proof.

(of 4.6) Without loss of generality we may suppose $X$ is geometrically
connected. Choose some $U\subset X$ which is open affine, such that
$U=\text{Spec}(A)$ possesses local coordinates; choose a coordinatized lift of
Frobenius on $\mathfrak{U}$ as above. Then the above yields an isomorphism
$\epsilon_{\mathfrak{U}}:\Phi^{!}\omega_{\mathfrak{U}}\tilde{\to}W\omega_{U}$
which respects the Frobenius structure and the structure of a
$\mathcal{\widehat{D}}_{W(X)}$-module. Choose and fix one such isomorphism.

Now let $V\subset U$ be another such open affine; pick a Frobenius lift $\Psi$
on $\mathfrak{V}\subset\mathfrak{U}$. Then we have
$\epsilon_{\mathfrak{V}}:\Psi^{!}\omega_{\mathfrak{V}}\tilde{\to}W\omega_{V}$,
and so we obtain the isomorphism
$\epsilon_{\mathfrak{U}}\circ\epsilon_{\mathfrak{V}}^{-1}$ on $W\omega_{V}$.
This isomorphism commutes with $F^{!}$. But it is easy to see that the set of
such maps is simply $\mathbb{Z}_{p}^{\times}$; in particular the isomorphism
$\Phi^{!}\omega_{\mathfrak{U}}\tilde{\to}\Psi^{!}\omega_{\mathfrak{V}}$ must
respect the $\widehat{\mathcal{D}}_{W(X)}$-module structure. Furthermore, we
may rescale the map $\epsilon_{\mathfrak{V}}$ and obtain
$\epsilon_{\mathfrak{U}}\circ\epsilon_{\mathfrak{V}}^{-1}=1$ on
$\omega_{dRW(V)}^{n}$. Since $X$ is geometrically connected any open affine
$W\subset X$ intersects $U$ nontrivially and the result follows directly. ∎

Now we are ready to discuss the left-right interchange. Given the right
$\mathcal{\widehat{D}}_{W(X)}^{(m)}$-module structure on $W\omega_{X}$, and
the left $\mathcal{\widehat{D}}_{W(X)}^{(m)}$-module structure on
$\mathcal{O}_{W(X)}$, we see that
$\mathcal{H}om_{W(k)}(\mathcal{O}_{W(X)},W\omega_{X})$ acquires the structure
of a
$(\mathcal{\widehat{D}}_{W(X)}^{(m),\text{opp}},\mathcal{\widehat{D}}_{W(X)}^{(m),\text{opp}})$-bimodule,
while $\mathcal{H}om_{W(k)}(W\omega_{X},\mathcal{O}_{W(X)})$ acquires the
structure of a
$(\mathcal{\widehat{D}}_{W(X)}^{(m)},\mathcal{\widehat{D}}_{W(X)}^{(m)})$-bimodule.

######  Proposition 4.10.

1) Consider the sheaf
$W\omega_{X}\widehat{\otimes}_{\mathcal{O}_{W(X)}}\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)}$,
which is the completion of
$W\omega_{X}\otimes_{\mathcal{O}_{W(X)}}\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)}$
along the filtration999Here, $G^{j}$ denotes the standard filtration on
$W\omega_{X}$ coming from de Rham-Witt theory
$\\{V^{i}(\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)})\otimes
G^{j}(W\omega_{X})\\}_{i+j\geq r}$. There is a natural injective map

$W\omega_{X}\widehat{\otimes}_{\mathcal{O}_{W(X)}}\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)}\to\mathcal{H}om_{W(k)}(\mathcal{O}_{W(X)},W\omega_{X})$

the image of which is a
$(\mathcal{\widehat{D}}_{W(X)}^{(m),\text{opp}},\mathcal{\widehat{D}}_{W(X)}^{(m),\text{opp}})$-bi-
sub-module; therefore
$W\omega_{X}\widehat{\otimes}_{\mathcal{O}_{W(X)}}\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)}$
is itself a
$(\mathcal{\widehat{D}}_{W(X)}^{(m),\text{opp}},\mathcal{\widehat{D}}_{W(X)}^{(m),\text{opp}})$-bimodule.

2) Consider the sheaf
$\tilde{\mathcal{H}om}_{\mathcal{O}_{W(X)}}(W\omega_{X},\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)})$
where $\tilde{\mathcal{H}om}$ denotes those morphisms which take
$G^{i}(W\omega_{X})$ to
$V^{i}(\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)})$. There is a natural
injective map

$\tilde{\mathcal{H}om}_{\mathcal{O}_{W(X)}}(W\omega_{X},\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)})\to\mathcal{H}om_{W(k)}(W\omega_{X},\mathcal{O}_{W(X)})$

the image of which is a
$(\mathcal{\widehat{D}}_{W(X)}^{(m)},\mathcal{\widehat{D}}_{W(X)}^{(m)})$-bi-
sub-module; therefore
$\tilde{\mathcal{H}om}_{\mathcal{O}_{W(X)}}(W\omega_{X},\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)})$
is itself a
$(\mathcal{\widehat{D}}_{W(X)}^{(m)},\mathcal{\widehat{D}}_{W(X)}^{(m)})$-bimodule.

###### Proof.

$1)$ The map is given as the completion of the map

$W\omega_{X}\otimes_{\mathcal{O}_{W(X)}}\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)}\to\mathcal{H}om_{W(k)}(\mathcal{O}_{W(X)},W\omega_{X})$

which takes $\delta\otimes P\to\delta\cdot P$ (where
$P\in\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)}$ is regarded as an
endomorphism of $\mathcal{O}_{W(X)}$). To verify the properties of this map,
we work locally. Assume $X=\text{Spec}(A)$, $\Phi:\mathcal{A}\to W(A)$ and
$\pi$ are given. Then as
$\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)}=\Phi^{*}\mathcal{D}_{\mathcal{A}}^{(m)}\widehat{\otimes}_{\mathcal{D}_{\mathcal{A}}^{(m)}}\Phi^{!}\mathcal{D}_{\mathcal{A}}^{(m)}$
we start by considering
$W\omega_{A}\widehat{\otimes}_{W(A)}\Phi^{*}\mathcal{D}_{\mathcal{A}}^{(m)}$,
the completion of
$W\omega_{A}\otimes_{W(A)}\Phi^{*}\mathcal{D}_{\mathcal{A}}^{(m)}$ along
$\\{V^{i}(W\omega_{A})\otimes_{W(A)}V^{j}(\Phi^{*}\mathcal{D}_{\mathcal{A}}^{(m)})\\}_{i+j\geq
r}$. . We have

$W\omega_{A}\widehat{\otimes}_{W(A)}\Phi^{*}\mathcal{D}_{\mathcal{A}}^{(m)}\tilde{=}W\omega_{A}\widehat{\otimes}_{\mathcal{A}}\mathcal{D}_{\mathcal{A}}^{(m)}$

where on the right the completion is along
$\\{V^{i}(W\omega_{A})\otimes_{\mathcal{A}}p^{j}\mathcal{D}_{\mathcal{A}}^{(m)})\\}_{i+j\geq
r}$. We then have

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

where the last map is given by the completion of $\delta\otimes
Q\to\delta\cdot Q$. From the fact that $W\omega_{A}$ is an inverse limit of
free $\mathcal{A}$-modules, this is clearly injective, and we have that its
image is closed under the right action of $\mathcal{\widehat{D}}_{W(A)}^{(m)}$
(the proof of this is extremely similar to 3.2). As
$\Phi^{!}\mathcal{D}_{\mathcal{A}}^{(m)}\subset\text{$\tilde{Hom}$}_{W(k)}(W(A),\mathcal{A})$
is preserved under the right action of $\mathcal{\widehat{D}}_{W(A)}^{(m)}$,
the map

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

defined by composition has image closed under the action of
$(\mathcal{\widehat{D}}_{W(A)}^{(m),\text{opp}},\mathcal{\widehat{D}}_{W(A)}^{(m),\text{opp}})$.

$2)$ This is extremely similar to $1)$, using instead that

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

where the last map is given by noting

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

and then embedding that module into
$\text{$\tilde{Hom}$}_{W(k)}(W\omega_{A},\mathcal{A})$ by composing maps. As
above the image is closed under the left action of
$\mathcal{\widehat{D}}_{W(A)}^{(m)}$, and the result follows. ∎

From this we deduce

######  Corollary 4.11.

The functors

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

and

$\otimes_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}^{L}\tilde{\mathcal{H}om}_{\mathcal{O}_{W(X)}}(W\omega_{X},\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)})$

give inverse equivalences of categories from left accessible to right
accessible $\mathcal{\widehat{D}}_{W(X)}^{(m)}$-modules.

###### Proof.

First let us examine the local situation.

We have

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

where, in the first line,
$W\omega_{A}\widehat{\otimes}_{\mathcal{A}}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$
is the completion of
$W\omega_{A}\otimes_{\mathcal{A}}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$
along $\\{V^{i}(W\omega_{A})\otimes
p^{j}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}\\}$; the second isomorphism is
from
$W\omega_{A}\tilde{=}\tilde{\text{Hom}}_{\mathcal{A}}(W(A),\omega_{\mathcal{A}})$,
and the third and fourth are from the fact that $\omega_{\mathcal{A}}$ is
locally free over $\mathcal{A}$.

It follows that

$(W\omega_{A}\widehat{\otimes}_{W(A)}\mathcal{\widehat{D}}_{W(A),\text{acc}}^{(m)})\otimes_{\mathcal{\widehat{D}}_{W(A)}^{(m)}}\Phi^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}\tilde{\to}W\omega_{A}\widehat{\otimes}_{W(A)}\Phi^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}$

and therefore

$(W\omega_{A}\widehat{\otimes}_{W(A)}\mathcal{\widehat{D}}_{W(A),\text{acc}}^{(m)})\otimes_{\mathcal{\widehat{D}}_{W(A)}^{(m)}}^{L}\Phi^{*}\mathcal{M}^{\cdot}\tilde{=}(W\omega_{A}\widehat{\otimes}_{W(A)}\Phi^{*}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)})\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}}^{L}\mathcal{M}^{\cdot}$
$(\omega_{\mathcal{A}}\otimes_{\mathcal{A}}\Phi^{!}\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)})\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathcal{A}}^{(m)}}^{L}\mathcal{M}^{\cdot}=\Phi^{!}(\omega_{\mathcal{A}}\otimes_{\mathcal{A}}\mathcal{M}^{\cdot})$

so this functor agrees with the usual left-right interchange over
$\mathcal{A}$. Now, the functor
$(W\omega_{X}\widehat{\otimes}_{\mathcal{O}_{W(X)}}\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)})\otimes_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}$
admits an adjoint; namely
$R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m),\text{op}}}((W\omega_{X}\widehat{\otimes}_{\mathcal{O}_{W(X)}}\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)}))$,
and an extremely similar argument shows that it also preserves the
subcategories of accessible modules; as it must be the inverse locally, we see
that it is so globally as well.

The proof for the functor
$\otimes_{\mathcal{\widehat{D}}_{W(A)}^{(m)}}^{L}\tilde{\mathcal{H}om}_{\mathcal{O}_{W(X)}}(W\omega_{X},\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)})$
is essentially identical. ∎

###  4.2. Operations on modules: Pull-Back

Throughout this section let $m\geq 0$. Let
$\varphi:\mathfrak{X}\to\mathfrak{Y}$ be a morphism of smooth formal schemes
over $W(k)$. Recall that Berthelot in [5], section 3.2, has shown that
$\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m)}:=\varphi^{*}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)})$
carries the structure of a left
$\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}$-module (by $\varphi^{*}$ we mean
the $p$-adically completed pullback). By definition
$\varphi^{*}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)}$ carries the structure
of a right $\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)})$-module.
This, in turn allows one to define the functor $\varphi^{*}$ via

$L\varphi^{*}(\mathcal{M}):=\varphi^{*}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(j)}\widehat{\otimes}_{\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}})}^{L}\varphi^{-1}(\mathcal{M})$

as usual, the completion means the cohomological completion. One sets
$\varphi^{!}:=L\varphi^{*}[d_{X/Y}]$ (where
$d_{X/Y}=\text{dim}(X)-\text{dim}(Y)$).

Now, if $\varphi:X\to Y$, we wish to define the analogous pullback functor
$L\varphi^{*}$ from accessible $\mathcal{\widehat{D}}_{W(Y)}^{(m)}$-modules to
accessible $\mathcal{\widehat{D}}_{W(X)}^{(m)}$-modules. Recall that, by the
functoriality of the Witt vectors, $\varphi$ give rise to a morphism of ringed
spaces $W\varphi:(X,\mathcal{O}_{WX)})\to(Y,\mathcal{O}_{W(Y)})$. Thus we have
a map $W\varphi^{\\#}:W\varphi^{-1}(\mathcal{O}_{W(Y)})\to\mathcal{O}_{W(X)}$;
there is also an induced map
$W\varphi^{\\#}:W\varphi^{-1}(\mathcal{O}_{W(Y)}/p^{r})\to\mathcal{O}_{W(X)}/p^{r}$.
We shall construct a suitable sheaf of
$(\mathcal{\widehat{D}}_{W(X)}^{(m)},W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)}))$
bimodules; before doing so, let us remark on a slight generalization of the
notion of accessibility.

Starting in characteristic $p$, we note that there is an exact, conservative
functor

$\mathcal{Q}\to\mathcal{B}_{X}^{(m)}\otimes_{\mathcal{D}_{X}^{(m)}}\mathcal{Q}\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(m)})}\varphi^{-1}(\mathcal{B}_{Y}^{(m),r})$

from the category of
$(\mathcal{D}_{X}^{(m)},\varphi^{-1}(\mathcal{D}_{Y}^{(m)}))$-bimodules to the
category of
$(\mathcal{D}_{W(X)}^{(m)}/p,W\varphi^{-1}(\mathcal{D}_{W(Y)}^{(m)})/p)$-bimodules,
with right adjoint given by

$\mathcal{P}\to\mathcal{B}_{X}^{(m),r}\otimes_{\mathcal{D}_{W(X)}^{(m)}/p}\mathcal{P}\otimes_{W\varphi^{-1}(\mathcal{D}_{W(Y)}^{(m)})/p)}W\varphi^{-1}(\mathcal{B}_{Y}^{(m)})$

We see (exactly as in 3.16) that there is a full embedding of categories

$D((\mathcal{D}_{X}^{(m)},\varphi^{-1}(\mathcal{D}_{Y}^{(m)}))-\text{bimod})\to
D((\mathcal{D}_{W(X)}^{(m)}/p,W\varphi^{-1}(\mathcal{D}_{W(Y)}^{(m)})/p)-\text{bimod})$

whose image we call the accessible
$(\mathcal{D}_{W(X)}^{(m)}/p,W\varphi^{-1}(\mathcal{D}_{W(Y)}^{(m)})/p)$-bimodules.

######  Definition 4.12.

A complex $\mathcal{P}^{\cdot}\in
D_{cc}((\mathcal{\widehat{D}}_{W(X)}^{(m)},W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)}))-\text{bimod})$
is said to be accessible if $\mathcal{P}^{\cdot}\otimes_{W(k)}^{L}k$ is
accessible as a complex of
$(\mathcal{D}_{W(X)}^{(m)}/p,W\varphi^{-1}(\mathcal{D}_{W(Y)}^{(m)})/p)$-bimodules.

Exactly as in Theorem 3.18, we have that $\mathcal{P}^{\cdot}$ is accessible
iff, locally, it can be written as

$(\Phi_{1}^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)})\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}^{L}\mathcal{Q}^{\cdot}\widehat{\otimes}_{\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)})}^{L}\varphi^{-1}(\Phi_{2}^{!}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)})$

where $\Phi_{1}$ and $\Phi_{2}$ are coordinatized lifts of Frobenius.
Furthermore, we have the natural functor

$\mathcal{P}^{\cdot}\to\mathcal{P}_{\text{acc}}^{\cdot}:=\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}^{L}\mathcal{P}^{\cdot}\widehat{\otimes}_{W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})}^{L}W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})$

which takes any complex of bimodules to an accessible complex, and which is
the identity on accessible complexes.

######  Definition 4.13.

For any morphism $\varphi:X\to Y$, we have the sheaf of
$(\mathcal{\widehat{D}}_{W(X)}^{(m)},W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)}))$-bimodules
$\widehat{\mathcal{D}}_{W(X)\to W(Y)}^{(m)}$ constructed in 2.33 Then we
define

$\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}:=\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}^{L}\widehat{\mathcal{D}}_{W(X)\to
W(Y)}^{(m)}\widehat{\otimes}_{W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})}^{L}W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y),\text{acc}}^{(m)})$

a sheaf of accessible
$(\mathcal{\widehat{D}}_{W(X)}^{(m)},W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)}))$-bimodules
on $X$.

By the local description of $\widehat{\mathcal{D}}_{W(X),\text{acc}}^{(m)}$
(given right above 4.5) we see that when $\varphi$ is the identity map we have
$\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}=\widehat{\mathcal{D}}_{W(X),\text{acc}}^{(m)}$

Then we have the

######  Definition 4.14.

Let $\mathcal{M}^{\cdot}\in
D_{\text{acc}}(\mathcal{\widehat{D}}_{W(Y)}^{(m)}-\text{mod})$. We define

$L(W\varphi)^{*}(\mathcal{M}^{\cdot}):=\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}\widehat{\otimes}_{W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})}^{L}W\varphi^{-1}(\mathcal{M}^{\cdot})\in
D(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$

Similarly we define, for any $\mathcal{M}^{\cdot}\in
D(\mathcal{\widehat{D}}_{W(Y)}^{(m)}/p^{r}-\text{mod})$,

$L(W\varphi)^{*}(\mathcal{M}^{\cdot}):=\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}/p^{r}\widehat{\otimes}_{W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)}/p^{r})}^{L}W\varphi^{-1}(\mathcal{M}^{\cdot})\in
D(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod})$

In order to get control over this definition, we need to compare with the
usual pullback functor in the presence of local lifts. Suppose for a moment
that $X=\text{Spec}(A)$ and $Y=\text{Spec}(B)$ are affine, each possessing
local coordinates, as well as coordinatized Frobenius lifts; let
$\Phi_{1}:\mathcal{O}_{\mathfrak{X}}\to\mathcal{O}_{W(X)}$ and
$\Phi_{2}:\mathcal{O}_{\mathfrak{Y}}\to\mathcal{O}_{W(Y)}$ be the associated
morphism, with projections $\pi_{1}$ and $\pi_{2}$, respectively. Suppose we
have a lift $\varphi:\mathfrak{X}\to\mathfrak{Y}$.

######  Proposition 4.15.

Let
$\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}$
denote the $p$-adic completion of the
$(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)},\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)}))$
bimodule locally generated by $\pi_{1}\circ
W\varphi^{\\#}\circ\varphi^{-1}(\Phi_{2}):\varphi^{-1}(\mathcal{O}_{\mathfrak{Y}})\to\mathcal{O}_{\mathfrak{X}}$.
We have

$\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}\tilde{=}\Phi_{1}^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}^{L}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}\widehat{\otimes}_{\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)})}^{L}\varphi^{-1}(\Phi_{2}^{!}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)})$

i.e., the derived completed tensor product is concentrated in degree $0$(and
is equal to the usual $p$-adic completion of the tensor product). In
particular, if $\varphi\circ\Phi_{1}=\Phi_{2}\circ\varphi$ then we obtain

$\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}\tilde{=}\Phi_{1}^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}^{L}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m)}\widehat{\otimes}_{\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)})}^{L}\varphi^{-1}(\Phi_{2}^{!}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)})$

###### Proof.

The first statement is equivalent to

$\Phi_{1}^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\otimes_{\widehat{\mathcal{D}}_{W(X)}^{(m)}}\widehat{\mathcal{D}}_{W(X)\to
W(Y)}^{(m)}\otimes_{W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})}W\varphi^{-1}(\Phi_{2}^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)})\tilde{=}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}$

The composition of morphisms produces a map

(4.4)
$\Phi_{1}^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\otimes_{\widehat{\mathcal{D}}_{W(X)}^{(m)}}\widehat{\mathcal{D}}_{W(X)\to
W(Y)}^{(m)}\otimes_{W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})}W\varphi^{-1}(\Phi_{2}^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)})\to\mathcal{H}om_{W(k)}(\varphi^{-1}(\mathcal{O}_{\mathfrak{Y}}),\mathcal{O}_{\mathfrak{X}})$

As the left hand side is the summand of $\widehat{\mathcal{D}}_{W(X)\to
W(Y)}^{(m)}$ consisting of maps whose image is contained
$\pi_{1}(\mathcal{O}_{W(X)})$, and which vanish on the complement of
$\varphi^{-1}(\Phi_{2}(\mathcal{O}_{\mathfrak{Y}}))$ in $\mathcal{O}_{W(Y)}$,
this map is clearly injective. We shall show that the image is equal to
$\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}$.

By 2.31,
$\Phi_{1}^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\otimes_{\widehat{\mathcal{D}}_{W(X)}^{(m)}}\widehat{\mathcal{D}}_{W(X)\to
W(Y)}^{(m)}\otimes_{W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})}W\varphi^{-1}(\Phi_{2}^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)})$
is the sheaf whose sections over $\mathfrak{X}$ are of the form

(4.5)
$\sum_{I}(\sum_{r=0}^{\infty}\sum_{J_{I}}\pi_{1}F^{-r}(\alpha_{J_{I}})\cdot
W\varphi^{\\#}(\\{\partial\\}_{J_{I}/p^{r}})\\{\partial\\}^{I}\pi_{2}$

where the notation is as in 2.31.

Now, write

$\\{\partial\\}_{J_{I}/p^{r}}\\{\partial\\}^{I}\pi_{2}=\sum_{(K,s)}p^{s}T^{K/p^{s}}\Phi_{2}(b_{K})$

for $b_{K}\in\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)}(\mathfrak{Y})$, where
$s<r$ implies $b_{K}\in
p^{r-s}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)}(\mathfrak{Y})$.

Then

$\pi_{1}F^{-r}(\alpha_{J_{I}})\cdot
W\varphi^{\\#}(\\{\partial\\}_{J_{I}/p^{r}})\\{\partial\\}^{I}\pi_{2}=\sum_{(K,s)}\pi_{1}\circ
F^{-r}(\alpha_{J_{I}})\cdot W\varphi^{\\#}\circ
p^{s}T^{K/p^{s}}\cdot\Phi_{2}(b_{K})$

As $p^{r}T^{J/p^{r}}=V^{r}(T^{J})$ and $W\varphi^{\\#}$ commutes with $V$, we
see that the above sum is equal to

$\sum_{(K,s)}\pi_{1}F^{-r}(\alpha_{J_{I}})\cdot(V^{r}(W\varphi^{\\#}(T^{J})))\circ(W\varphi^{\\#}\circ\Phi_{2})(b_{K})$

Let $\\{X_{1},\dots X_{n}$ be local coordinates on $\mathcal{A}$. Then we can
write

$(V^{r}(W\varphi^{\\#}(T^{J})))\circ(W\varphi^{\\#}\circ\Phi_{2})(b_{J})=\sum_{(L,m)}p^{m}X^{L/m}\cdot(W\varphi^{\\#}\circ\Phi_{2})(b_{L})$

and $m<r$ implies $b_{L}\in
p^{r-m}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)}(\mathfrak{Y})$. Now, we have

$\sum_{(L,m)}\pi_{1}F^{-r}(\alpha_{J_{I}})\cdot
p^{m}X^{L/m}\cdot(W\varphi^{\\#}\circ\Phi_{2})(b_{L})=\sum_{(L,m)}c_{L}\pi_{1}(W\varphi^{\\#}\circ\Phi_{2})(b_{L})$

where $c_{L}\in p^{M}\mathcal{A}$ with $M=\text{max}\\{m,r\\}$, we see that
this sum is $p$-adically convergent, and therefore

$\pi_{1}F^{-r}(\alpha_{J_{I}})\cdot
W\varphi^{\\#}(\\{\partial\\}_{J_{I}/p^{r}})\\{\partial\\}^{I}\pi_{2}=\sum_{(L,m),(J,r)}c_{L}\cdot\pi_{1}\circ(W\varphi^{\\#}\circ\Phi_{2})(b_{J})$

which is contained in the $p$-adic completion of the
$(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}(\mathfrak{X}),\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)}(\mathfrak{Y}))$
bimodule generated by $\pi_{1}\circ W\varphi^{\\#}\circ\Phi_{2}$; from the
above conditions on $a_{I},b_{J}$ we also see that
$\pi_{1}F^{-r}(\alpha_{J_{I}})\cdot
W\varphi^{\\#}(\\{\partial\\}_{J_{I}/p^{r}})\\{\partial\\}^{I}\pi_{2}$ is
contained in
$p^{R}\cdot\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}$
where $R=\text{max}\\{r,v\\}$ where $v$ is the least natural number such that
$\alpha_{J_{I}}\in V^{v}(W(A))$.

Therefore the image of the sum 4.5 is contained in
$\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}$
as well. Thus the image of the map 4.4 is contained in
$\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}$.
It is surjective, as the image of (4.4) is already a
$(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)},\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)}))$
bi-submodule containing $\pi_{1}\circ
W\varphi^{\\#}\circ\varphi^{-1}(\Phi_{2})$; and so we have the desired result.

Finally, if $\varphi\circ\Phi_{1}=\Phi_{2}\circ\varphi$ we have $\pi_{1}\circ
W\varphi^{\\#}\circ\varphi^{-1}(\Phi_{2})=\varphi^{\\#}$ and so the last
sentence of the proposition follows. ∎

Now, for some $r\geq 1$ (including $r=\infty$) let us suppose we are given a
lift of $\varphi$ to $\varphi:\mathfrak{X}_{r}\to\mathfrak{Y}_{r}$. We say
that this map $\varphi$ is _locally compatible with a Frobenius lift_ if,
locally on $\mathfrak{X}_{r}$ and $\mathfrak{Y}_{r}$, we can find lifts
$\mathfrak{X}$ and $\mathfrak{Y}$ of $\mathfrak{X}_{r}$ and $\mathfrak{Y}_{r}$
and a lift $\varphi:\mathfrak{X}\to\mathfrak{Y}$, which commutes with some
coordinatized lifts of Frobenius $\Phi_{1}$ on $\mathfrak{X}$ and $\Phi_{2}$
on $\mathfrak{Y}$. Then

######  Corollary 4.16.

Let $\varphi:\mathfrak{X}_{r}\to\mathfrak{Y}_{r}$ be locally compatible with a
Frobenius lift (when $m=0$ and $p=2$, we suppose $r=1$). Let
$\mathcal{N}^{\cdot}\in D(\text{Mod}(\mathcal{D}_{\mathfrak{Y}_{r}}))$. Then
there is an isomorphism

$LW\varphi^{*}(\mathcal{B}_{\mathfrak{Y}_{r}}^{(m)}\otimes_{\mathcal{D}_{\mathfrak{Y}_{r}}^{(m)}}^{L}\mathcal{N}^{\cdot})\tilde{\to}\mathcal{B}_{\mathfrak{X}_{r}}\otimes_{\mathcal{D}_{\mathfrak{X}_{r}}^{(m)}}^{L}L\varphi^{*}(\mathcal{N}^{\cdot})$

where on the right hand side the pullback is the pullback in the category of
$\mathcal{D}_{\mathfrak{X}_{r}}^{(m)}$-modules. The analogous statement holds
when $r=\infty$, i.e., for $\varphi:\mathfrak{X}\to\mathfrak{Y}$.

###### Proof.

We have the isomorphism

$\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}\tilde{=}\Phi_{1}^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}^{L}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m)}\widehat{\otimes}_{\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)})}^{L}\varphi^{-1}(\Phi_{2}^{!}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)})$

for $\Phi\circ\varphi=\Phi\circ\varphi$ (c.f. 4.15); taking reduction mod
$p^{r}$ this proves the result locally on $\mathfrak{X}_{r}$ and
$\mathfrak{Y}_{r}$. If $\Phi_{1}^{\prime},\Phi^{\prime}_{2}$ are different
local lifts, by 3.7 the isomorphism
$\Phi_{2}^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)}\tilde{\to}(\Phi_{2}^{\prime})^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)}$
is compatible with reduction mod $p^{r}$ (and similarly for $\Phi_{1}$), and
the result follows; when $p=2$ and $n=1$ we appeal to 5.15 for the required
gluing statement. ∎

To make better use of this statement, we note the:

######  Lemma 4.17.

For $\mathcal{M}^{\cdot}\in D(\mathcal{\widehat{D}}_{W(Y)}^{(m)}-\text{mod})$
we have

$L(W\varphi)^{*}(\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k\tilde{\to}L(W\varphi)^{*}(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)$

where the functor on the right denotes the pullback in the category of
$\mathcal{\widehat{D}}_{W(Y)}^{(m)}/p$ modules. Similarly, if
$\mathcal{M}^{\cdot}\in
D(\mathcal{\widehat{D}}_{W(Y)}^{(m)}/p^{n}-\text{mod})$ for some $n\geq 1$,
then

$L(W\varphi)^{*}(\mathcal{M}^{\cdot})\otimes_{W_{n}(k)}^{L}k\tilde{\to}L(W\varphi)^{*}(\mathcal{M}^{\cdot}\otimes_{W_{n}(k)}^{L}k)$

###### Proof.

As $\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})$ and
$\widehat{\mathcal{D}}_{W(X)\to W(Y),\text{acc}}^{(m)}$ are flat over $W(k)$
we have

$(\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}\widehat{\otimes}_{\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})}^{L}\varphi^{-1}(\mathcal{M}^{\cdot}))\otimes_{W(k)}^{L}k$
$\tilde{\to}(\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}/p)\widehat{\otimes}_{\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})/p}^{L}(\varphi^{-1}(\mathcal{M}^{\cdot})\widehat{\otimes}_{W(k)}^{L}k)$

which implies the first result. The statement about $\mathcal{M}^{\cdot}\in
D(\mathcal{\widehat{D}}_{W(Y)}^{(m)}/p^{n}-\text{mod})$ is similar. ∎

Combining these two results yields immediately

######  Proposition 4.18.

Let $\varphi:X\to Y$ be locally compatible with a lift of Frobenius. Then
$L(W\varphi)^{*}$ takes
$D_{\text{acc}}(\mathcal{\widehat{D}}_{W(Y)}^{(m)}-\text{mod})$ to
$D_{\text{acc}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$. The same
holds mod $p^{r}$, i.e., for
$D_{\text{acc}}(\mathcal{\widehat{D}}_{W(Y)}^{(m)}/p^{r}-\text{mod})$.

To generalize this to all morphisms, we need some information about
compositions:

######  Lemma 4.19.

Let $\varphi:X\to Y$ and $\psi:Y\to Z$. There is a natural map

$\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}\otimes_{W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)})}W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)\to
W(Z),\text{acc}}^{(m)})\to\widehat{\mathcal{D}}_{W(X)\to
W(Z),\text{acc}}^{(m)}$

of
$(\widehat{\mathcal{D}}_{W(X)}^{(m)},W(\psi\circ\varphi)^{-1}(\widehat{\mathcal{D}}_{W(X)}^{(m)}))$-bimodules.

###### Proof.

The composition of morphisms yields a map

$\widehat{\mathcal{D}}_{W(X)\to
W(Y)}^{(m)}\otimes_{\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)})}\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)\to
W(Z)}^{(m)})\to\mathcal{H}om_{W(k)}((\varphi\circ\psi)^{-1}\mathcal{O}_{W(Z)},\mathcal{O}_{W(X)})$

and one sees directly- working locally and use the commutation relations of
Lemma 2.24-that the image of this map is contained in
$\widehat{\mathcal{D}}_{W(X)\to W(Z)}^{(m)}$. Now, by definition we have

$\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}=\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}^{L}\widehat{\mathcal{D}}_{W(X)\to
W(Y)}^{(m)}\widehat{\otimes}_{W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})}^{L}W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y),\text{acc}}^{(m)})$

and so the natural maps
$\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)}\to\mathcal{\widehat{D}}_{W(X)}^{(m)}$
(and
$\mathcal{\widehat{D}}_{W(Y),\text{acc}}^{(m)}\to\mathcal{\widehat{D}}_{W(Y)}^{(m)}$)
yield a natural map

$\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}\to\widehat{\mathcal{D}}_{W(X)\to W(Y)}^{(m)}$

of
$(\widehat{\mathcal{D}}_{W(X)}^{(m)},W\varphi{}^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)}))$-bimodules;
and the analogous fact holds for $\widehat{\mathcal{D}}_{W(Y)\to
W(Z),\text{acc}}^{(m)}$. So, we have

$\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}\otimes_{W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)})}W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)\to
W(Z),\text{acc}}^{(m)})$ $\to\widehat{\mathcal{D}}_{W(X)\to
W(Y)}^{(m)}\otimes_{W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)})}W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)\to
W(Z)}^{(m)})\to\widehat{\mathcal{D}}_{W(X)\to W(Z)}^{(m)}$

And, since $\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}\otimes_{W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)})}W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)\to
W(Z),\text{acc}}^{(m)})$ is an accessible
$(\widehat{\mathcal{D}}_{W(X)}^{(m)},W(\psi\circ\varphi)^{-1}(\widehat{\mathcal{D}}_{W(X)}^{(m)}))$
bimodule, by applying the functor
$\mathcal{P}^{\cdot}\to\mathcal{P}_{\text{acc}}^{\cdot}$ for
$(\widehat{\mathcal{D}}_{W(X)}^{(m)},W(\psi\circ\varphi)^{-1}(\widehat{\mathcal{D}}_{W(X)}^{(m)}))$
bimodules, we obtain a natural map

$\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}\otimes_{W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)})}W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)\to
W(Z),\text{acc}}^{(m)})\to\widehat{\mathcal{D}}_{W(X)\to
W(Z),\text{acc}}^{(m)}$

as required. ∎

Now, combining this with the previous result gives:

######  Corollary 4.20.

1) With notation as in 4.15, we have an isomorphism of bimodules
$\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}\otimes_{W(k)}^{L}k\tilde{=}\widehat{\mathcal{D}}_{X\to
Y}^{(m)}$.

2) Suppose $\psi:Y\to Z$ is another map of smooth affine varieties with local
coordinates. Then the map of the previous lemma induces an isomorphism

$\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}\otimes_{W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)})}^{L}W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)\to
W(Z),\text{acc}}^{(m)})\tilde{\to}\widehat{\mathcal{D}}_{W(X)\to
W(Z),\text{acc}}^{(m)}$

of
$(\widehat{\mathcal{D}}_{W(X)}^{(m)},W(\psi\circ\varphi)^{-1}(\widehat{\mathcal{D}}_{W(X)}^{(m)}))$-bimodules.

3) There is an isomorphism of functors $L(W\varphi)^{*}\circ
L(W\psi)^{*}\tilde{\to}LW(\psi\circ\varphi)^{*}$ from
$D_{\text{acc}}(\mathcal{\widehat{D}}_{W(Z)}^{(m)}-\text{mod})$ to
$D_{\text{acc}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$.

###### Proof.

There is a map
$\mathcal{O}_{\mathfrak{X}}\otimes\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)})\to\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}$
given by sending a tensor $a\otimes\varphi^{-1}(P)$ to $a\cdot\pi_{1}\circ
W\varphi^{\\#}\circ\varphi^{-1}(\Phi_{2}\cdot P)$, passing to the $p$-adic
completion yields a map

$\varphi^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)}\to\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}$

and we shall show that this map is an isomorphism. The surjectivity follows
from (the proof of) 4.15, where we in fact showed that every term of the form
(4.5) is a sum of terms of the form $a\cdot\pi_{1}\circ
W\varphi^{\\#}\circ\varphi^{-1}(\Phi_{2}\cdot P)$. As for the injectivity, we
may apply the functor of reduction mod $p$ to obtain a map

$\varphi^{*}\mathcal{D}_{Y}^{(m)}\to\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}/p\to\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}/p$

the latter map is injective as
$\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}$
is a summand of $\widehat{\mathcal{D}}_{W(X)\to W(Y),\text{acc}}^{(m)}$. On
the other hand we have maps

$\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}/p\to\widehat{\mathcal{D}}_{W(X)\to
W(Y)}^{(m)}/p\to\varphi^{*}\mathcal{D}_{Y}^{(m)}$

where the first map is the canonical one and the second is given by the
quotient by $I^{(1)}$ (c.f. 2.32). It is not hard to see that the composition
of these maps is the identity on $\varphi^{*}\mathcal{D}_{Y}^{(m)}$; therefore
$\varphi^{*}\mathcal{D}_{Y}^{(m)}\to\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}/p$
is injective. As
$\varphi^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)}\to\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}$
is a surjection of torsion-free $p$-adically complete sheaves, by applying
$\otimes_{W(k)}^{L}k$ we see that this map is an isomorphism. Furthermore, the
composition

$\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}/p\to\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}/p\to\varphi^{*}\mathcal{D}_{Y}^{(m)}$

is easily seen to be a map of
$(\mathcal{D}_{X}^{(m)},\varphi^{-1}(\mathcal{D}_{Y}^{(m)}))$ where
$\mathcal{D}_{X}^{(m)}$ acts via the identification
$\varphi^{*}\mathcal{D}_{Y}^{(m)}\tilde{=}\mathcal{D}_{X\to Y}^{(m)}$ (this
follows readily in local coordinates). Thus $1)$ is proved.

As for $2)$, the map in question is obtained from the previous lemma by noting
that

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

Using the identifications

$\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}\tilde{=}\Phi_{1}^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}^{L}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}\widehat{\otimes}_{\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)})}^{L}\varphi^{-1}(\Phi_{2}^{!}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)})$

and

$W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)\to
W(Z),\text{acc}}^{(m)})\tilde{=}W\varphi^{-1}(\Phi_{2}^{*}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)}}^{L}\widehat{\mathcal{D}}_{\mathfrak{Y}\to\mathfrak{Z}}^{(m),\Phi_{2},\Phi_{3}}\widehat{\otimes}_{\psi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Z}}^{(m)})}^{L}\psi^{-1}(\Phi_{3}^{!}\widehat{\mathcal{D}}_{\mathfrak{Z}}^{(m)}))$

yields an isomorphism

$\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}\otimes_{W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)})}^{L}W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)\to
W(Z),\text{acc}}^{(m)})$
$\tilde{\to}\Phi_{1}^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}^{L}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}\widehat{\otimes}_{\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)})}^{L}\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}\to\mathfrak{Z}}^{(m),\Phi_{2},\Phi_{3}}\widehat{\otimes}_{\psi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Z}}^{(m)})}^{L}\psi^{-1}(\Phi_{3}^{!}\widehat{\mathcal{D}}_{\mathfrak{Z}}^{(m)}))$
$\tilde{\to}\Phi_{1}^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}^{L}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}\widehat{\otimes}_{\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)})}^{L}\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}\to\mathfrak{Z}}^{(m),\Phi_{2},\Phi_{3}}))\widehat{\otimes}_{(\varphi\circ\psi)^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Z}}^{(m)})}^{L}(\varphi\circ\psi)^{-1}(\Phi_{3}^{!}\widehat{\mathcal{D}}_{\mathfrak{Z}}^{(m)}))$

So the map of the previous lemma yields a map

$\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}\widehat{\otimes}_{\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)})}^{L}\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}\to\mathfrak{Z}}^{(m),\Phi_{2},\Phi_{3}})\to\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Z}}^{(m),\Phi_{1},\Phi_{3}}$

which is seen to be an isomorphism by applying $\otimes_{W(k)}^{L}k$ and part
$1)$; so part $2)$ follows as well.

3) Let $\mathcal{M}^{\cdot}\in
D_{\text{acc}}(\mathcal{\widehat{D}}_{W(Y)}^{(m)}-\text{mod})$. Then by part
$2)$ we have

$LW(\psi\circ\varphi)^{*}\mathcal{M}^{\cdot}=\widehat{\mathcal{D}}_{W(X)\to
W(Z),\text{acc}}^{(m)}\otimes_{W(\psi\circ\varphi)^{-1}\mathcal{\widehat{D}}_{W(Z)}^{(m)}}^{L}W(\psi\circ\varphi)^{-1}\mathcal{M}^{\cdot}$
$\tilde{\to}\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}\otimes_{W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)})}^{L}W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)\to
W(Z),\text{acc}}^{(m)})\otimes_{W\varphi^{-1}\circ
W\psi^{-1}(\mathcal{\widehat{D}}_{W(Z)}^{(m)})}^{L}W\varphi^{-1}\circ
W\psi{}^{-1}(\mathcal{M}^{\cdot})$ $\tilde{\to}\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}\otimes_{W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)})}^{L}W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)\to
W(Z),\text{acc}}^{(m)}\otimes_{W\psi^{-1}(\mathcal{\widehat{D}}_{W(Z)}^{(m)})}^{L}W\psi{}^{-1}(\mathcal{M}^{\cdot}))$
$\tilde{\to}LW\varphi^{*}(LW\psi^{*}\mathcal{M}^{\cdot})$

as required. ∎

Thus we may conclude

######  Corollary 4.21.

1) For any $\varphi$, if $\mathcal{M}^{\cdot}\in
D_{\text{acc}}(\mathcal{\widehat{D}}_{W(Y)}^{(m)}-\text{mod})$, we have
$L\varphi^{*}(\mathcal{M}^{\cdot})\in
D_{\text{acc}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$. The same holds
for $\mathcal{M}^{\cdot}\in
D_{\text{acc}}(\mathcal{\widehat{D}}_{W(Y)}^{(m)}/p^{r}-\text{mod})$.

2) The result of 4.16 holds for an arbitrary morphism $\varphi$.

###### Proof.

1) Factor $\varphi:X\to Y$ as $\varphi_{1}:X\to Z$ followed by
$\varphi_{2}:Z\to Y$, where $\varphi$ is a closed immersion and $\varphi_{2}$
is smooth. By the previous result, it suffices to check that each of
$L(W\varphi_{1})^{*}$ and $L(W\varphi_{2})^{*}$ preserve accessibility; as
they are each locally compatible with a lift of Frobenius this follows from
4.18.

2) This is similar to $1)$\- we break up $\varphi=\varphi_{2}\circ\varphi_{1}$
and use 4.16 . ∎

To finish off this section, let’s note the compatibility with the pullback for
crystals. Namely,

######  Proposition 4.22.

Let $r\geq 1$, and let $\epsilon$ be the functor of Theorem 3.29. Then for any
smooth morphism $\varphi:X\to Y$ smooth, and any $\mathcal{M}^{\cdot}\in
D_{\text{qcoh}}(\text{Crys}_{W_{r}(k)}(X))$, we have

$LW\varphi^{*}\circ\epsilon\tilde{\to}\epsilon\circ
L\varphi_{\text{crys}}^{*}$

where $L\varphi_{\text{crys}}^{*}$ is the pullback in the derived category of
crystals.

###### Proof.

As $\varphi$ is smooth, $\varphi^{*}$ is exact, and so it suffices to prove
this when $\mathcal{M}$ is concentrated in a single degree. In that case, we
have that, for a local lift $\varphi:\mathfrak{X}_{r}\to\mathfrak{Y}_{r}$ is
compatible with a Frobenius lift; and so $LW\varphi^{*}$ corresponds to the
usual pullback under 4.16. This shows immediately that the pullback preserves
the local nilpotence condition; furthermore, this is exactly the definition of
the pullback in the category of crystals, and the proposition follows. ∎

###  4.3. Operations on modules: Pushforward

As discussed in the introduction, the key to defining a pushforward in (the
usual) $\mathcal{D}$-module theory is the definition of a transfer bimodule
$\mathcal{D}_{X\leftarrow Y}$. This is constructed as follows: we already have
a $(\mathcal{D}_{X},\varphi^{-1}(\mathcal{D}_{Y}))$ bimodule
$\mathcal{D}_{X\to Y}$; we can apply the left-right interchange simultaneously
to left $\mathcal{D}_{X}$-modules and to right
$\varphi^{-1}(\mathcal{D}_{Y})$-modules to obtain

$\mathcal{D}_{Y\leftarrow
X}:=\omega_{X}\otimes_{\mathcal{O}_{X}}\mathcal{D}_{X\to
Y}\otimes_{\varphi^{-1}(\mathcal{O}_{Y})}\varphi^{-1}(\omega_{Y}^{-1})$

which is now a $(\varphi^{-1}(\mathcal{D}_{Y}),\mathcal{D}_{X})$-bimodule. In
particular, if $Y$ is a point then $\mathcal{D}_{Y\leftarrow X}=\omega_{X}$
with its canonical right $\mathcal{D}$-module structure.

If we now have $\varphi:\mathfrak{X}\to\mathfrak{Y}$ a morphism of smooth
formal schemes, and we fix some $m\geq 0$, an identical procedure defines
$\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(m)}$, and then
the pushforward is defined101010This is not quite Berthelot’s definition; as
he works with sheaves over $W_{r}(k)$ and then takes an inverse limit; whereas
we bypass this by using the cohomological completion. The two notions agree on
a large subclass of objects, for instance on coherent $\mathcal{D}$-modules as

$\int_{\varphi}\mathcal{M}^{\cdot}:=R\varphi_{*}(\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(m)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}}\mathcal{M}^{\cdot})$

We can utilize the same procedure to define
$\widehat{\mathcal{D}}_{W(Y)\leftarrow W(X),\text{acc}}^{(m)}$.

######  Definition 4.23.

Let $\varphi:X\to Y$. The transfer bimodule
$\widehat{\mathcal{D}}_{W(Y)\leftarrow W(X),\text{acc}}^{(m)}$ is the sheaf of
$(W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)}),\mathcal{\widehat{D}}_{W(X)}^{(m)})$
bimodules defined as:

$(W\omega_{X}\widehat{\otimes}_{\mathcal{O}_{W(X)}}\mathcal{\widehat{D}}_{W(X),\text{acc}}^{(m)})\otimes_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}\otimes_{W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})}W\varphi^{-1}(\tilde{\mathcal{H}om}_{\mathcal{O}_{W(Y)}}(W\omega_{Y},\mathcal{\widehat{D}}_{W(Y),\text{acc}}^{(m)}))$

With this in hand, one makes the

######  Definition 4.24.

Let $\mathcal{M}^{\cdot}\in
D_{\text{acc}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod})$ (for
$r\geq 1$) Then for any morphism $\varphi:X\to Y$ we define

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

Next, let $\mathcal{M}^{\cdot}\in
D_{\text{acc}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$. Then we define

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

where as above $\widehat{\otimes}$ denotes the derived completion.

In fact these two definitions agree on their overlap:

######  Remark 4.25.

Let $\mathcal{M}^{\cdot}\in
D_{\text{acc}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod})$, and
regard $\mathcal{M}^{\cdot}$ as an element of
$D_{\text{acc}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$ via the
obvious inclusion. Then we have an isomorphism

$(\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X)}^{(m)}/p^{r})\otimes_{\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}}^{L}\mathcal{M}^{\cdot}\tilde{\to}\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X)}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot}$

of complexes of $\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})$-modules. To
see this, note that $\widehat{\mathcal{D}}_{W(Y)\leftarrow W(X)}^{(m)}$ is
itself $p$-torsion free. So, to show the above, we let $\mathcal{F}^{\cdot}$
be a flat resolution of $\widehat{\mathcal{D}}_{W(Y)\leftarrow W(X)}^{(m)}$ in
the category of right-$\mathcal{\widehat{D}}_{W(X)}^{(m)}$-modules. Then, as
$p^{r}$ annihilates each term of $\mathcal{M}^{\cdot}$, we have

$\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X)}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot}=\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X)}^{(m)}\otimes_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot}$

and this is computed by

$\mathcal{F}^{\cdot}\otimes_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}\mathcal{M}^{\cdot}\tilde{\to}\mathcal{F}^{\cdot}/p^{r}\otimes_{\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}}\mathcal{M}^{\cdot}$

and since $\widehat{\mathcal{D}}_{W(X)\leftarrow W(Y)}^{(m)}$ is flat over
$W(k)$, we have that $\mathcal{F}^{\cdot}/p^{r}$ is a flat resolution of
$\widehat{\mathcal{D}}_{W(X)\leftarrow W(Y)}^{(m)}/p^{r}$ over
$\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}$, whence the result.

As for pullback, the main tool to study this morphism comes from giving an
explicit expression for it locally. Indeed, combining 4.23 with 4.15 we obtain

######  Proposition 4.26.

Let
$\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(m),\Phi_{1},\Phi_{2}}$
denote the
$(\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)}),\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)})$-bimodule
obtained from
$\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}$
(as defined in 4.15) via simultaneously applying the left right interchange of
$\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}$-modules and
$\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)})$-modules. We have

$\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(m)}\tilde{=}\varphi^{-1}(\Phi_{2}^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)})\widehat{\otimes}_{\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)})}^{L}\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(m),\Phi_{1},\Phi_{2}}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}^{L}\Phi_{1}^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}$

and this tensor product is concentrated in homological degree $0$. In
particular, if $\varphi\circ\Phi_{1}=\Phi_{2}\circ\varphi$ then we obtain

$\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(m)}\tilde{=}\varphi^{-1}(\Phi_{2}^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)})\widehat{\otimes}_{\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)})}^{L}\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}^{L}\Phi_{1}^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}$

Applying the fact that
$\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m),\Phi_{1},\Phi_{2}}/p\tilde{=}\mathcal{D}_{X\to
Y}^{(m)}$ (by 4.20, part $1$), we obtain from this the

######  Corollary 4.27.

For any $\varphi:X\to Y$ there is an isomorphism

$\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(m)}/p\tilde{\to}\varphi^{-1}(\mathcal{B}_{Y}^{(m)})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(m)})}\mathcal{D}_{Y\leftarrow
X}^{(m)}\otimes_{\mathcal{D}_{X}^{(m)}}\mathcal{B}_{X}^{(m),r}$

From this we deduce

######  Theorem 4.28.

Let $\varphi:X\to Y$ be an arbitrary morphism.

1) Then the natural map

$R(W\varphi)_{*}(W\varphi^{-1}(\mathcal{B}_{Y}^{(m)})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(m)})}^{L}\mathcal{D}_{X\leftarrow
Y}^{(m)}\otimes_{\mathcal{D}_{X}^{(m)}}^{L}\mathcal{N}^{\cdot})_{\text{acc}}\to\mathcal{B}_{Y}^{(m)}\otimes_{\mathcal{D}_{Y}^{(m)}}^{L}R\varphi_{*}(\mathcal{D}_{Y\leftarrow
X}^{(m)}\otimes_{\mathcal{D}_{X}^{(m)}}^{L}\mathcal{N}^{\cdot})$

is an isomorphism for any $\mathcal{N}^{\cdot}\in
D(\mathcal{D}_{X}^{(m)}-\text{mod})$.

2) Let $\mathcal{N}^{\cdot}\in D(\mathcal{D}_{X}^{(m)}-\text{mod})$. Then
there is a natural isomorphism

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

###### Proof.

By general nonsense there is a functorial morphism

$R(W\varphi)_{*}(W\varphi^{-1}(\mathcal{B}_{Y}^{(m)})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(m)})}^{L}\mathcal{M}^{\cdot})\to\mathcal{B}_{Y}^{(m)}\otimes_{\mathcal{D}_{Y}^{(m)}}^{L}R\varphi_{*}(\mathcal{M}^{\cdot})$

for any $\mathcal{M}^{\cdot}\in
D(\varphi^{-1}(\mathcal{D}_{Y}^{(m)})-\text{mod})$, c.f. [Stacks], tag 0B54.
As the right hand side is clearly accessible, applying the functor
$()_{\text{acc}}$ to both sides gives a morphism

(4.6)
$R(W\varphi)_{*}(W\varphi^{-1}(\mathcal{B}_{Y}^{(m)})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(m)})}^{L}\mathcal{M}^{\cdot})_{\text{acc}}\to\mathcal{B}_{Y}^{(m)}\otimes_{\mathcal{D}_{Y}^{(m)}}^{L}R\varphi_{*}(\mathcal{M}^{\cdot})$

To show that is is an isomorphism it suffices to show that it is so after
applying
$R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(Y)}^{(m)}/p}(\mathcal{B}_{Y}^{(m)},)$.
We have

$R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(Y)}^{(m)}/p}(\mathcal{B}_{Y}^{(m)},R(W\varphi)_{*}(W\varphi^{-1}(\mathcal{B}_{Y}^{(m)})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(m)})}^{L}\mathcal{M}^{\cdot})_{\text{acc}})$
$\tilde{\to}R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(Y)}^{(m)}/p}(\mathcal{B}_{Y}^{(m)},R(W\varphi)_{*}(W\varphi^{-1}(\mathcal{B}_{Y}^{(m)})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(m)})}^{L}\mathcal{M}^{\cdot}))$
$\tilde{\to}R\varphi_{*}R\mathcal{H}om_{\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)}/p)}(W\varphi^{-1}(\mathcal{B}_{Y}^{(m)}),\varphi^{-1}(\mathcal{B}_{Y}^{(m)})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(m)})}^{L}\mathcal{M}^{\cdot})$
$\tilde{\to}R\varphi_{*}(\mathcal{M}^{\cdot})$

where the first isomorphism is by the definition of accessibility, and the
second is by the adjunction of $W\varphi^{-1}$ and $R\varphi_{*}$. It follows
that the map (4.6) is an isomorphism, and $1)$ follows by setting
$\mathcal{M}^{\cdot}=\mathcal{D}_{X\leftarrow
Y}^{(m)}\otimes_{\mathcal{D}_{X}^{(m)}}^{L}\mathcal{N}^{\cdot}$. Now $2)$
follows from the description of the transfer bimodule given directly above. ∎

Now let’s discuss the base change functor $\otimes_{W(k)}^{L}k$. The situation
is very nice here:

######  Lemma 4.29.

There is, for any $\mathcal{M}^{\cdot}\in
D_{\text{acc}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$, an isomorphism

$(\int_{W\varphi}\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k\tilde{=}\int_{W\varphi}(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)$

where on the right we take the pushforward in the category
$D_{\text{acc}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p-\text{mod})$.

###### Proof.

Again by [Stacks], tag 0B54 there is a map

$R(W\varphi)_{*}(\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X)}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k\to
R(W\varphi)_{*}((\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X)}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k)$

which is an isomorphism because $k$ is a perfect complex over $W(k)$. However,
we have

$(\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X)}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k$
$\tilde{=}(\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X)}^{(m)}/p)\widehat{\otimes}_{\mathcal{\widehat{D}}_{W(X)}^{(m)}/p}^{L}(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)$

as both $\widehat{\mathcal{D}}_{W(X)\leftarrow W(Y)}^{(m)}$ and
$\widehat{\mathcal{D}}_{W(X)}^{(m)}$ are flat over $W(k)$. Further, as the
functor $()_{\text{acc}}$ commutes with $\otimes_{W(k)}^{L}k$, applying
$()_{\text{acc}}$ to both sides yields the result. ∎

This yields

######  Corollary 4.30.

Suppose $\mathcal{M}^{\cdot}\in
D_{\text{acc}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$, and let
$\varphi:X\to Y$, $\psi:Y\to Z$. There is a functorial morphism

$\int_{W\psi}\int_{W\varphi}\mathcal{M^{\cdot}}\to\int_{W(\psi\circ\varphi)}\mathcal{M}^{\cdot}$

If $\mathcal{M}^{\cdot}\in
D_{\text{qcoh}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$, then this map
is an isomorphism. The same holds for $\mathcal{M}^{\cdot}\in
D_{\text{qcoh}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod})$, for the
pushforward of $\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}$-modules.

###### Proof.

The existence of the map is a little exercise in base changing. Namely, from
4.20 we obtain the isomorphism

$W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Z)\leftarrow
W(Y),\text{acc}}^{(m)})\otimes_{W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)})}^{L}\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(m)}\tilde{\to}\widehat{\mathcal{D}}_{W(Z)\leftarrow
W(X),\text{acc}}^{(m)}$

So we have (by [Stacks], tag 0B54)

$RW\psi_{*}(\widehat{\mathcal{D}}_{W(Z)\leftarrow
W(Y),\text{acc}}^{(m)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(Y)}^{(m)}}R(W\varphi)_{*}(\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X)}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot}))$
$\to RW\psi_{*}\circ
RW\varphi_{*}(W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Z)\leftarrow
W(Y),\text{acc}}^{(m)})\widehat{\otimes}_{W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)})}(\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X)}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot}))$
$\tilde{\to}RW(\psi\circ\varphi)_{*}(\widehat{\mathcal{D}}_{W(Z)\leftarrow
W(X),\text{acc}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot})$

Further, we have

$\int_{W\psi}\int_{W\varphi}\mathcal{M}^{\cdot}=RW\psi_{*}(\widehat{\mathcal{D}}_{W(Z)\leftarrow
W(Y),\text{acc}}^{(m)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(Y)}^{(m)}}\int_{W\varphi}\mathcal{M}^{\cdot})_{\text{acc}}$
$=RW\psi_{*}(\widehat{\mathcal{D}}_{W(Z)\leftarrow
W(Y),\text{acc}}^{(m)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(Y)}^{(m)}}R(W\varphi)_{*}(\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X)}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot})_{\text{acc}})_{\text{acc}}$
$\to RW\psi_{*}(\widehat{\mathcal{D}}_{W(Z)\leftarrow
W(Y),\text{acc}}^{(m)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(Y)}^{(m)}}R(W\varphi)_{*}(\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X)}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot}))_{\text{scc}}$

vis the natural map $\mathcal{N}_{\text{acc}}^{\cdot}\to\mathcal{N}^{\cdot}$
in $\widehat{\mathcal{D}}_{W(Y)}^{(m)}-\text{mod}$. Then, applying
$()_{\text{acc}}$ in $\widehat{\mathcal{D}}_{W(Z)}^{(m)}$-mod to the previous
map, we obtain a map

$\int_{W\psi}\int_{W\varphi}\mathcal{M}^{\cdot}\to\tilde{\to}RW(\psi\circ\varphi)_{*}(\widehat{\mathcal{D}}_{W(Z)\leftarrow
W(X),\text{acc}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot})_{\text{acc}}=\int_{W(\psi\circ\varphi)}\mathcal{M}^{\cdot}$

To show its an isomorphism, apply $\otimes_{W(k)}^{L}k$ and the previous lemma
to reduce to the case of $\mathcal{D}_{X}^{(m)}$-modules in positive
characteristic. There, we have

$\int_{\psi}\int_{\varphi}\mathcal{M^{\cdot}\tilde{\to}\int_{\psi\circ\varphi}\mathcal{M}^{\cdot}}$

for any $\mathcal{M}^{\cdot}\in
D_{\text{qcoh}}(\mathcal{D}_{X}^{(m)}-\text{mod})$ (c.f. [17], lemma 7.7 for
this statement), whence the result for $\mathcal{M}^{\cdot}\in
D_{\text{qcoh}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$. To obtain the
result for $\mathcal{M}^{\cdot}\in
D_{\text{qcoh}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}-\text{mod})$, apply
4.25. ∎

To round things out, we deduce the following compatibility:

######  Corollary 4.31.

1) Let $\varphi:\mathfrak{X}_{r}\to\mathfrak{Y}_{r}$ where, if $m=0$ and
$r>1$, we assume $p\neq 2$ (we allow $r=\infty$ here). Let
$\mathcal{N}^{\cdot}\in D(\mathcal{D}_{\mathfrak{X}_{r}}^{(m)}-\text{mod})$.
Then there is a natural isomorphism

$\int_{\varphi}(\mathcal{B}_{\mathfrak{X}_{r}}^{(m)}\otimes_{\mathcal{D}_{\mathfrak{X_{r}}}^{(m)}}^{L}\mathcal{N}^{\cdot})\tilde{\to}\mathcal{B}_{\mathfrak{Y}_{r}}^{(m)}\otimes_{\mathcal{D}_{\mathfrak{Y}_{r}}^{(m)}}^{L}\int_{\varphi}\mathcal{N}^{\cdot}$

###### Proof.

This follows by breaking $\varphi$ up into a smooth morphism and a closed
embedding just as in 4.21. ∎

Finally, we’d like to discuss some further specific properties of the
pushforward when $\varphi$ satisfies extra conditions. First, we specialize to
the case where $\varphi:X\to Y$ is a smooth morphism. In this case, there is
an adjunction between the pullback and the pushforward; more precisely, we
have:

######  Corollary 4.32.

Let $\varphi:X\to Y$ be smooth of relative dimension $d$; let
$\mathcal{M}^{\cdot}\in
D_{\text{acc}}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$ and
$\mathcal{N}^{\cdot}\in
D_{\text{acc}}(\mathcal{\widehat{D}}_{W(Y)}^{(m)}-\text{mod}))$. Then there is
an isomorphism of functors

$RW\varphi_{*}R\underline{\mathcal{H}om}_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}(L(W\varphi)^{*}\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})[d]\tilde{\to}R\underline{\mathcal{H}om}{}_{\mathcal{\widehat{D}}_{W(Y)}^{(m)}}(\mathcal{N}^{\cdot},\int_{W\varphi}\mathcal{M}^{\cdot})$

In particular the functors $LW\varphi^{*}[d]=W\varphi^{!}$ and
${\displaystyle\int_{\varphi}}$ form an adjoint pair on accessible modules.
The analogous statement holds for accessible
$\mathcal{\widehat{D}}_{W(X)}^{(m)}/p^{r}$ modules for any $r\geq 1$.

We will deduce the this theorem from the analogous fact for
$\mathcal{\widehat{D}}^{(m)}$-modules. As in that case, the key point is to
prove the

######  Proposition 4.33.

For any smooth morphism $\varphi:X\to Y$ of relative dimension $d$ there is an
isomorphism of
$(\widehat{\mathcal{D}}_{W(Y)}^{(m)},\widehat{\mathcal{D}}_{W(X)}^{(m)})$
bimodules

$R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}(\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)},\widehat{\mathcal{D}}_{W(X),\text{acc}}^{(m)})_{Y-\text{acc}}\tilde{=}\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(m)}[-d]$

where on the left hand side $()_{Y-\text{acc}}$ refers to the functor of
accessibalization applied in the category of left
$\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)})$-modules.

Before doing so, we recall that the analogous isomorphism

$R\mathcal{H}om_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)})\tilde{\to}\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(m)}[-d]$

is the basic point in proving this adjunction for
$\widehat{\mathcal{D}}^{(m)}$-modules; when $m=0$ this can be proved via the
de Rham resolution, and when $m>0$ it follows via Frobenius descent.

###### Proof.

Let us begin in the case where $X$ and $Y$ are affine, with smooth lifts
$\mathfrak{X}$ and $\mathfrak{Y}$. Let $\Phi_{1}$ and $\Phi_{2}$ be lifts of
Frobenius on $\mathfrak{X}$ and $\mathfrak{Y}$, respectively, chosen so that
$\Phi_{2}\circ\varphi=\varphi\circ\Phi_{1}$ (this is possible as $\varphi$ is
smooth).This means that, according to 4.26

$\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(m)}\tilde{=}\varphi^{-1}(\Phi_{2}^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)})\otimes_{\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(m)})}^{L}\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(m)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}^{L}\Phi_{1}^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}$

As both sides are, by definition, accessible over
$\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)})$, we may construct the
isomorphism by applying the functor
$R\mathcal{H}om_{W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)})}(W\varphi^{-1}(\Phi_{2}^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)}),?)$
to both sides. On the left hand side, this yields

$R\mathcal{H}om_{W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)})}(W\varphi^{-1}(\Phi_{2}^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)}),R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}(\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)},\widehat{\mathcal{D}}_{W(X),\text{acc}}^{(m)}))$
$\tilde{\to}R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}(\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}\widehat{\otimes}_{W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)})}^{L}W\varphi^{-1}(\Phi_{2}^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)}),\widehat{\mathcal{D}}_{W(X),\text{acc}}^{(m)}))$
$\tilde{\to}R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}(\Phi_{1}^{*}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}}^{L}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m)},\widehat{\mathcal{D}}_{W(X),\text{acc}}^{(m)}))$
$\tilde{\to}R\mathcal{H}om_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m)},\Phi_{1}^{!}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)})$
$\tilde{\to}R\mathcal{H}om_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(m)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)})\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}}^{L}\Phi_{1}^{!}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}$
$\tilde{\to}\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(m)}\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}}^{L}\Phi_{1}^{!}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}[-d]$

where we used hom tensor adjunction for the first isomorphism, 4.15 for the
second, the construction of $\widehat{\mathcal{D}}_{W(X),\text{acc}}^{(m)}$
for the third, the fact that
$\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(m)}$ is isomorphic to
as perfect complex of
$\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}$-modules111111As
$\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}$ is locally of finite homological
dimension for the fourth, and isomorphism directly above for the fifth. Thus
we obtain the isomorphism of the proposition when $X$ and $Y$ are affine by
the description of $\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(m)}$ given just above.

For the general case, we note that $\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(m)}$ is an accessible
$(\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)}),\widehat{\mathcal{D}}_{W(X)}^{(m)})$
bimodule, and so we have

$\mathcal{E}nd_{(\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)}),\widehat{\mathcal{D}}_{W(X)}^{(m)})}(\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(m)})\tilde{\to}\mathcal{E}nd_{(\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(m)}),\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)})}(\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(m)})=W(k)_{\mathfrak{X}}$

where $W(k)_{\mathfrak{X}}$ refers to the locally constant sheaf with sections
$W(k)$. As this sheaf is flasque, we see that the the locally defined
isomorphisms constructed above must glue to a globally defined isomorphism
(which is unique up to rescale by an element of $W(k)$). ∎

Given this, let’s give the

###### Proof.

(of 4.32) (following [27], Theorem 4.40). We have

$R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}(LW\varphi^{*}\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})[d]$
$=R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}(\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)}\otimes_{W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)})}^{L}\varphi^{-1}(\mathcal{N}^{\cdot}),\mathcal{M}^{\cdot})[d]$
$\tilde{\to}R\mathcal{H}om_{W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})}(W\varphi^{-1}(\mathcal{N}^{\cdot}),R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}(\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)},\mathcal{M}^{\cdot}))[d]$

Now, there is the obvious natural map

$R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}(\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)},\widehat{\mathcal{D}}_{W(X)}^{(m)})\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot}\to
R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}(\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)},\mathcal{M}^{\cdot})$

for any $\mathcal{M}^{\cdot}\in
D(\widehat{\mathcal{D}}_{W(X)}^{(m)}-\text{mod})$. Via the right action of
$W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)})$ on
$\widehat{\mathcal{D}}_{W(X)\to W(Y),\text{acc}}^{(m)}$, this is a morphism
complexes of of left
$W\varphi^{-1}(\widehat{\mathcal{D}}_{W(Y)}^{(m)})$-modules. Thus, applying
the functors
$R\mathcal{H}om_{W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})}(W\varphi^{-1}(\mathcal{N}^{\cdot}),)$
and $RW\varphi_{*}$ we obtain a morphism

$RW\varphi_{*}R\mathcal{H}om_{W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})}(W\varphi^{-1}(\mathcal{N}^{\cdot}),R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}(\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)},\widehat{\mathcal{D}}_{W(X)}^{(m)})\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot})[d]$
$\to
RW\varphi_{*}R\mathcal{H}om_{W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})}(W\varphi^{-1}(\mathcal{N}^{\cdot}),R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}(\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)},\mathcal{M}^{\cdot}))[d]$
$\tilde{=}R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}(LW\varphi^{*}\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})[d]$

of complexes of $\widehat{\mathcal{D}}_{W(Y)}^{(m)}$-modules. Now, as
$\mathcal{N}^{\cdot}$ is accessible over $\mathcal{\widehat{D}}_{W(Y)}^{(m)}$,
we have

$R\mathcal{H}om_{W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})}(W\varphi^{-1}(\mathcal{N}^{\cdot}),R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}(\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)},\widehat{\mathcal{D}}_{W(X)}^{(m)})\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot})[d]$
$=R\mathcal{H}om_{W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})}(W\varphi^{-1}(\mathcal{N}^{\cdot}),R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}(\widehat{\mathcal{D}}_{W(X)\to
W(Y),\text{acc}}^{(m)},\widehat{\mathcal{D}}_{W(X)}^{(m)})_{\text{acc}}\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot})[d]$
$\tilde{\to}R\mathcal{H}om_{W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})}(W\varphi^{-1}(\mathcal{N}^{\cdot}),\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(m)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot})$

by the previous proposition. Furthermore

$RW\varphi_{*}R\mathcal{H}om_{W\varphi^{-1}(\mathcal{\widehat{D}}_{W(Y)}^{(m)})}(W\varphi^{-1}(\mathcal{N}^{\cdot}),\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(m)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot})$
$\tilde{\to}R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(Y)}^{(m)}}(\mathcal{N}^{\cdot},RW\varphi_{*}(\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(m)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot}))$
$\tilde{\to}R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(Y)}^{(m)}}(\mathcal{N}^{\cdot},\int_{W\varphi}\mathcal{M}^{\cdot})$

where we again use that $\mathcal{N}^{\cdot}$ is accessible in the last line;
this follows from

${\displaystyle\int_{W\varphi}\mathcal{M}^{\cdot}=R\varphi_{*}(\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(m)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(m)}}^{L}\mathcal{M}^{\cdot})_{\text{acc}}}$

Summing up, we’ve obtained a functorial map

$R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(Y)}^{(m)}}(\mathcal{N}^{\cdot},\int_{W\varphi}\mathcal{M}^{\cdot})\to
R\mathcal{H}om_{\mathcal{\widehat{D}}_{W(X)}^{(m)}}(LW\varphi^{*}\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})[d]$

To check it is an isomorphism, we may (by cohomological completeness) apply
$\otimes_{W(k)}^{L}k$, and then use Lemma 4.29 to reduce the statement to the
(known) adjunction for $\mathcal{D}^{(m)}$-modules. ∎

Combining this with the compatibility of pullback on crystals, we deduce:

######  Corollary 4.34.

Let $\mathcal{M}^{\cdot}\in D^{+}(\text{Qcoh}(\text{Crys}(X,W_{r}(k)))$, and
let $\epsilon(\mathcal{M}^{\cdot})$ denote the associated element of
$D_{\text{qcoh,}\text{acc}}^{+}(\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}-\text{mod})$.

Then there is an isomorphism

$\int_{\varphi}\epsilon(\mathcal{M}^{\cdot})[-d]\tilde{\to}\epsilon(R\varphi_{*,\text{crys}}(\mathcal{M}^{\cdot}))$

in $D_{\text{qcoh}}^{+}(\mathcal{\widehat{D}}_{W(Y)}^{(0)}/p^{r}-\text{mod})$.

###### Proof.

By definition, the functors

$L\varphi_{\text{crys}}^{*}:D^{+}(\text{Qcoh}(\text{Crys}(Y,W_{r}(k)))\to
D^{+}(\text{Qcoh}(\text{Crys}(X,W_{r}(k)))$

and $R\varphi_{*,\text{crys}}:D^{+}(\text{Qcoh}(\text{Crys}(X,W_{r}(k)))\to
D^{+}(\text{Qcoh}(\text{Crys}(Y,W_{r}(k)))$ form an adjoint pair. As the
functor $\epsilon$ is compatible with pullback, we simply have to check that
${\displaystyle\int_{\varphi}}$ preserves the image; i.e., that it sends
quasicoherent complexes of nilpotent modules over
$\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}$ to quasicoherent complexes of
nilpotent modules over $\mathcal{\widehat{D}}_{W(Y)}^{(0)}/p^{r}$. By the fact
that nilpotency (for quasicoherent modules) can be characterized as being
supported on a certain closed subset (of the spectrum of the center of the
sheaf of differential operators), this can be checked after applying
$\otimes_{W(k)}^{L}k$. But then, applying Theorem 4.28, this follows from the
analogous fact for $\mathcal{D}_{X}^{(0)}$-modules, where it is a well known
theorem of Katz (using the language of the Gauss-Manin connection, it is
proved in [30], section $7$, or, in the language of $\mathcal{D}$-modules, in
[34], section 2.5). ∎

Finally, we have the following important fact when $\varphi$ is proper:

######  Theorem 4.35.

Let $\varphi:X\to Y$ be a proper morphism. Then
${\displaystyle\int_{\varphi}}$ takes
$D_{\text{coh}}^{b}(\mathcal{\widehat{D}}_{W(X)}^{(m)}-\text{mod})$ to
$D_{\text{coh}}^{b}(\mathcal{\widehat{D}}_{W(Y)}^{(m)}-\text{mod})$.

###### Proof.

Applying [28], theorem 1.6.3, we see that, a complex $\mathcal{N}^{\cdot}\in
D(\mathcal{\widehat{D}}_{W(Y)}^{(m)}-\text{mod})$ is contained in
$D_{\text{coh}}^{b}(\mathcal{\widehat{D}}_{W(Y)}^{(m)}-\text{mod})$ iff
$\mathcal{N}^{\cdot}\otimes_{W(k)}^{L}k$ is contained in
$D_{\text{coh}}^{b}(\mathcal{\widehat{D}}_{W(Y)}^{(m)}/p-\text{mod})$. So it
suffices to show that

$\int_{\varphi}:D(\mathcal{\widehat{D}}_{W(X)}^{(m)}/p-\text{mod})\to
D(\mathcal{\widehat{D}}_{W(Y)}^{(m)}/p-\text{mod})$

takes $D_{\text{coh }}^{b}$ to $D_{\text{coh}}^{b}$; this, in turn, follows
via Theorem 4.28 from the analogous fact for $\mathcal{D}^{(m)}$-modules. ∎

###  4.4. The de Rham-Witt resolution

In this section we’ll recall the (relative) de Rham-Witt resolution and
explain the connection between the pushforward functor introduced above and
the (relative) de Rham-Witt complex.

Throughout this section, fix a number $r\geq 1$; we shall work mod $p^{r}$.
According to [25], and [32] for the relative case, there is attached to any
smooth morphism $\varphi:X\to Y$ the relative de Rham-Witt complex
$W\Omega_{X/Y}^{\cdot}$. It is a dg algebra inside quasicoherent sheaves on
$W(X)$, it is complete with respect to a canonical filtration which we shall
denote $G(W\Omega_{X/Y}^{\cdot})$, and the quotient
$W\Omega_{X/Y}^{\cdot}/G^{1}\tilde{\to}\Omega_{X/Y}^{\cdot}$. In particular
zeroth term of $W\Omega_{X/Y}^{\cdot}$ is $W(X)$, equipped with its usual
$V^{i}$-filtration.

Suppose $Y=\text{Spec}(k)$, and that $X$ is affine and admits local
coordinates, and we fix a coordinatized lift of $F$. Then there is an
inclusion $\Omega_{\mathfrak{X}_{r}}^{i}\to W\Omega_{X}^{i}/p^{r}$ which is
compatible with the differential, and with the action of
$\mathcal{O}_{\mathfrak{X}_{r}}$ (via
$\Phi:\mathcal{O}_{\mathfrak{X}_{r}}\to\mathcal{O}_{W(X)}/p^{r}$).

We begin by reviewing a basic result of Etesse, in a form which is useful for
us:

######  Theorem 4.36.

([20]) Let $\mathcal{M}$ be a quasicoherent crystal on $X$, over $W_{r}(k)$,
and let $\tilde{\mathcal{M}}$ be the associated sheaf on $W(X)_{p^{r}=0}$.
This sheaf admits an integrable de Rham-Witt connection

$\nabla:\tilde{\mathcal{M}}\to\tilde{\mathcal{M}}\widehat{\otimes}_{\mathcal{O}_{W(X)}/p^{r}}W\Omega_{X}^{1}/p^{r}$

which is continuous with respect to the natural topologies on both sides.

Suppose $X$ is affine with local coordinates. Let $\mathfrak{X}$ be a smooth
lift of $X$ with coordinatized lift of Frobenius $F$. Then $\mathcal{M}$
defines a unique sheaf with integrable nilpotent connection
$\mathcal{M}^{\prime}$ on $\mathfrak{X}_{n}$, and we have

$\tilde{\mathcal{M}}\tilde{=}\widehat{\Phi}^{*}\mathcal{M}^{\prime}$

with the connection given by

$\nabla(p^{m}T^{I/p^{m}}\cdot m)=d(p^{m}T^{I/p^{m}})m+p^{m}T^{I/p^{m}}dm$

for $m\in\mathcal{M}^{\prime}$; here, we are regarding
$dm\in\mathcal{M}^{\prime}\otimes\Omega_{\mathfrak{X}_{r}}^{1}\subset\mathcal{M}^{\prime}\otimes_{\mathcal{O}_{\mathfrak{X}_{r}}}W\Omega_{X}^{1}/p^{r}\subset\widehat{\Phi}^{*}\mathcal{M}^{\prime}\widehat{\otimes}_{\mathcal{O}_{W(X)}/p^{r}}W\Omega_{X}^{1}/p^{r}$.
The same results also hold for pro-objects in the category
$\text{Qcoh(Crys}(X,W_{r}(k))$.

In other words, the theorem tells us that the explicitly defined de Rham-Witt
connection on $\widehat{\Phi}^{*}\mathcal{F}^{\prime}$ is in fact independent
of the choice of $\Phi$. Let’s apply this to
$\mathcal{F}=(\widehat{\mathcal{D}}_{W(X),\text{crys}}^{(0)}/p^{r})_{\text{c-acc}}$
the completion (along the left action of $V^{i}(\mathcal{O}_{W(X)}/p^{r})$) of
$\widehat{\mathcal{D}}_{W(X),\text{crys},\text{acc}}/p^{r}$. This is as
inverse limit of quasicoherent crystals on $X$ over $W_{r}(k)$, and so the
theorem applies to it. From this we conclude:

######  Corollary 4.37.

Let $(\widehat{\mathcal{D}}_{W(X)}/p^{r})_{c-\text{acc}}$ be the completion
(along the left action of $V^{i}(\mathcal{O}_{W(X)}/p^{r})$) of
$\widehat{\mathcal{D}}_{W(X),\text{acc}}/p^{r}$ There is a unique continuous,
integrable de Rham-Witt connection $\nabla$ on
$(\widehat{\mathcal{D}}_{W(X)}/p^{r})_{c-\text{acc}}$ satisfying the
following: suppose $U\subset X$ is open affine with local coordinates. Let
$\mathfrak{U}$ be a smooth lift of $U$ with coordinatized lift of Frobenius
$\Phi$. Then we have the isomorphism

$(\widehat{\mathcal{D}}_{W(U)}^{(0)}/p^{r})_{c-\text{acc}}\tilde{=}\widehat{\Phi}^{*}\Phi^{!}\mathcal{D}_{\mathfrak{U}_{r}}^{(0)}$

and therefore a continuous integrable de Rham-Witt connection on
$\widehat{\Phi}^{*}\Phi^{!}\mathcal{D}_{\mathfrak{U}_{r}}^{(0)}$ which is
induced from the natural connection on
$\Phi^{!}\mathcal{D}_{\mathfrak{U}_{r}}^{(0)}$ (it is a left
$\mathcal{D}_{\mathfrak{U}_{r}}^{(0)}$-module by definition). Then the
connection $\nabla$ agrees with this induces connection under the isomorphism.

###### Proof.

By the previous theorem this is true after completing along the central ideal
defining the nilpotent support condition. As the natural completion map

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

is injective, we see that the de Rham-Witt connection on
$(\widehat{\mathcal{D}}_{W(X)}/p^{r})_{c-\text{acc}}$ which is locally defined
by the isomorphism
$(\widehat{\mathcal{D}}_{W(U)}^{(0)}/p^{r})_{c-\text{acc}}\tilde{=}\widehat{\Phi}^{*}\Phi^{!}\mathcal{D}_{\mathfrak{U}_{r}}^{(0)}$
is in fact independent of the choice of $\Phi$. Therefore this connection
glues to a globally defined connection on
$(\widehat{\mathcal{D}}_{W(X)}/p^{r})_{c-\text{acc}}$ as required. ∎

From this follows

######  Corollary 4.38.

Let
$\mathcal{M}\in\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}-\text{mod}_{\text{acc}}$.
Then there is an unique, continuous, integrable de Rham-Witt connection on
$\mathcal{\widehat{M}}$ defined via

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

where the second arrow is induced by the de Rham-Witt connection of the
previous corollary, and the right action of
$\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r}$ on
$(\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r})_{c-\text{acc}}\widehat{\otimes}_{\mathcal{O}_{W(X)}}W\Omega_{X}^{1}/p^{r}$
is via the right action of $\widehat{\mathcal{D}}_{W(X)}/p^{r}$ on
$(\widehat{\mathcal{D}}_{W(X)}/p^{r})_{c-\text{acc}}$. This connection agrees
with the “obvious” connection on
$\mathcal{\widehat{M}}=\widehat{\Phi}^{*}(\mathcal{N})$ for any affine open
with lift of Frobenius $\Phi$.

In order to show that the (relative) de Rham-Witt complex computes the
pushforward, we need to define the (relative) de-Rham-Witt resolution. Let
$\varphi:X\to Y$ be a smooth morphism of relative dimension $d$. Via the
natural quotient map $W\Omega_{X}^{1}\to W\Omega_{X/Y}^{1}$, the de Rham-Witt
connection on $(\widehat{\mathcal{D}}_{W(X)}/p^{r})_{c-\text{acc}}$ defines an
integrable connection

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

and so we have the associated relative de Rham-Witt complex

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

concentrated in degrees $\\{0,\dots,d\\}$. We also define the object
$(\widehat{\mathcal{D}}_{W(Y)\leftarrow W(X)}^{(0)}/p^{r})_{\text{c-acc}}$ to
be the completion of $(\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(0)}/p^{r})$ along the filtration induced by
$\\{W\varphi^{-1}(V^{i}(\mathcal{O}_{W(Y)}/p^{r})\\}$. We note that since
$\varphi$ is smooth, the right $\mathcal{D}_{X}^{(0)}$-module
$\mathcal{D}_{Y\leftarrow X}^{(0)}$ is coherent. This implies (via the
description of 4.26) that $(\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X)}^{(0)}/p^{r})_{\text{c-acc}}$ is already complete along the filtration
induced from $\\{V^{i}(\mathcal{O}_{W(X)}/p^{r})\\}$. When $Y$ is a point, we
have

$(\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X)}^{(0)}/p^{r})_{\text{c-acc}}=W\omega_{X}/p^{r}$

Then we have

######  Theorem 4.39.

There is a quasi-isomorphism

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

###### Proof.

First, we’ll construct a map

$(\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r})_{c-\text{acc}}\widehat{\otimes}_{\mathcal{O}_{W(X)/p^{r}}}W\omega_{X/Y}/p^{r}\to(\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X)}^{(0)}/p^{r})_{\text{c-acc}}$

and show that the resulting augmented complex is exact. This map is
constructed as follows: via the composition of morphisms there is a map

$\mathcal{E}nd_{W(k)}(W\omega_{X})\times\mathcal{H}om_{\mathcal{O}_{W(X)}}(\varphi^{*}(W\omega_{Y}),W\omega_{X})\to\mathcal{H}om_{W(k)}(\varphi^{*}(W\omega_{Y}),W\omega_{X})$

which, via the embeddings
$W\omega_{X/Y}\to\mathcal{H}om_{W(k)}(\varphi^{-1}(W\omega_{Y}),W\omega_{X})$
and $\widehat{\mathcal{D}}_{W(X)}^{(0)}\to\mathcal{E}nd_{W(k)}(W\omega_{X})$
yields a map

$\widehat{\mathcal{D}}_{W(X)}^{(0)}\otimes_{\mathcal{O}_{W(X)/p^{r}}}W\omega_{X/Y}\to\mathcal{H}om_{W(k)}(\varphi^{*}(W\omega_{Y}),W\omega_{X})$

and, after completion and accessiblization, one sees by looking in local
coordinates that the image of this map is contained in
$\text{$\lim$}_{r}\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{c-acc}}^{(0)}/p^{r}$ (when $Y$ is a point this map is just the
action map coming from the right action of
$\widehat{\mathcal{D}}_{W(X)}^{(0)}$ on $W\omega_{X}$).

Now, to show the required exactness it suffices to work after applying
$\otimes_{W_{r}(k)}^{L}k$. Working locally, we have an isomorphism

$(\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r})_{c-\text{acc}}\widehat{\otimes}_{\mathcal{O}_{W(X)/p^{r}}}W\Omega_{X/Y}^{i}/p^{r}\tilde{=}\Phi^{!}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}/p^{r})\widehat{\otimes}_{\mathcal{O}_{\mathfrak{X}}/p^{r}}W\Omega_{X/Y}^{i}/p^{r}$
$:=\lim_{\leftarrow}(\Phi^{!}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}/p^{r}))/F^{m}\otimes{}_{\mathcal{O}_{\mathfrak{X}}/p^{r}}W\Omega_{X/Y}^{i}/p^{r}$

and, since
$(\Phi^{!}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}/p^{r}))/F^{m}$ is
locally free over $\mathcal{O}_{\mathfrak{X}}/p^{r}$, we see that this sheaf
is flat over $W_{r}(k)$. Therefore it suffices to analyze the complex

$(\widehat{\mathcal{D}}_{W(X)}^{(0)}/p)_{c-\text{acc}}\widehat{\otimes}_{\mathcal{O}_{W(X)/p}}W\Omega_{X/Y}^{\cdot}/p$

Again working locally, we have the isomorphism

$(\widehat{\mathcal{D}}_{W(X)}^{(0)}/p)_{c-\text{acc}}\widehat{\otimes}_{\mathcal{O}_{W(X)/p}}W\Omega_{X/Y}^{\cdot}/p$
$\tilde{\to}R\lim_{\leftarrow}(\Phi_{m}^{!}(\mathcal{D}_{X}^{(0)}))\otimes_{\mathcal{O}_{X}}W\Omega_{X/Y}^{\cdot}/p$

(where
$\Phi_{m}^{!}(\mathcal{D}_{X}^{(0)})=\Phi^{!}(\mathcal{D}_{X}^{(0)})/V^{m}(\mathcal{O}_{W(X)}/p)$).
In fact we shall show that

$\mathcal{H}^{i}(\Phi_{m}^{!}(\mathcal{D}_{X}^{(0)}))\otimes_{\mathcal{O}_{\mathfrak{X}/p}}W\Omega_{X/Y}^{\cdot}/p)=0$

for $i<d$ and

$\mathcal{H}^{d}(\Phi_{m}^{!}(\mathcal{D}_{X}^{(0)}))\otimes_{\mathcal{O}_{\mathfrak{X}/p}}W\Omega_{X/Y}^{\cdot}/p)\tilde{\to}\Phi_{m}^{!}(W\varphi^{-1}(\mathcal{O}_{W(Y)}/p)\widehat{\otimes}_{\varphi^{-1}(\mathcal{O}_{Y})}\mathcal{D}_{Y\leftarrow
X}^{(0)})$

where, in the right hand side, $\Phi_{m}^{!}$ refers to
$\otimes_{\mathcal{D}_{X}^{(0)}}\Phi_{m}^{!}(\mathcal{D}_{X}^{(0)})$ via the
right $\mathcal{D}_{X}^{(0)}$-module structure on $\mathcal{D}_{Y\leftarrow
X}^{(0)}$, and the completion is with respect to the $V$-adic completion of
$\mathcal{O}_{W(Y)}$. As each tower
$\\{\Phi_{m}^{!}(\mathcal{D}_{X}^{(0)}))\otimes_{\mathcal{O}_{\mathfrak{X}/p}}W\Omega_{X/Y}^{\cdot}/p\\}_{m}$
is a tower of quasicoherent sheaves satisfying the Mittag-Leffler condition,
we may take the inverse limit over $m$ (c.f. [28], lemma 1.1.6), so the result
then follows directly from 4.26.

To prove the required equalities, let $(\mathcal{M},\nabla)$ be any integrable
connection on $X$. Then, as proved above, there is an induced de Rham-Witt
connection on $\widehat{\Phi}^{*}\mathcal{M}$, and therefore a de Rham-Witt
complex

$\widehat{\Phi}^{*}\mathcal{M}\widehat{\otimes}_{\mathcal{O}_{W(X)/p}}W\Omega_{X/Y}^{\cdot}/p\tilde{\to}\mathcal{M}\otimes_{\mathcal{O}_{X}}W\Omega_{X/Y}^{\cdot}/p$

We note that the quotient by
$\mathcal{M}\otimes_{\mathcal{O}_{X}}V(W\Omega_{X/Y}^{\cdot}/p)$ induces a map

$\mathcal{M}\otimes_{\mathcal{O}_{X}}W\Omega_{X/Y}^{\cdot}/p\to\mathcal{M}\otimes_{\mathcal{O}_{X}}(\Omega_{X/Y}^{\cdot}\widehat{\otimes}_{\varphi^{-1}(\mathcal{O}_{Y})}W\varphi^{-1}(\mathcal{O}_{W(Y)}/p))$

If $\mathcal{M}$ is a vector bundle with nilpotent connection, this map
induces an isomorphism

$\mathcal{H}^{i}(\mathcal{M}\otimes_{\mathcal{O}_{X}}W\Omega_{X/Y}^{\cdot}/p)\tilde{\to}\mathcal{H}^{i}(\mathcal{M}\otimes_{\mathcal{O}_{X}}\Omega_{X/Y}^{\cdot})\widehat{\otimes}_{\varphi^{-1}(\mathcal{O}_{Y})}W\varphi^{-1}(\mathcal{O}_{W(Y)}/p)$

(via Cartier descent, this follows formally from the case $\mathcal{O}_{X}$,
where it a basic computation, c.f., [32], section 3.2); more precisely it is
induced from

$\mathcal{H}^{i}(\mathcal{M}\otimes_{\mathcal{O}_{X}}W_{j}\Omega_{X/Y}^{\cdot}/p)\tilde{\to}\mathcal{H}^{i}(\mathcal{M}\otimes_{\mathcal{O}_{X}}\Omega_{X/Y}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{O}_{Y})}W\varphi^{-1}(\mathcal{O}_{W_{j}(Y)}/p)$

by taking the inverse limit over $j$.

Now, the quotient of $\mathcal{D}_{X}^{(0)}$ along any power of the central
ideal $\mathcal{I}_{1}$ defining the nilpotence condition, is such a vector
bundle with nilpotent connection. Thus we obtain for each $m,j,s$

$\mathcal{H}^{i}(\Phi_{m}^{!}(\mathcal{D}_{X}^{(0)}/\mathcal{I}^{s})\otimes_{\mathcal{O}_{X}}W_{j}\Omega_{X/Y}^{\cdot}/p)\tilde{\to}\mathcal{H}^{i}(\Phi_{m}^{!}(\mathcal{D}_{X}^{(0)}/\mathcal{I}^{s})\otimes_{\mathcal{O}_{X}}\Omega_{X/Y}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{O}_{Y})}W\varphi^{-1}(\mathcal{O}_{W_{j}(Y)}/p)$

and, as everything in question is a finite right
$\mathcal{D}_{X}^{(0)}$-module we may take the inverse limit over $s$ and
obtain

$\mathcal{H}^{i}(\Phi_{m}^{!}(\mathcal{D}_{X,\text{crys}}^{(0)})\otimes_{\mathcal{O}_{X}}W_{j}\Omega_{X/Y}^{\cdot}/p)\tilde{\to}\mathcal{H}^{i}(\Phi_{m}^{!}(\mathcal{D}_{X,\text{crys}}^{(0)})\otimes_{\mathcal{O}_{X}}\Omega_{X/Y}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{O}_{Y})}W\varphi^{-1}(\mathcal{O}_{W_{j}(Y)}/p)$

this vanishes for $i<d$ and is isomorphic to
$\Phi_{m}^{!}(W\varphi^{-1}(\mathcal{O}_{W_{j}(Y)}/p)\otimes_{\varphi^{-1}(\mathcal{O}_{Y})}\mathcal{D}_{Y\leftarrow
X,\text{crys}}^{(0)})$ when $i=d$. This shows that both the kernel and the
cokernel of

$\mathcal{H}^{i}(\Phi_{m}^{!}\mathcal{D}_{X}^{(0)}\otimes_{\mathcal{O}_{X}}W_{j}\Omega_{X/Y}^{\cdot}/p)\tilde{\to}W\varphi^{-1}(\mathcal{O}_{W_{j}(Y)}/p)\otimes_{\varphi^{-1}(\mathcal{O}_{Y})}\mathcal{H}^{i}(\Phi_{m}^{!}\mathcal{D}_{X}^{(0)}\otimes_{\mathcal{O}_{X}}\Omega_{X/Y}^{\cdot})$

are supported, as modules over
$\mathcal{Z}(\mathcal{D}_{X}^{(0)})\tilde{=}\mathcal{O}_{T^{*}X^{(1)}}$, away
from the zero section.

To show that the kernel and cokernel are actually $0$, we argue as follows:
passing to the algebraic closure of $k$, we consider a closed point $x\in
T^{*}X^{(1)}$. As $X$ is affine and has local coordinates, we have that $x$ is
contained in the subscheme defined by the ideal
$(\partial_{1}-a_{1}^{p},\partial_{2}-a_{2}^{p},\dots,\partial_{n}-a_{n}^{p})$
for some $a_{i}\in k$. The algebra of global sections
$\Gamma(\mathcal{D}_{X}^{(0)})$ possesses an automorphism $\chi$ which
preserves $\Gamma(\mathcal{O}_{X})$ and sends $\partial_{i}$ to
$\partial_{i}-a_{i}$ (this is clear from the defining relations on
$\mathcal{D}_{X}^{(0)}$). This automorphism preserves
$\mathcal{Z}(\mathcal{D}_{X}^{(0)})$, and the associated action on
$T^{*}X^{(1)}$ interchanges the zero section $X^{(1)}$ with the subscheme
defined by
$(\partial_{1}-a_{1}^{p},\partial_{2}-a_{2}^{p},\dots,\partial_{n}-a_{n}^{p})$.

Now, we may repeat the above argument for $(\mathcal{D}_{X}^{(0)})^{\chi}$,
the $\mathcal{D}_{X}^{(0)}$-module whose left and right module structure are
twisted by the action of $\chi$. This shows that the kernel and cokernel of

$\mathcal{H}^{i}(\Phi_{m}^{!}\mathcal{D}_{X}^{(0)}\otimes_{\mathcal{O}_{X}}W_{j}\Omega_{X/Y}^{\cdot}/p)\tilde{\to}W\varphi^{-1}(\mathcal{O}_{W_{j}(Y)}/p)\otimes_{\varphi^{-1}(\mathcal{O}_{Y})}\mathcal{H}^{i}(\Phi_{m}^{!}\mathcal{D}_{X}^{(0)}\otimes_{\mathcal{O}_{X}}\Omega_{X/Y}^{\cdot})$

are supported away from the subscheme defined by the ideal
$(\partial_{1}-a_{1}^{p},\partial_{2}-a_{2}^{p},\dots,\partial_{n}-a_{n}^{p})$.
As this is true for all $(a_{1},\dots,a_{n})$ in $k$, we see that in fact the
support of the kernel and cokernel are empty, as required. ∎

Now we want to compare the (relative) de Rham-Witt cohomology with the
relative pushforward constructed above. To do so, we need the following basic
fact:

######  Theorem 4.40.

Fix $r\geq 1$ and let $\mathcal{M}^{\cdot}$ be a bounded accessible complex
over $\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r}$; and let
$\widehat{\mathcal{M}}^{\cdot}$ be its derived completion (as discussed above
3.25). Consider the functor

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

where on the right we define

$\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{$c-$acc}}^{(0)}/p^{r}\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r}}^{L}\mathcal{\widehat{M}}^{\cdot}$
$:=\text{holim}_{i}(W\varphi^{-1}((\mathcal{O}_{W(Y)}/p^{r})/V^{i})\otimes_{W\varphi^{-1}((\mathcal{O}_{W(Y)}/p^{r})}^{L}\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{$c-$acc}}^{(0)}/p^{r}\otimes_{\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r}}^{L}\mathcal{\widehat{M}}^{\cdot})$

Then there is a canonical isomorphism

$\text{holim}_{i}((\mathcal{O}_{W(Y)}/p^{r})/V^{i})\otimes_{\mathcal{O}_{W(Y)}/p^{r}}^{L}\int_{W\varphi}\mathcal{M}^{\cdot}\tilde{\to}\widehat{\int}_{W\varphi}\mathcal{M}^{\cdot}$

###### Proof.

There is a canonical map

$\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(0)}/p^{r}\otimes_{\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r}}^{L}\mathcal{M}^{\cdot}\to\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{$c-$acc}}^{(0)}/p^{r}\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r}}^{L}\mathcal{\widehat{M}}^{\cdot}$

which induces a map

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

Work locally on $Y$; choose a coordinatized lift of Frobenius and an
associated $\Phi$. We will show that the complex
${\displaystyle\widehat{\int}_{W\varphi}\mathcal{M}^{\cdot}}$ is a bounded
complex which is quasi-isomorphic to a complex of the form
$\widehat{\Phi}\mathcal{K}^{\cdot}$. This shows that the map above factors
through $\int_{W\varphi}\mathcal{M}^{\cdot}$, we then show that in fact
$\Phi^{*}\mathcal{K}^{\cdot}\tilde{\to}{\displaystyle\int_{W\varphi}\mathcal{M}^{\cdot}}$;
this immediately implies the result.

Begin by assuming that $X$ is also affine. As $\varphi$ is smooth it is
locally compatible with a lift of Frobenius, which we also call $\Phi$ and so
by 4.26 we have

$\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{acc}}^{(m)}/p^{r}\tilde{=}\varphi^{-1}(\Phi^{*}\mathcal{D}_{\mathfrak{Y_{r}}}^{(0)})\widehat{\otimes}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y_{r}}}^{(0)})}^{L}\mathcal{D}_{\mathfrak{Y_{r}}\leftarrow\mathfrak{X}_{r}}^{(0)}\otimes_{\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}}^{L}\Phi^{!}\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}$

where the $\widehat{\otimes}$ on the left indicates completion with respect to
$W\varphi^{-1}(V^{i}(\mathcal{O}_{W(X)}/p^{r}))$. Writing
$\mathcal{\widehat{M}}^{\cdot}=\widehat{\Phi}^{*}\mathcal{N}^{\cdot}$ we see
that

$\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{$c-$acc}}^{(0)}/p^{r}\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r}}^{L}\mathcal{\widehat{M}}^{\cdot}$
$\tilde{=}\varphi^{-1}(\Phi^{*}\mathcal{D}_{\mathfrak{Y_{r}}}^{(0)})\widehat{\otimes}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y_{r}}}^{(0)})}^{L}\mathcal{D}_{\mathfrak{Y_{r}}\leftarrow\mathfrak{X}_{r}}^{(0)}\otimes_{\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}}^{L}\mathcal{N}^{\cdot}$

Now,
$\mathcal{D}_{\mathfrak{Y_{r}}\leftarrow\mathfrak{X}_{r}}^{(0)}\otimes_{\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}}^{L}\mathcal{N}^{\cdot}$
is a bounded complex of
$\varphi^{-1}(\mathcal{D}_{\mathfrak{Y_{r}}}^{(0)})$-modules; if we represent
it by a bounded complex $\mathcal{F}^{i}$, then the $i$th term of the above is
given by

$\varphi^{-1}(\Phi^{*}\mathcal{D}_{\mathfrak{Y_{r}}}^{(0)})\widehat{\otimes}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y_{r}}}^{(0)})}\mathcal{F}^{i}$

which is simply an infinite product of copies of $\mathcal{F}^{i}$. So we see
easily that

$R\varphi_{*}(\varphi^{-1}(\Phi^{*}\mathcal{D}_{\mathfrak{Y_{r}}}^{(0)})\widehat{\otimes}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y_{r}}}^{(0)})}\mathcal{F}^{i})\tilde{\to}\Phi^{*}\mathcal{D}_{\mathfrak{Y_{r}}}^{(0)}\widehat{\otimes}_{\mathcal{D}_{\mathfrak{Y_{r}}}^{(0)}}R\varphi_{*}(\mathcal{F}^{i})$

and the analogous result follows easily for a bounded complex (by induction on
the cohomological length), so that we have

$R\varphi_{*}(\varphi^{-1}(\Phi^{*}\mathcal{D}_{\mathfrak{Y_{r}}}^{(0)})\widehat{\otimes}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y_{r}}}^{(0)})}^{L}\mathcal{D}_{\mathfrak{Y_{r}}\leftarrow\mathfrak{X}_{r}}^{(0)}\otimes_{\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}}^{L}\mathcal{N}^{\cdot})\tilde{\to}\Phi^{*}\mathcal{D}_{\mathfrak{Y_{r}}}^{(0)}\widehat{\otimes}_{\mathcal{D}_{\mathfrak{Y_{r}}}^{(0)}}^{L}R\varphi_{*}(\mathcal{D}_{\mathfrak{Y_{r}}\leftarrow\mathfrak{X}_{r}}^{(0)}\otimes_{\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}}^{L}\mathcal{N}^{\cdot})$

This shows that both
${\displaystyle\widehat{\int}_{W\varphi}\mathcal{M}^{\cdot}}$ is quasi-
isomorphic to a complex of the form $\widehat{\Phi}\mathcal{K}^{\cdot}$ and
that
$\Phi^{*}\mathcal{K}^{\cdot}\tilde{\to}{\displaystyle\int_{W\varphi}\mathcal{M}^{\cdot}}$,
which is what we wanted.

To get the result for a general $X$, simply cover it with affines; and use the
fact that if $X=U\cup V$ then there is a distinguished triangle

$\mathcal{M}^{\cdot}\to(j_{U})_{*}\mathcal{M}^{\cdot}\oplus(j_{V})_{*}\mathcal{M}^{\cdot}\to(j_{U\cap
V})_{*}\mathcal{M}^{\cdot}$

where $j_{U},j_{V},$$j_{U\cap V}$ denote the inclusions from those open
subsets, respectively; this allows one to do induction on the number of open
affines and deduce the result. ∎

Putting it all together, we conclude:

######  Corollary 4.41.

Let $\varphi:X\to Y$ be smooth, and let
$\mathcal{M}\in\mathcal{D}_{W(X)}^{(0)}/p^{r}-\text{mod}_{\text{acc,qcoh}}$.
Let
$\mathcal{\widehat{M}}\widehat{\otimes}_{\mathcal{O}_{W(X)}/p}W\Omega_{X/Y}^{\cdot}$
be the associated de Rham-Witt complex. There is an isomorphism of sheaves on
$W(Y)_{p^{r}=0}$

$\widehat{\int}_{W\varphi}\mathcal{M}[d]\tilde{\to}RW\varphi_{*}(\mathcal{\widehat{M}}\widehat{\otimes}_{\mathcal{O}_{W(X)}/p}W\Omega_{X/Y}^{\cdot})$

In particular, if $\mathcal{M}$ is nilpotent and quasicoherent, and
$\mathcal{\widehat{M}}=\widehat{\epsilon}(\mathcal{N})$ for a quasicoherent
crystal $\mathcal{N}$, then for each $i\geq 0$ we have

$\widehat{\epsilon}(R^{i}\varphi_{*,\text{crys}}\mathcal{N})\tilde{\to}R^{i}W\varphi_{*}(\mathcal{M}\widehat{\otimes}_{\mathcal{O}_{W(X)}/p}W\Omega_{X/Y}^{\cdot})$

###### Proof.

The second sentence follows from the first by 4.34. To prove the first, we
shall invoke the previous two results. First, by Theorem 4.39 there is a
quasi-isomorphism

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

Now, the lemma below ensures that, for each $i$,
$(\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r})_{c-\text{acc}}\widehat{\otimes}_{\mathcal{O}_{W(X)/p^{r}}}W\Omega_{X/Y}^{i}/p^{r}$
is flat over $\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r}$ (in fact, locally, it
is an inverse limit of a surjective system of projective modules of the form
$\Phi^{*}\mathcal{D}_{\mathfrak{X}_{r}}^{(0)}$). So on the right hand side we
actually have the complex whose terms are

$(\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r})_{c-\text{acc}}\widehat{\otimes}_{\mathcal{O}_{W(X)/p^{r}}}W\Omega_{X/Y}^{i}/p^{r}\otimes_{\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r}}^{L}\mathcal{\widehat{M}}$

which we can complete (with respect to the natural filtration on the tensor
product) and thus get a map to the complex whose terms are

$(\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r})_{c-\text{acc}}\widehat{\otimes}_{\mathcal{O}_{W(X)/p^{r}}}W\Omega_{X/Y}^{i}/p^{r}\otimes_{\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r}}^{L}\mathcal{\widehat{M}}\tilde{\to}W\Omega_{X/Y}^{i}/p^{r}\widehat{\otimes}_{\mathcal{O}_{W(X)/p^{r}}}\mathcal{\widehat{\mathcal{M}}}$

(the last isomorphism is because $\mathcal{M}$ is accessible). Summing up,
we’ve obtained a map

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

To apply the previous result, we have to take the derived completion with
respect to $\\{W\varphi^{-1}V^{i}(\mathcal{O}_{W(Y)}/p^{r})\\}$ (i.e.,
applying holim to
$W\varphi^{-1}((\mathcal{O}_{W(Y)}/p^{r})/V^{i})\otimes_{W\varphi^{-1}((\mathcal{O}_{W(Y)}/p^{r})}^{L}$).
However, it is not difficult to see that
$W\Omega_{X/Y}^{\cdot}/p^{r}\widehat{\otimes}_{\mathcal{O}_{W(X)/p^{r}}}\mathcal{\widehat{\mathcal{M}}}$
is already derived complete with respect to
$\\{W\varphi^{-1}V^{i}(\mathcal{O}_{W(Y)}/p^{r})\\}$ (for instance, one may
work locally and relate the $\\{V^{i}(\mathcal{O}_{W(Y)}/p^{r})\\}$ to the
filtration $\\{F^{i}(\mathcal{O}_{W(Y)}/p^{r})\\}$ as in the proof of 3.25).
Thus we actually obtain a map

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

The result follows if we can show that it is a quasi-isomorphism. Working
locally, we can assume $\mathcal{M}=\Phi^{*}\mathcal{N}$. We begin in the case
$r=1$. In that case, we remark that if $\mathcal{M}^{\cdot}$ is any complex of
accessible quasicoherent modules, by applying the above construction to each
$\mathcal{M}^{i}$ and taking the total complex, we obtain a morphism

(4.8) $\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{$c-$acc}}^{(0)}/p\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(0)}/p}^{L}\mathcal{\widehat{M}}^{\cdot}\to
W\Omega_{X/Y}^{\cdot}/p\widehat{\otimes}_{\mathcal{O}_{W(X)/p}}^{L}\mathcal{\widehat{\mathcal{M}}}^{\cdot}$

(this works as $\mathcal{M}$ is accessible and $W\Omega_{X/Y}^{\cdot}/p$ is
flat over $\mathcal{O}_{X}$ by the lemma below) and so for each $j$ we obtain
an induced map

$\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{$c-$acc}}^{(0)}/p\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(0)}/p}^{L}\mathcal{\widehat{M}}^{\cdot}\to
W_{j}\Omega_{X/Y}^{\cdot}/p\widehat{\otimes}_{\mathcal{O}_{W_{j}(X)/p}}^{L}\mathcal{M}_{j}^{\cdot}$

(where
$\mathcal{M}_{j}^{\cdot}=\mathcal{M}^{\cdot}\otimes_{\mathcal{O}_{W(X)}/p}^{L}(\mathcal{O}_{W(X)}/p/V^{j})$.
Consider the set $\mathcal{S}$ of complexes $\mathcal{M}^{\cdot}$ for which
this map is a quasi-isomorphism for all $j$. By the previous theorem we have

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

So that, by Theorem 4.39
$\mathcal{M}=\Phi^{*}\mathcal{D}_{X}^{(0)}\in\mathcal{S}$. It is also clear
that $\mathcal{S}$ is closed under cones, arbitrary sums, and summands. But
since it contains $\Phi^{*}\mathcal{D}_{X}^{(0)}$, it must therefore121212By
the fact that $\Phi^{*}\mathcal{D}_{X}^{(0)}$, along with all of its shifts,
are set of compact generators for
$D_{\text{acc,qcoh}}(\widehat{\mathcal{D}}_{W(Y)}^{(0)}/p^{r}-\text{mod})$
contain any quasi-coherent accessible complex; taking the complex to be in a
single degree proves the result when $r=1$. It then follows, by induction on
the cohomological length, that (4.8) is an isomorphism for bounded complexes.

Now let $r\geq 1$. To prove that (4.7) is an isomorphism, we regard both sides
as complexes of $W(k)$-modules and apply $\otimes_{W(k)}^{L}k$ to both sides.
By the argument of 4.25 (where the role of
$\widehat{\mathcal{D}}_{W(Y)\leftarrow W(X)}^{(m)}$ is played by
${\displaystyle\lim_{r}\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{$c-$acc}}^{(0)}/p^{r}}$), we have

$(\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{$c-$acc}}^{(0)}/p^{r}\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(0)}/p^{r}}^{L}\mathcal{\widehat{M}})\otimes_{W(k)}^{L}k\tilde{\to}\widehat{\mathcal{D}}_{W(Y)\leftarrow
W(X),\text{$c-$acc}}^{(0)}/p\widehat{\otimes}_{\widehat{\mathcal{D}}_{W(X)}^{(0)}/p}^{L}(\mathcal{\widehat{M}}\otimes_{W(k)}^{L}k)$

and similarly for
$W\Omega_{X/Y}^{\cdot}/p^{r}\widehat{\otimes}_{\mathcal{O}_{W(X)/p^{r}}}\mathcal{\widehat{\mathcal{M}}}$.
As the complex $(\mathcal{\widehat{M}}\otimes_{W(k)}^{L}k)$ is bounded, the
result follows from (4.8) for bounded complexes. ∎

In the proof, we needed the

######  Lemma 4.42.

Suppose $A$ and $B$ are smooth $k$-algebras so that $B\to A$ is smooth. Let
$A$ posses local coordinates, and let $\Phi:\mathcal{A}\to W(A)$. For each
$i\geq 0$, the module $W\Omega_{A/B}^{i}$ is (faithfully) flat over
$\mathcal{A}$. Further, for each $j\geq 0$ the module
$W_{j}\Omega_{A/B}^{i}/p$ is finite flat over $A$.

###### Proof.

It is enough to show that $W\Omega_{A/B}^{i}/p$ is faithfully flat over $A$,
where $A$ acts via the embedding $\Phi:A\to W(A)/p$; and therefore it is
enough to show that second statement of the lemma. As $W\Omega_{A/B}^{i}/p$ is
local for the etale topology, we can assume that $A=B[T_{1},\dots T_{n}]$;
and, by extending the ground field, that $k$ is algebraically closed.

The module $W\Omega_{A/B}^{i}/p$ is, by definition, the inverse limit of
$W_{j}\Omega_{A/B}^{i}/p$, the quotient of $W\Omega_{A/B}^{i}/p$ by the module
$V^{j}(W\Omega_{A/B}^{i}/p):=V^{j}(W\Omega_{A/B}^{i})/p$. Each
$\text{gr}_{j}(W\Omega_{A}^{i}/p)=V^{j}(W\Omega_{A/B}^{i}/p)/V^{j+1}(W\Omega_{A/B}^{i}/p)$
is a module over

$(W(A)/p)/V^{j+1}(W(A)/p)=W_{j+1}(A)/p$

In general this action does not factor through the quotient
$W_{j+1}(A)/p/V(W_{j+1}(A)/p)=A$. We therefore refine this filtration by
defining

$\text{gr}_{j,t}(W\Omega_{A}^{i}/p)=V^{t}(W(A)/p)\cdot\text{gr}_{j}(W\Omega_{A}^{i}/p)$

for $0\leq t\leq j+1$. Then the action of $W_{j+1}(A)/p$ on each
$\text{gr}_{j,t}(W\Omega_{A}^{i}/p)/\text{gr}_{j,t+1}(W\Omega_{A}^{i}/p)$
factors through $A$; and this action of $A$ agrees with the action of $A$ on
$\text{gr}_{j,t}(W\Omega_{A}^{i}/p)/\text{gr}_{j,t+1}(W\Omega_{A}^{i}/p)$
defined via $\Phi_{1}$. Thus
$(W\Omega_{A/B}^{i}/p)/V^{j}(W\Omega_{A/B}^{i}/p)$ is filtered by finitely
many $A$-modules
$\text{gr}_{j,t}(W\Omega_{A/B}^{i}/p)/\text{gr}_{j,t+1}(W\Omega_{A/B}^{i}/p)$
and so it suffices to show each one is projective over $A$.

To do that, note that any automorphism $\sigma$ of $A$ which preserves $B$
yields an isomorphism of $A$-modules

$\sigma^{*}\text{gr}_{j,t}(W\Omega_{A/B}^{i}/p)/\text{gr}_{j,t+1}(W\Omega_{A/B}^{i}/p)\tilde{\to}\text{gr}_{j,t}(W\Omega_{A/B}^{i}/p)/\text{gr}_{j,t+1}(W\Omega_{A/B}^{i}/p)$

Indeed, let $\sigma:W_{n+1}(A)/p\to W_{n+1}(A)/p$ denote the induced
isomorphism. Then we have
$\sigma^{*}\text{gr}_{j}(W\Omega_{A/B}^{i}/p)\tilde{\to}\text{gr}_{j}(W\Omega_{A/B}^{i}/p)$
by the functoriality of the de Rham Witt construction. The result then follows
for $\text{gr}_{j,t}(W\Omega_{A}^{i}/p)$ from the fact that the automorphism
$\sigma$ preserves $V^{t}(W(A)/p)$. As the action of automorphisms of
$A=B[T_{1},\dots T_{n}]$ is transitive on closed points in each fibre over
$\text{Spec}(B)$, we see that the finite $A$-module
$\text{gr}_{j,t}(W\Omega_{A/B}^{i}/p)/\text{gr}_{j,t+1}(W\Omega_{A/B}^{i}/p)$
is necessarily projective (the geometric fibres all have the same rank) and so
the result follows. ∎

##  5\. The algebra $\widehat{\mathcal{U}}$, and applications

In this chapter we give an alternate construction of accessibility on the
scheme $W(X)/p$ via an algebra $\widehat{\mathcal{U}}$, which is a kind of
enveloping algebra for derivations of $\mathcal{O}_{W(X)}/p$. This allows us
to prove 3.16in all positive characteristics, in particular avoiding Theorem
3.6. As mentioned above, the key to the proof is the fact that $W(A)/p\to A$
is a square zero extension; the argument is a really as rephrasing of
Grothendieck’s fundamental result which says that a flat connection extends
uniquely over a square zero extension (c.f. [7], the introduction to chapter 2
for a nice discussion of this). Once this is done, we turn to a deeper study
of the relationship between de Rham-Witt connections and accessible
$\mathcal{D}_{W(X)}^{(0)}$-modules. Working mod $p$ we will use
$\widehat{\mathcal{U}}$ to show that there is (essentially) an equivalence
between the two. Then we will lift this mod $p^{r}$ and in particular prove
Theorem 1.15.

###  5.1. The algebra $\widehat{\mathcal{U}}$.

Until further notice $A$ is a smooth $k$-algebra which possesses local
coordinates. We begin by constructing an algebra of differential operators
over $W(A)/p$, which is closely related to
$\mathcal{\widehat{D}}_{W(A)}^{(0)}/p$, and which we will use to analyze the
pull-back from $\mathcal{D}_{A}^{(0)}$-modules. Let us set some notation:

######  Definition 5.1.

For each $j\geq 1$ let $\mathcal{I}_{j}=\text{ker}(W(A)/p\to
W_{j}(A)/p)=V^{j}(W(A)/p)$; and set $\mathcal{I}_{0}=W(A)/p$. Define
$\mathcal{T}^{\prime}_{W(X)/p}:=\\{\partial\in\text{Der}_{\text{cont}}(W(A)/p)|\partial(\mathcal{I}_{j})\subset\mathcal{I}_{j}\phantom{i}\text{for
all}\phantom{i}j\\}$.

Clearly $\mathcal{T}^{\prime}_{W(A)/p}$ is a Lie algebras under the natural
bracket of derivations. However, we have more:

######  Lemma 5.2.

Let $\mathcal{T}^{\prime}_{W(A)/p}(1)$ be the sub-sheaf of Lie-algebras of
$\mathcal{T}^{\prime}_{W(A)/p}$ consisting of derivations whose image lies in
$\mathcal{I}_{1}$.

1) There is a natural map $\mathcal{T}^{\prime}_{W(A)/p}\to\mathcal{T}_{A}$.
The kernel of this map is $\mathcal{T}^{\prime}_{W(A)/p}(1)$.

2) For $\partial_{1},\partial_{2}\in\mathcal{T}^{\prime}_{W(A)/p}(1)$, the
composition $\partial_{1}\circ\partial_{2}\in\mathcal{T}^{\prime}_{W(A)/p}(1)$
as well. Thus $\mathcal{T}^{\prime}_{W(A)/p}(1)$ carries the structure of a
sheaf of (non-unital) algebras.

###### Proof.

1) Since any $\delta\in\mathcal{T}^{\prime}_{W(A)/p}$ takes $\mathcal{I}_{1}$
to itself, it induces a derivation
$\bar{\delta}:(W(A)/p)/\mathcal{I}_{1}\to(W(A)/p)/\mathcal{I}_{1}$. But since
$(W(A)/p)/\mathcal{I}_{1}=A$ we obtain the result.

2) Since $\mathcal{I}_{1}$ is a square zero ideal, if
$\partial_{i}\in\mathcal{T}^{\prime}_{W(A)/p}(1)$ for $i=1,2$, then
$\partial_{1}(a)\partial_{2}(b)=0$ for all $a$ and $b$. Therefore, we have

$\partial_{1}\partial_{2}(ab)=\partial_{1}(b\partial_{2}(a)+a\partial_{2}(b))=b\partial_{1}\partial_{2}(a)+a\partial_{1}\partial_{2}(b)$

as claimed. ∎

Now we recall the definition of a Lie-Rinehart enveloping algebra. Let $S$ be
an algebra over a commutative ring $R$. Denote by $\mathcal{T}_{S}$ the module
of $R$-derivations on $S$, and suppose that $\mathfrak{a}$is a Lie algebra
which is also an $S$-module, equipped with a morphism
$\rho:\mathfrak{a}\to\mathcal{T}_{Y}$ which is both $S$-linear and a morphism
of Lie algebras, and such that

$[a,fb]=f[a,b]+\rho(a)(f)\cdot b$

for all $a,b$ in $\mathfrak{a}$ and $f\in S$. In this situation we have the
sheaf of universal enveloping algebras $\mathcal{U}(\mathfrak{a},\rho)$ (or
simply $\mathcal{U}(\mathfrak{a})$ if $\rho$ is understood) as constructed in
[35], c.f. also [BB] section 1.4. Namely, $\mathcal{U}(\mathfrak{a},\rho)$ is
the $S$-algebra generated by $\mathfrak{a}$ and satisfying the relations:

$f\cdot a=fa$

for $f\in S$ and $a\in\mathfrak{a}$; here, the left hand side denotes the
multiplication in $\mathcal{U}(\mathfrak{a},\rho)$ and the right hand side
denotes the action of $S$ on $\mathfrak{a}$; next we demand

$a_{1}a_{2}-a_{2}a_{1}=[a_{1},a_{2}]$

for $a_{1},a_{2}\in\mathfrak{a}$, where the bracket on the right denotes the
Lie-algebra bracket, and finally

$a\cdot f-f\cdot a=\rho(a)(f)$

for $f\in S$ and $a\in\mathfrak{a}$.

To get control of this object, one typically makes the assumption that
$\mathfrak{a}$ is a projective $S$-module (c.f., e.g. [1] for examples of
this), which allows one to give an explicit description of
$\mathcal{U}(\mathfrak{a},\rho)$. However, this assumption does not hold in
the situations of interest in this paper; instead, we make the following:

######  Definition 5.3.

The $W(A)/p$-module $\mathcal{T}^{\prime}_{W(A)/p}$ is a sub-Lie-algebra of
the Lie algebra of all derivations $\mathcal{T}_{W(A)/p}$; it is therefore a
Lie algebroid via the inclusion map
$\rho:\mathcal{T}^{\prime}_{W(A)/p}\to\mathcal{T}{}_{W(A)/p}$. Define the
algebra $\mathcal{U}^{\prime}_{W(A)/p}$ to be quotient of
$\mathcal{U}(\mathcal{T}^{\prime}_{W(A)/p})$ by the two-sided ideal generated
by

$\partial_{1}\cdot\partial_{2}-\partial_{1}\circ\partial_{2}$

for all $\partial_{1},\partial_{2}\in\mathcal{T}^{\prime}_{W(X)/p}(1)$ (here
we are using that $\partial_{1}\circ\partial_{2}$ is a derivation by Lemma
5.2). This algebra is filtered by the two sided ideals
$\\{\mathcal{K}^{(i)}\\}$, where $\mathcal{K}^{(i)}$ is generated by
$V^{i}(W(A)/p)$ and
$\\{\partial\in\mathcal{T}^{\prime}_{W(X)/p}|\partial(W(A)/p)\subset\mathcal{I}_{i}\\}$.

In order to discuss local coordinates on this algebra, we choose a lift of
Frobenius $F:\mathcal{A}\to\mathcal{A}$, along with the corresponding
$\Phi:\mathcal{A}\to W(A)$. The reduction mod $p$ of $\Phi$ is a splitting of
the natural reduction map $W(A)/p\to A$. Until further notice, we fix such a
map and we regard $A\subset W(A)/p$; in fact this puts $A\subset W_{r+1}(A)/p$
for each $r$, and we have

######  Lemma 5.4.

The inclusion $A\subset W_{r+1}(A)/p$ makes $W_{r+1}(A)/p$ a free $A$-module.
Furthermore, there is an isomorphism of $A$-modules

$W_{r+1}(A)/p\tilde{=}A\oplus\bigoplus_{i=1}^{r}K_{i}/K_{i+1}$

###### Proof.

In fact, one may check easily that a basis of $W_{r+1}(A)/p$ over $A$ is given
by

$1\cup\\{V^{i}(T^{I})\\}_{1\leq i\leq r}$

where, for a fixed $i$, the multi-index $I$ ranges over
$\\{(i_{1},\dots,i_{n})\\}$ such that each $i_{j}$ is contained in
$\\{1,\dots,p^{i}\\}$ and at least one $i_{j}$ is not divisible by $p$. This
directly implies both statements. ∎

Using the isomorphism of the previous lemma we have

######  Lemma 5.5.

There is an isomorphism of Lie algebras

$\mathcal{T}^{\prime}_{W(A)/p}\tilde{=}\text{Der}_{A}(A,W(A)/p)\oplus\tilde{\text{Hom}}_{A}(\mathcal{I}_{1})$

where on the right we have filtration-preserving $A$-linear maps on
$\mathcal{I}_{1}$. Equivalently,

$\mathcal{T}^{\prime}_{W(A)/p}\tilde{=}\text{Der}_{A}(A,W_{1}(A)/p)\oplus\mathcal{T}^{\prime}_{1}$

where $\mathcal{T}^{\prime}_{1}$ consists of derivations in
$\mathcal{T}^{\prime}_{W(A)/p}$ which vanish on $A$.

###### Proof.

Let $\partial\in\mathcal{T}^{\prime}_{W(A)/p}$. Then the restriction of
$\partial$ to $A$ is a derivation from $A$ into $W(A)/p$. Further, if
$\delta:A\to W(A)/p$ is any derivation, we may extend it to an element of
$\mathcal{T}^{\prime}_{W(A)/p}$ by setting $\delta$ to be zero on each term of
the form $V^{i}(T^{I})$ where, as above, each index $i_{j}$ in $I$ is
contained in $\\{1,\dots,p^{i}\\}$ and at least one $i_{j}$ is not divisible
by $p$. Thus we have

$\mathcal{T}^{\prime}_{W(A)/p}\tilde{=}\text{Der}_{A}(A,W_{1}(A)/p)\oplus\mathcal{T}^{\prime}_{1}$

where $\mathcal{T}^{\prime}_{1}$ consists of derivations in
$\mathcal{T}^{\prime}_{W(A)/p}$ which vanish on $A$ which proves the second
claim of the lemma.

To show that this implies the first claim, let $\phi\in\mathcal{T}_{1}$. Then
for any $a\in A$ and $b\in\mathcal{I}_{1}$ we have

$\phi(ab)=a\phi(b)+\phi(a)b=a\phi(b)$

so that $\phi:\mathcal{I}_{1}\to\mathcal{I}_{1}$ is an $A$-linear map.
Conversely, since $\mathcal{I}_{1}$ is a square-zero ideal, any $A$-linear map
$\phi:\mathcal{I}_{1}\to\mathcal{I}_{1}$, extended to all of $W(A)/p$ by
setting $\phi(A)=0$, is a derivation of $W(A)/p$. Since derivations in
$\mathcal{T}^{\prime}_{W(A)/p}$ are required to preserve the filtration given
by the $\\{\mathcal{I}_{j}\\}$, one deduces the first claim of the lemma. ∎

From this we deduce

######  Corollary 5.6.

Let $\\{\partial_{1},\dots,\partial_{n}\\}$ be a set of coordinate derivations
on $A$. Then every element of $\mathcal{U}^{\prime}_{W(A)/p}$ may be written
as

$\sum_{J}b_{J}\partial^{J}+\phi$

where $b_{J}\in W(A)/p$, $\phi$ is contained in the left ideal generated by
$\mathcal{T}_{1}$, and the sum is finite. The analogous statement holds in
$\mathcal{\widehat{U}}^{\prime}_{W(A)/p}$; namely, any element thereof may be
written as

$\sum_{J}b_{J}\partial^{J}+\phi$

where $b_{J}\in W(A)/p$, $\phi$ is contained in the completed left ideal
generated by $\mathcal{T}_{1}$, and $b_{J}\to 0$ in $W(A)/p$ as
$|J|\to\infty$.

###### Proof.

Let $x\in\mathcal{U}^{\prime}_{W(A)/p}$. By definition $x$ can be written as a
sum of terms of the form $\alpha_{1}\cdots\alpha_{m}$ where each $\alpha_{i}$
is either an element of $W(A)/p$, one of the $\partial_{i}$, or
$\phi\in\mathcal{T}_{1}$ (here we use the fact that any element of
$\text{Der}_{A}(A,W_{1}(A)/p)$ is a sum of elements of the form
$a_{i}\partial_{i}$ with $a_{i}\in W(A)/p$). Commuting the $\alpha_{i}$ past
one another and using Lemma 5.5, and then noting that $\mathcal{T}_{1}$ is
closed under products, we deduce the existence of a form as above. The case of
$\mathcal{\widehat{U}}^{\prime}_{W(A)/p}$ follows by taking completion. ∎

We now construct the fundamental bimodule
$W(A)/p\otimes_{A}\mathcal{D}_{A}^{(0)}$:

######  Proposition 5.7.

There is a left action of $\mathcal{U}^{\prime}_{W(A)/p}$ on
$W(A)/p\otimes_{A}\mathcal{D}_{A}^{(0)}$ which makes
$W(A)/p\otimes_{A}\mathcal{D}_{A}^{(0)}$ into a
$(\mathcal{U}^{\prime}_{W(A)/p},\mathcal{D}_{A}^{(0)})$ bimodule.

Let $\Phi^{*}\mathcal{D}_{A}^{(0)}$ be the completion of
$W(A)/p\otimes_{A}\mathcal{D}_{A}^{(0)}$ along the filtration
$\mathcal{I}_{j}\otimes_{A}\mathcal{D}_{A}^{(0)}$. There is a left action of
$\widehat{\mathcal{U}}^{\prime}_{W(A)/p}$ on $\Phi^{*}\mathcal{D}_{A}^{(0)}$
making it into a
$(\mathcal{\widehat{U}}^{\prime}_{W(A)/p},\mathcal{D}_{A}^{(0)})$ bimodule.

###### Proof.

In Lemma 5.2 above, we constructed a surjection
$\mathcal{T}_{W(A)/p}^{\prime}\to\mathcal{T}_{A}$. By the functoriality of the
Lie-Rinehart construction, we obtain a map
$\mathcal{U}(\mathcal{T}^{\prime}_{W(A)/p})\to\mathcal{U}(\mathcal{T}_{A})=\mathcal{D}_{A}^{(0)}$.
As the ideal $\mathcal{T}^{\prime}_{W(A)/p}(1)$ is contained in the kernel of
this map, it necessarily factors through $\mathcal{U}^{\prime}_{W(A)/p}$; so
we obtain $Q:\mathcal{U}^{\prime}_{W(A)/p}\to\mathcal{D}_{A}^{(0)}$.

Now consider a derivation $\partial\in\mathcal{T}_{W(A)/p}^{\prime}$. If we
restrict $\partial$ to $A$ we obtain an element of $\text{Der}_{A}(A,W(A)/p)$;
so we may write

$\partial|_{A}=Q(\partial)+\sum_{i=1}^{n}\epsilon_{i}\partial_{i}$

where $\epsilon_{i}\in\mathcal{I}_{1}$ and $Q(\partial)\in\text{Der}_{A}(A,A)$
is the application of the algebra map $Q$ to $\partial$. (Here $\partial_{i}$
stands for the extension of the canonical coordinate derivation to $W(A)/p$
which vanishes on terms of the form $p^{r}T^{I/p^{r}}$ as above).

Then for $a\in W(A)/p$ and $P\in\mathcal{D}_{A}^{(0)}$ we set

$\partial\cdot(a\otimes P)=\partial(a)\otimes P+a\otimes
Q(\partial)P+\sum_{i=1}^{n}a\epsilon_{i}\otimes\partial_{i}P$

We claim that this defines an action of $\mathcal{U}^{\prime}_{W(A)/p}$ on
$W(A)/p\otimes_{A}\mathcal{D}_{A}^{(0)}$. To see this, let $b\in A$, and $a$
and $P$ as above. Then we have

$\partial(ab\otimes P)=\partial(ab)\otimes P+ab\otimes
Q(\partial)P+\sum_{i=1}^{n}ab\epsilon_{i}\otimes\partial_{i}P$
$=\partial(a)b\otimes P+a\partial(b)\otimes P+ab\otimes
Q(\partial)P+\sum_{i=1}^{n}ab\epsilon_{i}\otimes\partial_{i}P$
$=\partial(a)\otimes bP+a\otimes
Q(\partial)(b)P+\sum_{i=1}^{n}a\epsilon_{i}\otimes\partial_{i}(b)P+a\otimes
bQ(\partial)P+\sum_{i=1}^{n}a\epsilon_{i}\otimes b\partial_{i}P$

where we used
${\displaystyle\partial(b)=Q(\partial)(b)+\sum_{i=1}^{n}\epsilon_{i}\partial_{i}(b)}$.
Further, using the equalities $Q(\partial)(b)P+bQ(\partial)P=Q(\partial)bP$
and $\partial_{i}(b)P+b\partial_{i}P=\partial_{i}bP$, we see that the previous
line is equal to

$\partial(a)\otimes bP+a\otimes
Q(\partial)bP+\sum_{i=1}^{n}a\epsilon_{i}\otimes\partial_{i}bP=\partial(a\otimes
bP)$

Therefore each derivation defines an endomorphism of
$W(A)/p\otimes_{A}\mathcal{D}_{A}^{(0)}$; one sees directly that this defines
an action of a left action of $\mathcal{U}^{\prime}_{W(A)/p}$ on
$W(A)/p\otimes_{A}\mathcal{D}_{A}^{(0)}$. Furthermore, let $b\in A$ and
$\partial_{i}$ a coordinate derivation on $A$. Then

$\partial\cdot(a\otimes Pb)=\partial(a)\otimes Pb+a\otimes
Q(\partial)Pb+\sum_{i=1}^{n}a\epsilon_{i}\otimes\partial_{i}Pb$
$=\partial\cdot(a\otimes P)b$

and similarly $\partial\cdot(a\otimes P\partial_{i})=\partial\cdot(a\otimes
P)\partial_{i}$. So the left action of $\mathcal{U}^{\prime}_{W(A)/p}$
commutes with the natural right action of $\mathcal{D}_{A}^{(0)}$, as
required. The second sentence follows from the first by simply completing both
objects along their natural filtrations. ∎

Then we have

######  Corollary 5.8.

There is an isomorphism

$\Phi^{*}\mathcal{D}_{A}^{(0)}\tilde{\leftarrow}\widehat{\mathcal{U}}^{\prime}_{W(A)/p}/\widehat{\mathcal{U}}^{\prime}_{W(A)/p}\cdot\mathcal{T}^{\prime}_{1}$

where $\widehat{\mathcal{U}}^{\prime}_{W(A)/p}\cdot\mathcal{T}^{\prime}_{1}$
denotes the completion of the left ideal generated by
$\mathcal{T}^{\prime}_{1}$.

###### Proof.

The map is given by sending an element
$x\in\widehat{\mathcal{U}}^{\prime}_{W(A)/p}$ to $x\cdot(1\otimes 1)$. Writing

$x=\sum_{J}b_{J}\partial^{J}+\phi$

as in 5.6, we see

$x\cdot(1\otimes 1)=(\sum_{J}b_{J}\partial^{J}+\phi)\cdot(1\otimes 1)$
$=\sum_{J}b_{j}\otimes\partial^{J}$

(the last equality follows from the definition of the action in 5.7). As the
last sum is zero iff all $b_{J}$ are $0$, the result follows directly. ∎

Now suppose we are in the presence of two distinct coordinatized lifts of
Frobenius $\Phi_{1},\Phi_{2}$ on $\mathcal{A}$, with two associated embeddings
which we now label $\iota_{1},\iota_{2}:A\to W(A)/p$. Then we have

######  Corollary 5.9.

There is a canonical isomorphism
$\Phi_{1}^{*}\mathcal{D}_{A}^{(0)}\tilde{\to}\Phi_{2}^{*}\mathcal{D}_{A}^{(0)}$
of bimodules. If given a third lift $\Phi_{3}$, then the cocycle condition on
isomorphisms is satisfied.

###### Proof.

Slightly modifying the above notation, we let
$\mathcal{T}^{\prime}_{1,\iota_{1}}$ denote the set of derivations in
$\mathcal{T}^{\prime}$ which vanish on $\iota_{1}(A)$, and similarly for
$\mathcal{T}^{\prime}_{1,\iota_{2}}$. We need to provide a canonical
isomorphism

(5.1)
$\widehat{\mathcal{U}}^{\prime}_{W(A)/p}/\widehat{\mathcal{U}}^{\prime}_{W(A)/p}\cdot\mathcal{T}^{\prime}_{1,\iota_{1}}\tilde{\to}\widehat{\mathcal{U}}^{\prime}_{W(A)/p}/\widehat{\mathcal{U}}^{\prime}_{W(A)/p}\cdot\mathcal{T}^{\prime}_{1,\iota_{2}}$

Now, the fact that the kernel of $W(A)/p\to A$ is a square zero ideal implies
that $\iota_{12}=i_{1}-i_{2}:A\to W(A)/p$ is a derivation. We can therefore
consider $\iota_{12}$ as a derivation $\iota_{2}(A)\to W(A)/p$, and then we
extend this to a derivation $\tilde{\iota}_{12}:W(A)/p\to W(A)/p$ as in the
proof of 5.7, so that $\tilde{\iota}_{12}(p^{r}T^{I/p^{r}})=0$, where $T$
refers to the coordinates for $\Phi_{2}$. Then we can consider
$\tilde{\iota}_{12}$ as an element of $\mathcal{T}^{\prime}$ and therefore as
an element of $\widehat{\mathcal{U}}^{\prime}_{W(A)/p}$. The element
$1+\tilde{\iota}_{12}$ is a unit in $\widehat{\mathcal{U}}^{\prime}_{W(A)/p}$,
with inverse $1-\tilde{\iota}_{12}$ (this follows from the relation
$\partial_{1}\cdot\partial_{2}-\partial_{1}\circ\partial_{2}$ for all
$\partial_{1},\partial_{2}\in\mathcal{T}^{\prime}_{W(A)/p}(1)$). Therefore
there is an isomorphism

$\widehat{\mathcal{U}}^{\prime}_{W(A)/p}\xrightarrow{\cdot(1+\tilde{\iota}_{12})}\widehat{\mathcal{U}}^{\prime}_{W(A)/p}$

of left $\widehat{\mathcal{U}}^{\prime}_{W(A)/p}$-modules. It clearly takes
$\widehat{\mathcal{U}}^{\prime}_{W(A)/p}\cdot\mathcal{T}^{\prime}_{1,\iota_{1}}$
to
$\widehat{\mathcal{U}}^{\prime}_{W(A)/p}\cdot\mathcal{T}^{\prime}_{1,\iota_{2}}$
(and the inverse map $\cdot(1-\tilde{\iota}_{12}$ takes
$\widehat{\mathcal{U}}^{\prime}_{W(A)/p}\cdot\mathcal{T}^{\prime}_{1,\iota_{2}}$
to
$\widehat{\mathcal{U}}^{\prime}_{W(A)/p}\cdot\mathcal{T}^{\prime}_{1,\iota_{1}}$).
Therefore we deduce an isomorphism (5.1), and we need to check that this map
intertwines the associated right $\mathcal{D}_{A}^{((0)}$-module structures;
the cocycle condition is clear from the definition of the isomorphism.

To see this, we shall show

$x\cdot\iota_{1}(a)(1+\tilde{\iota}_{12})=x\cdot(1+\tilde{\iota}_{12})\iota_{2}(a)$

for all $x\in\widehat{\mathcal{U}}^{\prime}_{W(A)/p}$ and $a\in A$ and

$x\cdot\delta_{1}(1+\tilde{\iota}_{12})=x\cdot(1+\tilde{\iota}_{12})\delta_{2}$

for all $x\in\widehat{\mathcal{U}}^{\prime}_{W(A)/p}$ ; here the notation is
as follows let $\delta$ be a derivation of $A$. Then $\delta_{1}$ and
$\delta_{2}$ refer to elements of $\mathcal{T}^{\prime}_{W(A)/p}$ , so that
$\delta_{1}$ preserves $\iota_{1}(A)$, and
$\delta_{1}|_{\iota_{1}(A)}=\delta$, and $\delta_{2}$ preserves $\iota_{2}(A)$
and $\delta_{2}|_{\iota_{2}(A)}=\delta$ and so that
$\delta_{1}|_{\mathcal{I}_{1}}=\delta_{2}|_{\mathcal{I}_{1}}$ (it is easy to
see that it is always possible to choose such $\delta_{1},\delta_{2}$, for a
given $\delta$). As $\mathcal{D}_{A}^{((0)}$ is generated by $A$ and
derivations, this implies the result by the previous corollary.

Now let us verify these equations; for the first, we have

$x\cdot(1+\tilde{\iota}_{12})\iota_{2}(a)=x\iota_{2}(a)+x\tilde{\iota}_{12}\cdot\iota_{2}(a)$
$=x\iota_{2}(a)+x\tilde{\iota}_{12}\cdot\iota_{2}(a)$

and, as
$\tilde{\iota}_{12}\cdot\iota_{2}(a)=\iota_{2}(a)\cdot\tilde{\iota}_{12}+\tilde{\iota}_{12}(\iota_{2}(a))=\iota_{2}(a)\cdot\tilde{\iota}_{12}+i_{1}(a)-\iota_{2}(a)$
we see that the above term equals

$x\iota_{2}(a)\cdot\tilde{\iota}_{12}+x\cdot i_{1}(a)$

Further,
$\iota_{2}(a)\cdot\tilde{\iota}_{12}=\iota_{1}(a)\cdot\tilde{\iota}_{12}$, as
the difference $\iota_{1}(a)-\iota_{2}(a)\in\mathcal{I}_{1}$ and therefore
annihilates $\tilde{\iota}_{12}$ (as $\tilde{\iota}_{12}$ takes values in
$\mathcal{I}_{1}$ and this ideal has square zero). Thus this term is equal to

$x\iota_{1}(a)\cdot\tilde{\iota}_{12}+x\cdot i_{1}(a)$

as required.

For the second equation, we set
$\epsilon=[\tilde{\iota}_{12},\delta_{2}]\in\mathcal{T}_{A}^{\prime}$ and note