# Revisiting Dwork cohomology: Visibility and divisibility of Frobenius
eigenvalues in rigid cohomology

Daqing Wan  Department of Mathematics, University of California, Irvine, CA
92697-3875 USA<EMAIL_ADDRESS>and  Dingxin Zhang  Yau Mathematical
Sciences Center, Tsinghua University, Beijing 100086 China.
<EMAIL_ADDRESS>
###### Abstract.

We study Frobenius eigenvalues of the compactly supported rigid cohomology of
a variety defined over a finite field of $q$ elements via Dwork’s method. A
couple of arithmetic consequences will be drawn from this study. As the first
application, we show that the zeta functions for finitely many related affine
varieties are capable of witnessing all Frobenius eigenvalues of the rigid
cohomology of the variety up to Tate twist. This result does not seem to be
known for $\ell$-adic cohomology. As the second application, we prove several
$q$-divisibility lower bounds for Frobenius eigenvalues of the rigid
cohomology of the variety in terms of the multi-degrees of the defining
equations. These divisibility bounds for rigid cohomology are generally better
than what is suggested from the best known divisibility bounds in $\ell$-adic
cohomology, both before and after the middle cohomological dimension.

###### Contents

  1. 1 Introduction
  2. 2 Dwork cohomology
  3. 3 Comparison theorem: hypersurface case
  4. 4 Comparison theorem: general case
  5. 5 The Artin–Hasse presentation
  6. 6 Visibility of Frobenius eigenvalues
  7. 7 Some algebra
  8. 8 Divisibility of Frobenius eigenvalues

##  1\. Introduction

Dwork [dwork:zeta-function-hypersurfaces-1] engineered a cohomology theory in
order to study the zeta function of a projective hypersurface over a finite
field $\mathbf{F}_{q}$ of $q$ elements with characteristic $p$. In this paper,
we revisit his construction, and prove a comparison theorem between an
overconvergent version of Dwork cohomology and rigid cohomology. We use the
comparison theorem to study some problems on Frobenius eigenvalues of affine
varieties in rigid cohomology.

Let us postpone the statement of the comparison theorem until Section 2, and
start by stating some of its arithmetic consequences.

### Visibility of Frobenius eigenvalues

The first application concerns with the _visibility_ of Frobenius eigenvalues
in the zeta function of an affine variety. For an algebraic variety $Z$ over
$\mathbf{F}_{q}$, its zeta function is

$\zeta_{Z}(t)=\exp\left\\{\sum_{m=1}^{\infty}\frac{|Z(\mathbf{F}_{q^{m}})|}{m}t^{m}\right\\}.$

Weil conjectured, and Dwork [dwork:rationality-zeta-function] proved, that
$\zeta_{Z}(t)$ is a rational function.

By trace formulae in rigid cohomology (due to Étesse and Le Stum [etesse-le-
stum:l-functions-associated-to-overconvergent-f-isocrystals-1]) and
$\ell$-adic cohomology (see, e.g., [deligne:sga4.5, Rapport, 4–6]), the zeta
function is an alternating product

($\ast$) $\zeta_{Z}(t)=\prod_{i=0}^{2\dim
Z}\det\left(1-t\cdot\mathrm{Frob}_{q}\mid\mathrm{H}^{i}_{c}(Z)\right)^{(-1)^{i+1}}.$

Here, $\mathrm{H}^{i}_{c}(Z)$ could mean either Berthelot’s compactly
supported rigid cohomology $\mathrm{H}^{i}_{\mathrm{rig},c}(Z)$, or compactly
supported $\ell$-adic cohomology
$\mathrm{H}^{i}_{c}(Z_{\overline{\mathbf{F}}_{q}},\mathbf{Q}_{\ell})$
($\overline{\mathbf{F}}_{q}$ is a fixed algebraic closure of $\mathbf{F}_{q}$,
$Z_{\overline{\mathbf{F}}_{q}}=Z\otimes_{\mathbf{F}_{q}}\overline{\mathbf{F}}_{q}$,
$\ell\neq p$ is a prime number). The finite dimensionality of
$\mathrm{H}^{i}_{c}(Z)$ also yields a cohomological proof for the rationality
of the zeta function. The above trace formula implies that the reciprocal
roots and poles of $\zeta_{Z}(t)$ are among the Frobenius eigenvalues of these
cohomology theories. However, the converse is not true in general.

If $Z$ is _smooth and proper_ over $\mathbf{F}_{q}$, then the Weil conjecture,
proved by Deligne [deligne:weil1]*Théorème 1.6 (for rigid cohomology, use
Katz–Messing [katz-messing:consequence-riemann-hypothesis]), implies that the
Frobenius eigenvalues of $\mathrm{H}^{i}_{c}(Z)$ are algebraic integers having
archimedean length $q^{i/2}$ (with respect to any embedding
$\mathbf{Q}_{\ell}\hookrightarrow\mathbf{C}$). Therefore the denominator and
numerator of the right hand side of ($\ast$ ‣ 1) do not have common factors.
In this case, the zeta function alone can recover the Frobenius eigenvalues.

Without the smooth proper condition, the linear factors of the determinants in
($\ast$ ‣ 1) could cancel. If a cancellation happens, $\zeta_{Z}(t)$ may not
be capable of witnessing all the Frobenius eigenvalues, not even up to Tate
twist. Consider the following example. Let $X$ be a general nonsingular cubic
curve in $\mathbf{A}^{2}_{\mathbf{F}_{q}}$, and
$Y=\mathbf{A}^{2}_{\mathbf{F}_{q}}\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}X$
its complement. Then the zeta function of the affine variety $Z=X\amalg Y$
equals that of $\mathbf{A}^{2}_{\mathbf{F}_{q}}$, namely $(1-q^{2}t)^{-1}$. On
the other hand, there exist Frobenius eigenvalues of
$\mathrm{H}^{1}_{c}(Z)=\mathrm{H}^{1}_{c}(X)\oplus\mathrm{H}^{1}_{c}(Y)$ of
length $q^{1/2}$.

Our first theorem asserts that, if we are willing to take the _defining
equations_ of an affine variety $Z$ into the consideration, then we can
_recover all the Frobenius eigenvalues of $Z$ up to Tate twist_ from zeta
functions of finitely many varieties related to $Z$. In order to give the
precise statement, let us introduce some terminologies.

######  Definition.

Let $\Gamma=\\{\Gamma_{a}(t),\Gamma_{b}(t),\ldots\\}\subset
1+t\mathbf{C}_{p}[\\![t]\\!]$ be a collection of $p$-adic meromorphic function
on $\mathbf{C}_{p}$ (i.e., fractions of $p$-adic entire functions). We say a
$p$-adic number
$\gamma\in\mathbf{C}_{p}\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}\\{0\\}$
is _visible_ in $\Gamma$, if $\Gamma_{a}(\gamma^{-1})=0$ or $\infty$ for some
$\Gamma_{a}\in\Gamma$. We say $\gamma$ is _weakly visible_ in $\Gamma$, if
there exists $m\in\mathbf{Z}$ such that $q^{m}\gamma$ is visible in $\Gamma$.

Now let $f_{1},\ldots,f_{r}\in\mathbf{F}_{q}[x_{1},\ldots,x_{n}]$ be a
collection of polynomials. For every subset $I\subset\\{1,2,\ldots,r\\}$, set
$Z_{I}=\operatorname{Spec}\mathbf{F}_{q}[x_{1},\ldots,x_{n}]/(f_{i}:i\in
I)\subset\mathbf{A}^{n}_{\mathbf{F}_{q}}$, and
$Z_{I}^{\ast}=Z_{I}\cap\mathbf{G}_{\mathrm{m}}^{n}$. Write
$Z=Z_{\\{1,2,\ldots,r\\}}$.

######  Theorem 1.1.

Any Frobenius eigenvalue of $\mathrm{H}^{\bullet}_{\mathrm{rig},c}(Z)$ is
weakly visible in the finite set
$\\{\zeta_{Z_{I}^{\ast}}(t):{I\subset\\{1,2,\ldots,r\\}}\\}$.

The theorem is already interesting when
$Z\subset\mathbf{A}^{n}_{\mathbf{F}_{q}}$ is an affine hypersurface. In this
special situation, it asserts that if a Frobenius eigenvalue $\lambda$ is
canceled in the zeta function, then either it equals $q^{m}$ for some $m$, or
there must exist a reciprocal root or reciprocal pole of
$\zeta_{Z\cap\mathbf{G}_{\mathrm{m}}^{n}}(t)$ that equals a Tate twist of
$\lambda$. Thus the zeta function $\zeta_{Z\cap\mathbf{G}_{\mathrm{m}}^{n}}$
alone can recover all the Frobenius eigenvalues of
$\mathrm{H}^{\bullet}_{\mathrm{rig},c}(Z)$ up to Tate twist.

In view of the “motivic” philosophy, the same result should also hold for
$\ell$-adic cohomology. But our method, $p$-adic in nature, depends upon an
explicit chain model of rigid cohomology, which does not work in the
$\ell$-adic context.

### Divisibility of Frobenius eigenvalues

The second part of the applications deals with $q$-_divisibility_ as algebraic
integers of Frobenius eigenvalues of affine and projective varieties.

Let $f_{1},\ldots,f_{r}\in\mathbf{F}_{q}[x_{1},\ldots,x_{n}]$ be a collection
of polynomials. Write $d_{j}=\deg f_{j}$. Without loss of generality, we will
assume that all the degrees $d_{j}$ are positive. By rearranging we could and
will assume $d_{1}\geq d_{2}\geq\cdots\geq d_{r}$. Let

$Z=\operatorname{Spec}\mathbf{F}_{q}[x_{1},\ldots,x_{n}]/(f_{1},\ldots,f_{r})$

be the vanishing scheme of these polynomials in
$\mathbf{A}^{n}_{\mathbf{F}_{q}}$. For integer $j\geq 0$, we define a non-
negative integer

$\mu_{j}(n;d_{1},\cdots,d_{r})=j+\max\left\\{0,\left\lceil\frac{n-j-\sum_{i=1}^{r}d_{i}}{d_{1}}\right\rceil\right\\}.$

Recall that the classical Ax–Katz theorem [ax:q-divisible, katz:on-theorem-of-
ax] states that all the reciprocal roots and poles of the zeta function of $Z$
are divisible by $q^{\mu_{0}(n;d_{1},\ldots,d_{r})}$ as algebraic integers.
This divisibility was later upgraded to a divisibility on Frobenius
eigenvalues on $\ell$-adic cohomology: Esnault and Katz [esnault-
katz:cohomological-divisibility] shows that the Frobenius eigenvalues of
$\mathrm{H}^{\bullet}_{c}(Z_{\overline{\mathbf{F}}_{q}},\mathbf{Q}_{\ell})$
are divisible by $q^{\mu_{0}(n;d_{1},\ldots,d_{r})}$, furthermore, for $j\geq
0$, the Frobenius eigenvalues of
$\mathrm{H}^{n-1+j}_{c}(Z_{\overline{\mathbf{F}}_{q}},\mathbf{Q}_{\ell})$ are
divisible by $q^{\mu_{j}(n;d_{1},\ldots,d_{r})}$.

The theorem of Esnault and Katz does not give the most optimal bound when
$r>1$. Recently, Esnault and the first author [esnault-wan:divisibility]
obtained a better bound beyond the middle cohomological dimension. They showed
that any Frobenius eigenvalue of $\mathrm{H}^{\dim
Z+j}_{c}(Z_{\overline{\mathbf{F}}_{q}},\mathbf{Q}_{\ell})$ is divisible by
$q^{\mu_{j}(n;d_{1},\ldots,d_{r})}$, in the ring of algebraic integers, for
all integers $j$ satisfying $0\leq j\leq\dim Z$. This theorem simultaneously
improves the results of [esnault-katz:cohomological-divisibility] and
Deligne’s integrality theorem [sga7-2]*Exposé XXI, §5 beyond the middle
cohomological dimension. Their proof only uses some formal properties of a
cohomology theory, one of which (a certain Gysin lemma) is not yet available
in rigid cohomology.

Since Frobenius eigenvalues are supposed to be “motivic”, the rigid
cohomological analogue of the above divisibility theorems for $\ell$-adic
cohomology should also hold, as already pointed out in Esnault–Katz [esnault-
katz:cohomological-divisibility]. In fact, the same axiomatic formal proof
works if one assumes the conjectural weak Lefschetz theorem and the related
Gysin lemma in rigid cohomology.

Since these results are not available in rigid cohomology yet, we would like
to pursue the divisibility for rigid cohomology using a different method, via
Dwork’s $p$-adic theory, refining and upgrading the chain level approach in
[wan:poles] to Dwork cohomology, and then by comparison with rigid cohomology.

Somewhat surprisingly, the final bound we obtain is sharper than what is
anticipated by the $\ell$-adic theorems in [sga7-2, esnault-
katz:cohomological-divisibility, esnault-wan:divisibility]. To state it, let
us define

$d_{i}^{*}=\begin{cases}d_{i},&\text{ if }1\leq i\leq n-\dim Z;\\\ 1,&\text{
if }i>n-\dim Z,\text{ and }d_{i}=d_{1};\\\ 0,&\text{ if }i>n-\dim Z,\text{ and
}d_{i}<d_{1}.\end{cases}$

For integer $j\geq 0$, define another non-negative integer

$\nu_{j}(n;d_{1},\ldots,d_{r})=j+\max\left\\{0,\left\lceil\frac{n-j-\sum_{i=1}^{r}d_{i}^{*}}{d_{1}}\right\rceil\right\\}.$

Note that $d_{i}\geq d_{i}^{*}$, thus
$\nu_{j}(n;d_{1},\ldots,d_{r})\geq\mu_{j}(n;d_{1},\ldots,d_{r})$; also the
numbers $\nu_{j}(n;d_{1},\ldots,d_{r})$ form an increasing sequence in $j$. In
the complete intersection case, namely when $n-\dim Z=r$, one checks that
$d_{i}^{*}=d_{i}$ and
$\nu_{j}(n;d_{1},\ldots,d_{r})=\mu_{j}(n;d_{1},\ldots,d_{r})$.

We state our divisibility results as two theorems, as the bound looks
different, depending on whether the cohomological degree is at least $\dim Z$
or at most $\dim Z$. Together, they cover all cohomological degrees.

######  Theorem 1.2 (Divisibility beyond middle dimension).

Let notation be as above. For every $0\leq j\leq\dim Z$,

  * •

the Frobenius eigenvalues of $\mathrm{H}^{\dim Z+j}_{\mathrm{rig},c}(Z)$ are
divisible by $q^{\nu_{j}(n;d_{1},\ldots,d_{r})}$ in the ring of algebraic
integers;

  * •

the Frobenius eigenvalues of $\mathrm{H}^{\dim
Z+1+j}_{\mathrm{rig},c}(\mathbf{A}^{n}_{\mathbf{F}_{q}}\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}Z)$
are divisible by $q^{\nu_{j}(n;d_{1},\ldots,d_{r})}$ in the ring of algebraic
integers.

######  Remarks.

(a) The second item is the consequence of the first, thanks to the long exact
sequence for compactly supported cohomology.

(b) For any separated variety $Z$ over $\mathbf{F}_{q}$, the Frobenius
eigenvalues of $\mathrm{H}^{\ast}_{\mathrm{rig},c}(Z)$ are always algebraic
integers. When $Z$ is smooth proper, we use the Weil conjecture and the
integrality of the zeta function, see [katz-messing:consequence-riemann-
hypothesis]*Theorem 1. If $Z$ is proper but possibly singular, we can produce
a proper hypercovering using smooth proper varieties by alteration [de-
jong:alteration], and then apply cohomological descent [tsuzuki:cohomological-
descent-rigid-cohomology]. If $Z$ is not proper, we can embed $Z$ into a
proper variety $\overline{Z}$, and conclude by using the assertion for proper
varieties and the long exact sequence

$\cdots\to\mathrm{H}^{i}_{\mathrm{rig},c}(Z)\to\mathrm{H}^{i}_{\mathrm{rig}}(\overline{Z})\to\mathrm{H}^{i}_{\mathrm{rig}}(\overline{Z}\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}Z)\to\cdots.$

Our method is capable of seeing this too, see p. 8.

Recall that in the $\ell$-adic situation, beyond the middle dimension, the
theorem of Esnault and the first author says that the divisibility of
Frobenius eigenvalues for $\mathrm{H}^{\dim
Z+j}_{c}(Z_{\overline{\mathbf{F}}_{q}},\mathbf{Q}_{\ell})$ is by
$q^{\mu_{j}(n;d_{1},\ldots,d_{r})}$. Since
$\nu_{j}(n;d_{1},\ldots,d_{r})\geq\mu_{j}(n;d_{1},\ldots,d_{r})$, Theorem 1.2
establishes an improved $p$-adic companion of the $\ell$-adic theorem of
Esnault and the first author. In the complete intersection case, they are
exact analogue of each other. If $Z$ is not a complete intersection, the
divisibility bound in Theorem 1.2 can be strictly better.

What about before the middle cohomological dimension? In this range, the only
known divisibility for $\ell$-adic cohomology is the theorem of Esnault–Katz
which says that the divisibility is by $q^{\mu_{0}(n;d_{1},\ldots,d_{r})}$. We
have an improved $p$-adic companion in this range as well.

Since $Z$ is cut out by $r$ equations, $\mathrm{H}^{i}_{\mathrm{rig},c}(Z)$
and $\mathrm{H}^{i}_{c}(Z_{\overline{\mathbf{F}}_{q}},\mathbf{Q}_{\ell})$ all
vanish if $i<n-r$ (see Lemma 8.1). So, we will assume that $i\geq n-r$. If
$n-r=\dim Z$, i.e., $Z$ is a complete intersection, Theorem 1.2 already covers
all the possible cohomological degrees $i$ such that
$\mathrm{H}^{i}_{\mathrm{rig},c}(Z)$ is nontrivial. However, if $Z$ is not a
complete intersection, $\mathrm{H}^{i}_{\mathrm{rig},c}(Z)$ and
$\mathrm{H}^{i}_{c}(Z_{\overline{\mathbf{F}}_{q}},\mathbf{Q}_{\ell})$ could be
nonzero for $n-r\leq i<\dim Z$. A novelty of our approach is that we can
provide improved divisibility information of Frobenius eigenvalues in these
degrees as well, of course, only for rigid cohomology.

######  Theorem 1.3 (Divisibility before middle dimension).

Let notation be as above. For every $0\leq m\leq\dim Z-(n-r)$, the Frobenius
eigenvalues of $\mathrm{H}^{n-r+m}_{\mathrm{rig},c}(Z)$ are divisible by
$q^{\epsilon_{m}(n;d_{1},\ldots,d_{r})}$ in the ring of algebraic integers,
where

$\epsilon_{m}(n;d_{1},\ldots,d_{r})=\max\left\\{0,\left\lceil\frac{n-(d_{1}+\cdots+d_{r-m}+d_{r-m+1}^{*}+\cdots+d_{r}^{*})}{d_{1}}\right\rceil\right\\}.$

The Frobenius eigenvalues of
$\mathrm{H}^{n-r+1+m}_{\mathrm{rig},c}(\mathbf{A}^{n}_{\mathbf{F}_{q}}\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}Z)$
are divisible by $q^{\epsilon_{m}(n;d_{1},\ldots,d_{r})}$ in the ring of
algebraic integers as well.

The numbers $\epsilon_{m}(n;d_{1},\ldots,d_{r})$ ($m=0,1,\ldots,\dim Z-(n-r)$)
form an increasing sequence in the closed interval
$[\mu_{0}(n;d_{1},\ldots,d_{r}),\nu_{0}(n;d_{1},\ldots,d_{r})]$, the smallest
one $\epsilon_{0}(n;d_{1},\ldots,d_{r})=\mu_{0}(n;d_{1},\ldots,d_{r})$ being
responsible for the Ax–Katz theorem and the Esnault–Katz theorem; and we have
$\epsilon_{\dim Z-(n-r)}(n;d_{1},\ldots,d_{r})=\nu_{0}(n;d_{1},\ldots,d_{r})$.

Theorems 1.2, 1.3 have projective analogues. For a closed subvariety $Z$ of
$\mathbf{P}^{n}_{\mathbf{F}_{q}}$, set
$\mathrm{H}^{\ast}_{\mathrm{rig}}(Z)_{\mathrm{prim}}=\operatorname{Coker}(\mathrm{H}^{\ast}_{\mathrm{rig}}(\mathbf{P}_{\mathbf{F}_{q}}^{n})\to\mathrm{H}^{\ast}_{\mathrm{rig}}(Z))$.

######  Theorem 1.4.

Let $f_{1},\ldots,f_{r}\in\mathbf{F}_{q}[x_{0},\ldots,x_{n}]$ be homogeneous
polynomials of positive degrees $d_{1}\geq\cdots\geq d_{r}$. Let $Z$ be the
vanishing scheme of $f_{1},\ldots,f_{r}$ in $\mathbf{P}^{n}_{\mathbf{F}_{q}}$.
Then, for $0\leq j\leq\dim Z$,

  * •

the Frobenius eigenvalues of $\mathrm{H}^{\dim
Z+j}_{\mathrm{rig}}(Z)_{\mathrm{prim}}$ are divisible, as algebraic integers,
by $q^{\nu_{j}(n+1;d_{1},\ldots,d_{r})}$;

  * •

the Frobenius eigenvalues of $\mathrm{H}^{\dim
Z+1+j}_{\mathrm{rig},c}(\mathbf{P}^{n}_{\mathbf{F}_{q}}\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}Z)$
are divisible, as algebraic integers, by
$q^{\nu_{j}(n+1;d_{1},\ldots,d_{r})}$.

For $0\leq m\leq\dim Z-(n-r)$,

  * •

the Frobenius eigenvalues of
$\mathrm{H}^{n-r+m}_{\mathrm{rig}}(Z)_{\mathrm{prim}}$ are divisible, as
algebraic integers, by $q^{\epsilon_{m}(n+1;d_{1},\ldots,d_{r})}$;

  * •

the Frobenius eigenvalues of
$\mathrm{H}^{n-r+1+m}_{\mathrm{rig},c}(\mathbf{P}^{n}_{\mathbf{F}_{q}}\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}Z)$
are divisible, as algebraic integers, by
$q^{\epsilon_{m}(n+1;d_{1},\ldots,d_{r})}$.

These divisibility theorems in rigid cohomology now raise new questions for
$\ell$-adic cohomology and Hodge theory, through the “motivic” philosophy.

  * •

The Frobenius eigenvalues of $\ell$-adic cohomology groups of affine or
projective $Z$ should also satisfy the divisibility stated in Theorems 1.2,
1.3 and 1.4.

  * •

For complex affine or projective varieties, the numbers
$\nu_{j}(n;d_{1},\ldots,d_{r})$ and $\epsilon_{m}(n;d_{1},\ldots,d_{r})$
should give lower bounds for Hodge levels of compactly supported singular
cohomology of complex varieties cut out by a set of polynomial equations of
degrees $d_{1},\ldots,d_{r}$.

Our method, purely analytic, is certainly not applicable for $\ell$-adic
cohomology. But the chain level considerations may be useful for the problem
on Hodge levels.

The paper is organized as follows. Sections 2 and 5 do not contain original
contents, we simply state the comparison theorem relating Dwork cohomology and
rigid cohomology, and recall some basics of the Dwork crystal. Sections 3 and
4 are devoted to the proof of the comparison theorem. In sections 6–8, we
prove Theorems 1.1–1.4.

##  2\. Dwork cohomology

######  Notation.

We fix the following notation. Let $k$ be a perfect field of characteristic
$p>0$. Let $V$ be the complete discrete valuation ring $W(k)[\zeta_{p}]$,
$\zeta_{p}$ being a primitive $p$th root of unity. Thus there exists a
uniformizer $\pi$ satisfying $\pi^{p-1}+p=0$. Let $K$ be the field of
fractions of $V$, which is an ultrametric field. The ring $V$ admits an
automorphism $\sigma$ which fixes $\pi$ and satisfies
$\pi\mid(\sigma(c)-c^{p})$.

Throughout this paper, we consider rigid cohomology over the base field $K$.
For an algebraic variety $X$ over $k$, and an overconvergent F-isocrystal
$\mathcal{E}$ on $X$, $\mathrm{H}^{i}_{\mathrm{rig}}(X,\mathcal{E})$ means
$\mathrm{H}^{i}_{\mathrm{rig}}(X/K,\mathcal{E})$, which is a $K$-vector space.
According to Kedlaya [kedlaya:finiteness-rigid-cohomology], all the rigid
cohomology groups considered in the sequel are finite dimensional.

### Dwork crystal

On the structure sheaf of the rigid analytic affine line
$\mathbf{A}^{1,\mathrm{an}}_{K}$ (with coordinate $z$), we define an
integrable connection $\nabla_{\pi}$ by the formula

$\nabla_{\pi}\xi=\mathrm{d}\xi+\pi\xi\mathrm{d}z.$

The connection is overconvergent. It is equipped with a Frobenius structure

$F\colon\xi\mapsto\xi^{\sigma}(z^{p})\cdot\theta(z)^{-1}$

where $\theta(z)$ is the Dwork exponential

$\theta(z)=\exp(\pi z-\pi z^{p}).$

It is well-known that the radius of convergence of $\theta$ is
$|p|^{-\frac{p-1}{p^{2}}}$, thus overconvergent. The pair $(\nabla_{\pi},F)$
defines an overconvergent F-isocrystal on $\mathbf{A}^{1}_{k}$, called the
_Dwork crystal_. See [le-stum:rigid-cohomology]*§4.2.1 and §8.3 for more
details. The dual isocrystal of $\mathcal{L}_{\pi}$ is $\mathcal{L}_{-\pi}$.

### Dwork cohomology

Let $\overline{R}$ be a smooth finite type $k$-algebra, $g\in\overline{R}$.
Then by taking inverse image, we get an overconvergent F-isocrystal
$g^{\ast}\mathcal{L}_{\pi}$ on $\overline{R}$. The rigid cohomology of
$\operatorname{Spec}\overline{R}$ with coefficient in $\mathcal{L}_{\pi}$ will
be called the “overconvergent Dwork cohomology” associated to the function
$g$.

Let us present this overconvergent F-isocrystal using the method of Monsky and
Washnitzer. According to Elkik [elkik:solutions-of-equations-with-
coefficients-in-henselian-ring], there is a smooth $V$-algebra
$R_{\mathrm{int}}$, such that $R_{\mathrm{int}}\otimes_{V}k=\overline{R}$. Let
$\widehat{g}\in R_{\mathrm{int}}$ be a lift of $g$. Let
$R_{\mathrm{int}}^{\dagger}$ be the weak completion of $R_{\mathrm{int}}$
([monsky-washnitzer:formal-cohomology-1]*Definition 1.1), and let
$R^{\dagger}=R^{\dagger}_{\mathrm{int}}\otimes_{V}K$. Then the integrable
connection on $R^{\dagger}$ given by

$\widehat{g}^{\ast}\nabla_{\pi}\colon
R^{\dagger}\to\Omega^{1}_{R^{\dagger}/K},\quad\xi\mapsto\mathrm{d}\xi+\pi\xi\mathrm{d}\widehat{g}$

is a presentation of $g^{\ast}\mathcal{L}_{\pi}$. The cohomology of the de
Rham complex of $\widehat{g}^{\ast}\nabla_{\pi}$ equals
$\mathrm{H}^{\bullet}_{\mathrm{rig}}(\operatorname{Spec}\overline{R},g^{\ast}\mathcal{L}_{\pi})$,
the overconvergent Dwork cohomology of $g$.

### Dwork cohomology and cohomology with support

Let us consider the following special situation:
$\overline{R}=\overline{A}[T_{1},\ldots,T_{r}]$ for some smooth finite type
$k$-algebra $\overline{A}$, and $g=\sum_{i=1}^{r}T_{i}f_{i}$ for some
$f_{1},\ldots,f_{r}\in\overline{A}$. In the next two sections, we will prove
the following theorem.

######  Theorem 2.1.

Let $Z$ be the affine variety
$\operatorname{Spec}(\overline{A}/(f_{1},\ldots,f_{r}))$. Then there is a
natural isomorphism

$\mathrm{H}^{\bullet}_{\mathrm{rig}}(\operatorname{Spec}(\overline{A}[T_{1},\ldots,T_{r}]),g^{\ast}\mathcal{L}_{\pi})\simeq\mathrm{H}^{\bullet}_{\mathrm{rig},Z}(\operatorname{Spec}(\overline{A})).$

which is compatible with Frobenius actions.

The theorem can be formulated in a more general fashion, namely
$\operatorname{Spec}\overline{A}$ can be replaced by a smooth $k$-variety $X$,
$\mathbf{A}^{r}_{\overline{A}}$ by a vector bundle over $X$, and
$(f_{1},\ldots,f_{r})$ by a section of the bundle. But we will not need this
added generality.

######  Remarks.

(a) Dwork cohomology originated in Dwork’s study of zeta functions of
projective hypersurfaces [dwork:zeta-function-hypersurfaces-1, dwork:zeta-
function-hypersurface-2]. Dwork used Banach spaces rather than weakly
completed algebras in the sense of Monsky–Washnitzer. His construction was
systematically generalized in the context of toric exponential sums by
Adolphson and Sperber [adolphson-sperber:exponential-sums-newton-polyhedra]
(still using Banach spaces instead of weakly completed versions). Apparently,
the nice properties of overconvergent Dwork cohomology was first studied by
Monsky [monsky:formal-cohomology-3].

(b) Dwork and Adolphson–Sperber proved the finiteness of the cohomology spaces
they considered, when the function $g$ is “Newton nondegenerate and
convenient”. By contrast, their overconvergent variant, the “overconvergent
Dwork cohomology”, is always finite dimensional, thanks to the finiteness
theorem of Kedlaya [kedlaya:finiteness-rigid-cohomology].

(c) If in the definition of Dwork cohomology one uses finite type rings
instead of using weakly completed algebra or Banach algebras, one gets the so-
called “algebraic Dwork cohomology”. The algebraic analogue of Theorem 2.1 is
well-known: it is proved by N. Katz [katz:differential-equations-satisfied-by-
period-matrices] for hypersurfaces; by Adolphson–Sperber [adolphson-
sperber:dwork-cohomology] for _smooth_ complete intersections in a smooth
affine variety; by Dimca et al. [dimca-maaref-sabbah-saito:dwork-cohomology-
and-d-modules] and Baldassarri–D’Agnolo [baldassarri-d-agnolo:dwork-
cohomology-algebraic-d-modules] in general.

##  3\. Comparison theorem: hypersurface case

We begin by introducing some notation in order to explicate the recipes we
need.

  * •

Let $\overline{A}$ be a smooth $k$-algebra of pure dimension $n$. Fix an
element $f\in\overline{A}$ that is not constant. Let $g=Tf\in\overline{A}[T]$,
regarded as a regular function
$g\colon\mathbf{A}^{1}\times_{k}\operatorname{Spec}\overline{A}\to\mathbf{A}^{1}_{k}$.

  * •

Fix a smooth $V$-algebra $A_{\mathrm{int}}$ such that $A_{\mathrm{int}}/\pi
A_{\mathrm{int}}=\overline{A}$. Fix $\widehat{f}\in A_{\mathrm{int}}$ such
that $\widehat{f}\equiv f$ modulo $\pi$. Fix a presentation
$A_{\mathrm{int}}\cong V[x_{1},\ldots,x_{N}]/(u_{1},\ldots,u_{m})$ of
$A_{\mathrm{int}}$.

  * •

For a positive real number $\rho>1$, let

$\textstyle
A_{\mathrm{int},\rho}=V\langle\frac{x_{1}}{\rho},\ldots,\frac{x_{N}}{\rho}\rangle/(u_{1},\ldots,u_{m}),$

where

$\textstyle
V\langle\frac{x_{1}}{\rho},\ldots,\frac{x_{n}}{\rho}\rangle=\left\\{\sum\limits_{\alpha\in\mathbf{N}^{n}}a_{\alpha}x^{\alpha}:|a_{\alpha}|\rho^{|\alpha|}\to
0,\text{ as }|\alpha|\to\infty.\right\\}$

For $\rho_{1}>\rho_{2}>1$ we have a restriction homomorphism
$A_{\mathrm{int},\rho_{1}}\to A_{\mathrm{int},\rho_{2}}$. Let
$A^{\dagger}_{\mathrm{int}}=\mathop{\operatorname{colim}}\limits_{\rho\to
1^{+}}A_{\mathrm{int},\rho}$ be the weak completion of $A_{\mathrm{int}}$.

  * •

Let $A_{\rho}=A_{\mathrm{int},\rho}\otimes_{V}K$, which is a Banach algebra,
and
$A^{\dagger}=A_{\mathrm{int}}^{\dagger}\otimes_{V}K=\mathop{\operatorname{colim}}\limits_{\rho\to
1^{+}}A_{\rho}$.

Let $\mathbf{D}^{+}(0;\rho)$ be the rigid analytic space associated to the
disk in $\mathbf{A}^{1,\mathrm{an}}_{K}$ of radius $\rho$ centered at $0$.
Then $A_{\rho}$ is the ring of rigid analytic functions on the affinoid space
$(\operatorname{Spec}A)^{\mathrm{an}}\cap\mathbf{D}^{+}(0;\rho)^{n}$. Let
$|\cdot|$, or sometimes $|\cdot|_{\rho}$, to denote the Banach norm of
$A_{\rho}$ obtained from the quotient
$V\langle\frac{x_{1}}{\rho},\ldots,\frac{x_{n}}{\rho}\rangle\to
A_{\mathrm{int},\rho}$.

  * •

For a positive real number $\eta>1$, let $C_{\rho}^{\geq\eta^{-1}}$ denote the
Banach algebra of rigid analytic functions on the affinoid subspace

$\\{x\in(\operatorname{Spec}A)^{\mathrm{an}}\cap\mathbf{D}^{+}(0;\rho)^{n}:|\widehat{f}(x)|\geq\eta^{-1}\\}.$

Its norm is denoted by $|\cdot|_{C_{\rho}^{\geq\eta^{-1}}}$. Clearly,
$|\widehat{f}^{-1}|_{C_{\rho}^{\geq\eta^{-1}}}\leq\eta$. Let
$C^{\dagger}=\mathop{\operatorname{colim}}\limits_{\begin{subarray}{c}\eta\to
1^{+}\\\ \rho\to 1^{+}\end{subarray}}C_{\rho}^{\geq\eta^{-1}}$. The ring
$C^{\dagger}$ furnishes as a $p$-adic incarnation of $\overline{A}[f^{-1}]$.

As Banach spaces, $A_{\rho}$ is a closed subspace of
$C_{\rho}^{\geq\eta^{-1}}$.

  * •

Let $B_{\rho,\eta}$ be the quotient space $C_{\rho}^{\geq\eta^{-1}}/A_{\rho}$,
which is also a Banach space over $K$. An element of$B_{\rho,\eta}$ can be
expressed as a of formal sum

(3.1) $\sum_{m=1}^{\infty}c_{m}\widehat{f}^{-m},\quad c_{m}\in
A_{\rho},\quad\text{such that }|c_{m}|_{\rho}\eta^{m}\to 0\text{ as
}m\to\infty.$

Define
$B^{\dagger}=\mathop{\operatorname{colim}}\limits_{\begin{subarray}{c}\eta\to
1^{+}\\\ \rho\to 1^{+}\end{subarray}}B_{\rho,\eta}$. This is the $p$-adic
analogue of the “algebraic local cohomology”. In the language of rigid
geometry, we have

$B^{\dagger}[-1]=R\Gamma_{Z}^{\dagger}(\mathcal{O}).$

See [le-stum:rigid-cohomology]*Proposition 5.2.21.

The spaces $A_{\rho}$, $C_{\rho}^{\geq\eta^{-1}}$, $A^{\dagger}$,
$C^{\dagger}$, $B_{\rho,\eta}$ and $B^{\dagger}$ are equipped with actions
from differential operators (of finite order). Locally, supposing we have a
fixed étale coordinate of $A_{\mathrm{int}}$ given by $x_{1},\ldots,x_{n}$,
then the spaces above are equipped with mutually commuting derivations

$E_{i}=\frac{\partial}{\partial x_{i}},\quad i=1,2,\ldots,n.$

In this situation, the $(i-n)$th cohomology of the Koszul complex
$\mathrm{Kos}(B^{\dagger};E_{1},\ldots,E_{n})$

$\cdots\to(B^{\dagger})^{\binom{n}{2}}\to\underset{\bullet}{(B^{\dagger})^{\oplus
n}}\to B^{\dagger}$

(here the dot is indicating the item is placed in cohomological degree $0$) is
by definition the $i$th rigid cohomology of $\operatorname{Spec}\overline{A}$
with supports in the hypersurface $Z$.

The rigid cohomology of $\operatorname{Spec}\overline{A}$ is computed by the
cohomology of the shifted Koszul complex
$\mathrm{Kos}(A^{\dagger};E_{1},\ldots,E_{n})[-n]$; and the rigid cohomology
of
$\operatorname{Spec}\overline{A}\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}Z$
is computed by the shifted Koszul complex
$\mathrm{Kos}(C^{\dagger};E_{1},\ldots,E_{n})[-n]$.

For a positive real number $\eta>1$, let
$A_{\rho}\langle\frac{T}{\eta}\rangle$ be the Banach algebra

$\textstyle\left\\{\sum\limits_{m\in\mathbf{N}}a_{m}T^{m}:|a_{m}|_{\rho}\eta^{m}\to
0\text{ as }m\to\infty,\ a_{m}\in A_{\rho}\right\\}$

equipped with the $\eta$-Gauss norm. Define $A^{\dagger}\langle
T\rangle^{\dagger}=\mathop{\operatorname{colim}}\limits_{\begin{subarray}{c}\rho\to
1^{+}\\\ \eta\to
1^{+}\end{subarray}}A_{\rho}\left\langle\frac{T}{\eta}\right\rangle$.

The algebras $A_{\rho}\langle\frac{T}{\eta}\rangle$ and
$A^{\dagger}\langle{T}\rangle^{\dagger}$ both underlie the integrable
connection $\mathrm{d}+\pi\mathrm{d}g$. When expressed in terms of the étale
coordinate $(x_{1},\ldots,x_{n},T)$ the connection is determined by $(d+1)$
mutually commuting differential operators

(3.2) $\displaystyle D_{i}$ $\displaystyle=\frac{\partial}{\partial x_{i}}-\pi
T\frac{\partial\widehat{f}}{\partial x_{i}}=\exp(\pi
T\widehat{f})\circ\frac{\partial}{\partial x_{i}}\circ\exp(-\pi
T\widehat{f}),\quad i=1,2,\ldots,n;$ $\displaystyle D$
$\displaystyle=\frac{\partial}{\partial T}-\pi\widehat{f}=\exp(\pi
T\widehat{f})\circ\frac{\partial}{\partial T}\circ\exp(-\pi T\widehat{f}).$

Under an étale coordinate, the $i$th overconvergent Dwork cohomology is then
the $(i-(n+1))$th cohomology of the Koszul complex
$\mathrm{Kos}(A^{\dagger}\langle T\rangle^{\dagger};D_{1},\ldots,D_{n},D)$;

$\cdots\to\left(A^{\dagger}\langle
T\rangle^{\dagger}\right)^{\oplus\binom{n+1}{2}}\to\left(A^{\dagger}\langle
T\rangle^{\dagger}\right)^{\oplus(n+1)}\to\underset{\bullet}{A^{\dagger}\langle
T\rangle^{\dagger}}.$

Now we can state the main theorem of this section.

######  Theorem 3.3.

There is a natural quasi-isomorphism

$B^{\dagger}\to\left[A^{\dagger}\langle
T\rangle^{\dagger}\xrightarrow{D}\underset{\bullet}{A^{\dagger}\langle
T\rangle^{\dagger}}\right]$

Moreover, in any étale coordinate system, the residual actions of the
differential operators $D_{1},\ldots,D_{n}$ on the right hand side and the
actions $E_{1},\ldots,E_{n}$ on the left hand side agree.

### Proof of Theorem 3.3

For any pair of numbers $\rho>1$ and $\eta>1$, form the following commutative
ladder of exact sequences

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In the diagram, $B_{\rho,\eta}\langle\frac{T}{\eta}\rangle$ is defined to be
the Banach space of formal sums

$\textstyle\left\\{\sum\limits_{m=0}^{\infty}b_{m}T^{m}:b_{m}\in
B_{\rho,\eta},|b_{m}|\eta^{m}\to 0\text{ as }m\to\infty\right\\},$

equipped with the $\eta$-norm; the middle differential $\widetilde{D}$ is
still defined as in (3.2); the vertical map $D_{B}$ is induced by the
commutativity of the left square.

Taking colimit with respect to $\rho\to 1^{+}$ and $\eta\to 1^{+}$ gives an
“overconvergent” version of the ladder (3.4)

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The idea of the proof of Theorem 3.3 can be explained as follows. First, we
show the middle column of (3.4′) is exact. We will do so by proving the middle
column of (3.4) is exact. see Lemma 3.7. Second, we show that the kernel of
the right column is isomorphic to $B^{\dagger}$, see Lemma 3.13. The theorem
will follow from the snake lemma.

######  Lemma 3.5.

For any $\rho>1$, $\eta>1$, the map $D\colon
A_{\rho}\langle\frac{T}{\eta}\rangle\to A_{\rho}\langle\frac{T}{\eta}\rangle$
is injective.

###### Proof.

It suffices to show

$\textstyle\exp(\pi
T\widehat{f})=\sum\limits_{m=0}^{\infty}\frac{\pi^{m}\widehat{f}^{m}}{m!}T^{m}\notin
A_{\rho}\langle\frac{T}{\eta}\rangle$

Since the $1$-norm of $\widehat{f}$ is $1$ (the reduction of $\widehat{f}$ is
nonzero), $|\widehat{f}|_{\rho}\geq 1$. Since $|\pi|^{m}\sim|m!|$, and
$\eta>1$, we have

(3.6)
$\frac{|\pi|^{m}}{|m!|}\cdot|\widehat{f}^{m}|_{\rho}\cdot\eta^{m}\to\infty.$

The lemma is proved. ∎

######  Lemma 3.7.

For any $\rho\geq 1,\eta>1$, the map $\widetilde{D}$ in (3.4) is bijective.

###### Proof.

The same reasoning as in the proof of Lemma 3.5 shows that the map
$\widetilde{D}$ is injective. Write $\widetilde{D}=\frac{\partial}{\partial
T}-\pi\widehat{f}$. It is tempting to prove the surjectivity of
$\widetilde{D}$ by showing the operator norm of $\partial/\partial T$ is less
than the operator norm of the multiplication-by-$\pi\widehat{f}$ map, so by
perturbation the surjectivity of $\widetilde{D}$ will be a consequence of that
of multiplication-by-$\pi\widehat{f}$. Unfortunately, this will only work for
the specific radius $\eta=|\pi|^{-1}$. To fix this, we will prove the
surjectivity of $\frac{1}{e!}\widetilde{D}^{e}$ for some $e>1$, which will
imply the surjectivity of $\widetilde{D}$. For $e$ large, the perturbation
argument will work. This technique is learned from [robba-christol:p-adic-
differential-equations]*Proposition 10.1.3.

For a linear operator $\Psi$ on
$C^{\geq\eta^{-1}}_{\rho}\langle\frac{T}{\eta}\rangle$, we denote by
$\|\Psi\|$ the operator norm of $\Psi$. For a series $u\in
C^{\geq\eta^{-1}}_{\rho}\langle\frac{T}{\eta}\rangle$, we still denote by $u$
the multiplication operator $v\mapsto uv$. In this case, $\|u\|$ equals $|u|$.

Recall formally $\widetilde{D}=\exp(\pi
T\widehat{f})\circ\frac{\partial}{\partial T}\circ\exp(-\pi T\widehat{f})$.
Then

$\displaystyle\frac{1}{e!}\widetilde{D}^{e}$ $\displaystyle=\exp(\pi
T\widehat{f})\circ\frac{1}{e!}\left(\frac{\partial}{\partial
T}\right)^{m}\circ\exp(-\pi T\widehat{f})$
$\displaystyle=\sum_{i=0}^{e}\frac{1}{(e-i)!}\frac{\partial^{e-i}\exp(-\pi
T\widehat{f})}{\partial T^{e-i}}\frac{1}{i!}\left(\frac{\partial}{\partial
T}\right)^{i}.$
$\displaystyle=\sum_{i=0}^{e}\frac{1}{(e-i)!}(-\pi\widehat{f})^{e-i}\cdot\frac{1}{i!}\left(\frac{\partial}{\partial
T}\right)^{i}$ $\displaystyle=\frac{1}{e!}(-\pi\widehat{f})^{e}+R_{e}.$

Claim. If $e$ is sufficiently large, the operator norm of $R_{e}$ is strictly
smaller than $\|\frac{1}{e!}(-\pi\widehat{f})^{e}\|$ on the Banach space
$C^{\geq\eta^{-1}}\langle\frac{T}{\eta}\rangle$.

Granting the claim, let us prove the lemma. Noticing that the operator
$\frac{1}{e!}(-\pi\widehat{f})^{e}$ is invertible, and the operator norm of
its inverse is precisely the inverse of its operator norm, we apply the
perturbation theorem in functional analysis [robba-christol:p-adic-
differential-equations]*Proposition 7.2.3 to conclude that the operator
$\frac{1}{e!}\widetilde{D}^{e}$ is also surjective. This proves the theorem.

Let us now prove $\|R_{e}\|<\|\frac{1}{e!}(-\pi\widehat{f})^{e}\|$ for $e$
large. Using equation (3.6), for any $\eta>1$, there exists $e$ such that

(3.8)
$\left|\frac{\pi^{e}}{e!}\widehat{f}^{e}\right|_{\rho}\eta^{e}>\max_{1\leq
i\leq
e}\left|\frac{\pi^{e-i}}{(e-i)!}\widehat{f}^{e-i}\right|_{\rho}\eta^{e-i}.$

Moreover, the norm of $\widehat{f}$ in
$C_{\rho}^{\geq\eta^{-1}}\langle\frac{T}{\eta}\rangle$, which equals the
supremum of $\widehat{f}$ on the “annulus”
$\\{x\in\mathbf{D}^{+}(0;\rho)^{n}:|\widehat{f}(x)|\geq\eta^{-1}\\}$, also
equals the $\rho$-norm of $\widehat{f}$. On the other hand, for any $u\in
C^{\geq\eta^{-1}}\langle\frac{T}{\eta}\rangle$, we have

(3.9) $\left|\frac{1}{i!}\left(\frac{\partial}{\partial
T}\right)^{i}u\right|\leq|u|\eta^{-i}.$

Combining (3.8) and (3.9), we see

$\|R_{e}\|\leq\sup_{1\leq i\leq
e}\left|\frac{\pi^{e-i}}{(e-i)!}\right|\cdot|\widehat{f}^{e-i}|_{\rho}\cdot\eta^{-i}<\left\|\frac{1}{e!}(\pi\widehat{f})^{e}\right\|.$

The claim is proved. ∎

######  Lemma 3.10.

For any $\rho>1,\eta>1$, we have
$\operatorname{Coker}D\simeq\operatorname{Ker}D_{B}$ in (3.4).

###### Proof.

Apply the snake lemma to (3.4) and use Lemma 3.7. ∎

######  Lemma 3.11.

In (3.4), we have $\operatorname{Ker}D_{B}\simeq B_{\rho,\eta}$ for any
$\rho>1,\eta>1$.

###### Proof.

For an element $b\in B_{\rho,\eta}$, denote by $|b|_{B_{\rho,\eta}}$ its
Banach norm obtained from the quotient structure. See (3.1).

Let $\sum_{m=0}^{\infty}b_{m}T^{m}$ be an element in the kernel of $D_{B}$.
Then, by induction, we can write

$b_{m}=\frac{(\widehat{f}\pi)^{m}}{m!}b_{0}.$

In other words, the series is completely determined by $b_{0}\in
B_{\rho,\eta}$.

It remains to show that for any $b\in B_{\rho,\eta}$, the sum
$\sum\frac{(\widehat{f}\pi)^{m}}{m!}bT^{m}$ makes sense in
$B_{\rho,\eta}\langle\frac{T}{\eta}\rangle$. We need to show:

$\frac{|\pi|^{m}}{|m!|}\cdot\Big{|}\widehat{f}^{m}\cdot
b\Big{|}_{B_{\rho,\eta}}\eta^{m}\to 0.$

By definition, we may represent any element $b$ of $B_{\rho,\eta}$ as a sum
$\sum_{j=1}^{\infty}c_{j}\widehat{f}^{-j}$ as in (3.1), such that
$|c_{m}|_{\rho}\eta^{m}$ converges to zero. The product $\widehat{f}^{m}b$ is
then represented by the formal sum
$\sum_{j=m+1}^{\infty}c_{j}\widehat{f}^{m-j}$.

By definition of the quotient norm, we have

$|b|_{B_{\rho,\eta}}\leq\sup_{m}|c_{m}|_{\rho}\eta^{m}.$

Since $|\widehat{f}^{-1}|_{C^{\geq\eta^{-1}}_{\rho}}\leq\eta$, the property of
quotient norm yields
$|(\widehat{f}^{-1})^{j-m}|_{B_{\rho,\eta}}\leq\eta^{j-m}$. Thus

$|\widehat{f}^{m}\cdot
b|_{B_{\rho,\eta}}\cdot\eta^{m}\leq\sup_{j>m}|c_{j}|_{\rho}\eta^{j}.$

Since $|c_{m}|_{\rho}\eta^{m}$ converges to zero,
$\sup_{j>m}|c_{j}|_{\rho}\eta^{j}$ converges to zero as $m\to\infty$ as well.
We win by using the asymptotic property $|\pi|^{m}\sim|m!|$. ∎

Let $B^{\dagger}\langle T\rangle^{\dagger}$ be the space obtained from
$B_{\rho,\eta}\langle\frac{T}{\eta}\rangle$ by taking colimit with respect to
$\eta\to 1^{+}$ and $\rho\to 1^{+}$. The derivations $D_{B}$ extend to a
derivation

$D_{B}^{\dagger}\colon B^{\dagger}\langle T\rangle^{\dagger}\to
B^{\dagger}\langle T\rangle^{\dagger}.$

Since taking filtered colimit with respect to vector spaces is an exact
operator, Lemma 3.10 implies that

(3.12) $\operatorname{Coker}(D\colon A^{\dagger}\langle T\rangle^{\dagger}\to
A^{\dagger}\langle
T\rangle^{\dagger})\simeq\operatorname{Ker}D_{B}^{\dagger}.$

Taking colimit $\rho\to 1^{+}$, $\eta\to 1^{+}$ in Lemma 3.11 gives the
following lemma.

######  Lemma 3.13.

There is a natural isomorphism
$B^{\dagger}\simeq\operatorname{Ker}D_{B}^{\dagger}$.∎

The module $B^{\dagger}\langle T\rangle^{\dagger}$ also receives the action of
the differential operators $D_{i}$ defined as in (3.2), which commute with
$D_{B}^{\dagger}$. These operators induce an action on the space $B^{\dagger}$
via the isomorphism provided by Lemma 3.13. We now verify that these actions
agree with the usual action by $E_{i}$. This will complete the proof of
Theorem 3.3.

######  Lemma 3.14.

The isomorphism provided by Lemma 3.13 takes the operator $D_{i}$ to $E_{i}$.

###### Proof.

Each $b\in B^{\dagger}$ corresponds to the formal sum
$\sum\frac{1}{m!}(\pi\widehat{f})^{m}bT^{m}$, which will be denoted by
$\mathrm{Init}(b)$. It suffices to prove $\mathrm{Init}(\partial b/\partial
x_{i})$ equals the constant term of $D_{i}(\mathrm{Init}(b))$. But this is a
rather simple computation. ∎

##  4\. Comparison theorem: general case

Generalizing Theorem 3.3 to the general case will only use general nonsense.
Let us first set up the notation.

######  Notation.

Let $\overline{A}$ be a smooth $k$-algebra. Let $f_{1},\ldots,f_{r}$ be a
collection of elements in $\overline{A}$. Let $Z_{i}$ be the hypersurface cut
out by $f_{i}$ in $\operatorname{Spec}\overline{A}$, and
$Z=\bigcap_{i=1}^{r}Z_{i}$. We _do not_ assume $Z$ is a complete intersection
of $r$ equations; its codimension is usually smaller than $r$. Let
$g(x,T)=\sum_{i=1}^{r}T_{i}f_{i}(x)$.

For each $i$, choose a lift $\widehat{f}_{i}\in A_{\mathrm{int}}$ of $f_{i}$.
In the algebra

$A^{\dagger}\langle T_{1},\ldots,T_{r}\rangle^{\dagger},$

we have a lift $\widehat{g}=\sum_{i=1}^{r}T_{i}\widehat{f}_{i}$ of $g$.

Let us explain how to compute the overconvergent Dwork cohomology
$\mathrm{H}_{\mathrm{rig}}^{\ast}(\operatorname{Spec}\overline{A}\times\mathbf{A}^{r}_{k},g^{\ast}\mathcal{L}_{-\pi})$.
The algebra $A^{\dagger}\langle T_{1},\ldots,T_{r}\rangle^{\dagger}$ is
equipped with an integrable connection given by
$\mathrm{d}+\pi\mathrm{d}\widehat{g}$.

When an étale coordinate is given, the de Rham complex of this integrable
connection can be expressed as a Koszul complex of the following commuting
differential operators:

$D_{i}=\frac{\partial}{\partial
x_{i}}-\pi\sum_{j=1}^{r}T_{j}\frac{\partial\widehat{f}_{j}}{\partial
x_{i}},\quad i=1,2,\ldots,n$

as well as

(4.1) $D^{(j)}=\frac{\partial}{\partial T_{j}}-\pi\widehat{f}_{j},\quad
j=1,2,\ldots,r.$

Then the overconvergent Dwork cohomology
$\mathrm{H}^{i}_{\mathrm{rig}}(\operatorname{Spec}\overline{A}\times_{k}\mathbf{A}^{r}_{k},g^{\ast}\mathcal{L}_{-\pi})$
is just the $(i-(n+r))$th cohomology of the Koszul complex

$\mathrm{Kos}(A^{\dagger}\langle
T_{1},\ldots,T_{r}\rangle^{\dagger};D^{(1)},\ldots,D^{(r)},D_{1},\ldots,D_{n}).$

Temporarily ignoring the operators $D_{i}$, let us compute the partial Koszul
complex

(4.2) $\mathrm{Kos}(A^{\dagger}\langle
T_{1},\ldots,T_{r}\rangle^{\dagger};D^{(1)},\ldots,D^{(r)}).$

Note that this is the weakly completed tensor product of the Koszul complexes:

$\displaystyle\mathrm{Kos}(A^{\dagger}\langle
T_{1},\ldots,T_{r}\rangle^{\dagger};D^{(1)},\ldots,D^{(r)})$ $\displaystyle=$
$\displaystyle\operatorname{Kos}(A^{\dagger}\langle
T_{1}\rangle,D^{(1)})\mathop{\overset{\mathbf{L}}{\underset{A^{\dagger}}{\otimes}}{\\!}^{\dagger}}\cdots\mathop{\overset{\mathbf{L}}{\underset{A^{\dagger}}{\otimes}}{\\!}^{\dagger}}\operatorname{Kos}(A^{\dagger}\langle
T_{r}\rangle,D^{(r)}).$

Recall that if $R=\mathop{\operatorname{colim}}\limits_{\rho\to 1}R_{\rho}$ is
a weakly completed, finitely generated algebra, and
$M=\mathop{\operatorname{colim}}\limits_{\rho\to 1}M_{\rho}$,
$N=\mathop{\operatorname{colim}}\limits_{\rho\to 1}N_{\rho}$ are finitely
generated $R$-modules which are “overconvergent”, then
$M\otimes^{\dagger}_{R}N$ is defined as the colimit of the completed tensor
products

$\mathop{\operatorname{colim}}_{\rho\to
1}M_{\rho}\mathop{\widehat{\otimes}}_{R_{\rho}}N_{\rho}.$

When $M$ and $N$ are $R$-algebras, the weakly completed product is also an
$R$-algebra.

Now we can state the theorem that generalizes Theorem 3.3 to a general
subariety $Z$.

######  Theorem 4.3.

There is a natural quasi-isomorphism between the partial Koszul complex

$\mathrm{Kos}(A^{\dagger}\langle
T_{1},\ldots,T_{r}\rangle^{\dagger};D^{(1)},\ldots,D^{(r)})$

and $R\Gamma_{Z_{1}\cap\cdots\cap Z_{r}}^{\dagger}(\mathcal{O})[r]$, the
“derived incarnation” of the rigid cohomology of
$\operatorname{Spec}\overline{A}$ supported in $Z$. In addition, this quasi-
isomorphism is compatible with the derivations on $A^{\dagger}$.

###### Proof.

By Lemmas 3.5, 3.12, and 3.13, we conclude that this Koszul complex is
computed by the weakly completed tensor product

$\widetilde{B}^{\dagger}=B_{1}^{\dagger}\mathop{\overset{\mathbf{L}}{\underset{A^{\dagger}}{\otimes}}{\\!}^{\dagger}}\cdots\mathop{\overset{\mathbf{L}}{\underset{A^{\dagger}}{\otimes}}{\\!}^{\dagger}}B_{r}^{\dagger}.$

Here $B_{i}^{\dagger}$ is the space of overconvergent hyperfunctions along
$Z_{i}$. Indeed, for two closed subvarieties $Y_{1}$ and $Y_{2}$ of a variety
$X$, one has [le-stum:rigid-cohomology]*Proposition 5.2.4(iii) and projection
formula

$R\Gamma_{Y_{1}}^{\dagger}\otimes R\Gamma_{Y_{2}}^{\dagger}=R\Gamma_{Y_{1}\cap
Y_{2}}^{\dagger}.$

Ergo, $\widetilde{B}^{\dagger}\simeq R\Gamma_{Z_{1}\cap\cdots\cap
Z_{r}}^{\dagger}(\mathcal{O})[r]$.

Then, it remains to check, as in Lemma 3.14, that the derivations induced by
$D_{1},\ldots,D_{n}$ on this tensor product via the isomorphism agree with
$E_{1},\ldots,E_{n}$ respectively. By definition, the shifted Koszul complex
of $\widetilde{B}^{\dagger}$ with respect to $E_{1},\ldots,E_{n}$ _is_ the
rigid cohomology of $\operatorname{Spec}\overline{A}$ supported on
$Z=Z_{1}\cap\cdots\cap Z_{r}$. See [le-stum:rigid-cohomology]*Definition 5.2.3
for the definition of cohomology with support (sheaf level); and [le-
stum:rigid-cohomology]*Proposition 5.2.21 for the result on rigid cohomology
(in our circumstance, $E=(\mathcal{O},\mathrm{d})$ is the trivial connection).
∎

Taking the de Rham cohomology functor on $A^{\dagger}$, with respect to the
$R\Gamma_{Z}(\mathcal{O})$ and the partial Koszul complex
$\mathrm{Kos}(A^{\dagger}\langle
T_{1},\ldots,T_{r}\rangle^{\dagger};D^{(1)},\ldots,D^{(r)})$ gives the
following comparison theorem of rigid cohomology spaces.

######  Corollary 4.4.

There is a natural isomorphism

$u\colon\mathrm{H}^{i}_{\mathrm{rig}}(\operatorname{Spec}\overline{A}\times_{k}\mathbf{A}^{r}_{k},g^{*}\mathcal{L}_{-\pi})\xrightarrow{\sim}\mathrm{H}^{i}_{\mathrm{rig},Z}(\operatorname{Spec}(\overline{A})).$

Taking Poincaré duality yields an isomorphism

$v\colon\mathrm{H}^{i}_{\mathrm{rig},c}(Z)\xrightarrow{\sim}\mathrm{H}^{i+2r}_{\mathrm{rig},c}(\operatorname{Spec}\overline{A}\times_{k}\mathbf{A}^{r}_{k},g^{*}\mathcal{L}_{\pi}).$

Our next objective is to discuss the Frobenius actions on the two rigid
cohomology spaces. To this end, recall $\sigma\colon V\to V$ is a ring
homomorphism such that $\pi$ divides $\sigma(c)-c^{p}$ in $V$, and
$\sigma(\pi)=\pi$. Denote also by $\sigma$ its extension to $K$. Both the
rigid cohomology with support in $Z$ and the overconvergent Dwork cohomology
admit $\sigma$-semilinear Frobenius actions.

######  Theorem 4.5.

Then the isomorphism $u$ is an isomorphism of $K$-vector spaces with
$\sigma$-linear semilinear maps.

###### Proof.

Let $\phi\colon A^{\dagger}\to A^{\dagger}$ be a lift of the Frobenius. It
induces a Frobenius operator on the spaces $B^{\dagger}_{i}$,
$i=1,2,\ldots,r$, and on the tensor product $\widetilde{B}^{\dagger}$. The
operator on $\widetilde{B}^{\dagger}$ induces the Frobenius on the rigid
cohomology $\mathrm{H}^{i}_{\mathrm{rig},Z}(\operatorname{Spec}\overline{A})$.

Extend the operator $\phi$ to $A^{\dagger}\langle
T_{1},\ldots,T_{r}\rangle^{\dagger}$ by

$\textstyle\phi\left(\sum\limits_{I\subset\mathbf{N}^{r}}a_{I}(x)T^{I}\right)=\sum\limits_{I\subset\mathbf{N}^{r}}\phi(a_{I}(x))T^{pI}.$

The Frobenius operator on the (inverse image of the) Dwork crystal
$g^{\ast}\mathcal{L}$ is presented by a $\phi$-semilinear map

$\varphi\colon A^{\dagger}\langle
T_{1}\rangle^{\dagger}\mathop{\overset{\mathbf{L}}{\underset{A^{\dagger}}{\otimes}}{\\!}^{\dagger}}\cdots\mathop{\overset{\mathbf{L}}{\underset{A^{\dagger}}{\otimes}}{\\!}^{\dagger}}A^{\dagger}\langle
T_{r}\rangle^{\dagger}\to A^{\dagger}\langle
T_{1}\rangle^{\dagger}\mathop{\overset{\mathbf{L}}{\underset{A^{\dagger}}{\otimes}}{\\!}^{\dagger}}\cdots\mathop{\overset{\mathbf{L}}{\underset{A^{\dagger}}{\otimes}}{\\!}^{\dagger}}A^{\dagger}\langle
T_{r}\rangle^{\dagger},$

which can be chosen as
$\varphi(\xi)=\phi(\xi)\cdot\bigotimes_{i=1}^{r}\theta_{i}(x,T_{i})$, where
$\theta_{i}(x,T_{i})$ is the overconvergent series $\exp(\pi
T_{i}\widehat{f}_{i}-\pi T_{i}^{p}\phi(\widehat{f}_{i}))$. We must check that
under the isomorphism given in Lemma 3.13, the operator $\varphi$ is
transformed to the action of $\phi$. It suffices to check this on each
individual factor.

Recall the map $b\mapsto\mathrm{Init}(b)$ defined in the proof of Lemma 3.14.
For $b\in B_{i}^{\dagger}$, $\mathrm{Init}(b)=\sum
b\cdot\frac{\pi^{m}\widehat{f}_{i}^{m}}{m!}T^{m}_{i}\in B^{\dagger}\langle
T_{i}\rangle$ fall in the kernel of $D_{B_{i}}^{\dagger}$. Then

$\varphi(\mathrm{Init}(b))=\theta_{i}(x,T_{i})\cdot\phi(b)\cdot\sum\frac{\pi^{m}\phi(\widehat{f}_{i})^{m}}{m!}T^{pm}_{i}.$

Since the constant term of $\theta_{i}(x,T_{i})$ is $1$, the constant term of
$\varphi(\mathrm{Init}(b))$ equals $\phi(b)$. Since it is still horizontal, it
has to be equal to $\mathrm{Init}(\phi(b))$. This completes the verification.
∎

The theorem also implies a result on cohomology with compact supports by
taking Poincaré duality. Before stating it let us recall some standard
notation about Tate twists.

For an integer $w$, let $K(w)$ be the $1$-dimensional vector space over $K$
equipped with a $\sigma$-semilinear map $x\mapsto p^{-w}\sigma(x)$. For a
finite dimensional $K$-vector space $M$ equipped with a $\sigma$-semilinear
map, we use $M(w)$ to denote the tensor product $M\otimes_{K}K(w)$, whose
$\sigma$-semilinear map is twisted by $p^{-w}$. Note that $(M(w))^{\ast}\simeq
M^{\ast}(-w)$ as spaces equipped with $\sigma$-semilinear maps.

######  Corollary 4.6.

Then the isomorphism $v$ of Corollary 4.4 induces an isomorphism

$\mathrm{H}^{i}_{\mathrm{rig},c}(Z)(-r)\xrightarrow{\sim}\mathrm{H}^{i+2r}_{\mathrm{rig},c}(\mathbf{A}^{r}_{\overline{A}},g^{\ast}\mathcal{L}_{\pi}).$

###### Proof.

Poincaré duality in rigid cohomology (see [le-stum:rigid-cohomology]*Corollary
8.3.14, and [kedlaya:finiteness-rigid-cohomology]*Theorem 1.2.3; note the
latter article ignores the Tate twist) asserts that, for a nonsingular,
_connected_ variety $X$ of pure dimension $d$ over $k$, and an overconvergent
$F$-isocrystal $E$ on $X$, the pairing

$\mathrm{H}^{i}_{\mathrm{rig},c}(X,E)\otimes_{K}\mathrm{H}^{2d-i}_{\mathrm{rig}}(X,E^{\vee})\to\mathrm{H}^{2d}_{\mathrm{rig},c}(X)\simeq
K(-d)$

is a perfect pairing. In other words,

$\mathrm{H}^{i}_{\mathrm{rig},c}(X,E^{\vee})\simeq(\mathrm{H}^{2d-i}_{\mathrm{rig}}(X,E)(d))^{\ast}.$

Similarly, for any closed subvariety $Z$ of $X$, we have a perfect pairing

$\mathrm{H}^{i}_{\mathrm{rig},Z}(X)\otimes_{K}\mathrm{H}^{2d-i}_{\mathrm{rig},c}(Z)\to
K(-d).$

Applying these facts to the overconvergent Dwork cohomology yields

$\displaystyle\mathrm{H}^{i+2r}_{\mathrm{rig},c}(\mathbf{A}^{r}_{\overline{A}},g^{\ast}\mathcal{L}_{\pi})$
$\displaystyle\simeq(\mathrm{H}^{2n-i}_{\mathrm{rig}}(\mathbf{A}^{r}_{\overline{A}},g^{\ast}\mathcal{L}_{-\pi})(n+r))^{\ast}$
(by Theorem 4.5)
$\displaystyle\simeq(\mathrm{H}^{2n-i}_{\mathrm{rig},Z}(\operatorname{Spec}\overline{A})(n+r))^{\ast}=(\mathrm{H}^{2n-i}_{\mathrm{rig},Z}(\operatorname{Spec}\overline{A})(n))^{\ast}(-r)$
$\displaystyle\simeq\mathrm{H}^{i}_{\mathrm{rig},c}(Z)(-r).$

This completes the proof. ∎

##  5\. The Artin–Hasse presentation

From this section onward, we assume in addition that $k=\mathbf{F}_{q}$ is a
finite field, where $q=p^{a}$ is a power of $p$. Thus
$K=W(\mathbf{F}_{q})(\zeta_{p})$ is the unramified extension of
$\mathbf{Q}_{p}(\zeta_{p})$ of degree $a$.

Let $g\in k[x_{1},\ldots,x_{N}]$ be a polynomial. The main purpose of this
section is to explain another presentation of the Dwork crystal
$g^{\ast}\mathcal{L}_{\pi}$ in terms of the Artin–Hasse exponentials.

To begin with, we define a set of operators which will play the role of
“inverse Frobenii”. The reader is referred to Dwork’s article [dwork:zeta-
function-hypersurfaces-1]*§4 for the assertions made in this paragraph. Recall
that the Artin–Hasse exponential is defined by

$E(z)=\exp\left\\{\sum_{m=0}^{\infty}\frac{z^{p^{m}}}{p^{m}}\right\\}.$

Then $E(z)\in\mathbf{Z}_{p}[\\![z]\\!]$. Let $\gamma$ be a root of the power
series $\sum_{m=0}^{\infty}\frac{z^{p^{m}}}{p^{m}}$ satisfying
$|\gamma|=|\pi|$ (the existence of $\gamma$ is proven by examining the Newton
polygon, for example). Let $\vartheta(z)=E(\gamma t)$. Since
$E\in\mathbf{Z}_{p}[\\![z]\\!]$, $\vartheta(z)$ defines a rigid analytic
function on the open disk $\\{|z|<|p|^{\frac{1}{p-1}}\\}$ bounded by $1$. For
each choice of $\gamma$, it is well-known that

(5.1) $\Psi(t)=\vartheta(1)^{\mathrm{Tr}_{\mathbf{F}_{q}/\mathbf{F}_{p}}(t)}$

is a nontrivial additive character on $\mathbf{F}_{q}$.

For $b>0$, let $B(b)$ be the Banach space of analytic functions on the closed
polydisk
$\mathbf{D}^{+}(0;|p|^{-b})^{N}=\\{|z_{i}|\leq|p|^{-b}\\}\subset\mathbf{A}^{N,\mathrm{an}}_{K}$.
Then $B=\mathop{\operatorname{colim}}\limits_{b\to 0^{+}}B(b)$ equals the
Monsky–Washnitzer algebra in $N$ variables.

For each subset $I$ of $S=\\{1,2,\ldots,N\\}$, let $x^{I}$ be the product
$\prod_{i\in I}x_{i}$, and let $B_{I}=x^{I}B$, which is a closed subspace of
$B$. Similarly one can define $B_{I}(b)$.

Next we define operators which operate on $B_{I}$. Write
$g(x)=\sum_{u}a_{u}x^{u}$. Let $A_{u}$ be the Teichmüller lift of $a_{u}$. Set
$\widehat{g}(x)=\sum A_{u}x^{u}$. Define

$G(x)=\prod_{u}\vartheta(A_{u}x^{u}).$

Then it is easy to see that $G$ falls in $B(b)$ for any $b<\frac{1}{d(p-1)}$,
where $d$ is the degree of the polynomial $g$.

Define an operator $\psi\colon B(b)\to B(pb)$ by

$\psi\left(\sum a_{w}x^{w}\right)=\sum a_{pw}x^{w}.$

For $0<b<\frac{p}{d(p-1)}$, we define an endomorphism $\alpha_{0}$ of $B(b)$
by

(5.2) $\alpha_{0}\colon B(b)\hookrightarrow B(p^{-1}b)\xrightarrow{\cdot
G}B(p^{-1}b)\xrightarrow{\sigma^{-1}\circ\psi}B(b).$

Then $\alpha_{0}$ is $\mathbf{Q}_{p}(\zeta_{p})$-linear, and is
$\sigma^{-1}$-semilinear over $K$. Its $a$th iteration $\alpha_{0}^{a}$, which
is $K$-linear, is denoted by $\alpha$.

The operators $\alpha_{0}$ and $\alpha$ both preserve the spaces $B_{I}(b)$.
Passing to colimit, $\alpha_{0}$ and $\alpha$ induce $\sigma^{-1}$-semilinear
resp. linear endomorphisms on the spaces $B_{I}$. We will informally refer to
$\alpha$ as the “inverse Frobenius” operator, although it is, on the chain
level, only a left inverse.

######  Remark 5.3.

For suitable $b$, the decomposition (5.2) implies that operators
$\alpha|_{B_{I}(b)}$ are completely continuous. By Monsky’s theorem
[monsky:formal-cohomology-3]*Theorem 2.1, each $\alpha|_{B_{I}}$ is then a
nuclear operator, and its Fredholm determinant $\det(1-t\cdot\alpha\mid
B_{I})$ is a $p$-adic entire function. Moreover, if $b$ is not too large, the
infinite matrices of $\alpha$ on all the spaces $B_{I}(b)$ are similar (one
sees this by choosing suitable scalar multiples of monomials as orthonormal
basis). It follows that the Fredholm determinant $\det(1-t\cdot\alpha\mid
B_{I})$ equals $\det(1-t\cdot\alpha\mid B_{I}(b))$.

Next, let us define another integrable connection similar to the integrable
connection presenting the overconvergent F-isocrystal
$g^{\ast}\mathcal{L}_{\pi}$. We call this complex the “Artin–Hasse
presentation”.

As the reader will see, compared with the usual connection presenting
$g^{\ast}\mathcal{L}_{\pi}$, the connection operator of the Artin–Hasse
presentation are transcendental; and is only defined on a proper subspace of
entire rigid analytic affine line. However, it has the virtue that its
“inverse Frobenius” operator has better convergence property, which is crucial
for the chain level argument of §8.

The reader is referred to the articles [dwork:zeta-function-hypersurfaces-1,
adolphson-sperber:chevalley-warning, adolphson-sperber:exponential-sums-
newton-polyhedra] for more discussions (including motivations) about the
following definitions.

Write $\gamma_{\ell}=\sum_{i=0}^{\ell}\frac{\gamma^{p^{i}}}{p^{i}}$,
$h(t)=\sum_{\ell=0}^{\infty}\gamma_{\ell}t^{p^{\ell}}$, and

(5.4)
$H(x)=\sum_{\ell=0}^{\infty}\gamma_{\ell}\widehat{f}^{\sigma^{\ell}}(x^{p^{\ell}}).$

Then $H\in B(b)$ for any $b<\frac{1}{d(p-1)}$, and
$x_{i}\frac{\partial}{\partial x_{i}}H\in B(b)$ for any
$0<b<\frac{p}{d(p-1)}$.

Let

(5.5) $D_{i}=x_{i}\frac{\partial}{\partial
x_{i}}+x_{i}\frac{\partial}{\partial x_{i}}H.$

Then one checks that $D_{i}\circ D_{j}=D_{j}\circ D_{i}$, and

(5.6) $\alpha\circ D_{i}=qD_{i}\circ\alpha.$

Let
$\widetilde{\nabla}_{g}=\sum_{i=1}^{N}D_{i}\cdot\frac{\mathrm{d}x_{i}}{x_{i}}$.
Then $\widetilde{\nabla}_{g}$ is an integrable connection on the rigid
analytic closed disk
$\mathbf{D}^{+}(0;|p|^{-b})^{N}=\\{|z_{i}|\leq|p|^{-b}\\}$.

The space of $i$-forms on the rigid analytic closed disk can be written as a
direct sum

$\Omega^{i}_{\mathbf{D}^{+}(0;|p|^{-b})^{N}}\cong\bigoplus_{|I|=i}B_{I}(b)\frac{\mathrm{d}x^{I}}{x^{I}}$

where $\frac{\mathrm{d}x^{I}}{x^{I}}$ the wedge power of
$\frac{\mathrm{d}x_{i}}{x_{i}}$, in the increasing order. Thus the de Rham
complex of $\widetilde{\nabla}_{g}$ can be written as:

(5.7)
$\underset{\bullet}{B(b)}\to\bigoplus_{|I|=1}B_{I}(b)\frac{\mathrm{d}x^{I}}{x^{I}}\to\bigoplus_{|I|=2}B_{I}(b)\frac{\mathrm{d}x^{I}}{x^{I}}\to\cdots\to
B_{\\{1,2,\ldots,N\\}}(b)\frac{\mathrm{d}x_{1}\wedge\cdots\wedge\mathrm{d}x_{N}}{x_{1}\cdots
x_{N}}.$

Regarding the monomial forms $\mathrm{d}x^{I}/x^{I}$ as a “placeholders”, and
suppressing them in (5.7) above, we can express the de Rham complex of
$\widetilde{\nabla}_{g}$ in a simpler form:

(5.7′)
$\underset{\bullet}{B(b)}\to\bigoplus_{|I|=1}B_{I}(b)\to\bigoplus_{|I|=2}B_{I}(b)\to\cdots.$

######  Remark 5.8.

The operators $D_{i}$ (5.5) are defined using the “toric convention”. Their
“affine” counterparts are $x_{i}^{-1}D_{i}$. Using the affine convention, the
complex (5.7′) is simply the shifted Koszul complex
$\mathrm{Kos}(B(b);x_{1}^{-1}D_{1},\ldots,x_{N}^{-N}D_{N})[-N]$. The reason
that we use the “toric” convention is because the following “inverse Frobenius
operator” can be defined more naturally.

On (5.7′), the operator

$\bigoplus_{|I|=i}q^{-i}\alpha|_{B_{I}(b)}\colon\bigoplus_{|I|=i}B_{I}(b)\to\bigoplus_{|I|=i}B_{I}(b)$

acts (thus, in terms of (5.7),
$\alpha(\mathrm{d}x^{I}/x^{I})=q^{-i}\mathrm{d}x^{I}/x^{I}$). Thanks to the
relation (5.6), these operators form a chain map of the de Rham complex
(5.7′).

Letting $b\to 0^{+}$ in (5.7′) yields an overconvergent de Rham complex

(5.9)
${\underset{\bullet}{B}}$${\bigoplus\limits_{|I|=1}B_{I}}$${\bigoplus\limits_{|I|=2}B_{I}}$${\cdots}$${B}$${\bigoplus\limits_{|I|=1}B_{I}}$${\bigoplus\limits_{|I|=2}B_{I}}$${\cdots}$$\scriptstyle{\alpha}$$\scriptstyle{\sum
q^{-1}\alpha|_{B_{I}}}$$\scriptstyle{\sum q^{-2}\alpha|_{B_{I}}}$

equipped with “inverse Frobenii”.

The following lemma of Peigen Li shows that the Artin–Hasse presentation is
indeed a presentation of the Dwork crystal.

######  Lemma 5.10 ([li_peigen:adolphson-sperber-vs-rigid]*Proposition 2.1).

The cohomology of the complex (5.9) is isomorphic to the overconvergent Dwork
cohomology of $g^{\ast}\mathcal{L}_{\pi}$. The induced action of
$q^{-*}\sum_{|I|=*}\alpha|_{B_{I}}$ on cohomology equals the inverse of the
Frobenius on the overconvergent Dwork cohomology.

###### Indication of proof.

The de Rham complex of the ordinary presentation of the Dwork crystal also
looks like (5.9), except the latter has a different set of derivations. The
upshot is that the differentials of the de Rham complex of the ordinary
presentation are the exponential twists

$\exp(\pi\widehat{g})\circ\mathrm{d}\circ\exp(-\pi\widehat{g}),$

whereas the differentials of (5.9) are given by different exponential twists

$\textstyle\exp(\sum_{u}h(A_{u}x^{u})\circ\mathrm{d}\circ\exp(-\sum_{u}h(A_{u})x^{u})$

(the series $h$ was defined in 5.4). The same remark applies to the respective
“inverse Frobenii”.

Formally, multiplying the “untwist-then-retwist” function

(5.11) $\textstyle R=\exp(\sum_{u}h(A_{u}x^{u})-\pi\widehat{g})$

gives an isomorphism from “usual” de Rham complex to (5.9) and it commutes
with the inverse Frobenii. In loc. cit., Peigen Li showed $R$ is indeed
overconvergent. Therefore, the “untwist-then-retwist” operation is indeed
well-defined. ∎

##  6\. Visibility of Frobenius eigenvalues

######  Notation.

  * •

Let $\Psi\colon\mathbf{F}_{q}\to\mathbf{C}_{p}^{\ast}$ be the nontrivial
additive character defined by
$\Psi(x)=\vartheta(1)^{\operatorname{Tr}_{\mathbf{F}_{q}/\mathbf{F}_{p}}(x)}$,
see (5.1).

  * •

Let $f_{1},\ldots,f_{r}\in\mathbf{F}_{q}[x_{1},\ldots,x_{n}]$ be a collection
of polynomials.

  * •

For every subset $I\subset\\{1,2,\ldots,r\\}$, set

$Z_{I}=\operatorname{Spec}\mathbf{F}_{q}[x_{1},\ldots,x_{n}]/(f_{i}:i\in I),$

and $Z_{I}^{\ast}=Z_{I}\cap\mathbf{G}_{\mathrm{m}}^{n}$. Write
$Z=Z_{\\{1,2,\ldots,r\\}}$.

  * •

In Section 5, set $N=n+r$, and let
$g=\sum_{i=1}^{r}x_{n+i}f_{i}\in\mathbf{F}_{q}[x_{1},\ldots,x_{n+r}]$.

The purpose of this section is to prove Theorem 1.1, that is, we want to show
that the Frobenius eigenvalues of $Z$ are weakly visible in the zeta functions
of $\zeta_{Z_{I}^{\ast}}$.

The idea of the proof of Theorem 1.1 can be explained as follows:

  1. (1)

Renaming $T_{i}$ by $x_{n+i}$, Corollary 4.6, applying to $g$, shows that the
overconvergent Dwork cohomology associated to $g$ computes the rigid
cohomology of $Z$. Hence the Frobenius eigenvalues of $Z$ can be manifestly
computed using the operator $\alpha$. This is our starting point of the proof
of Theorem 1.1.

  2. (2)

As we have seen, the overconvergent Dwork cohomology of $g$ admits an explicit
chain model (5.9). The chain level operators $\alpha|_{B_{I}}$ are “nuclear
operators”. Since cohomology spaces are subquotients of $\bigoplus B_{I}$,
Monsky’s spectral theory (see Lemmas 6.1, 6.2) implies that the cohomological
eigenvalues are also “eigenvalues” of $\bigoplus\alpha|_{B_{I}}$. In the
latter context, “eigenvalue” should be interpreted as the reciprocal roots of
the Fredholm determinant of $\alpha$. Thus, Frobenius eigenvalues are visible
in the Fredholm determinants $\det(1-t\alpha\mid B_{I})$.

  3. (3)

The spaces $B_{I}$ are all subspaces of $B$, on which $\alpha$ also operates.
Thus $\det(1-t\alpha\mid B)$ witnesses all the “chain level eigenvalues” of
$\alpha|_{B_{I}}$. The Dwork trace formula, applying to $\alpha|_{B}$, equates
an alternating product of $\det(1-t\cdot q^{m}\alpha\mid B)$ with an
alternating product of zeta functions (see Theorem 6.8 and Lemma 6.9). Möbius
inversion (see the formulae in Definition 6.5) then allows us to represent
$\det(1-t\alpha\mid B)$ as an infinite product of zeta functions. The
Frobenius eigenvalues, which are visible in the Fredholm determinant, are
thereby weakly visible in the zeta functions.

Let us carry out the details. To show the Fredholm determinants
$\det(1-t\alpha\mid B_{I})$ contain all the information of Frobenius
eigenvalues, we need the following lemma, extracted from [monsky:formal-
cohomology-3]*Theorem 1.4.

######  Lemma 6.1.

Let $X$ be a variable. Let
$M^{\prime}\xrightarrow{f}M\xrightarrow{g}M^{\prime\prime}$ be a complex of
nuclear $K[X]$-modules in the sense of [monsky:formal-cohomology-3]*Definition
1.4. Then $H=\operatorname{Ker}g/\operatorname{Im}f$ is also nuclear, and
$\det(1-t\cdot X\mid H)$ is a factor of $\det(1-t\cdot X\mid M)$.

###### Proof.

We follow Monsky’s notation [monsky:formal-cohomology-3]*§1. By
[monsky:formal-cohomology-3]*Theorem 1.4, $H$ is a nuclear $K[X]$-module. It
follows that for any bounded subset ([monsky:formal-cohomology-3]*Definition
1.3) $S\subset K[X]\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}XK[X]$, we
have a decomposition $H=N(S,H)\oplus F(S,H)$ ([monsky:formal-
cohomology-3]*Theorem 1.1). By [monsky:formal-cohomology-3]*Proof of Theorem
1.4, eighth line, $N(S,H)$ is a subquotient of $N(S,M)$. As both $N(S,H)$ and
$N(S,M)$ are finite dimensional, $\det(1-tX\mid N(S,H))$ divides
$\det(1-tX\mid N(S,M))$. Taking limit by letting $S$ run through all bounded
subsets, we conclude that $\det(1-tX\mid H)$ is a factor of $\det(1-tX\mid
M)$. ∎

######  Lemma 6.2.

The product $\prod_{|I|=i}\det(1-t\cdot q^{-i}\alpha\mid B_{I})$ contains
$\det(1-tF^{-1}\mid\mathrm{H}^{i}_{\mathrm{rig}}(\mathbf{A}^{n+r},g^{*}\mathcal{L}_{-\pi}))$
as a factor.

###### Proof.

By [monsky:formal-cohomology-3]*Theorem 1.6 and the discussions in (5.3), all
summands $B_{I}$ of (5.9) are nuclear $K[X]$-modules, where the action of $X$
coming from that of $\alpha$. The lemma then follows from Lemma 6.1. ∎

######  Lemma 6.3.

For each $I\subset S$, $\det(1-t\cdot q^{-i}\alpha\mid B_{I})$ is a $p$-adic
entire function. Moreover, the polynomial
$\det(1-tF\mid\mathrm{H}^{n-r+j}_{\mathrm{rig},c}(Z))$ is a factor of

$\prod\limits_{|I|=n+r-j}\det(1-t\cdot q^{j-r}\alpha\mid B_{I}).$

###### Proof.

As $\alpha$ is nuclear, the series $\det(1-t\cdot q^{-i}\alpha\mid B_{I})$ is
a $p$-adic entire function by Monsky’s trace formula. Now we have the
following rather straightforward computation:

$\displaystyle\det(1-tF^{-1}\mid\mathrm{H}^{i}_{\mathrm{rig}}(\mathbf{A}^{n+r},g^{*}\mathcal{L}_{-\pi}))$
$\displaystyle=$
$\displaystyle\det(1-tF^{-1}\mid\mathrm{H}_{\mathrm{rig},c}^{2n+2r-i}(\mathbf{A}^{n+r},g^{*}\mathcal{L}_{\pi})^{\ast}(-n-r))$
$\displaystyle=$
$\displaystyle\det(1-tq^{n+r}F^{-1}\mid\mathrm{H}_{\mathrm{rig},c}^{2n+2r-i}(\mathbf{A}^{n+r},g^{*}\mathcal{L}_{\pi})^{*})$
$\displaystyle=$
$\displaystyle\det(1-tq^{-n-r}F\mid\mathrm{H}_{\mathrm{rig},c}^{2n+2r-i}(\mathbf{A}^{n+r},g^{*}\mathcal{L}_{\pi}))$
$\displaystyle=$
$\displaystyle\det(1-tq^{-n}F\mid\mathrm{H}_{\mathrm{rig},c}^{2n-i}(Z)).$

The last step used Corollary 4.6. Making a change of variable $2n-i=n-r+j$,
Lemma 6.2 implies that $\det(1-tF\mid\mathrm{H}^{n-r+j}_{\mathrm{rig},c}(Z))$
is a factor of $\prod\limits_{|I|=n+r-j}\det(1-t\cdot q^{j-r}\alpha\mid
B_{I})$. ∎

######  Corollary 6.4.

Any Frobenius eigenvalue of $\mathrm{H}^{\ast}_{\mathrm{rig},c}(Z)$ is weakly
visible in the $p$-adic entire function $\det(1-t\alpha\mid B)$.

###### Proof.

By Lemma 6.3, The Frobenius eigenvalue of
$\mathrm{H}^{2n-i}_{\mathrm{rig},c}(Z)$ is weakly visible in the product
$\prod\limits_{|I|=i}\det(1-t\alpha\mid B_{I})$. The corollary follows since
$B_{I}$ is a nuclear submodule of $B$. ∎

Next, let us explain how to read off the Fredholm determinants from the zeta
functions.

######  Definition 6.5.

Following Dwork, we introduce an operation $\delta$ on the set
$1+t\mathbf{C}_{p}[\\![t]\\!]$ of formal power series with constant term one:

$\delta(\Gamma(t))\overset{\mathrm{def}}{=\\!=}\frac{\Gamma(t)}{\Gamma(qt)}.$

The endomorphism $\delta$ is invertible, and its inverse reads

$\delta^{-1}(\Gamma(t))=\prod_{i=0}^{\infty}\Gamma(q^{i}t).$

######  Lemma 6.6.

If $\Gamma(t)\in 1+t\mathbf{C}_{p}[\\![t]\\!]$ is an entire function, then
$\delta^{-1}\Gamma(t)$ is also an entire function.

###### Proof.

Let $\Gamma(t)=\prod_{j=1}^{\infty}(1-\gamma_{i}t)$ be its infinite product
expansion, where the reciprocal zero $\gamma_{j}$ approches zero as $j$ goes
to infinity. Then

$\delta^{-1}(\Gamma(t))=\prod_{i=0}^{\infty}\prod_{j=1}^{\infty}(1-q^{i}\gamma_{j}t)$

is also such an infinity product whose reciprocal zero approches zero. ∎

######  Lemma 6.7.

Assume that $\Gamma\in 1+t\mathbf{C}_{p}[\\![t]\\!]$ is a $p$-adic meromorphic
function on $\mathbf{C}_{p}$, and $\lambda\in\mathbf{C}_{p}$ is weakly visible
in $\Gamma$. Then $\lambda$ is weakly visible in $\delta(\Gamma)$.

###### Proof.

Let $Z(t)=\delta(\Gamma)$. Then $\Gamma(t)=\prod_{i=0}^{\infty}Z(q^{i}t)$.
Write $Z(t)=u(t)/v(t)$, where $u,v$ are entire functions, without common
zeros. Then

$\Gamma(t)=\prod_{i=0}^{\infty}\frac{u(q^{i}t)}{v(q^{i}t)}=\frac{\delta^{-1}(u(t))}{\delta^{-1}(v(t))}.$

By Lemma 6.6, $\delta^{-1}u(t)$ and $\delta^{-1}v(t)$ are entire. If
$q^{m}\lambda$ is a reciprocal zero of $\Gamma(t)$. Then $1-q^{m}\lambda t$
must be a factor of the infinite product
$\delta^{-1}(u(t))=\prod_{i}u(q^{i}t)$. Hence $q^{m}\lambda$ is a reciprocal
zero of $u(q^{i}t)$ for some $i$, i.e., $q^{m-i}\lambda$ is a reciprocal zero
of $Z(t)$. Thus $\lambda$ is weakly visible in $Z(t)$. The polar case is
similar. ∎

At this point, we recall the Dwork trace formula [dwork:rationality-zeta-
function]*Lemma 2, p. 637. The following overconvergent version is due to
Monsky [monsky:formal-cohomology-3]*Theorem 5.3.

######  Theorem 6.8 (Dwork trace formula).

For each positive integer $m$, define the $m$-th toric exponential sum

$S^{\ast}_{m}(g)=\sum\limits_{x\in\mathbf{G}_{\mathrm{m}}^{n+r}(\mathbf{F}_{q^{m}})}(\Psi\circ\operatorname{Tr}_{\mathbf{F}_{q^{m}}/\mathbf{F}_{q}})(g(x)).$

Then

$L^{*}(t)\overset{\mathrm{def}}{=\\!=}\exp\left\\{\sum_{m=1}^{\infty}S_{m}^{\ast}(g)\frac{t^{m}}{m}\right\\}=\left\\{\delta^{n}\big{(}\det(1-t\alpha\mid
B)\big{)}\right\\}^{(-1)^{n-1}}.$

Combining Corollary 6.4, Lemma 6.7, and Theorem 6.8, we infer that any
Frobenius eigenvalue of $\mathrm{H}^{\bullet}_{\mathrm{rig},c}(Z)$ is weakly
visible in $L^{\ast}(t)$.

Our next objective is to relate the function $L^{\ast}(t)$ to the zeta
functions of $Z_{I}$.

######  Lemma 6.9.

We have

$L^{*}(t)=\prod_{J\subset\\{1,2,\ldots,r\\}}\zeta_{Z^{*}_{J}}(q^{|J|}t)^{(-1)^{r-|J|}}.$

###### Proof.

For any subset $J$ of $\\{1,2,\ldots,r\\}$, let $\mathbf{A}^{J}$ denote the
affine space with coordinates $(y_{j})_{j\in J}$, and $g_{J}$ the regular
function $\sum_{j\in J}y_{j}f_{j}(x)$. The orthogonality of characters implies

$S_{J}\overset{\text{def}}{=\\!=}\sum\limits_{x\in\mathbf{G}_{\mathrm{m}}^{n}(\mathbf{F}_{q})}\sum_{y\in\mathbf{A}^{J}(\mathbf{F}_{q})}\Psi(g_{J}(x,y))=q^{|J|}|Z^{*}_{J}(\mathbf{F}_{q})|$

Since $\mathbf{A}^{n}=\mathbf{G}_{\mathrm{m}}^{n}\cup D$, where $D$ is the
union of all coordinate hyperplanes, and since $D$ admits the following
standard semisimplicial resolution

${\cdots}$${\coprod\limits_{\begin{subarray}{c}J\subset\\{1,2,\ldots,r\\}\\\
|J|=r-2\end{subarray}}\mathbf{A}^{J}}$${\coprod\limits_{\begin{subarray}{c}J\subset\\{1,2,\ldots,r\\}\\\
|J|=r-1\end{subarray}}\mathbf{A}^{J}}$${D,}$

we deduce from the inclusion-exclusion that

$\displaystyle\sum_{x\in\mathbf{G}_{\mathrm{m}}^{n+r}(\mathbf{F}_{q})}\Psi(g(x))$
$\displaystyle=\sum_{J\subset\\{1,2,\ldots,r\\}}(-1)^{r-|J|}S_{J}$
$\displaystyle=\sum_{J\subset\\{1,2,\ldots,r\\}}(-1)^{r-|J|}q^{|J|}|Z_{J}^{*}(\mathbf{F}_{q})|.$

Replacing $\mathbf{F}_{q}$ by $\mathbf{F}_{q^{m}}$ and $\Psi$ by
$\Psi\circ\mathrm{Tr}_{\mathbf{F}_{q^{m}}/\mathbf{F}_{q}}$ in the above
calculation, and exponentiating, we get the desired result. ∎

###### Proof of Theorem 1.1.

By Corollary 6.4, the Frobenius eigenvalues of
$\mathrm{H}^{\ast}_{\mathrm{rig},c}(Z)$ are weakly visible in
$\det(1-t\alpha\mid B)$. By Lemma 6.7 and Theorem 6.8, the Frobenius
eigenvalues are weakly visible in $L^{\ast}(t)$. By Lemma 6.9, they are weakly
visible in $\\{\zeta_{Z_{J}^{*}}(t):J\subset\\{1,2,\ldots,r\\}\\}$. ∎

######  Remark 6.10.

The same proof also works for subvarieties in $\mathbf{G}_{\mathrm{m}}^{n}$.
That is, if $\\{f_{1},\ldots,f_{r}\\}$ is a set of Laurent polynomials, and
$Z_{I}$ is the vanishing scheme of $(f_{i})_{i\in I}$, then the Frobenius
eigenvalues of $Z_{\\{1,2,\ldots,r\\}}$ is weakly visible in the set
$\\{\zeta_{Z_{I}}(t):I\subset\\{1,2,\ldots,r\\}\\}$.

##  7\. Some algebra

The goal of this section is to prove a technical lemma, Lemma 7.3, that will
be used in the proof of Theorem 1.2. Throughout this section, we assume that
$k$ is an algebraically closed field. The common zero locus, in
$\mathbf{A}_{k}^{n}$, of a collection of polynomials $f_{1},\ldots,f_{r}\in
k[x_{1},\ldots,x_{n}]$ will be denoted by $Z(f_{1},\ldots,f_{r})$.

######  Lemma 7.1.

Suppose that $Z(g_{1},\ldots,g_{m})$ is a set-theoretic complete intersection
in $\mathbf{A}^{n}_{k}$, and

(7.2) $\dim Z(g_{1},...,g_{m})>\dim
Z(g_{1},\ldots,g_{m},f_{m+1},\ldots,f_{r}).$

Then, there are constants $c_{m+1},\ldots,c_{r}$ in $k$ such that

  * •

if $f_{m+1}$ does not vanish on $Z(g_{1},\ldots,g_{m})$, $c_{m+1}=1$,

  * •

if $f_{m+1}$ vanishes identically on $Z(g_{1},\ldots,g_{m})$, $c_{m+1}=0$, and

  * •

$Z(g_{1},\ldots,g_{m},c_{m+1}f_{m+1}+\cdots+c_{r}f_{r})$ is a set theoretic
complete intersection in $\mathbf{A}^{n}_{k}$.

###### Proof.

Denote the irreducible components of the variety $Z(g_{1},...,g_{m})$ by
$D_{1},\ldots,D_{h}$. These components have the same dimension by the
unmixedness theorem. If $f_{m+1}$ vanishes (identically) on all components
$D_{1},\ldots,D_{h}$, then $\dim Z(g_{1},...,g_{m},f_{m+1})=\dim
Z(g_{1},...,g_{m})$. In this case, we set $c_{m+1}=0$, drop $f_{m+1}$ from our
list, and consider the shorter list
$\\{g_{1},\ldots,g_{m},f_{m+2},\ldots,f_{r}\\}$ which still satisfies the
condition (7.2).

Hence, without loss of generality, we may assume that $f_{m+1}$ does not
identically vanish on $D_{1},\ldots,D_{h_{1}}$ with $h_{1}>0$, but vanishes on
$D_{h_{1}+1},\ldots,D_{h}$. If $h_{1}=h$, then

$\dim Z(g_{1},\ldots,g_{m},f_{m+1})<\dim Z(g_{1},\ldots,g_{m}).$

Since the variety $Z(g_{1},\ldots,,g_{m},f_{m+1})$ is non-empty, this forces
that

$\dim Z(g_{1},\ldots,g_{m},f_{m+1})=\dim Z(g_{1},\ldots,g_{m})-1,$

that is, $Z(g_{1},...,g_{m},f_{m+1})$ is a set theoretic complete
intersection.

Assume now $h_{1}<h$. By condition (7.2), there is another polynomial among
$\\{f_{m+2},\ldots,f_{r}\\}$, say $f_{m+2}$, which does not vanish identically
on all of $D_{h_{1}+1},\ldots,D_{h}$. Without loss of generality, we may
assume that $f_{m+2}$ dos not vanish on $D_{h_{1}+1}$, $\ldots$,
$D_{h_{1}+h_{2}}$ with $h_{2}>0$, but vanishes on
$D_{h_{1}+h_{2}+1},\ldots,D_{h}$.

Claim. There is a non-zero constant $c$ in $k$ such that $f_{m+1}+cf_{m+2}$
does not vanish on $D_{1},\ldots,D_{h_{1}},\ldots,D_{h_{1}+h_{2}}$.

###### Proof of Claim.

Because $f_{m+1}$ vanishes on $D_{h_{1}+1},\ldots,D_{h_{1}+h_{2}}$, and
$f_{m+2}$ does not, for any non-zero constant $c$ in $k$, the polynomial
$f_{m+1}+cf_{m+2}$ does not vanish on $D_{h_{1}+1},\ldots,D_{h_{1}+h_{2}}$.
For each $i=1,\ldots,h_{1}$, we can choose $x_{i}$ in $D_{i}$ such that
$f_{m+1}(x_{i})$ is non-zero as $f_{m+1}$ is not identically zero on $D_{i}$.
Choose non-zero constant $c$ in $k$ such that none of the $h_{1}$ numbers

$f_{m+1}(x_{i})+cf_{m+2}(x_{i}),i=1,\ldots,h_{1}$

is zero: one simply chooses any non-zero $c$ in $k$ such that $c$ is not among
the $h_{1}$ numbers $\\{-f_{m+1}(x_{i})/f_{m+2}(x_{i}),i=1,\ldots,h_{1}\\}$,
which is possible since $k$ is an infinite field. The claim is proved. ∎

Repeating the above procedure, we see there are constants
$c_{m+1},\ldots,c_{r}$ in $k$ such that the linear combination

$g_{m+1}=c_{m+1}f_{m+1}+c_{m+2}f_{m+2}+\cdots+c_{r}f_{r}$

does not vanish identically on the component $D_{i}$ for $i=1,\ldots,h$. It
follows that $Z(g_{1},\ldots,g_{m},g_{m+1})$ is a set-theoretical complete
intersection. ∎

######  Lemma 7.3.

Let $f_{1},\ldots,f_{r}\in k[x_{1},\ldots,x_{n}]$ be a collection of
polynomials. Set $d_{i}=\deg f_{i}$. Assume that $d_{1}\geq
d_{2}\geq\cdots\geq d_{r}$. Let $Z=Z(f_{1},\ldots,f_{r})$. Then there exists a
new sequence of polynomials $g_{1},\ldots,g_{r}\in k[x_{1},\ldots,x_{r}]$ such
that

  1. (1)

$Z(g_{1},\ldots,g_{r})=Z$,

  2. (2)

$\deg g_{i}\leq d_{i}$,

  3. (3)

$Z(g_{1},\ldots,g_{n-\dim Z})$ is a set-theoretic complete intersection.

###### Proof.

Applying Lemma 7.1 repeatedly gives rise to a new sequence of polynomials
$g_{1},\ldots,g_{r}\in k[x_{1},\ldots,x_{n}]$, satisfying the following:

  * •

$g_{1}=f_{1}$, $g_{m}=f_{m}$ if $m>n-\dim Z$;

  * •

there exists an upper-triangular square matrix
$B=(b_{\alpha\beta})_{1\leq\alpha,\beta\leq r}$ with entries in $k$, whose
diagonal entries are either $0$ or $1$, such that

(7.4) $\begin{bmatrix}g_{1}\\\ \vdots\\\
g_{r}\end{bmatrix}=B\cdot\begin{bmatrix}f_{1}\\\ \vdots\\\
f_{r}\end{bmatrix};$

and

  * •

$Z(g_{1},\ldots,g_{n-\dim Z})$ is a set-theoretic complete intersection.

Thus the condition (3) is ensured. By construction, $\deg g_{i}\leq d_{i}$ for
any $i=1,2,\ldots,r$. The condition (2) is checked.

Proof of (1). Since $g_{1},\ldots,g_{r}$ are $k$-linear combinations of
$f_{1},\ldots,f_{r}$, $Z$ is contained in $Z(g_{1},\ldots,g_{r})$. Let us
prove $Z(g_{1},\ldots,g_{r})\subset Z$.

If the $j$th diagonal entry of $B$ is zero, we say $j$ is a “jumping” index.
By Lemma 7.1, for each jumping $j$, $f_{j}$ vanishes identically on
$Z(g_{1},\ldots,g_{j-1})$; hence $f_{j}$ vanishes identically on
$Z(g_{1},\ldots,g_{r})$ as well.

It remains to show that if $\beta$ is not a jumping index, $f_{\beta}(Q)=0$
for any $Q\in Z(g_{1},\ldots,g_{r})$. For each jumping $j$, remove the $j$th
row and $j$th column from the matrix $B$. The resulting matrix $C$ is upper
triangular, and its diagonal entries are all $1$. In particular, $C$ is
invertible. Evaluating (7.4) at $Q$, using the vanishing of jumping $f_{j}$ at
$Q$, we see that, for any non-jumping index $\alpha$, we have

$0=g_{\alpha}(Q)=\sum_{\beta\text{ non-jumping}}b_{\alpha\beta}f_{\beta}(Q),$

The matrix associated with the above system of linear equations is the
invertible matrix $C$. Thus $f_{\beta}(Q)=0$ for any non-jumping $\beta$. This
concludes the proof. ∎

##  8\. Divisibility of Frobenius eigenvalues

We retain the notation made in Section 6. Recall the situation: we are given a
collection of polynomials
$f_{1},\ldots,f_{r}\in\mathbf{F}_{q}[x_{1},\ldots,x_{n}]$, and denote by

$Z=\operatorname{Spec}\mathbf{F}_{q}[x_{1},\ldots,x_{n}]/(f_{1},\ldots,f_{r})$

the vanishing scheme of $f_{1},\ldots,f_{r}$. By rearranging the order, we
shall assume $d_{1}\geq\cdots\geq d_{r}$, where $d_{i}=\deg f_{i}$. The
codimension $n-\dim Z$ of $Z$ is denoted by $c$.

The following lemma should be well-known. It shows that the vanishing of
compactly supported cohomology of $Z$ can be controlled by the number of
defining equations of $Z$. So Theorems 1.2 and 1.3 cover all nontrivial
cohomology degrees. In its statement, $\mathrm{H}^{i}_{c}(Z)$ could either be
$\mathrm{H}^{i}_{\mathrm{rig},c}(Z)$ or
$\mathrm{H}^{i}_{c}(Z_{\overline{\mathbf{F}}_{q}},\mathbf{Q}_{\ell})$.

######  Lemma 8.1.

Let $Y$ be a nonsingular affine variety of dimension $n$. Let
$f_{1},\ldots,f_{r}\in\Gamma(Y,\mathcal{O}_{Y})$ be regular functions on $Y$.
Let $Z$ be the common zero locus of $f_{1},\ldots,f_{r}$ in $Y$. Then
$\mathrm{H}^{i}_{c}(Z)=0$ for $i<n-r$.

###### Proof.

We have a long exact sequence

$\cdots\to\mathrm{H}^{i}_{c}(Y)\to\mathrm{H}^{i}_{c}(Z)\to\mathrm{H}^{i+1}_{c}(Y\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}Z)\to\mathrm{H}^{i+1}_{c}(Y)\to\cdots.$

If $i<n-1$, then $\mathrm{H}^{i}_{c}(Y)=0$ by smoothness of $Y$, Poincaré
duality, and Artin vanishing. Thus, it suffices to prove
$\mathrm{H}^{i}_{c}(Y\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}Z)=0$ for
$i<n-r+1$.

Write
$Y\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}Z=\bigcup_{i=r}^{r}U_{i}$,
where $U_{i}=Y\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}\\{f_{i}=0\\}$.
Then for $I\subset\\{1,2,\ldots,r\\}$, $\bigcap_{i\in
I}U_{i}=Y\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}\\{\prod_{i\in
I}f_{i}=0\\}$. By Mayer–Vietoris, we have a spectral sequence

$E_{1}^{-a,b}=\bigoplus_{\operatorname{Card}I=a+1}\mathrm{H}_{c}^{b}(U_{I})\Rightarrow\mathrm{H}^{b-a}_{c}(Y\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}Z).$

Since each $U_{I}$ is a smooth affine variety of dimension $n$, by Poincaré
duality and Artin vanishing again,
$\mathrm{H}^{i+1}_{c}(Y\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}U_{I})=0$
if $i<n-1$. It follows that

$E^{-a,b}_{1}\neq 0\Longrightarrow\begin{cases}b\geq n,\text{ and }\\\ a\leq
r-1,\end{cases}\Longrightarrow b-a\geq n-r+1.$

Ergo,
$\mathrm{H}^{i}_{c}(Y\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}Z)=0$ if
$i<n-r+1$. ∎

######  Remarks.

(a) The above same argument also works for the Betti cohomology of an
algebraic variety $Z$ defined by the vanishing of $r$ regular functions on a
smooth affine variety $Y$ over $\mathbf{C}$.

(b) The lemma for rigid cohomology also follows from Corollary 4.4 and
Poincaré duality with coefficients.

The remainder of this section is devoted to the proofs of Theorems 1.2 and
1.3. The upshot is that, by Lemma 6.2 and Lemma 6.3, we can reduce the
“cohomological divisibility problem” to a “chain level divisibility problem”.
Thus we will only need to study the divisibility of the reciprocal roots of
the Fredholm determinants $\det(1-t\alpha\mid B_{I})$. Then, by some “yoga”,
we could further reduce ourselves to estimating the slopes (i.e., $q$-orders)
of these reciprocal roots, which we achieve in Lemma 8.3.

Unlike Section 6, where only a qualitative understanding to the Dwork complex
and the operator $\alpha$ is needed, Lemma 8.3 requires some quantitative
analysis for $\alpha$, due to Adolphson–Sperber [adolphson-sperber:chevalley-
warning]. In the work of Adolphson and Sperber, the operator $\alpha$ acts on
some Banach spaces smaller than $B_{I}$; we shall use the overconvergent
variants $B^{\prime}_{I}$ of these spaces (which will have the same effect,
see Remark 8.5 below).

######  Definition.

Let $\Delta$ be the Newton polyhedron of $g$ at infinity, that is, the convex
closure of $0\in\mathbf{R}^{n+r}$ and $\\{u\in\mathbf{N}^{n+r}:\text{the
coefficient of }x^{u}\text{ in }g\text{ is nonzero}\\}$. Let $C(\Delta)$ be
the smallest conical region spanned by $\Delta$. For any subset
$I\subset\\{1,2,\ldots,n+r\\}$, set

$B^{\prime}_{I}=\left\\{\textstyle\sum A_{u}x^{u}\in B_{I}:A_{u}=0\text{ if
}u\notin C(\Delta)\cap\mathbf{Z}^{n+r}\right\\}.$

It is clear that $B^{\prime}_{I}$ is an $\alpha$-stable linear subspace of
$B_{I}$.

The theorem of Adolphson and Sperber concerns with the lower bound of slopes
of reciprocal roots of $\det(1-t\alpha\mid B^{\prime}_{I})$. However, we need
the information about $\det(1-t\alpha\mid B_{I})$. Fortunately, the following
lemma shows that the Fredholm determinant of $\alpha|_{B_{I}}$ and that of
$\alpha|_{B^{\prime}_{I}}$ are the same.

######  Lemma 8.2.

The Fredholm determinant of the induced endomorphism on the quotient space
$B_{I}/B^{\prime}_{I}$ is $1$.

###### Proof.

By Lemma 6.1, $\alpha$ induces a nuclear operator on the quotient space
$B_{I}/B^{\prime}_{I}$. It is enough to prove that $\mathrm{Tr}(\alpha^{m}\mid
B_{I}/B_{I}^{\prime})=0$ for all positive integers $m$. A monomial basis for
$B_{I}/B_{I}^{\prime}$ consists of $x^{u}$, where $u$ runs over all lattice
points in $\mathbf{N}^{n+r}$ which are not in $C(\Delta)$. In particular, $u$
is non-zero. By definition of $\alpha$, one computes that

$\alpha^{m}(x^{u})=\sum_{v}b_{v}x^{(u+v)/q^{m}},$

where $v$ runs over $C(\Delta)\cap\mathbf{Z}^{n+r}$, and $b_{v}=0$ if
$q^{m}\nmid u+v$. If $u=(u+v)/q^{m}$, then $v=(q^{m}-1)u$ will not be in the
cone generated by $\Delta$, a contradiction. This proves that the diagonal
entries of $\alpha^{m}$ with respect to the monomial basis $\\{x^{u}\\}$ are
all zero. Consequently, $\mathrm{Tr}(\alpha^{m}\mid B_{I}/B_{I}^{\prime})=0$.
∎

We introduce the following

######  Notation.

For a subset $I$ of $\\{1,2,\ldots,n+r\\}$, let
$I^{\prime}=I\cap\\{1,2,\ldots,n\\}$, and
$I^{\prime\prime}=I\cap\\{n+1,\ldots,n+r\\}$.

######  Lemma 8.3.

Let $\lambda$ be a reciprocal root of the $p$-adic entire function
$\det(1-t\alpha\mid B_{I})$ Then

$\operatorname{ord}_{q}\lambda\geq\frac{1}{d_{1}}\left(|I^{\prime}|+\sum_{i\in
I^{\prime\prime}}(d_{1}-d_{i})\right)=\frac{1}{d_{1}}\left(|I|+\sum_{i\in
I^{\prime\prime}}(d_{1}-d_{i}-1)\right),$

where $\operatorname{ord}_{q}$ is the $p$-adic valuation normalized so that
$\operatorname{ord}_{q}(q)=1$.

###### Proof.

Since the problem only concerns with the Fredholm determinant, we can replace
$B_{I}$ by $B^{\prime}_{I}$ thanks to Lemma 8.2.

As we shall explain in Remark 8.5, the computation made in [adolphson-
sperber:chevalley-warning] works equally well with their Banach spaces
$L_{I}(b)$ (to be recalled in Remark 8.5) replaced by their overconvergent
cousins $B^{\prime}_{I}$. Thus, by [adolphson-sperber:chevalley-
warning]*Proposition 4.2, a lower bound of $\operatorname{ord}_{q}\lambda$ is
given by the quantity

$w_{I}=\min\\{w(u):u\in C(\Delta)\cap\mathbf{N}^{n+r},u_{i}>0,\forall i\in
I\\},$

where $w(u)$ is Adolphson–Sperber’s weight function on the cone $C(\Delta)$
which in the present case is given by
$w(u_{1},...,u_{n+r})=u_{n+1}+\cdots+u_{n+r}$.

Thus, it suffices to show

$w_{I}\geq\frac{1}{d}\left(|I^{\prime}|+\sum_{i\in
I^{\prime\prime}}(d-d_{i})\right).$

The numbers $y_{1},\ldots,y_{n+r}$ are subject to the following constraints:

(8.4) $\begin{cases}y_{i}\geq 1,\forall i\in I,\\\ y_{1}+\cdots+y_{n}\leq
d_{1}y_{n+1}+\cdots+d_{r}y_{n+r}.\end{cases}$

Write

$\xi_{i}=\begin{cases}y_{i}-1,&i\in I;\\\ y_{i},&i\notin I.\end{cases}$

Then (8.4) is equivalent to

(8.4′) $\begin{cases}\xi_{i}\geq 0,\forall i=1,2,\ldots,n+r;\\\
\sum_{i=1}^{r}d_{i}\xi_{i}\geq\sum_{i=1}^{n}\xi_{i}+|I^{\prime}|-\sum_{i\in
I^{\prime\prime}}d_{i}\end{cases}$

It follows that

$\displaystyle
y_{n+1}+\cdots+y_{n+r}=\sum_{i=1}^{r}\xi_{i+r}+|I^{\prime\prime}|$
$\displaystyle=$
$\displaystyle\frac{1}{d_{1}}\left(\sum_{i=1}^{r}(d_{1}-d_{i})\xi_{i+r}+\sum_{i=1}^{r}d_{i}\xi_{i+r}\right)+|I^{\prime\prime}|$
$\displaystyle\text{[apply \eqref{eq:linear-prog-prime}]}~{}\geq$
$\displaystyle\frac{1}{d_{1}}\left(|I^{\prime}|-\sum_{i\in
I^{\prime\prime}}d_{i}\right)+|I^{\prime\prime}|=\frac{1}{d_{1}}\left(|I^{\prime}|+\sum_{i\in
I^{\prime\prime}}(d_{1}-d_{i})\right).$

This completes the proof. ∎

######  Remark 8.5.

Let us indicate how to transplant Adolphson–Sperber’s result to the spaces
$B^{\prime}_{I}$. In [adolphson-sperber:chevalley-warning], all the
calculations took place in some $p$-adic Banach spaces $L(b)$ and $L_{I}(b)$,
where

$L(b)=\left\\{\sum_{u\in C(\Delta)\cap\mathbf{Z}^{n+r}}A_{u}x^{u}:\exists
c\in\mathbf{R},\mathrm{ord}_{p}(A_{u})\geq bw(u)+c\right\\}$

($\mathrm{ord}_{p}$ being the $p$-adic valuation normalized by
$\operatorname{ord}_{p}(p)=1$), and $L_{I}(b)=x^{I}L(b)$.

The estimate for $\operatorname{ord}_{q}\lambda$ can be obtained from the
estimate for the slopes of the Fredholm determinant $\det(1-t\alpha_{0}\mid
L_{I}(b))$ of the semilinear opeartor $\alpha_{0}$ by a standard argument. For
$b=\frac{p}{p-1}$, loc. cit. proved that the first slope of
$\det(1-t\alpha_{0}\mid L(b))$ is at least $w_{I}$. For all
$0<b\leq\frac{p}{p-1}$, we have the factorization of $\alpha_{0}$ as in (5.2)

$L_{I}(b)\xrightarrow{\iota}L_{I}(b/p)\xrightarrow{G}L_{I}(b/p)\xrightarrow{\sigma^{-1}\circ\psi}L_{I}(b).$

Thus $\alpha_{0}|_{L_{I}(b)}$ is nuclear for any such $b$. Moreover, the
matrices of $\alpha_{0}|_{L_{I}(b)}$ with respect to the
$\mathbf{Q}_{p}(\zeta_{p})$-bases $\\{x^{u}\\}$ of $L_{I}(b)$ are all similar.
Hence the Fredholm determinants for all $b\leq\frac{p}{p-1}$ are the same.
Since $B^{\prime}_{I}=\mathop{\operatorname{colim}}\limits_{b\to 0^{+}}L(b)$,
the Fredholm determinant for $\alpha_{0}|_{B^{\prime}_{I}}$ also equals that
of $\alpha_{0}|_{L_{I}(\frac{p}{p-1})}$. This gives the desired bound
$\operatorname{ord}_{q}\lambda\geq w_{I}$.

With these preparations handy, we can now step into the proof of Theorems 1.2
and 1.3.

### Step 1: Reduction

Recall that $n-\dim Z$ is denoted by $c$. By Lemma 7.3, there exists a finite
extension $k^{\prime}$ of $\mathbf{F}_{q}$, and a collection of polynomials
$g_{1},\ldots,g_{r}$, such that

  * •

$\deg g_{1}=d_{1}$, $\deg g_{i}\leq d_{i}$;

  * •

$Z$ is the common zero locus of $g_{1},\ldots,g_{r}$, and

  * •

$\operatorname{Spec}k^{\prime}[x_{1},\ldots,x_{n}]/(g_{1},\ldots,g_{c})$ is a
set-theoretic complete intersection of dimension $\dim Z$.

Since the conclusion of Theorem 1.2 is not sensitive to the base field, and
since we have,

$\displaystyle\nu_{j}(n;\deg g_{1},\ldots,\deg g_{r})$
$\displaystyle\geq\nu_{j}(n;d_{1},\ldots,d_{r})$
$\displaystyle\epsilon_{m}(n;\deg g_{1},\ldots,\deg g_{r})$
$\displaystyle\geq\epsilon_{m}(n;d_{1},\ldots,d_{r}),$

it suffices to prove the theorems with $\mathbf{F}_{q}$ replaced by
$k^{\prime}$ and $f_{i}$ replaced by $g_{i}$. Thus, it suffices to prove
Theorems 1.2 and 1.3 under the following additional hypothesis:

(8.6) The scheme
$\operatorname{Spec}\mathbf{F}_{q}[x_{1},\ldots,x_{n}]/(f_{1},\ldots,f_{c})$
has dimension equal to $\dim Z$.

### Step 2: Slope estimates

Let us first prove that the numbers $\nu_{j}(n;d_{1},\ldots,d_{r})$ and
$\epsilon_{m}(n;d_{1},\ldots,d_{r})$ provide lower bounds of the $q$-order of
the Frobenius eigenvalues of $\mathrm{H}^{\ast}_{\mathrm{rig},c}(Z)$. Later we
will bootstrap this bound to a bound of $q$-divisibility of algebraic numbers.

Rewrite the overconvergent Dwork complex (5.9) as the total complex of the
following double complex (in order to save ink, we have omitted the monomials
$\mathrm{d}x^{I}/x^{I}$ in the expression):

(8.7) ${\bigoplus\limits_{\begin{subarray}{c}|I^{\prime}|=n\\\
|I^{\prime\prime}|=0\end{subarray}}B_{I}}$${\cdots}$${\bigoplus\limits_{\begin{subarray}{c}|I^{\prime}|=n\\\
|I^{\prime\prime}|=c\end{subarray}}B_{I}}$${\cdots}$${\bigoplus\limits_{\begin{subarray}{c}|I^{\prime}|=n\\\
|I^{\prime\prime}|=r\end{subarray}}B_{I}}$${{\vdots}}$${{\vdots}}$${{\vdots}}$${\bigoplus\limits_{\begin{subarray}{c}|I^{\prime}|=0\\\
|I^{\prime\prime}|=0\end{subarray}}B_{I}}$${\cdots}$${\bigoplus\limits_{\begin{subarray}{c}|I^{\prime}|=0\\\
|I^{\prime\prime}|=c\end{subarray}}B_{I}}$${\cdots}$${\bigoplus\limits_{\begin{subarray}{c}|I^{\prime}|=0\\\
|I^{\prime\prime}|=r\end{subarray}}B_{I}}$

In the diagram, the horizontal differentials are induced by
$D_{r+1},\ldots,D_{n}$, and the vertical ones are induced by
$D_{1},\ldots,D_{n}$ (see (5.5)).

The following lemma shows that the $0$th, $1$st, $\ldots$, and $(c-1)$st
columns of the $E_{1}$-page of the spectral sequence associated to the double
complex (8.7) are all zero.

######  Lemma 8.8.

For each $0\leq i\leq n$, the $i$th row of (8.7) is exact in cohomology degree
$0,1,\ldots,c-1$.

###### Proof.

For the proof of the lemma, the “affine convention” mentioned in Remark 5.8 is
more convenient.

Let $A^{\dagger}=K\langle x_{1},\ldots,x_{n}\rangle^{\dagger}$. Then the
zeroth row of the double complex, up to a shift, is a tensor product (over
$A^{\dagger}$) of

(I) $\mathrm{Kos}(A^{\dagger}\langle
x_{n+1},\ldots,x_{n+c}\rangle^{\dagger};D^{(1)},\ldots,D^{(n+c)})$

and

(II) $\mathrm{Kos}(A^{\dagger}\langle
x_{n+c+1},\ldots,x_{n+r}\rangle^{\dagger};D^{(n+c+1)},\ldots,D^{(n+r)}),$

where $D^{(j)}$ is the differential operator

$D^{(j)}=\frac{\partial}{\partial x_{n+j}}-\frac{\partial H}{\partial
x_{n+j}},\quad j=1,2,\ldots,r,$

that $x_{n+j}D^{(j)}=D_{n+j}$, see Remark 5.8.

By the Künneth forumla, it suffices to prove the Koszul complex I is acyclic
except in top degree. Using Li’s lemma, Lemma 5.10 (specifically, use the
untwist-then-retwist series (5.11) in the proof), we see it does not matter
whether we use the differentials above or the operators in (4.1). Ergo, the
complex (I) computes
$R\Gamma^{\dagger}_{\\{f_{1}=\cdots=f_{c}=0\\}}(\mathcal{O})[c]$ by Theorem
4.3. Under Hypothesis (8.6), $\\{f_{1}=\cdots=f_{c}=0\\}$ is a set-theorretic
complete intersection. It is a standard fact in rigid cohomology that this
complex is acyclic except in degree $0$, the top degree (we are unable to find
a reference; a proof is provdied below, see Lemma 8.9). The desired acyclicity
is proved.

The $i$th row is equal to a direct sum of $\binom{n}{i}$ copies the $0$th row.
By the argument above, we see this row computes
$R\Gamma_{\\{f_{1}=\cdots=f_{c}=0\\}}^{\dagger}(\Omega^{i}_{\mathbf{A}^{n,\mathrm{an}}_{K}})$.
Thus the acyclicity of the zeroth row implies the acyclicity of the $i$th row.
∎

The following lemma was used in the above proof.

######  Lemma 8.9.

Let $k$ be a perfect field of characteristic $p>0$. Let $\overline{A}$ be a
smooth $k$-algebra, lifting smoothly to a smooth $V$-algebra
$A_{\mathrm{int}}$. Let $F_{1},\ldots,F_{r}$ be a regular sequence in
$A_{\mathrm{int}}$. Let $Z_{i}$ be the vanishing locus of $F_{i}$ modulo the
uniformizer. Then $R\Gamma_{Z_{1}\cap\cdots\cap Z_{r}}^{\dagger}(\mathcal{O})$
is acyclic excpt in degree $r$.

###### Proof.

Let $A$ be the weak completion of $A_{\mathrm{int}}$. Let $J_{e}$ be the ideal
of $A$ generated by $F_{1}^{e},\ldots,F_{r}^{e}$. The system
$(A/J_{e})_{e=1}^{\infty}$ admits a system of Kosuzl resolutions

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