Emiliano Ambrosi and Domenico Valloni

Let $k$ be an algebraically closed field of characteristic $p \geq 0$ and $V$ be a faithful $k$-rational representation of a finite $\ell$-group $G$, where $\ell$ is a prime number. The Noether problem asks whether $V/G$ is a stably rational variety. While if $\ell=p$ it is well-known that $V/G$ is always rational, when $\ell\neq p$, Saltman and then Bogomolov constructed $\ell$-groups for which $V/G$ is not stably rational. Hence, the geometry of  $V/G$ depends heavily on  the characteristic of the field. We show that for all the groups $G$ constructed by Saltman and Bogomolov, one cannot interpolate between the Noether problem in characteristic $0$ and $p$. More precisely, we show that it does not exist a complete valuation ring $R$ of mixed characteristic $(0,p)$ and a smooth proper $R$-scheme $X\rightarrow \Spec(R)$ whose special fiber and generic fiber are both stably birational to $V/G$. The proof combines the integral $p$-adic Hodge theoretic results of Bhatt-Morrow-Scholze with the study of indefinitely closed differential forms in positive characteristic.

§ INTRODUCTION

§.§ Noether problem

Let $G$ be a finite group and let $k$ be an algebraically closed field of characteristic $p\geq 0$. Recall that two (irreducible) $k$-varieties $X,Y$ are said to be stably birational if $X\times \mathbb P_k^n$ is birational to $Y\times \mathbb P_k^m$ for some $n,m\in \mathbb N$. If $X$ is stably birational to a point, we say that $X$ is stably rational. We write $\Stab(k)$ for the set of varieties up to the stably birational equivalence relation and for a $k$-variety $X$ we write $[X]_k\in \Stab(k)$ for its class in $\Stab(k)$.

Let $G\rightarrow GL(V)$ a faithful (finite-dimensional) representation of $G$. It is well known (<cit.>) that $X:=V/G$ is a normal unirational variety whose stably birational class $[G]_k$ does not depend on the chosen faithful representation of $G$. Knowing more about $[G]_k$ is a fundamental question in algebraic geometry that dates back to Emmy Noether (<cit.>), when she asked whether $[G]_k=[ \{ \Spec(k)\} ]_k$, which is now known as the Noether's problem for $(G,k)$.

§.§.§ Dependence on the field

We remark that when $k$ is not algebraically closed, there might be arithmetic obstructions to the Noether problem. For instance, Fischer theorem <cit.> asserts that $V/G$ is always rational when $G$ is cyclic and $k=\overline \Q$, whereas Swan's original counterexample <cit.> to the Noether problem uses $G = \Z/47 \Z$ and $k = \Q$. We will focus on the case of an $\ell$-group $G$, where $\ell$ is a prime number, and $k$ is algebraically closed. In this case, the Noether's problem for $G$ heavily depends on $p \coloneqq \mathrm{char}(k)$.
Indeed, while a well-known result (see e.g. <cit.> and <cit.>) states that $[G]_k=[\{\Spec(k)\}]_k$ if $\ell=p$, there are many $\ell$-groups $G$ with $\ell\neq p$ for which the Noether problem has a negative answer.

The most relevant for our purposes are the ones constructed by Saltman in <cit.> and Bogomolov (refining Saltman's techniques) in <cit.>. Both these examples are over $\C$, and use the non-triviality of the unramified Bruaer group to deduce that the quotient variety is not stably rational. Other examples using the third unramified cohomology have been constructed more recently by Peyre in <cit.>.

Since many techniques to study rationality problems are based on specialization methods both in equicharacteristic and in mixed characteristic (see e.g. <cit.>, <cit.>, <cit.>) and the geometry of $[G]_k$ heavily depends on $p$, it seems interesting to understand whether one can specialize the Noether's problem for a group $G$ from characteristic zero to positive characteristic. More precisely, let $R$ be a complete valuation ring of mixed characteristic ($0$,$p$) with algebraically closed fraction field $K$ and residue field $k$.

Let $G$ be a $p$-group. We say that the Noether problem for $G$ has good-reduction at $p$ if there exists a smooth projective scheme $X/R$ whose generic fibre has stable birational class $[G]_K$ and whose special fibre has stable birational class $[G]_k$ (and hence is stably rational) for some $R$ as above.

When $[G]_K=[\Spec(K)]_K$, one can take $\Pp^n_R$ as a smooth projective model, so the question can be interesting only when $[G]_K\neq [\Spec(K)]_K$. As shown in Section <ref>, adapting the constructions in <cit.> and following a suggestion of Colliot-Théléne, irrational varieties can have good rational reduction modulo $p$. Hence, understanding for which $p$-groups $G$ the Noether problem has good-reduction at $p$ is a non-trivial problem.

As a corollary of our main Theorem <ref>, we get the following non-existence result.

For all the $p$-groups $G$ constructed in <cit.> and in <cit.> the Noether problem does not have good reduction at $p$.

If one considers $\Pp^n_R$ together with a $R$-linear action of $G$ such that the induced action on $\Pp^n_k$ is faithful, Corollary <ref> implies that one cannot resolve the singularities of $\Pp^n_R/G$ over $R$ in all these cases.

§.§ Stably Birational invariants

The main ingredient to prove Corollary <ref> is the study certain stably-birational invariants. To introduced them, let $X$ be a smooth proper variety over an algebraically closed field $L$.

§.§.§ Extremal Hodge number

By <cit.> and the Künneth formula for de-Rham cohomology, the extremal Hodge number of $X$

$$h^{0,i}:=\dim(H^{i}(X,\mathcal O_X)) \quad h^{i,0}:=\dim(H^{0}(X,\Omega^i_X))$$

are stably birational invariants[More precisely, the birational invariance of $h^{i,0}$ follows from Hartogs' lemma, while the birational invariance $h^{0,i}$ follows from the Hodge symmetry in characteristic $0$ and from the main result of <cit.> in positive characteristic. To pass from birational to stable birational invariance, one uses the Künneth formuma for de-Rham cohomology.]. These can be used to obstruct the existence of families of Kummer surfaces in mixed characteristic $(0,2)$ leading to results similar to  Corollary <ref>. More precisely, let $R$ be a complete DVR of mixed characteristic $(0,2)$ and $ A\rightarrow \Spec(R)$ be an abelian scheme of relative dimension $2$ with supersingular special fiber $A_k$.
By <cit.>, the Kummer surface $\Kum(A_k)$ is  a rational variety, hence $H^0(\Kum(A_k),\Omega^2)=0$,  while $\Kum(A_K)$ is a K3-surfaces, hence $H^0(\Kum(A_K),\Omega^2)=K$. Hence, by semicontinuity of coherent cohomology, there does not exist a smooth proper $R$-scheme $Y\rightarrow \Spec(R)$ such that $[Y_k]_k=[\Kum(A_k)]_k$ and $[Y_K]_K=[\Kum(A_K)]_K$.

On the positive side, let us point out that a recent result of Lazda-Skorobogatov (<cit.>) shows that if $A_k$ is not supersingular then such a model exists. So the extremal Hodge numbers of $\Kum(A_k)$ are the only obstruction to the construction of a mixed characteristic family interpolating the Kummer construction.

§.§.§ Artin-Mumford invariant

These arguments cannot be used to study smooth proper varieties birational to $V/G$, since all of their extremal Hodge numbers vanish in characteristic $0$.

Let $\Br(X)$ be the Brauer group of $X$ and $\Br(X)_{div}$ the maximal divisible subgroup of $\Br(X)$. Let $\ell$ be a prime different from the characteristic of $L$. By the stably birational invariance of the Brauer group, also the group $H_{\et}^3(X,\Z_\ell)[\ell]\simeq \Br(X)/\Br(X)_{div}[\ell]$ is a stable birational invariant for smooth proper varieties. It is usually called the $\ell$-adic Artin-Mumford $AM_{\ell}([X]_L)$ of $[X]_L\in SB(L)$.

The way in which Saltman (and then Bogomolov) proved that $[G]_L \neq [\{\Spec(L)\}]_L$, for $L$ of characteristic $0$, was by showing that $\AM_{\ell}([G]_L)\neq 0$ for his particular choice of $G$. Our main result is that the presence of such obstruction to stable rationality also prevents the specialization to a smooth projective, stably rational variety in characteristic $p$:

Let $R$ be a complete valuation ring of mixed characteristic ($0$,$p$) with algebraically closed fraction field $K$ and residue field $k$. Let $X\rightarrow R$	be a smooth proper $R$-variety whose special fiber is stably rational. Then $H^3(X_{K},\Z_p)[p]=0$

In particular, $H^3(X_{\C},\Z)[p]=0$ for any embedding $K \subset \C$. Corollary <ref> follows then directly from Theorem <ref> and the discussion above it.

§.§ Integral p-adic Hodge theory

To state the second main result of the paper, we explain our strategy to prove Theorem <ref>, of which we retain the notation. For every prime $\ell\neq p$, the smooth proper base change theorem shows that

$$H^3(X_{K},\Z_{\ell})[\ell]\simeq H^3(X_{k},\Z_{\ell})[\ell]=0,$$

where the last equality follows from the stable birational invariance of $H^3(X_{k},\Z_{\ell})[\ell]$ and the assumption on $X_k$.

For $p$-adic coefficients, the smooth proper base change theorem does not hold except in a few special cases, but it can be replaced with the recent developments in integral $p$-adic Hodge theory by  Bhatt-Morrow-Scholze. By  <cit.>, one has the following inequality

\begin{equation}\label{eq : inequality}
\dim_k(H^3_{\crys}(X_{k})[p])\geq \dim_{\mathbb F_p}(H^3_{\et}(X_{K},\mathbb Z_p)[p]),
\end{equation}

where $H^3_{\crys}(X_{k}):=H^3_{\crys}(X_{k}/W)$ is the third integral crystalline cohomology group of $X_{k}$, and which will play the role of the semicontinuity theorem used in the Kummer surface examples (Section <ref>).

§.§.§ Torsion in crystalline cohomology

Thanks to (<ref>), to prove Theorem <ref> it would be enough to show that $\dim_k(H^3_{\crys}(X_k)[p])$ is a stably birational invariant. While this would follow easily from a strong form of resolution of singularities (in particular, weak factorization of birational maps) it is unclear how to prove it without assuming it. The main issue is that crystalline cohomology with integral coefficient is very badly behaved for open varieties, so the standard techniques used in characteristic zero do not work anymore.

Nevertheless, we prove something slightly weaker, but which is strong enough to imply Theorem <ref>.

Let $k$ be an algebraically closed field of characteristic $p$ and $X$ a smooth proper $k$-variety.
Assume that

* $h^{0,i}_X=h_X^{i,0}=0$ for $1\leq i\leq 3$,

* $\Br(X)=0$.

Then $H^3_{\crys}(X/W)[p]=0$.

Since, as already mentioned, the assumptions of Theorem <ref> are satisfied if $X$ is stably rational (or, more generally, if $X$ admits a decomposition of the diagonal) Theorem <ref> and the inequality (<ref>) imply Theorem <ref>.

The crucial point in the proof of Theorem <ref> is to show that, under the hypothesis of the theorem, the cycles class map

$$\Cl_{\dR}:\NS(X)\otimes k\rightarrow H^2_{\dR}(X)$$

is an isomorphism, hence that a version of the Hodge conjecture for divisors holds. To prove this, the main idea to use indefinitely closed differential forms to cut out a well-behaved piece of the second de-Rham cohomology group of $X$. Under the assumption of Theorem <ref>, we then show that there are enough of these forms to generate the whole cohomology and, combining this with the theory of the de-Rham Witt complex, we get Theorem <ref>.

§.§ Acknowledgement

The authors are grateful to Stefan Schreieder for having introduced the second named author to the Noether problem, to Giuseppe Ancona for many interesting discussions and comments on a preliminary version of the article, and to Colliot-Théléne for pointing us out
the example in Section <ref> .

§ PROOF OF THEOREM -

In this section we prove Theorem <ref> and so we retain the assumption and the notation therein.
For $a,b,n\in \mathbb N$, set

$$h^{a,b}:=\dim_k(H^b(X,\Omega^a_X))\quad \text{and}\quad h^n_{\dR}:=\dim_k(H^n_{\dR}(X)).$$

§.§ Preliminary reductions

We start observing that the universal coefficient theorem gives an exact sequence

$$0\rightarrow H^2_{\crys}(X/W)/p\rightarrow H^2_{\dR}(X)\rightarrow  H^3_{\crys}(X/W)[p]\rightarrow 0,$$

hence it is enough to show that the injection $H^2_{\crys}(X/W)/p\hookrightarrow H^2_{\dR}(X)$ is surjective.
To prove this, we use the commutative diagram

\begin{equation}\label{eq : commutative cycles}
\begin{tikzcd}
\NS(X)\otimes W\arrow{r}{\Cl_{\crys}}\arrow[two heads]{d} & H^2_{\crys}(X/W)\arrow{d}\\
\NS(X)\otimes k\arrow{r}{\Cl_{\dR}} & H^2_{\dR}(X),\\
\end{tikzcd}
\end{equation}

in which the horizontal arrows are the cycle class maps for crystalline and de-Rham cohomology and which shows that it is enough to prove that

* $\Cl_{\crys}:\NS(X)\otimes W\xrightarrow{\simeq} H^2_{\crys}(X/W)$ is an isomorphism and

* $\dim_k(\NS(X)\otimes k)\geq h^2_{\dR}(X)$.

While it would be enough to prove directly that $\Cl_{\dR}:\NS(X)\otimes k\rightarrow H^2_{\dR}(X)$ is surjective, we cannot prove it without passing through crystalline cohomology to show that it is injective and then compare the dimensions. We remark that the equality of the dimensions  of $\NS(X)\otimes k$ and $H^2_{\dR}(X)$ is not enough to guarantee that the cycle class map is surjective or injective, has shown by the example of supersingular K3 surfaces.

§.§ Proof of (1)

§.§.§ Preliminaries on the De-Rham Witt complex

Before starting with the proof of (1), we recall what we need from the theory of the de-Rham Witt complex, which will be the main tool to interpolate between the Néron-Severi group of $X$, its integral crystalline cohomology groups, and its flat cohomology.
In <cit.> it is defined a $k$-linear complex

$$ W\Omega_X^{\bullet}: \quad W\Omega^0_X=W\mathcal O_X \xrightarrow{d}W\Omega^{1}_X\xrightarrow{d} \dots \xrightarrow{d} W\Omega^{d-1}_X\xrightarrow{d} W\Omega^{d}_X$$

of quasi coherent $W\mathcal O_X$-modules such that

$$\mathbb H^n(W\Omega_X^{\bullet})\simeq H^n_{\crys}(X),$$

see <cit.>. If $X$ is clear from the context, we remove the subscript and we write just $W\Omega^i$

We let $W\Omega^{\geq i}$ to be the naive  truncation of the complex at $i$, so that there is a short exact sequence

\begin{equation}\label{eq : truncation}
0\rightarrow W\Omega^{\geq 1}\rightarrow W\Omega^{\bullet}\rightarrow W\mathcal O_X\rightarrow 0
\end{equation}

of complexes.
Let $\sigma$ be the lift of the Frobenius of $k$ to $W(k)$.
The complex $W \Omega^{\bullet}_X$ and the sheaves  $W \Omega^{i}_X$ are endowed with a natural $\sigma$-semilinear endomorphism[In <cit.>, $\mathbf{F}$ is denoted with F, but we prefer this notation to make clearer the difference between $ \mathbf{F}$ and $F$] $ \mathbf{F}:W\Omega^{\bullet}_X\rightarrow W\Omega^{\bullet}_X$, see <cit.>.
We write
$F:W\Omega^{\bullet}_X\rightarrow W\Omega^{\bullet}_X$ for the morphism  which is $p^i \mathbf{F}$ is degree $i$ and $F'$ for the one which is $p^{i-1} \mathbf{F}$ in degree $i$,  see <cit.> and <cit.> respectively.

By <cit.>, there is an exact sequence

\begin{equation}\label{eq: derhamwitt vs flat}
\dots \rightarrow  H_{\fl}^i(X,\mathbb Z_p(1))\rightarrow H^i(X,W\Omega^{\geq 1})\xrightarrow{1-F'} H^i(X,W\Omega^{\geq 1}) \rightarrow \dots
\end{equation}

$$H_{\fl}^i(X,\mathbb Z_p(1)):=\varprojlim_{n}H_{\fl}^i(X,\mathbb \mu_{p^n}).$$

§.§.§ Proof of (1)

The cycle class map $\NS(X)\rightarrow H^2_{\crys}(X)$ factors trough the morphism $\mathbb H^1(X,W\Omega^{\geq 1})\rightarrow\mathbb H^2(X,W\Omega^{\bullet})\simeq H^2_{\crys}(X)$ induced by the inclusion $W\Omega^{\geq 1}\subseteq W\Omega^{\bullet}$ and the Kummer exact sequence for the flat topology

\begin{equation}\label{eq : kummer}
0\rightarrow \mu_{p^n}\rightarrow \mathbb G_m\xrightarrow{p^n} \mathbb G_m\rightarrow 0
\end{equation}

induces  (<cit.>) an injection
$$\Cl_{\fl}:\NS(X)\otimes \mathbb Z_p \hookrightarrow H^2_{\fl}(X,\mathbb Z_p(1))$$

fitting into a commutative diagram

\begin{equation}\label{eq : diagramcycleclassmap}
\begin{tikzcd}[column sep=small]
& & \NS(X)\otimes \Z_p\arrow{d}{\Cl_{\fl}}\arrow[hook]{r} & \NS(X)\otimes W(k)\arrow{d}{\Cl_{\crys}}\arrow{rd}{\Cl_{\crys}}\\
& & H^2_{\fl}(X,\Z_p(1))\arrow{r}& \mathbb H^2(X,W\Omega^{\geq 1})\arrow{r} &  \mathbb H^2_{\crys}(X),
\end{tikzcd}
\end{equation}

in which the bottom left arrow is induced from the exact sequence (<ref>) and bottom right one is induced by the exact sequence (<ref>).

Since $0=Br(X)\simeq H^2_{\fl}(X,\mathbb G_m)$, the exact sequence (<ref>) shows that
$\Cl_{\fl}:\NS(X)\otimes \mathbb Z_p \hookrightarrow H^2_{\fl}(X,\mathbb Z_p(1))$
is an isomorphism.
Hence, it is enough to show that both the natural maps

$$H_{\fl}^2(X,\mathbb Z_p(1))\otimes W\xrightarrow{} \mathbb H^2(X,W\Omega^{\geq 1})\xrightarrow{} \mathbb H^2(X,W\Omega^{\bullet})(\simeq H^2_{\crys}(X))$$

are isomorphisms. To be able to prove this, we will prove the following slightly stronger facts:

(a) the map $H_{\fl}^2(X,\mathbb Z_p(1))\otimes W\xrightarrow{} \mathbb H^2(X,W\Omega^{\geq 1})$ is an isomorphism

(b) the map $\mathbb H^i(X,W\Omega^{\geq 1})\rightarrow \mathbb H^i(X,W\Omega^{\bullet})$ is an isomorphism for $1\leq i\leq 2$

Prove first $(b)$ and then $(a)$, since knowing $(b)$ will be important to prove $(a)$.

(b) Using the exact sequence

$$0\rightarrow W\Omega^{\geq 1}_X\rightarrow W\Omega^{\bullet}_X\rightarrow W\mathcal O_X\rightarrow 0,$$

we reduce to show that
$H^i(X,W\mathcal O_X)=0$ for $1\leq i\leq 2$. By <cit.>, there exists an exact sequence

$$0\rightarrow W\mathcal O_X\xrightarrow{V} W\mathcal O_X\rightarrow \mathcal O_X\rightarrow 0,$$

where $V$ is the Verschiebung morphism.
So, since $H^i(X,\mathcal O_X)=0$ for $1\leq i\leq 3$, the morphism $V:H^i(X,W\mathcal O_X)\xrightarrow{V} H^i(X,W\mathcal O_X)$ is an isomorphism for $1\leq i\leq 2$.
On the other hand, by <cit.>, $H^i(X,W\mathcal O_X)$ is topologically separated for the topology induced by $V$, hence, for $1\leq i\leq 2$,

$$0=\bigcap_n V^n(H^i(X,W\mathcal O_X))=\bigcap_n H^i(X,W\mathcal O_X)=H^i(X,W\mathcal O_X).$$

By point (b), the $W(k)$-module $\mathbb H^2(X,W\Omega^{\geq 1})\simeq \mathbb H^2(X,W\Omega^{\bullet})$ is finitely generated, hence by <cit.> and the exact sequence
(<ref>) it is enough to show that

$$\Ker(1-F')=H^2_{\fl}(X,\Z_p(1))\quad \text{and}\quad F':\mathbb H^2(X,W\Omega^{\geq 1})\rightarrow \mathbb H^2(X,W\Omega^{\geq 1}) \text{ is an isomorphism}.$$

Since, by $(b)$, also $\mathbb H^1(X,W\Omega^{\geq 1})$ is a finitely generated $W(k)$-module, the morphism $1-F':\mathbb H^1(X,W\Omega^{\geq 1})\rightarrow \mathbb H^1(X,W\Omega^{\geq 1})$ is surjective by <cit.>, so that the equality on the left follows from the exact sequence (<ref>)
We are left to show that $F':\mathbb H^2(X,W\Omega^{\geq 1})\rightarrow \mathbb H^2(X,W\Omega^{\geq 1}) $ is an isomorphism.
Since the natural map
$\mathbb H^2(X,W\Omega^{\geq 1})\rightarrow \mathbb H^2(X,W\Omega^{\bullet})$ is an isomorphism by point (b), this would follow from  <cit.>, once we verify its assumptions. To this end, we have to check that

* $H^0(X,Z^1_X)=H^0(X,\Omega^1_X)$, where $Z^1_X\subseteq \Omega^1_X$ is the subsheaf of closed forms, which holds since $H^0(X,\Omega^1_X)=0$ by assumption;

* $H^2_{\crys}(X/W)[p]=0$, which holds since $h^{0,1}=h^{1,0}=0=H^1_{\dR}(X)$ by assumption, so that the universal coefficients theorem implies $H^2_{\crys}(X/W)[p]=0$;

* the F-crystals $H^2_{\crys}(X/W)$ is purely of slope $1$, which holds since, thanks to vanishing of $\Br(X)$, the Picard rank of $X$ is equal to  $b_2(X)=\dim_K(H^2_{\crys}(X/W)\otimes \Q)$, hence the cycle class map
$\NS(X)\otimes K\rightarrow H^2_{\crys}(X)\otimes \Q$ is surjective.

This concludes the proof of (1).

§.§ Proof of (2)

In this subsection, to avoid double subscripts, we will write

§.§.§ Preliminary reduction

Let $\Omega^i_{\log}(X)\subseteq \Omega^i(X)$ be the abelian subsheaf generated by $ \mathrm{dlog}(x_1)\wedge  \mathrm{dlog}(x_2)\dots \wedge  \mathrm{dlog}(x_i)$, where the $x_i$'s are local sections of $\mathcal O_X^*$. By <cit.>, there is a natural isomorphism
$\Omega^1_{\log}(X)\simeq \mathcal O^*_X/(\mathcal O_X^*)^{p}$, hence, by
there is a natural isomorphism

$$H^2_{\fl}(X,\mu_p)\simeq  H^1(X,\Omega^1_{\log}(X)).$$

On the other hand, by the Kummer exact sequence (<ref>) and the assumption on $\Br(X)$, one has $\NS(X)\otimes \Z/p\Z\simeq H^2_{\fl}(X,\mu_p)$.
Hence it is enough to prove that

$$	\dim(H^1(X,\Omega^1_{\log}(X)\otimes k))\geq h^2_{\dR}.$$

The Hodge to de-Rham spectral sequence

$$E_1^{a,b}:=H^b(X,\Omega^{a})\Rightarrow H^2_{\dR}(X)$$

together with the assumption $h^{0,1}=h^{1,0}=0$, shows that
$h^{1,1}\geq h^2_{\dR}$, hence it is enough to show that

\begin{equation}\label{eq : final equality}
\dim(H^1(X,\Omega^1_{\log}(X))\otimes k)=h^{1,1}.
\end{equation}

The main problem in proving  (<ref>) is that it is unclear how to understand the cokernel of the inclusion
$\Omega^1_{\log}(X)\subseteq \Omega^1(X)$ so we need to interpret in a different way  $\Omega^1_{\log}(X)$ to give it a better description in terms of differential forms.
This will be done in the next section, by using a result of Raynaud (Theorem <ref>) on indefinitely closed differential forms.

§.§.§ Indefinitely closed differential forms

In this section, we mainly follow <cit.>.
Consider the commutative cartesian diagram

XrdF_X/k[bend left]rrdF_X[bend right]rdd

X^(1)drF_k×[phantom]dr     Xd

krF_k    k

in which $F_k$, $F_X$ are the absolute Frobenii and $F_{X/k}$ is the relative Frobenius of $X/k$.

$$Z^i(X):=\Ker(d:\Omega^i(X)\rightarrow\Omega^{i+1}(X));\quad B^i(X):=\Image(d:\Omega^{i-1}(X)\rightarrow\Omega^{i}(X));\quad
\calH^i(X)=\frac{Z^i(X)}{B^i(X)}.$$

Then one has (<cit.>) a canonical isomorphism of graded $\mathcal O_{X^{(1)}}$-algebras

\begin{equation}\label{eq : cartier}
C^{-1}_{X/k}:\Omega^{\bullet}(X^{(1)})\simeq \calH^{\bullet}(X)
\end{equation}

called the Cartier isomorphism, so that there exists a canonical exact sequence

$$0\rightarrow B^{\bullet}(X)\rightarrow Z^{\bullet}(X)\xrightarrow{C_{X/k}}\Omega^{\bullet}(X^{(1)})\rightarrow 0.$$

We now define, for every integer $n\geq 0$, abelian subsheaves of $\Omega^{\bullet}_X$ by induction on $n$.
For $n=0,1$ set

$$B^i_0(X):=0, \quad B^i_1(X):=B^i(X), \quad Z^i_0(X):=0,\quad  Z^i_1(X):=Z^i(X),$$

and for $n>1$ we let $B^{i}_{n+1}(X)$ and $Z_{n+1}^i$ be defined by following commutative cartesian diagrams

B_n+1^i(X)[hook]rrd[phantom]rrd       Z^i(X)[two heads]d    Z_n+1^i(X)[hook]rrd[phantom]rrd       Z^i(X)[two heads]d

B_n^i(X^(1))[hook]r   Ω^i(X^(1))rC_X/k^-1    ℋ^i(X),   Z_n^i(X^(1))[hook]r   Ω^i(X^(1))rC_X/k^-1    ℋ^i(X),

and we call $B^i_n(X)\subseteq Z^i_n(X)$ the sheaf of $i$-form that are $n$-exact and $n$-closed respectively.
By construction (see <cit.>) there is a chain of inclusions

$$0\subseteq B_1^i(X)\subseteq B_2^i(X)\subseteq \dots \subseteq B_n^i(X)\subseteq \dots \subseteq Z^i_{n+1}(X) \subseteq Z^i_{n}(X)\subseteq \dots \subseteq Z^i_{2}(X)\subseteq Z^i_{1}(X)\subseteq \Omegai(X)$$

Finally, set

$$\bigcup_{n\in \mathbb N} B^i_{n}:=B^i_{\infty}(X)\subseteq \bigcap_{n\in \mathbb N}Z^i_{n}(X):=Z^i_{\infty}(X)$$

which we call the sheaves of $i$-form that are indefinitely-exact and indefinitely-closed respectively.

Since $k$ is perfect, there is a Cartier morphism $C_X:Z^i(X)\rightarrow \Omegai(X)$ (see e.g. <cit.> which, by <cit.>, sends $Z^i_{\infty}(X)$ to itself, hence it induces an endomorphism of $C_X$ of $Z^i_{\infty}(X)$. By <cit.>, there is a canonical decomposition

\begin{equation}\label{decomposition}
Z^i_{\infty}(X)=B^i_{\infty}(X)\oplus (Z^i_{\infty}(X))_{ss},
\end{equation}

where  $(Z^i_{\infty}(X))_{ss}$ is characterized by the fact that for every open affine subset $U\subseteq X$ the vector space $(Z^i_{\infty}(X))_{ss}(U)$ identifies with union of the finite-dimensional subspaces of $Z^i_{\infty}(U)$ stable under $C$ and on which $C$ is an automorphism.

With this notation, we can recall from  <cit.> the main result we will use.

There is a natural isomorphism of sheaves

$$\Omega^i_{\log}(X)\otimes k\simeq  (Z^i_{\infty}(X))_{ss}.$$

§.§.§ Proof of  (-

We can now prove (<ref>).
Thanks to Theorem <ref> it is enough to prove that

\begin{equation}\label{eq : equalitydimension}
\dim_k(H^1(X, Z^1_{\infty}(X))_{ss}))=h^{1,1}.
\end{equation}

To conclude to proof, we show that there is a chain of isomorphisms:

We prove first (i), then (iii) and then (ii).

(i) By taking the long exact sequence associated with the short exact sequence

$$0\rightarrow \mathcal O_X\xrightarrow{(-)^p} \mathcal O_X\xrightarrow{d} B^1(X)\rightarrow 0,$$

one sees that the assumptions on $X$ imply that

\begin{equation}\label{eq : vanishing B1}
H^i(X,B^1(X))=0 \quad  \text{for}\quad   0\leq i\leq 2.
\end{equation}

Then the exact sequence

$$0\rightarrow B^1(X)\rightarrow Z^1(X)\xrightarrow{C} \Omega^1(X^{(1)})\rightarrow 0$$

shows that

$$H^1(X,Z^1(X))\simeq H^1(X^{(1)},\Omega^1(X^{(1)}))\simeq H^1(X,\Omega^1(X))),$$

where the last equality follows from the perfectness of $k$.

(iii) Thanks to the decomposition (<ref>), one has

$$H^1(X,Z^1_{\infty}(X))=H^1(X,B^1_{\infty}(X))\oplus H^1(X,(Z^i_{\infty}(X))_{ss}),$$

so that it is enough to show that

$$B^1_{\infty}(X)=\bigcup_{n\in \mathbb N} B^1_{n}(X)=\varinjlim_n B^1_{n}(X) $$

one has
$$H^1(X,B^1_{\infty}(X))=\varinjlim_n H^1(X,B^1_{n}(X)),$$

hence it is enough to show that $H^i(X,B_n^1(X))=0$ for every $n$ and $0\leq i\leq 2$. We prove this by induction on $n$, the case $n=1$ being (<ref>). So assuming that $n>1$ and $H^i(X,B_m^1(X))=0$ for every $m\leq n$ and $0\leq i\leq 2$ we have to show that $H^{i}(X,B_{n+1}^1(X))=0$. By <cit.>, one has a commutative diagram with exact rows

\begin{equation}\label{inductionbn}
\begin{tikzcd}
0\arrow{r} & B_1^2(X)\arrow{r}\arrow[equal]{d} & B_n^2(X)\arrow{r}{C^{-1}}\arrow{d} & B_{n-1}^2(X)\arrow{r}\arrow{d} & 0\\
0\arrow{r} & B_1^2(X)\arrow{r} & B_{n+1}^2(X)\arrow{r}{C^{-1}} & B_{n}^2(X)\arrow{r} & 0,
\end{tikzcd}
\end{equation}

which, taking the associated long exact sequence, shows the induction step.

$$Z^1_{\infty}(X)=\bigcap_{n\in \mathbb N} Z^1_{n}(X)=\varprojlim_n Z^1_{n}(X), $$

by <cit.> there is a canonical exact sequence

$$0\rightarrow \varprojlim_n\text{}^1(H^0(X,Z_n^1(X)))\rightarrow H^1(X,  Z^1_{\infty}(X))\rightarrow \varprojlim_n H^1(X,Z^1_{n}(X))\rightarrow 0.$$

Since $H^0(X,Z_n^1(X)))\subseteq H^0(X,\Omega^1(X)))=0$ by assumption,  it is enough to prove that the map $H^1(X,Z^1_{n+1}(X))\rightarrow H^1(X,Z^1_{n}(X))$ induced by the inclusion $Z^1_{n}(X)\subseteq Z^1_{n-1}(X)$ is an isomorphism.
To prove this, we first remark that, by <cit.> there is an exact sequence

$$0\rightarrow B^1_n(X)\rightarrow Z^1_{n+1}(X)\xrightarrow{C} Z^1_n(X)\rightarrow 0,$$
hence, by (<ref>), one gets $\dim(H^1(X,Z_{n+1}(X)))=\dim(H^1(X,Z_{n}(X)))$, so that we are reduced to show that $H^1(X,Z^1_{n+1}(X))\rightarrow H^1(X,Z^1_{n}(X))$ induced by the inclusion $Z^1_{n+1}(X)\subseteq Z^1_{n}(X)$ is injective.
By <cit.>, the inclusion $Z^1_{n+1}(X)\subseteq Z^1_{n}(X)$ sits in an exact sequence

$$0\rightarrow Z^1_{n+1}(X)\rightarrow Z^1_{n}(X)\rightarrow B^2_{n+1}(X)/B^2_{n}(X)\rightarrow 0,$$

so that it is enough to show that $H^0(B^2_{n+1}(X)/B^2_{n}(X))=0$.
The diagram (<ref>) shows that

$$B_{n+1}^2(X)/B_{n}^2(X)\simeq B_{n}^2(X)/B_{n-1}^2(X)\simeq B_{1}^2(X)/B_{0}^2(X)\simeq B^2(X).$$

$$H^0(X,B_{n+1}^2(X)/B_{n}^2(X))\simeq H^0(X,B^2(X))\subseteq H^0(X,\Omega^2(X))=0.$$

This concludes the proof of $(2)$ and hence of Theorem <ref>.

§ A STABLY IRRATIONAL VARIETY REDUCING TO A RATIONAL VARIETY

Let $R$ be the ring of integers of $K:=\C_p$ and $k$ its residue field.
In this last section, we show how to construct, for every $p\gg 0$, examples of smooth projective schemes $X/R$ such that $X_K$ is not stably-rational and such that $X_k$ is rational, as it was suggested to us by Colliot-Thélène.
The construction uses and it based on the analogous construction in <cit.> of a family of smooth proper varieties over the complex number with stably irrational general fiber but with some rational fiber.

§.§ A general lemma

We begin with a general lemma, which reduces the construction of examples to the construction of mixed characteristic families with easier-to-check properties.
Let $B/R$ be smooth with geometrically integral fibres and $X \rightarrow B$ a smooth projective family of varieties.

Assume that there exists a point $b\in B(\C_p)$ such that $X_b$ is not stably-rational. Then, for every $a\in B(k)$ there exists a lift $b'\in B(R)$ of $a$ such that $X_{b'}$ is not stably rational.

By <cit.>, the set

$$ B(\C_p)_{r} \coloneqq \{ b \in B(\C_p) \colon X_b \text{ is stably-rational} \}$$

is a countable union of closed subvarieties. Define now $B(\C_p)_{nr} \coloneqq B(\C_p) \setminus B(\C_p)_{r}$. By the assumption on $b$, the set $B(\C_p)_{r}$ is the countable union of proper closed subvarieties.

Since, by Hensel lemma, the map $\pi:B(R) \rightarrow B(k)$ is surjective we can choose a lift $b''$ of $a$. The set $\pi^{-1}(a)\subseteq B(R) \subseteq B(K)$ is an open neighborhood of $b''$ in $B(K)$.  Since $B(\C_p)_{r}$ is the countable union of proper closed subvarieties, we can apply <cit.> to deduce that there exists a $b'\in B(\C_p)_{nr}\cap \pi^{-1}(a)$. This concludes the proof.

§.§ An example

By Lemma <ref>, to construct a smooth projective scheme $X/R$ such that $X_K$ is not stably-rational and such that $X_k$ is rational, it is enough to construct a family $X\rightarrow B$ over $R$ such that there exist points $b\in B(\mathbb C_p)$ and $a\in B(k)$ such that $X_b$ is stably irrational and $X_a$ is rational.  Such a family can be constructed using directly Hassett, Pirutka, and Tschinkel example <cit.>, see also <cit.> and <cit.>. We give some details.

Let $X'\rightarrow Z$ be the universal family of quadric bundles over $\mathbb P^{2}_{\Q}$ given in $\mathbb P_\Q^3\times \mathbb P_\Q^2$ by a bihomogeneous form of bidegree $(2, 2)$. After choosing coordinates $x,y,z$ and $U,V,W,T$ on $\mathbb P_\Q^3\times \mathbb P_\Q^2$, the variety $Z$ identifies with the space of bihomogeneous forms $F = F(x,y,z,U,V,W,T)$ of bidegree $(2,2)$ in $\Pp^2_\Q \times \Pp^3_\Q$ which are symmetric quadratic forms in the variables $U,V,W,T$, since any such $F$ determines a quadratic bundle over $\Pp^2_\Q$ via the projection $\Pp^2_\Q \times \Pp^3_\Q \rightarrow \Pp^2_\Q$. In turn, these forms are given by a $4 \times 4$ symmetric matrix $A = (a_{i,j})_{1 \leq i,j \leq 4}$ where each entry $a_{i,j} = a_{i,j}(x,y,z)$ is a homogeneous polynomial of degree $2$.

By the arguments in  <cit.> and  Bertini theorem, there exists a dense open Zariski $B_{\Q} \subset Z$ such that the restriction of the family $X_{\Q}\rightarrow B_{\Q}$ to $B_{\Q}$ parametrizes conic bundle flat over $\Pp^2_\Q$ and with smooth total space.

By spreading out, this construction extends to give a smooth family $X \rightarrow B$ over $\mathbb{Z}[1/n]$ for $n$ big enough whose base change to $\Q$ identifies with $X_{\Q}\rightarrow B_{\Q}$. By the main result of <cit.>, the general fiber of $X(\mathbb C)\rightarrow B(\mathbb C)$ is not stably rational, hence, for every $p$, there exists a $b\in B(\mathbb C_p)$ such that $Y_b$ is not stably rational. So, we are left to show that for $p\gg 0$, there exists a $a\in B(\overline {\mathbb F}_{p})$ such that $X_a$ is rational.

Using Bertini, there exists of a rational point $r\in B(\Q)$ such that the corresponding $4 \times 4$ symmetric matrix $A = (a_{i,j})_{1 \leq i,j \leq 4}$ has $a_{1,1}=0$. By spreading out we can choose $p\gg 0$ such that $r \in B(\Q)$ extends to a point $\widetilde a \in B(\Z_p)$ whose reduction $a$ modulo $p$ defines a flat conic bundle $X_a\rightarrow  \mathbb P^2_{\mathbb F_p}$ with smooth total space, whose associated matrix has $a_{1,1}=0$.
Since $a_{1,1}=0$, the morphism $X_a\rightarrow \mathbb P^2_{\mathbb F_p}$ has a rational section, hence $X_a$ is rational. This concludes the construction of a smooth proper scheme over $R$ with rational special fiber and stably irrational generic fiber.

Emiliano Ambrosi, University of StrasbourgE-mail address:

Domenico Valloni, University of HannoverE-mail address:


# The Hamiltonian for Entangled States Cannot Be Additive

Kent A. Peacock  Department of Philosophy, University of Lethbridge
4401 University Drive, Lethbridge, Alberta, Canada. T1K 3M4
<EMAIL_ADDRESS>
###### Abstract

The assumption that the system Hamiltonian for entangled states is additive is
widely used in orthodox quantum no-signalling arguments. It is shown that
additivity implies a contradiction with the assumption that the system being
studied is entangled.

##  1 Introduction: Is there a ‘Bell Telephone’?

A large literature exists on the question of whether it is possible to use
quantum nonlocality to signal superluminally in a controllable fashion. The
conventional wisdom is that this is out of the question in quantum mechanics
as it is currently formulated (e.g., [5, 3, 16, 8, 15, 4]), although a
dissenting literature exists [1, 13, 9, 11, 14].

Many no-controllable signalling (NCS) arguments depend, directly or
indirectly, upon the assumption that the Hamiltonian for entangled states is
of a local or additive form. The aim of this note is to show that this
assumption can be undercut by a very simple _reductio ad absurdum_ argument.

An example of a NCS argument that uses the additivity assumption appears a
widely-cited paper by A. Shimony [16]. Shimony considers a two-particle system
(which presumably could be entangled) and _assumes without argument_ that if
we label the particles $A$ and $B$, the Hamiltonian for the system of two
particles (not including interactions it may have with measuring devices) has
the form

$H_{AB}=H_{A}\otimes\mathbb{I}_{B}+\mathbb{I}_{A}\otimes H_{B},$ (1)

where $\mathbb{I}_{B}$ is identity in $B$’s subspace, and correspondingly for
$A$. Shimony also assumes that the measurement procedure applied to one
particle (say $A$) acts only locally on $A$. Under these strong locality
assumptions Shimony shows that the time evolution operator associated with the
measurement process, $U(t)=\exp{(-H_{AB}t/\hbar)}$, acts only trivially on the
other particle; nothing done to the first particle can affect the local
statistics measureable on the second. It is well known that changing the
relative detector parameter in entangled states affects the _correlation_
between results taken on $A$ and $B$; thus one can encode a message in the
correlations, a fact which is the basis for quantum cryptography. But this can
be read only by having both local sequences of results, and thus it does not
constitute a method of controllable nonlocal signalling as such.

An interesting feature of Shimony’s argument is that it does not use the
collapse hypothesis; rather, the argument takes it that a measurement acting
on one particle will evolve the system in a unitary fashion. The question of
signalling does not turn on whether the wave function collapses, therefore,
but on getting the correct description for the dynamics of the system
consisting of an apparatus coupled to an entangled multiparticle state.

An approach similar to Shimony’s is taken by M. Redhead [15, §4.6], and D.
Dieks also presents a NCS argument that depends upon the assumption that a
Hamiltonian can be assigned to the individual particles in an entangled state
[3]. The question now is whether the additivity assumption is justified.

##  2 A Reductio of the Additivity Assumption

We show here that there is a straightforward reductio ad absurdum argument
against the additivity assumption for entangled states. To begin with, we cite
two facts about the mathematics of entangled states.

Rule 1: It is well known that subsystems of a tensor product (entangled) state
cannot be pure states. As Cohen-Tannoudji et al. put it,

> ”[A]n interaction between the two systems transforms an initial state which
> is a product into one which is no longer a product: any interaction between
> two systems therefore introduces, in general, correlations between them.
> …This question is very important since, in general, every physical system
> has interacted with others in the past … _it is not possible to associate a
> state vector $|\phi(1)\rangle$ [a pure state] with system (1) alone_.”
> [emphasis added] [2, p. 293].

Subsystems of a tensor product state must be described as mixed
states—classical probability distributions over sets of pure states.

Rule 2: To say that an observable is associated with a state $|\psi\rangle$ is
to say that the observable has a set of eigenstates which define a basis for
the state space in which $|\psi\rangle$ lives [2, Chap II.D].

Now suppose that it is possible to associate a Hamiltonian $H_{1}$ with a
particle $p_{1}$ belonging to an entangled multiparticle system. By Rule 2, it
would be possible to write the state of that particle as an expansion of the
form

$|\phi(p_{1})\rangle=\sum_{i}c_{i}|e_{i}\rangle$ (2)

where $\\{|e_{i}\rangle\\}$ are the energy eigenstates of the particle $p_{1}$
with respect to $H_{1}$, and $c_{i}$ are complex coefficients. _Any expression
of this form is a pure state_ , since it is a linear combination of pure
states (the presumed local energy eigenstates of $H_{1}$). But by Rule 1,
because $p_{1}$ is taken to be a member of an entangled state, it cannot, by
itself, itself be represented as a pure state—otherwise, it would not be
entangled. Hence, there cannot exist a Hamiltonian that can be associated with
$p_{1}$ in this way.

In sum, the assumption of additivity (1) contradicts the assumption that the
particles are entangled. Thus, any no-signalling argument that depends upon
the assumption of additivity (or any other equivalent expression of dynamic
localizability) is merely a demonstration of no-signalling for _product
states_ , which _do_ have additive dynamics.

Just as the state of a particle in an entangled state is a mixture (a
classical probability distribution over possible pure states), the possible
energies for individual particles in entangled states must be a classical
probability distribution over possible energy states. As Cohen-Tannoudji et
al. observe,

> “Just as with vectors, there exist operators in $\mathscr{E}$ [a tensor
> product space] which are not tensor products of an operator in
> $\mathscr{E}_{1}$ and an operator of $\mathscr{E}_{2}$.” [2, p. 157]

The system Hamiltonian for an entangled tensor product space belonging to a
multiparticle system must be such a global or nonlocal operator.

The energy of one particle in an entangled system must in general, therefore,
depend nonlocally on what is done to the other remote particles in the system.
As Shimony himself correctly emphasized [16], whether or not this dependency
is locally controllable is an important but distinct question. However, the
reductio argument outlined here rules out any NCS argument depending upon a
prior assumption of the dynamic locality of the subsystem particles.

##  3 Dynamics of Entangled States in Quantum Information Theory

What, then, is the correct description of the dynamics of entangled states? In
1933, W. Pauli stated,

> “An additive decomposition of the Hamiltonian into independent summands
> corresponds [_entsprich_], therefore, to a product decomposition of the
> wavefunction into independent factors.” [12, 7]

Unfortunately, Pauli did not directly address the question of what would
pertain to the dynamics of _nonfactorizable_ multiparticle states and so it is
unclear, from his exposition alone, whether his entsprich should be read as
“if and only if”.

In fact, it is a commonplace in recent literature on quantum information
theory that the Hamiltonians for entangled states themselves contain cross-
terms and thus cannot in general be expressed in additive form. For example,
Dür et al. (2002) give the general Hamiltonian for an entangled (Bell) state
of two particles as,

$H_{AB}=\sum_{i=1}^{3}\alpha_{i}\sigma_{i}^{A}\otimes\mathbb{I}_{B}+\sum_{j=1}^{3}\beta_{j}\mathbb{I}_{A}\otimes\sigma_{j}^{B}+\sum_{i,j=1}^{3}\gamma_{i,j}\sigma_{i}^{A}\otimes\sigma_{j}^{B},$
(3)

where the $\sigma_{i}$ are the Pauli matrices. This expansion contains
irreducible cross-terms as shown, and therefore cannot be represented in
purely additive form. Thus, again, we see that any NCS argument that depends
upon Shimony’s additivity assumption (1) cannot apply without qualification to
entangled states, which are the only sort of states of interest in the
signalling question.

The cross-terms in (3) suggest that the full energy spectrum of entangled
states must contain terms that are not local to the individual components of
the system, but which (like the energies of electron orbitals in atoms) are
properties of the system as a whole. An interferometric experiment by Lee et
al. [10, 6] directly demonstrates the existence of nonlocal energy states in
certain kinds of entangled systems. In this experiment, two diamond chips 30
cm apart are demonstrably put into the same phonon state. This experimental
finding alone indicates that the additivity assumption for entangled states
cannot be generally correct.

##  4 Conclusion and Discussion

In sum: if the Hamiltonian of a multiparticle system is additive (entirely
local to the particles), then the system cannot be entangled—since otherwise
it would be impossible to assign a spectrum of pure states to the component
particles as in (2). If this view is correct, then any NCS arguments that rely
on the additivity assumption amount merely to unobjectionable demonstrations
of NCS for product states. This observation, by itself, does not show that
controllable quantum signalling is possible, but it shows that a large class
of arguments commonly cited against that possibility are simply inapplicable
to the only kind of systems (entangled states) for which the question of
signalling arises.

The other major route that has been used to argue for NCS is via
microcausality (the assumption that measurement operations on spacelike
separate particles in a multiparticle system always commute) [11]. The
argument presented in this note does not fully address the question of
microcausality. We simply note the following points. If operations on
individual spacelike separate particles in a multiparticle system act only
locally on their respective particles, then it is readily shown that they
commute in their action on the system as a whole, as one would expect. On the
other hand, if operators are global in the sense indicated by Cohen-Tannoudji
et al., it is not immediately apparent that they can be guaranteed to commute
in their action on an entangled state considered _as a whole_ [13]. This
important question demands further study.

##  5 Acknowledgements

The author thanks the University of Lethbridge and the Social Sciences and
Humanities Research Council of Canada for support. Thanks also to the
following for helpful discussion: J.R. Brown, M.B. Brown, S. Das, B. Hepburn,
and W. Dür. None of these people or organizations are responsible for any
errors in this paper.

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