X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})}\in
D(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)}))$ where $\mathcal{D}_{Y\leftarrow
X}^{(0,1)}$ is defined analogously to
$\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}$.

4) if $\mathfrak{Y}=\text{Specf}(W(k))$, then we denote
${\displaystyle\mathbb{H}_{\mathcal{G}}^{\cdot}(\mathcal{M}^{\cdot}):=\int_{\varphi}\mathcal{M}^{\cdot}}$
for any $\mathcal{M}^{\cdot}\in
D_{cc}(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}))$.

Similarly, there are push-forwards in the category of right
$\mathcal{\widehat{D}}^{(0,1)}$-modules defined by
${\displaystyle\int_{\varphi}\mathcal{M}_{r}^{\cdot}:=R\varphi_{*}(\mathcal{M}_{r}^{\cdot}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})}$
for $\mathcal{M}_{r}^{\cdot}\in
D(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}))^{\text{op}}$;
clearly the left-right interchange intertwines the two pushforwards. Similar
remarks apply to a morphism $\varphi:X\to Y$ over $k$.

We begin by recording some basic compatibilities; for these note that we have
the transfer bimodule
$\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0)}:=\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}/(v-1)$
in the category of $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$-modules, and
$\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(1)}:=(\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}/(f-1))^{\widehat{}}$
(here the $()^{\widehat{}}$ denotes $p$-adic completion, which is the same as
cohomological completion in this case by 5.5). One may therefore define
${\displaystyle\int_{\varphi,0}\mathcal{M}^{\cdot}:=R\varphi_{*}(\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)}}^{L}\mathcal{M}^{\cdot})}$
for $\mathcal{M}^{\cdot}\in\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$ and
${\displaystyle\int_{\varphi,1}\mathcal{M}^{\cdot}:=R\varphi_{*}(\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(1)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)}}^{L}\mathcal{M}^{\cdot})}$
for $\mathcal{M}^{\cdot}\in\widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)}$. As in
the case of the pullback, this is not quite Berthelot’s definition of these
functors; because he uses the more traditional $\text{R}\lim$. However, they
do agree in important cases, such as when $\varphi$ is proper and
$\mathcal{M}^{\cdot}$ is coherent.

We have

######  Proposition 7.2.

Let $\mathcal{M}^{\cdot}\in
D(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}))$.

1)
${\displaystyle(\int_{\varphi}\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k\tilde{=}\int_{\varphi}(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)}$
in the category $D(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$.

2)
$(\int_{\varphi}\mathcal{M}^{\cdot})^{-\infty}\tilde{=}(\int_{\varphi,0}\mathcal{M}^{\cdot,-\infty})$
where the pushforward on the right is defined as
$R\varphi_{*}(\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)}}^{L}\mathcal{M}^{\cdot,-\infty})$.

3) If $\mathcal{M}^{\cdot}\in
D_{coh}^{b}(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}))$, then
$\widehat{((\int_{\varphi}\mathcal{M}^{\cdot})^{\infty})}\tilde{=}\int_{\varphi,1}\widehat{(\mathcal{M}^{\cdot,\infty})}$
where both uses of $\widehat{}$ denote derived completion.

###### Proof.

1) We have

$\int_{\varphi}\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k=R\varphi_{*}(\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k$
$\tilde{=}R\varphi_{*}((\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k)$

(since $k$ is a perfect complex over $W(k)$, this is a special case of the
projection formula where we consider $X$ and $Y$ as ringed spaces with the
locally constant sheaf of rings $W(k)$; c.f. [Stacks], tag 0B54). We have the
isomorphism

$k\otimes_{W(k)}^{L}\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\tilde{=}\mathcal{D}_{Y\leftarrow
X}^{(0,1)}$

since $\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}$ is
a $p$-torsion-free sheaf; and so

$R\varphi_{*}((\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k)\tilde{=}R\varphi_{*}(k\otimes_{W(k)}^{L}(\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}))$
$R\varphi_{*}(k\otimes_{W(k)}^{L}(\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}))\tilde{=}R\varphi_{*}(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot})$

where we used that for any complex $\mathcal{N}^{\cdot}$ we have
$k\otimes_{W(k)}^{L}\mathcal{N}^{\cdot}\tilde{=}k\otimes_{W(k)}^{L}\widehat{\mathcal{N}^{\cdot}}$
(c.f. 2.10). Now, we have

$R\varphi_{*}(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot})\tilde{=}\int_{\varphi}\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}(\mathcal{D}_{X}^{(0,1)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot})$

But since
$\mathcal{D}_{X}^{(0,1)}=\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}/p$ we
have
$\mathcal{D}_{X}^{(0,1)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}\tilde{=}\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k$
and the result follows.

2) For any complex we have
$\mathcal{M}^{\cdot,-\infty}=\mathcal{M}^{\cdot}\otimes_{D(W(k))}^{L}(D(W(k))/(v-1))$.
Thus the proof is an easier variant of that of $1)$, replacing
$\otimes_{W(k)}^{L}k$ with $\otimes_{D(W(k))}^{L}D(W(k))/(v-1)$.

3) We have

$\mathcal{K}^{\cdot}\to\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1),\infty}\to\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(1)}$

where $\mathcal{K}^{\cdot}$ is a complex of
$\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1),\infty}[p^{-1}]$-modules; indeed,
$\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1),\infty}$ is
a $p$-torsion-free sheaf whose completion is exactly
$\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(1)}$ (c.f. 5.5
and 3.3).Thus there is a distinguished triangle

$\mathcal{K}^{\cdot}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\infty}}^{L}\mathcal{M}^{\cdot,\infty}\to\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1),\infty}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\infty}}^{L}\mathcal{M}^{\cdot,\infty}\to\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(1)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\infty}}^{L}\mathcal{M}^{\cdot,\infty}$

and the term on the left is a complex of
$\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1),\infty}[p^{-1}]$-modules. Thus
the derived completion of
$\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1),\infty}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\infty}}^{L}\mathcal{M}^{\cdot,\infty}$
is isomorphic to the derived completion of
$\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(1)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\infty}}^{L}\mathcal{M}^{\cdot,\infty}$.

Further, we have

$\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(1)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\infty}}^{L}\mathcal{M}^{\cdot,\infty}\tilde{=}\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(1)}\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)}}^{L}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\infty}}^{L}\mathcal{M}^{\cdot,\infty})$

And, since $\mathcal{M}^{\cdot}\in
D_{coh}^{b}(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}))$, we
have (by 2.14) that
$\widehat{\mathcal{M}^{\cdot,\infty}}\tilde{=}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\infty}}^{L}\mathcal{M}^{\cdot,\infty}$
as modules over $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)}$. Therefore we
obtain

$\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1),\infty}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\infty}}^{L}\mathcal{M}^{\cdot,\infty}\tilde{=}\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)}}^{L}\widehat{(\mathcal{M}^{\cdot,\infty})}$

and so, taking $R\varphi_{*}$ yields

$R\varphi_{*}(\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1),\infty}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\infty}}^{L}\mathcal{M}^{\cdot,\infty})\tilde{\to}\int_{\varphi,1}\widehat{(\mathcal{M}^{\cdot,\infty})}$

But the term on the left is isomorphic to the derived completion of
${\displaystyle\int_{\varphi}\mathcal{M}^{\cdot,\infty}}$ by 2.12. ∎

Now we will discuss the relationship between the $\mathcal{D}_{X}^{(0,1)}$
pushforward and the push-forwards over $\mathcal{R}(\mathcal{D}_{X}^{(1)})$
and $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$. As usual we’ll work with
the functors
$\text{$\mathcal{M}$}^{\cdot}\to\mathcal{R}(\mathcal{D}_{X}^{(1)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}\tilde{=}k[f]\otimes_{D(k)}^{L}\mathcal{M}^{\cdot}$
and
$\text{$\mathcal{M}$}^{\cdot}\to\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}\tilde{=}k[v]\otimes_{D(k)}^{L}\mathcal{M}^{\cdot}$
which take $D(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$ to
$D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)})))$ and
$D(\mathcal{G}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})))$, respectively
(as in 4.22).

Both of the algebras $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ and
$\mathcal{R}(\mathcal{D}_{X}^{(0)})$ possess transfer bimodules associated to
any morphism $\varphi:X\to Y$, and hence are equipped with a push-pull
formalism. In the case of $\mathcal{R}(\mathcal{D}_{X}^{(0)})$ this is well
known (c.f., e.g [Sa], chapter $1$), while in the case of
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ this theory is developed in
[OV], in the language of filtered derived categories. We shall proceed using
the push-pull formalism for $\mathcal{D}_{X}^{(0,1)}$-modules that we have
already developed, and discuss the relations with the other theories in
section 7.1 below.

######  Definition 7.3.

Let $\varphi:X\to Y$ be a morphism. We define a
$(\varphi^{-1}\mathcal{R}(\mathcal{D}_{Y}^{(1)}),\mathcal{R}(\mathcal{D}_{X}^{(1)}))$
bimodule $\mathcal{R}_{Y\leftarrow X}^{(1)}:=\mathcal{D}_{Y\leftarrow
X}^{(0,1)}/v$. Define a
$(\varphi^{-1}\mathcal{\overline{R}}(\mathcal{D}_{Y}^{(1)}),\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))$
bimodule $\mathcal{R}_{Y\leftarrow X}^{(1)}:=\mathcal{D}_{Y\leftarrow
X}^{(0,1)}/f$. Define
${\displaystyle\int_{\varphi,1}}\mathcal{M}^{\cdot}=R\varphi_{*}(\mathcal{R}_{Y\leftarrow
X}^{(1)}\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}^{L}\mathcal{M}^{\cdot})$
on the category $\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)}))$, and
analogously ${\displaystyle\int_{\varphi,0}}$ for
$\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})$-modules. As above, there is
also a push-forward for right modules defined by
${\displaystyle\int_{\varphi,1}}\mathcal{M}_{r}^{\cdot}=R\varphi_{*}(\mathcal{M}_{r}^{\cdot}\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}^{L}\mathcal{R}_{X\to
Y}^{(1)})$, and analogously for right
$\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})$-modules.

We have the basic compatibility:

######  Proposition 7.4.

If $\mathcal{M}^{\cdot}\in D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$,
then we have

${\displaystyle\mathcal{R}(\mathcal{D}_{Y}^{(1)})\otimes_{\mathcal{D}_{Y}^{(0,1)}}^{L}\int_{\varphi}\mathcal{M}^{\cdot}\tilde{=}\int_{\varphi,1}(\mathcal{R}(\mathcal{D}_{X}^{(1)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})}$

The analogous result holds for
$\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})$.

###### Proof.

We have

$\mathcal{R}(\mathcal{D}_{Y}^{(1)})\otimes_{\mathcal{D}_{Y}^{(0,1)}}^{L}\int_{\varphi}\mathcal{M}^{\cdot}=\mathcal{R}(\mathcal{D}_{Y}^{(1)})\otimes_{\mathcal{D}_{Y}^{(0,1)}}^{L}R\varphi_{*}(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})$
$\tilde{=}R\varphi_{*}(\varphi^{-1}(\mathcal{R}(\mathcal{D}_{Y}^{(1)}))\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}))$

(we will prove this last isomorphism in the lemma directly below). We have the
isomorphism

$\varphi^{-1}(\mathcal{R}(\mathcal{D}_{Y}^{(1)}))\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\tilde{=}\mathcal{R}_{Y\leftarrow X}^{(1)}$

which is proved in the same way as (LABEL:transfer-iso-1) above. Therefore

$R\varphi_{*}(\varphi^{-1}(\mathcal{R}(\mathcal{D}_{Y}^{(1)}))\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}))\tilde{=}R\varphi_{*}(\mathcal{R}_{Y\leftarrow
X}^{(1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})$
$\tilde{=}R\varphi_{*}(\mathcal{R}_{Y\leftarrow
X}^{(1)}\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}^{L}\mathcal{R}(\mathcal{D}_{X}^{(1)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})=\int_{\varphi,1}(\mathcal{R}(\mathcal{D}_{X}^{(1)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})$

as claimed. The proof for the case of
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ is essentially identical. ∎

In the previous proof we used the

######  Lemma 7.5.

Let $\mathcal{M}^{\cdot}\in D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$,
and $\mathcal{N}^{\cdot}\in
D_{qcoh}(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)})^{\text{opp}})$. Then there is an
isomorphism

$\mathcal{N}^{\cdot}\otimes_{\mathcal{D}_{Y}^{(0,1)}}^{L}R\varphi_{*}(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})\tilde{=}R\varphi_{*}(\varphi^{-1}(\mathcal{N}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}))$

###### Proof.

(c.f. the proof of [N], proposition 5.3). First, we construct a canonical map

$\mathcal{N}^{\cdot}\otimes_{\mathcal{D}_{Y}^{(0,1)}}^{L}R\varphi_{*}(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})\to
R\varphi_{*}(\varphi^{-1}(\mathcal{N}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}))$

as follows: one may replace $\mathcal{N}^{\cdot}$ with a complex of $K$-flat
graded $\mathcal{D}_{Y}^{(0,1)}$-modules, $\mathcal{F}^{\cdot}$. Choosing a
quasi-isomorphism $\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}\tilde{\to}\mathcal{I}^{\cdot}$,
a $K$-injective complex of graded
$\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})$-modules, one obtains the quasi-
isomorphism

$\mathcal{N}^{\cdot}\otimes_{\mathcal{D}_{Y}^{(0,1)}}^{L}R\varphi_{*}(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})\tilde{\to}\mathcal{F}^{\cdot}\otimes_{\mathcal{D}_{Y}^{(0,1)}}\varphi_{*}\mathcal{I}^{\cdot}$

Then there is the obvious isomorphism

$\mathcal{F}^{\cdot}\otimes_{\mathcal{D}_{Y}^{(0,1)}}\varphi_{*}\mathcal{I}^{\cdot}\tilde{\to}\varphi_{*}(\varphi^{-1}(\mathcal{F}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}\mathcal{I}^{\cdot})$

and a canonical map

$\varphi_{*}(\varphi^{-1}(\mathcal{F}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}\mathcal{I}^{\cdot})\to
R\varphi_{*}((\varphi^{-1}(\mathcal{F}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}\mathcal{I}^{\cdot}))$
$\tilde{\to}R\varphi_{*}(\varphi^{-1}(\mathcal{N}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}))$

Thus we obtain the canonical map

$\mathcal{N}^{\cdot}\otimes_{\mathcal{D}_{Y}^{(0,1)}}^{L}R\varphi_{*}(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})\to
R\varphi_{*}(\varphi^{-1}(\mathcal{N}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}))$

this map exists for all $\mathcal{N}^{\cdot}\in
D(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)})^{\text{op}})$ and
$\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$. To check
whether it is an isomorphism, we may work locally on $Y$ and suppose that $Y$
is affine from now on.

To prove this, we proceed in a similar manner to the proof of the projection
formula for quasi-coherent sheaves, in the general version of [N], proposition
5.3. Fix $\mathcal{M}^{\cdot}\in
D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$. For any
$\mathcal{N}^{\cdot}\in
D_{qcoh}(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)})^{\text{opp}})$, we claim that
$\varphi^{-1}(\mathcal{N}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})$ is quasi-
isomorphic to a complex in $D_{qcoh}(D(\mathcal{O}_{X}))$. To see this, we
observe that any quasicoherent $\mathcal{D}_{X}^{(0,1)}$-module $\mathcal{M}$
is a quotient of the $\mathcal{D}_{X}^{(0,1)}$-module
$\mathcal{D}_{X}^{(0,1)}\otimes_{D(\mathcal{O}_{X})}\mathcal{M}$ (where the
$\mathcal{D}_{X}^{(0,1)}$-module is via the action on the left hand factor on
the tensor product). It follows that any bounded-above complex in
$D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$ is quasi-isomorphic to a
complex, whose terms are of the form
$\mathcal{D}_{X}^{(0,1)}\otimes_{D(\mathcal{O}_{X})}\mathcal{M}$ for quasi-
coherent $\mathcal{M}$. Therefore any complex in
$D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$ is a homotopy colimit of such
complexes. Therefore $\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}$ is quasi-
isomorphic to a complex of quasicoherent $D(\mathcal{O}_{X})$-modules. In
addition, since $Y$ is affine, $\mathcal{N}^{\cdot}$ is quasi-isomorphic to a
$K$-projective complex of $\mathcal{D}_{Y}^{(0,1)}$-modules; in particular, a
complex whose terms are projective $\mathcal{D}_{Y}^{(0,1)}$-modules. It
follows that
$\varphi^{-1}(\mathcal{N}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})$ is quasi-
isomorphic to a complex in $D_{qcoh}(D(\mathcal{O}_{X}))$ as claimed.

Now, since $R\varphi_{*}$ commutes with arbitrary direct sums on
$D_{qcoh}(D(\mathcal{O}_{X}))$ (by [N], lemma 1.4), we see that both sides of
arrow commute with arbitrary direct sums (over objects in
$D_{qcoh}(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)})^{\text{opp}})$); so the set of
objects on which the arrow is an isomorphism is closed under arbitrary direct
sums. Since $Y$ is affine, the category
$D_{qcoh}(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)})^{\text{op}})$ is generated by
the compact objects $\\{\mathcal{D}_{Y}^{(0,1)}[i]\\}_{i\in\mathbb{Z}}$;
therefore (as in the proof of [N], lemma 5.3), it actually suffices to show
that the arrow is an isomorphism on $\mathcal{D}_{Y}^{(0,1)}$ itself, but this
is obvious. ∎

This type of projection formula is so useful that we will record here a minor
variant:

######  Lemma 7.6.

Let $\varphi:\mathfrak{X}\to\mathfrak{Y}$ be a morphism. Let
$\mathcal{M}^{\cdot}\in
D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}))$ and
$\mathcal{N}^{\cdot}\in
D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1),\text{opp}}))$,
such that $\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k\in
D_{qcoh}(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)}))$ and
$\mathcal{N}^{\cdot}\otimes_{W(k)}^{L}k\in
D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)})^{\text{opp}})$. Then we have

$(\int_{\varphi}\mathcal{N}^{\cdot})\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}\tilde{\to}R\varphi_{*}(\mathcal{N}^{\cdot}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}L\varphi^{*}\mathcal{M}^{\cdot})$

The analogous statement holds for $\mathcal{M}^{\cdot}\in
D_{qcoh}(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)}))$ and $\mathcal{N}^{\cdot}\in
D_{coh}^{b}(\mathcal{G}(\mathcal{D}_{X}^{(0,1),\text{opp}}))$; as well as for
the Rees algebras $\mathcal{R}(\mathcal{D}^{(1)})$ and
$\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})$.

###### Proof.

We have that
${\displaystyle\int_{\varphi}\mathcal{N}^{\cdot}=R\varphi_{*}(\mathcal{N}^{\cdot}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})}$.
As in the proof of 7.5, there is a morphism

(7.1)
$R\varphi_{*}(\mathcal{N}^{\cdot}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}\to
R\varphi_{*}(\mathcal{N}^{\cdot}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}\widehat{\otimes}_{\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})}^{L}\varphi^{-1}(\mathcal{M}^{\cdot}))$

Indeed, one constructs the map

$R\varphi_{*}(\mathcal{N}^{\cdot}\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})\otimes_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}\to
R\varphi_{*}(\mathcal{N}^{\cdot}\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}\otimes_{\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})}^{L}\varphi^{-1}(\mathcal{M}^{\cdot}))$

exactly as above; and then passes to the cohomological completion.

Since
$L\varphi^{*}\mathcal{M}^{\cdot}=\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}\widehat{\otimes}_{\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})}^{L}\varphi^{-1}(\mathcal{M}^{\cdot})$
by definition, the result will follow if (LABEL:adunction) is an isomorphism.
To prove that, apply $\otimes_{W(k)}^{L}k$ and quote the previous result. The
proof in the case of the Rees algebras is completely analogous. ∎

Here is an important application of these ideas:

######  Lemma 7.7.

Let $\varphi:\mathfrak{X}\to\mathfrak{Y}$ and
$\psi:\mathfrak{Y}\to\mathfrak{Z}$ be morphisms. There is a canonical map

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

for any $\mathcal{M}^{\cdot}\in
D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$, which is
an isomorphism if $\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k\in
D_{qoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$. If $\varphi:X\to Y$ and
$\psi:Y\to Z$ are morphisms, we have the analogous statements in
$D(\mathcal{G}(\mathcal{D}_{Z}^{(0,1)}))$.

###### Proof.

As in 5.11, we have an isomorphism

$\varphi^{-1}(\mathcal{D}_{\mathfrak{Z\leftarrow\mathfrak{Y}}}^{(0,1)})\widehat{\otimes}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}}^{(0,1)})}^{L}\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\tilde{=}\mathcal{\widehat{D}}_{\mathfrak{Z}\leftarrow\mathfrak{X}}^{(0,1)}$

as
$((\psi\circ\varphi)^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Z}}^{(0,1)}),\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})$
bimodules. Then we have

$\int_{\psi}\circ\int_{\varphi}\mathcal{M}^{\cdot}=R\psi_{*}(\mathcal{D}_{\mathfrak{Z\leftarrow\mathfrak{Y}}}^{(0,1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}^{L}R\varphi_{*}(\mathcal{D}_{\mathfrak{Y\leftarrow\mathfrak{X}}}^{(0,1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}))$
$\to
R\psi_{*}R\varphi_{*}(\varphi^{-1}(\mathcal{D}_{\mathfrak{Z\leftarrow\mathfrak{Y}}}^{(0,1)})\widehat{\otimes}_{\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})}^{L}\mathcal{D}_{\mathfrak{Y\leftarrow\mathfrak{X}}}^{(0,1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot})$
$\tilde{\to}R(\psi\circ\varphi)_{*}(\mathcal{\widehat{D}}_{\mathfrak{Z}\leftarrow\mathfrak{X}}^{(0,1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot})=\int_{\psi\circ\varphi}\mathcal{M}^{\cdot}$

where the first arrow is constructed as in 7.6 and the second isomorphism is
given above. Applying the functor $\otimes_{W(k)}^{L}k$ and using 2.12, part
$1)$, we reduce to proving the analogous statement for $\varphi:X\to Y$ and
$\psi:Y\to Z$; where it follows exactly as in 7.5. ∎

We shall also need results relating the pushforwards when
$\mathcal{M}\in\mathcal{G}(\mathcal{D}_{X}^{(0,1)})$ is already annihilated by
$f$ (or $v$):

######  Proposition 7.8.

Suppose $\mathcal{M}\in\mathcal{G}(\mathcal{D}_{X}^{(0,1)})$ satisfies
$v\mathcal{M}=0$. Then ${\displaystyle\int_{\varphi}\mathcal{M}}$ is contained
in the image of the functor
$D(\mathcal{R}(\mathcal{D}_{X}^{(1)})-\text{mod})\to
D(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$. In fact, there is an isomorphism of
graded sheaves of $\mathcal{O}_{X}[f,v]$-modules

$R\varphi_{*}(\mathcal{R}_{Y\leftarrow
X}^{(1)}\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}^{L}\mathcal{M})\tilde{=}R\varphi_{*}(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M})$

In other words, the pushforward of $\mathcal{M}$, regarded as a module over
$\mathcal{R}(\mathcal{D}_{X}^{(1)})$, agrees with its pushforward as a
$\mathcal{D}_{X}^{(0,1)}$-module. The analogous result hold when
$f\mathcal{M}=0$.

###### Proof.

This is an immediate consequence of 4.21 ∎

As a consequence of these results, we obtain:

######  Theorem 7.9.

Let $\varphi:X\to Y$ be a morphism. Then, for each of the algebras
$\mathcal{R}(\mathcal{D}_{X}^{(1)})$,
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$, and
$\mathcal{D}_{X}^{(0,1)}$, the pushforward along $\varphi$ takes
$D_{qcoh}^{b}$ to $D_{qcoh}^{b}$. If $\varphi$ is proper, then the pushforward
along $\varphi$ takes $D_{coh}^{b}$ to $D_{coh}^{b}$.

###### Proof.

Let us start with the statement, that the pushforward takes $D_{qcoh}$ to
$D_{qcoh}$ in all of these cases. For this, we can argue as in the proof of
7.5: namely, one may assume $Y$ is affine, and then if $\mathcal{M}^{\cdot}\in
D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$, we may replace
$\mathcal{M}^{\cdot}$ by a homotopy colimit of
$\mathcal{D}_{X}^{(0,1)}$-modules of the form
$\mathcal{D}_{X}^{(0,1)}\otimes_{\mathcal{O}_{X}[f,v]}\mathcal{M}$, for quasi-
coherent $\mathcal{M}$. Therefore $\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}$ is quasi-
isomorphic to a complex of quasicoherent $\mathcal{O}_{X}[f,v]$-modules, which
implies that the cohomology sheaves of its pushforward are quasi-coherent
$\mathcal{O}_{Y}[f,v]$-modules; and therefore quasi-coherent
$\mathcal{D}_{Y}^{(0,1)}$-modules. The same argument works for
$\mathcal{R}(\mathcal{D}_{X}^{(1)})$ and
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$.

To prove the boundedness, we can factor $\varphi$ as a closed immersion (the
graph $X\to X\times Y$) followed by the projection $X\times Y\to Y$, and,
applying 7.7, we see that it suffices to consider separately the case of a
closed immersion and the case of a smooth morphism. For a closed immersion
$\iota:X\to Y$ we have that the bimodule $\mathcal{D}_{X\to Y}^{(0)}$ is
locally free over $\mathcal{D}_{X}^{(0,1)}$ (this elementary fact will be
checked below in 8.11) and so the tensor product
$\otimes_{\mathcal{D}_{X}^{(0,1)}}\mathcal{D}_{X\to Y}^{(0)}$ takes
quasicoherent sheaves to quasicoherent sheaves.

Now, if $X\to Y$ is smooth, we have by (the proof of) 5.15, that
$\mathcal{D}_{Y\leftarrow X}^{(0,1)}$ is a coherent
$\mathcal{D}_{X}^{(0,1),\text{opp}}$-module. Further, since it is locally the
reduction mod $p$ of a standard module, it is rigid, so that by 4.22 it is
locally of finite homological dimension; and the result follows directly. Thus
we see that ${\displaystyle\int_{\varphi}}$ is bounded on
$D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$, the same holds for the
pushforward on $D_{qcoh}(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)})))$ and
$D_{qcoh}(\mathcal{G}(\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})))$ by the
previous proposition.

Now suppose $\varphi$ is proper. Let us say that a right
$\mathcal{D}_{X}^{(0,1)}$-module is induced if it is of the form
$\mathcal{F}\otimes_{D(\mathcal{O}_{X})}\mathcal{D}_{X}^{(0,1)}$ for some
coherent $\mathcal{F}$ over $D(\mathcal{O}_{X})$. In this case we have

$\int_{\varphi}\mathcal{F}\otimes_{D(\mathcal{O}_{X})}\mathcal{D}_{X}^{(0,1)}=R\varphi_{*}(\mathcal{F}\otimes_{D(\mathcal{O}_{X})}\mathcal{D}_{X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{D}_{X\to
Y}^{(0,1)})$
$\tilde{=}R\varphi_{*}(\mathcal{F}\otimes_{D(\mathcal{O}_{X})}^{L}\mathcal{D}_{X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\varphi^{*}(\mathcal{D}_{Y}^{(0,1)}))\tilde{\to}R\varphi_{*}(\mathcal{F}\otimes_{D(\mathcal{O}_{X})}^{L}\varphi^{*}(\mathcal{D}_{Y}^{(0,1)}))$
$\tilde{\to}R\varphi_{*}(\mathcal{F})\otimes_{D(O_{Y})}^{L}\mathcal{D}_{Y}^{(0,1)}$

Thus the result is true for any induced module. If $\mathcal{M}$ is an
arbitrary coherent right $\mathcal{D}_{X}^{(0,1)}$-module, then, as a
quasicoherent sheaf over $\mathcal{O}_{X}[f,v]$, it is the union of its
$\mathcal{O}_{X}[f,v]$ coherent sub-sheaves. Selecting such a subsheaf which
generates $\mathcal{M}$ as a $\mathcal{D}_{X}^{(0,1)}$-module, we obtain a
short exact sequence

$0\to\mathcal{K}\to\mathcal{F}\otimes_{D(\mathcal{O}_{X})}\mathcal{D}_{X}^{(0,1)}\to\mathcal{M}\to
0$

where $\mathcal{K}$ is also coherent. Since the functor
${\displaystyle\int_{\varphi}}$ is concentrated in homological degrees $\leq
d_{X/Y}$ for all coherent $\mathcal{D}_{X}^{(0,1)}$-modules, we can now deduce
the coherence of ${\displaystyle\mathcal{H}^{i}(\int_{\varphi}\mathcal{M})}$
by descending induction on $i$. This proves the result for
$\mathcal{D}_{X}^{(0,1)}$-modules, and we can deduce the result for
$\mathcal{R}(\mathcal{D}_{X}^{(1)})$,
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$-modules by again invoking 7.8.
∎

From this and the formalism of cohomological completion (specifically, 4.3),
we deduce

######  Corollary 7.10.

Let $\varphi:\mathfrak{X}\to\mathfrak{Y}$ be proper. Then
${\displaystyle\int_{\varphi}}$ takes
$D_{coh}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ to
$D_{coh}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}))$. If
$\mathcal{M}^{\cdot}\in
D_{coh,F^{-1}}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$
then ${\displaystyle\int_{\varphi}\mathcal{M}}\in
D_{coh,F^{-1}}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}))$.

###### Proof.

The first part follows immediately from the proceeding theorem by applying
$\otimes_{W(k)}^{L}k$. The second part follows from 7.2 (part $3)$), as well
as Berthelot’s theorem that
${\displaystyle\int_{\varphi,0}F^{*}\tilde{\to}F^{*}\int_{\varphi,1}}$ (c.f.
[B2], section 3.4, and also 7.13 below). ∎

###  7.1. Push-forwards for $\mathcal{R}(\mathcal{D}_{X}^{(1)})$ and
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$.

In this section we take a close look at the theory over $k$. In particular, we
study the pushforwards of modules over $\mathcal{R}(\mathcal{D}_{X}^{(1)})$
and $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$, and compare them with
more traditional filtered pushforwards found in the literature. For
$\mathcal{R}(\mathcal{D}_{X}^{(1)})$ and
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ modules themselves, we will
construct the analogue of the relative de Rham resolution. This will allow us
to exhibit an adjunction between ${\displaystyle\int_{\varphi}}$ and
$\varphi^{\dagger}$ when $\varphi$ is smooth.

We begin with $\mathcal{R}(\mathcal{D}_{X}^{(1)})$, where we can reduce
everything to the more familiar situation of
$\mathcal{R}(\mathcal{D}_{X}^{(0)})$-modules using the fact that
$\mathcal{R}(\mathcal{D}_{X}^{(1)})$ is Morita equivalent to
$\mathcal{R}(\mathcal{D}_{X}^{(0)})$ (c.f. 4.19).

Let $\varphi:X\to Y$. Recall that Laumon constructed in [L] the push-forward
in the filtered derived category of $\mathcal{D}_{X}^{(0)}$-modules (with
respect to the symbol filtration); essentially, his work upgrades the bimodule
$\mathcal{D}_{Y\leftarrow X}^{(0)}$ to a filtered
$(\varphi^{-1}(\mathcal{D}_{Y}^{(0)}),\mathcal{D}_{X}^{(0)})$-bimodule via

$F_{i}(\mathcal{D}_{Y\leftarrow
X}^{(0)}):=\varphi^{-1}(F_{i}(\mathcal{D}_{Y}^{(0)})\otimes_{\mathcal{O}_{Y}}\omega_{Y}^{-1})\otimes_{\varphi^{-1}(\mathcal{O}_{Y})}\omega_{X}$

(c.f. [L], formula 5.1.3); then one may define ${\displaystyle\int_{\varphi}}$
via the usual formula, but using the tensor product and push-forward in the
filtered derived categories. On the other hand, we can apply the Rees
construction to the above filtered bimodule to obtain
$\mathcal{R}(\mathcal{D}_{Y\leftarrow X}^{(0)})$, a graded
$(\varphi^{-1}(\mathcal{R}(\mathcal{D}_{Y}^{(0)})),\mathcal{R}(\mathcal{D}_{X}^{(0)}))$
bimodule, which (again by the usual formula) yields a push-forward functor
${\displaystyle\int_{\varphi}}:D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(0)})))\to
D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{Y}^{(0)})))$, and we have the following
evident compatibility:

######  Lemma 7.11.

Let $\mathcal{M}^{\cdot}\in D((\mathcal{D}_{X}^{(0)},F)-\text{mod})$. Then we
have

$\mathcal{R}(\int_{\varphi}\mathcal{M}^{\cdot})\tilde{\to}\int_{\varphi}\mathcal{R}(\mathcal{M}^{\cdot})$

In particular, the Hodge-to-deRham spectral sequence for
${\displaystyle\int_{\varphi}\mathcal{M}^{\cdot}}$ degenerates at $E_{1}$ iff
each of the sheaves
${\displaystyle\mathcal{H}^{i}(\int_{\varphi}\mathcal{R}(\mathcal{M}^{\cdot}))}$
is torsion-free over the Rees parameter $f$.

Next, we relate this to the pull-back and push-forward for
$\mathcal{R}(\mathcal{D}_{X}^{(1)})$ modules; starting with the analogous
statement for pull-back:

######  Lemma 7.12.

Let $\varphi:X\to Y$ and suppose $\mathcal{M}^{\cdot}\in
D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(0)})))$. Then $L\varphi^{*}\circ
F_{Y}^{*}\mathcal{M}^{\cdot}\tilde{=}F_{X}^{*}\circ
L\varphi^{*}\mathcal{M}^{\cdot}$. Here, the pullback on the left is in the
category of $\mathcal{R}(\mathcal{D}^{(1)})$-modules, while the pullback on
the right is in the category of $\mathcal{R}(\mathcal{D}^{(0)})$-modules.

###### Proof.

Since $\varphi\circ F_{X}=F_{Y}\circ\varphi$ we have

$L\varphi^{*}\circ F_{Y}^{*}\mathcal{M}^{\cdot}\tilde{\to}F_{X}^{*}\circ
L\varphi^{*}\mathcal{M}^{\cdot}$

as (graded) $\mathcal{O}_{X}$-modules; we need to check that this map
preserves the $\mathcal{R}(\mathcal{D}_{X}^{(1)})$-module structures on both
sides.This question is local, so we may suppose $X=\text{Spec}(B)$ and
$Y=\text{Spec}(A)$ both posses local coordinates. Further, by taking a K-flat
resolution of $\mathcal{M}^{\cdot}$ we may suppose that
$\mathcal{M}^{\cdot}=\mathcal{M}$ is concentrated in a single degree. Now, as
an $\mathcal{R}(\mathcal{D}_{X}^{(1)})$-module, $F_{Y}^{*}\mathcal{M}$
possesses the structure of a connection with $p$-curvature $0$, and so the
induced connection on $\varphi^{*}F_{Y}^{*}\mathcal{M}$ also has $p$-curvature
$0$; and the kernel of this connection is equal to
$(\varphi^{(1)})^{*}\mathcal{M}^{(1)}\subset\varphi^{*}F_{Y}^{*}\mathcal{M}$
(here $\mathcal{M}^{(1)}$ denotes $\sigma^{*}\mathcal{M}$ where
$\sigma:X^{(1)}\to X$ is the natural isomorphism of schemes). Note that
$\mathcal{M}^{(1)}$ possesses the action of
$\mathcal{R}(\mathcal{D}_{Y^{(1)}}^{(0)})$ (c.f. 4.20).

Let $\\{\partial_{i}\\}_{i=1}^{n}$ be coordinate derivations on $X$. Then the
action of $\partial_{i}^{[p]}$ on $\varphi^{*}F_{Y}^{*}\mathcal{M}$ is given
(by 5.8) by first restricting $\partial_{i}^{[p]}$ to a differential operator
$\varphi^{-1}(\mathcal{O}_{Y})\to\mathcal{O}_{X}$, writing the resulting
operator as

$\sum_{j=1}^{r}b_{j}^{p}\partial_{j}^{[p]}+\sum_{J}b_{J}\partial^{J}$

(where $\\{\partial_{j}\\}_{j=1}^{r}$ are coordinate derivations on $Y$, and
$b_{j},b_{J}\in B$) and then letting $\partial_{i}^{[p]}$ act as

$\sum_{j=1}^{r}b_{j}^{p}\partial_{j}^{[p]}+\sum_{J}b_{J}\partial^{J}$

therefore, the action of $\partial_{i}^{[p]}$ preserves
$(\varphi^{(1)})^{*}\mathcal{M}^{(1)}$ and it acts there as
${\displaystyle\partial_{i}^{[p]}(1\otimes
m)=\sum_{j=1}^{r}b_{j}^{p}\cdot\partial_{j}^{[p]}(m)}$. But the action of
$\\{\partial_{j}^{[p]}\\}$ on $\mathcal{M}^{(1)}$ defines the action of
$\mathcal{R}(\mathcal{D}_{X^{(1)}}^{(0)})$ on $\mathcal{M}^{(1)}$, and this
formula simply defines the pullback from
$\mathcal{R}(\mathcal{D}_{Y^{(1)}}^{(0)})$ to
$\mathcal{R}(\mathcal{D}_{X^{(1)}}^{(0)})$-modules; in other words,
$(\varphi^{(1)})^{*}\mathcal{M}^{(1)}=((\varphi)^{*}\mathcal{M})^{(1)}$ where
$\varphi^{*}\mathcal{M}$ is the usual pullback of
$\mathcal{R}(\mathcal{D}^{(0)})$-modules. Thus we see that
$\varphi^{*}F_{Y}^{*}\mathcal{M}=F_{X}^{*}((\varphi)^{*}\mathcal{M})$ as
$\mathcal{R}(\mathcal{D}_{X}^{(1)})$-modules, as desired. ∎

Now we discuss push-forward:

######  Theorem 7.13.

Let $\mathcal{M}^{\cdot}$ be a complex of graded
$\mathcal{R}(\mathcal{D}_{X}^{(1)})$-modules, and via 4.19 write
$\mathcal{M}^{\cdot}\tilde{=}F_{X}^{*}\mathcal{N}^{\cdot}$, where
$\mathcal{N}^{\cdot}$ is a complex of graded
$\mathcal{R}(\mathcal{D}_{X}^{(0)})$-modules. There is an isomorphism

$\int_{\varphi,1}\mathcal{M}^{\cdot}\tilde{\to}F_{X}^{*}\int_{\varphi}\mathcal{N}^{\cdot}$

where ${\displaystyle\int_{\varphi}\mathcal{N}}^{\cdot}$ is the pushforward of
$\mathcal{N}^{\cdot}$ over $\mathcal{R}(\mathcal{D}_{X}^{(0)})$.

###### Proof.

(following [B2], theoreme 3.4.4) By left-right interchange it is equivalent to
prove the right-handed version

$\int_{\varphi,1}F_{X}^{!}\mathcal{N}^{\cdot}\tilde{\to}F_{Y}^{!}\int_{\varphi}\mathcal{N}^{\cdot}$

for any $\mathcal{N}^{\cdot}\in
D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(0)})^{\text{opp}}))$.

We have

$\int_{\varphi,1}F_{X}^{!}(\mathcal{N}^{\cdot})=R\varphi_{*}(F_{X}^{!}(\mathcal{N}^{\cdot})\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}^{L}\mathcal{R}_{X\to
Y}^{(1)})=R\varphi_{*}(\mathcal{N}^{\cdot}\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(0)})}^{L}F_{X}^{!}(\mathcal{R}(\mathcal{D}_{X}^{(0)}))\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}^{L}\mathcal{R}_{X\to
Y}^{(1)})$

Now, recall

$\mathcal{R}_{X\to
Y}^{(1)}=\varphi^{*}\mathcal{R}(\mathcal{D}_{X}^{(1)})\tilde{=}\varphi^{*}F_{Y}^{*}F_{Y}^{!}\mathcal{R}(\mathcal{D}_{Y}^{(1)})\tilde{=}F_{X}^{*}\varphi^{*}F_{Y}^{!}\mathcal{R}(\mathcal{D}_{Y}^{(1)})$

where the second isomorphism is 4.27, and the third is by the lemma above;
note that this isomorphism preserves the natural right
$\varphi^{-1}(\mathcal{R}(\mathcal{D}_{Y}^{(1)}))$ -module structures on both
sides. It follows (c.f. 4.26, part $2)$, and 4.27) that

$F_{X}^{!}(\mathcal{R}(\mathcal{D}_{X}^{(0)}))\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}^{L}\mathcal{R}_{X\to
Y}^{(1)}\tilde{=}\varphi^{*}F_{Y}^{!}\mathcal{R}(\mathcal{D}_{Y}^{(0)})$

(as
$(\mathcal{R}(\mathcal{D}_{X}^{(1)}),\varphi^{-1}(\mathcal{R}(\mathcal{D}_{Y}^{(1)}))$
bimodules). Therefore

$R\varphi_{*}(\mathcal{N}^{\cdot}\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(0)})}^{L}F^{!}(\mathcal{R}(\mathcal{D}_{X}^{(0)}))\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}^{L}\mathcal{R}_{X\to
Y}^{(1)})\tilde{=}R\varphi_{*}(\mathcal{N}^{\cdot}\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(0)})}^{L}\varphi^{*}F_{Y}^{!}\mathcal{R}(\mathcal{D}_{Y}^{(0)}))$
$\tilde{=}\int_{\varphi}\mathcal{N}^{\cdot}\otimes_{\mathcal{R}(\mathcal{D}_{Y}^{(0)})}^{L}F_{Y}^{!}\mathcal{R}(\mathcal{D}_{Y}^{(0)}))$

where the last line is 7.6. However,

$\int_{\varphi}\mathcal{N}^{\cdot}\otimes_{\mathcal{R}(\mathcal{D}_{Y}^{(0)})}^{L}F_{Y}^{!}\mathcal{R}(\mathcal{D}_{Y}^{(0)}))=F_{Y}^{!}\int_{\varphi}\mathcal{N}^{\cdot}$

whence the result. ∎

To exploit this result, we recall that the formalism of de Rham cohomology
applies to $\mathcal{R}(\mathcal{D}_{X}^{(0)})$:

######  Proposition 7.14.

Let $\varphi:X\to Y$ be smooth of relative dimension $d$. Then the induced
connection
$\nabla:\mathcal{D}_{X}^{(0)}\to\mathcal{D}_{X}^{(0)}\otimes\Omega_{X/Y}^{1}(1)$
is a morphism of filtered right $\mathcal{D}_{X}^{(0)}$-modules (with respect
to the symbol filtration; the symbol $(1)$ denotes a shift in the filtration).
The associated de Rham complex

$\mathcal{D}_{X}^{(0)}\to\mathcal{D}_{X}^{(0)}\otimes\Omega_{X/Y}^{1}(1)\to\mathcal{D}_{X}^{(0)}\otimes\Omega_{X/Y}^{2}(2)\to\dots\to\mathcal{D}_{X}^{(0)}\otimes\Omega_{X/Y}^{d}(d)$

is exact except at the right most term, where the cokernel is
$\mathcal{D}_{Y\leftarrow X}^{(0)}(d)$ (as a filtered module).

After applying the left-right swap and a shift in the filtration, we obtain
the Spencer complex

$\mathcal{D}_{X}^{(0)}\otimes\mathcal{T}_{X/Y}^{d}(-d)\to\mathcal{D}_{X}^{(0)}\otimes\mathcal{T}_{X/Y}^{d-1}(-d+1)\to\dots\to\mathcal{D}_{X}^{(0)}\otimes\mathcal{T}_{X/Y}(-1)\to\mathcal{D}_{X}^{(0)}$

of left filtered $\mathcal{D}_{X}^{(0)}$-modules, which is exact except at the
right most term, and the cokernel is $\mathcal{D}_{X\to Y}^{(0)}$ (as a
filtered module). Applying the Rees functor yields a complex

$\mathcal{R}(\mathcal{D}_{X}^{(0)})\otimes\mathcal{T}_{X/Y}^{d}(-d)\to\mathcal{R}(\mathcal{D}_{X}^{(0)})\otimes\mathcal{T}_{X/Y}^{d-1}(-d+1)\to\dots\to\mathcal{R}(\mathcal{D}_{X}^{(0)})\otimes\mathcal{T}_{X/Y}(-1)\to\mathcal{R}(\mathcal{D}_{X}^{(0)})$

which is exact except at the right most term, and the cokernel is
$\mathcal{R}(\mathcal{D}_{X\to Y}^{(0)})$.

The proof of this is identical to that of the corresponding result in
characteristic zero ([K], proposition 4.2); one notes that the associated
graded is a Koszul resolution. Applying this resolution in the definition of
the filtered push-forward, one deduces

######  Corollary 7.15.

Let $\varphi:X\to Y$ be smooth of relative dimension $d$. Let $\mathcal{M}$ be
a filtered $\mathcal{D}_{X}^{(0)}$-module (with respect to the symbol
filtration). Then there is an isomorphism

$\int_{\varphi}\mathcal{M}[-d]\tilde{=}R\varphi_{*}(\mathcal{M}(-d)\xrightarrow{\nabla}\mathcal{M}\otimes\Omega_{X/Y}^{1}(1-d)\xrightarrow{\nabla}\mathcal{M}\otimes\Omega_{X/Y}^{2}(2)\xrightarrow{\nabla}\dots\xrightarrow{\nabla}\mathcal{M}\otimes\Omega_{X/Y}^{d})$

in the filtered derived category of $\mathcal{O}_{Y}$-modules.

In fact, with a little more work, one can show that, for any $i$, the
$\mathcal{D}_{Y}^{(0)}$-module structure on the sheaf
${\displaystyle\mathcal{H}^{i}(\int_{\varphi}\mathcal{M})}$ is given by the
Gauss-Manin connection (c.f., e.g. [DMSS], proposition 1.4). Thus the push-
forward for $\mathcal{R}(\mathcal{D}_{X}^{(1)})$-modules is exactly the
“Frobenius pullback of Gauss-Manin.”

As another corollary, we have

######  Corollary 7.16.

Let $\varphi:X\to Y$ be smooth of relative dimension $d$.

1) There is an isomorphism
$R\underline{\mathcal{H}om}{}_{\mathcal{R}(\mathcal{D}_{X}^{(0})}(\mathcal{R}(\mathcal{D}_{X\to
Y}^{(0)}),\mathcal{R}(\mathcal{D}_{X}^{(0)}))\tilde{=}\mathcal{R}(\mathcal{D}_{Y\leftarrow
X}^{(0)})(d)[-d]$ as
$(\varphi^{-1}(\mathcal{R}(\mathcal{D}_{Y}^{(0)})),\mathcal{R}(\mathcal{D}_{X}^{(0)}))$
bimodules.

2) There is an isomorphism of functors

$R\varphi_{*}R\underline{\mathcal{H}om}_{\mathcal{R}(\mathcal{D}_{X}^{(0)})}(\varphi^{\dagger}\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})\tilde{\to}R\underline{\mathcal{H}om}{}_{\mathcal{R}(\mathcal{D}_{Y}^{(0)})}(\mathcal{N}^{\cdot},\int_{\varphi}\mathcal{M}^{\cdot}(d))$

for any $\mathcal{N}^{\cdot}\in
D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{Y}^{(0)})))$ and any
$\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(0)})))$.
The analogous isomorphism holds for
$\mathcal{R}(\mathcal{D}_{X}^{(1)})$-modules.

###### Proof.

Part $1)$ follows directly from the previous proposition; compare [K],
propositions 4.2 and 4.19. Then $2)$ follows from $1)$, as in [K], Theorem
4.40 (we’ll give the argument below in a slightly different context in 7.24)
Finally, the statement for $\mathcal{R}(\mathcal{D}_{X}^{(1)})$-modules
follows from the Frobenius descent (7.12 and 7.13). ∎

Next we are going to give the analogue of these results for the push-forward
of $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$-modules; and compare with
the constructions of [OV], section 3.4. We start with the analogues of the de
Rham resolution and the adjunction for smooth morphisms. Although
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ does possess a canonical flat
connection, the resulting (relative) de Rham complex is not a resolution of a
transfer bimodule. Instead, we consider the action of the center

$\mathcal{Z}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))\tilde{=}\mathcal{O}_{T^{*}X^{(1)}}[v]$

The action map
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X^{(1)}}}\mathcal{T}_{X^{(1)}}(-1)\to\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$
yields (by dualizing) a map

$\Theta:\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\to\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X^{(1)}}}\Omega_{X^{(1)}}^{1}(1)$

which makes $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ into a Higgs sheaf
over $X^{(1)}$. In particular we have $\Theta\circ\Theta=0$ and so we can form
the complex
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X^{(1)}}}\Omega_{X^{(1)}}^{i}(i)$
with the differential induced from $\Theta$. In addition, we can form the
analogue of the Spencer complex, whose terms are
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X^{(1)}}}\mathcal{T}_{X^{(1)}}^{i}(-i)$.

Now let $\varphi:X\to Y$ be a smooth morphism of relative dimension $d$. Let
$X_{Y}^{(1)}\to Y$ be the base change of this morphism over the absolute
Frobenius on $Y$. Then we can perform the above constructions for
$\Omega_{X_{Y}^{(1)}/Y}^{i}$ instead of $\Omega_{X^{(1)}}^{i}$. We have

######  Lemma 7.17.

The complex
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X_{Y}^{(1)}}}\mathcal{T}_{X_{Y}^{(1)}/Y}^{i}(-i)$
is exact except at the right-most term. The image of the map

$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X_{Y}^{(1)}}}\mathcal{T}_{X_{Y}^{(1)}/Y}(-1)\to\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$

is the central ideal $\mathcal{J}$ generated by
$\mathcal{T}_{X_{Y}^{(1)}/Y}\subset\mathcal{Z}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))$.
The cokernel of the map,
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})/\mathcal{J}$, carries the
structure of a right $\mathcal{D}_{X/Y}^{(0)}$-module, of $p$-curvature zero;
this action commutes with the natural left
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$-module structure on
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})/\mathcal{J}$. The cokernel of
the associated map
$(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})/\mathcal{J})\otimes_{\mathcal{O}_{X/Y}}\mathcal{T}_{X/Y}\to\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})/\mathcal{J}$
is isomorphic to $\mathcal{\overline{R}}_{X\to Y}^{(0)}$.

The complex
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X_{Y}^{(1)}}}\Omega_{X_{Y}^{(1)}/Y}^{i}(i)$
is exact except at the right-most term. The cokernel of the map

$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X_{Y}^{(1)}}}\Omega_{X_{Y}^{(1)}/Y}^{d-1}(d-1)\to\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X_{Y}^{(1)}}}\Omega_{X_{Y}^{(1)}/Y}^{d}(d)$

denoted $\mathcal{K}_{X/Y}$, carries the structure of a left
$\mathcal{D}_{X/Y}^{(0)}$-module, of $p$-curvature zero; this action commutes
with the natural right $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$-module
structure on $\mathcal{K}_{X/Y}$. The kernel of the associated connection on
$\mathcal{K}_{X/Y}$ is isomorphic to $\mathcal{\overline{R}}_{Y\leftarrow
X}^{(0)}(d)$.

###### Proof.

Choose local coordinates on $X$ for which $\text{Der}(\mathcal{O}_{X})$ is the
free module on $\\{\partial_{1},\dots\partial_{n}\\}$ and
$\text{Der}_{\mathcal{O}_{Y}}(\mathcal{O}_{X})=\\{\partial_{n-d+1},\dots\partial_{n}\\}$.
Then the complex under consideration is simply the Koszul complex for the
elements $\\{\partial_{n-d+1}^{p},\dots,\partial_{n}^{p}\\}$, which proves the
exactness statements. Furthermore, as the elements
$\\{\partial_{n-d+1}^{p},\dots,\partial_{n}^{p}\\}$ are central in
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$, we see that
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})/\mathcal{J}$ has the structure
of a left and right $\mathcal{D}_{X}^{(0)}$-module (we are here using the fact
that $\mathcal{D}_{X}^{(0)}$ is the degree $0$ part of
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$). Now, we have

$\mathcal{\overline{R}}_{X\to
Y}^{(0)}=\text{coker}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X_{Y}^{(1)}}}\mathcal{T}_{X/Y}(-1)\to\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))$
$=\text{coker}((\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})/\mathcal{J})\otimes_{\mathcal{O}_{X/Y}}\mathcal{T}_{X/Y}\to\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})/\mathcal{J})$

since $X\to Y$ is smooth. The second statement follows similarly. ∎

Now we can give the analogue of 7.16. It reads:

######  Corollary 7.18.

Let $\varphi:X\to Y$ be smooth of relative dimension $d$.

1) There is an isomorphism
$R\underline{\mathcal{H}om}{}_{\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0})}(\mathcal{\overline{R}}(\mathcal{D}_{X\to
Y}^{(0)}),\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)}))\tilde{=}\mathcal{\overline{R}}_{Y\leftarrow
X}^{(0)}(d)[-d]$ as
$(\varphi^{-1}(\mathcal{\overline{R}}(\mathcal{D}_{Y}^{(0)})),\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))$
bimodules.

2) There is an isomorphism of functors

$R\varphi_{*}R\underline{\mathcal{H}om}_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(\varphi^{\dagger,(0)}\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})\tilde{\to}R\underline{\mathcal{H}om}{}_{\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})}(\mathcal{N}^{\cdot},\int_{\varphi,0}\mathcal{M}^{\cdot}(d))$

for any $\mathcal{N}^{\cdot}\in
D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{Y}^{(0)})))$ and any
$\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(0)})))$.

###### Proof.

As in the proof of 7.24 below, $2)$ follows formally from $1)$. To see $1)$,
we note that for any
$\mathcal{N}\in\mathcal{G}(\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)}))$ the
complex
$R\underline{\mathcal{H}om}{}_{\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0})}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})/\mathcal{J},\mathcal{N})$
can be considered a complex of left $\mathcal{D}_{X/Y}^{(0)}$-modules with
$p$-curvature $0$ (as
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})/\mathcal{J}$ is a right
$\mathcal{D}_{X/Y}^{(0)}$-module of $p$-curvature zero, and this action
commutes with the left $\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0})$-action).
As Cartier descent for $\mathcal{D}_{X/Y}^{(0)}$-modules of $p$-curvature $0$
is an exact functor, applying the previous lemma we obtain

$R\underline{\mathcal{H}om}{}_{\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0})}(\mathcal{\overline{R}}(\mathcal{D}_{X\to
Y}^{(0)}),\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)}))\tilde{=}R\underline{\mathcal{H}om}{}_{\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0})}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})/\mathcal{J},\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)}))^{\nabla}$
$(\mathcal{K}_{X/Y}[-d])^{\nabla}=\mathcal{\overline{R}}_{Y\leftarrow
X}^{(0)}(d)[-d]$

as desired. ∎

Now we’ll give the relation of our pushforward to the constructions of [OV],
section 3.4. We recall that to any morphism $\varphi:X\to Y$ we may attach the
diagram

$T^{*}X\xleftarrow{d\varphi}X\times_{Y}T^{*}Y\xrightarrow{\pi}T^{*}Y$

and we use the same letters to denote the products of these morphisms with
$\mathbb{A}^{1}$; We have the following analogue of [BezB], proposition 3.7
(c.f. also [OV], theorem 3.11)

######  Lemma 7.19.

There is an equivalence of graded Azumaya algebras
$(d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\sim(\pi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})$.

###### Proof.

Consider the (graded) Azumaya algebra
$\mathcal{A}:=(d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{(X\times_{Y}T^{*}Y)^{(1)}}[v]}(\pi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})^{\text{opp}}$
on $(X\times_{Y}T^{*}Y)^{(1)}\times\mathbb{A}^{1}$. It is enough to find a
(graded) splitting module for $\mathcal{A}$; i.e., a graded
$\mathcal{A}$-module which is locally free of rank
$p^{\text{dim}(X)+\text{dim}(Y)}$ over
$\mathcal{O}_{(X\times_{Y}T^{*}Y)^{(1)}}[v]$.

The graded
$(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}),\varphi^{-1}(\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})))$
bimodule $\overline{\mathcal{R}}_{X\to
Y}:=\varphi^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})$ inherits the
structure of an $\mathcal{A}$-module; we claim it is locally free over
$\mathcal{O}_{(X\times_{Y}T^{*}Y)^{(1)}}[v]$ of the correct rank. This can be
checked after inverting $v$ and setting $v=0$; upon inverting $v$ it becomes
(via the isomorphisms
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})[v^{-1}]\tilde{=}\mathcal{D}_{X}^{(0)}[v,v^{-1}]$
and
$\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})[v^{-1}]\tilde{=}\mathcal{D}_{Y}^{(0)}[v,v^{-1}]$)
a direct consequence of [BezB], proposition 3.7. After setting $v=0$ we have
$\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})/v=\text{gr}(\mathcal{D}_{Y}^{(0)})=\overline{\mathcal{D}}_{Y}^{(0)}\otimes_{\mathcal{O}_{Y^{(1)}}}\mathcal{O}_{T^{*}Y^{(1)}}$;
and similarly for $X$. Therefore

$\varphi^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})/v\tilde{=}\mathcal{O}_{X}\otimes_{\varphi^{-1}(\mathcal{O}_{Y})}\varphi^{-1}(\overline{\mathcal{D}}_{Y}^{(0)}\otimes_{\mathcal{O}_{Y^{(1)}}}\mathcal{O}_{T^{*}Y^{(1)}})$
$\tilde{=}\mathcal{O}_{X}\otimes_{\varphi^{-1}(\mathcal{O}_{Y})}\varphi^{-1}(\overline{\mathcal{D}}_{Y}^{(0)})\otimes_{\varphi^{-1}(\mathcal{O}_{Y^{(1)}})}\varphi^{-1}(\mathcal{O}_{T^{*}Y^{(1)}})$

But $\varphi^{-1}(\overline{\mathcal{D}}_{Y}^{(0)})$ is locally free of rank
$p^{\text{dim}(Y)}$ over $\varphi^{-1}(\mathcal{O}_{Y})$, and
$\mathcal{O}_{X}$ is locally free of rank $p^{\text{dim}(X)}$ over
$\mathcal{O}_{X^{(1)}}$; so
$\varphi^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})/v$ is locally free
of rank $p^{\text{dim}(X)+\text{dim}(Y)}$ over
$\mathcal{O}_{(X\times_{Y}T^{*}Y)^{(1)}}$ as claimed. ∎

Next, we have the following straightforward:

######  Lemma 7.20.

Let $\varphi:X\to Y$ be smooth. Then $d\varphi$ is a closed immersion, and we
may regard the algebra
$(d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ as a
(graded) central quotient of $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$.
The obvious functor
$(d\varphi)_{*}:D(\mathcal{G}((d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})))\to
D(\mathcal{G}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})))$ admits a right
adjoint $(d\varphi)^{!}$ defined by $\mathcal{M}^{\cdot}\to
R\underline{\mathcal{H}om}_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}((d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}),\mathcal{M}^{\cdot})$.

Therefore we obtain

######  Corollary 7.21.

Let
$C:D(\mathcal{G}((d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})))\to
D(\mathcal{G}((\pi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})))$
denote the equivalence of categories resulting from 7.19. Then, when
$\varphi:X\to Y$ is smooth of relative dimension $d$, there is an isomorphism
of functors

$\int_{\varphi,0}\tilde{\to}R\pi_{*}^{(1)}\circ
C\circ(d\varphi^{(1)})^{!}[-d]:D(\mathcal{G}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})))\to
D(\mathcal{G}(\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})))$

Therefore, the functor ${\displaystyle\int_{\varphi,0}}[d]$ agrees, under the
application of the Rees functor, with the pushforward of conjugate-filtered
derived categories constructed in [OV], section 3.4.

###### Proof.

(in the spirit of [OV], proposition 3.12) We have, for any
$\mathcal{M}^{\cdot}\in
D(\mathcal{G}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})))$,

$C\circ(d\varphi)^{!}(\mathcal{M}^{\cdot})=C\circ
R\underline{\mathcal{H}om}_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}((d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}),\mathcal{M}^{\cdot})$
$\tilde{=}\underline{\mathcal{H}om}_{(d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(\varphi^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)}),R\underline{\mathcal{H}om}_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}((d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}),\mathcal{M}^{\cdot}))$

Since $\varphi^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})$ is locally
projective over
$(d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$, this is
canonically isomorphic to

$R\underline{\mathcal{H}om}_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(\varphi^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})\otimes_{(d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}),\mathcal{M}^{\cdot}))$
$=R\underline{\mathcal{H}om}_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(\varphi^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)}),\mathcal{M}^{\cdot}))$

so that

$R\pi_{*}^{(1)}\circ
C\circ(d\varphi^{(1)})^{!}(\mathcal{M}^{\cdot})\tilde{\to}R\pi_{*}^{(1)}R\underline{\mathcal{H}om}_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(\varphi^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)}),\mathcal{M}^{\cdot}))$
$\tilde{=}R\varphi_{*}R\underline{\mathcal{H}om}_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(\varphi^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)}),\mathcal{M}^{\cdot}))$

But the right-hand functor is canonically isomorphic to
${\displaystyle\int_{\varphi,0}}$ by smooth adjunction (7.18). ∎

From this description it follows directly (c.f. [OV], lemma 3.18) that (up to
a renumbering) the spectral sequence associated to the filtration on
${\displaystyle\int_{\varphi}\mathcal{O}_{X}}$ agrees with the usual conjugate
spectral sequence; i.e., the “second spectral sequence” for
$R\varphi_{dR,*}(\mathcal{O}_{X})$ as discussed in [Ka].

###  7.2. Adjunction for a smooth morphism, base change, and the projection
formula

In this section, we prove adjunction for a smooth morphism
$\varphi:\mathfrak{X}\to\mathfrak{Y}$ and the projection formula for an
arbitrary morphism; as consequences we obtain the smooth base change and the
and the Kunneth formula, in fairly general contexts. To start off, let us
recall:

######  Proposition 7.22.

For a smooth morphism $\varphi:\mathfrak{X}\to\mathfrak{Y}$ there is an
isomorphism of sheaves
$\mathcal{R}\mathcal{H}om_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)})\tilde{=}\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0)}[-d_{X/Y}]$.

This is proved identically to the analogous fact for $\mathcal{D}_{X}^{(0)}$
and $\mathcal{R}(\mathcal{D}_{X}^{(0)})$-modules, as discussed above in 7.16.

######  Proposition 7.23.

For a smooth morphism $\varphi:\mathfrak{X}\to\mathfrak{Y}$ of relative
dimension $d$, there is an isomorphism
$\mathcal{R}\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})\tilde{=}\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}(d)[-d]$

###### Proof.

We have

$\mathcal{R}\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$
$\tilde{=}\mathcal{R}\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})\otimes_{D(W(k))}^{L}(W(k)[f,v]/(v-1)$
$\tilde{=}\mathcal{R}\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}\otimes_{D(W(k))}^{L}(W(k)[f,v]/(v-1),\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}\otimes_{D(W(k))}^{L}(W(k)[f,v]/(v-1))$
$\tilde{=}R\mathcal{H}om_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)})$

and

$\mathcal{R}\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{D}_{X}^{(0,1)}\tilde{=}\mathcal{R}\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})\otimes_{W(k)}^{L}k$
$\tilde{=}R\mathcal{H}om_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{D}_{X\to
Y}^{(0,1)},\mathcal{D}_{X}^{(0,1)})$

By the same token, we have

$R\underline{\mathcal{H}om}{}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{D}_{X\to
Y}^{(0,1)},\mathcal{D}_{X}^{(0,1)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{R}(\mathcal{D}_{X}^{(1)})\tilde{=}R\underline{\mathcal{H}om}{}_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}(\mathcal{R}_{X\to
Y}^{(1)},\mathcal{R}(\mathcal{D}_{X}^{(1)}))$

and the analogous statement for
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$. Applying the smooth
adjunction for $\mathcal{R}(\mathcal{D}_{X}^{(1)})$-modules (7.16, part $2)$)
to the case where $\mathcal{N}^{\cdot}=\mathcal{R}(\mathcal{D}_{Y}^{(1)})$ and
$\mathcal{M}^{\cdot}=\mathcal{R}(\mathcal{D}_{X}^{(1)})$, we have an
isomorphism

$R\underline{\mathcal{H}om}{}_{\mathcal{R}_{X}}(\mathcal{R}_{X\to
Y}^{(1)},\mathcal{R}(\mathcal{D}_{X}^{(1)}))\tilde{=}\mathcal{R}_{Y\leftarrow
X}^{(1)}(d)[-d]$

and by 7.18 we have

$R\underline{\mathcal{H}om}{}_{\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0})}(\mathcal{\overline{R}}(\mathcal{D}_{X\to
Y}^{(0)}),\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)}))\tilde{=}\mathcal{\overline{R}}_{Y\leftarrow
X}^{(0)}(d)[-d]$

Furthermore, using the relative de Rham resolution for
$\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$-modules (or, equivalently, the
previous proposition) we have
$\mathcal{R}\mathcal{H}om_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)})\tilde{=}\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0)}[-d_{X/Y}]$.

On the other hand, we have the short exact sequence

$\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})\to\mathcal{D}_{X}^{(0,1)}\to\mathcal{R}(\mathcal{D}_{X}^{(1)})(-1)$

which by 4.21 yields the distinguished triangle

$R\underline{\mathcal{H}om}{}_{\bar{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(\mathcal{\overline{R}}_{X\to
Y},\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)}))\to
R\underline{\mathcal{H}om}{}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{D}_{X\to
Y}^{(0,1)},\mathcal{D}_{X}^{(0,1)})$ $\to
R\underline{\mathcal{H}om}{}_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}(\mathcal{R}_{X\to
Y},\mathcal{R}(\mathcal{D}_{X}^{(1)}))(-1)$

which implies that
$R\underline{\mathcal{H}om}{}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{D}_{X\to
Y}^{(0,1)},\mathcal{D}_{X}^{(0,1)})$ is concentrated in a single homological
degree (namely $d$). So, since
$R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})$
is cohomologically complete, we see that
$\mathcal{H}^{d}(R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$
is $p$-torsion-free and concentrated in degree $0$. We also see, by 4.3, that
this module is coherent over $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$
(each of $\mathcal{R}_{Y\leftarrow X}^{(1)}$ and
$\mathcal{\overline{R}}_{Y\leftarrow X}^{(0)}$ are coherent, since $X\to Y$ is
smooth). Further, since
$R\underline{\mathcal{H}om}{}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{D}_{X\to
Y}^{(0,1)},\mathcal{D}_{X}^{(0,1)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{R}(\mathcal{D}_{X}^{(1)})$
and $R\underline{\mathcal{H}om}{}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{D}_{X\to
Y}^{(0,1)},\mathcal{D}_{X}^{(0,1)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})$
are concentrated in degree $d$ as well, we see that
$\text{im}(f)=\text{ker}(v)$ and $\text{im}(v)=\text{ker}(f)$ on
$\mathcal{H}^{d}(R\underline{\mathcal{H}om}{}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{D}_{X\to
Y}^{(0,1)},\mathcal{D}_{X}^{(0,1)}))$ (by 4.8). Furthermore, the distinguished
triangle above now yields the short exact sequence

$\mathcal{\overline{R}}_{Y\leftarrow
X}^{(0)}(d)\to\mathcal{H}^{d}(R\underline{\mathcal{H}om}{}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{D}_{X\to
Y}^{(0,1)},\mathcal{D}_{X}^{(0,1)}))\to\mathcal{R}_{Y\leftarrow X}^{(1)}(d-1)$

and since $\mathcal{R}_{Y\leftarrow X}^{(1)}$ is $f$-torsion-free, we see that
$\text{im}(v)=\text{ker}(f)=\mathcal{\overline{R}}_{Y\leftarrow X}^{(0)}(d)$
and so
$\mathcal{H}^{d}(R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$
satisfies the conditions of 4.9. So we may conclude that the module
$\mathcal{H}^{d}(R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$
is standard. Furthermore, we see that the grading on
$\mathcal{\overline{R}}_{Y\leftarrow X}^{(0)}(d)$ is zero in degrees $<-d$ and
is nontrivial in degree $-d$ and above. Therefore, the index (as defined
directly below 4.4) is $d$. Since we identified
$\mathcal{H}^{d}(R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})^{-\infty})$
with $\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0)}$ we see
that

$\mathcal{H}^{d_{X/Y}}(R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})^{i-d})=\\{m\in\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0)}[p^{-1}]|p^{i}m\in\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0)}\\}$

which is exactly the definition of
$\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}(d)$. ∎

From this one deduces

######  Corollary 7.24.

Let $\varphi:\mathfrak{X}\to\mathfrak{Y}$ be smooth of relative dimension $d$;
let $\mathcal{M}^{\cdot}\in
D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ and
$\mathcal{N}^{\cdot}\in
D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}))$. Then there
is an isomorphism of functors

$R\varphi_{*}R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\varphi^{\dagger}\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})\tilde{\to}R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}(\mathcal{N}^{\cdot},\int_{\varphi}\mathcal{M}^{\cdot}(d))$

In particular, if $\varphi$ is also proper, then since both
$\varphi^{\dagger}$ and ${\displaystyle\int_{\varphi}}$ preserve
$D_{coh}^{b}$, we obtain that these functors form an adjoint pair on
$D_{coh}^{b}$.

Further, the analogous isomorphism for $\varphi:X\to Y$ holds, and in this
situation the functors are adjoint on $D_{qcoh}^{b}$ in this setting (even if
$\varphi$ is not proper).

###### Proof.

(following [K], Theorem 4.40). We have

$R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\varphi^{\dagger}\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})\tilde{\to}R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}\widehat{\otimes}_{\varphi^{-1}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}^{L}\varphi^{-1}\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})[d]$
$\tilde{\to}R\underline{\mathcal{H}om}{}_{\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})}(\varphi^{-1}\mathcal{N}^{\cdot},R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\mathcal{M}^{\cdot}))[d]$

To prove the last isomorphism, one may reduce mod $p$, and then apply 2.7
(part $1$), noting that $\mathcal{D}_{X\to Y}^{(0,1)}$ is faithfully flat over
$\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})$.

Further, we have

$R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\mathcal{M}^{\cdot})\tilde{\leftarrow}R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}\tilde{=}\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}(d)[-d]$

where the first isomorphism again follows by reduction mod $p$ and then
applying the fact that $\mathcal{D}_{X\to Y}^{(0,1)}$ is (locally) isomorphic
to a bounded complex of projective $\mathcal{D}_{X}^{(0,1)}$-modules (by 4.22)
and the second isomorphism is the previous proposition. Applying this to the
previous isomorphism we obtain

$R\underline{\mathcal{H}om}{}_{\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})}(\varphi^{-1}\mathcal{N}^{\cdot},R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\mathcal{M}^{\cdot}))[d]\tilde{=}R\underline{\mathcal{H}om}{}_{\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})}(\varphi^{-1}\mathcal{N}^{\cdot},\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}(d))$

Then applying $R\varphi_{*}$ we obtain

$R\varphi_{*}R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\varphi^{\dagger}\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})$
$\tilde{=}R\varphi_{*}R\underline{\mathcal{H}om}{}_{\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})}(\varphi^{-1}\mathcal{N}^{\cdot},\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}(d))$
$\tilde{\to}R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}(\mathcal{N}^{\cdot},R\varphi_{*}(\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}(d)))\tilde{\to}R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}(\mathcal{N}^{\cdot},\int_{\varphi}\mathcal{M}^{\cdot}(d))$

where the final isomorphism, is the adjunction between $\varphi^{-1}$ and
$R\varphi_{*}$. One applies analogous reasoning for $\varphi:X\to Y$. ∎

Now we prove the projection formula, and then give the the smooth base change
and Kunneth formulas in this context. We start with

######  Theorem 7.25.

(Projection Formula) Let $\varphi:\mathfrak{X}\to\mathfrak{Y}$ be a morphism.
Let $\mathcal{M}^{\cdot}\in
D_{cc}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ and
$\mathcal{N}^{\cdot}\in
D_{cc}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}))$, be
such that $\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k\in
D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$ and
$\mathcal{N}^{\cdot}\otimes_{W(k)}^{L}k\in
D_{qcoh}(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)}))$. Then we have

$\int_{\varphi}(L\varphi^{*}(\mathcal{N}^{\cdot})\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}})}^{L}\mathcal{M}^{\cdot})\tilde{\to}\mathcal{N}^{\cdot}\otimes_{D(\mathcal{O}_{\mathfrak{Y}})}^{L}\int_{\varphi}\mathcal{M}^{\cdot}$

The proof works essentially the same way as the complex analytic one (c.f.
[Bj], theorem 2.3.19). In particular, we use 7.6, as well as the tensor
product juggling lemma 6.9

###### Proof.

By the left-right interchange it suffices to prove

$\int_{\varphi}(\mathcal{M}_{r}^{\cdot}\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}})}^{L}L\varphi^{*}\mathcal{N}^{\cdot})\tilde{=}\int_{\varphi}(\mathcal{M}_{r}^{\cdot})\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{Y}})}^{L}\mathcal{N}^{\cdot}$

where
$\mathcal{M}_{r}^{\cdot}=\omega_{\mathfrak{X}}\otimes_{\mathcal{O}_{\mathfrak{X}}}\mathcal{M}^{\cdot}$.
We have

$\int_{\varphi}(\mathcal{M}_{r}^{\cdot})\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{Y}})}^{L}\mathcal{N}^{\cdot}\tilde{=}\int_{\varphi}(\mathcal{M}_{r}^{\cdot})\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)}}^{L}(\mathcal{N}^{\cdot}\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{Y}})}^{L}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)})$
$\tilde{=}R\varphi_{*}(\mathcal{M}_{r}^{\cdot}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}L\varphi^{*}(\mathcal{N}^{\cdot}\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{Y}})}^{L}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)}))\tilde{=}R\varphi_{*}(\mathcal{M}_{r}^{\cdot}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}L\varphi^{*}(\mathcal{N}^{\cdot})\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}})}^{L}\varphi^{*}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)}))$
$\tilde{=}R\varphi_{*}((\mathcal{M}_{r}^{\cdot}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}L\varphi^{*}(\mathcal{N}^{\cdot}))\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}})}^{L}\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})\tilde{=}R\varphi_{*}((\mathcal{M}_{r}^{\cdot}\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}})}^{L}L\varphi^{*}(\mathcal{N}^{\cdot}))\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})$
$=\int_{\varphi}(\mathcal{M}_{r}^{\cdot}\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}})}^{L}L\varphi^{*}\mathcal{N}^{\cdot})$

as claimed; note that the second isomorphism is 7.6 which uses the assumption
on $\mathcal{M}^{\cdot}$ and $\mathcal{N}^{\cdot}$. ∎

Now we turn to the smooth base change. Consider the fibre square of smooth
formal schemes

$\begin{CD}\mathfrak{X}_{\mathfrak{Z}}@>{\tilde{\psi}}>{}>\mathfrak{X}\\\
@V{}V{\tilde{\varphi}}V@V{}V{\varphi}V\\\
\mathfrak{Z}@>{\psi}>{}>\mathfrak{Y}\end{CD}$

where the bottom row $\psi:\mathfrak{Z}\to\mathfrak{Y}$ is smooth of relative
dimension $d$.

We have also the analogous square for smooth varieties over $k$.

######  Theorem 7.26.

Suppose that $\mathcal{M}^{\cdot}\in
D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ and
$\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k\in
D_{qcoh}^{b}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$. There is an isomorphism

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

inside $D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{Z}}^{(0,1)}))$.
The analogous statement holds for smooth varieties over $k$.

###### Proof.

By the adjunction for
${\displaystyle(\tilde{\psi}^{\dagger},\int_{\tilde{\psi}}(d))}$ there is a
morphism of functors

$\int_{\varphi}\to\int_{\varphi}\circ\int_{\tilde{\psi}}(\tilde{\psi})^{\dagger}(d)\tilde{=}\int_{\psi}\circ\int_{\tilde{\varphi}}(\tilde{\psi})^{\dagger}(d)$

where the last isomorphism follows from the composition of push-forwards
(7.7). Now, applying the adjunction for
${\displaystyle(\psi^{\dagger},\int_{\psi}(d))}$, we obtain a morphism

$\psi^{\dagger}\int_{\varphi}\to\int_{\tilde{\varphi}}(\tilde{\psi})^{\dagger}$

After applying $\otimes_{W(k)}^{L}k$ we obtain the analogous map over $k$. So
it suffices to show that the map is an isomorphism for varieties over $k$.
Furthermore, working locally on $Z$, we reduce to the case where the map
$\psi:Z\to Y$ factors as an etale morphism $Z\to Z^{\prime}$ followed by a
projection $Z^{\prime}\tilde{=}Y\times\mathbb{A}^{d}\to Y$. In the case of an
etale morphism, the functor ${\displaystyle\int_{\varphi}}$ agrees with
$R\varphi_{*}$, so the result follows from the usual flat base change for
quasicoherent sheaves. In the case of the projection, we have

$\int_{\tilde{\varphi}}(\tilde{\psi})^{\dagger}\mathcal{M}^{\cdot}\tilde{=}\int_{\text{id}\times\varphi}D(\mathcal{O}_{\mathbb{A}_{k}^{d}})\boxtimes\mathcal{M}^{\cdot}[d]\tilde{=}D(\mathcal{O}_{\mathbb{A}_{k}^{d}})\boxtimes\int_{\varphi}\mathcal{M}^{\cdot}[d]\tilde{=}\psi^{\dagger}\mathcal{M}^{\cdot}$

where the second isomorphism follows directly from the definition of the
pushforward; this implies the result in this case. ∎

From this we deduce the Kunneth formula:

######  Corollary 7.27.

Let $\mathcal{M}^{\cdot}\in
D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ and
$\mathcal{N}^{\cdot}\in
D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}))$, so that
$\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k\in
D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$ and
$\mathcal{N}^{\cdot}\otimes_{W(k)}^{L}k\in
D_{qcoh}(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)}))$. Then there is an isomorphism

$\mathbb{H}_{\mathcal{G}}^{\cdot}(\mathcal{M}^{\cdot}\boxtimes\mathcal{N}^{\cdot})\tilde{=}\mathbb{H}_{\mathcal{G}}^{\cdot}(\mathcal{M}^{\cdot})\widehat{\otimes}_{W(k)[f,v]}^{L}\mathbb{H}_{\mathcal{G}}^{\cdot}(\mathcal{N}^{\cdot})$

(where $\mathbb{H}_{\mathcal{G}}^{\cdot}$ is defined in 7.1)The analogous
statement holds for complexes in
$D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$ and
$D_{qcoh}(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)}))$.

This is a formal consequence of the projection formula and the smooth base
change (compare, e.g. [Di], corollary 2.3.30).

##  8\. Operations on Gauges: Duality

In this section we study the duality functor on
$D_{coh}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ (and
on $D_{coh}^{b}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$. Although neither
$\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ nor $\mathcal{D}_{X}^{(0,1)}$
have finite homological dimension, we shall show (using 4.21) that there is a
well-behaved duality functor $\mathbb{D}$ which takes bounded complexes of
coherent modules to bounded complexes of coherent modules. Further, under
suitable conditions this functor commutes with push-forward, in the following
sense:

######  Theorem 8.1.

Let $\varphi:\mathfrak{X}\to\mathfrak{Y}$ be either a smooth proper morphism
or a projective morphism. Then there is an isomorphism of functors

$\int_{\varphi}\mathbb{D}_{\mathfrak{X}}\tilde{\to}\mathbb{D}_{\mathfrak{Y}}\int_{\varphi}$

The analogous statement holds for either a smooth proper or a projective
morphism $\varphi:X\to Y$. In particular; when $\varphi$ is smooth proper the
functors $(\int_{\varphi},\varphi^{\dagger})$ form an adjoint pair on
$D_{coh}^{b}$.

The proof, which will essentially occupy this section of the paper, is
somewhat unsatisfactory. The key point is to construct a trace morphism

$\text{tr}:\int_{\varphi}D(\mathcal{O}_{\mathfrak{X}})[d_{X}]\to
D(\mathcal{O}_{\mathfrak{Y}})[d_{Y}]$

When $\varphi$ is smooth proper this is done by first constructing the map in
$\mathcal{D}^{(0)}$ and $\mathcal{D}^{(1)}$ modules (using the Hodge to de
Rham spectral sequence), and then deducing its existence for
$\mathcal{D}^{(0,1)}$-modules. When $\varphi$ is a closed immersion the
construction of the trace follows from a direct consideration of the structure
of ${\displaystyle\int_{\varphi}}$ (the transfer bimodule is easy to describe
in this case). For a projective $\varphi$ one defines the trace by breaking up
the map into an immersion followed by a projection. Presumably there is a way
to construct the trace for all proper morphisms at once, but I have been
unable to find it.

To kick things off, we need to define the duality functor and show that it has
finite homological dimension.

######  Definition 8.2.

Let $\mathcal{M}^{\cdot}\in
D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$. We define
$\mathbb{D}_{\mathfrak{X}}(\mathcal{M}^{\cdot}):=\omega_{\mathfrak{X}}^{-1}\otimes_{\mathcal{O}_{\mathfrak{X}}}R\underline{\mathcal{H}om}(\mathcal{M}^{\cdot},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})[d_{X}]\in
D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ (where we
have used the natural right
$\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$-module structure on
$R\underline{\mathcal{H}om}(\mathcal{M}^{\cdot},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})$).

The same formula defines $\mathbb{D}_{X}$ for a smooth variety $X$ over $k$;
and in the analogous way we define the duality functors for
$\mathcal{R}(\mathcal{D}_{X}^{(1)})$ and
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$.

This is really a duality on the category of coherent modules:

######  Proposition 8.3.

Suppose $\mathcal{M}^{\cdot}\in
D_{coh}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ then
$\mathbb{D}_{\mathfrak{X}}(\mathcal{M}^{\cdot})\in
D_{coh}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$.
Further, the natural transformation
$\mathcal{M}^{\cdot}\to\mathbb{D}_{\mathfrak{X}}\mathbb{D}_{\mathfrak{X}}\mathcal{M}^{\cdot}$
is an isomorphism.

The same result holds for a smooth variety $X$ over $k$.

###### Proof.

By reduction mod $p$ it suffices to prove the result for $X$. Using 4.21, and
the fact that
$\text{ker}(f:\mathcal{D}_{X}^{(0,1)}\to\mathcal{D}_{X}^{(0,1)})\tilde{=}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})(1)$
and
$\text{ker}(v:\mathcal{D}_{X}^{(0,1)}\to\mathcal{D}_{X}^{(0,1)})\tilde{=}\mathcal{R}(\mathcal{D}_{X}^{(1)})(-1)$
one reduces to proving the analogous result for
$\mathcal{R}(\mathcal{D}_{X}^{(1)})$ and
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$. But these algebras have
finite homological dimension, as noted above, and the results follow at once.
∎

###  8.1. Duality for a smooth proper morphism

Now we turn to defining the trace morphism, and proving the duality, for a
smooth proper map $\mathfrak{X}\to\mathfrak{Y}$ of relative dimension $d$. In
this case the usual Grothendieck duality theory gives us a canonical morphism

$\text{tr}:R^{d}\varphi_{*}(\omega_{\mathfrak{X}/\mathfrak{Y}})\to\mathcal{O}_{\mathfrak{Y}}$

Now consider $\mathcal{O}_{\mathfrak{X}}$ as a module over
$\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$. As the pushforward
${\displaystyle\int_{\varphi,0}\mathcal{O}_{\mathfrak{X}}}$ can be computed by
the relative de Rham complex, looking at the Hodge-to-de Rham spectral
sequence in degree $2d$ yields an isomorphism of
$\mathcal{O}_{\mathfrak{Y}}$-modules

$\mathcal{H}^{d}(\int_{\varphi,0}\mathcal{O}_{\mathfrak{X}})\tilde{=}R^{d}\varphi_{*}(\omega_{\mathfrak{X}/\mathfrak{Y}})$

so; composing with the trace morphism above, we obtain a map

$\text{tr}:\mathcal{H}^{d}(\int_{\varphi,0}\mathcal{O}_{\mathfrak{X}})\to\mathcal{O}_{\mathfrak{Y}}$

of $\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0)}$-modules.

Now consider $\varphi:\mathfrak{X}_{n}\to\mathfrak{Y}_{n}$, the reduction mod
$p^{n}$ of $\varphi$ for each $n\geq 0$. Repeating the argument, we can
construct

$\text{tr}:\mathcal{H}^{d}(\int_{\varphi,0}\mathcal{O}_{\mathfrak{X}_{n}})\to\mathcal{O}_{\mathfrak{Y}_{n}}$

and, in fact, the inverse limit of these maps is the trace constructed above.
In this setting, the de Rham complex
$\Omega_{\mathfrak{X}_{n}/\mathfrak{Y}_{n}}^{\cdot}$ has the structure of a
complex of coherent sheaves over the scheme $W_{n}(\mathcal{O}_{X^{(n)}})$
(here we are identifying the underlying topological spaces of
$\mathfrak{X}_{n}$ and $W_{n}(\mathcal{O}_{X^{(n)}})$). Thus we may also
consider the second spectral sequence for the pushforward of this complex, and
we obtain an isomorphism

$R^{d}\varphi_{*}(\text{coker}(d:\Omega_{\mathfrak{X}_{n}/\mathfrak{Y}_{n}}^{d-1}\to\omega_{\mathfrak{X}_{n}/\mathfrak{Y}_{n}}))\tilde{\to}R^{d}\varphi_{*}(\omega_{\mathfrak{X}/\mathfrak{Y}})$

or, equivalently,

$R^{d}\varphi_{*}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0)}\otimes_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0)}}\mathcal{O}_{\mathfrak{X}_{n}})\tilde{\to}R^{d}\varphi_{*}(\omega_{\mathfrak{X}_{n}/\mathfrak{Y}_{n}})$

Now we consider the the pushforward of $\mathcal{O}_{\mathfrak{X}_{n}}$, in
the category of $\mathcal{D}_{\mathfrak{X}_{n}}^{(1)}$-modules. By the
commutativity of Frobenius descent with push-forward ([B2], theoreme 3.4.4),
we have

$\int_{\varphi,1}\mathcal{O}_{\mathfrak{X}_{n}}\tilde{=}\int_{\varphi,1}F^{*}\mathcal{O}_{\mathfrak{X}_{n}}\tilde{\to}F^{*}\int_{\varphi,0}\mathcal{O}_{\mathfrak{X}_{n}}$

Therefore we obtain a trace map

$\text{tr}:\mathcal{H}^{d}(\int_{\varphi,1}\mathcal{O}_{\mathfrak{X}_{n}})\to\mathcal{O}_{\mathfrak{Y}_{n}}$

in the category of $\mathcal{D}_{\mathfrak{Y}_{n}}^{(1)}$-modules; and, using
the second spectral sequence for the pushforward as above, we have

$R^{d}\varphi_{*}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(1)}\otimes_{\mathcal{D}_{\mathfrak{X}_{n}}^{(1)}}\mathcal{O}_{\mathfrak{X}_{n}})\tilde{\to}\mathcal{H}^{d}(\int_{\varphi,1}\mathcal{O}_{\mathfrak{X}_{n}})$

Using these maps, we construct a trace for
$\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}$-modules:

######  Lemma 8.4.

There is a canonical morphism

$\text{tr}:R^{d}\varphi_{*}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)}\otimes{}_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}}D(\mathcal{O}_{\mathfrak{X}_{n}}))\to
D(\mathcal{O}_{\mathfrak{Y}_{n}})$

which has the property that the map
$\text{tr}^{\infty}:R^{d}\varphi_{*}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)}\otimes{}_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}}D(\mathcal{O}_{\mathfrak{X}_{n}}))^{\infty}\to
D(\mathcal{O}_{\mathfrak{Y}_{n}}){}^{\infty}$ agrees with the trace map for
$\mathcal{D}_{\mathfrak{X}_{n}}^{(1)}$-modules constructed above; and the map
$\text{tr}^{-\infty}:R^{d}\varphi_{*}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)}\otimes{}_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}}D(\mathcal{O}_{\mathfrak{X}_{n}}))^{-\infty}\to
D(\mathcal{O}_{\mathfrak{Y}_{n}})^{-\infty}$ agrees with the trace map for
$\mathcal{D}_{\mathfrak{X}_{n}}^{(0)}$-modules constructed above. We have the
analogous statement for a proper morphism $\varphi:X\to Y$, as well as in the
categories of $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$-modules and
$\mathcal{R}(\mathcal{D}_{X}^{(1)})$-modules.

This map yields a trace map in the derived category:

$\text{tr}:\int_{\varphi}D(\mathcal{O}_{\mathfrak{X}_{n}})[d]\to
D(\mathcal{O}_{\mathfrak{Y}_{n}})$

Upon taking the inverse limit over $n$, we obtain a map

$\text{tr}:\int_{\varphi}D(\mathcal{O}_{\mathfrak{X}})[d]\to
D(\mathcal{O}_{\mathfrak{Y}})$

###### Proof.

We begin with the case $n=1$; i.e., $\mathfrak{X}_{n}=X$ and
$\mathfrak{Y}_{n}=Y$. We claim that the
$\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})$-gauge $\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes{}_{\mathcal{D}_{X}^{(0,1)}}D(\mathcal{O}_{X})$ satisfies the
property that $v$ is an isomorphism in degrees $0$ and below and $f$ is an
isomorphism in degrees $1$ and above. This can be checked in local
coordinates, where we have the isomorphism

$\mathcal{D}_{Y\leftarrow
X}^{(0,1)}=\mathcal{J}\backslash\mathcal{D}_{X}^{(0,1)}$

where $\mathcal{J}$ is the right ideal generated by
$\\{\partial_{n-d+1},\dots,\partial_{n},\partial_{n-d+1}^{[p]},\dots,\partial_{n}^{[p]}\\}$.
In degrees below $0$, the elements
$\\{\partial_{n-d+1}^{[p]},\dots,\partial_{n}^{[p]}\\}$ act trivially; so that

$(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes{}_{\mathcal{D}_{X}^{(0,1)}}D(\mathcal{O}_{X}))^{i}=\mathcal{O}_{X}/(\partial_{n-d+1},\dots,\partial_{n})$

for all $i\leq 0$. On the other hand we have

$(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes{}_{\mathcal{D}_{X}^{(0,1)}}D(\mathcal{O}_{X}))^{i}=\mathcal{O}_{X}/(\partial_{n-d+1},\dots,\partial_{n},\partial_{n-d+1}^{[p]},\dots,\partial_{n}^{[p]})$

for $i>0$; and the claim about $f$ and $v$ follows immediately. As the functor
$R^{d}\varphi_{*}$ commutes with direct sums, we see that the gauge

$R^{d}(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes{}_{\mathcal{D}_{X}^{(0,1)}}D(\mathcal{O}_{X}))$

has the same property: $v$ is an isomorphism in degrees $0$ and below and $f$
is an isomorphism in degrees $1$ and above. Thus we may define

$\text{tr}:R^{d}(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes{}_{\mathcal{D}_{X}^{(0,1)}}D(\mathcal{O}_{X}))^{i}\to\mathcal{O}_{Y}$

for any $i$ as follows: if $i\leq 0$ we have
$v_{-\infty}:R^{d}(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes{}_{\mathcal{D}_{X}^{(0,1)}}D(\mathcal{O}_{X}))^{i}\tilde{=}R^{d}\varphi_{*}(\mathcal{D}_{Y\leftarrow
X}^{(0)}\otimes_{\mathcal{D}_{X}^{(0)}}\mathcal{O}_{X})$ and so we define the
trace as the composition $\text{tr}\circ v_{-\infty}$, where here tr denotes
the trace for $\mathcal{D}_{X}^{(0)}$-modules constructed above. If $i>0$ we
have $f_{\infty}:R^{d}(\mathcal{D}_{Y\leftarrow
X}^{(0,1)}\otimes{}_{\mathcal{D}_{X}^{(0,1)}}D(\mathcal{O}_{X}))^{i}\tilde{=}R^{d}\varphi_{*}(\mathcal{D}_{Y\leftarrow
X}^{(1)}\otimes_{\mathcal{D}_{X}^{(1)}}\mathcal{O}_{X})$ and so we define the
trace as the composition $\text{tr}\circ f_{\infty}$, where here tr denotes
the trace for $\mathcal{D}_{X}^{(1)}$-modules constructed above. In a similar
way, we construct the trace map in the categories of
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$-modules and
$\mathcal{R}(\mathcal{D}_{X}^{(1)})$-modules.

Now we consider $\mathfrak{X}_{n}$ for $n>1$. Since the functor $\varphi_{*}$
has homological dimension $d$ (on the category of quasicoherent sheaves), we
have that
$(R^{d}\varphi_{*}\mathcal{F})\otimes_{W(k)}k\tilde{=}(R^{d}\varphi_{*}\mathcal{F}\otimes_{W(k)}^{L}k)$
for any $\mathcal{F}\in\text{Qcoh}(\mathfrak{X}_{n})$. So, by Nakayama’s lemma
and the result of the previous paragraph, we see that $f$ is onto in degrees
$1$ and above while $v$ is onto in degrees $0$ and below; by the coherence of
the sheaves involved we see that these maps are isomorphisms for $i<<0$ and
$i>>0$. Since the target of the trace map,
$D(\mathcal{O}_{\mathfrak{Y}_{n}})$, has the property that $v$ is an
isomorphism in degrees $0$ and below and $f$ is an isomorphism in degrees $1$
and above, we may define the trace map in the exact same way as above. ∎

######  Remark 8.5.

If $\varphi:\mathfrak{X}\to\mathfrak{Y}$ and
$\psi:\mathfrak{Y}\to\mathfrak{Z}$, then the trace map for the composition
satisfies
${\displaystyle\text{tr}_{\psi\circ\varphi}=\text{tr}_{\psi}\circ\int_{\psi}\text{tr}_{\varphi}}$.
This follows from the analogous result for the trace map in coherent sheaf
theory.

Now, following the usual method of algebraic $\mathcal{D}$-module theory (c.f.
[HTT], theorem 2.7.2), we have

######  Proposition 8.6.

There is a canonical morphism

$\int_{\varphi}\mathbb{D}_{\mathfrak{X}}\mathcal{M}^{\cdot}\to\mathbb{D}_{\mathfrak{Y}}\int_{\varphi}\mathcal{M}^{\cdot}$

for any $\mathcal{M}^{\cdot}\in
D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$. The same
holds for $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$
when we have a proper map $\varphi:X\to Y$. Further, these maps are compatible
under application of $\otimes_{W(k)}^{L}k$.

###### Proof.

We have

$\int_{\varphi}\mathbb{D}_{\mathfrak{X}}\mathcal{M}^{\cdot}=R\varphi_{*}(R\underline{\mathcal{H}om}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\mathcal{M}^{\cdot},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})\widehat{\otimes}_{\mathcal{O}_{\mathfrak{Y}}}^{L}\omega_{\mathfrak{Y}}^{-1}[d_{X}]$
$=R\varphi_{*}(R\underline{\mathcal{H}om}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\mathcal{M}^{\cdot},\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})\widehat{\otimes}_{\mathcal{O}_{\mathfrak{Y}}}^{L}\omega_{\mathfrak{Y}}^{-1}[d_{X}]$

while

$\mathbb{D}_{\mathfrak{Y}}\int_{\varphi}\mathcal{M}^{\cdot}=R\underline{\mathcal{H}om}_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}(\int_{\varphi}\mathcal{M}^{\cdot},\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})\widehat{\otimes}_{\mathcal{O}_{\mathfrak{Y}}}^{L}\omega_{\mathfrak{Y}}^{-1}[d_{Y}]$

To construct a canonical map between these complexes, we begin by considering
${\displaystyle\int_{\varphi}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}}$.
By
$\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}=L\varphi^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}$,
we may apply 7.25 to obtain

$\int_{\varphi}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}=\int_{\varphi}L\varphi^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}\tilde{\to}(\int_{\varphi}D(\mathcal{O}_{\mathfrak{X}}))\widehat{\otimes}_{\mathcal{O}_{\mathfrak{Y}}[f,v]}^{L}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}$

so applying the trace map yields a canonical morphism

$\int_{\varphi}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}[d]\to\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}$

and since $d=d_{X}-d_{Y}$ we have

$\int_{\varphi}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}[d_{X}]\to\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}[d_{Y}]$

Then we have

$R\varphi_{*}(R\underline{\mathcal{H}om}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\mathcal{M}^{\cdot},\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})[d_{X}]$
$\to
R\varphi_{*}(R\underline{\mathcal{H}om}_{\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})}(\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot},\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})[d_{X}]$
$\to
R\underline{\mathcal{H}om}_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}(R\varphi_{*}(\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}),R\varphi_{*}(\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}))[d_{X}]$
$=R\underline{\mathcal{H}om}_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}(\int_{\varphi}\mathcal{M}^{\cdot},\int_{\varphi}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}[d_{X}])\to
R\underline{\mathcal{H}om}_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}(\int_{\varphi}\mathcal{M}^{\cdot},\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}[d_{Y}])$

where the last map is the trace. Combining with the above description yields
the canonical map

$\int_{\varphi}\mathbb{D}_{\mathfrak{X}}\mathcal{M}^{\cdot}\to\mathbb{D}_{\mathfrak{Y}}\int_{\varphi}\mathcal{M}^{\cdot}$

as desired; the case of a proper map $\varphi:X\to Y$ is identical. ∎

Now we turn to

######  Theorem 8.7.

The canonical map
$\int_{\varphi}\mathbb{D}_{\mathfrak{X}}\mathcal{M}^{\cdot}\to\mathbb{D}_{\mathfrak{Y}}\int_{\varphi}\mathcal{M}^{\cdot}$
is an isomorphism for $\mathcal{M}^{\cdot}\in
D_{coh}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$. The
same is true for a proper map $\varphi:X\to Y$.

The proof of this result will make use of several auxiliary results. First, we
recall a basic computation for pushforwards of
$\mathcal{R}(\mathcal{D}_{X}^{(0)})$-modules; as in the previous section we
have the diagram

$T^{*}X\xleftarrow{d\varphi}X\times_{Y}T^{*}Y\xrightarrow{\pi}T^{*}Y$

and the result reads

######  Lemma 8.8.

Let $\mathcal{M}^{\cdot}\in
D_{coh}^{b}(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(0)})))$. There is an
isomorphism

$(\int_{\varphi}\mathcal{M}^{\cdot})\otimes_{k[f]}^{L}k\tilde{=}R\pi_{*}((d\varphi)^{!}(\mathcal{M}^{\cdot}\otimes_{k[f]}^{L}k))$

inside $D_{coh}^{b}(T^{*}Y)$; in this formula $d\varphi^{!}$ is the
extraordinary inverse image in coherent sheaf theory.

This is a result of Laumon, (c.f. [L], construction 5.6.1). For a proof in the
Rees algebra language, see [CDK], corollary 3.9.

Next, we need the following Grothendieck duality statement for Azumaya
algebras:

######  Lemma 8.9.

Let $X$ and $Y$ be smooth varieties over $k$ and let $\pi:X\to Y$ be a smooth
proper morphism, of relative dimension $d$. Suppose that $\mathcal{A}_{Y}$ be
an Azumaya algebra on $Y$. Set $\mathcal{A}_{X}=\pi^{*}\mathcal{A}_{Y}$, an
Azumaya algebra on $X$. Then there is a trace map
$R^{d}\pi_{*}(\mathcal{A}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X})\to\mathcal{A}_{Y}\otimes_{\mathcal{O}_{Y}}\omega_{Y}$
which induces, for any $\mathcal{M}^{\cdot}\in
D_{coh}^{b}(\mathcal{A}_{X}-\text{mod})$ a functorial isomorphism

$R\pi_{*}R\mathcal{H}om_{\mathcal{A}_{X}}(\mathcal{M}^{\cdot},\mathcal{A}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X})[d]\tilde{\to}R\mathcal{H}om_{\mathcal{A}_{Y}}(R\pi_{*}\mathcal{M}^{\cdot},\mathcal{A}_{Y}\otimes_{\mathcal{O}_{Y}}\omega_{Y})$

inside $D_{coh}^{b}(\mathcal{O}_{Y}-\text{mod})$.

###### Proof.

Via the projection formula we have

$R\pi_{*}(\mathcal{A}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X})=R\pi_{*}(\pi^{*}\mathcal{A}_{Y}\otimes_{\mathcal{O}_{X}}\omega_{X})\tilde{\to}\mathcal{A}_{Y}\otimes_{\mathcal{O}_{Y}}^{L}R\pi_{*}(\omega_{X})$

so the usual trace $\text{tr}:R^{d}\pi_{*}(\omega_{X})\to\omega_{Y}$ induces a
trace
$\text{tr}:R^{d}\pi_{*}(\mathcal{A}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X})\to\mathcal{A}_{Y}\otimes_{\mathcal{O}_{Y}}\omega_{Y}$.
Since $\pi$ has homological dimension $d$, we have
$R\pi_{*}(\mathcal{A}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X})[d]\to
R^{d}\pi_{*}(\mathcal{A}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X})$ so that
there is a map

$R\pi_{*}(\mathcal{A}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X})[d]\to\mathcal{A}_{Y}\otimes_{\mathcal{O}_{Y}}\omega_{Y}$

Thus we obtain

$R\pi_{*}R\mathcal{H}om_{\mathcal{A}_{X}}(\mathcal{M}^{\cdot},\mathcal{A}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X})[d]\to
R\mathcal{H}om_{\mathcal{A}_{Y}}(R\pi_{*}\mathcal{M}^{\cdot},R\pi_{*}(\mathcal{A}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X})[d])$
$\to
R\mathcal{H}om_{\mathcal{A}_{Y}}(R\pi_{*}\mathcal{M}^{\cdot},\mathcal{A}_{Y}\otimes_{\mathcal{O}_{Y}}\omega_{Y})$

for any $\mathcal{M}^{\cdot}\in D_{coh}^{b}(\mathcal{A}_{X}-\text{mod})$. To
prove that this map is an isomorphism, we can can work in the etale (or flat)
topology on $Y$ and so assume that $\mathcal{A}_{Y}$ is split; i.e.,
$\mathcal{A}_{Y}=\mathcal{E}nd(\mathcal{E}_{Y})$ for some vector bundle
$\mathcal{E}_{Y}$. This implies
$\mathcal{A}_{X}=\mathcal{E}nd(\mathcal{E}_{X})$ where
$\mathcal{E}_{X}=\pi^{*}\mathcal{E}_{Y}$. Then for any $\mathcal{M}^{\cdot}\in
D_{coh}^{b}(\mathcal{A}_{X}-\text{mod})$ we have
$\mathcal{M}^{\cdot}=\mathcal{E}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{N}^{\cdot}$
for a complex $\mathcal{N}^{\cdot}\in
D_{coh}^{b}(\mathcal{O}_{X}-\text{mod})$. Therefore

$R\pi_{*}R\mathcal{H}om_{\mathcal{A}_{X}}(\mathcal{M}^{\cdot},\mathcal{A}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X})[d]\tilde{=}R\pi_{*}R\mathcal{H}om_{\mathcal{A}_{X}}(\mathcal{E}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{N}^{\cdot},\mathcal{E}_{X}\otimes_{\mathcal{O}_{X}}(\mathcal{E}_{X}^{*}\otimes_{\mathcal{O}_{X}}\omega_{X}))[d]$
$\tilde{=}R\pi_{*}R\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{N}^{\cdot},\mathcal{E}_{X}^{*}\otimes_{\mathcal{O}_{X}}\omega_{X})[d]\tilde{=}R\pi_{*}R\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{M}^{\cdot},\omega_{X}[d])$
$\tilde{\to}R\mathcal{H}om_{\mathcal{O}_{Y}}(R\pi_{*}\mathcal{M}^{\cdot},\omega_{Y})\tilde{=}R\mathcal{H}om_{\mathcal{A}_{Y}}(\mathcal{E}_{Y}\otimes_{\mathcal{O}_{Y}}R\pi_{*}\mathcal{M}^{\cdot},\mathcal{E}_{Y}\otimes_{\mathcal{O}_{Y}}\omega_{Y})$
$\tilde{\to}R\mathcal{H}om_{\mathcal{A}_{Y}}(R\pi_{*}\mathcal{M}^{\cdot},\mathcal{A}_{Y}\otimes_{\mathcal{O}_{Y}}\omega_{Y})$

where the isomorphism
$R\pi_{*}R\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{M}^{\cdot},\omega_{X}[d])\tilde{\to}R\mathcal{H}om_{\mathcal{O}_{Y}}(R\pi_{*}\mathcal{M}^{\cdot},\omega_{Y})$
is Grothendieck duality for coherent sheaves. ∎

Now we can proceed to the

###### Proof.

(of 8.7) By applying $\otimes_{W(k)}^{L}k$ we reduce to the characteristic $p$
situation of a smooth proper morphism $\varphi:X\to Y$. By induction on the
cohomological length, we may suppose that $\mathcal{M}^{\cdot}$ is
concentrated in a single degree; i.e.,
$\mathcal{M}^{\cdot}=\mathcal{M}\in\mathcal{G}(\mathcal{D}_{X}^{(0,1)})$. Then
$\mathcal{M}$ admits a short exact sequence

$\mathcal{M}_{0}\to\mathcal{M}\to\mathcal{M}_{1}$

where
$\mathcal{M}_{0}\in\mathcal{G}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))$
and $\mathcal{M}_{1}\in\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)}))$. By
4.21 and 7.8, we see that is suffices to prove the analogous statements in
$D_{coh}^{b}(\mathcal{G}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})))$ and
$D_{coh}^{b}(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)})))$. By Frobenius
descent (7.13), one sees that it suffices to prove the result for
$D_{coh}^{b}(\mathcal{G}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})))$ and
$D_{coh}^{b}(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(0)})))$. These two
cases require similar, but slightly different techniques; we begin with the
case of $\mathcal{R}(\mathcal{D}_{X}^{(0)})$. In this case, since the grading
on $\mathcal{R}(\mathcal{D}_{X}^{(0)})$ is concentrated in degrees $\geq 0$,
the graded Nakayama lemma applies and so it suffices to prove that the map is
an isomorphism after applying $\otimes_{k[f]}^{L}k$; i.e., we have to prove

$R\pi_{*}(d\varphi)^{!}R\mathcal{H}om_{\mathcal{O}_{T^{*}X}}((\mathcal{M}\otimes_{k[f]}^{L}k),\omega_{T^{*}X})[d]$
$\tilde{\to}R\mathcal{H}om_{\mathcal{O}_{T^{*}Y}}(R\pi_{*}((d\varphi)^{!}(\mathcal{M}\otimes_{k[f]}^{L}k),\omega_{T^{*}Y})$

Since $d\varphi$ is a closed immersion of smooth schemes, we have

$(d\varphi)^{!}R\mathcal{H}om_{\mathcal{O}_{T^{*}X}}((\mathcal{M}\otimes_{k[f]}^{L}k),\omega_{T^{*}X})[d]$
$\tilde{=}R\mathcal{H}om_{\mathcal{O}_{X\times_{Y}T^{*}Y}}((d\varphi)^{!}(\mathcal{M}\otimes_{k[f]}^{L}k),(d\varphi)^{!}\omega_{T^{*}X})[d]$

Furthermore,
$(d\varphi)^{!}\omega_{T^{*}X}=\omega_{X\times_{Y}T^{*}Y}\tilde{=}\pi^{!}(\omega_{T^{*}Y})$.
Therefore

$R\pi_{*}R\mathcal{H}om_{\mathcal{O}_{T^{*}X}}((d\varphi)^{!}(\mathcal{M}\otimes_{k[f]}^{L}k),(d\varphi)^{!}\omega_{T^{*}X})[d]$
$\tilde{=}R\pi_{*}R\mathcal{H}om_{\mathcal{O}_{X\times_{Y}T^{*}Y}}((d\varphi)^{!}(\mathcal{M}\otimes_{k[f]}^{L}k),\pi^{!}\omega_{T^{*}Y}[d])$
$\tilde{\to}R\mathcal{H}om_{\mathcal{O}_{T^{*}Y}}(R\pi_{*}((d\varphi)^{!}(\mathcal{M}\otimes_{k[f]}^{L}k),\omega_{T^{*}Y})$

where the last isomorphism is induced by the trace for
$X\times_{Y}T^{*}Y\xrightarrow{\pi}T^{*}Y$, i.e., it is given by Grothendieck
duality for $\pi$; this proves the result for
$D_{coh}^{b}(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(0)})))$.

In order to handle
$D_{coh}^{b}(\mathcal{G}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})))$, we
apply a similar technique, but working directly101010As the grading on
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ in unbounded, the graded
Nakayama lemma does not apply with the Azumaya algebra
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$. Since the morphism $\varphi$
is smooth, we can make use of 7.21 and work with the functor
$R\pi_{*}^{(1)}\circ C\circ(d\varphi^{(1)})^{!}$. We therefore have to prove

$R\pi_{*}^{(1)}\circ
C\circ(d\varphi^{(1)})^{!}R\mathcal{H}om_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(\mathcal{M}^{\cdot},\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))\otimes_{\mathcal{O}_{X}}\omega_{X}^{-1}[d]$
$\tilde{\to}R\mathcal{H}om_{\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})}(R\pi_{*}^{(1)}\circ
C\circ(d\varphi^{(1)})^{!}\mathcal{M}^{\cdot},\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))\otimes_{\mathcal{O}_{Y}}\omega_{Y}^{-1}$

We proceed as above. We have an isomorphism

$C\circ(d\varphi^{(1)})^{!}R\mathcal{H}om_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(\mathcal{M}^{\cdot},\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))\otimes_{\mathcal{O}_{X}}\omega_{X}^{-1}$
$\tilde{=}R\mathcal{H}om_{(\pi^{(1)})^{*}(\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)}))}(C\circ(d\varphi^{(1)})^{!}\mathcal{M}^{\cdot},C\circ(d\varphi^{(1)})^{!}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X}}\omega_{X}^{-1}))$

Applying the definition of $(d\varphi^{(1)})^{!}$ and $C$, one deduces

$C\circ(d\varphi^{(1)})^{!}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X}}\omega_{X}^{-1})\tilde{=}(\pi^{(1)})^{*}(\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})\otimes_{\mathcal{O}_{Y}}\omega_{Y}^{-1})\otimes_{\mathcal{O}_{(X\times_{Y}T^{*}Y)^{(1)}}}\omega_{(X\times_{Y}T^{*}Y)^{(1)}}$

Therefore

$R\mathcal{H}om_{\pi^{*}(\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)}))}(C\circ(d\varphi^{(1)})^{!}\mathcal{M}^{\cdot},C\circ(d\varphi^{(1)})^{!}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X}}\omega_{X}^{-1}))[d]$
$\tilde{\to}R\mathcal{H}om_{\pi^{*}(\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)}))}(C\circ(d\varphi^{(1)})^{!}\mathcal{M}^{\cdot},(\pi^{(1)})^{*}(\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})\otimes_{\mathcal{O}_{Y}}\omega_{Y}^{-1})\otimes_{\mathcal{O}_{(X\times_{Y}T^{*}Y)^{(1)}}}\omega_{(X\times_{Y}T^{*}Y)^{(1)}})$
$\tilde{\to}R\mathcal{H}om_{\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})}(R\pi_{*}^{(1)}\circ
C\circ(d\varphi^{(1)})^{!}\mathcal{M}^{\cdot},\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{Y}}\omega_{Y}^{-1})$

where the last isomorphism follows from 8.9; this proves the result for
$\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$. ∎

This implies, by an identical argument to theorem 2.7.3 of [HTT]:

######  Corollary 8.10.

There is a functorial isomorphism

$R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{Y}}^{(0,1)}}(\int_{\varphi}\mathcal{M}^{\cdot},\mathcal{N}^{\cdot})\tilde{\to}\varphi_{*}R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{X}}^{(0,1)}}(\mathcal{M}^{\cdot},\varphi^{\dagger}\mathcal{N}^{\cdot})$

for all $\mathcal{M}^{\cdot}\in
D_{coh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{X}}^{(0,1)}))$ and
$\mathcal{N}^{\cdot}\in
D_{coh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{Y}}^{(0,1)}))$.

###  8.2. Duality for a Projective morphism

Now we turn to constructing the trace map in the case where
$\varphi:\mathfrak{X}\to\mathfrak{Y}$ is a closed embedding, of relative
dimension $d$. In this case the pushforward is fairly easy to describe:

######  Lemma 8.11.

Let $\varphi:\mathfrak{X}_{n}\to\mathfrak{Y}_{n}$ be the reduction to
$W_{n}(k)$ of the closed embedding $\varphi$. Then the transfer bimodule
$\mathcal{D}_{\mathfrak{X}_{n}\to\mathfrak{Y}_{n}}^{(0,1)}$ is locally free
over $\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}$ and is coherent over
$\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1),\text{opp}}$. Thus the functor
$\int_{\varphi}^{0}:\mathcal{G}_{coh}(\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)})\to\mathcal{G}_{coh}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})$
is exact.

###### Proof.

Working locally, we can assume that $\mathfrak{X}_{n}=\text{Spec}(B_{n})$,
$\mathfrak{Y}_{n}=\text{Spec}(A_{n})$, and $A_{n}$ admits local coordinates
$\\{x_{1},\dots,x_{n}\\}$ for which $B_{n}=A_{n}/(x_{1},\dots,x_{m})$. Then

$\Gamma(\mathcal{D}_{\mathfrak{X}_{n}\to\mathfrak{Y}_{n}}^{(0,1)})=\mathcal{D}_{A_{n}}^{(0,1)}/I\cdot\mathcal{D}_{A_{n}}^{(0,1)}$

is coherent over $\mathcal{D}_{A_{n}}^{(0,1),\text{opp}}$. Now, 3.10 implies
that $\mathcal{D}_{A_{n}}^{(0,1)}$ is free over $D(A_{n})$ (c.f. also the
proof of 3.11), with basis given by the set
$\\{\partial^{I}(\partial^{[p]})^{J}\\}$, where $I=(i_{1},\dots,i_{n})$ is a
multi-index with $0\leq i_{j}\leq p-1$ for all $j$ and $J$ is any multi-index
with entries $\geq 0$. So
$\mathcal{D}_{A_{n}}^{(0,1)}/I\cdot\mathcal{D}_{A_{n}}^{(0,1)}$ is free over
$\mathcal{D}_{B_{n}}^{(0,1)}$with basis given by
$\\{\partial^{I}(\partial^{[p]})^{J}\\}$, where $I=(i_{1},\dots,i_{m})$ is a
multi-index with $0\leq i_{j}\leq p-1$ for all $j$ and $J=(j_{1},\dots,j_{m})$
is any multi-index with entries $\geq 0$. ∎

Now we can proceed to analyze this functor, and the pullback
$\varphi^{\dagger}$, in exactly the same way as is done in the usual algebraic
$\mathcal{D}$-module theory. In this case, the existence of the trace map is
essentially deduced from the duality. To start off, we have

######  Proposition 8.12.

Let $\varphi:\mathfrak{X}_{n}\to\mathfrak{Y}_{n}$ be as above. Define
$\varphi^{\sharp}(\mathcal{M}^{\cdot}):=R\underline{\mathcal{H}om}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)},\varphi^{-1}(\mathcal{M}^{\cdot}))$.
Then there is an isomorphism of functors
$\varphi^{\dagger}\tilde{=}\varphi^{\sharp}$ on
$D(\mathcal{G}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}))$.

###### Proof.

This is very similar to [HTT], propositions 1.5.14 and 1.5.16. One first shows

$R\underline{\mathcal{H}om}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1),\text{opp}})}(\mathcal{D}_{\mathfrak{X}_{n}\to\mathfrak{Y}_{n}}^{(0,1)},\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}))\tilde{=}\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)}[-d]$

by using the Koszul complex to write a locally free resolution for
$\mathcal{D}_{\mathfrak{X}_{n}\to\mathfrak{Y}_{n}}^{(0,1)}$ over
$\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1),\text{opp}})$; note that
by the left-right interchange this implies

$R\underline{\mathcal{H}om}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)},\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}))\tilde{=}\mathcal{D}_{\mathfrak{X}_{n}\to\mathfrak{Y}_{n}}^{(0,1)}[-d]$

Then, we have

$\varphi^{\dagger}(\mathcal{M}^{\cdot})=\mathcal{D}_{\mathfrak{X}_{n}\to\mathfrak{Y}_{n}}^{(0,1)}\otimes_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})}^{L}\varphi^{-1}(\mathcal{M}^{\cdot})[-d]$
$\tilde{=}R\underline{\mathcal{H}om}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)},\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}))\otimes_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})}^{L}\varphi^{-1}(\mathcal{M}^{\cdot})$
$\tilde{\to}R\underline{\mathcal{H}om}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)},\varphi^{-1}(\mathcal{M}^{\cdot}))$

where the last isomorphism uses the fact that
$\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)}$ admits,
locally, a finite free resolution over
$\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})$. ∎

In turn, this implies

######  Corollary 8.13.

We have a functorial isomorphism

$R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}}(\int_{\varphi}\mathcal{M}^{\cdot},\mathcal{N}^{\cdot})\tilde{\to}\varphi_{*}R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}}(\mathcal{M}^{\cdot},\varphi^{\dagger}\mathcal{N}^{\cdot})$

for all $\mathcal{M}^{\cdot}\in
D_{qcoh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}))$ and
$\mathcal{N}^{\cdot}\in
D_{qcoh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}))$.

###### Proof.

(Just as in [HTT], proposition 1.5.25). By the previous proposition, it
suffices to prove the result for $\varphi^{\sharp}$ instead of
$\varphi^{\dagger}$. To proceed, note that we have the local cohomology
functor $\mathcal{N}^{\cdot}\to
R\Gamma_{\mathfrak{X}_{n}}(\mathcal{N}^{\cdot})$ which takes
$\mathcal{N}^{\cdot}\in
D_{qcoh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}))$ to
$D_{qcoh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}))$. We have

$R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}}(\int_{\varphi}\mathcal{M}^{\cdot},\mathcal{N}^{\cdot})=R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}}(\varphi_{*}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)}\otimes_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}),\mathcal{N}^{\cdot})$
$\tilde{=}R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}}(\varphi_{*}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)}\otimes_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}),R\Gamma_{\mathfrak{X}_{n}}(\mathcal{N}^{\cdot}))$
$\tilde{=}\varphi_{*}(R\underline{\mathcal{H}om}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})}(\varphi^{-1}(\varphi_{*}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)}\otimes_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}}^{L}\mathcal{M}^{\cdot})),\varphi^{-1}(R\Gamma_{\mathfrak{X}_{n}}(\mathcal{N}^{\cdot})))$
$\tilde{=}\varphi_{*}(R\underline{\mathcal{H}om}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)}\otimes_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}),\varphi^{-1}(R\Gamma_{\mathfrak{X}_{n}}(\mathcal{N}^{\cdot})))$
$\tilde{=}\varphi_{*}R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}}(\mathcal{M}^{\cdot},R\underline{\mathcal{H}om}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)},\varphi^{-1}(R\Gamma_{\mathfrak{X}_{n}}(\mathcal{N}^{\cdot}))))$
$\tilde{=}\varphi_{*}R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}}(\mathcal{M}^{\cdot},\varphi^{\sharp}\mathcal{N}^{\cdot})$

where, in both the second isomorphism and the last, we have used the existence
of an exact triangle

$R\Gamma_{\mathfrak{X}_{n}}(\mathcal{N}^{\cdot})\to\mathcal{N}^{\cdot}\to\mathcal{K}^{\cdot}$

where $\mathcal{K}^{\cdot}\in
D_{qcoh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}))$ is
isomorphic to
$Rj_{*}(\mathcal{N}^{\cdot}|_{\mathfrak{Y}_{n}\backslash\mathfrak{X}_{n}})$;
here $j:\mathfrak{Y}_{n}\backslash\mathfrak{X}_{n}\to\mathfrak{Y}_{n}$ is the
inclusion. In particular, we have that
$R\underline{\mathcal{H}om}(\mathcal{C}^{\cdot},\mathcal{K}^{\cdot})=0$ for
any $\mathcal{C}^{\cdot}$ supported along $\mathfrak{X}_{n}$. ∎

######  Corollary 8.14.

There is a canonical map

$\text{tr}:\int_{\varphi}\mathcal{O}_{\mathfrak{X}_{n}}[d_{X}]\to\mathcal{O}_{\mathfrak{Y}_{n}}[d_{Y}]$

After taking inverse limit, we obtain a trace map
${\displaystyle\text{tr}:\int_{\varphi}\mathcal{O}_{\mathfrak{X}}[d_{X}]\to\mathcal{O}_{\mathfrak{Y}}[d_{Y}]}$.
If $\psi:\mathfrak{Y}\to\mathfrak{Z}$ is a smooth morphism,

We also have a trace map

$\text{tr}:\int_{\varphi}\mathcal{O}_{\mathfrak{X}}[d_{X}]\to\mathcal{O}_{\mathfrak{Y}}[d_{Y}]$

given by taking the inverse limit of the above maps; the same compatibility
holds for this trace as well.

###### Proof.

The previous corollary gives an adjunction
${\displaystyle\int_{\varphi}\varphi^{\dagger}\to\text{Id}}$. Since
$\varphi^{\dagger}(\mathcal{O}_{\mathfrak{Y}_{n}})=\mathcal{O}_{\mathfrak{X}_{n}}[d_{X}-d_{Y}]$
we obtain the trace map via this adjunction. ∎

Now, by factoring an arbitrary projective morphism as a closed immersion
followed by a smooth projective map, we obtain by composing the trace maps a
trace map for an arbitrary projective morphism. Arguing as in the classical
case (c.f. [H], section 2.10), we see that this map is independent of the
choice of the factorization. Therefore we obtain

######  Theorem 8.15.

Let $\varphi:\mathfrak{X}\to\mathfrak{Y}$ be a projective morphism. Then we
have a functorial morphism

$\int_{\varphi}\mathbb{D}_{\mathfrak{X}}\mathcal{M}^{\cdot}\to\mathbb{D}_{\mathfrak{Y}}\int_{\varphi}\mathcal{M}^{\cdot}$

which is an isomorphism for $\mathcal{M}^{\cdot}\in
D_{coh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{X}}^{(0,1)}))$. Further, we
have a functorial isomorphism

$R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{Y}}^{(0,1)}}(\int_{\varphi}\mathcal{M}^{\cdot},\mathcal{N}^{\cdot})\tilde{\to}\varphi_{*}R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{X}}^{(0,1)}}(\mathcal{M}^{\cdot},\varphi^{\dagger}\mathcal{N}^{\cdot})$

for all $\mathcal{M}^{\cdot}\in
D_{coh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{X}}^{(0,1)}))$ and
$\mathcal{N}^{\cdot}\in
D_{coh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{Y}}^{(0,1)}))$.

##  9\. Applications

In this section we put things together and give the statement and proof of our
generalization of Mazur’s theorem for a mixed Hodge module. We begin with a
brief review of the pushforward operation in the world of mixed Hodge modules.

Let $X_{\mathbb{C}}$ be a smooth complex variety, and suppose that
$(\mathcal{M}_{\mathbb{C}},F^{\cdot},\mathcal{K}_{\mathbb{Q}},W_{\cdot})$ is a
mixed Hodge module on $X_{\mathbb{C}}$. We won’t attempt to recall a complete
definition here, instead referring the reader to[S1],[S2], and the excellent
survey [Sch]. We will only recall that $\mathcal{M}_{\mathbb{C}}$ is a
coherent $\mathcal{D}$-module which comes equipped with a good filtration
$F^{\cdot}$, a weight filtration $W_{\cdot}$, and $\mathcal{K}_{\mathbb{Q}}$
is a perverse sheaf defined over $\mathbb{Q}$ which corresponds to
$\mathcal{M}_{\mathbb{C}}$ under the Riemann-Hilbert correspondence. In this
paper, our attention is on the filtration $F^{\cdot}$ and we will mostly
suppress the other aspects of the theory. For the sake of notational
convenience, we will denote simply by $\mathcal{O}_{X_{\mathbb{C}}}$ the mixed
Hodge module whose underlying filtered $\mathcal{D}$-module is
$\mathcal{O}_{X_{\mathbb{C}}}$ with its trivial filtration:
$F^{i}(\mathcal{O}_{X_{\mathbb{C}}})=\mathcal{O}_{X_{\mathbb{C}}}$ for all
$i\geq 0$, while $F^{i}(\mathcal{O}_{X_{\mathbb{C}}})=0$ for $i<0$.

Now let $\varphi:X_{\mathbb{C}}\to Y_{\mathbb{C}}$ be a morphism of smooth
complex varieties. By Nagata’s compatification theorem, combined with
Hironaka’s resolution of singularities, we can find an open immersion
$j:X_{\mathbb{C}}\to\overline{X}_{\mathbb{C}}$ into a smooth variety, whose
compliment is a normal crossings divisor, and a proper morphism
$\overline{\varphi}:\overline{X}_{\mathbb{C}}\to Y_{\mathbb{C}}$, with
$\varphi=\overline{\varphi}\circ j$.

Then, the following is one of the main results of [S2] (c.f. theorem 4.3 and
theorem 2.14)

######  Theorem 9.1.

Let $\varphi,\overline{\varphi},j$ be morphisms as above.

1) There is a mixed Hodge module
$(j_{\star}(\mathcal{M}_{\mathbb{C}}),F^{\cdot}j_{*}\mathcal{K}_{\mathbb{Q}},W_{\cdot})$,
whose underlying $\mathcal{D}$-module agrees with the usual pushforward of
$\mathcal{D}$-modules under $j$. This defines an exact functor
$j_{\star}:\text{MHM}(X_{\mathbb{C}})\to\text{MHM}(\overline{X}_{\mathbb{C}})$.

2) There is an object of $D^{b}(\text{MHM}(Y_{\mathbb{C}}))$,
$R\overline{\varphi}_{\star}(j_{\star}(\mathcal{M}_{\mathbb{C}}),F^{\cdot}j_{*}\mathcal{K}_{\mathbb{Q}},W_{\cdot})$,
whose underlying complex of filtered $\mathcal{D}$-modules agrees with
${\displaystyle\int_{\overline{\varphi}}(j_{\star}\mathcal{M}_{\mathbb{C}})}$.
This object of $D^{b}(\text{MHM}(Y_{\mathbb{C}}))$ is, up to isomorphism,
independent of the choice of factorization $\varphi=\overline{\varphi}\circ
j$. Furthermore, the filtration on this complex is strict.

The reason for stating the theorem this way is that, if $\varphi$ is not
proper, the filtered pushforward ${\displaystyle\int_{\varphi}}$ of filtered
$\mathcal{D}$-modules does not agree with the pushforward of mixed Hodge
modules. The issue appears already if $Y_{\mathbb{C}}$ is a point and
$\mathcal{M}_{\mathbb{C}}=\mathcal{O}_{X_{\mathbb{C}}}$. In that case, the
pushforward $R\varphi_{\star}$ returns111111up to a homological shift, and a
re-indexing of the Hodge filtration Deligne’s Hodge cohomology of
$X_{\mathbb{C}}$, while ${\displaystyle{\displaystyle\int_{\varphi}}}$ returns
the de Rham cohomology of $X_{\mathbb{C}}$ equipped with the naive Hodge-to-de
Rham filtration; these disagree, e.g., if $X_{\mathbb{C}}$ is affine.

The construction of the extension $j_{\star}(\mathcal{M}_{\mathbb{C}})$ is, in
general, quite deep, and relies on the detailed study of the degenerations of
Hodge structures given in [S] and [CKS]. However, when
$\mathcal{M}_{\mathbb{C}}=\mathcal{O}_{X_{\mathbb{C}}}$ is the trivial mixed
Hodge module, one can be quite explicit:

######  Lemma 9.2.

Let $j:X_{\mathbb{C}}\to\overline{X}_{\mathbb{C}}$ be an open immersion of
smooth varieties, whose compliment is a normal crossings divisor
$D_{\mathbb{C}}$. Let $x\in X_{\mathbb{C}}$ be a point, about which
$D_{\mathbb{C}}$ is given by the equation $\\{x_{1}\cdots x_{j}=0\\}$. Then as
filtered $\mathcal{D}$-modules we have
$j_{\star}\mathcal{O}_{X_{\mathbb{C}}}=(j_{*}(\mathcal{O}_{X_{\mathbb{C}}}),F^{\cdot})$
where
$F^{l}(j_{*}(\mathcal{O}_{X_{\mathbb{C}}})):=F^{l}(\mathcal{D}_{X_{\mathbb{C}}})\cdot(x_{1}\cdots
x_{j})^{-1}$.

In particular, $F^{l}(j_{*}(\mathcal{O}_{X_{\mathbb{C}}}))$ is spanned over
$\mathcal{O}_{X_{\mathbb{C}}}$ by terms of the form $x_{1}^{-(i_{1}+1)}\cdots
x_{j}^{-(i_{j}+1)}$ where ${\displaystyle\sum_{t=1}^{j}i_{t}\leq l}$.

For a proof, see [MP1], section 8. This implies that the Hodge cohomology of
$X_{\mathbb{C}}$, as an object in the filtered derived category of vector
spaces, can be computed as
${\displaystyle\int_{\overline{\varphi}}j_{\star}\mathcal{O}_{X_{\mathbb{C}}}(d)[d]}$
where $\overline{\varphi}:\overline{X}_{\mathbb{C}}\to\\{*\\}$. Of course,
this can be checked directly by comparing the log de Rham complex with the de
Rham complex of a the filtered $\mathcal{D}$-module
$j_{\star}\mathcal{O}_{X_{\mathbb{C}}}$.

Combining 4.10with 7.10 gives:

######  Proposition 9.3.

1) Let $\varphi:\mathfrak{X}\to\mathfrak{Y}$ be a projective morphism, and let
$\mathfrak{D}\subset\mathfrak{X}$ be a (possibly empty) normal crossings
divisor. Let ${\displaystyle j_{\star}D(\mathcal{O}_{\mathfrak{X}})}$ be the
gauge of 4.35 on $\mathfrak{X}$. Suppose that each
$\mathcal{H}^{i}({\displaystyle\int_{\varphi}(j_{\star}\mathcal{O}_{\mathfrak{X}})^{-\infty}})$
is a $p$-torsion-free $\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0)}$-module, and
that each $\mathcal{H}^{i}({\displaystyle(\int_{\varphi}{\displaystyle
j_{\star}D(\mathcal{O}_{\mathfrak{X}}})}\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[f])$
is $f$-torsion-free. Then each
$\mathcal{H}^{i}{\displaystyle(\int_{\varphi}{\displaystyle
j_{\star}D(\mathcal{O}_{\mathfrak{X}}}}))$ is a standard gauge on
$\mathfrak{Y}$.

2) Let ${\displaystyle
j_{!}D(\mathcal{O}_{\mathfrak{X}}):=\mathbb{D}_{\mathfrak{X}}j_{\star}D(\mathcal{O}_{\mathfrak{X}})}$.
The same conclusion holds for $j_{!}D(\mathcal{O}_{\mathfrak{X}})$.

When $\mathfrak{Y}$ is a point this recovers the log-version of Mazur’s
theorem, as discussed in Ogus’ paper [O].

Now let $R$ be a finite type algebra over $\mathbb{Z}$ so that there exists
smooth (over $R$) models $X_{R},Y_{R}$ for $X_{\mathbb{C}}$ and
$Y_{\mathbb{C}}$, respectively, and a projective morphism $\varphi:X_{R}\to
Y_{R}$ whose base change to $\mathbb{C}$ is the original morphism. We may
suppose the divisor $D_{\mathbb{C}}$ is defined over $R$ as well.

Let $\mathcal{D}_{X_{R}}^{(0)}$ be the level zero differential operators over
$X_{R}$, equipped with the symbol filtration; let the associated Rees algebra
be $\mathcal{R}(\mathcal{D}_{X_{R}}^{(0)})$ (as usual we will use $f$ for the
Rees parameter). Since $\text{Rees}(j_{*}\mathcal{O}_{U_{\mathbb{C}}})$ is a
coherent $\mathcal{R}(\mathcal{D}_{X_{\mathbb{C}}})$-module, we can by generic
flatness choose a flat model for
$\text{Rees}(j_{*}\mathcal{O}_{U_{\mathbb{C}}})$; in fact, we can describe it
explicitly as follows: if $D_{R}$ is given, in local coordinates, by
$\\{x_{1}\cdots x_{j}=0\\}$, then we may consider

$\mathcal{D}_{X_{R}}^{(0)}\cdot x_{1}^{-1}\cdots x_{j}^{-1}\subset
j_{*}\mathcal{O}_{U_{R}}$

with the filtration inherited from the symbol filtration on
$\mathcal{D}_{X_{R}}^{(0)}$. The Rees module of this filtered
$\mathcal{D}_{X_{R}}^{(0)}$-module is a flat $R$-model for
$\text{Rees}(j_{*}\mathcal{O}_{U_{\mathbb{C}}})$. Let us call this sheaf
${\displaystyle j_{\star}\mathcal{O}_{U_{R}}[f]}$; we will denote the
associated filtered $\mathcal{D}_{X_{R}}^{(0)}$-module by ${\displaystyle
j_{\star}\mathcal{O}_{U_{R}}}$. Then, localizing $R$ if necessary, we have
that

$\int_{\varphi}j_{\star}\mathcal{O}_{U_{R}}[f]$

is an $f$-torsion-free complex inside
$D_{coh}^{b}(\mathcal{D}_{Y_{R}}^{(0)}-\text{mod})$ (since it becomes
$f$-torsion-free after base change to $\mathbb{C}$, as remarked above). By
generic flatness, we may also suppose (again, localizing $R$ if necessary),
that each cohomology sheaf
${\displaystyle\mathcal{H}^{i}(\int_{\varphi}j_{\star}\mathcal{O}_{U_{R}})}$
is flat over $R$. Let $k$ be a perfect field of characteristic $p>0$, for
which there is a morphism $R\to W(k)$ (so that $R/p\to k$)121212If we extend
$R$ so that it is smooth over $\mathbb{Z}$, then any map $R/p\to k$ lifts to
$R\to W(k)$. Then, combining this discussion with the previous proposition, we
obtain

######  Corollary 9.4.

Let $\mathfrak{X}$ be the formal completion of $X_{R}\times_{R}W(k)$, and
similarly for $\mathfrak{Y}$. Then each gauge
$\mathcal{H}^{i}({\displaystyle(\int_{\varphi}{\displaystyle
j_{\star}D(\mathcal{O}_{\mathfrak{X}})}}))$ is a standard, coherent,
$F^{-1}$-gauge on $\mathfrak{Y}$. There is an isomorphism

$\mathcal{H}^{i}(({\displaystyle(\int_{\varphi}{\displaystyle
j_{\star}\mathcal{O}_{\mathfrak{X}}[f,v]}})\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[f])\tilde{\to}F^{*}\mathcal{H}^{i}(\int_{\varphi}j_{\star}\mathcal{O}_{U_{R}}[f]\otimes_{R}^{L}k)$

in $\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)}))$. In particular, the Hodge
filtration on
${\displaystyle\mathcal{H}^{i}({\displaystyle(\int_{\varphi}{\displaystyle
j_{\star}D(\mathcal{O}_{\mathfrak{X}}})}))^{\infty}/p}$ is the Frobenius
pullback of the Hodge filtration on
${\displaystyle\mathcal{H}^{i}(\int_{\varphi}j_{\star}\mathcal{O}_{U_{R}}\otimes_{R}^{L}k)}$.
The same holds if we replace ${\displaystyle
j_{\star}D(\mathcal{O}_{\mathfrak{X}}})$ by ${\displaystyle
j_{!}D(\mathcal{O}_{\mathfrak{X}}})$. The same statement holds for the
pushforward of
${\displaystyle\mathcal{H}^{i}(\int_{\varphi}j_{\star}\mathcal{O}_{U_{R}})}$
under another proper morphism $\psi:Y\to Z$.

###### Proof.

The displayed isomorphism follows immediately from 7.13. Since
${\displaystyle\int_{\varphi}j_{\star}\mathcal{O}_{U_{R}}[f]}$ has $f$-torsion
free cohomology sheaves, which are also flat over $R$, we deduce that
${\displaystyle{\displaystyle((\int_{\varphi}{\displaystyle
j_{\star}\mathcal{O}_{\mathfrak{X}}[f,v]}})\otimes_{W(k)}^{L}k})\otimes_{D(k)}^{L}k[f]$
has $f$-torsion free cohomology sheaves. Comparing the description of the
Hodge filtration on ${\displaystyle({\displaystyle
j_{\star}\mathcal{O}_{\mathfrak{X}}[f,v]}})^{\infty}/p$ with the result of
9.2, the result now follows from 4.31. ∎

Let us give some first applications of these results.

Suppose that $X_{\mathbb{C}}$ is an arbitrary (possibly singular) quasi-
projective variety. Let $V_{\mathbb{C}}$ be a smooth quasi-projective variety