R}\Gamma(\mathfrak{P},\mathcal{M})=\mathbf{R}\Gamma(\mathfrak{P},\underset{m}{\mathrm{colim}}\,\mathcal{M}^{(m)}\otimes_{{\mathbb{Z}}}{\mathbb{Q}})=\underset{m}{\mathrm{colim}}\,\mathbf{R}\Gamma(\mathfrak{P},\mathcal{M}^{(m)})\otimes_{{\mathbb{Z}}}{\mathbb{Q}}.$

Before proving that (5.3.1) is an isomorphism, I need to show that ${\bf
L}\hat{\mathrm{sp}}^{*}$ is compatible with open immersions.

######  5.3.1 Lemma.

Let $j:\mathfrak{U}\rightarrow\mathfrak{P}$ denote an open immersion of flat
formal schemes. Then the natural map

$j^{-1}\mathbf{L}\hat{\mathrm{sp}}_{\mathfrak{P}}^{*}\rightarrow\mathbf{L}\hat{\mathrm{sp}}_{\mathfrak{U}}^{*}j^{-1}$

is an isomorphism of functors $\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathcal{O}_{\mathfrak{P}})\rightarrow{\bf
D}(\mathcal{O}_{\mathfrak{U}_{K}}^{+})$.

###### Proof.

It suffices to prove the analogous statement with $\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathcal{O}_{\mathfrak{P}})$ replaced by ${\bf
D}_{\mathrm{qc}}(\mathcal{O}_{\mathfrak{P}})$. The functor
$\mathfrak{P}^{\prime}\mapsto\mathfrak{P}^{\prime}\times_{\mathfrak{P}}\mathfrak{U}$
from admissible blowups of $\mathfrak{P}$ to those of $\mathfrak{U}$ is
cofinal, so I can compute

$\displaystyle\mathbf{L}\hat{\mathrm{sp}}_{\mathfrak{U}}^{*}\mathbf{L}\hat{j}^{*}\mathcal{M}$
$\displaystyle=\underset{\pi\colon\mathfrak{U}^{\prime}\rightarrow\mathfrak{U}}{\mathrm{colim}}\,\mathrm{sp}_{\mathfrak{U}^{\prime}}^{-1}\mathbf{L}\hat{\pi}^{*}\mathbf{L}\hat{j}^{*}\mathcal{M}$
$\displaystyle=\underset{\pi\colon\mathfrak{U}^{\prime}\rightarrow\mathfrak{U}}{\mathrm{colim}}\,\mathrm{sp}_{\mathfrak{U}^{\prime}}^{-1}j^{-1}\mathbf{L}\hat{\pi}^{*}\mathcal{M}$
$\displaystyle=\underset{\pi\colon\mathfrak{P}^{\prime}\rightarrow\mathfrak{P}}{\mathrm{colim}}\,\mathrm{sp}_{\pi^{-1}(\mathfrak{U})}^{-1}j^{-1}\mathbf{L}\hat{\pi}^{*}\mathcal{M}$
$\displaystyle=j^{-1}\underset{\pi\colon\mathfrak{P}^{\prime}\rightarrow\mathfrak{P}}{\mathrm{colim}}\,\mathrm{sp}_{\mathfrak{P}^{\prime}}^{-1}\mathbf{L}\hat{\pi}^{*}\mathcal{M}$
$\displaystyle=j^{-1}\mathbf{L}\hat{\mathrm{sp}}^{*}_{\mathfrak{P}}\mathcal{M}$

using Lemma 5.1.6 for the second (canonical) isomorphism. ∎

######  5.3.2 Theorem.

Let $u\colon\mathfrak{P}\rightarrow\mathfrak{Q}$ be a flat morphism of flat
formal schemes. Then

${\bf L}\hat{\mathrm{sp}}_{\mathfrak{Q}}^{*}\circ{\bf R}u_{*}\rightarrow{\bf
R}u_{*}\circ{\bf L}\hat{\mathrm{sp}}^{*}_{\mathfrak{P}}$

is an isomorphism of functors $\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathcal{O}_{\mathfrak{P}})\rightarrow{\bf
D}(\mathcal{O}_{\mathfrak{Q}_{K}})$.

###### Proof.

Since both $\mathfrak{P}$ and $\mathfrak{P}_{K}$ are quasi-compact, cohomology
commutes with filtered colimits. It therefore suffices to prove the analogous
statement with $\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathcal{O}_{\mathfrak{P}})$ replaced by ${\bf
D}_{\mathrm{qc}}(\mathcal{O}_{\mathfrak{P}})$.

First suppose that $\mathfrak{Q}=\mathrm{Spf}\left(\mathcal{V}\right)$ is a
point. Then the base change morphism amounts to a morphism

$\mathbf{R}\Gamma(\mathfrak{P},\mathcal{M})\rightarrow\mathbf{R}\Gamma(\mathfrak{P}_{K},\mathbf{L}\hat{\mathrm{sp}}^{*}_{\mathfrak{P}}\mathcal{M})$

in ${\bf D}(K)$. To prove that this is an isomorphism, recall from [Hub94, §4]
that, as topological spaces,

$\mathfrak{P}_{K}=\underset{I_{\mathfrak{P}}}{\mathrm{lim}}\,\mathfrak{P}^{\prime}.$

In this case, since each topological space $\mathfrak{P}^{\prime}$ is coherent
and sober, and the transition maps are quasi-compact, it follows from [FK18,
Chapter 0, Proposition 3.1.19] that the map

$\underset{\mathfrak{P}^{\prime}}{\mathrm{colim}}\,\mathbf{R}\Gamma(\mathfrak{P}^{\prime},\mathbf{L}\hat{\pi}^{*}\mathcal{M})\rightarrow\mathbf{R}\Gamma(\mathfrak{P}_{K},\mathbf{L}\hat{\mathrm{sp}}^{*}\mathcal{M})$

is an isomorphism. In fact, the result in [FK18] is stated for sheaves, rather
than complexes, but it is straightforward to extend it to complexes using
$K$-injective resolutions. It now simply suffices to observe that, by
Proposition 5.1.13, the map

$\mathbf{R}\Gamma(\mathfrak{P},\mathcal{M})\rightarrow\mathbf{R}\Gamma(\mathfrak{P}^{\prime},\mathbf{L}\hat{\pi}^{*}\mathcal{M})$

is an isomorphism for all $\mathfrak{P}^{\prime}$.

For the general case, it suffices to show that the given map induces an
isomorphism on derived global sections for any quasi-compact open
$V\subset\mathfrak{Q}_{K}$. To prove this, I can replace $\mathfrak{Q}$ by an
admissible blowup without changing either side, thus I can assume that
$V=\mathfrak{V}_{K}$ for an open subscheme $\mathfrak{V}\subset\mathfrak{Q}$.
By Lemma 5.3.1, I can then replace $\mathfrak{Q}$ by $\mathfrak{V}$, and hence
assume that $V=\mathfrak{Q}_{K}$. In this case, the base change map fits into
the following commutative diagram.

$\textstyle{\mathbf{R}\Gamma(\mathfrak{Q}_{K},{\bf R}u_{*}{\bf
L}\hat{\mathrm{sp}}^{*}_{\mathfrak{P}}\mathcal{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf
R}\Gamma(\mathfrak{P}_{K},{\bf
L}\hat{\mathrm{sp}}^{*}_{\mathfrak{P}}\mathcal{M})}$$\textstyle{\mathbf{R}\Gamma(\mathfrak{Q}_{K},{\bf
L}\hat{\mathrm{sp}}^{*}_{\mathfrak{Q}}{\bf
R}u_{*}\mathcal{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf
R}\Gamma(\mathfrak{Q},{\bf
R}u_{*}\mathcal{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf
R}\Gamma(\mathfrak{P},\mathcal{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The two horizontal maps are clearly isomorphisms, and the two vertical maps
are isomorphisms by the already treated case
$\mathfrak{Q}=\mathrm{Spf}\left(\mathcal{V}\right)$. Hence the top left
diagonal map is an isomorphism. ∎

It’s worth explicitly stating the special case
$\mathfrak{Q}=\mathrm{Spf}\left(\mathcal{V}\right)$.

######  5.3.3 Corollary.

Let $\mathfrak{P}$ be a flat formal scheme, and
$\mathcal{M}\in\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathcal{O}_{\mathfrak{P}})$. Then the natural
map

${\bf R}\Gamma(\mathfrak{P},\mathcal{M})\rightarrow{\bf
R}\Gamma(\mathfrak{P}_{K},{\bf
L}\hat{\mathrm{sp}}_{\mathfrak{P}}^{*}\mathcal{M})$

is an isomorphism in ${\bf D}(K)$.

Eventually, I will want to upgrade Theorem 5.3.2 to include
$\mathscr{D}$-module structures, and in order to do so I will need a slightly
different description of the functor ${\bf R}u_{*}$ that works under the
additional assumption that $u$ is flat. Restricting
$\mathcal{B}_{\mathfrak{P}}$ along the functor $I_{\mathfrak{Q}}\rightarrow
I_{\mathfrak{P}}$ gives rise to a diagram
$\mathcal{B}_{\mathfrak{P}/\mathfrak{Q}}\colon
I_{\mathfrak{Q}}\rightarrow\bf{FSch}$ which by the flatness hypothesis on $u$
sends an admissible blowup $\mathfrak{Q}^{\prime}\rightarrow\mathfrak{Q}$ to
the admissible blowup
$\mathfrak{Q}^{\prime}\times_{\mathfrak{Q}}\mathfrak{P}\rightarrow\mathfrak{P}$.
This gives rise to a category $\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathcal{O}_{\mathcal{B}_{\mathfrak{P}/\mathfrak{Q}}})$
of inductive systems of quasi-coherent complexes on
$\mathcal{B}_{\mathfrak{P}/\mathfrak{Q}}$ as above, and restriction along
$I_{\mathfrak{Q}}\rightarrow I_{\mathfrak{P}}$ induces a ‘forgetful’ functor

$\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathcal{O}_{\mathcal{B}_{\mathfrak{P}}})\rightarrow\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathcal{O}_{\mathcal{B}_{\mathfrak{P}/\mathfrak{Q}}})$

Of course, the morphism of diagrams
$u\colon\mathcal{B}_{\mathfrak{P}/\mathfrak{Q}}\rightarrow\mathcal{B}_{\mathfrak{Q}}$
induces a functor

${\bf R}u_{*}\colon\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathcal{O}_{\mathcal{B}_{\mathfrak{P}/\mathfrak{Q}}})\rightarrow\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathcal{O}_{\mathcal{B}_{\mathfrak{Q}}}).$

######  5.3.4 Lemma.

The diagram

$\textstyle{\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathcal{O}_{\mathcal{B}_{\mathfrak{P}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\bf
R}u_{*}}$$\textstyle{\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathcal{O}_{\mathcal{B}_{\mathfrak{P}/\mathfrak{Q}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\bf
R}u_{*}}$$\textstyle{\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathcal{O}_{\mathcal{B}_{\mathfrak{Q}}})}$

commutes upto $2$-isomorphism.

###### Proof.

This is simply a matter of unwinding the definitions. ∎

I end this section with a brief discussion of the compatibility of ${\bf
L}\hat{\mathrm{sp}}^{*}$ with tensor products. If $\mathfrak{P}$ is a flat
formal scheme, and $\pi:\mathfrak{P}^{\prime}\rightarrow\mathfrak{P}$ is an
admissible blowup, there is a natural map

$\mathbf{L}\hat{\pi}^{*}\mathcal{M}\otimes_{\mathcal{O}_{\mathfrak{P}^{\prime}}}^{\mathbf{L}}\mathbf{L}\hat{\pi}^{*}\mathcal{N}\rightarrow\mathbf{L}\hat{\pi}^{*}\mathcal{M}\widehat{\otimes}_{\mathcal{O}_{\mathfrak{P}^{\prime}}}^{\mathbf{L}}\mathbf{L}\hat{\pi}^{*}\mathcal{N}=\mathbf{L}\hat{\pi}^{*}(\mathcal{M}\widehat{\otimes}_{\mathcal{O}_{\mathfrak{P}}}^{\mathbf{L}}\mathcal{N}).$

Applying $\mathrm{sp}^{-1}_{\mathfrak{P}^{\prime}}$ gives

$\mathrm{sp}^{-1}_{\mathfrak{P}^{\prime}}\mathbf{L}\hat{\pi}^{*}\mathcal{M}\otimes_{\mathrm{sp}^{-1}_{\mathfrak{P}^{\prime}}\mathcal{O}_{\mathfrak{P}^{\prime}}}^{\mathbf{L}}\mathrm{sp}^{-1}_{\mathfrak{P}^{\prime}}\mathbf{L}\hat{\pi}^{*}\mathcal{N}\rightarrow\mathrm{sp}^{-1}_{\mathfrak{P}^{\prime}}\mathbf{L}\hat{\pi}^{*}(\mathcal{M}\widehat{\otimes}_{\mathcal{O}_{\mathfrak{P}}}^{\mathbf{L}}\mathcal{N}),$

and passing to the colimit in $\mathfrak{P}^{\prime}$ gives a map

$\mathbf{L}\hat{\mathrm{sp}}^{*}\mathcal{M}\otimes_{\mathcal{O}_{\mathfrak{P}^{+}_{K}}}^{\mathbf{L}}\mathbf{L}\hat{\mathrm{sp}}^{*}\mathcal{N}\rightarrow\mathbf{L}\hat{\mathrm{sp}}^{*}(\mathcal{M}\widehat{\otimes}_{\mathcal{O}_{\mathfrak{P}}}^{\mathbf{L}}\mathcal{N})$

in ${\bf D}(\mathcal{O}^{+}_{\mathfrak{P}_{K}})$. Tensoring with
${\mathbb{Q}}$ therefore gives, for any $\mathcal{M},\mathcal{N}\in{\bf
D}_{{\mathbb{Q}},\mathrm{qc}}(\mathcal{O}_{\mathfrak{P}})$, a map

(5.3.2)
$\mathbf{L}\hat{\mathrm{sp}}^{*}\mathcal{M}\otimes_{\mathcal{O}_{\mathfrak{P}_{K}}}^{\mathbf{L}}\mathbf{L}\hat{\mathrm{sp}}^{*}\mathcal{N}\rightarrow\mathbf{L}\hat{\mathrm{sp}}^{*}(\mathcal{M}\widehat{\otimes}_{\mathcal{O}_{\mathfrak{P}}}^{\mathbf{L}}\mathcal{N})$

in ${\bf D}(\mathcal{O}_{\mathfrak{P}_{K}})$. Passing to the colimit gives a
similar map whenever $\mathcal{M},\mathcal{N}\in\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathcal{O}_{\mathfrak{P}})$, I leave it to
the reader to give a precise construction of this map. In general (5.3.2)
won’t be an isomorphism, however, I will show later on that this will be the
case whenever $\mathcal{M},\mathcal{N}$ underly overholonomic
$\mathscr{D}^{\dagger}_{\mathfrak{P}{\mathbb{Q}}}$-modules.

###  5.4. Functions with overconvergent singularities

Now suppose that $\mathfrak{P}$ is smooth, and consider a divisor $D\subset
P$. In this situation, Berthelot has defined in [Ber02, §4.4] the
$\mathcal{O}_{\mathfrak{P}}$-algebra
$\widehat{\mathcal{B}}_{\mathfrak{P}}(D,r)$ for any $r\in{\mathbb{N}}$.
Locally, if there exists some $f\in\mathcal{O}_{\mathfrak{P}}$ such that
$D=P\cap V(f)$, then

$\widehat{\mathcal{B}}_{\mathfrak{P}}(D,r)=\frac{\mathcal{O}_{\mathfrak{P}}\langle
X\rangle}{(f^{r}X-p)}.$

######  5.4.1 Lemma.

$\widehat{\mathcal{B}}_{\mathfrak{P}}(D,r)$ is quasi-coherent as a complex of
$\mathcal{O}_{\mathfrak{P}}$-modules.

###### Proof.

Quasi-coherence can be checked locally, by Remark 5.1.7, so I can assume that
$\mathfrak{P}$ is affine and that $D\subset P$ is the vanishing locus of a
single function $f\in\mathcal{O}_{\mathfrak{P}}$, which is a non-zero divisor
in $\mathcal{O}_{P}$. In this case

$\widehat{\mathcal{B}}_{\mathfrak{P}}(D,r)=\frac{\mathcal{O}_{\mathfrak{P}}\langle
X\rangle}{(f^{r}X-p)}$

is $p$-torsion-free, and hence flat over $\mathcal{O}_{\mathfrak{P}}$. Thus

$\mathcal{O}_{P_{n}}\otimes^{\mathbf{L}}_{\mathcal{O}_{\mathfrak{P}}}\frac{\mathcal{O}_{\mathfrak{P}}\langle
X\rangle}{(f^{r}X-p)}\cong\mathcal{O}_{P_{n}}\otimes_{\mathcal{O}_{\mathfrak{P}}}\frac{\mathcal{O}_{\mathfrak{P}}\langle
X\rangle}{(f^{r}X-p)}=\frac{\mathcal{O}_{P_{n}}[X]}{(f^{r}X-p)}$

is quasi-coherent, and the transition maps
$\frac{\mathcal{O}_{P_{n+1}}[X]}{(f^{r}X-p)}\rightarrow\frac{\mathcal{O}_{P_{n}}[X]}{(f^{r}X-p)}$
are surjective. Hence

$\mathbf{R}\underset{n}{\mathrm{lim}}\,\frac{\mathcal{O}_{P_{n}}[X]}{(f^{r}X-p)}\cong\underset{n}{\mathrm{lim}}\,\frac{\mathcal{O}_{P_{n}}[X]}{(f^{r}X-p)}\cong\frac{\mathcal{O}_{\mathfrak{P}}\langle
X\rangle}{(f^{r}X-p)}$

as required. ∎

Berthelot then defines
$\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}}(D):=\widehat{\mathcal{B}}_{\mathfrak{P}}(D,p^{m+1})$,
which form an inductive system as $m$ varies. This gives rise to an object

$\widehat{\mathcal{B}}^{(\bullet)}_{\mathfrak{P}}(D)\in\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathcal{O}_{\mathfrak{P}}).$

######  5.4.2 Theorem.

Let $\mathfrak{P}$ be a smooth formal scheme, $D\subset P$ a divisor, and
$U:=P\setminus D$. Suppose that $\mathcal{M}\in\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathcal{O}_{\mathfrak{P}})$. Then the
morphism

$\mathcal{M}\rightarrow\widehat{\mathcal{B}}^{(\bullet)}_{\mathfrak{P}}(D)\widehat{\otimes}^{{\bf
L}}_{\mathcal{O}_{\mathfrak{P}}}\mathcal{M}$

induces an isomorphism

$j_{U}^{\dagger}{\bf
L}\hat{\mathrm{sp}}^{*}\mathcal{M}\overset{\cong}{\longrightarrow}{\bf
L}\hat{\mathrm{sp}}^{*}(\widehat{\mathcal{B}}^{(\bullet)}_{\mathfrak{P}}(D)\widehat{\otimes}^{\bf
L}_{\mathcal{O}_{\mathfrak{P}}}\mathcal{M})$

in ${\bf D}^{(}\mathcal{O}_{\mathfrak{P}_{K}})$.

###### Proof.

Set $r_{m}=p^{m+1}$, $\lambda_{m}:=p^{-1/r_{m}}$, let
$\left]D\right[_{\lambda_{m}}\subset\mathfrak{P}_{K}$ be the open tube of
radius $\lambda_{m}$, $\overline{D}_{m}$ its closure in $\mathfrak{P}_{K}$,
and write $V_{m}:=\mathfrak{P}_{K}\setminus\overline{D}_{m}$. Let
$j_{m}:V_{m}\rightarrow\mathfrak{P}_{K}$ denote the inclusion. Thus, if $D$ is
defined by $f=0$ inside $P$, for some $f\in\mathcal{O}_{\mathfrak{P}}$, then
$V_{m}$ is defined inside $\mathfrak{P}_{K}$ by
$\left|f\right|\geq\lambda_{m}$. Therefore, for any complex $\mathscr{K}$ on
$\mathfrak{P}_{K}$,

$j_{U}^{\dagger}\mathscr{K}\overset{\cong}{\longrightarrow}\underset{m}{\mathrm{colim}}\,{\bf
R}j_{m*}j_{m}^{-1}\mathscr{K}.$

The map

${\bf L}\hat{\mathrm{sp}}^{*}\mathcal{M}\rightarrow{\bf
L}\hat{\mathrm{sp}}^{*}(\widehat{\mathcal{B}}^{(\bullet)}_{\mathfrak{P}}(D)\widehat{\otimes}^{{\bf
L}}_{\mathcal{O}_{\mathfrak{P}}}\mathcal{M})$

is the colimit of the maps

${\bf L}\hat{\mathrm{sp}}^{*}\mathcal{M}\rightarrow{\bf
L}\hat{\mathrm{sp}}^{*}(\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}}(D)\widehat{\otimes}^{{\bf
L}}_{\mathcal{O}_{\mathfrak{P}}}\mathcal{M}).$

It therefore suffices to show that

${\bf L}\hat{\mathrm{sp}}^{*}\mathcal{M}\rightarrow{\bf
L}\hat{\mathrm{sp}}^{*}(\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}}(D)\widehat{\otimes}^{{\bf
L}}_{\mathcal{O}_{\mathfrak{P}}}\mathcal{M})$

factors through an isomorphism

${\bf R}j_{m*}j_{m}^{-1}{\bf L}\hat{\mathrm{sp}}^{*}\mathcal{M}\rightarrow{\bf
L}\hat{\mathrm{sp}}^{*}(\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}}(D)\widehat{\otimes}^{{\bf
L}}_{\mathcal{O}_{\mathfrak{P}}}\mathcal{M})$

(which is then necessarily unique). To prove this, consider the admissible
blowup

$\pi\colon\mathfrak{P}^{\prime}:=\mathrm{Bl}_{(p,f^{r_{m}})}\mathfrak{P}\rightarrow\mathfrak{P}.$

There exists a unique open subset
$j_{m}\colon\mathfrak{V}_{m}^{\prime}\rightarrow\mathfrak{P}^{\prime}$ such
that
$V_{m}=\mathrm{sp}^{-1}_{\mathfrak{P}^{\prime}}(\mathfrak{V}_{m}^{\prime})$.
Now set $\mathcal{M}^{\prime}:={\bf L}\hat{\pi}^{*}\mathcal{M}$ and
$\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}^{\prime}}(D):={\bf
L}\hat{\pi}^{*}\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}}(D)$, thus what I
need to show is that the map

${\bf
L}\hat{\mathrm{sp}}^{*}_{\mathfrak{P}^{\prime}}\mathcal{M}^{\prime}\rightarrow{\bf
L}\hat{\mathrm{sp}}^{*}_{\mathfrak{P}^{\prime}}(\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}^{\prime}}(D)\widehat{\otimes}^{{\bf
L}}_{\mathcal{O}_{\mathfrak{P}^{\prime}}}\mathcal{M}^{\prime}),$

factors through an isomorphism

${\bf R}j_{m*}j_{m}^{-1}{\bf
L}\hat{\mathrm{sp}}^{*}_{\mathfrak{P}^{\prime}}\mathcal{M}^{\prime}\rightarrow{\bf
L}\hat{\mathrm{sp}}^{*}_{\mathfrak{P}^{\prime}}(\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}^{\prime}}(D)\widehat{\otimes}^{{\bf
L}}_{\mathcal{O}_{\mathfrak{P}^{\prime}}}\mathcal{M}^{\prime})$

(again, this isomorphism is then necessarily unique). Applying Theorem 5.3.2,
it therefore suffices to prove that the map

$\mathcal{M}^{\prime}\rightarrow\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}^{\prime}}(D)\widehat{\otimes}^{{\bf
L}}_{\mathcal{O}_{\mathfrak{P}^{\prime}}}\mathcal{M}^{\prime}$

factors through an isomorphism

${\bf
R}j_{m*}j_{m}^{-1}\mathcal{M}^{\prime}\rightarrow\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}^{\prime}}(D)\widehat{\otimes}^{{\bf
L}}_{\mathcal{O}_{\mathfrak{P}^{\prime}}}\mathcal{M}^{\prime}$

in $\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathcal{O}_{\mathfrak{P}^{\prime}})$. By
Lemma 5.1.10 I can reduce to the case
$\mathcal{M}^{\prime}=\mathcal{O}_{\mathfrak{P}^{\prime}}$, in other words I
need to show that

$\mathcal{O}_{\mathfrak{P}^{\prime}}\rightarrow\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}^{\prime}}(D)$

factors through an isomorphism

${\bf
R}j_{m*}\mathcal{O}_{\mathfrak{V}_{m}^{\prime}}\rightarrow\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}^{\prime}}(D)$

in ${\bf D}_{{\mathbb{Q}},\mathrm{qc}}(\mathcal{O}_{\mathfrak{P}^{\prime}})$.
To see this, I will construct the inverse isogeny

$\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}^{\prime}}(D)\rightarrow{\bf
R}j_{m*}\mathcal{O}_{\mathfrak{V}_{m}^{\prime}}.$

Locally on $\mathfrak{P}$,

$\mathfrak{P}^{\prime}=\mathbf{Proj}_{\mathcal{O}_{\mathfrak{P}}}\left(\frac{\mathcal{O}_{\mathfrak{P}}[X,Y]}{(f^{r_{m}}X-pY)}\right),$

where $D=P\cap V(f)$. This is covered by the two open formal subschemes

$\displaystyle\mathfrak{V}_{m}^{\prime}$
$\displaystyle=\mathrm{Spf}\left(\frac{\mathcal{O}_{\mathfrak{P}}\langle
X\rangle}{(f^{r_{m}}X-p)}\right)$ $\displaystyle\mathfrak{U}$
$\displaystyle=\mathrm{Spf}\left(\frac{\mathcal{O}_{\mathfrak{P}}\langle
Y\rangle}{(f^{r_{m}}-pY)}\right)$

In particular, locally on $\mathfrak{P}$, $\mathfrak{V}_{m}^{\prime}=D(Y)$ is
the complement of a hypersurface in $\mathfrak{P}^{\prime}$, and hence ${\bf
R}j_{m*}\mathcal{O}_{\mathfrak{V}^{\prime}_{m}}=j_{m*}\mathcal{O}_{\mathfrak{V}^{\prime}_{m}}$
is a module (and not just a complex). Thus I am allowed to work locally on
$\mathfrak{P}$, and therefore assume that there is indeed such an $f$.

I can then calculate

$\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}^{\prime}}(D)\\!\mid_{\mathfrak{V}^{\prime}_{m}}={\bf
R}\underset{n}{\mathrm{lim}}\,\frac{\mathcal{O}_{P_{n}}[T]}{(f^{r_{m}}T-p)}\otimes^{{\bf
L}}_{\mathcal{O}_{P_{n}}}\frac{\mathcal{O}_{P_{n}}[X]}{(f^{r_{m}}X-p)}.$

Since $f$ is not a zero divisor in $\mathcal{O}_{P_{n}}$, a direct calculation
then shows that

$\mathcal{O}_{P_{n}}[X]\overset{f^{r_{m}}X-p}{\longrightarrow}\mathcal{O}_{P_{n}}[X]$

is a flat resolution of $\frac{\mathcal{O}_{P_{n}}[X]}{(f^{r_{m}}X-p)}$. Thus

$\frac{\mathcal{O}_{P_{n}}[T]}{(f^{r_{n}}T-p)}\otimes^{{\bf
L}}_{\mathcal{O}_{P_{n}}}\frac{\mathcal{O}_{P_{n}}[X]}{(f^{r_{m}}X-p)}=\left[\frac{\mathcal{O}_{P_{n}}[X,T]}{(f^{r_{m}}T-p)}\overset{f^{r_{m}}X-p}{\longrightarrow}\frac{\mathcal{O}_{P_{n}}[X,T]}{(f^{r_{m}}T-p)}\right].$

Since each $\frac{\mathcal{O}_{P_{n}}[X,T]}{(f^{r_{m}}T-p)}$ is a quasi-
coherent $\mathcal{O}_{P_{n}}$-module, and the transition maps

$\frac{\mathcal{O}_{P_{n+1}}[X,T]}{(f^{r_{m}}T-p)}\rightarrow\frac{\mathcal{O}_{P_{n}}[X,T]}{(f^{r_{m}}T-p)}$

are surjective,

${\bf
R}\underset{n}{\mathrm{lim}}\,\frac{\mathcal{O}_{P_{n}}[X,T]}{(f^{r_{m}}T-p)}=\underset{n}{\mathrm{lim}}\,\frac{\mathcal{O}_{P_{n}}[X,T]}{(f^{r_{m}}T-p)}=\frac{\mathcal{O}_{\mathfrak{P}}\langle
X,T\rangle}{(f^{r_{m}}T-p)}.$

Thus

$\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}^{\prime}}(D)\\!\mid_{\mathfrak{V}^{\prime}_{m}}=\left[\frac{\mathcal{O}_{\mathfrak{P}}\langle
X,T\rangle}{(f^{r_{m}}T-p)}\overset{f^{r_{m}}X-p}{\longrightarrow}\frac{\mathcal{O}_{\mathfrak{P}}\langle
X,T\rangle}{(f^{r_{m}}T-p)}\right].$

A direct calculation shows that this map is injective, thus

$\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}^{\prime}}(D)\\!\mid_{\mathfrak{V}^{\prime}_{m}}=\frac{\mathcal{O}_{\mathfrak{P}}\langle
X,T\rangle}{(f^{r_{m}}T-p,f^{r_{m}}X-p)}.$

On $\mathfrak{V}^{\prime}_{m}$, the factorisation

$\mathcal{O}_{\mathfrak{P}^{\prime}}\rightarrow{\bf
R}j_{m*}\mathcal{O}_{\mathfrak{V}^{\prime}_{m}}\rightarrow\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}^{\prime}}(D)$

I am after is simply the inclusion

$\frac{\mathcal{O}_{\mathfrak{P}}\langle
X\rangle}{(f^{r_{m}}X-p)}\rightarrow\frac{\mathcal{O}_{\mathfrak{P}}\langle
T,X\rangle}{(f^{r_{m}}T-p,f^{r_{m}}X-p)}.$

The inverse map

$\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}^{\prime}}(D)\rightarrow{\bf
R}j_{m*}\mathcal{O}_{\mathfrak{V}^{\prime}_{m}}$

is then the multiplication

$\frac{\mathcal{O}_{\mathfrak{P}}\langle
T,X\rangle}{(f^{r_{m}}T-p,f^{r_{m}}X-p)}\rightarrow\frac{\mathcal{O}_{\mathfrak{P}}\langle
X\rangle}{(f^{r_{m}}X-p)}.$

which is surjective, with kernel $(T-X)$ annihilated by $p$. Similarly, on
$\mathfrak{U}$,

$\displaystyle\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}^{\prime}}(D)\\!\mid_{\mathfrak{U}}$
$\displaystyle={\bf
R}\underset{n}{\mathrm{lim}}\,\frac{\mathcal{O}_{P_{n}}[T]}{(f^{r_{m}}T-p)}\otimes^{{\bf
L}}_{\mathcal{O}_{P_{n}}}\frac{\mathcal{O}_{P_{n}}[Y]}{(f^{r_{m}}-pY)}$
$\displaystyle=\frac{\mathcal{O}_{\mathfrak{P}}\langle
T,Y\rangle}{(f^{r_{m}}T-p,f^{r_{m}}-pY)}.$

The inverse map

$\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}^{\prime}}(D)\rightarrow{\bf
R}j_{m*}\mathcal{O}_{\mathfrak{V}^{\prime}_{m}}$

is therefore the map

$\frac{\mathcal{O}_{\mathfrak{P}}\langle
T,Y\rangle}{(f^{r_{m}}T-p,f^{r_{m}}-pY)}\rightarrow\frac{\mathcal{O}_{\mathfrak{P}}\langle
Y,Y^{-1}\rangle}{(f^{r_{m}}-pY)}$

given by $T\mapsto Y^{-1}$. Again, this map is surjective, with kernel
$(1-TY)$ annihilated by $p$.

Putting this together, I have constructed a map

$\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}^{\prime}}(D)\rightarrow
j_{m*}\mathcal{O}_{\mathfrak{V}^{\prime}_{m}}$

which is surjective, and whose kernel is annihilated by $p$. It follows that
it is an isogeny, and hence induces an isomorphism

$\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}^{\prime}}(D)\overset{\cong}{\longrightarrow}{\bf
R}j_{m*}\mathcal{O}_{\mathfrak{V}^{\prime}_{m}}$

in ${\bf D}_{{\mathbb{Q}},\mathrm{qc}}(\mathcal{O}_{\mathfrak{P}^{\prime}})$.
To identify this with the inverse of an isomorphism

${\bf
R}j_{m*}\mathcal{O}_{\mathfrak{V}^{\prime}_{m}}\overset{\cong}{\longrightarrow}\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}^{\prime}}(D)$

factoring the natural map

$\mathcal{O}_{\mathfrak{P}^{\prime}}\overset{\cong}{\longrightarrow}\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}^{\prime}}(D)$

I can therefore restrict to $\mathfrak{V}^{\prime}_{m}$, in which case
everything has already been worked out explicitly. ∎

##  6\. Rigidification of $\mathscr{D}^{\dagger}$-modules and constructibility

Suppose now that the formal scheme $\mathfrak{P}$ is smooth over
$\mathcal{V}$. Thus Berthelot defined in [Ber02, §4.2] the category

$\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\widehat{\mathscr{D}}^{(\bullet)}_{\mathfrak{P}})$

of inductive systems of quasi-coherent complexes of
$\widehat{\mathscr{D}}^{(\bullet)}_{\mathfrak{P}}$-modules, up to isogeny. In
this section, my goal is to upgrade ${\bf L}\hat{\mathrm{sp}}^{*}$ to a
functor

${\bf L}\hat{\mathrm{sp}}^{*}\colon\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\widehat{\mathscr{D}}_{\mathfrak{P}}^{(\bullet)})\rightarrow{\bf
D}(\mathscr{D}_{\mathfrak{P}_{K}}),$

and prove that if $\mathcal{M}$ is an overholonomic complex of Frobenius type,
then ${\bf L}\hat{\mathrm{sp}}^{*}\mathcal{M}$ is a constructible complex on
$\mathfrak{P}$.

###  6.1. $\boldsymbol{\mathscr{D}^{(k,0)}}$-modules on admissible blowups

The functor ${\bf L}\hat{\mathrm{sp}}^{*}_{\mathfrak{P}}$ passes through the
category of quasi-coherent complexes of $\mathcal{O}$-modules on the diagram
of all admissible blowups of $\mathfrak{P}$, and the same will be therefore
need to be true for its $\mathscr{D}$-module enhancement. In trying to do this
directly, one quickly runs into the following problem: if
$\pi\colon\mathfrak{P}^{\prime}\rightarrow\mathfrak{P}$ is an admissible
blowup of a smooth formal scheme $\mathfrak{P}$, then
$\pi^{-1}\mathscr{D}_{\mathfrak{P}}^{(0)}$ does not necessarily act on
$\mathcal{O}_{\mathfrak{P}^{\prime}}$. Thus
$\pi^{*}\mathscr{D}_{\mathfrak{P}}^{(0)}$ does not have any natural ring
structure over which the pullback $\pi^{*}\mathcal{M}$ of a
$\mathscr{D}^{(0)}_{\mathfrak{P}}$-module $\mathcal{M}$ could be a module.

To get around this problem, I will need to use the rings of differential
operators with congruence level from [HSS21]. In that article, the authors
define, for any $k\geq 0$, a subring

$\mathscr{D}^{(k,0)}_{\mathfrak{P}}\subset\mathscr{D}^{(0)}_{\mathfrak{P}}$

consisting of ‘differential operators of congruence level $k$’. In terms of
local co-ordinates $z_{1},\ldots,z_{d}$, with corresponding derivations
$\partial_{1},\ldots,\partial_{d}$, there is the usual description

$\mathscr{D}^{(0)}_{\mathfrak{P}}=\left\\{\left.\sum_{\underline{i}=(i_{1},\ldots,i_{d})\in{\mathbb{N}}^{d}}a_{\underline{i}}\partial_{1}^{i_{1}}\ldots\partial_{d}^{i_{d}}\;\right|\;a_{\underline{i}}\in\mathcal{O}_{\mathfrak{P}},\;\text{only
finitely many }\neq 0\right\\},$

and then $\mathscr{D}^{(k,0)}_{\mathfrak{P}}$ consists of those elements for
which
$a_{\underline{i}}\in\mathfrak{m}^{k\left|\underline{i}\right|}=\mathfrak{m}^{k(i_{1}+\ldots+i_{d})}$.
In particular,
$\mathscr{D}^{(k,0)}_{\mathfrak{P}{\mathbb{Q}}}=\mathscr{D}^{(0)}_{\mathfrak{P}{\mathbb{Q}}}$.

If $\pi\colon\mathfrak{P}^{\prime}\rightarrow\mathfrak{P}$ is an admissible
blowup, then
$\pi^{-1}\mathscr{D}^{(k,0)}_{\mathfrak{P}{\mathbb{Q}}}=\pi^{-1}\mathscr{D}^{(0)}_{\mathfrak{P}{\mathbb{Q}}}$
acts on $\mathcal{O}_{\mathfrak{P}^{\prime}{\mathbb{Q}}}$. Since
$\mathfrak{P}^{\prime}$ is flat over $\mathcal{V}$, it therefore makes sense
to ask whether or not this induces an action of
$\pi^{-1}\mathscr{D}^{(k,0)}_{\mathfrak{P}}$ on
$\mathcal{O}_{\mathfrak{P}^{\prime}}$. In [HSS21, Corollary 2.1.15] the
authors show that this will be the case for all sufficiently large $k$. In
particular, this will be true if $\mathfrak{P}^{\prime}={\rm
Bl}_{\mathcal{I}}(\mathfrak{P})$ and $\varpi^{k}\in\mathcal{I}$.

######  6.1.1 Definition.

If $\pi\colon\mathfrak{P}^{\prime}\rightarrow\mathfrak{P}$ is an admissible
blowup, with $\mathfrak{P}$ smooth, define
$k_{\mathfrak{P}^{\prime}}:=\min\\{k\mid\pi^{-1}\mathscr{D}^{(k,0)}_{\mathfrak{P}}\text{
acts on }\mathcal{O}_{\mathfrak{P}^{\prime}}\\}$. If
$\rho\colon\mathfrak{P}^{\prime\prime}\rightarrow\mathfrak{P}^{\prime}$ is a
morphism of admissible blowups, define
$k_{\rho}:=\max\\{k_{\mathfrak{P}^{\prime}},k_{\mathfrak{P}^{\prime\prime}}\\}$.

Thus, for any $k\geq k_{\mathfrak{P}^{\prime}}$ it is possible to form the
sheaf of rings
$\mathscr{D}^{(k,0)}_{\mathfrak{P}^{\prime}}:=\pi^{*}\mathscr{D}^{(k,0)}_{\mathfrak{P}}$.
If $\mathcal{M}$ is a $\mathscr{D}^{(0)}_{\mathfrak{P}}$-module, then
$\pi^{*}\mathcal{M}$ is naturally a
$\mathscr{D}^{(k,0)}_{\mathfrak{P}^{\prime}}$-module, for all $k\geq
k_{\mathfrak{P}^{\prime}}$. More generally, if
$\rho\colon\mathfrak{P}^{\prime\prime}\rightarrow\mathfrak{P}^{\prime}$ is a
morphism of admissible blowups, and $\mathcal{M}$ is a
$\mathscr{D}^{(k_{\mathfrak{P}^{\prime}},0)}_{\mathfrak{P}^{\prime}}$-module,
then $\rho^{*}\mathcal{M}$ is naturally a
$\mathscr{D}^{(k,0)}_{\mathfrak{P}^{\prime}}$-module, for all $k\geq
k_{\rho}$.

######  6.1.2 Definition.

Let $\mathcal{B}_{\mathfrak{P}}$ denote the diagram of admissible blowups of
$\mathfrak{P}$. A (left)
$\mathscr{D}^{(0)}_{\mathcal{B}_{\mathfrak{P}}}$-module consists of:

  * •

for each admissible blowup $\mathfrak{P}^{\prime}\rightarrow\mathfrak{P}$, a
(left)
$\mathscr{D}^{(k_{\mathfrak{P}^{\prime}},0)}_{\mathfrak{P}^{\prime}}$-module
$\mathcal{M}_{\mathfrak{P}^{\prime}}$;

  * •

for each morphism $\rho\colon\mathfrak{P}^{\prime}\rightarrow\mathfrak{P}$, a
morphism
$\chi_{\rho}\colon\rho^{*}\mathcal{M}_{\mathfrak{P}^{\prime}}\rightarrow\mathcal{M}_{\mathfrak{P}^{\prime\prime}}$
of (left) $\mathscr{D}^{(k_{\rho},0)}_{\mathfrak{P}^{\prime\prime}}$-modules,

subject to the condition that, for any commutative triangle

---
$\textstyle{\mathfrak{P}^{\prime\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{\rho\circ\tau}$$\textstyle{\mathfrak{P}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{\mathfrak{P}^{\prime}}$

of admissible blowups,
$\chi_{\rho\circ\tau}=\chi_{\tau}\circ\tau^{*}\chi(\rho)$.

Note that the condition
$\chi_{\rho\circ\tau}=\chi_{\tau}\circ\tau^{*}\chi(\rho)$ can be checked on
the underlying $\mathcal{O}$-modules, hence does not depend on the congruence
level of any of the terms. There is, of course, an analogous category of
$\mathscr{D}^{(0)}_{\mathcal{B}_{P_{\bullet}}}$-modules, consisting of
projective systems of
$\mathscr{D}^{(k_{\mathfrak{P}^{\prime}})}_{P^{\prime}_{n}}$-modules on each
$\mathfrak{P}^{\prime}$,

The major difficulty in working with this category comes from the fact that
the varying rings
$\mathscr{D}^{(k_{\mathfrak{P}}^{\prime},0)}_{\mathfrak{P}^{\prime}}$ do _not_
form a sheaf of rings on $\mathcal{B}_{\mathfrak{P}}$, thus a
$\mathscr{D}^{(0)}_{\mathcal{B}_{\mathfrak{P}}}$-module is not strictly
speaking a module over a sheaf of rings on a site in any natural way. Despite
this, it is nevertheless easy to extend the formalism of $K$-flat and
$K$-injective resolutions to the category of
$\mathscr{D}^{(0)}_{\mathcal{B}_{\mathfrak{P}}}$-modules, and therefore do
homological algebra as if we were just working in the category of modules over
a sheaf of rings on $\mathcal{B}_{\mathfrak{P}}$. For example:

######  6.1.3 Lemma.

The category of left $\mathscr{D}^{(0)}_{\mathcal{B}_{\mathfrak{P}}}$-modules
is an abelian category with enough injectives.

###### Proof.

∎

I will denote by ${\bf D}(\mathscr{D}^{(0)}_{\mathcal{B}_{\mathfrak{P}}})$ the
derived category of $\mathscr{D}^{(0)}_{\mathcal{B}_{\mathfrak{P}}}$-modules,
and ${\bf D}_{\mathrm{qc}}(\mathscr{D}^{(0)}_{\mathcal{B}_{\mathfrak{P}}})$
its full subcategory on quasi-coherent objects, that is, objects $\mathcal{M}$
such that each $\mathcal{M}_{\mathfrak{P}^{\prime}}$ is quasi-coherent as a
complex of $\mathcal{O}_{\mathfrak{P}^{\prime}}$-modules. For such objects,
the maps $\chi_{\rho}$ extend to morphisms

${\bf
L}\hat{\rho}^{*}\mathcal{M}_{\mathfrak{P}^{\prime}}\rightarrow\mathcal{M}_{\mathfrak{P}^{\prime\prime}}$

of quasi-coherent complexes of $\mathscr{D}^{(k_{\rho},0)}$-modules. There is
of course a similar category ${\bf
D}_{\mathrm{qc}}(\mathscr{D}^{(0)}_{\mathcal{B}^{(\bullet)}_{\mathfrak{P}}})$
of inductive systems of such objects, as well as its localisation
$\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathscr{D}^{(0)}_{\mathcal{B}_{\mathfrak{P}}})$.
Exactly as in the case of $\mathcal{O}_{\mathfrak{P}}$-modules, I can then
define a functor

${\bf L}\hat{\pi}^{*}\colon{\bf
D}_{\mathrm{qc}}(\mathscr{D}^{(0)}_{\mathfrak{P}^{(\bullet)}})\rightarrow{\bf
D}_{\mathrm{qc}}(\mathscr{D}^{(0)}_{\mathcal{B}^{(\bullet)}_{\mathfrak{P}}}).$

Since
$\mathscr{D}^{(k_{\mathfrak{P}}^{\prime},0)}_{\mathfrak{P}^{\prime}{\mathbb{Q}}}=\mathrm{sp}_{\mathfrak{P}^{\prime}*}\mathscr{D}_{\mathfrak{P}_{K}}$,
I can also define a functor

${\bf L}\mathrm{sp}^{*}\colon{\bf
D}_{\mathrm{qc}}(\mathscr{D}^{(0)}_{\mathcal{B}^{(\bullet)}_{\mathfrak{P}}})\rightarrow{\bf
D}(\mathscr{D}_{\mathfrak{P}}),$

which, roughly speaking, applies $\mathrm{sp}^{-1}_{\mathfrak{P}^{\prime}}$ on
each admissible blowup $\mathfrak{P}^{\prime}$, then takes the colimit over
both $I_{\mathfrak{P}}^{\rm op}$ and ${\mathbb{N}}$, and then finally tensors
with ${\mathbb{Q}}$.

######  6.1.4 Definition.

The functor

${\bf L}\hat{\mathrm{sp}}^{*}={\bf
L}\hat{\mathrm{sp}}^{*}_{\mathfrak{P}}\colon\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\widehat{\mathscr{D}}^{(\bullet)}_{{\mathfrak{P}}})\rightarrow{\bf
D}(\mathscr{D}_{\mathfrak{P}_{K}})$

is defined to be the composite

${\bf
D}_{\mathrm{qc}}(\widehat{\mathscr{D}}^{(\bullet)}_{\mathfrak{P}})\overset{\rm
forget}{\longrightarrow}{\bf
D}_{\mathrm{qc}}(\mathscr{D}^{(0)}_{\mathfrak{P}^{(\bullet)}})\overset{{\bf
L}\hat{\pi}^{*}}{\longrightarrow}{\bf
D}_{\mathrm{qc}}(\mathscr{D}^{(0)}_{\mathcal{B}^{(\bullet)}_{\mathfrak{P}}})\overset{{\bf
L}\mathrm{sp}^{*}}{\longrightarrow}{\bf D}(\mathscr{D}_{\mathfrak{P}_{K}}).$

I then define the shifted functor
$\mathrm{sp}^{!}=\mathrm{sp}_{\mathfrak{P}}^{!}:={\bf
L}\hat{\mathrm{sp}}^{*}[-\dim\mathfrak{P}]$.

It is straightforward to check that ${\bf L}\hat{\mathrm{sp}}^{*}$ (and
therefore the shifted version $\mathrm{sp}^{!}$) descends to the localisation
$\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\widehat{\mathscr{D}}^{(\bullet)}_{{\mathfrak{P}}})$
of ${\bf D}_{\mathrm{qc}}(\widehat{\mathscr{D}}^{(\bullet)}_{\mathfrak{P}})$.

###  6.2. Compatibility with de Rham pushforwards

Now let $u\colon\mathfrak{P}\rightarrow\mathfrak{Q}$ be a smooth morphism of
smooth formal $\mathcal{V}$-schemes. Since $u$ is in particular flat, the
strict transform of an admissible blowup
$\mathfrak{Q}^{\prime}\rightarrow\mathfrak{Q}$ is just the fibre product
$\mathfrak{Q}^{\prime}\times_{\mathfrak{Q}}\mathfrak{P}$.

######  6.2.1 Lemma.

Let $u\colon\mathfrak{P}\rightarrow\mathfrak{Q}$ be a smooth morphism of
smooth formal schemes, and $\mathfrak{Q}^{\prime}\rightarrow\mathfrak{Q}$ an
admissible blowup. Then
$k_{\mathfrak{Q}^{\prime}\times_{\mathfrak{Q}}\mathfrak{P}}\leq
k_{\mathfrak{Q}^{\prime}}$.

###### Proof.

Straightforward calculation. ∎

I now consider the functor $I_{\mathfrak{Q}}\rightarrow I_{\mathfrak{P}}$
given by taking the fibre product over $\mathfrak{Q}$ with $\mathfrak{P}$, and
restrict the diagram $\mathcal{B}_{\mathfrak{P}}$ along this functor to
produce a diagram $\mathcal{B}_{\mathfrak{P}/\mathfrak{Q}}\colon
I_{\mathfrak{Q}}\rightarrow{\bf FSch}$ taking $\mathfrak{Q}^{\prime}$ to
$\mathfrak{Q^{\prime}}\times_{\mathfrak{Q}}\mathfrak{P}$. I then consider the
following variant of Definition 6.1.2.

######  6.2.2 Definition.

A (left) $\mathscr{D}^{(0)}_{\mathcal{B}_{\mathfrak{P}/\mathfrak{Q}}}$-module
consists of:

  * •

for each admissible blowup $\mathfrak{Q}^{\prime}\rightarrow\mathfrak{Q}$, a
(left)
$\mathscr{D}^{(k_{\mathfrak{Q}^{\prime}},0)}_{\mathfrak{Q}^{\prime}\times_{\mathfrak{Q}}\mathfrak{P}}$-module
$\mathcal{M}_{\mathfrak{Q}^{\prime}}$;

  * •

for each morphism
$\rho\colon\mathfrak{Q}^{\prime\prime}\rightarrow\mathfrak{Q}^{\prime}$, a
morphism
$\chi_{\rho}\colon\rho^{*}\mathcal{M}_{\mathfrak{Q}^{\prime}}\rightarrow\mathcal{M}_{\mathfrak{Q}^{\prime\prime}}$
of (left)
$\mathscr{D}^{(k_{\rho},0)}_{\mathfrak{Q}^{\prime\prime}\times_{\mathfrak{Q}}\mathfrak{P}}$-modules,

subject to the condition that, for any commutative triangle

---
$\textstyle{\mathfrak{Q}^{\prime\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{\rho\circ\tau}$$\textstyle{\mathfrak{Q}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{\mathfrak{Q}^{\prime}}$

of admissible blowups,
$\chi_{\rho\circ\tau}=\chi_{\tau}\circ\tau^{*}\chi(\rho)$.

I can then define the transfer module
$\mathscr{D}^{(0)}_{\mathcal{B}_{\mathfrak{P}}\rightarrow\mathcal{B}_{\mathfrak{Q}}}$
as a left $\mathscr{D}^{(0)}_{\mathcal{B}_{\mathfrak{P}/\mathfrak{Q}}}$-module
by taking its restriction to each
$\pi\colon\mathfrak{Q}^{\prime}\times_{\mathfrak{Q}}\mathfrak{P}\rightarrow\mathfrak{P}$
to be
$\pi^{*}\mathscr{D}^{(k_{\mathfrak{Q}^{\prime}},0)}_{\mathfrak{P}\rightarrow\mathfrak{Q}}$.
The usual side switching operations therefore give a right
$\mathscr{D}^{(0)}_{\mathcal{B}_{\mathfrak{P}/\mathfrak{Q}}}$-module
$\mathscr{D}^{(0)}_{\mathcal{B}_{\mathfrak{Q}}\leftarrow\mathcal{B}_{\mathfrak{P}}}$,
which also comes with the usual structure of a ‘left
$u^{-1}\mathscr{D}^{(0)}_{\mathcal{B}_{\mathfrak{Q}}}$-module on
$\mathcal{B}_{\mathfrak{P}/\mathfrak{Q}}$’, in a sense that I leave to the
reader to make precise. I can therefore define a functor

$\displaystyle u_{+}\colon{\bf
D}(\mathscr{D}^{(0)}_{\mathcal{B}^{(\bullet)}_{\mathfrak{P}/\mathfrak{Q}}})$
$\displaystyle\rightarrow{\bf
D}(\mathscr{D}^{(0)}_{\mathcal{B}^{(\bullet)}_{\mathfrak{Q}}})$
$\displaystyle\mathcal{M}$ $\displaystyle\mapsto{\bf
R}u_{*}(\mathscr{D}^{(0)}_{\mathcal{B}_{\mathfrak{Q}}\leftarrow\mathcal{B}_{\mathfrak{P}}}\otimes^{{\bf
L}}_{\mathscr{D}^{(0)}_{\mathcal{B}_{\mathfrak{P}}}}\mathcal{M})$

in the usual way. Finally, I define

$u_{+}\colon{\bf
D}(\mathscr{D}^{(0)}_{\mathcal{B}^{(\bullet)}_{\mathfrak{P}}})\rightarrow{\bf
D}(\mathscr{D}^{(0)}_{\mathcal{B}^{(\bullet)}_{\mathfrak{Q}}})$

by composing with the‘restriction’ functor ${\bf
D}(\mathscr{D}^{(0)}_{\mathcal{B}^{(\bullet)}_{\mathfrak{P}}})\rightarrow{\bf
D}(\mathscr{D}^{(0)}_{\mathcal{B}^{(\bullet)}_{\mathfrak{P}/\mathfrak{Q}}})$,
which exists by Lemma 6.2.1.

Since $u$ is smooth, the Spencer resolutions for
$\mathscr{D}^{(0)}_{\mathfrak{Q}\leftarrow\mathfrak{P}}$ and
$\widehat{\mathscr{D}}^{(m)}_{\mathfrak{Q}\leftarrow\mathfrak{P}}$ show that
the natural map

$\mathscr{D}^{(0)}_{\mathfrak{Q}\leftarrow\mathfrak{P}}\otimes_{\mathscr{D}^{(0)}_{\mathfrak{P}}}\widehat{\mathscr{D}}^{(m)}_{\mathfrak{P}}\rightarrow\widehat{\mathscr{D}}^{(m)}_{\mathfrak{Q}\leftarrow\mathfrak{P}}$

is an isogeny [Ber02, §3.5.5]. Hence the diagram

$\textstyle{\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\widehat{\mathscr{D}}^{(\bullet)}_{\mathfrak{P}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{+}}$$\textstyle{\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathscr{D}^{(0)}_{\mathfrak{P}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{+}^{(0)}}$$\textstyle{\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\widehat{\mathscr{D}}^{(\bullet)}_{\mathfrak{Q}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathscr{D}^{(0)}_{\mathfrak{Q}})}$

commutes up to natural isomorphism. Now using the facts that
$\mathscr{D}^{(0)}_{\mathcal{B}_{\mathfrak{Q}}\leftarrow\mathcal{B}_{\mathfrak{P}},{\mathbb{Q}}}=\pi^{*}\mathscr{D}^{(0)}_{\mathfrak{Q}\leftarrow\mathfrak{P},{\mathbb{Q}}}$,
and
$\mathrm{sp}^{-1}_{\mathcal{B}_{\mathfrak{P}/\mathfrak{Q}}}\mathscr{D}^{(0)}_{\mathcal{B}_{\mathfrak{Q}}\leftarrow\mathcal{B}_{\mathfrak{P}},{\mathbb{Q}}}=\mathscr{D}_{\mathfrak{Q}_{K}\leftarrow\mathfrak{P}_{K}}$,
I obtain a natural base change map

${\bf L}\hat{\mathrm{sp}}_{\mathfrak{Q}}^{*}\circ u^{(0)}_{+}\rightarrow
u_{+}\circ{\bf L}\hat{\mathrm{sp}}_{\mathfrak{P}}^{*}$

of functors $\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathscr{D}^{(0)}_{\mathfrak{P}})\rightarrow{\bf
D}(\mathscr{D}_{\mathfrak{Q}_{K}})$. Note that since $u$ is smooth, say of
relative dimension $d$, the functor $u_{+}$ on the RHS here is what I have
previously called ${\bf R}u_{\mathrm{dR}*}[d]$.

######  6.2.3 Proposition.

Let $u\colon\mathfrak{P}\rightarrow\mathfrak{Q}$ be a smooth morphism of
smooth formal schemes of relative dimension $d$, and
$\mathcal{M}\in\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\mathscr{D}^{(0)}_{\mathfrak{P}})$. Then the
base change map

${\bf
L}\hat{\mathrm{sp}}^{*}_{\mathfrak{Q}}u_{+}^{(0)}\mathcal{M}\rightarrow\mathbf{R}u_{\mathrm{dR}*}{\bf
L}\hat{\mathrm{sp}}^{*}_{\mathfrak{P}}\mathcal{M}[d]$

is an isomorphism. Thus the diagram

$\textstyle{\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\widehat{\mathscr{D}}^{(\bullet)}_{\mathfrak{P}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{+}}$$\scriptstyle{\mathrm{sp}_{\mathfrak{P}}^{!}}$$\textstyle{{\bf
D}(\mathscr{D}_{\mathfrak{P}_{K}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\bf
R}u_{\mathrm{dR}*}[2d]}$$\textstyle{\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\widehat{\mathscr{D}}^{(\bullet)}_{\mathfrak{Q}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{sp}_{\mathfrak{Q}}^{!}}$$\textstyle{{\bf
D}(\mathscr{D}_{\mathfrak{Q}_{K}})}$

commutes up to natural isomorphism.

###### Proof.

By taking the stupid filtration on

$\mathscr{D}^{(0)}_{\mathfrak{Q}\leftarrow\mathfrak{P}}\otimes^{\bf
L}_{\mathscr{D}_{\mathfrak{P}}^{(0)}}\mathcal{M}=\Omega^{\bullet}_{\mathfrak{P}/\mathfrak{Q}}\otimes_{\mathcal{O}_{\mathfrak{P}}}\mathcal{M},$

I cam reduce to showing that the base change map

${\bf L}\hat{\mathrm{sp}}^{*}_{\mathfrak{Q}}{\bf
R}u_{*}(\Omega^{n}_{\mathfrak{P}/\mathfrak{Q}}\widehat{\otimes}^{\bf
L}_{\mathcal{O}_{\mathfrak{P}}}\mathcal{M})\rightarrow\mathbf{R}u_{*}{\bf
L}\hat{\mathrm{sp}}^{*}_{\mathfrak{P}}(\Omega^{n}_{\mathfrak{P}/\mathfrak{Q}}\widehat{\otimes}^{\bf
L}_{\mathcal{O}_{\mathfrak{P}}}\mathcal{M})$

is an isomorphism for each fixed $n$. Thanks to Lemma 5.3.4, this follows from
Theorem 5.3.2. ∎

For example, taking $\mathfrak{Q}=\mathrm{Spf}\left(\mathcal{V}\right)$ shows
that the natural map

${\bf R}\Gamma_{\mathrm{dR}}(\mathfrak{P},\mathcal{M})\rightarrow{\bf
R}\Gamma_{\mathrm{dR}}(\mathfrak{P}_{K},{\bf
L}\hat{\mathrm{sp}}^{*}_{\mathfrak{P}}\mathcal{M})$

is an isomorphism in ${\bf D}(K)$. Of course, this case could equally well be
deduced slightly more directly from Theorem 5.3.2.

###  6.3. Constructibility of $\bm{\mathrm{sp}^{!}}$

I now come to the first main result concerning $\mathrm{sp}^{!}$, namely that
it sends dual constructible overholonomic complexes on $\mathfrak{P}$ to
constructible isocrystals on $\mathfrak{P}_{K}$.

######  6.3.1 Theorem.

Let $\mathfrak{P}$ be a smooth formal scheme, and
$\mathcal{M}\in\mathbf{DCon}_{F}(\mathfrak{P})\subset{\bf
D}^{b}_{\mathrm{hol},F}(\mathfrak{P})$. Then
$\mathrm{sp}^{!}\mathcal{M}\in\mathbf{Isoc}_{\mathrm{cons},F}(\mathfrak{P})\subset{\bf
D}(\mathscr{D}_{\mathfrak{P}_{K}})$ is a constructible isocrystal on
$\mathfrak{P}$. If $\mathcal{M}$ is supported on some locally closed subscheme
$X\hookrightarrow P$, then so is $\mathrm{sp}^{!}\mathcal{M}$.

Before diving into the proof, I need a preparatory lemma, and a special case.

######  6.3.2 Lemma.

Let

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\mathfrak{P}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{P}}$

be a morphism of l.p. frames such that:

  * •

$u$ is smooth and proper, of relative dimension $d$; $g$ is proper, and $f$ is
finite étale;

  * •

$X$ is locally (on $Y$) the complement of a hypersurface in $Y$.

Let $\mathcal{M}\in\mathbf{DCon}_{F}(\mathfrak{P})$ be a dual constructible
module supported $X$. If
$\mathrm{sp}^{!}_{\mathfrak{P}^{\prime}}\mathbf{R}\underline{\Gamma}^{\dagger}_{X^{\prime}}u^{!}\mathcal{M}$
is a constructible isocrystal supported on $X^{\prime}$, then
$\mathrm{sp}_{\mathfrak{P}}^{!}\mathcal{M}$ is a constructible isocrystal
supported on $X$.

######  6.3.3 Remark.

Of course, if I am considering $\mathcal{M}$ as an object of
$\mathbf{DCon}_{F}(X,Y)$, then
$\mathbf{R}\underline{\Gamma}^{\dagger}_{X^{\prime}}u^{!}\mathcal{M}$ is
simply $f^{!}\mathcal{M}\in\mathbf{DCon}_{F}(X^{\prime},Y^{\prime})$.

###### Proof.

The question is local on $\mathfrak{P}$, so I can therefore suppose that
$\mathfrak{P}$ is affine and $X=D(s)$ for some
$s\in\Gamma(Y,\mathcal{O}_{Y})$. I am therefore in the situation of Theorem
3.4.2. Write
$\mathcal{N}:=\mathbf{R}\underline{\Gamma}^{\dagger}_{X^{\prime}}u^{!}\mathcal{M}$,
this is a dual constructible module on $\mathfrak{P}^{\prime}$. By Proposition
6.2.3,

$\mathrm{sp}^{!}_{\mathfrak{P}}u_{+}\mathcal{N}={\bf
R}u_{\mathrm{dR}*}\mathrm{sp}^{!}_{\mathfrak{P}^{\prime}}\mathcal{N}[2d]$

where $\mathrm{sp}^{!}_{\mathfrak{P}^{\prime}}\mathcal{N}$ is a constructible
isocrystal on $\mathfrak{P}^{\prime}$, supported on $X^{\prime}$. I then
consider the commutative diagram

$\textstyle{\left]X^{\prime}\right[_{\mathfrak{P}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left]f\right[}$$\textstyle{\mathfrak{P}^{\prime}_{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{\left]X\right[_{\mathfrak{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathfrak{P}_{K}}$

Since $u$ is proper, I can identify ${\bf R}u_{\mathrm{dR}!}={\bf
R}u_{\mathrm{dR}*}$ as functors

${\bf D}^{b}(\mathscr{D}_{\mathfrak{P}^{\prime}_{K}})\rightarrow{\bf
D}^{b}(\mathscr{D}_{\mathfrak{P}_{K}}).$

Setting
$\mathscr{F}:=\mathrm{sp}^{!}_{\mathfrak{P}^{\prime}}\mathcal{N}|_{\left]X^{\prime}\right[_{\mathfrak{P}^{\prime}}}\in\mathbf{Isoc}_{\mathrm{cons}}(X^{\prime},Y^{\prime},\mathfrak{P}^{\prime})$,
I therefore have

${\bf
R}u_{\mathrm{dR}*}\mathrm{sp}^{!}_{\mathfrak{P}^{\prime}}\mathcal{N}[2d]=i_{!}{\bf
R}\\!\left]f\right[_{\mathrm{dR}!}\mathscr{F}[2d].$

Now applying Theorem 3.4.2 I deduce that
$\mathrm{sp}^{!}_{\mathfrak{P}}u_{+}\mathcal{N}$ is a constructible isocrystal
supported on $X$. Since $\mathcal{M}$ is a direct summand of
$u_{+}\mathcal{N}$ (see Remark 4.5.6), I can conclude. ∎

######  6.3.4 Lemma.

Let $\mathfrak{P}$ be a smooth formal scheme, $D\subset P$ a divisor, and let
$j\colon U:=P\setminus D\rightarrow P$ be the complementary open immersion.
Suppose that $\mathscr{F}\in\mathbf{Isoc}_{F}(U,P)$, and let
$\mathcal{M}=\mathrm{sp}_{U!}\mathscr{F}\in\mathbf{DCon}_{F}(\mathfrak{P})$.161616See
§4.3 and Example 4.4.6. Then $\mathrm{sp}^{!}\mathcal{M}\cong
j_{*}\mathscr{F}\in\mathbf{Isoc}_{\mathrm{cons}}(\mathfrak{P})$.

###### Proof.

Given the shifts involved, an equivalent way of stating the conclusion is that

${\bf
L}\hat{\mathrm{sp}}_{\mathfrak{P}}^{*}\widetilde{\mathrm{sp}}_{U+}\mathscr{F}\cong
j_{*}\mathscr{F}$

in ${\bf D}^{b}(\mathscr{D}_{\mathfrak{P}_{K}})$. Also, since
$j_{*}\mathscr{F}$ is a module (not just a complex) the claim is local on
$\mathfrak{P}$, which I may therefore assume to be affine. I can also assume
that there exists $f\in\mathcal{O}_{\mathfrak{P}}$ which is a non-zero divisor
in $\mathcal{O}_{P}$, such that $D=V(f)\cap P$.

Let $j_{m}\colon V_{m}\rightarrow\mathfrak{P}_{K}$ be as in the proof of
Theorem 5.4.2, recall that $V_{m}$ is affinoid. For $m$ large enough,
$\mathscr{F}$ extends to a vector bundle with integrable connection
$\mathscr{F}_{m}$ on $V_{m}$, and

$j_{*}\mathscr{F}\overset{\cong}{\longrightarrow}\underset{m}{\mathrm{colim}}\,j_{m*}\mathscr{F}_{m}\overset{\cong}{\longrightarrow}\underset{m}{\mathrm{colim}}\,{\bf
R}j_{m*}\mathscr{F}_{m}.$

In this case, $\widetilde{\mathrm{sp}}_{U+}\mathscr{F}$ is explicitly defined
in [Ber96b, §4.4] by showing that each

$\mathrm{sp}_{\mathfrak{P}*}j_{m*}\mathscr{F}_{m}\overset{\cong}{\longrightarrow}{\bf
R}\mathrm{sp}_{\mathfrak{P}*}{\bf R}j_{m*}\mathscr{F}_{m}$

is a coherent $\hat{\mathcal{B}}_{\mathfrak{P}}^{(m)}(D)$-module, for which
the $\mathscr{D}_{\mathfrak{P}}^{(0)}$-module structure extends uniquely to a
continuous $\widehat{\mathscr{D}}_{\mathfrak{P}}^{(m)}$-module structure
compatible with that on $\hat{\mathcal{B}}_{\mathfrak{P}}^{(m)}(D)$. Then

$\widetilde{\mathrm{sp}}_{U+}\mathscr{F}=\\{\mathrm{sp}_{\mathfrak{P}*}j_{m*}\mathscr{F}_{m}\\}_{m\in{\mathbb{N}}}\in\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\widehat{\mathscr{D}}^{(\bullet)}_{\mathfrak{P}}).$

It therefore suffices to show that

(6.3.1) ${\bf L}\hat{\mathrm{sp}}_{\mathfrak{P}}^{*}{\bf
R}\mathrm{sp}_{\mathfrak{P}*}{\bf
R}j_{m*}\mathscr{F}_{m}\overset{\cong}{\longrightarrow}{\bf
R}j_{m*}\mathscr{F}_{m}=j_{m*}\mathscr{F}_{m}$

in ${\bf D}^{b}(\mathscr{D}_{\mathfrak{P}_{K}})$. Indeed, if this is the case
then everything in sight is just a module, rather than a complex, and so
showing compatibility in $m$ is straightforward. To prove (6.3.1), let
$\pi\colon\mathfrak{P}^{\prime}\rightarrow\mathfrak{P}$ be an admissible
blowup such that the open immersion $j_{m}\colon
V_{m}\rightarrow\mathfrak{P}_{K}$ arises from an open immersion
$j_{m}\colon\mathfrak{V}_{m}^{\prime}\rightarrow\mathfrak{P}^{\prime}$ of
formal schemes. I then claim that the natural morphism

${\bf L}\hat{\pi}^{*}{\bf R}\mathrm{sp}_{\mathfrak{P}*}{\bf
R}j_{m*}\mathscr{F}_{m}\rightarrow{\bf
R}\mathrm{sp}_{\mathfrak{P}^{\prime}*}{\bf R}j_{m*}\mathscr{F}_{m}$

is an isogeny. To see this, since $V_{m}$ is affinoid, and $\mathscr{F}_{m}$
is a vector bundle on $V_{m}$, it is in fact a direct summand of a trivial
vector bundle. Hence I can reduce to the case
$\mathscr{F}_{m}=\mathcal{O}_{V_{m}}$. In this case, the given map can be
identified with

${\bf
L}\hat{\pi}^{*}\widehat{\mathcal{B}}^{(m)}_{\mathfrak{P}}(D)\rightarrow{\bf
R}j_{m*}\mathcal{O}_{\mathfrak{V}_{m}^{\prime}},$

which was shown to be an isogeny during the proof of Theorem 5.4.2. I can now
compute

$\displaystyle{\bf L}\hat{\mathrm{sp}}_{\mathfrak{P}}^{*}{\bf
R}\mathrm{sp}_{\mathfrak{P}*}{\bf R}j_{m*}\mathscr{F}_{m}$
$\displaystyle\overset{\cong}{\longrightarrow}{\bf
L}\hat{\mathrm{sp}}_{\mathfrak{P}^{\prime}}^{*}{\bf L}\hat{\pi}^{*}{\bf
R}\mathrm{sp}_{\mathfrak{P}*}{\bf R}j_{m*}\mathscr{F}_{m}$
$\displaystyle\overset{\cong}{\longrightarrow}{\bf
L}\hat{\mathrm{sp}}_{\mathfrak{P}^{\prime}}^{*}{\bf
R}\mathrm{sp}_{\mathfrak{P}^{\prime}*}{\bf R}j_{m*}\mathscr{F}_{m}$
$\displaystyle\overset{\cong}{\longrightarrow}{\bf R}j_{m*}{\bf
L}\hat{\mathrm{sp}}_{\mathfrak{V}_{m}^{\prime}}^{*}{\bf
R}\mathrm{sp}_{\mathfrak{V}^{\prime}_{m}*}\mathscr{F}_{m}$
$\displaystyle\overset{\cong}{\longrightarrow}{\bf R}j_{m*}\mathscr{F}_{m}$

where the first isomorphism follows simply from the definitions, the second is
the isogeny proved above, the third follows from Theorem 5.3.2, and the last
from the fact that $\mathscr{F}_{m}$ is a locally free
$\mathcal{O}_{V_{m}}$-module. ∎

######  6.3.5 Remark.

The particular isomorphism $\mathrm{sp}^{!}\mathcal{M}\cong j_{*}\mathscr{F}$
constructed can be identified as follows: if we set
$\mathfrak{U}=\mathfrak{P}\setminus D$, then on $\mathfrak{U}_{K}$ it is
simply the shift of the counit
$\mathrm{sp}^{*}_{\mathfrak{U}}\mathrm{sp}_{\mathfrak{U}*}\mathscr{F}\rightarrow\mathscr{F}$
of the adjunction between $\mathrm{sp}^{*}$ and $\mathrm{sp}_{*}$ (note that
since $\mathcal{M}\\!\\!\mid_{\mathfrak{U}}$ is locally projective, ${\bf
L}\hat{\mathrm{sp}}^{*}_{\mathfrak{U}}=\mathrm{sp}^{*}_{\mathfrak{U}}$ in this
case).

###### Proof of Theorem 6.3.1.

The question is local on $\mathfrak{P}$, which I can therefore assume to be
affine, in particular $\mathfrak{P}$ admits a locally closed immersion into a
smooth and proper formal $\mathcal{V}$-scheme. This then means that every
frame encountered during the proof will be an l.p. frame. By dévissage, that
is, by Proposition 4.3.3, every
$\mathcal{M}\in\mathbf{DCon}_{F}(\mathfrak{P})$ is an iterated extension of
objects of the form $\mathrm{sp}_{X!}\mathscr{F}$ with $X\hookrightarrow P$ a
smooth locally closed subscheme, with closure $Y$ in $P$, and
$\mathscr{F}\in\mathbf{Isoc}_{F}(X,Y)$. I can therefore assume that
$\mathcal{M}$ is of this form.

Now, by de Jong’s alterations [dJ96], there exists a projective, generically
étale morphism $g\colon Y^{\prime}\rightarrow Y$ with $Y^{\prime}$ smooth. By
Noetherian induction on $X$, I can therefore assume that $g^{-1}(X)\rightarrow
X$ is finite étale, and that $X$ is the complement of a hypersurface in $Y$. I
then have a morphism of frames of the form

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\mathfrak{P}^{\prime}=\widehat{{\mathbb{P}}}^{d}_{\mathfrak{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{P}}$

with left hand square Cartesian, $Y^{\prime}$ smooth, $f$ finite étale, and
$u$ the projection. Appealing to Lemma 6.3.2, I can replace
$(X,Y,\mathfrak{P})$ by $(X^{\prime},Y^{\prime},\mathfrak{P}^{\prime})$, in
other words I can also assume that $Y$ is smooth. Of course, I have now lost
the assumption that $\mathfrak{P}$ is affine, but I can then further localise
on $\mathfrak{P}$ to restore it.

Since $\mathfrak{P}$ is affine, and $Y$ is smooth, there exists a closed
immersion of smooth formal schemes $\mathfrak{Y}\hookrightarrow\mathfrak{P}$
lifting the closed immersion $Y\hookrightarrow P$. Now applying Lemma 6.3.2 to
the morphism of frames

$\textstyle{\mathfrak{Y}\times_{\mathcal{V}}\mathfrak{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathfrak{P}}$

(where the diagonal map
$Y\rightarrow\mathfrak{Y}\times_{\mathcal{V}}\mathfrak{P}$ is the product of
the two given immersions, and the right hand vertical map is the second
projection), it suffices to prove the claim with $\mathfrak{P}$ replaced by
$\mathfrak{Y}\times_{\mathcal{V}}\mathfrak{P}$. In other words, I can assume
that the closed immersion $\mathfrak{Y}\rightarrow\mathfrak{P}$ is a section
of a smooth morphism $\pi\colon\mathfrak{P}\rightarrow\mathfrak{Y}$.

Let $c$ be the codimension of $\mathfrak{Y}$ in $\mathfrak{P}$. By further
localising on $\mathfrak{P}$ if necessary, I can choose functions
$t_{1}\ldots,t_{c}\in\Gamma(\mathfrak{P},\mathcal{O}_{\mathfrak{P}})$ such
that $\mathfrak{Y}=V(t_{1},\ldots,t_{c})$. For any subset
$I\subset\\{1,\ldots,c\\}$ set $\mathfrak{U}_{I}:=\cap_{i\in I}D(t_{i})$. I
also set
$\mathcal{N}:=\pi^{!}\pi_{+}\mathcal{M}\in\mathbf{DCon}_{F}(\mathfrak{P})$,
thus, by the discussion in §4.3,
$\mathcal{N}=\mathrm{sp}_{\pi^{-1}(X)!}\pi^{*}\mathscr{F}$ where
$\pi^{*}\mathscr{F}\in\mathbf{Isoc}_{F}(\pi^{-1}(X),P)$. Let $j$ also denote
the inclusion $\pi^{-1}(X)\rightarrow P$, and write
$\mathscr{G}:=j_{*}\pi^{*}\mathscr{F}\in\mathbf{Isoc}_{\mathrm{cons},F}(\mathfrak{P})$.

There is then the following exact sequence of dual constructible
$\mathscr{D}^{\dagger}$-modules on $\mathfrak{P}$:

$0\rightarrow\mathcal{M}\rightarrow\mathcal{N}\rightarrow\bigoplus_{i}\mathbf{R}\underline{\Gamma}_{U_{i}}^{\dagger}\mathcal{N}\rightarrow\ldots\rightarrow\mathbf{R}\underline{\Gamma}_{U_{\\{1,\ldots,c\\}}}^{\dagger}\mathcal{N}\rightarrow
0,$

whose exactness can be seen by applying the t-exact and conservative family of
functors $\mathbf{R}\underline{\Gamma}^{\dagger}_{x}$, for $x\in P$ a closed
point, together with the inclusion-exclusion principle.

It follows from Theorem 5.4.2 and Lemma 6.3.4 that applying
$\mathrm{sp}_{\mathfrak{P}}^{!}$ to all except the first term in this exact
sequence results in the complex

$\mathscr{G}\rightarrow\bigoplus_{i}j_{U_{i}}^{\dagger}\mathscr{G}\rightarrow\ldots\rightarrow
j_{U_{\\{1,\ldots,c\\}}}^{\dagger}\mathscr{G}$

of $\mathscr{D}_{\mathfrak{P}_{K}}$-modules. I therefore deduce that

$\mathrm{sp}_{\mathfrak{P}}^{!}\mathcal{M}\overset{\cong}{\longrightarrow}\left[\mathscr{G}\rightarrow\bigoplus_{i}j_{U_{i}}^{\dagger}\mathscr{G}\rightarrow\ldots\rightarrow
j_{U_{\\{1,\ldots,c\\}}}^{\dagger}\mathscr{G}\right]\overset{\cong}{\longrightarrow}i_{!}i^{*}\mathscr{G},$

by using [Ber96a, Proposition 2.1.8]. This completes the proof. ∎

###  6.4. Consequences

I now gather several important consequences of Theorem 6.3.1, starting with:

######  6.4.1 Corollary.

Let $\mathfrak{P}$ be a smooth formal scheme. Then $\mathrm{sp}^{!}$ induces a
functor

$\mathrm{sp}^{!}\colon{\bf
D}^{b}_{\mathrm{hol},F}(\mathfrak{P})\rightarrow{\bf
D}^{b}_{\mathrm{cons},F}(\mathfrak{P})$

which is t-exact for the dual constructible t-structure on the source, and the
natural t-structure on the target.

Examining the proof of Theorem 6.3.1, I have also shown the following.

######  6.4.2 Corollary.

Let $\mathfrak{P}$ be a smooth formal scheme, $i\colon X\rightarrow P$ be a
smooth, locally closed subscheme, with closure $Y$,
$\mathscr{F}\in\mathbf{Isoc}_{F}(X,Y)$, and
$\mathcal{M}=\mathrm{sp}_{X!}\mathscr{F}\in\mathbf{DCon}_{F}(\mathfrak{P})$.
Then
$\mathrm{sp}^{!}\mathcal{M}\overset{\cong}{\longrightarrow}i_{!}\mathscr{F}\in\mathbf{Isoc}_{\mathrm{cons},F}(\mathfrak{P})$.

######  6.4.3 Remark.

It’s not straightforward to do explicitly, but the particular isomorphism
$\mathrm{sp}^{!}\mathcal{M}\overset{\cong}{\longrightarrow}i_{!}\mathscr{F}\in\mathbf{Isoc}_{\mathrm{cons},F}(\mathfrak{P})$
thus constructed can be identified by following through the various steps in
the proof of Theorem 6.3.1 and reducing to the case considered in Lemma 6.3.4.

The functor $\mathrm{sp}^{!}$ is compatible with many of the cohomological
functors introduced so far. Explicitly, if $U\hookrightarrow P\hookleftarrow
Z$ are complementary open and closed immersions, and $\mathcal{M}\in{\bf
D}^{b}_{\mathrm{hol},F}(\mathfrak{P})$, then repeatedly applying Theorem 5.4.2
gives rise to an isomorphism

$j_{U}^{\dagger}\mathrm{sp}^{!}\mathcal{M}\overset{\cong}{\longrightarrow}\mathrm{sp}^{!}\mathcal{M}(^{\dagger}Z)$

in ${\bf D}^{b}_{\mathrm{cons},F}(\mathfrak{P})$, which is natural in
$\mathcal{M}$. Moreover, this is compatible with the unit of the two
adjunctions, in the sense that the diagram

$\textstyle{\mathrm{sp}^{!}\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{j_{U}^{\dagger}\mathrm{sp}^{!}\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{sp}^{!}\mathcal{M}(^{\dagger}Z)}$

commutes. Similarly, there exists an isomorphism

${\bf
R}\underline{\Gamma}_{Z}^{\dagger}\mathrm{sp}^{!}\mathcal{M}\overset{\cong}{\longrightarrow}\mathrm{sp}^{!}{\bf
R}\underline{\Gamma}_{Z}^{\dagger}\mathcal{M},$

natural in $\mathcal{M}$, which is again compatible with the counit of the two
adjunctions in the sense that the diagram

$\textstyle{{\bf
R}\underline{\Gamma}_{Z}^{\dagger}\mathrm{sp}^{!}\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{sp}^{!}\mathcal{M}}$$\textstyle{\mathrm{sp}^{!}{\bf
R}\underline{\Gamma}_{Z}^{\dagger}\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

commutes. This implies that for any locally closed subscheme $i\colon
X\rightarrow P$, there is a canonical isomorphism

$\mathrm{sp}^{!}{\bf R}\underline{\Gamma}_{X}^{\dagger}\mathcal{M}\cong
i_{!}i^{-1}\mathrm{sp}^{!}\mathcal{M}$

in ${\bf D}^{b}_{\mathrm{cons},F}(\mathfrak{P})$.

If now $u\colon\mathfrak{P}\rightarrow\mathfrak{Q}$ is a smooth and proper
morphism of smooth formal schemes, of relative dimension $d$, and
$\mathcal{M}\in{\bf D}^{b}_{\mathrm{hol},F}(\mathfrak{P})$, then Proposition
6.2.3 provides a canonical isomorphism

$\mathrm{sp}_{\mathfrak{Q}}^{!}u_{+}\mathcal{M}\overset{\cong}{\longrightarrow}{\bf
R}u_{\mathrm{dR}*}\mathrm{sp}_{\mathfrak{P}}^{!}\mathcal{M}[2d]$

inside ${\bf D}^{b}(\mathscr{D}_{\mathfrak{Q}})$, in particular the RHS is in
fact in ${\bf D}^{b}_{\mathrm{cons},F}(\mathfrak{Q})$.

Compatibility of $\mathrm{sp}^{!}$ with $u^{!}$ (and therefore with tensor
product) will need to wait until later on, but I can at least record a special
case for now. Consider a morphism of l.p. frames

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\mathfrak{P}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{P}}$

such that $u$ is smooth and proper, of relative dimension $d$, $g$ is proper,
and $f$ is finite étale. By the support claim in Theorem 6.3.1, I can view
$\mathrm{sp}^{!}_{\mathfrak{P}}$ as a functor

${\bf D}^{b}_{\mathrm{hol},F}(X,Y,\mathfrak{P})\rightarrow{\bf
D}^{b}_{\mathrm{cons},F}(X,Y,\mathfrak{P}),$

and similarly for $\mathrm{sp}^{!}_{\mathfrak{P}^{\prime}}$. Thanks to Theorem
3.4.2, the isomorphisms

$\mathrm{sp}_{\mathfrak{P}}^{!}u_{+}\mathcal{M}\overset{\cong}{\longrightarrow}{\bf
R}u_{\mathrm{dR}*}\mathrm{sp}_{\mathfrak{P}^{\prime}}^{!}\mathcal{M}[2d]$

induce an isomorphism

$\chi_{f}\colon\mathrm{sp}_{\mathfrak{P}}^{!}\circ
f_{+}\overset{\cong}{\longrightarrow}{\bf
R}\\!\left]f\right[_{\mathrm{dR}*}\circ\mathrm{sp}_{\mathfrak{P}^{\prime}}^{!}$

of functors

${\bf
D}^{b}_{\mathrm{hol},F}(X^{\prime},Y^{\prime},\mathfrak{P}^{\prime})\rightarrow{\bf
D}^{b}_{\mathrm{cons},F}(X,Y,\mathfrak{P}).$

The adjunctions $(f^{!},f_{+})$ and $(f^{*},{\bf
R}\\!\left]f\right[_{\mathrm{dR}*})$ mean that I can define a morphism

$\psi_{f}\colon
f^{*}\circ\mathrm{sp}_{\mathfrak{P}}^{!}\rightarrow\mathrm{sp}_{\mathfrak{P}^{\prime}}^{!}\circ
f^{!}$

of functors

${\bf D}^{b}_{\mathrm{hol},F}(X,Y,\mathfrak{P})\rightarrow{\bf
D}^{b}_{\mathrm{cons},F}(X^{\prime},Y^{\prime},\mathfrak{P}^{\prime})$

by taking the adjoint of the morphism

$\displaystyle\mathrm{sp}_{\mathfrak{P}}^{!}$ $\displaystyle\rightarrow{\bf
R}\\!\left]f\right[_{\mathrm{dR}*}\circ\mathrm{sp}_{\mathfrak{P}^{\prime}}^{!}\circ
f^{!}$
$\displaystyle\overset{\chi_{f}^{-1}}{\longrightarrow}\mathrm{sp}_{\mathfrak{P}}^{!}\circ
f_{+}\circ f^{!}$

obtained by applying $\mathrm{sp}_{\mathfrak{P}}^{!}$ to the morphism ${\rm
id}\rightarrow f_{+}\circ f^{!}$.

######  6.4.4 Proposition.

The morphism $\psi_{f}$ is an isomorphism, and gives rise to a commutative
diagram

$\textstyle{{\rm
Hom}(f^{!}\mathcal{M},\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\psi_{f}}$$\textstyle{{\rm
Hom}(\mathcal{M},f_{+}\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi_{f}}$$\textstyle{{\rm
Hom}(f^{*}\mathrm{sp}_{\mathfrak{P}}^{!}\mathcal{M},\mathrm{sp}_{\mathfrak{P}^{\prime}}^{!}\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{{\rm
Hom}(\mathrm{sp}_{\mathfrak{P}}^{!}\mathcal{M},{\bf
R}\\!\left]f\right[_{\mathrm{dR}*}\mathrm{sp}_{\mathfrak{P}^{\prime}}^{!}\mathcal{N})}$

for any $\mathcal{M}\in{\bf D}^{b}_{\mathrm{hol},F}(X,Y)$, $\mathcal{N}\in{\bf
D}^{b}_{\mathrm{hol},F}(X^{\prime},Y^{\prime})$.

###### Proof.

The fact that the diagram commutes just follows from the definition of
$\psi_{f}$ via adjunction, the key point is to show that $\psi_{f}$ is an
isomorphism. But again, by compatibility of $\mathrm{sp}^{!}$ with
$\underline{{\bf R}\Gamma}^{\dagger}$, this reduces to the case when $X$ is a
point, which is straightforward. ∎

##  7\. Logarithmic variations

Before I can prove my main result, I will need some elementary logarithmic
versions of some of the results proved in previous sections. I have not tried
especially hard to frame these results in the most general possible content,
but have instead concentrated on giving quick proofs of the specific results
that I need.

###  7.1. Log convergent isocrystals

The general theory of log convergent isocrystals was developed in [Shi02], I
will recall here some of the fundamental definitions and results that I need.
Throughout I will equip the bases $\mathrm{Spec}\left(k\right)$ and
$\mathrm{Spf}\left(\mathcal{V}\right)$ with the trivial log structure.

A fine log variety (over $k$) will mean a fine log scheme over $k$ whose
underlying $k$-scheme is a variety. If $X$ is smooth variety over $k$, and
$D\subset X$ is a normal crossings divisor, I will let $M(D)$ denote the log
structure on $X$ associated to $D$, and $(X,M(D))$ the corresponding log
scheme. I will use similar notation $(\mathfrak{X},M(\mathfrak{D}))$ in the
case of a relative crossings divisor $\mathfrak{D}\subset\mathfrak{X}$ inside
a smooth formal scheme over $\mathcal{V}$.

######  7.1.1 Definition.

Let $(X,M)$ be a fine log variety. Then an _enlargement_ of $(X,M)$ is a fine
log scheme $(Z,M_{Z})\rightarrow(X,M)$, together with an exact closed
immersion $(Z,M_{Z})\rightarrow(\mathfrak{T},M_{\mathfrak{T}})$ into a fine
log formal scheme over $\mathcal{V}$, such that $T_{\rm red}\subset Z$.

There is the obvious notion of a morphism of enlargements of $(X,M)$.

######  7.1.2 Definition.

A convergent log isocrystal on $(X,M)$ consists of the following data:

  1. (1)

for every enlargement $(\mathfrak{T},M_{\mathfrak{T}})$ of $(X,M)$, a coherent
$\mathcal{O}_{\mathfrak{T}{\mathbb{Q}}}$-module $\mathscr{F}_{\mathfrak{T}}$;

  2. (2)

for every morphism of enlargements
$g\colon(\mathfrak{T}^{\prime},M_{\mathfrak{T}^{\prime}})\rightarrow(\mathfrak{T},M_{\mathfrak{T}})$
a ‘transition’ isomorphism
$g^{*}\mathscr{F}_{\mathfrak{T}}\rightarrow\mathscr{F}_{\mathfrak{T}^{\prime}}$
of $\mathcal{O}_{\mathfrak{T}^{\prime}{\mathbb{Q}}}$-modules,

The transition isomorphisms are required to satisfy an appropriate cocycle
condition. In fact, Shiho considers two variations, where $\mathscr{F}$ is
considered as a sheaf on $\mathfrak{T}$ with respect to the Zariski and étale
topologies. He then shows in [Shi02, Proposition 2.1.21] these give rise to
equivalent theories.

There is an obvious notion of a morphism of convergent log isocrystals on
$(X,M)$, and I will denote the resulting category by
$\mathbf{Isoc}^{\circ}(X,M)$.

######  7.1.3 Remark.

My choice of notation here reflects the fact that I am using
$\mathbf{Isoc}(X)$, and not $\mathbf{Isoc}^{\dagger}(X)$, to denote the
category of overconvergent isocrystals on a variety $X$. Thus, when the log
structure $M$ is trivial, $\mathbf{Isoc}^{\circ}(X)$ is the category of
convergent isocrystals on $X$, which has previously been denoted
$\mathbf{Isoc}(X,X)$.

As with the case of ordinary (that is, non-logarithmic) convergent
isocrystals, these objects can be described in terms of their ‘realisations’
in the world of rigid analytic geometry. Following Shiho, suppose that there
is an exact closed immersion

$(X,M)\rightarrow(\mathfrak{P},L)$

of fine log formal schemes over $\mathcal{V}$, such that $(\mathfrak{P},L)$ is
log smooth over $\mathcal{V}$.171717In fact, Shiho works in much greater
generality than this, but this special case is technically simpler and will
suffice for me. Then the tube $\left]X\right[_{\mathfrak{P}}$ admits a natural
log structure coming from the log structure induced by $L$ on
$\mathfrak{P}_{K}$, and is log smooth over $K$. In [Shi02, §2.1,§2.2] Shiho
constructs a functor

$\displaystyle\mathbf{Isoc}^{\circ}(X,M)$
$\displaystyle\rightarrow\mathbf{Coh}_{\nabla}(\left]X\right[_{\mathfrak{P}},L)$
$\displaystyle\mathscr{F}$
$\displaystyle\mapsto\mathscr{F}_{\left]X\right[_{\mathfrak{P}}}$

from the category of log convergent isocrystals on $(X,M)$ to the category
$\mathbf{Coh}_{\nabla}(\left]X\right[_{\mathfrak{P}},L)$ of coherent
$\mathcal{O}_{\left]X\right[_{\mathfrak{P}}}$-modules with integrable log
connection. In general, it is not clear whether or not this ‘realisation on
$(\mathfrak{P},L)$’ functor is fully faithful, in order to obtain full
faithfulness it is necessary to impose local freeness conditions on convergent
isocrystals.

######  7.1.4 Definition.

Let $(X,M)$ be a fine log variety over $k$, and $\mathscr{F}$ a convergent log
isocrystals on $(X,M)$. Then $\mathscr{F}$ is locally free if, for all
enlargements $(\mathfrak{T},M_{\mathfrak{T}})$ of $(X,M)$, the coherent
$\mathcal{O}_{\mathfrak{T}{\mathbb{Q}}}$-module $\mathscr{F}_{\mathfrak{T}}$
is locally projective.

It is then a consequence of [Shi02, Corollary 2.3.9] that the restriction of
the realisation functor to locally free isocrystals is fully faithful.
Moreover, if we are given $\mathscr{F}\in\mathbf{Isoc}^{\circ}(X,M)$ we can
actually test whether or not it is locally free by taking its realisation on
$(\mathfrak{P},L)$ (this follows from [Shi02, Proposition 2.2.7]).

####  7.1.5. Frobenius structures

Since log convergent isocrystals are clearly functorial in $(X,M)$, there is a
well-defined Frobenius pullback functor
$F^{*}\colon\mathbf{Isoc}^{\circ}(X,M)\rightarrow\mathbf{Isoc}^{\circ}(X,M)$,
which makes it possible to talk about Frobenius structures on convergent log
isocrystals. The category of convergent log $F$-isocrystals on $(X,M)$, that
is, the category of convergent log isocrystals on $(X,M)$ equipped with a
Frobenius structure, will be denoted $F\text{-}\mathbf{Isoc}^{\circ}(X,M)$. In
certain cases, the presence of a Frobenius structure is enough to guarantee
local freeness.

######  7.1.6 Lemma.

Let $X$ be a smooth variety, and $D\subset X$ a normal crossings divisor. Then
every convergent log $F$-isocrystal $\mathscr{F}$ on $(X,M(D))$ is locally
free.

###### Proof.

It suffices to prove the claim over some étale cover of $X$, so I can
therefore assume that the pair $(X,D)$ lifts to a pair
$(\mathfrak{X},\mathfrak{D})$ consisting of a smooth formal scheme
$\mathfrak{X}$ over $\mathcal{V}$ and a strict relative normal crossings
divisor $\mathfrak{D}\subset\mathfrak{X}$. In fact, I can assume that
$\mathfrak{X}=\mathrm{Spf}\left(R\right)$ is affine and connected, and that
there exists an étale morphism
$\mathfrak{X}\rightarrow\widehat{{\mathbb{A}}}^{d}_{\mathcal{V}}$, with co-
ordinates $z_{1},\ldots,z_{d}$ on the target, such that $\mathfrak{D}$ is (the
pullback of) the strict normal crossings divisor $V(z_{1}\ldots z_{c})$ for
some $c\leq d$. In particular, the standard lift of Frobenius on
$\widehat{{\mathbb{A}}}^{d}_{\mathcal{V}}$ extends to an endomorphism of
$\mathfrak{X}$ lifting the absolute ($q$-power) Frobenius on $X$. Let
$\sigma:\colon R\rightarrow R$ denote the ring homomorphism induced by the
Frobenius lift. It is then enough to prove the local freeness and of the
realisation of $\mathscr{F}$ on $\mathfrak{X}_{K}$.

The proof of this is a standard fitting ideal argument: it suffices to show
that the only ideals $I\subset R_{K}$ stable under $\sigma$, in the sense that
$\sigma(I)R_{K}=I$, are the zero and unit ideals. Setting $I^{+}=I\cap R$,
this is a $\pi$-adically saturated ideal of $R$, such that
$\sigma(I^{+})R\subset I^{+}$. Since $\sigma$ is finite flat, it follows that
$\sigma(I_{0})R$ is also saturated, hence in fact $\sigma(I^{+})R=I^{+}$.
Again, it suffices to show that the only any such ideals are the zero and unit
ideals. Let $I_{0}=I^{+}\otimes_{\mathcal{V}}k$ be the reduction of $I^{+}$
modulo $\pi$, this is therefore a Frobenius stable ideal in $R_{k}$, in the
sense that $F(I_{0})R_{k}=I_{0}$, and it is enough to show that the only such
ideals are the zero and unit ideals.

But if $F(I_{0})R_{k}=I_{0}$ then $I_{0}=I_{0}^{n}$ for all $n\geq 1$, and so
$I_{0}=\bigcap_{n}I_{0}^{n}$. Since $R_{k}$ is smooth, connected and
Noetherian, it follows from Krull’s intersection theorem [Eis95, Corollary
5.4] that either $I_{0}=\\{0\\}$ or $I_{0}=R$. ∎

###  7.2. A vanishing result in log rigid cohomology

The main result I will need in log rigid cohomology is a certain vanishing
result, appearing as Theorem 7.2.1 below. It will take a little bit of effort
to setup and formulate this result, but this is mostly an issue of keeping
track of notation, the underlying geometry is actually rather simple.

Anyway, Theorem 7.2.1 will apply in the situation when I have a smooth formal
scheme $\mathfrak{P}$, together with a strict relative normal crossings
divisor $\mathfrak{D}\subset\mathfrak{P}$. In this case, I will let
$\mathfrak{P}^{\sharp}$ denote the log scheme
$(\mathfrak{P},M(\mathfrak{D}))$. It’s generic fibre
$\mathfrak{P}_{K}^{\sharp}$ is therefore a logarithmic analytic variety over
$K$, and I will let
$\mathscr{D}_{\mathfrak{P}^{\sharp}_{K}}\subset\mathscr{D}_{\mathfrak{P}_{K}}$
denote the corresponding sheaf of logarithmic (algebraic) differential
operators. For example, if $z_{1},\ldots,z_{d}$ are local co-ordinates on
$\mathfrak{P}$, such that $\mathfrak{D}$ is defined by $z_{1}\ldots z_{c}=0$,
and $\partial_{1},\ldots,\partial_{d}$ are the corresponding derivations, then
$\mathscr{D}_{\mathfrak{P}^{\sharp}_{K}}$ is generated as a
sub-$\mathcal{O}_{\mathfrak{P}_{K}}$-algebra of
$\mathscr{D}_{\mathfrak{P}_{K}}$ by
$z_{1}\partial_{1},\ldots,z_{c}\partial_{c},\partial_{c+1},\ldots,\partial_{d}$.
In particular, any constructible isocrystal on $\mathfrak{P}$ can be viewed a
$\mathscr{D}_{\mathfrak{P}^{\sharp}_{K}}$-module via restriction of scalars
along
$\mathscr{D}_{\mathfrak{P}_{K}^{\sharp}}\rightarrow\mathscr{D}_{\mathfrak{P}_{K}}$.

Let $\mathfrak{D}_{1},\ldots,\mathfrak{D}_{c}$ denote the irreducible
components of $\mathfrak{D}$, with special fibres $D_{1},\ldots,D_{c}$. For
each $J\subset\\{1,\ldots,c\\}$, set

$\displaystyle\mathfrak{D}_{J}=\cap_{j\in J}\mathfrak{D}_{j},\;\;$
$\displaystyle\;\;D_{J}=\mathfrak{D}_{J,k}=\cap_{j\in J}D_{j}$
$\displaystyle\mathfrak{D}^{(J)}=\cup_{j\notin J}\mathfrak{D}_{j},\;\;$
$\displaystyle\;\;D^{(J)}=\mathfrak{D}_{k}^{(J)}=\cup_{j\notin J}D_{j}.$

The log structure $M\mathfrak{D})$ on $\mathfrak{P}$ can be pulled back to
each $\mathfrak{D}_{J}$, giving rise to a log formal scheme
$\mathfrak{D}_{J}^{\sharp}$. Similarly, I let
$\left]D_{J}\right[_{\mathfrak{P}^{\sharp}}$ denote the tube of $D_{J}$
equipped with the log structure obtained by pulling back the log structure
$M(\mathfrak{D}_{K})$ from $\mathfrak{P}_{K}$ of the log structure
$M(\mathfrak{D}_{K})$. I will let
$\mathscr{D}_{\left]D_{J}\right[_{\mathfrak{P}}^{\sharp}}$ denote the ring of
(algebraic) log differential operators on
$\left]D_{J}\right[_{\mathfrak{P}}^{\sharp}$, or in other words the
restriction of $\mathscr{D}_{\mathfrak{P}_{K}^{\sharp}}$ to
$\left]D_{J}\right[_{\mathfrak{P}}$.

There is also a second is natural log structure on $\mathfrak{D}_{J}$, namely
that induced by the strict normal crossings divisor

$\mathfrak{D}_{J}\cap\mathfrak{D}^{(J)}\subset\mathfrak{D}_{J}.$

I will denote the resulting log formal scheme by
$\mathfrak{D}_{J}^{\flat}=(\mathfrak{D}_{J},M(\mathfrak{D}_{J}\cap\mathfrak{D}^{(J)}))$,
and denote the ring of (algebraic) log differential operators on its generic
fibre $\mathfrak{D}_{J,K}^{\flat}$ by
$\mathscr{D}_{\mathfrak{D}_{J,K}^{\flat}}$.

Now, if $\mathscr{F}$ is a locally free log convergent isocrystal on the
special fibre $P^{\sharp}$ of $\mathfrak{P}^{\sharp}$, I can take its
realisation as a locally free $\mathcal{O}_{\mathfrak{P}_{K}}$-module with
integrable log connection, which I will also denote by $\mathscr{F}$. There is
then an induced integrable log connection on the twist
$\mathscr{F}(\mathfrak{D}_{K})$ of $\mathscr{F}$, coming from the canonical
log connection on the line bundle
$\mathcal{O}_{\mathfrak{P}_{K}}(\mathfrak{D}_{K})$. In particular,
$\mathscr{F}(\mathfrak{D}_{K})$ is a
$\mathscr{D}_{\mathfrak{P}_{K}^{\sharp}}$-module, which can then be restricted
to any tube $\left]D_{J}\right[_{\mathfrak{P}}$.

######  7.2.1 Theorem.

Let $\mathscr{F}\in F\text{-}\mathbf{Isoc}^{\circ}(P^{\sharp})$ be a log
convergent $F$-isocrystal on $P^{\sharp}$, and let
$\mathscr{G}\in\mathbf{Isoc}_{\mathrm{cons}}(\mathfrak{P})$ a constructible
isocrystal on $\mathfrak{P}$. Then, for any $J\subset\\{1,\ldots,c\\}$,

${\bf R}{\rm
Hom}_{\mathscr{D}_{\left]D_{J}\right[^{\sharp}_{\mathfrak{P}}}}(\mathscr{F}(\mathfrak{D}_{K}),\mathscr{G})=0.$

######  7.2.2 Remark.

The crucial observation of the proof is that the presence of a Frobenius
structure forces $\mathscr{F}$ to have nilpotent residues along
$\mathfrak{D}_{K}$, which then implies that the residues of the twist
$\mathscr{F}(\mathfrak{D}_{K})$ along $\mathfrak{D}_{K}$ are in fact
_isomorphisms_.

###### Proof.

The question is local on $\mathfrak{P}$, so I can assume that there exists an
étale map $\mathfrak{P}\rightarrow\widehat{{\mathbb{A}}}^{d}_{\mathcal{V}}$,
with co-ordinates $z_{1},\ldots,z_{d}$ on the target, such that
$\mathfrak{D}=V(z_{1}\ldots z_{c})$ for some $c\leq d$, and
$\mathfrak{D}_{J}=V(z_{1},\ldots,z_{b})$ for $b=\\#J\leq c$. The functions
$z_{b+1},\ldots,z_{d}$ induce an étale map
$\mathfrak{D}_{J}\rightarrow\widehat{{\mathbb{A}}}^{d-b}_{\mathcal{V}}$, and
adding in the remaining functions $z_{1},\ldots,z_{b}$ gives an étale map
$\mathfrak{D}_{J}\times_{\mathcal{V}}\widehat{{\mathbb{A}}}^{b}_{\mathcal{V}}\rightarrow\widehat{{\mathbb{A}}}^{d}_{\mathcal{V}}$.
I now let $\mathfrak{P}^{\prime}$ be the fibre product

$\textstyle{\mathfrak{P}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{\mathfrak{D}_{J}\times_{\mathcal{V}}\widehat{{\mathbb{A}}}^{b}_{\mathcal{V}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{{\mathbb{A}}}^{d}_{\mathcal{V}}.}$

The closed immersion $\mathfrak{D}_{J}\rightarrow\mathfrak{P}$ canonically
extends to a closed immersion
$\mathfrak{D}_{J}\rightarrow\mathfrak{P}^{\prime}$ whose composition with the
projection
$\mathfrak{P}^{\prime}\rightarrow\mathfrak{D}_{J}\times_{\mathcal{V}}\widehat{{\mathbb{A}}}^{b}_{\mathcal{V}}$
is the identity on the first factor, and the zero section on the second. This
gives rise to a section of the étale map
$u^{-1}(\mathfrak{D}_{J})\rightarrow\mathfrak{D}_{J}$, so there exists an open
subspace of $\mathfrak{P}^{\prime}$ on which
$u^{-1}(\mathfrak{D}_{J})\overset{\cong}{\longrightarrow}\mathfrak{D}_{J}$.

The upshot of all of this is that after replacing $\mathfrak{P}$ by this open
subspace of $\mathfrak{P}^{\prime}$, I can assume that there exists a smooth
morphism $\mathfrak{P}\rightarrow\mathfrak{D}_{J}$ of which the canonical
inclusion is a section. Hence by the weak fibration theorem, there exists an
isomorphism

$\left]D_{J}\right[_{\mathfrak{P}}\overset{\cong}{\longrightarrow}{\mathbb{D}}^{b}_{\mathfrak{D}_{J,K}}(0;1^{-})$

identifying $\mathfrak{D}_{J,K}\rightarrow\left]D_{J}\right[_{\mathfrak{P}}$
is identified with the zero section of
${\mathbb{D}}^{b}_{\mathfrak{D}_{J,K}}(0;1^{-})$. In fact, this isomorphism
can be upgraded to include log structures as follows. As above, let
$\mathfrak{D}_{J}^{\flat}$ denote the log formal scheme given by
$\mathfrak{D}_{J}$ equipped with the log structure coming from the strict
normal crossings divisor $\mathfrak{D}_{J,K}\cap\mathfrak{D}_{K}^{(J)}$, and
let $\mathfrak{D}_{J,K}^{\flat}$ denote its special fibre. If
$z_{1},\ldots,z_{b}$ are co-ordinates on ${\mathbb{D}}^{b}_{K}(0;1^{-})$, then
${\mathbb{D}}^{b}_{K}(0;1^{-})$ is equipped with the log structure coming from
the strict normal crossings divisor $V(z_{1}\ldots z_{b})$. Then the above
isomorphism can be upgraded to an isomorphism

$\left]D_{J}\right[^{\sharp}_{\mathfrak{P}}\overset{\cong}{\longrightarrow}\mathfrak{D}_{J,K}^{\flat}\times_{K}{\mathbb{D}}^{b}_{K}(0;1^{-})$

of log analytic varieties. Let $\pi$ denote the first projection
$\left]D_{J}\right[_{\mathfrak{P}}^{\sharp}\rightarrow\mathfrak{D}_{J,K}^{\flat}$.

I now want to apply [Shi10, Proposition 2.12] to show that
$\mathscr{F}|_{\left]D_{J}\right[_{\mathfrak{P}}}$ is an iterated extension of
pullbacks of locally free log integrable connections on
$\mathfrak{D}_{J,K}^{\flat}$ along $\pi$, at least after possibly localising
to ensure that $\mathfrak{D}_{J}$ is affine and connected. For Shiho’s result
to apply, I need to check two conditions: firstly, that the supremum norm on
$\Gamma(\mathfrak{D}_{J,K},\mathcal{O}_{\mathfrak{D}_{J,K}})$ is
multiplicative, and secondly, that $\mathscr{F}$ has nilpotent residues along
$\mathfrak{D}_{K}$. The first of these simply follows from the fact that
$\mathfrak{D}_{J}$ is smooth over $\mathcal{V}$, and the second from the fact
that $\mathscr{F}$ has a Frobenius structure.

Therefore, I can assume that there exists a locally free log integrable
connection $\mathscr{F}_{0}$ on $\mathfrak{D}_{J,K}^{\flat}$ such that
$\mathscr{F}|_{\left]D_{J}\right[_{\mathfrak{P}}}=\pi^{*}\mathscr{F}_{0}$. In
a similar vein, it follows from Theorem 3.2.1 there exists a constructible
isocrystal $\mathscr{G}_{0}$ on $\mathfrak{D}_{J}$ such that

$\mathscr{G}|_{\left]D_{J}\right[_{\mathfrak{P}}}=\pi^{*}\mathscr{G}_{0}.$

I now set
$\mathscr{H}_{0}:=\mathscr{G}_{0}\otimes_{\mathcal{O}_{\mathfrak{D}_{J,K}}}\mathscr{F}_{0}^{\vee}(-\mathfrak{D}_{J,K}\cap\mathfrak{D}_{K}^{(J)})$,
which is thus a $\mathscr{D}_{\mathfrak{D}^{\flat}_{J,K}}$-module, and write
$\mathscr{T}$ for the divisor $V(z_{1}\ldots
z_{b})\subset\left]D_{J}\right[_{\mathfrak{P}}$ arising via pullback from
${\mathbb{D}}^{b}_{K}(0;1^{-})$. Now I can calculate

$\displaystyle{\bf R}{\rm
Hom}_{\mathscr{D}_{\left]D_{J}\right[^{\sharp}_{\mathfrak{P}}}}(\mathscr{F}(\mathfrak{D}_{K}),\mathscr{G})$
$\displaystyle={\bf R}{\rm
Hom}_{\mathscr{D}_{\left]D_{J}\right[^{\sharp}_{\mathfrak{P}}}}(\mathcal{O}_{\left]D_{J}\right[_{\mathfrak{P}}}(\mathscr{T}),\pi^{*}\mathscr{H}_{0})$
$\displaystyle={\bf R}{\rm
Hom}_{\mathscr{D}_{\left]D_{J}\right[^{\sharp}_{\mathfrak{P}}}}(\mathcal{O}_{\left]D_{J}\right[_{\mathfrak{P}}},\pi^{*}\mathscr{H}_{0}\otimes_{\mathcal{O}_{\left]D_{J}\right[_{\mathfrak{P}}}}\mathcal{O}_{\left]D_{J}\right[_{\mathfrak{P}}}(-\mathscr{T}))$
$\displaystyle={\bf R}{\rm
Hom}_{\mathfrak{D}_{J,K}^{\flat}}(\mathcal{O}_{\mathfrak{D}_{J,K}},{\bf
R}\pi_{\mathrm{dR}*}(\pi^{*}\mathscr{H}_{0}\otimes_{\mathcal{O}_{\left]D_{J}\right[_{\mathfrak{P}}}}\mathcal{O}_{\left]D_{J}\right[_{\mathfrak{P}}}(-\mathscr{T}))).$

I will show that in fact

${\bf
R}\pi_{\mathrm{dR}*}(\pi^{*}\mathscr{H}_{0}\otimes_{\mathcal{O}_{\left]D_{J}\right[_{\mathfrak{P}}}}\mathcal{O}_{\left]D_{J}\right[_{\mathfrak{P}}}(-\mathscr{T}))=0$

which clearly suffices. To show this, note that since $\mathscr{H}_{0}$ is a
constructible locally free $\mathcal{O}_{\mathfrak{D}_{J,K}}$-module,
Proposition 2.1.6 implies that, after replacing

$\pi\colon\left]D_{J}\right[_{\mathfrak{P}}={\mathbb{D}}^{b}_{\mathfrak{D}_{J,K}}(0;1^{-})\rightarrow\mathfrak{D}_{J,K}$

by the projection

$\pi_{\rho}\colon{\mathbb{D}}^{b}_{\mathfrak{D}_{J,K}}(0;\rho)\rightarrow\mathfrak{D}_{J,K}$

from the closed polydisc of radius $\rho<1$, the projection formula

${\bf
R}\pi_{\rho\mathrm{dR}*}(\pi^{*}\mathscr{H}_{0}\otimes_{\mathcal{O}_{\left]D_{J}\right[_{\mathfrak{P}}}}\mathcal{O}_{\left]D_{J}\right[_{\mathfrak{P}}}(-\mathscr{T}))=\mathscr{H}_{0}\otimes^{{\bf
L}}_{\mathcal{O}_{\mathfrak{D}_{J,K}}}{\bf
R}\pi_{\rho\mathrm{dR}*}\mathcal{O}_{\left]D_{J}\right[_{\mathfrak{P}}}(-\mathscr{T}).$

holds. Since

${\bf
R}\pi_{\mathrm{dR}*}(\pi^{*}\mathscr{H}_{0}\otimes_{\mathcal{O}_{\left]D_{J}\right[_{\mathfrak{P}}}}\mathcal{O}_{\left]D_{J}\right[_{\mathfrak{P}}}(-\mathscr{T}))={\bf
R}\underset{\rho}{\mathrm{lim}}\,{\bf
R}\pi_{\rho\mathrm{dR}*}(\pi^{*}\mathscr{H}_{0}\otimes_{\mathcal{O}_{\left]D_{J}\right[_{\mathfrak{P}}}}\mathcal{O}_{\left]D_{J}\right[_{\mathfrak{P}}}(-\mathfrak{T}_{K})),$

it is enough to show that the pro-system

${\bf
R}\pi_{\rho\mathrm{dR}*}\mathcal{O}_{\left]D_{J}\right[_{\mathfrak{P}}}(-\mathscr{T})$

has zero transition maps. Letting

$\pi_{\rho^{-}}\colon{\mathbb{D}}^{b}_{\mathfrak{D}_{J,K}}(0;\rho^{-})\rightarrow\mathfrak{D}_{J,K}$

denote the projection from the open polydisc, it is then enough to show that

${\bf
R}\pi_{\rho^{-}\mathrm{dR}*}\mathcal{O}_{\left]D_{J}\right[_{\mathfrak{P}}}(-\mathscr{T})=0.$

To see this, I can replace the log structure on $\mathfrak{D}_{J,K}$ by the
trivial one, and the log structure on
${\mathbb{D}}^{b}_{\mathfrak{D}_{J,K}}(0;\rho^{-})$ by that coming from the
strict normal crossings divisor $\mathscr{T}=V(z_{1}\ldots z_{b})$. Replacing
$\mathfrak{D}_{J,K}$ be an arbitrary smooth analytic variety $V$ makes it
possible to argue via induction and reduce to the case $b=1$, that is, the
case of the projection

$\pi_{\rho^{-}}\colon{\mathbb{D}}^{1}_{V}(0;\rho^{-})\rightarrow V.$

for a smooth analytic variety $V$. The target here has the trivial log
structure, and the source the log structure coming from the zero section.
Thus, arguing locally on $V$, and appealing to Theorems A and B for non-
Archimedean quasi-Stein spaces [Kie67, Satz 2.4], I can reduce to calculating
the cohomology of the complex

$zR\left\\{\rho^{-1}z\right\\}\overset{z\partial_{z}}{\longrightarrow}zR\left\\{\rho^{-1}z\right\\}$

where $R$ is an affinoid algebra over $K$, and $R\left\\{\rho^{-1}z\right\\}$
is the ring of functions on the open disc of radius $\rho$ over $R$. It is
then a straightforward calculation that this cohomology vanishes. ∎

###  7.3. Rigidification of log $\boldsymbol{\mathscr{D}^{\dagger}}$-modules

For the logarithmic analogue of the $\mathscr{D}^{\dagger}$-module side of the
picture, I will stick to log formal schemes coming from strict normal
crossings divisors on smooth formal schemes. Thus I will let $\mathfrak{P}$ be
a smooth formal scheme, $\mathfrak{D}\subset\mathfrak{P}$ a strict normal
crossings divisor relative to $\mathcal{V}$, and
$\mathfrak{P}^{\sharp}=(\mathfrak{P},M(\mathfrak{D}))$ the corresponding log
formal scheme. There is therefore the logarithmic analogue
$\mathscr{D}^{\dagger}_{\mathfrak{P}^{\sharp}{\mathbb{Q}}}$ of Berthelot’s
ring of overconvergent differential operators, as defined in [Mon02] and
[Car09a]. As in the non-logarithmic case,

$\mathscr{D}^{\dagger}_{\mathfrak{P}^{\sharp}{\mathbb{Q}}}:=\underset{m}{\mathrm{colim}}\,\widehat{\mathscr{D}}^{(m)}_{\mathfrak{P}^{\sharp}}\otimes_{{\mathbb{Z}}}{\mathbb{Q}}$

where $\widehat{\mathscr{D}}^{(m)}_{\mathfrak{P}^{\sharp}}$ is the $p$-adic
completion of the sheaf of level $m$ differential operators on
$\mathfrak{P}^{\sharp}$. The isogeny category $\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\widehat{\mathscr{D}}^{(\bullet)}_{\mathfrak{P}^{\sharp}})$
of ind-complexes of
$\widehat{\mathscr{D}}^{(\bullet)}_{\mathfrak{P}^{\sharp}}$-modules is defined
exactly as in the non-logarithmic case, and actually taking the colimit
induces an equivalence

$\underrightarrow{{\bf
LD}}^{b}_{{\mathbb{Q}},\mathrm{coh}}(\widehat{\mathscr{D}}^{(\bullet)}_{\mathfrak{P}^{\sharp}})\overset{\cong}{\longrightarrow}{\bf
D}^{b}_{\mathrm{coh}}(\mathscr{D}^{\dagger}_{\mathfrak{P}^{\sharp}{\mathbb{Q}}}),$

see [CT12, §1.2]. The method of §6 works _mutatis mutandis_ in the logarithmic
case to define

$\mathrm{sp}^{!}=\mathrm{sp}^{!}_{\mathfrak{P}^{\sharp}}={\bf
L}\hat{\mathrm{sp}}_{\mathfrak{P}^{\sharp}}^{*}[-\dim\mathfrak{P}]\colon\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\widehat{\mathscr{D}}^{(\bullet)}_{\mathfrak{P}^{\sharp}})\rightarrow{\bf
D}(\mathscr{D}_{\mathfrak{P}_{K}^{\sharp}}),$

where $\mathfrak{P}_{K}^{\sharp}$ is the generic fibre of
$\mathfrak{P}^{\sharp}$, considered as a log analytic variety. It follows from
the construction that $\mathrm{sp}^{!}$ commutes with pullback along the
morphism $\mathfrak{P}^{\sharp}\rightarrow\mathfrak{P}$. The approach of §6
can also be followed through in the logarithmic case to prove the following.

######  7.3.1 Proposition.

For any $\mathcal{M}\in\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\widehat{\mathscr{D}}^{(\bullet)}_{\mathfrak{P}^{\sharp}})$,
${\bf L}\hat{\mathrm{sp}}^{*}_{\mathfrak{P}^{\sharp}}$ induces a base change
isomorphism

${\bf R}\Gamma_{{\rm
log}\text{-}\mathrm{dR}}(\mathfrak{P}^{\sharp},\mathcal{M})\overset{\cong}{\longrightarrow}{\bf
R}\Gamma_{{\rm log}\text{-}\mathrm{dR}}(\mathfrak{P}_{K}^{\sharp},{\bf
L}\hat{\mathrm{sp}}_{\mathfrak{P}}^{*}\mathcal{M})$

which is natural in $\mathcal{M}$.

######  7.3.2 Remark.

The definition of ${\bf R}\Gamma_{{\rm
log}\text{-}\mathrm{dR}}(\mathfrak{P}^{\sharp},\mathcal{M})$ is analogous to
the non-logarithmic case, namely if
$\mathcal{M}=\\{\mathcal{M}^{(m)}\\}_{m\in{\mathbb{N}}}$ then

${\bf
R}\Gamma_{\log\text{-}\mathrm{dR}}(\mathfrak{P}^{\sharp},\mathcal{M})=\underset{m}{\mathrm{colim}}\,{\bf
R}\Gamma_{\log\text{-}\mathrm{dR}}(\mathfrak{P}^{\sharp},\mathcal{M}^{(m)})\otimes_{{\mathbb{Z}}}{\mathbb{Q}}.$

In will also need a very rudimentary logarithmic version of the specialisation
functors considered in §4.3. Suppose, then, that $\mathscr{F}$ is a locally
free convergent log isocrystal on the special fibre $P^{\sharp}$ of
$\mathfrak{P}^{\sharp}$, viewed as a module with integrable logarithmic
connection on $\mathfrak{P}_{K}^{\sharp}$. Then
$\mathrm{sp}_{\mathfrak{P}*}\mathscr{F}={\bf
R}\mathrm{sp}_{\mathfrak{P}*}\mathscr{F}$ is naturally an
$\mathcal{O}_{\mathfrak{P}{\mathbb{Q}}}$-coherent
$\mathscr{D}^{\dagger}_{\mathfrak{P}^{\sharp}{\mathbb{Q}}}$-module, and is
moreover coherent as an
$\mathscr{D}^{\dagger}_{\mathfrak{P}^{\sharp}{\mathbb{Q}}}$-module by [Car09a,
Théorème 4.15]. As in the non-logarithmic case, it is straightforward to
verify that there is a canonical isomorphism

${\bf
L}\hat{\mathrm{sp}}^{*}_{\mathfrak{P}^{\sharp}}\mathrm{sp}_{\mathfrak{P}*}\mathscr{F}\overset{\cong}{\longrightarrow}\mathscr{F}$

of $\mathscr{D}_{\mathfrak{P}_{K}^{\sharp}}$-modules.

##  8\. The overconvergent Riemann–Hilbert correspondence

I can now finally prove the main result of this article.

######  8.0.1 Theorem.

Let $\mathfrak{P}$ be a smooth formal scheme. Then the functor

$\mathrm{sp}^{!}\colon{\bf
D}^{b}_{\mathrm{hol},F}(\mathfrak{P})\rightarrow{\bf
D}^{b}_{\mathrm{cons},F}(\mathfrak{P})$

is an equivalence of categories.

As in §3.2, this reduces, via dévissage, to the following derived full
faithfulness assertion.

######  8.0.2 Theorem.

Let $\mathfrak{P}$ be a smooth formal scheme, $j\colon X\rightarrow P$ a
locally closed subscheme, smooth over $k$, with closure $Y$ in $P$. Suppose
$\mathscr{F}\in F\text{-}\mathbf{Isoc}(X,Y)$, and set
$\mathcal{M}:=\mathrm{sp}_{X!}\mathscr{F}\in\mathbf{DCon}_{F}(\mathfrak{P})$.
Then, for any $\mathcal{N}\in\mathbf{DCon}_{F}(\mathfrak{P})$, the map

${\bf
R}\mathrm{Hom}_{\mathscr{D}^{\dagger}_{\mathfrak{P}{\mathbb{Q}}}}(\mathcal{M},\mathcal{N})\rightarrow{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}_{K}}}(\mathrm{sp}^{!}\mathcal{M},\mathrm{sp}^{!}\mathcal{N})$

is an isomorphism in ${\bf D}(K)$.

###### Proof.

To begin with, the question is local on $\mathfrak{P}$, which I can therefore
assume to be affine. As in the proof of Theorem 6.3.1, one consequence of this
is that every formal scheme appearing in the proof will admit a locally closed
embedding into a smooth and proper formal scheme.

The fact that $\mathcal{M}$ is supported on $Y$, together with compatibility
of $\mathrm{sp}^{!}$ with the natural morphism

${\bf R}\underline{\Gamma}_{Y}^{\dagger}\rightarrow\mathrm{id}$

means that there is a commutative diagram

$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}^{\dagger}_{\mathfrak{P}{\mathbb{Q}}}}(\mathcal{M},\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}_{K}}}(\mathrm{sp}^{!}\mathcal{M},\mathrm{sp}^{!}\mathcal{N})}$$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}_{K}}}(\mathrm{sp}^{!}\mathcal{M},{\bf
R}\underline{\Gamma}_{Y}^{\dagger}\mathrm{sp}^{!}\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}^{\dagger}_{\mathfrak{P}{\mathbb{Q}}}}(\mathcal{M},{\bf
R}\underline{\Gamma}_{Y}^{\dagger}\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}_{K}}}(\mathrm{sp}^{!}\mathcal{M},\mathrm{sp}^{!}{\bf
R}\underline{\Gamma}_{Y}^{\dagger}\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\cong}$

Hence, replacing $\mathcal{N}$ with $\underline{{\bf
R}\Gamma}^{\dagger}\mathcal{N}$, I can assume that $\mathcal{N}$ is supported
on $Y$.

Now, the semistable reduction theorem for $F$-isocrystals [Ked11, Theorem
2.4.4] shows that there exists a morphism of pairs
$(f,g)\colon(\widetilde{X},\widetilde{Y})\rightarrow(X,Y)$ such that:

  * •

$\widetilde{Y}$ is smooth;

  * •

$g$ is projective and generically étale;

  * •

$\widetilde{X}=g^{-1}(X)$;

  * •

$\widetilde{D}:=\widetilde{Y}\setminus\widetilde{X}$ is a strict normal
crossings divisor in $\widetilde{Y}$;

  * •

$f^{*}\mathscr{F}\in F\text{-}\mathbf{Isoc}(\widetilde{X},\widetilde{Y})$
extends to a convergent log $F$-isocrystal on the log smooth log scheme
$(\widetilde{Y},M(\widetilde{D}))$.

Since $g$ is projective, I can extend $(f,g)$ to a morphism of frames

$\textstyle{\widetilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\widetilde{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\widetilde{\mathfrak{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{P}}$

where $u$ is smooth and projective. By Noetherian induction on $X$, I am free
to replace $X$ with an open subscheme, hence I can assume moreover that $f$ is
finite étale.

Now, by Proposition 4.5.4, $\mathcal{N}\in{\bf
D}^{b}_{\mathrm{hol},F}(X,Y,\mathfrak{P})\subset{\bf
D}^{b}_{\mathrm{hol},F}(\mathfrak{P})$ is a direct summand of
$f_{+}f^{!}\mathcal{N}$, so I can replace $\mathcal{N}$ by
$f_{+}f^{!}\mathcal{N}$. Then, by Proposition 6.4.4, combined with Proposition
6.2.3 and the preceding discussion, there is a commutative diagram

$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}^{\dagger}_{\mathfrak{P}{\mathbb{Q}}}}(\mathcal{M},f_{+}f^{!}\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}_{K}}}(\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{M},\mathrm{sp}^{!}_{\mathfrak{P}}f_{+}f^{!}\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}_{K}}}(\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{M},{\bf
R}\left]f\right[_{\mathrm{dR}*}\mathrm{sp}^{!}_{\widetilde{\mathfrak{P}}}f^{!}\mathcal{N})}$$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}^{\dagger}_{\widetilde{\mathfrak{P}}{\mathbb{Q}}}}(f^{!}\mathcal{M},f^{!}\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\widetilde{\mathfrak{P}}_{K}}}(\mathrm{sp}^{!}_{\widetilde{\mathfrak{P}}}f^{!}\mathcal{M},\mathrm{sp}^{!}_{\widetilde{\mathfrak{P}}}f^{!}\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\widetilde{\mathfrak{P}}_{K}}}(f^{*}\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{M},\mathrm{sp}^{!}_{\widetilde{\mathfrak{P}}}f^{!}\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\cong}$

I can therefore replace $(X,Y,\mathfrak{P})$ by
$(\widetilde{X},\widetilde{Y},\widetilde{\mathfrak{P}})$, in other words, I
can assume that $Y$ is smooth, that $D:=Y\setminus X$ is a strict normal
crossings divisor, and that $\mathscr{F}$ extends to a convergent log$F$\-
isocrystal $\mathscr{F}^{\sharp}$ on the log scheme $Y^{\sharp}:=(Y,M(D))$.

Further localising on $\mathfrak{P}$, I can assume that $Y\hookrightarrow P$
lifts to a closed immersion $i\colon\mathfrak{Y}\rightarrow\mathfrak{P}$ of
smooth formal schemes. Set
$\mathfrak{Q}=\mathfrak{P}\times_{\mathcal{V}}\mathfrak{Y}$, and let
$\pi\colon\mathfrak{Q}\rightarrow\mathfrak{P}$ denote the projection, thus
$\mathcal{M}\cong\pi_{+}\underline{{\bf
R}\Gamma}^{\dagger}_{Y}\pi^{!}\mathcal{M}$ and similarly for $\mathcal{N}$.
Since $\mathrm{sp}^{!}$ is compatible with de Rham pushforward, by Proposition
6.2.3, there is a commutative diagram

$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}^{\dagger}_{\mathfrak{P}{\mathbb{Q}}}}(\mathcal{M},\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}_{K}}}(\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{M},\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{N})}$$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}^{\dagger}_{\mathfrak{Q}{\mathbb{Q}}}}(\underline{{\bf
R}\Gamma}^{\dagger}_{Y}\pi^{!}\mathcal{M},\underline{{\bf
R}\Gamma}^{\dagger}_{Y}\pi^{!}\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{Q}_{K}}}(\mathrm{sp}^{!}_{\mathfrak{Q}}\underline{{\bf
R}\Gamma}^{\dagger}_{Y}\pi^{!}\mathcal{M},\mathrm{sp}^{!}_{\mathfrak{Q}}\underline{{\bf
R}\Gamma}^{\dagger}_{Y}\pi^{!}\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$

where the vertical maps are isomorphisms by Proposition 4.2.3 and Theorem
3.2.1 respectively. Hence I can replace $\mathfrak{P}$ by $\mathfrak{Q}$, and
by the same argument I can then replace $\mathfrak{Q}$ by $\mathfrak{Y}$. In
other words, I can assume that $Y=P$.

With one final localisation on $\mathfrak{P}$ if necessary, I can lift
$D\subset P$ to a strict normal crossings divisor
$\mathfrak{D}\subset\mathfrak{P}$. I now let
$\mathfrak{P}^{\sharp}:=(\mathfrak{P},M(\mathfrak{D}))$ denote the
corresponding formal log-scheme, and set $d=\dim\mathfrak{P}$. Note that in
this case $\mathscr{F}$ is a locally free
$\mathcal{O}_{\left]X\right[_{\mathfrak{P}}}$-module with connection, and
$\mathcal{M}=\mathrm{sp}_{\mathfrak{P}*}j_{*}\mathscr{F}[d]$, where $j\colon
X\rightarrow P$ is the given open immersion. The extension
$\mathscr{F}^{\sharp}$ of $\mathscr{F}$ is a coherent
$\mathcal{O}_{\mathfrak{P}_{K}}$-module with integrable log connection, and I
set $\mathcal{M}^{\sharp}:=\mathrm{sp}_{\mathfrak{P}*}\mathscr{F}^{\sharp}[d]$
to be the shift of the $\mathcal{O}_{\mathfrak{P}{\mathbb{Q}}}$-coherent
$\mathscr{D}^{\dagger}_{\mathfrak{P}^{\sharp}{\mathbb{Q}}}$-module associated
to $\mathscr{F}^{\sharp}$. Thanks to Lemma 7.1.6, $\mathscr{F}^{\sharp}$ is
locally free. Hence $\mathcal{M}^{\sharp}[-d]$ is a locally projective
$\mathcal{O}_{\mathfrak{P}{\mathbb{Q}}}$-module.

The natural morphism

$\mathscr{F}^{\sharp}(\mathfrak{D}_{K})\rightarrow j_{*}\mathscr{F}$

of $\mathscr{D}_{\mathfrak{P}^{\sharp}_{K}}$-modules induces a morphism

$\mathcal{M}^{\sharp}(\mathfrak{D})\rightarrow\mathcal{M}$

of $\mathscr{D}^{\dagger}_{\mathfrak{P}^{\sharp}{\mathbb{Q}}}$-modules, which
in turn induces an _isomorphism_

$\mathscr{D}^{\dagger}_{\mathfrak{P}{\mathbb{Q}}}\otimes^{{\bf
L}}_{\mathscr{D}^{\dagger}_{\mathfrak{P}^{\sharp}{\mathbb{Q}}}}\mathcal{M}^{\sharp}(\mathfrak{D})\rightarrow\mathcal{M}$

of $\mathscr{D}^{\dagger}_{\mathfrak{P}{\mathbb{Q}}}$-modules by [CT12,
Theorem 2.2.9]. This gives rise to the following commutative diagram

$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}^{\dagger}_{\mathfrak{P}{\mathbb{Q}}}}(\mathcal{M},\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}_{K}}}(\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{M},\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}_{K}}}(j_{*}\mathscr{F},\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}^{\dagger}_{\mathfrak{P}^{\sharp}{\mathbb{Q}}}}(\mathcal{M}^{\sharp}(\mathfrak{D}),\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}^{\sharp}_{K}}}(\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{M}^{\sharp}(\mathfrak{D}),\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}^{\sharp}_{K}}}(\mathscr{F}^{\sharp}(\mathfrak{D}_{K}),\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{N}).}$

It therefore suffices to show that the two maps

$\displaystyle{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}_{K}}}(j_{*}\mathscr{F},\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{N})$
$\displaystyle\rightarrow{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}^{\sharp}_{K}}}(\mathscr{F}^{\sharp}(\mathfrak{D}_{K}),\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{N})$
$\displaystyle{\bf
R}\mathrm{Hom}_{\mathscr{D}^{\dagger}_{\mathfrak{P}^{\sharp}{\mathbb{Q}}}}(\mathcal{M}^{\sharp}(\mathfrak{D}),\mathcal{N})$
$\displaystyle\rightarrow{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}^{\sharp}_{K}}}(\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{M}^{\sharp}(\mathfrak{D}),\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{N})$

are isomorphisms. The second is a relatively straightforward consequence of
the fact that the shift $\mathcal{M}^{\sharp}(\mathfrak{D})[-d]$ is a locally
projective $\mathcal{O}_{\mathfrak{P}{\mathbb{Q}}}$-module. Indeed, letting
$\mathcal{M}^{*}$ denote the $\mathcal{O}_{\mathfrak{P}{\mathbb{Q}}}$-dual of
$\mathcal{M}^{\sharp}(\mathfrak{D})$, then $\mathcal{M}^{*}[d]$ is a locally
projective $\mathcal{O}_{\mathfrak{P}{\mathbb{Q}}}$-module, and there is a
commutative diagram

$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}^{\dagger}_{\mathfrak{P}^{\sharp}{\mathbb{Q}}}}(\mathcal{M}^{\sharp}(\mathfrak{D}),\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}^{\sharp}_{K}}}(\mathrm{sp}_{\mathfrak{P}}^{!}\mathcal{M}^{\sharp}(\mathfrak{D}),\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}^{\dagger}_{\mathfrak{P}^{\sharp}{\mathbb{Q}}}}(\mathcal{O}_{\mathfrak{P}{\mathbb{Q}}},\mathcal{N}\otimes_{\mathcal{O}_{\mathfrak{P}{\mathbb{Q}}}}\mathcal{M}^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}^{\sharp}_{K}}}(\mathcal{O}_{\mathfrak{P}_{K}},\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{N}\otimes_{\mathcal{O}_{\mathfrak{P}_{K}}}\mathrm{sp}_{\mathfrak{P}}^{!}\mathcal{M}^{*}).}$

The bottom arrow is an isomorphism by Proposition 7.3.1, and therefore the top
arrow is also an isomorphism.

Finally, to see that

${\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}_{K}}}(j_{*}\mathscr{F},\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{N})\rightarrow{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}^{\sharp}_{K}}}(\mathscr{F}^{\sharp}(\mathfrak{D}_{K}),\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{N})$

is an isomorphism, I will (after some set-up) apply Theorem 7.2.1. First, note
that the support $\left]X\right[_{\mathfrak{P}}$ of $j_{*}\mathscr{F}$ is
disjoint from the divisor $\mathfrak{D}_{K}$. It therefore follows that the
‘restriction’ map

${\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}_{K}}}(j_{*}\mathscr{F},\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{N})\rightarrow{\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}^{\sharp}_{K}}}(j_{*}\mathscr{F},\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{N})$

is an isomorphism. Now write $i:D\rightarrow P$ for the natural inclusion,
thus the localisation exact sequence

$0\rightarrow
i_{!}i^{*}\mathscr{F}^{\sharp}(\mathfrak{D}_{K})\rightarrow\mathscr{F}^{\sharp}(\mathfrak{D}_{K})\rightarrow
j_{*}\mathscr{F}\rightarrow 0$

reduces to proving that

${\bf
R}\mathrm{Hom}_{\mathscr{D}_{\mathfrak{P}^{\sharp}_{K}}}(i_{!}i^{*}\mathscr{F}^{\sharp}(\mathfrak{D}_{K}),\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{N})={\bf
R}\mathrm{Hom}_{\mathscr{D}_{\left]D\right[^{\sharp}_{\mathfrak{P}}}}(\mathscr{F}^{\sharp}(\mathfrak{D}_{K}),\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{N})=0,$

where I have written $\left]D\right[^{\sharp}_{\mathfrak{P}}$ for
$\left]D\right[_{\mathfrak{P}}$ equipped with the log structure given by
pulling back the given log structure on $\mathfrak{P}_{K}^{\sharp}$. Let
$\mathfrak{D}_{1},\ldots,\mathfrak{D}_{n}$ denote the irreducible components
of $\mathfrak{D}$, with special fibres $D_{1},\ldots,D_{n}$. For each
$J\subset\\{1,\ldots,n\\}$ set

$\displaystyle\mathfrak{D}_{J}=\cap_{j\in J}\mathfrak{D}_{j},\;\;$
$\displaystyle\;\;D_{J}=\mathfrak{D}_{J,k}=\cap_{j\in J}D_{j}$
$\displaystyle\mathfrak{D}^{(J)}=\cup_{j\notin J}\mathfrak{D}_{j},\;\;$
$\displaystyle\;\;D^{(J)}=\mathfrak{D}_{k}^{(J)}=\cup_{j\notin J}D_{j}.$

Using the open cover of $\left]D\right[_{\mathfrak{P}}$ given by the various
tubes $\left]D_{j}\right[_{\mathfrak{P}}$, I can therefore reduce to showing
that

${\bf
R}\mathrm{Hom}_{\mathscr{D}_{\left]D_{J}\right[^{\sharp}_{\mathfrak{P}}}}(\mathscr{F}^{\sharp}(\mathfrak{D}_{K}),\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{N})=0$

for each $J\subset\\{1,\ldots,n\\}$. Theorem 6.3.1 says that
$\mathrm{sp}^{!}_{\mathfrak{P}}\mathcal{N}$ is a constructible isocrystal on
$\mathfrak{P}$, and $\mathscr{F}^{\sharp}$ is a log convergent $F$-isocrystal
on $P^{\sharp}$ by assumption. Hence the required vanishing is precisely the
content of Theorem 7.2.1. ∎

Combining this with Proposition 6.2.3, I obtain the following.

######  8.0.3 Corollary.

Let $u\colon\mathfrak{P}\rightarrow\mathfrak{Q}$ be a smooth and proper
morphism of smooth formal schemes. Then the functor ${\bf R}u_{\mathrm{dR}*}$
maps ${\bf D}^{b}_{\mathrm{cons},F}(\mathfrak{P})$ into ${\bf
D}^{b}_{\mathrm{cons},F}(\mathfrak{Q})$.

##  9\. Cohomological operations for constructible isocrystals

I can now show compatibility of $\mathrm{sp}^{!}$ with the various
cohomological functors for $\mathscr{D}^{\dagger}$-modules and constructible
isocrystals. This will then enable me to deduce an analogue of Theorem 8.0.1
for pairs and for varieties.

###  9.1. The case of formal schemes

I have already proved the compatibility of $\mathrm{sp}^{!}$ with de Rham
pushfoward along smooth morphisms $u\colon\mathfrak{P}\rightarrow\mathfrak{Q}$
of smooth formal schemes in Proposition 6.2.3. Similarly, if $\mathfrak{P}$ is
a smooth formal scheme, and $i\colon X\rightarrow P$ is a locally closed
immersion, I have already shown in §6.4 that the diagram

$\textstyle{{\bf
D}^{b}_{\mathrm{hol},F}(\mathfrak{P})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{sp}_{\mathfrak{P}}^{!}}$$\scriptstyle{\underline{{\bf
R}\Gamma}^{\dagger}_{X}}$$\textstyle{{\bf
D}^{b}_{\mathrm{cons},F}(\mathfrak{P})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{{\bf
R}\Gamma}^{\dagger}_{X}=i_{!}i^{-1}}$$\textstyle{{\bf
D}^{b}_{\mathrm{hol},F}(\mathfrak{P})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{sp}_{\mathfrak{P}}^{!}}$$\textstyle{{\bf
D}^{b}_{\mathrm{cons},F}(\mathfrak{P}).}$

commutes up to natural isomorphism, and that this is moreover compatible with
the natural ‘adjunctions’ when $X$ is either open or closed in $P$.

There are similar results for pullbacks and tensor products. For the case of
pullbacks, suppose that $u\colon\mathfrak{P}\rightarrow\mathfrak{Q}$ is a (not
necessarily smooth) morphism of smooth formal schemes, and that
$\mathcal{M}\in{\bf D}^{b}_{\mathrm{hol},F}(\mathfrak{Q})$. Then I can
construct a natural (in $\mathcal{M}$) morphism

$\psi_{u}\colon
u^{*}\mathrm{sp}_{\mathfrak{Q}}^{!}\mathcal{M}\rightarrow\mathrm{sp}_{\mathfrak{P}}^{!}u^{!}\mathcal{M}$

as follows. I first use the usual trick of taking the graph two divide into
two separate cases: the first when $u$ is smooth, and the second when $u$ is a
section of a smooth morphism $\pi\colon\mathfrak{Q}\rightarrow\mathfrak{P}$.

In the first case, when $u$ is smooth, of relative dimension $d$, say, giving
a morphism

$u^{*}\mathrm{sp}_{\mathfrak{Q}}^{!}\mathcal{M}\rightarrow\mathrm{sp}_{\mathfrak{P}}^{!}u^{!}\mathcal{M}$

is equivalent to giving a morphism

$\mathrm{sp}_{\mathfrak{Q}}^{!}\mathcal{M}\rightarrow{\bf
R}u_{\mathrm{dR}*}\mathrm{sp}_{\mathfrak{P}}^{!}u^{!}\mathcal{M}=\mathrm{sp}_{\mathfrak{Q}}^{!}u_{+}u^{!}\mathcal{M}[-2d].$

Since I am not assuming that $u$ is proper, the functor $u_{+}$ appearing on
the RHS should be viewed as taking values in $\underrightarrow{{\bf
LD}}^{b}_{{\mathbb{Q}},\mathrm{qc}}(\widehat{\mathscr{D}}^{(\bullet)}_{\mathfrak{Q}})$
It therefore suffices to construct a morphism

$\mathcal{M}\rightarrow u_{+}u^{!}\mathcal{M}[-2d]$

in $\underrightarrow{{\bf
LD}}^{b}_{{\mathbb{Q}},\mathrm{qc}}(\widehat{\mathscr{D}}^{(\bullet)}_{\mathfrak{Q}})$.
Via the projection formula (which in the generality required here follows from
[Car04, Proposition 1.2.27] as in [Car04, §2.1.4]) it suffices to consider the
case $\mathcal{M}=\mathcal{O}_{\mathfrak{Q}{\mathbb{Q}}}$. In this case
$u^{!}\mathcal{M}=\mathcal{O}_{\mathfrak{P}{\mathbb{Q}}}[d]$ and the morphism
I seek is one of the form

$\mathcal{O}_{\mathfrak{Q}{\mathbb{Q}}}\rightarrow
u_{+}\mathcal{O}_{\mathfrak{P}{\mathbb{Q}}}[-d].$

After actually taking the colimit in the level $m$, and tensoring with
${\mathbb{Q}}$ to land inside ${\bf
D}(\mathscr{D}^{\dagger}_{\mathfrak{Q}{\mathbb{Q}}})$, this will simply the
morphism

$\mathcal{O}_{\mathfrak{Q}{\mathbb{Q}}}\rightarrow{\bf
R}u_{\mathrm{dR}*}\mathcal{O}_{\mathfrak{P}{\mathbb{Q}}}$

into the zeroeth relative de Rham cohomology group of
$\mathcal{O}_{\mathfrak{P}{\mathbb{Q}}}$. To see that this morphism lifts to
$\underrightarrow{{\bf
LD}}^{b}_{{\mathbb{Q}},\mathrm{qc}}(\widehat{\mathscr{D}}^{(\bullet)}_{\mathfrak{Q}})$
simply amounts to making the same construction levelwise in $m$.

In the second case, suppose that $u$ is a section of a smooth morphism
$\pi\colon\mathfrak{Q}\rightarrow\mathfrak{P}$, of relative dimension $d$,
say. Then I can calculate

$\displaystyle\mathrm{sp}_{\mathfrak{P}}^{!}u^{!}\mathcal{M}$
$\displaystyle=\mathrm{sp}_{\mathfrak{P}}^{!}\pi_{+}u_{+}u^{!}\mathcal{M}$
$\displaystyle=\mathrm{sp}_{\mathfrak{P}}^{!}\pi_{+}\underline{{\bf
R}\Gamma}^{\dagger}_{P}\mathcal{M}$ $\displaystyle={\bf
R}\pi_{\mathrm{dR}*}\mathrm{sp}_{\mathfrak{Q}}^{!}\underline{{\bf
R}\Gamma}^{\dagger}_{P}\mathcal{M}[2d]$ $\displaystyle={\bf
R}\pi_{\mathrm{dR}*}\underline{{\bf
R}\Gamma}^{\dagger}_{P}\mathrm{sp}_{\mathfrak{Q}}^{!}\mathcal{M}[2d]$

by compatibility of $\mathrm{sp}^{!}$ with both de Rham pushfowards and the
functor $\underline{{\bf R}\Gamma}^{\dagger}_{P}$ of sections with
overconvergent support. It therefore suffices to construct a morphism

$u^{*}\rightarrow{\bf R}\pi_{\mathrm{dR}*}\underline{{\bf
R}\Gamma}^{\dagger}_{P}[2d]$

of functors from ${\bf D}^{b}_{\mathrm{cons},F}(\mathfrak{Q})$ to ${\bf
D}^{b}_{\mathrm{cons},F}(\mathfrak{P})$. Let

$\left]\rm
id\right[_{u}\colon\mathfrak{P}_{K}\rightarrow\left]P\right[_{\mathfrak{Q}},\;\;\;\;\left]u\right[_{\rm
id}\colon\left]P\right[_{\mathfrak{Q}}\rightarrow\mathfrak{Q}_{K}$

denote the natural inclusions, and

$\left]\rm
id\right[_{\pi}\colon\left]P\right[_{\mathfrak{Q}}\rightarrow\mathfrak{P}_{K}$

the restriction of $\pi$. Thus $\underline{{\bf
R}\Gamma}^{\dagger}_{P}=\left]u\right[_{\rm id!}\left]u\right[^{*}_{\rm id}$.

######  9.1.1 Lemma.

The ‘forget supports’ map ${\bf R}\left]\rm id\right[_{\pi!}\rightarrow{\bf
R}\pi_{*}\circ\left]u\right[_{\rm id!}$ is an isomorphism.

###### Proof.

Explicitly, if $\mathscr{I}$ is an injective sheaf on
$\left]P\right[_{\mathfrak{Q}}$, then $\left]\rm id\right[_{\pi!}\mathscr{I}$
consists of sections of $\mathscr{I}$ with compact support over
$\mathfrak{P}_{K}$, and $\pi_{*}\circ\left]u\right[_{\rm id!}\mathscr{I}$
consists of sections with compact support over $\mathfrak{Q}_{K}$. To show
that $\left]\rm id\right[_{\pi!}\mathscr{I}=\pi_{*}\circ\left]u\right[_{\rm
id!}\mathscr{I}$ therefore amounts to showing that if
$V\subset\mathfrak{P}_{K}$ is open, and $S\subset\left]\rm
id\right[_{\pi}^{-1}(V)$ is a closed subset, then $S\rightarrow V$ is proper
if and only if $S\rightarrow\pi^{-1}(V)$ is proper. Since $S\rightarrow V$ is
always partially proper, and $\pi^{-1}(V)\rightarrow V$ is quasi-compact, this
is clear.

To complete the proof, then, I need to show that $\left]u\right[_{\rm
id!}\mathscr{I}$ is a flasque sheaf on $\mathfrak{Q}_{K}$. But since
$\left]u\right[_{\rm id!}$ is partially proper, this was demonstrated in the
proof of [AL20, Corollary 3.4.7] (note that the additional hypothesis there
that the second morphism $g$ is partially proper is not used to show that
$f_{!}\mathscr{I}$ is flasque). ∎

I am therefore looking to construct a morphism

$u^{*}\rightarrow{\bf R}\left]\rm
id\right[_{\pi\mathrm{dR}!}\left]u\right[^{*}_{\rm id}[2d],$

and in fact an isomorphism between these two functors is provided by Remark
3.2.5. This therefore gives rise to an _isomorphism_

$u^{*}\mathrm{sp}_{\mathfrak{Q}}^{!}\mathcal{M}\overset{\cong}{\longrightarrow}\mathrm{sp}_{\mathfrak{P}}^{!}u^{!}\mathcal{M},$

which is natural in $\mathcal{M}$.

######  9.1.2 Proposition.

Let $u\colon\mathfrak{P}\rightarrow\mathfrak{Q}$ be a morphism of smooth
formal schemes, and $\mathcal{M}\in{\bf
D}^{b}_{\mathrm{hol},F}(\mathfrak{Q})$. Then the morphism

$\psi_{u}\colon
u^{*}\mathrm{sp}_{\mathfrak{Q}}^{!}\mathcal{M}\rightarrow\mathrm{sp}_{\mathfrak{P}}^{!}u^{!}\mathcal{M}$

is an isomorphism in ${\bf D}^{b}_{\mathrm{hol},F}(\mathfrak{P})$.

###### Proof.

To show that $\psi_{u}$ is an isomorphism, I can check that it is so after
taking the pullback along any closed immersion
$\mathrm{Spf}\left(\mathcal{V}^{\prime}\right)\rightarrow\mathfrak{P}$ where
$\mathcal{V}^{\prime}$ is the ring of integers in a finite extension of $K$.
After possibly making a finite base change (for which the result can be easily
checked), I can in fact take such closed immersions with
$\mathcal{V}^{\prime}=\mathcal{V}$. Thus the general case reduces to the
special case when $u$ is a section of a smooth morphism, in which case the
given morphism which has already been shown to be an isomorphism. ∎

To show compatibility of $\mathrm{sp}^{!}$ with tensor product, recall that in
§5.2 I constructed, for $\mathcal{M},\mathcal{N}\in\underrightarrow{{\bf
LD}}_{{\mathbb{Q}},\mathrm{qc}}(\widehat{\mathscr{D}}^{(\bullet)}_{\mathfrak{P}})$,
a morphism

$\mathbf{L}\hat{\mathrm{sp}}_{\mathfrak{P}}^{*}\mathcal{M}\otimes_{\mathcal{O}_{\mathfrak{P}_{K}}}^{\mathbf{L}}\mathbf{L}\hat{\mathrm{sp}}_{\mathfrak{P}}^{*}\mathcal{N}\rightarrow\mathbf{L}\hat{\mathrm{sp}}_{\mathfrak{P}}^{*}(\mathcal{M}\widehat{\otimes}_{\mathcal{O}_{\mathfrak{P}}}^{\mathbf{L}}\mathcal{N})$

in ${\bf D}(\mathcal{O}_{\mathfrak{P}_{K}})$. It is straightforward, though
rather tedious, to upgrade this to a morphism in ${\bf
D}(\mathscr{D}_{\mathfrak{P}_{K}})$. For $\mathcal{M},\mathcal{N}\in{\bf
D}^{b}_{\mathrm{hol},F}(\mathfrak{P})$, I can therefore view it as a morphism

(9.1.1)
$\mathrm{sp}^{!}\mathcal{M}\otimes_{\mathcal{O}_{\mathfrak{P}_{K}}}\mathrm{sp}_{\mathfrak{P}}^{!}\mathcal{N}\rightarrow\mathrm{sp}^{!}(\mathcal{M}\widetilde{\otimes}_{\mathcal{O}_{\mathfrak{P}}}\mathcal{N})$

in ${\bf D}^{b}_{\mathrm{cons},F}(\mathfrak{P})$.

######  9.1.3 Corollary.

The morphism (9.1.1) is an isomorphism.

###### Proof.

Again, this can be checked on stalks, in which case it follows from
Proposition 9.1.2 together with the fact that both pullback functors $u^{!}$
and $u^{*}$ commute with the relevant tensor products. ∎

###  9.2. The case of pairs and varieties

In Corollary 8.0.3, the fact that $u$ is proper implies that ${\bf
R}u_{\mathrm{dR}*}={\bf R}u_{\mathrm{dR}!}$, so I could have equally well
phrased the result in terms of compactly supported de Rham cohomology. It
turns out that this version is the one that generalises to the case of pairs.
In fact, I can use the extra flexibility of constructible isocrystals to
remove the need for the formal schemes involved to be immersible in smooth and
proper ones. To obtain a functor that is independent of the frame, I also need
to include a shift.

######  9.2.1 Corollary.

Let

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\mathfrak{P}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{P}}$

be a morphism of frames, such that $g$ is proper, $\mathfrak{P}$ is smooth
around $X$, and $u$ is smooth around $X^{\prime}$, of relative dimension $d$.

Then the functor ${\bf R}\\!\left]f\right[_{\mathrm{dR}!}[2d]$ maps ${\bf
D}^{b}_{\mathrm{cons},F}(X^{\prime},Y^{\prime})$ into ${\bf
D}^{b}_{\mathrm{cons},F}(X,Y)$. The resulting functor only depends on the
morphism of pairs $g\colon(X^{\prime},Y^{\prime})\rightarrow(X,Y)$. If $Y$ is
proper, then it only depends on the morphism of varieties $f\colon
X^{\prime}\rightarrow X$.

###### Proof.

Let me first consider the claim that ${\bf
R}\\!\left]f\right[_{\mathrm{dR}!}[2d]$ preserves constructible isocrystals.
The key point is that I can use Theorem 3.2.1 repeatedly to replace our
morphism of frames by one that is covered by Corollary 8.0.3. Indeed, the
question is local on $X^{\prime}$, which I can therefore assume to be affine,
in particular $f$ is quasi-projective. By Chow’s lemma, there is a morphism of
varieties $g^{\prime}\colon Y^{\prime\prime}\rightarrow Y^{\prime}$ such that
$g^{\prime-1}(X^{\prime})\overset{\cong}{\longrightarrow}X^{\prime}$, and
$g\circ g^{\prime}$ is projective (therefore $g^{\prime}$ is also projective).
Alternately embedding $Y^{\prime\prime}$ into either
$\widehat{{\mathbb{P}}}_{\mathfrak{P}}^{N}$ or
$\widehat{{\mathbb{P}}}_{\mathfrak{P}^{\prime}}^{N^{\prime}}$, and using
Theorem 3.2.1 repeatedly, I can replace the original morphism of frames by one
of the form

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{{\mathbb{P}}}_{\mathfrak{P}}^{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{P},}$

and therefore apply Corollary 8.0.3. Now the independence of $u$ (and $g$ when
$Y$ is proper) is a simple consequence of Theorem 3.2.1, using transitivity of
${\bf R}\\!\left]f\right[_{\mathrm{dR}!}[2d]$ and the usual trick of taking
products of frames. ∎

######  9.2.2 Definition.

If $(f,g)\colon(X^{\prime},Y^{\prime})\rightarrow(X,Y)$ is a morphism of
weakly realisable pairs, I denote by

${\bf R}f_{\flat}\colon{\bf
D}^{b}_{\mathrm{cons},F}(X^{\prime},Y^{\prime})\rightarrow{\bf
D}^{b}_{\mathrm{cons},F}(X,Y)$

the functor implicitly given by Corollary 9.2.1. If $f\colon
X^{\prime}\rightarrow X$ is a morphism of weakly realisable varieties, I
similarly denote

${\bf R}f_{\flat}\colon{\bf
D}^{b}_{\mathrm{cons},F}(X^{\prime})\rightarrow{\bf
D}^{b}_{\mathrm{cons},F}(X)$

the functor obtained by choosing a suitable compactification of $f$.

######  9.2.3 Remark.

Note that whenever $g$ is a closed immersion, and
$Y\hookrightarrow\mathfrak{P}$ is a closed immersion into a formal scheme
which is smooth around $X$, then ${\bf R}f_{\flat}$ can be identified with the
functor $f_{!}$ of extension by zero along the induced morphism of tubes

$f\colon\left]X^{\prime}\right[_{\mathfrak{P}}\rightarrow\left]X\right[_{\mathfrak{P}}.$

If $(X,Y)$ is a strongly realisable pair, and $(X,Y,\mathfrak{P})$ is an l.p.
frame enclosing it, I can also deduce from Proposition 6.2.3 and Theorem 8.0.1
that the composite (equivalence!)

${\bf D}^{b}_{\mathrm{hol},F}(X,Y)\subset{\bf
D}^{b}_{\mathrm{hol},F}(\mathfrak{P})\overset{\mathrm{sp}_{\mathfrak{P}}^{!}}{\longrightarrow}{\bf
D}^{b}_{\mathrm{cons},F}(\mathfrak{P})\overset{{\rm res}}{\longrightarrow}{\bf
D}^{b}_{\mathrm{cons},F}(X,Y,\mathfrak{P})={\bf D}^{b}_{\mathrm{cons},F}(X,Y)$

does not depend on $\mathfrak{P}$, and can therefore be denoted

$\mathrm{sp}_{X}^{!}\colon{\bf
D}^{b}_{\mathrm{hol},F}(X,Y)\overset{\cong}{\longrightarrow}{\bf
D}^{b}_{\mathrm{cons},F}(X,Y).$

Of course, there is a similar functor

$\mathrm{sp}_{X}^{!}\colon{\bf
D}^{b}_{\mathrm{hol},F}(X)\overset{\cong}{\longrightarrow}{\bf
D}^{b}_{\mathrm{cons},F}(X).$

whenever $X$ is a strongly realisable variety. Moreover, these functors are
t-exact for the dual constructible structure on the source, and the natural
t-structure on the target.

######  9.2.4 Corollary.

For any strongly realisable pair $(X,Y)$, $\mathrm{sp}_{X}^{!}$ induces an
equivalence of categories

$\mathrm{sp}_{X}^{!}\colon\mathbf{DCon}_{F}(X,Y)\overset{\cong}{\longrightarrow}\mathbf{Isoc}_{\mathrm{cons},F}(X,Y).$

For any strongly realisable variety $X$, $\mathrm{sp}_{X}^{!}$ induces an
equivalence of categories

$\mathrm{sp}_{X}^{!}\colon\mathbf{DCon}_{F}(X)\overset{\cong}{\longrightarrow}\mathbf{Isoc}_{\mathrm{cons},F}(X).$

######  9.2.5 Remark.

Given that the category $\mathbf{Isoc}_{F}(X)$ of locally free
(overconvergent) isocrystals on $X$ is the $p$-adic analogue of the category
$\mathbf{Loc}(X_{\mathrm{\acute{e}t}},{\mathbb{Q}}_{\ell})$ of $\ell$-adic
local systems on $X$, it may seem a little odd to see it appearing as a full
subcategory of $\mathbf{Isoc}_{\mathrm{cons},F}(X)$, which I am promoting here
as a $p$-adic analogue of the category
$\mathbf{DCon}(X_{\mathrm{\acute{e}t}},{\mathbb{Q}}_{\ell})$ of _dual_
constructible sheaves on $X$ (not the category of constructible sheaves on
$X$). But something similar does indeed happen in the $\ell$-adic theory.
There is a fully faithful embedding

$\displaystyle\mathbf{Loc}(X_{\mathrm{\acute{e}t}},{\mathbb{Q}}_{\ell})$
$\displaystyle\rightarrow\mathbf{DCon}(X_{\mathrm{\acute{e}t}},{\mathbb{Q}}_{\ell})$
$\displaystyle\mathscr{F}$ $\displaystyle\mapsto{\bf
D}_{X}(\mathscr{F}^{\vee})$

where $(-)^{\vee}$ is the dual functor for local systems, and ${\bf D}_{X}$ is
the Verdier dual functor. If $X$ is smooth, then ${\bf
D}_{X}(\mathscr{F}^{\vee})$ is simply a shift of $\mathscr{F}$ by $2\dim X$,
but if $X$ is singular this will generally _not_ be a shift of a constructible
sheaf. This at least goes some way to explaining why the proof of Theorem
10.0.2 below _has_ to proceed via duality.

######  9.2.6 Corollary.

Let $(f,g)\colon(X^{\prime},Y^{\prime})\rightarrow(X,Y)$ be a morphism of
strongly realisable pairs. Then the diagram

$\textstyle{{\bf
D}^{b}_{\mathrm{hol},F}(X,Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{sp}_{X}^{!}}$$\scriptstyle{f^{!}}$$\textstyle{{\bf
D}^{b}_{\mathrm{cons},F}(X,Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\textstyle{{\bf
D}^{b}_{\mathrm{hol},F}(X^{\prime},Y^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{sp}_{X^{\prime}}^{!}}$$\textstyle{{\bf
D}^{b}_{\mathrm{cons},F}(X^{\prime},Y^{\prime})}$

commutes up to natural isomorphism. If $g$ is proper, then so does

$\textstyle{{\bf
D}^{b}_{\mathrm{hol},F}(X^{\prime},Y^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{sp}_{X^{\prime}}^{!}}$$\scriptstyle{f_{+}}$$\textstyle{{\bf
D}^{b}_{\mathrm{cons},F}(X^{\prime},Y^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\bf
R}f_{\flat}}$$\textstyle{{\bf
D}^{b}_{\mathrm{hol},F}(X,Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{sp}_{X}^{!}}$$\textstyle{{\bf
D}^{b}_{\mathrm{cons},F}(X,Y)}$

###### Proof.

The statement for pushforwards follows from Proposition 6.2.3. For pullbacks,
it follows from Proposition 9.1.2 together with the fact that, for any smooth
formal scheme $\mathfrak{P}$, and any locally closed immersion $i\colon
X\rightarrow\mathfrak{P}$, there is an isomorphism

$i_{!}i^{-1}\mathrm{sp}_{\mathfrak{P}}^{!}\overset{\cong}{\longrightarrow}\mathrm{sp}_{\mathfrak{P}}^{!}{\bf
R}\underline{\Gamma}^{\dagger}_{X}.\qed$

There is a similar statement for morphisms of strongly realisable varieties.
Finally, I can show compatibility with tensor product using Corollary 9.1.3.

######  9.2.7 Corollary.

Let $(X,Y)$ be a strongly realisable pair. Then there is a natural isomorphism

$\mathrm{sp}_{X}^{!}(-)\otimes_{\mathcal{O}_{X}}\mathrm{sp}_{X}^{!}\overset{\cong}{\longrightarrow}\mathrm{sp}_{X}^{!}(-\widetilde{\otimes}_{\mathcal{O}_{X}}-)$

of functors

${\bf D}^{b}_{\mathrm{hol},F}(X,Y)\times{\bf
D}^{b}_{\mathrm{hol},F}(X,Y)\rightarrow{\bf D}^{b}_{\mathrm{cons},F}(X,Y).$

Naturally, there is an analogous statement for varieties.

######  9.2.8 Remark.

Via the analogy between $\mathbf{Isoc}_{\mathrm{cons},F}(X,Y)$ and
$\mathbf{DCon}(X_{\mathrm{\acute{e}t}},{\mathbb{Q}}_{\ell})$, the standard
tensor product on constructible isocrystals corresponds to the dual tensor
product

$\mathscr{F}\widetilde{\otimes}_{{\mathbb{Q}}_{\ell}}\mathscr{G}:={\bf
D}_{X}({\bf D}_{X}(\mathscr{F})\otimes_{{\mathbb{Q}}_{\ell}}{\bf
D}_{X}(\mathscr{G}))$

for dual constructible $\ell$-adic sheaves. Note that this is t-exact for the
dual constructible t-structure on ${\bf
D}^{b}_{c}(X_{\mathrm{\acute{e}t}},{\mathbb{Q}}_{\ell})$.

###  9.3. Verdier duality

Of the six cohomological functors $f_{+},f^{+},f^{!},f_{!},{\bf
D}_{X},\widetilde{\otimes}_{\mathcal{O}_{X}}$ defined on ${\bf
D}^{b}_{\mathrm{hol},F}$, we have seen how to interpret three in terms of
${\bf D}^{b}_{\mathrm{cons},F}$, namely $f_{+}$, $f^{!}$ and
$\widetilde{\otimes}_{\mathcal{O}_{X}}$. This begs the question of how to
interpret $f^{+},f_{!}$ and ${\bf D}_{X}$ in the theory of constructible
isocrystals. Via the usual relations amongst $f_{!},f_{+}$, $f^{!}$ and
$f^{+}$, this amounts to describing the duality functor ${\bf D}_{X}$.

This is a subtle question that I don’t know how to answer at the moment. The
main reason that I don’t yet have a good answer is the following: if