# Poisson catenarity in Poisson nilpotent algebras

K. R. Goodearl  Department of Mathematics
University of California
Santa Barbara, CA 93106
U.S.A<EMAIL_ADDRESS>and  S. Launois  School of Mathematics,
Statistics and Actuarial Science
University of Kent
Canterbury, Kent, CT2 7FS
UK<EMAIL_ADDRESS>
###### Abstract.

We prove that for the iterated Poisson polynomial rings known as Poisson
nilpotent algebras (or Poisson-CGL extensions), the Poisson prime spectrum is
catenary, i.e., all saturated chains of inclusions of Poisson prime ideals
between any two given Poisson prime ideals have the same length.

###### Key words and phrases:

Poisson algebra, Poisson polynomial ring, Poisson-Ore extension, Poisson
nilpotent algebra, Poisson-CGL extension, Poisson prime ideal, Poisson prime
spectrum, catenary

###### 2010 Mathematics Subject Classification:

Primary 17B63; Secondary 13C15

The research of the first named author was supported by US National Science
Foundation grant DMS-1601184. That of the second named author was supported by
EPSRC grant EP/N034449/1.

##  1\. Introduction

The aim of this paper is to study the Poisson prime spectra of Poisson
algebras. More precisely, we focus on the catenary property – that all
saturated chains of inclusions of Poisson prime ideals between any two fixed
Poisson prime ideals have the same length – for the large class of Poisson
algebras called Poisson nilpotent algebras. Many such algebras arise as
semiclassical limits of quantum algebras, e.g., coordinate rings of matrix
varieties or Schubert cells with natural Poisson structures. See [8, Section
2] and [5], for instance, for a number of examples.

Many properties of quantum algebras are known, or are conjectured to be,
reflected in parallel properties of the Poisson algebras that arise as their
semiclassical limits. Catenarity has been proved for the prime spectra of many
quantized coordinate rings and related algebras, and the authors recently
proved it for quantum nilpotent algebras [9]. The parallel result, for the
Poisson prime spectra of Poisson nilpotent algebras, is established here.

In order to provide further detail, a few definitions are in order.

###  1.1. Poisson nilpotent algebras

The algebras just named are iterated Poisson polynomial algebras
$A[x;\sigma,\delta]_{p}$ (see Definition 2.1), meaning that $A$ is a Poisson
algebra and $A[x;\sigma,\delta]_{p}$ denotes a polynomial ring $A[x]$ equipped
with a Poisson bracket that extends the one on $A$ and satisfies

$\\{x,a\\}=\sigma(a)x+\delta(a)\qquad\forall\,a\in A,$

where $\sigma$ and $\delta$ are suitable derivations on $A$. Given an iterated
Poisson polynomial algebra

(1.1)
$R:={\mathbb{K}}[x_{1}]_{p}[x_{2};\sigma_{2},\delta_{2}]_{p}\cdots[x_{N};\sigma_{N},\delta_{N}]_{p}\,,$

we set

$R_{k}:={\mathbb{K}}[x_{1},\dots,x_{k}]={\mathbb{K}}[x_{1}]_{p}[x_{2};\sigma_{2},\delta_{2}]_{p}\cdots[x_{k};\sigma_{k},\delta_{k}]_{p}$

for $k\in[[0,N]]$. In particular, $R_{0}={\mathbb{K}}$ and
$R_{1}={\mathbb{K}}[x_{1}]_{p}={\mathbb{K}}[x_{1};0,0]_{p}$.

######  Definition 1.1.

An iterated Poisson polynomial algebra $R$ as in (1.1) is called a _Poisson
nilpotent algebra_ or a _Poisson-CGL extension_ [11, Definition 5.1] if it is
equipped with a rational action of an algebraic torus ${\mathcal{H}}$ by
Poisson algebra automorphisms such that

  1. (i)

The elements $x_{1},\dots,x_{N}$ are ${\mathcal{H}}$-eigenvectors.

  2. (ii)

For every $k\in[[2,N]]$, the map $\delta_{k}$ on $R_{k-1}$ is locally
nilpotent.

  3. (iii)

For every $k\in[[1,N]]$, there exists
$h_{k}\in\operatorname{Lie}{\mathcal{H}}$ such that
$(h_{k}{\cdot})|_{R_{k-1}}=\sigma_{k}$ and the $h_{k}$-eigenvalue of $x_{k}$
is nonzero.

The main theorem of the paper is

######  Theorem 1.2.

If $R$ is a Poisson nilpotent algebra, then its Poisson prime spectrum
$\operatorname{P{.}Spec}R$ is catenary.

Here $\operatorname{P{.}Spec}R$ consists of the ideals of $R$ which are both
Poisson ideals and prime ideals (see Definition 2.2 and Lemma 2.3).

###  1.2. Notation and conventions

Throughout, all algebras will be unital commutative algebras over a fixed base
field ${\mathbb{K}}$ of characteristic zero.

##  2\. Background and basics

Recall that a _Poisson algebra_ (over ${\mathbb{K}}$) is a commutative
${\mathbb{K}}$-algebra equipped with a Poisson bracket $\\{-,-\\}$, that is, a
Lie bracket which is also a derivation in each variable (for the associative
product).

######  Definition 2.1.

Let $A$ be a Poisson algebra. A _Poisson derivation_ on $A$ is a
${\mathbb{K}}$-linear map $\sigma$ on $A$ which is a derivation with respect
to both the multiplication and the Poisson bracket, that is,

$\sigma(ab)=\sigma(a)b+a\sigma(b)\qquad\text{and}\qquad\sigma(\\{a,b\\})=\\{\sigma(a),b\\}+\\{a,\sigma(b)\\}\qquad\forall\,a,b\in
A.$

Suppose that $\delta$ is a _Poisson $\sigma$-derivation_ on $A$ (in the
terminology of [3, §1.1.2]), meaning that $\delta$ is a ${\mathbb{K}}$-linear
derivation and

(2.1)
$\delta(\\{a,b\\})=\\{\delta(a),b\\}+\\{a,\delta(b)\\}+\sigma(a)\delta(b)-\delta(a)\sigma(b)\qquad\forall\,a,b\in
A.$

By [14, Theorem 1.1] (after replacing $\sigma$ with $-\sigma$), the Poisson
structure on $A$ extends uniquely to a Poisson algebra structure on the
polynomial ring $A[x]$ such that

$\\{x,a\\}=\sigma(a)x+\delta(a)\qquad\forall\,a\in A.$

We write $A[x]=A[x;\sigma,\delta]_{p}$ to denote this situation, and we refer
to $A[x;\sigma,\delta]_{p}$ as a _Poisson polynomial algebra_ or a _Poisson-
Ore extension_. The Poisson structure on $A[x]$ extends uniquely to the
Laurent polynomial ring $A[x^{\pm 1}]$. We write $A[x^{\pm
1};\sigma,\delta]_{p}$ for this Poisson algebra and refer to it as a _Poisson
Laurent polynomial algebra_.

In either of the above cases, we omit $\delta$ from the notation if it is
zero, that is, we write $A[x;\sigma]_{p}$ and $A[x^{\pm 1};\sigma]_{p}$ for
$A[x;\sigma,0]_{p}$ and $A[x^{\pm 1};\sigma,0]_{p}$.

The converse part of [14, Theorem 1.1] will also be needed: If a polynomial
ring $A[x]$ supports a Poisson bracket such that $A$ is a Poisson subalgebra
and $\\{x,A\\}\subseteq A+Ax$, then $A[x]=A[x;\sigma,\delta]_{p}$ for suitable
$\sigma$ and $\delta$.

######  Definition 2.2.

Let $R$ be a Poisson algebra. The Poisson bracket on $R$ induces unique
Poisson brackets on any quotient of $R$ modulo a _Poisson ideal_ , meaning an
ideal $I$ such that $\\{R,I\\}\subseteq I$, and on any localization of $R$
with respect to a multiplicative set (e.g., [13, Proposition 1.7]).

A _Poisson automorphism_ of $R$ is any algebra automorphism which preserves
the Poisson bracket. We use the term _Poisson action_ to refer to an action of
a group on $R$ by Poisson automorphisms.

The _Poisson center_ of $R$ is the set $Z_{p}:=\\{z\in R\mid\\{z,-\\}\equiv
0\\}$, which is a subalgebra of $R$. A _Poisson-prime_ ideal is any proper
Poisson ideal $P$ of $R$ such that $(IJ\subseteq P\implies I\subseteq
P\;\text{or}\;J\subseteq P)$ for all Poisson ideals $I$ and $J$ of $R$.

######  Lemma 2.3.

[7, Lemma 1.1] Let $R$ be a Poisson algebra.

(a) Every prime ideal minimal over a Poisson ideal is a Poisson ideal.

(b) If $R$ is noetherian, every Poisson-prime ideal of $R$ is prime.

######  Remark.

If $R$ is a noetherian Poisson algebra, Lemma 2.3(b) implies that the Poisson-
prime ideals in $R$ are precisely the ideals which are both Poisson ideals and
prime ideals; in that case, the hyphen in the term “Poisson-prime” becomes
unnecessary.

The _Poisson prime spectrum_ of a Poisson algebra $R$ is the set
$\operatorname{P{.}Spec}R$ of all Poisson-prime ideals of $R$, equipped with
the natural Zariski topology (where the closed sets are the sets
$\\{P\in\operatorname{P{.}Spec}R\mid P\supseteq I\\}$ for Poisson ideals $I$
of $R$).

Suppose $R$ is a Poisson algebra and we have a Poisson action of a group
${\mathcal{H}}$ on $R$. One defines _${\mathcal{H}}$ -prime_ and
_${\mathcal{H}}$ -Poisson-prime_ ideals of $R$ in the same manner as Poisson-
prime ideals. Namely, an ${\mathcal{H}}$-prime (resp.,
${\mathcal{H}}$-Poisson-prime) ideal is any proper ${\mathcal{H}}$-stable
(resp., ${\mathcal{H}}$-stable Poisson) ideal $P$ of $R$ such that
$(IJ\subseteq P\implies I\subseteq P\;\text{or}\;J\subseteq P)$ for all
${\mathcal{H}}$-stable (resp., ${\mathcal{H}}$-stable Poisson) ideals $I$ and
$J$ of $R$. We denote the sets of ${\mathcal{H}}$-prime and
${\mathcal{H}}$-Poisson-prime ideals of $R$ by
${\mathcal{H}}{\text{-}}\operatorname{Spec}R$ and
${\mathcal{H}}\text{-}{\operatorname{P{.}Spec}}\,R$, respectively.

These concepts are related by an analog of Lemma 2.3 when ${\mathcal{H}}$ is
an algebraic torus and the action of ${\mathcal{H}}$ on $R$ is _rational_ ,
meaning that $R$ is spanned by ${\mathcal{H}}$-eigenvectors whose
${\mathcal{H}}$-eigenvalues ${\mathcal{H}}\rightarrow{\mathbb{K}}^{*}$ are
morphisms of algebraic groups.

######  Lemma 2.4.

[11, Lemma 4.3] Let $R$ be a noetherian Poisson algebra, equipped with a
rational Poisson action of a torus ${\mathcal{H}}$. Then any prime ideal of
$R$ minimal over an ${\mathcal{H}}$-stable Poisson ideal is itself an
${\mathcal{H}}$-stable Poisson ideal, and the ${\mathcal{H}}$-Poisson-prime
ideals of $R$ are exactly the ${\mathcal{H}}$-stable, Poisson, prime ideals of
$R$.

Lemma 2.4 allows a shortening of terminology: In the setting of the lemma,

(2.2)
${\mathcal{H}}\text{-}{\operatorname{P{.}Spec}}\,R=\\{\text{Poisson}\;{\mathcal{H}}\text{-prime
ideals
of}\;R\\}={\mathcal{H}}{\text{-}}\operatorname{Spec}R\cap\operatorname{P{.}Spec}R\cap\operatorname{Spec}R.$

This is immediate from the lemma together with the fact that the
${\mathcal{H}}$-prime ideals of $R$ coincide with the ${\mathcal{H}}$-stable
prime ideals [2, Proposition II.2.9].

######  Proposition 2.5.

Let $R$ be a ${\mathbb{K}}$-algebra equipped with a rational action of a
${\mathbb{K}}$-torus ${\mathcal{H}}$, and set
$\mathfrak{h}:=\operatorname{Lie}{\mathcal{H}}$.

(a) The differential of the ${\mathcal{H}}$-action gives an action of
$\mathfrak{h}$ on $R$ by derivations. If $R$ is a Poisson algebra and the
${\mathcal{H}}$-action is a Poisson action, then $\mathfrak{h}$ acts on $R$ by
Poisson derivations.

(b) The $\mathfrak{h}$-action on $R$ commutes with the ${\mathcal{H}}$-action.

(c) The $\mathfrak{h}$-eigenspaces in $R$ coincide with the
${\mathcal{H}}$-eigenspaces.

(d) Suppose $R^{\prime}$ is another ${\mathbb{K}}$-algebra, equipped with a
rational action of the torus ${\mathcal{H}}$ by ${\mathbb{K}}$-algebra
automorphisms. Then any ${\mathcal{H}}$-equivariant ${\mathbb{K}}$-algebra
homomorphism $\phi:R\rightarrow R^{\prime}$ is also
$\mathfrak{h}$-equivariant.

###### Proof.

(a)(b)(c) See [8, Section 1.2 and Lemmas 1.3, 1.4].

(d) Since $\phi$ is ${\mathcal{H}}$-equivariant, it is a homogeneous map with
respect to the $X({\mathcal{H}})$-gradings on $R$ and $R^{\prime}$. It follows
from [8, Lemma 1.3] that $\phi(h\cdot r)=h\cdot\phi(r)$ for all
$h\in\mathfrak{h}$ and all $X({\mathcal{H}})$-homogeneous elements $r\in R$,
and thus $\phi$ is $\mathfrak{h}$-equivariant. ∎

Whenever we have a ${\mathbb{K}}$-algebra $R$ equipped with a rational action
of a ${\mathbb{K}}$-torus ${\mathcal{H}}$, we assume that
$\operatorname{Lie}{\mathcal{H}}$ is acting on $R$ by the differential of the
${\mathcal{H}}$-action.

##  3\. Poisson catenarity

We address Poisson catenarity of general Poisson algebras in this section,
beginning with an example for which Poisson catenarity fails. We then
establish two general theorems. The first provides sufficient conditions for
the Poisson prime spectrum of a Poisson algebra to be catenary, and the second
provides a reduction of the key condition in the first theorem.

The following is a Poisson adaptation of an example of Bell and Sigurksson [1,
Example 2.10].

######  Example 3.1.

Let $R$ be the polynomial ring ${\mathbb{K}}[x,y,z,w]$. There is a Poisson
bracket on $R$ such that

$\displaystyle\\{x,y\\}$ $\displaystyle=0$ $\displaystyle\qquad\\{x,z\\}$
$\displaystyle=0$ $\displaystyle\qquad\\{y,z\\}$ $\displaystyle=0$
$\displaystyle\\{w,x\\}$ $\displaystyle=2yz$ $\displaystyle\qquad\\{w,y\\}$
$\displaystyle=x+y^{2}$ $\displaystyle\\{w,z\\}$ $\displaystyle=0.$

The restriction of $\\{w,-\\}$ to $A:={\mathbb{K}}[x,y,z]$ equals the
derivation $\delta:=2yz\frac{\partial}{\partial
x}+(x+y^{2})\frac{\partial}{\partial y}$ and $R=A[w;0,\delta]_{p}$.

By [1, Example 2.10], the $\delta$-invariant prime ideal $P:=Ax+Ay$ of $A$ has
$\delta$-height $1$, that is, $P$ does not properly contain any nonzero
$\delta$-invariant prime ideals of $A$. We claim that the Poisson prime ideal
$RP=Rx+Ry$ has height $1$ in $\operatorname{P{.}Spec}R$, that is, $RP$ does
not properly contain any nonzero Poisson prime ideals of $R$.

Suppose, to the contrary, that $RP$ properly contains a nonzero Poisson prime
ideal $Q$. Then $Q\cap A$ is a $\delta$-invariant prime ideal of $A$ properly
contained in $P$, whence $Q\cap A=0$. Consider the polynomial ring $T:=F[w]$
where $F:=\operatorname{{\mathrm{Fract}}}A$. As a localization of $R$, the
ring $T$ is a Poisson algebra, and $TQ$ is a proper nonzero Poisson ideal of
$T$. Note that the Poisson bracket vanishes on $F\times F$, since it vanishes
on $A\times A$. Let

$t=t_{0}+t_{1}w+\cdots+t_{n}w^{n}$

be a nonzero polynomial in $TQ$ of minimal degree, and note that $n>0$. Since
we may replace $t$ by $t_{n}^{-1}t$, there is no loss of generality in
assuming that $t_{n}=1$. Observe that

$\\{x,t\\}=nw^{n-1}\\{x,w\\}+[\text{lower terms}]=-2nyzw^{n-1}+[\text{lower
terms}],$

so that $\deg\\{x,t\\}=n-1$. Since $\\{x,t\\}\in TQ$, this contradicts the
minimality of $n$.

Therefore $RP$ has height $1$ in $\operatorname{P{.}Spec}R$, as claimed.
Consequently, the chain

$0<Rx+Ry<Rx+Ry+Rz$

in $\operatorname{P{.}Spec}R$ is saturated. However, there is also a saturated
chain

$0<Rz<Rx+Rz<Rx+Ry+Rz$

in $\operatorname{P{.}Spec}R$. Therefore $\operatorname{P{.}Spec}R$ is not
catenary.

There are two general theorems in the literature that have been used to
establish catenarity for the prime spectra of various quantum algebras – [10,
Theorem 1.6] and [16, Theorem 0.1]. A key hypothesis in both theorems is
_normal separation_ , a condition that requires a suitable supply of normal
elements in prime factor rings. An analogous condition, as follows, is
important in the Poisson setting.

######  Definition 3.2.

Let $R$ be a Poisson algebra.

An element $c\in R$ is called _Poisson-normal_ provided $\\{c,R\\}\subseteq
Rc$, that is, $Rc$ is a Poisson ideal of $R$.

The set $\operatorname{P{.}Spec}R$ is said to have _Poisson-normal separation_
provided that for any pair of distinct comparable prime ideals $P\subsetneq Q$
in $R$, the factor $Q/P$ contains a nonzero Poisson-normal element of $R/P$.

######  Theorem 3.3.

Let $R$ be a noetherian Poisson algebra such that $\operatorname{P{.}Spec}R$
has Poisson-normal separation. Then any saturated chain in
$\operatorname{P{.}Spec}R$ is also saturated as a chain in
$\operatorname{Spec}R$.

###### Proof.

We just need to show that if $P\subsetneq Q$ are Poisson prime ideals such
that no Poisson prime ideals lie strictly between $P$ and $Q$, then no prime
ideals lie strictly between $P$ and $Q$, that is, $\operatorname{ht}(Q/P)=1$.

By Poisson-normal separation, there exists $c\in Q\setminus P$ such that $c$
is Poisson-normal modulo $P$. Hence, $Rc+P$ is a Poisson ideal contained in
$Q$. Let $Q^{\prime}\subseteq Q$ be a prime ideal minimal over $Rc+P$. By
Lemma 2.3, $Q^{\prime}$ is a Poisson ideal, and our assumption on $P\subsetneq
Q$ implies that $Q^{\prime}=Q$. Thus $Q$ is minimal over $Rc+P$. Krull’s
Principal Ideal Theorem therefore implies $\operatorname{ht}(Q/P)=1$, as
required. ∎

We refer to a Poisson algebra as _affine_ when the underlying associative
algebra is affine (i.e., finitely generated as an algebra).

######  Corollary 3.4.

Let $R$ be an affine Poisson algebra. If $\operatorname{P{.}Spec}R$ has
Poisson-normal separation, then $\operatorname{P{.}Spec}R$ is catenary.

###### Proof.

Since $R$ is an affine commutative algebra, $\operatorname{Spec}R$ is catenary
(e.g., [4, Corollary 13.6]). Given $P\subsetneq Q$ in
$\operatorname{P{.}Spec}R$, any two saturated chains of Poisson prime ideals
from $P$ to $Q$ are also saturated as chains of ordinary prime ideals by
Theorem 3.3, so they have the same length due to catenarity of
$\operatorname{Spec}R$. ∎

######  Definition 3.5.

The _Poisson height_ of a Poisson prime ideal $P$ in a Poisson algebra $R$,
denoted $\operatorname{P-ht}P$, is the height of $P$ within the poset
$\operatorname{P{.}Spec}R$, that is, the supremum of the lengths of all chains
of Poisson prime ideals in $R$ descending from $P$.

######  Corollary 3.6.

Let $R$ be an affine Poisson algebra. If $\operatorname{P{.}Spec}R$ has
Poisson-normal separation, then $\operatorname{P-ht}P=\operatorname{ht}P$ for
all $P\in\operatorname{P{.}Spec}R$.

###### Proof.

Let $P\in\operatorname{P{.}Spec}R$. Any minimal prime ideal $Q$ contained in
$P$ is a Poisson prime by Lemma 2.3(a), and any saturated chain from $P$ to
$Q$ in $\operatorname{P{.}Spec}R$ is also saturated in $\operatorname{Spec}R$
by Theorem 3.3. Since $\operatorname{P{.}Spec}R$ and $\operatorname{Spec}R$
are both catenary, it follows that
$\operatorname{P-ht}P/Q=\operatorname{ht}P/Q$. Taking suprema over all such
$Q$ yields $\operatorname{P-ht}P=\operatorname{ht}P$. ∎

######  Remark 3.7.

In many quantum algebras $T$, catenarity of $\operatorname{Spec}T$ is
accompanied by _Tauvel’s height formula_ , which says that

$\operatorname{ht}P+\operatorname{GKdim}T/P=\operatorname{GKdim}T$

for all prime ideals $P$ of $T$, assuming that $T$ is a prime ring. An
analogous Poisson version for an affine Poisson domain $R$ would say that

(3.1) $\operatorname{P-ht}P+\operatorname{GKdim}R/P=\operatorname{GKdim}R$

for all Poisson prime ideals $P$ of $R$. If $\operatorname{P{.}Spec}R$ has
Poisson-normal separation, then (3.1) follows from Corollary 3.6, since
$\operatorname{GKdim}=\operatorname{K{.}dim}$ for commutative affine algebras
(e.g., [12, Theorem 4.5]) and
$\operatorname{ht}P+\operatorname{K{.}dim}R/P=\operatorname{K{.}dim}R$ for
prime ideals $P$ in such algebras $R$ (e.g., [4, Corollary 13.4]).

A more “fully Poisson” version of (3.1) would replace GK-dimension with
_Poisson GK-dimension_ as defined in [15, Definition 3.5]. However, the
resulting equations would be the same as (3.1), since, as is easily verified,
Poisson GK-dimension agrees with GK-dimension on affine Poisson algebras.

Our approach to the second main theorem of this section involves a
stratification of the Poisson prime spectrum of a Poisson algebra.

######  Definition 3.8.

Let $R$ be a ring and ${\mathcal{H}}$ a group acting on $R$ by ring
automorphisms. The _${\mathcal{H}}$ -core_ of an ideal $I$ of $R$ is the ideal
$(I:{\mathcal{H}}):=\bigcap_{h\in{\mathcal{H}}}h\cdot I$. That is,
$(I:{\mathcal{H}})$ is the largest ${\mathcal{H}}$-stable ideal of $R$
contained in $I$.

######  Lemma 3.9.

[7, Lemma 3.1] Let $R$ be a noetherian Poisson algebra equipped with a
rational Poisson action of an algebraic torus ${\mathcal{H}}$.

(a) If $I$ is a Poisson ideal of $R$, then $(I:{\mathcal{H}})$ is an
${\mathcal{H}}$-stable Poisson ideal.

(b) $(P:{\mathcal{H}})\in{\mathcal{H}}\text{-}{\operatorname{P{.}Spec}}\,R$
for all $P\in\operatorname{P{.}Spec}R$.

######  Definition 3.10.

Let $R$ be a noetherian Poisson algebra equipped with a rational Poisson
action of an algebraic torus ${\mathcal{H}}$. For
$J\in{\mathcal{H}}\text{-}{\operatorname{P{.}Spec}}\,R$, set

$\operatorname{P{.}Spec}_{J}R=\\{P\in\operatorname{P{.}Spec}R\mid(P:{\mathcal{H}})=J\\}.$

In view of Lemma 3.9(b), we obtain a partition

$\operatorname{P{.}Spec}R=\bigsqcup_{J\in{\mathcal{H}}\text{-}{\operatorname{P{.}Spec}}\,R}\operatorname{P{.}Spec}_{J}R,$

which is known as the _${\mathcal{H}}$ -stratification of
$\operatorname{P{.}Spec}R$_.

Let $\mathcal{E}_{J}$ denote the set of ${\mathcal{H}}$-eigenvectors in $R/J$.
Since $R/J$ is a domain (Lemma 2.4), $\mathcal{E}_{J}$ is multiplicatively
closed, and the localization $R_{J}:=(R/J)[\mathcal{E}_{J}^{-1}]$ is a
subalgebra of the quotient field $\operatorname{{\mathrm{Fract}}}R/J$. The
Poisson bracket on $R$ uniquely induces Poisson brackets on $R/J$ and $R_{J}$
(as well as on $\operatorname{{\mathrm{Fract}}}R/J$), so that these algebras
become Poisson algebras. Similarly, the action of ${\mathcal{H}}$ uniquely
induces actions on $R/J$ and $R_{J}$ by Poisson automorphisms, as well as on
$\operatorname{{\mathrm{Fract}}}R/J$, although the latter action is no longer
rational.

######  Theorem 3.11.

[7, Theorem 4.2] Let $R$ be a noetherian Poisson algebra equipped with a
rational Poisson action of an algebraic torus ${\mathcal{H}}$. Let
$J\in{\mathcal{H}}\text{-}{\operatorname{P{.}Spec}}\,R$.

(a) The algebra $R_{J}$ is a graded field with respect to its induced
$X({\mathcal{H}})$-grading, i.e., all nonzero homogeneous elements of $R_{J}$
are invertible.

(b) $\operatorname{P{.}Spec}_{J}R$ is homeomorphic to
$\operatorname{P{.}Spec}R_{J}$ via localization and contraction.

(c) $\operatorname{P{.}Spec}R_{J}$ is homeomorphic to
$\operatorname{Spec}Z_{p}(R_{J})$ via contraction and extension.

From this theorem we obtain a Poisson analog of the first conclusion of [6,
Theorem 5.3] as follows.

######  Definition 3.12.

Suppose $R$ is a Poisson algebra equipped with a Poisson action of a group
${\mathcal{H}}$. Adapting the notation of [6, §5.2], we say that
${\mathcal{H}}\text{-}{\operatorname{P{.}Spec}}\,R$ has _Poisson-normal
${\mathcal{H}}$-separation_ provided that for any proper inclusion
$P\subsetneq Q$ of Poisson ${\mathcal{H}}$-prime ideals of $R$, there exists a
Poisson-normal ${\mathcal{H}}$-eigenvector of $R/P$ which lies in $Q/P$.

######  Theorem 3.13.

Let $R$ be a noetherian Poisson algebra equipped with a rational Poisson
action of an algebraic torus ${\mathcal{H}}$. If
${\mathcal{H}}\text{-}{\operatorname{P{.}Spec}}\,R$ has Poisson-normal
${\mathcal{H}}$-separation, then $\operatorname{P{.}Spec}R$ has Poisson-normal
separation.

###### Proof.

Let $P\subsetneq Q$ be a proper inclusion of Poisson prime ideals of $R$. Set
$J:=(P:{\mathcal{H}})$ and $K:=(Q:{\mathcal{H}})$, which are members of
${\mathcal{H}}\text{-}{\operatorname{P{.}Spec}}\,R$ such that
$P\in\operatorname{P{.}Spec}_{J}R$ and $Q\in\operatorname{P{.}Spec}_{K}R$.
Obviously $J\subseteq K$.

If $J\neq K$, then by Poisson-normal ${\mathcal{H}}$-separation there is an
${\mathcal{H}}$-eigenvector $c\in K\setminus J$ which is Poisson-normal modulo
$J$. Obviously $c\in Q$ and $c$ is Poisson-normal modulo $P$. Since $Rc$ is an
${\mathcal{H}}$-invariant ideal not contained in $J$, we see that
$Rc\nsubseteq P$, that is, $c\notin P$. Thus in this case we are done.

Now assume that $J=K$. Since it is harmless to pass to $R/J$, there is no loss
of generality in assuming that $J=K=0$. We shall need the following fact:

(3.2) $Z_{p}(R_{0})\subseteq\\{ac^{-1}\mid a,c\in R\ \text{and}\ c\ \text{is a
Poisson-normal}\ {\mathcal{H}}\text{-eigenvector}\\}.$

First note that the induced ${\mathcal{H}}$-action on $R_{0}$ is rational,
equivalently, $R_{0}$ is $X({\mathcal{H}})$-graded. Since ${\mathcal{H}}$ acts
by Poisson automorphisms, the Poisson bracket on $R_{0}$ is
$X({\mathcal{H}})$-graded, that is,
$\\{(R_{0})_{u},(R_{0})_{v}\\}\subseteq(R_{0})_{u+v}$ for all $u,v\in
X({\mathcal{H}})$. It follows that an element $z\in R_{0}$ is Poisson-central
if and only if all its homogeneous components are Poisson-central. Hence, to
prove (3.2) it suffices to show that any homogeneous element $z\in
Z_{p}(R_{0})$ can be written in the desired form.

Given such a $z$, consider the nonzero ideal $I:=\\{r\in R\mid zr\in R\\}$.
Since $z$ is both homogeneous and Poisson-central, $I$ is an
${\mathcal{H}}$-stable Poisson ideal. If $z\in R$, we can write $z=z1^{-1}$ in
the desired form, so we may assume $z\notin R$, whence $I\neq R$. There are
prime ideals $P_{1},\dots,P_{m}$ minimal over $I$ such that $P_{1}P_{2}\cdots
P_{m}\subseteq I$. Lemma 2.4 implies that the $P_{i}$ are Poisson
${\mathcal{H}}$-prime ideals. By Poisson-normal ${\mathcal{H}}$-separation,
each $P_{i}$ contains a Poisson-normal ${\mathcal{H}}$-eigenvector $c_{i}$.
Hence, $c:=c_{1}c_{2}\cdots c_{m}$ is a Poisson-normal
${\mathcal{H}}$-eigenvector lying in $I$, and $z=ac^{-1}$ where $a=zc\in R$.
This establishes (3.2).

Since $J=K=0$, we have $P,Q\in\operatorname{P{.}Spec}_{0}R$. By Theorem
3.11(b)(c), $R_{0}P\cap Z_{p}(R_{0})\subsetneq R_{0}Q\cap Z_{p}(R_{0})$, so
there exists a Poisson-central element $z\in R_{0}Q\setminus R_{0}P$. Apply
(3.2) to write $z=ac^{-1}$ for some $a,c\in R$ with $c$ a Poisson-normal
${\mathcal{H}}$-eigenvector of $R$. Note that $a\in Q\setminus P$, since
$R_{0}Q\cap R=Q$ and $R_{0}P\cap R=P$. Given any $r\in R$, we have
$\\{c,r\\}=cs$ for some $s\in R$, whence

$\\{a,r\\}=\\{zc,r\\}=z\\{c,r\\}=zcs=as.$

Therefore $a$ is Poisson-normal in $R$, hence also Poisson-normal modulo $P$,
and the proof is complete ∎

##  4\. A construction of Poisson-normal elements

###  4.1. Basic assumptions

Assume throughout this section that $R=A[X;\sigma,\delta]_{p}$ is a _Poisson-
Cauchon extension_ in the sense of [11, Definition 4.4], that is, the Poisson
polynomial algebra $R$ is equipped with a rational Poisson action of a
${\mathbb{K}}$-torus ${\mathcal{H}}$ such that

  * •

The subalgebra $A$ is ${\mathcal{H}}$-stable, and $X$ is an
${\mathcal{H}}$-eigenvector.

  * •

$\delta$ is locally nilpotent.

  * •

There exists $h_{\circ}\in\mathfrak{h}:=\operatorname{Lie}{\mathcal{H}}$ such
that $(h_{\circ}\cdot)|_{A}=\sigma$ and the $h_{\circ}$-eigenvalue
$\lambda_{\circ}$ of $X$ is nonzero.

The third condition relies on $X$ being an $\mathfrak{h}$-eigenvector, which
follows from Proposition 2.5(c).

We write $\chi_{r}:{\mathcal{H}}\rightarrow{\mathbb{K}}^{*}$ to denote the
${\mathcal{H}}$-eigenvalue of an ${\mathcal{H}}$-eigenvector $r\in R$.

###  4.2. The Poisson Cauchon map

Set $\widehat{R}:=A[X^{\pm 1};\sigma,\delta]_{p}$. Since $\widehat{R}$ is a
localization of $R$ at a multiplicative set consisting of
${\mathcal{H}}$-eigenvectors, the action of ${\mathcal{H}}$ on $R$ extends
uniquely to a rational action on $\widehat{R}$ by ${\mathbb{K}}$-algebra
automorphisms. As is easily checked, this is a Poisson action of
${\mathcal{H}}$ on $\widehat{R}$. Let $\theta:A\rightarrow\widehat{R}$ be the
_Poisson Cauchon map_ defined by

(4.1)
$\theta(a)=\sum_{l=0}^{\infty}\frac{1}{l!}\left(\frac{-1}{\lambda_{\circ}}\right)^{l}\delta^{l}(a)X^{-l}.$

######  Proposition 4.1.

[8, Lemmas 3.4, 3.6, Theorem 3.7]; [11, Proposition 4.5, Corollary 4.6]

(a) $\theta$ is an injective Poisson algebra homomorphism.

(b) $\\{X,\theta(a)\\}=\theta\sigma(a)X$ for all $a\in A$.

(c) $\theta$ extends uniquely to an injective Poisson algebra homomorphism
$A[Y;\sigma]_{p}\rightarrow\widehat{R}$ with $\theta(Y)=X$, and then to a
Poisson algebra isomorphism $A[Y^{\pm 1};\sigma]_{p}\rightarrow\widehat{R}$.

(d) Set $B:=\theta(A)$ and $T:=\theta(A[Y;\sigma]_{p})\subseteq\widehat{R}$.
Then $T=B[X;\alpha]_{p}$ where $\alpha$ is the Poisson derivation of $B$
defined by $\alpha(\theta(a))=\theta(\sigma(a))$.

(e) The localization $T[X^{-1}]=B[X^{\pm 1};\alpha]_{p}$ coincides with
$\widehat{R}$.

(f) The ${\mathcal{H}}$-action on $A$ extends uniquely to a rational Poisson
action of ${\mathcal{H}}$ on $A[Y^{\pm 1};\sigma]_{p}$ such that
$\chi_{Y}=\chi_{X}$. With respect to this action, the isomorphism
$\theta:A[Y^{\pm 1};\sigma]_{p}\rightarrow\widehat{R}$ is
${\mathcal{H}}$-equivariant.

(g) $\alpha=(h_{\circ}\cdot)|_{B}$.

######  Lemma 4.2.

Let $a\in A$ be nonzero and let $s\in{\mathbb{Z}}_{\geq 0}$ be maximal such
that $\delta^{s}(a)\neq 0$. Then $s$ is minimal such that $\theta(a)X^{s}\in
R$.

###### Proof.

Since $\delta^{l}(a)=0$ for $l>s$, we have
$\theta(a)=\sum_{l=0}^{s}c_{l}\delta^{l}(a)X^{-l}$ for some nonzero
$c_{l}\in{\mathbb{K}}$. Thus $\theta(a)X^{s}\in R$.

Suppose that $s>0$ and $\theta(a)X^{t}\in R$ for some $t<s$. Then
$\theta(a)X^{s-1}\in R$, whence $\delta^{s}(a)X^{-1}\in R$. Consequently,
$\delta^{s}(a)\in A\cap RX=0$, which contradicts our choice of $s$. Therefore
$s$ is minimal such that $\theta(a)X^{s}\in R$. ∎

The following Poisson version of [9, Lemma 2.2] is adapted from the proof of
[11, Theorem 4.7].

######  Proposition 4.3.

Assume that $A$ is a domain. Let $a\in A$ be a Poisson-normal
${\mathcal{H}}$-eigenvector, and let $s\in{\mathbb{Z}}_{\geq 0}$ be maximal
such that $\delta^{s}(a)\neq 0$. Then the element $x:=\theta(a)X^{s}$ is a
Poisson-normal ${\mathcal{H}}$-eigenvector in $R$. In particular,
$\\{x,X\\}=-\eta xX$, where $\eta$ is the $\sigma$-eigenvalue of $a$.

###### Proof.

In view of Proposition 4.1, the element $b:=\theta(a)$ is a Poisson-normal
${\mathcal{H}}$-eigenvector in $B$, and the $h_{\circ}$-eigenvalue of $b$
equals that of $a$ (Proposition 2.5(d)), namely $\eta$. By Lemma 4.2, $s$ is
minimal such that $bX^{s}\in R$. This places $x\in R$, and clearly $x$ is an
${\mathcal{H}}$-eigenvector.

Since

(4.2) $\\{X,b\\}=\alpha(b)X=(h_{\circ}\cdot b)X=\eta bX,$

we see that $\\{x,X\\}=-\eta xX$. It follows also that $b$ is Poisson-normal
in $T$ and in $\widehat{R}$. In particular, $\widehat{R}b$ is a Poisson ideal
of $\widehat{R}$. Since $\widehat{R}b=\widehat{R}x$, the ideal

$I:=\widehat{R}x\cap R$

is a Poisson ideal of $R$. We show that $I=Rx$, which will prove that $x$ is
Poisson-normal in $R$. Obviously $I\supseteq Rx$.

Let $y\in I$. Then $y\in\widehat{R}b$ implies $yX^{u}\in Tb$ for some $u\geq
0$. Now $yX^{u}=cb$ for some $c\in T$, and $cX^{v}\in R$ for some $v\geq 0$.
Then

$yX^{u+v+s}=cbX^{v+s}=cX^{v}x\in Rx.$

Let $t\in{\mathbb{Z}}_{\geq 0}$ be minimal such that $yX^{t}\in Rx$, and write
$yX^{t}=rx$ for some $r\in R$.

We will show that $t=0$. Write

$r=\sum_{i\geq 0}r_{i}X^{i},\qquad y=\sum_{i\geq 0}y_{i}X^{i},\qquad
x=\sum_{i\geq 0}x_{i}X^{i}$

for some $r_{i},y_{i},x_{i}\in A$. In case $s=0$, we would have $x=b=a\in A$
and so $x_{0}=a\neq 0$. In case $s>0$, we would have

$x_{0}X^{-1}+\sum_{i\geq 1}x_{i}X^{i-1}=xX^{-1}=bX^{s-1}\notin R$

by the minimality of $s$, again yielding $x_{0}\neq 0$. Thus, $x_{0}\neq 0$ in
any case.

Observe that

$\sum_{i\geq 0}y_{i}X^{i+t}=yX^{t}=rx=\sum_{i\geq 0}r_{i}xX^{i}=\sum_{i,j\geq
0}r_{i}x_{j}X^{i+j}.$

If $t>0$, it would follow that $r_{0}x_{0}=0$, whence $r_{0}=0$. But then
$r=r^{\prime}X$ for some $r^{\prime}\in R$, and so

$yX^{t-1}=rxX^{-1}=r^{\prime}x\in Rx,$

contradicting the minimality of $t$. Therefore $t=0$.

Consequently, $y=rx$, proving that $I=Rx$ as desired. ∎

##  5\. Poisson-normal elements in Poisson-Cauchon extensions

Throughout this section, keep the assumptions of Section 4, so that
$R=A[X;\sigma,\delta]_{p}$ is a Poisson-Cauchon extension. Assume in addition
that $A$ is a noetherian domain.

###  5.1. ${\mathcal{H}}$-primes in Poisson-Cauchon extensions

######  Lemma 5.1.

(a) Every Poisson ${\mathcal{H}}$-prime ideal of $R$ contracts to a
$\delta$-stable Poisson ${\mathcal{H}}$-prime ideal of $A$.

(b) For any $\delta$-stable Poisson ${\mathcal{H}}$-prime ideal $P_{0}$ of
$A$, there are at most two Poisson ${\mathcal{H}}$-prime ideals of $R$ that
contract to $P_{0}$ in $A$. There is always at least one, namely $RP_{0}$.

###### Proof.

Recall that any ${\mathcal{H}}$-stable ideal of $R$ or $A$ is also
$\mathfrak{h}$-stable.

(a) If $Q$ is a Poisson ${\mathcal{H}}$-prime ideal of $R$, then clearly
$Q\cap A$ is a Poisson ${\mathcal{H}}$-prime ideal of $A$. Given $a\in Q\cap
A$, we have $\\{X,a\\}\in\\{X,Q\\}\subseteq Q$ and $\sigma(a)=h_{\circ}\cdot
a\in h_{\circ}\cdot Q\subseteq Q$, whence

$\delta(a)=\\{X,a\\}-\sigma(a)X\in Q$

and thus $\delta(a)\in Q\cap A$.

(b) Since the action of $\mathfrak{h}$ on $R$ is by Poisson derivations
(Proposition 2.5(a)), the action of $(h_{\circ}\cdot)$ on $R$ is, in
particular, a derivation. Moreover, by assumption
$(h_{\circ}\cdot)|_{A}=\sigma$ and $h_{\circ}\cdot X=\lambda_{\circ}X$ with
$\lambda_{\circ}\in{\mathbb{K}}^{*}$. Thus, [8, Proposition 1.2] implies that
there are at most two $(h_{\circ}\cdot)$-stable Poisson prime ideals of $R$
that contract to $P_{0}$. Consequently, there are at most two Poisson
${\mathcal{H}}$-prime ideals of $R$ that contract to $P_{0}$ in $A$.

Since $P_{0}$ is stable under both $\sigma=(h_{\circ}\cdot)$ and $\delta$, the
induced ideal $RP_{0}$ is stable under $\\{X,-\\}$, and hence $RP_{0}$ is a
Poisson ideal of $R$. It is clearly also an ${\mathcal{H}}$-prime ideal, and
it contracts to $P_{0}$ in $A$. ∎

We shall need

######  Observation 5.2.

It is immediate from the definition that the ${\mathcal{H}}$-core of a prime
ideal of $A$ (or $R$) is an ${\mathcal{H}}$-prime ideal. Invoking [2,
Proposition II.2.9], we conclude that

  1. (i)

If $P$ is a prime ideal of $A$ (or $R$), then its ${\mathcal{H}}$-core
$(P:{\mathcal{H}})$ is a prime and ${\mathcal{H}}$-prime ideal of $A$ (or
$R$).

It follows from (i) that

  1. (ii)

If $I$ is an ${\mathcal{H}}$-stable ideal of $A$ (or $R$), then all prime
ideals minimal over $I$ are ${\mathcal{H}}$-prime.

Set $A^{*}:=\operatorname{{\mathrm{Fract}}}A$ and $R^{*}:=A^{*}[X]$. Since
$A^{*}$ and $R^{*}$ are localizations of $A$ and $R$, the Poisson structures
on $A$ and $R$ extends uniquely to Poisson structure on $A^{*}$ and $R^{*}$.
As is easily checked, $A^{*}$ is a Poisson subalgebra of $R^{*}$, and
$\\{X,A^{*}\\}\subseteq A^{*}+A^{*}X$. Hence, $R^{*}$ is a Poisson polynomial
ring $A^{*}[X;\sigma^{*},\delta^{*}]_{p}$ for suitable $\sigma^{*}$,
$\delta^{*}$. In particular, $\sigma^{*}$ and $\delta^{*}$ are derivations on
$A^{*}$ extending $\sigma$ and $\delta$, so they are the unique extensions
determined by the quotient rule. We thus label these extensions $\sigma$ and
$\delta$ as well, and write $R^{*}=A^{*}[X;\sigma,\delta]_{p}$.

The actions of ${\mathcal{H}}$ on $A$ and $R$ extend uniquely to Poisson
actions on $A^{*}$ and $R^{*}$. However, these actions are not rational
(unless ${\mathcal{H}}$ acts trivially), so we cannot differentiate them to
obtain $\mathfrak{h}$-actions. On the other hand, the $\mathfrak{h}$-actions
on $A$ and $R$ do extend (uniquely) to actions on $A^{*}$ and $R^{*}$ by
derivations, via the quotient rule. It can be checked that the actions of
$\mathfrak{h}$ on $A^{*}$ and $R^{*}$ are by Poisson derivations, but we do
not require these actions here.

Let us say that the algebra $R^{*}$ is _${\mathcal{H}}$ -Poisson simple_ in
case the only ${\mathcal{H}}$-stable Poisson ideals of $R^{*}$ are $0$ and
$R^{*}$. Thus, in view of Lemma 2.4, $R^{*}$ is ${\mathcal{H}}$-Poisson simple
if and only if $0$ is the unique Poisson ${\mathcal{H}}$-prime ideal of $R$.

######  Proposition 5.3.

Suppose that $R^{*}$ is not ${\mathcal{H}}$-Poisson simple.

(a) There is a unique element $d\in A^{*}$ such that

  1. (i)

$d$ is $X({\mathcal{H}})$-homogeneous with $\alpha\cdot d=\chi_{X}(\alpha)d$
for all $\alpha\in{\mathcal{H}}$.

  2. (ii)

$\\{d,a\\}=\sigma(a)d+\delta(a)$ for all $a\in A^{*}$.

In particular, $X-d$ is an ${\mathcal{H}}$-eigenvector with the same
${\mathcal{H}}$-eigenvalue as $X$.

(b) $\sigma(d)=\lambda_{\circ}d$ and $\delta(d)=-\lambda_{\circ}d^{2}$.

(c) There is a unique nonzero Poisson ${\mathcal{H}}$-prime ideal in $R^{*}$,
namely $R^{*}(X-d)$.

(d) Let $I^{*}$ be a proper nonzero ${\mathcal{H}}$-stable Poisson ideal of
$R^{*}$, let $n$ be the minimum degree for nonzero elements of $I^{*}$, and
let $f=X^{n}+cX^{n-1}+\text{\rm[lower terms]}$, with $c\in A^{*}$, be a monic
element of $I^{*}$ with degree $n$. Then $n>0$ and $d=(-1/n)c$.

###### Proof.

Since $R^{*}$ is not ${\mathcal{H}}$-Poisson simple, it contains a proper
nonzero ${\mathcal{H}}$-stable Poisson ideal $I^{*}$. Let $n$, $f$, and $c$ be
as in part (d). For any $a\in A^{*}$, the ideal $I^{*}$ contains the
polynomial

$\\{f,a\\}-n\sigma(a)f=[n\delta(a)+\\{c,a\\}+c(n-1)\sigma(a)-n\sigma(a)c]X^{n-1}+\text{[lower
terms]},$

which must vanish due to the minimality of $n$, and so
$n\delta(a)+\\{c,a\\}=c\sigma(a)$. Hence, the element

$d:=\frac{-1}{n}c$

satisfies condition (a)(ii).

Set $Y:=X-d$. Then $\\{Y,a\\}=\sigma(a)Y$ for $a\in A^{*}$, so $R^{*}$ is a
Poisson polynomial algebra of the form $A^{*}[Y;\sigma]_{p}$. In particular,
$Y$ is Poisson-normal in $R^{*}$, so that $R^{*}Y$ is a Poisson ideal of
$R^{*}$. It is also a prime ideal.

Returning to the polynomial $f$, note that

$\displaystyle(\alpha\cdot f)-\chi_{X}(\alpha)^{n}f$
$\displaystyle=[(\alpha\cdot
c)\chi_{X}(\alpha)^{n-1}-\chi_{X}(\alpha)^{n}c]X^{n-1}+\text{[lower
terms]}\qquad\forall\,\alpha\in{\mathcal{H}}$ $\displaystyle(h_{\circ}\cdot
f)-n\lambda_{\circ}f$ $\displaystyle=[(h_{\circ}\cdot
c)+(n-1)c\lambda_{\circ}-n\lambda_{\circ}c]X^{n-1}+\text{[lower terms]}.$

Since these polynomials lie in $I^{*}$, the minimality of $n$ now implies that
$(\alpha\cdot c)=\chi_{X}(\alpha)c$ for $\alpha\in{\mathcal{H}}$ and
$h_{\circ}\cdot c=\lambda_{\circ}c$. Consequently, condition (a)(i) holds for
$d$ and $\sigma(d)=\lambda_{\circ}d$. Thus, $Y$ is an
${\mathcal{H}}$-eigenvector with the same ${\mathcal{H}}$-eigenvalue as $X$.
In particular, it follows that $R^{*}Y$ is ${\mathcal{H}}$-stable. Therefore
$R^{*}Y$ is a Poisson ${\mathcal{H}}$-prime ideal of $R^{*}$. Further,
$\delta(d)=\\{d,d\\}-\sigma(d)d=-\lambda_{\circ}d^{2}$, verifying part (b).

It only remains to prove the uniqueness statements in parts (a) and (c). Since
$A^{*}$ is a field, any nonzero Poisson ${\mathcal{H}}$-prime ideal $Q^{*}$ of
$R^{*}$ contracts to $0$ in $A^{*}$. Hence, $Q^{*}\cap R$ and $R^{*}Y\cap R$
are nonzero Poisson ${\mathcal{H}}$-prime ideals of $R$ that contract to $0$
in $A$. Lemma 5.1(b) implies that $Q^{*}\cap R=R^{*}Y\cap R$, and so
$Q^{*}=R^{*}Y$. This establishes part (c).

Finally, suppose that $d^{\prime}$ is an element of $A^{*}$ satisfying
conditions (i),(ii). If $Y^{\prime}:=X-d^{\prime}$, we observe as above that
$R^{*}Y^{\prime}$ is a Poisson ${\mathcal{H}}$-prime ideal of $R^{*}$. By the
uniqueness result of part (c), $R^{*}Y^{\prime}=R^{*}Y$. Since $Y^{\prime}$
and $Y$ are monic of degree $1$, they must be equal, whence $d^{\prime}=d$.
This completes the proof of part (a). ∎

Whenever $R^{*}$ is not ${\mathcal{H}}$-Poisson simple, we keep the notation
$d$ for the unique element of $A^{*}$ described in Proposition 5.3(a). Note
that $R^{*}=A^{*}[X-d;\sigma]_{p}$ in this case, and that the analogs of Lemma
5.1(a) and Observation 5.2(i),(ii) hold for $R^{*}$ and $A^{*}$.

######  Corollary 5.4.

If $R^{*}$ is not ${\mathcal{H}}$-Poisson simple, then $R^{*}(X-d)\cap R$ is
the unique nonzero Poisson ${\mathcal{H}}$-prime ideal of $R$ that contracts
to $0$ in $A$. Moreover, any ${\mathcal{H}}$-stable Poisson ideal of $R$ that
contracts to $0$ in $A$ is contained in $R^{*}(X-d)\cap R$.

###### Proof.

On one hand, $P^{*}:=R^{*}(X-d)$ is a nonzero Poisson ${\mathcal{H}}$-prime
ideal of $R^{*}$ that contracts to zero in $A^{*}$, whence $P^{*}\cap R$ is a
nonzero Poisson ${\mathcal{H}}$-prime ideal of $R$ that contracts to $0$ in
$A$. On the other hand, any nonzero Poisson ${\mathcal{H}}$-prime ideal $Q$ of
$R$ with $Q\cap A=0$ localizes to a nonzero Poisson ${\mathcal{H}}$-prime
ideal $R^{*}Q$ in $R^{*}$, whence $R^{*}Q=P^{*}$ and thus $Q=R^{*}Q\cap
R=P^{*}\cap R$.

Similarly, any ${\mathcal{H}}$-stable Poisson ideal $I$ of $R$ with $I\cap
R=0$ localizes to an ${\mathcal{H}}$-stable Poisson ideal $R^{*}I$ of $R^{*}$.
Since $I\cap R=0$, we must have $R^{*}I\neq R^{*}$, whence there is at least
one prime ideal $Q^{*}$ of $R^{*}$ minimal over $R^{*}I$. Then $Q^{*}$ is a
Poisson ${\mathcal{H}}$-prime ideal by Lemma 2.3(a) and Observation 5.2(ii),
whence $Q^{*}=P^{*}$. Therefore $I\subseteq R^{*}I\cap R\subseteq P^{*}\cap
R$. ∎

###  5.2. Some Poisson-normal ${\mathcal{H}}$-eigenvectors

We upgrade [11, Lemma 4.11(a)] in the same manner as [9, Lemma 2.3]:

######  Lemma 5.5.

Let $\delta$ be a derivation on a domain $C$ of characteristic zero, and
suppose $e,f\in C$ with $\delta(e)=ef$ or $\delta(e)=fe$. If there is some
$m\in{\mathbb{Z}}_{\geq 0}$ such that $\delta^{m}(e)=\delta^{m}(f)=0$, then
$\delta(e)=0$.

###### Proof.

It suffices to show that one of $e$ or $f$ is zero. Suppose not, and let
$s,t\in{\mathbb{Z}}_{\geq 0}$ be maximal such that
$\delta^{s}(e),\delta^{t}(f)\neq 0$. By Leibniz’ Rule,

$\delta^{s+t}(ef)=\sum_{i=1}^{s+t}\tbinom{s+t}{i}\delta^{i}(e)\delta^{s+t-i}(f)=\tbinom{s+t}{s}\delta^{s}(e)\delta^{t}(f)\neq
0,$

since $\binom{s+t}{s}\neq 0$ in characteristic zero. Similarly,
$\delta^{s+t}(fe)\neq 0$. But then $\delta^{s+t+1}(e)\neq 0$, due to the
assumption that $\delta(e)=ef$ or $\delta(e)=fe$. This contradicts the choice
of $s$, since $s+t+1>s$. ∎

######  Lemma 5.6.

Assume there is a nonzero Poisson ${\mathcal{H}}$-prime ideal $P$ in $R$ with
$P\cap A=0$. Let $a\in A$ be a Poisson-normal ${\mathcal{H}}$-eigenvector, and
let $s\in{\mathbb{Z}}_{\geq 0}$ be maximal such that $\delta^{s}(a)\neq 0$.

(a) If $s>0$, then $x:=\theta(a)X^{s}$ is a Poisson-normal
${\mathcal{H}}$-eigenvector in $R$ and $x\in P$. Moreover,
$d=(\lambda_{\circ}s)^{-1}a^{-1}\delta(a)$.

(b) Now assume that $a$ is the leading coefficient of some element of $P$ with
degree $1$. Then $a+P$ is a Poisson-normal ${\mathcal{H}}$-eigenvector in
$R/P$. Moreover, if also $s=0$, then $\delta\equiv 0$ and $P=RX$.

###### Proof.

Note that $a\notin P$, because $P\cap A=0$. The ideal $P$ localizes to a
nonzero Poisson ${\mathcal{H}}$-prime ideal $P^{*}$ of $R^{*}$ such that
$P^{*}\cap R=P$, and $P^{*}=R^{*}(X-d)$ by Proposition 5.3(c).

(a) By Proposition 4.3, $x$ is a Poisson-normal ${\mathcal{H}}$-eigenvector in
$R$. Now $Rx$ is a nonzero ${\mathcal{H}}$-stable Poisson ideal of $R$, and
$Rx\cap A=0$ because $\deg x=s>0$. By Corollary 5.4, $Rx\subseteq P$, whence
$x\in P$.

Note that $x=aX^{s}+cX^{s-1}+\text{[lower terms]}$, where
$c=-\lambda_{\circ}^{-1}\delta(a)$. The ideal $Rx$ localizes to a proper
nonzero ${\mathcal{H}}$-stable Poisson ideal $R^{*}x$ in $R^{*}$, and $s$ is
the minimum degree for nonzero elements of $R^{*}x$. Since $a^{-1}x$ is a
monic element of $R^{*}x$ with degree $s$, Proposition 5.3(d) implies that
$d=(-1/s)a^{-1}c=(\lambda_{\circ}s)^{-1}a^{-1}\delta(a)$.

(b) By hypothesis, $aX+c\in P$ for some $c\in A$. Then $X+a^{-1}c$ is a monic
element of $P^{*}$ with degree $1$. Since $P^{*}$ is proper, it contains no
nonzero elements of degree $0$. Hence, we again apply Proposition 5.3(d),
obtaining $d=-a^{-1}c$.

If $s=0$, then $\delta(a)=0$, whence $\delta^{m}(d)=-a^{-1}\delta^{m}(c)=0$
for some $m\in{\mathbb{Z}}_{\geq 0}$. Since $\delta(d)=-\lambda_{\circ}d^{2}$
(Proposition 5.3(b)), Lemma 5.5 implies that $\delta(d)=0$. It follows that
$d=0$, whence $\delta\equiv 0$ in this case. We then have $P=R^{*}X\cap R=RX$.
Moreover, $a=x$ and so

$\\{a,X\\}=-\eta aX\in Ra,$

where $\eta$ is the $\sigma$-eigenvalue of $a$ (Proposition 4.3). Thus, $a$ is
Poisson-normal in $R$, whence $a+P$ is Poison-normal in $R/P$.

Finally, assume that $s>0$. By part (a), we have

$a^{-1}c=-d=-(\lambda_{\circ}s)^{-1}a^{-1}\delta(a),$

whence $\delta(a)=-\lambda_{\circ}sc$. Since $aX+c\in P$, it follows that

$\\{X,a\\}=\eta aX-\lambda_{\circ}sc\equiv\eta
aX-\lambda_{\circ}s(-aX)=(\eta+\lambda_{\circ}s)Xa\pmod{P}.$

As $a$ is already Poisson-normal in $A$, we conclude that $a+P$ is Poisson-
normal in $R/P$. ∎

######  Proposition 5.7.

Assume that every nonzero Poisson ${\mathcal{H}}$-prime ideal of $A$ contains
a Poisson-normal ${\mathcal{H}}$-eigenvector.

If $P\subsetneq Q$ are Poisson ${\mathcal{H}}$-prime ideals of $R$ with $P\cap
A=0$, then there exists a Poisson-normal ${\mathcal{H}}$-eigenvector $u$ of
$R/P$ such that $u\in Q/P$.

###### Proof.

Recall that $Q\cap A$ is a $\delta$-stable Poisson ${\mathcal{H}}$-prime ideal
of $A$.

Assume first that $P\neq 0$. Then $0$ and $P$ are two Poisson
${\mathcal{H}}$-prime ideals of $R$ that contract to $0$ in $A$, so $Q\cap
A\neq 0$ by Lemma 5.1(b).

Now $P$ localizes to a nonzero Poisson ${\mathcal{H}}$-prime ideal $P^{*}$ in
$R^{*}$, and $P^{*}=R^{*}(X-d)$ by Proposition 5.3(c). Writing $d=bc^{-1}$ for
some $b,c\in A$ with $c\neq 0$, we have $cX-b=c(X-d)\in P^{*}\cap R=P$. Thus,
the ideal

$J:=\\{a\in A\mid aX+e\in P\ \text{for some}\ e\in A\\}$

is nonzero, as is then $J\cap(Q\cap A)=J\cap Q$. Note that $J\cap Q\neq A$,
since $Q\neq R$. Since $P$ is ${\mathcal{H}}$-stable, so is $J$. We also
observe that $J$ is a Poisson ideal of $A$. Namely, if $a\in J$ and $f\in A$,
then $aX+e\in P$ for some $e\in A$, whence $\\{aX+e,f\\}\in P$. Since

$\displaystyle\\{aX+e,f\\}$
$\displaystyle=\\{a,f\\}X+a\bigl{(}\sigma(f)X+\delta(f)\big{)}+\\{e,f\\}$
$\displaystyle=\\{a,f\\}X+\sigma(f)(aX+e)+a\delta(f)-\sigma(f)e+\\{e,f\\},$

we see that $\\{a,f\\}X+(a\delta(f)-\sigma(f)e+\\{e,f\\})\in P$, so that
$\\{a,f\\}\in J$ as required. Thus, $J\cap Q$ is an ${\mathcal{H}}$-stable
Poisson ideal of $A$.

There exist prime ideals $P_{1},\dots,P_{r}$ in $A$ minimal over $J\cap Q$
such that $P_{1}P_{2}\cdots P_{r}$ is contained in $J\cap Q$. Since $J\cap Q$
is an ${\mathcal{H}}$-stable Poisson ideal, each $P_{i}$ is a Poisson
${\mathcal{H}}$-prime ideal of $A$ (Lemma 2.3(a) and Observation 5.2(ii)). By
hypothesis, each $P_{i}$ contains a Poisson-normal ${\mathcal{H}}$-eigenvector
$a_{i}$, and thus $a:=a_{1}a_{2}\cdots a_{r}$ is a Poisson-normal
${\mathcal{H}}$-eigenvector of $A$ that lies in $J\cap Q$. Since $a$ is in
$J$, it is the leading coefficient of an element of $P$ of degree $1$. By
Lemma 5.6(b), the coset $u:=a+P$ is a Poisson-normal
${\mathcal{H}}$-eigenvector of $R/P$. Moreover, $u\in Q/P$ because $a\in Q$.

Now assume that $P=0$. If $Q\cap A\neq 0$, then by hypothesis $Q\cap A$
contains a Poisson-normal ${\mathcal{H}}$-eigenvector $a$ of $A$. Then
$\delta^{l}(a)\in Q\cap A$ for all $l\in{\mathbb{Z}}_{\geq 0}$, whence the
element $u:=\theta(a)X^{s}$ lies in $Q$, where $s\in{\mathbb{Z}}_{\geq 0}$ is
minimal such that $\theta(a)X^{s}\in R$. By Lemma 4.2 and Proposition 4.3, $u$
is a Poisson-normal ${\mathcal{H}}$-eigenvector in $R$.

Finally, suppose that $Q\cap A=0$. As above, the set

$J:=\\{a\in A\mid aX+e\in Q\ \text{for some}\ e\in A\\}$

is a nonzero ${\mathcal{H}}$-stable Poisson ideal of $A$. If $J=A$, then $1\in
J$, while if $J\neq A$, then $J$ contains a product of nonzero Poisson
${\mathcal{H}}$-prime ideals of $A$. In either case, there is a Poisson-normal
${\mathcal{H}}$-eigenvector $a$ of $A$ that lies in $J$. Let
$s\in{\mathbb{Z}}_{\geq 0}$ be maximal such that $\delta^{s}(a)\neq 0$.

If $s>0$, then by Proposition 4.3 and Lemma 5.6(a), the element
$u:=\theta(a)X^{s}$ is a Poisson-normal ${\mathcal{H}}$-eigenvector of $R$
that lies in $Q$. On the other hand, if $s=0$, Lemma 5.6(b) shows that
$\delta\equiv 0$ and $Q=RX$. In this case, $u:=X$ is a Poisson-normal
${\mathcal{H}}$-eigenvector of $R$ that lies in $Q$. ∎

###  5.3. Carrying Poisson-normal ${\mathcal{H}}$-separation from $A$ to $R$

Recall the concept of Poisson-normal ${\mathcal{H}}$-separation from
Definition 3.12.

######  Theorem 5.8.

If ${\mathcal{H}}\text{-}{\operatorname{P{.}Spec}}\,A$ has Poisson-normal
${\mathcal{H}}$-separation, then so does
${\mathcal{H}}\text{-}{\operatorname{P{.}Spec}}\,R$.

###### Proof.

Let $P\subsetneq Q$ be a proper inclusion of Poisson ${\mathcal{H}}$-prime
ideals of $R$. Then $P_{0}:=P\cap A$ is a $\delta$-stable Poisson
${\mathcal{H}}$-prime ideal of $A$ (Lemma 5.1(a)), and we may replace $A$,
$R$, $P$, $Q$ by $A/P_{0}$, $R/RP_{0}$, $P/RP_{0}$, $Q/RP_{0}$, respectively.
Thus, there is no loss of generality in assuming that $P\cap A=0$.

The hypothesis of Poisson-normal ${\mathcal{H}}$-separation now implies that
every nonzero Poisson ${\mathcal{H}}$-prime ideal of $A$ contains a Poisson-
normal ${\mathcal{H}}$-eigenvector of $A$. Therefore, by Proposition 5.7,
there exists a Poisson-normal ${\mathcal{H}}$-eigenvector $u$ of $R/P$ such
that $u\in Q/P$. This verifies Poisson-normal ${\mathcal{H}}$-separation in
${\mathcal{H}}\text{-}{\operatorname{P{.}Spec}}\,R$. ∎

##  6\. Proof of the Main Theorem

The Main Theorem 1.2 follows easily from Theorems 5.8 and 3.13 together with
Corollary 3.4. In fact, these results yield Poisson catenarity for a somewhat
larger class of algebras, as we now show. The following lemma will line up an
inductive step.

######  Lemma 6.1.

Let $R$ be a Poisson ${\mathbb{K}}$-algebra, equipped with a rational Poisson
action of an algebraic ${\mathbb{K}}$-torus ${\mathcal{H}}$, and let $A$ be an
${\mathcal{H}}$-stable Poisson subalgebra of $R$. Assume that there exist an
${\mathcal{H}}$-eigenvector $x\in R$ and an element
$h_{\circ}\in\mathfrak{h}:=\operatorname{Lie}{\mathcal{H}}$ such that

  1. (i)

The rule $\delta(a)=\\{x,a\\}-(h_{\circ}\cdot a)x$ defines a map $\delta$ from
$A$ to itself.

  2. (ii)

$\delta$ is locally nilpotent.

  3. (iii)

The $h_{\circ}$-eigenvalue of $x$ is nonzero.

Then there exists a Poisson-Cauchon extension
$R^{+}=A[X;(h_{\circ}\cdot)|_{A},\delta]_{p}$ relative to the same torus
${\mathcal{H}}$, and the identity map on $A$ extends to an
${\mathcal{H}}$-equivariant Poisson algebra homomorphism
$\phi:R^{+}\rightarrow R$ such that $\phi(X)=x$.

###### Proof.

Recall (Proposition 2.5(a)) that $\sigma:=(h_{\circ}\cdot)|_{A}$ is a Poisson
derivation on $A$. Since also $\\{x,-\\}$ is a derivation, the map $\delta$ is
a derivation on $A$. As shown in the proof of [8, Proposition 1.8], $\delta$
satisfies condition (2.1), and thus there exists a Poisson polynomial ring
$R^{+}=A[X;\sigma,\delta]_{p}$. By construction, the identity map on $A$
extends to a Poisson algebra homomorphism $\phi:R^{+}\rightarrow R$ such that
$\phi(X)=x$.

The action of ${\mathcal{H}}$ on $A$ extends to a rational action of
${\mathcal{H}}$ on $R^{+}$ – by ${\mathbb{K}}$-algebra automorphisms, at least
– such that $X$ is an ${\mathcal{H}}$-eigenvector with the same
${\mathcal{H}}$-eigenvalue as $x$. With respect to this action, $\phi$ is
${\mathcal{H}}$-equivariant. For $\alpha\in{\mathcal{H}}$ and $a\in A$, we
compute that

$\displaystyle\alpha\cdot\\{X,a\\}$
$\displaystyle=\alpha\cdot\bigl{(}\sigma(a)X+\delta(a)\bigr{)}=\alpha\cdot\bigl{(}(h_{\circ}\cdot
a)X+\\{x,a\\}-(h_{\circ}\cdot a)x\bigr{)}$
$\displaystyle=\bigl{(}h_{\circ}\cdot(\alpha\cdot
a)\bigr{)}\chi_{x}(\alpha)X+\\{\chi_{x}(\alpha)x,\,\alpha\cdot
a\\}-\bigl{(}h_{\circ}\cdot(\alpha\cdot
a)\bigr{)}\bigl{(}\chi_{x}(\alpha)x\bigr{)}$
$\displaystyle=\chi_{x}(\alpha)\bigl{[}\sigma(\alpha\cdot
a)X+\delta(\alpha\cdot a)\bigr{]}=\chi_{x}(\alpha)\\{X,\,\alpha\cdot
a\\}=\\{\alpha\cdot X,\,\alpha\cdot a\\},$

where we have used Proposition 2.5(b). Since also
$\alpha\cdot\\{a,b\\}=\\{\alpha\cdot a,\,\alpha\cdot b\\}$ for all $a,b\in A$,
it follows that $\alpha$ preserves the Poisson bracket on $R^{+}$. Therefore
the action of ${\mathcal{H}}$ on $R^{+}$ is a Poisson action.

By assumption (iii), $h_{\circ}\cdot x=\lambda_{\circ}x$ for some nonzero
$\lambda_{\circ}\in{\mathbb{K}}$. Since $X$ and $x$ are
${\mathcal{H}}$-eigenvectors with the same ${\mathcal{H}}$-eigenvalue, they
are also $\mathfrak{h}$-eigenvectors with the same $\mathfrak{h}$-eigenvalue,
by Proposition 2.5(c)(d). In particular, $h_{\circ}\cdot X=\lambda_{\circ}X$.

Therefore $R^{+}$ is a Poisson-Cauchon extension relative to ${\mathcal{H}}$.
∎

######  Theorem 6.2.

Let $R$ be a Poisson ${\mathbb{K}}$-algebra, equipped with a rational Poisson
action of an algebraic ${\mathbb{K}}$-torus ${\mathcal{H}}$. Assume that $R$
is generated (as a ${\mathbb{K}}$-algebra) by ${\mathcal{H}}$-eigenvectors
$x_{1},\dots,x_{N}$, and that there exist
$h_{1},\dots,h_{N}\in\mathfrak{h}:=\operatorname{Lie}{\mathcal{H}}$ such that

  1. (i)

$\\{x_{i},x_{j}\\}-(h_{i}\cdot x_{j})x_{i}\in{\mathbb{K}}\langle
x_{1},\dots,x_{i-1}\rangle$ for $N\geq i>j\geq 1$.

  2. (ii)

For all $i\in[[2,N]]$, the map $a\mapsto\\{x_{i},a\\}-(h_{i}\cdot a)x_{i}$ on
${\mathbb{K}}\langle x_{1},\dots,x_{i-1}\rangle$ is locally nilpotent.

  3. (iii)

For all $i\in[[1,N]]$, the $h_{i}$-eigenvalue of $x_{i}$ is nonzero.

Then $\operatorname{P{.}Spec}R$ has Poisson-normal separation and is catenary.

######  Remark.

For $i\in[[2,N]]$, the map
$\delta_{i}:=\bigl{(}a\mapsto\\{x_{i},a\\}-(h_{i}\cdot a)x_{i}\bigr{)}$ in
hypothesis (ii) is a derivation from ${\mathbb{K}}\langle
x_{1},\dots,x_{i-1}\rangle$ to $R$, because $\\{x_{i},-\\}$ and $(h_{i}\cdot)$
are derivations. In view of hypothesis (i), it follows that $\delta_{i}$ maps
${\mathbb{K}}\langle x_{1},\dots,x_{i-1}\rangle$ to itself. Thus, hypothesis
(ii) says that $\delta_{i}$ is a locally nilpotent derivation on
${\mathbb{K}}\langle x_{1},\dots,x_{i-1}\rangle$.

###### Proof.

We show that ${\mathcal{H}}\text{-}{\operatorname{P{.}Spec}}\,R$ has Poisson-
normal ${\mathcal{H}}$-separation. Once that is established, Theorem 3.13 will
imply that $\operatorname{P{.}Spec}R$ has Poisson-normal separation, and then
Corollary 3.4 will imply that $\operatorname{P{.}Spec}R$ is catenary.

It suffices to show that Poisson-normal ${\mathcal{H}}$-separation holds in
${\mathcal{H}}\text{-}{\operatorname{P{.}Spec}}\,(R/P)$ for any factor algebra
$R/P$ where $P\in{\mathcal{H}}\text{-}{\operatorname{P{.}Spec}}\,R$. These
factor algebras are also Poisson ${\mathbb{K}}$-algebras, ${\mathcal{H}}$ acts
rationally on them by Poisson automorphisms, and they are generated by
${\mathcal{H}}$-eigenvectors satisfying conditions corresponding to (i)–(iii).
Thus, there is no loss of generality in assuming that $R$ is a domain.

We now prove Poisson-normal ${\mathcal{H}}$-separation for
${\mathcal{H}}\text{-}{\operatorname{P{.}Spec}}\,R$ by induction on $N$,
starting with the case $N=0$. In that case, $R={\mathbb{K}}$ and the condition
holds trivially. Now let $N>0$ and assume that
${\mathcal{H}}\text{-}{\operatorname{P{.}Spec}}\,$ of the algebra
$A:={\mathbb{K}}\langle x_{1},\dots,x_{N-1}\rangle$ has Poisson-normal
${\mathcal{H}}$-separation. Note that $A$ is stable under the
${\mathcal{H}}$-action and hence also under the $\mathfrak{h}$-action.

In view of Lemma 6.1, there exist a Poisson-Cauchon extension
$R^{+}:=A[X;\sigma,\delta]_{p}$ relative to ${\mathcal{H}}$ and a surjective
${\mathcal{H}}$-equivariant Poisson algebra homomorphism
$\phi:R^{+}\rightarrow R$.

Since $A$ is a noetherian domain and
${\mathcal{H}}\text{-}{\operatorname{P{.}Spec}}\,A$ has Poisson-normal
${\mathcal{H}}$-separation, Theorem 5.8 implies that
${\mathcal{H}}\text{-}{\operatorname{P{.}Spec}}\,R^{+}$ has Poisson-normal
${\mathcal{H}}$-separation. Therefore
${\mathcal{H}}\text{-}{\operatorname{P{.}Spec}}\,R$ has Poisson-normal
${\mathcal{H}}$-separation, completing the induction step. ∎

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# Kardar-Parisi-Zhang type dynamics with periodic tilt dependence of the
propagation velocity in 1+1 dimensions

Peter Grassberger  JSC, FZ Jülich, D-52425 Jülich, Germany

###### Abstract

We consider the evolution of interfaces with a diffusive term and a
generalized Kardar-Parisi-Zhang (KPZ) non-linearity, which results in a
propagation velocity that depends periodically on the tilt of the interface.
Using large scale simulations of a model class with these properties in 1+1
dimensions, we show that the fluctuations are in general still in the KPZ
universality class, but a new universality class seems to appear in the limit
of weak non-linearity. We argue that this is the typical behavior of any
interface model with periodic tilt dependence.

Interfaces are everywhere. And in general non-equilibrium situations, they
usually move and fluctuate Family ; Meakin ; Barabasi . In the present letter,
we will only consider one-dimensional interfaces in two-dimensional media.
Maybe the most simple model for a moving and fluctuating interface is the one
described by the Edwards-Wilkinson (EW) equation edwards

$\frac{\partial h(x,t)}{\partial t}=v+\nu\nabla^{2}h(x,t)+\eta(x,t).$ (1)

Here, $h(x,t)$ is the height of the interface (assumed to be a univalued
continuous function, $v$ is the propagation velocity of flat interfaces in the
absence of noise, and $\eta(x,t)$ is Gaussian white noise. Due to the latter,
the Laplacian of $h$ is strictly spoken not well defined, but this can be
easily cured. Although the same is much less obvious for the next more
complicated model, the Kardar-Parisi-Zhang (KPZ) equation KPZ

$\frac{\partial h(x,t)}{\partial
t}=v+\nu\nabla^{2}h(x,t)+\frac{\lambda}{2}(\nabla h)^{2}+\eta(x,t),$ (2)

also in this case all derivatives can be interpreted in a mathematically
rigorous way hairer .

One important difference between the EW and KPZ models is that the former
applies to interfaces in equilibria, while the latter is a genuinely non-
equilibrium model, without detailed balance. Also, solving the EW equation is
simple by means of Fourier transformation (the EW equation is linear), while
this is not at all true for KPZ. Indeed, the KPZ equation in higher dimensions
is still not well understood, but rigorous studies in 1+1 dimensions have
brought us closer to a complete understanding in recent years Spohn ; Takeuchi
.

Most of the complications of the KPZ model as compared to the EW model follow
from the fact that the propagation velocity of a tilted straight interface
depends on the tilt angle, which is not true for EW. Assume an isotropic
medium, and a straight horizontal interface that moves (in the absence of
noise) upward with velocity $v$. In a rotated frame, where the axes make an
angle $\phi$ with the original ones, the interface is tilted, but can be still
viewed as moving up, in the direction of the new $z$-axis. But the velocity
will then be $v/\cos(\phi)$ instead of $v$.

For curved interfaces this has additional consequences: A hat-convex interface
moving upward will widen, while a cup-convex interface with become narrower,
until it finally develops a cusp singularity. For the full KPZ model with
noise – and with non-linearity $\lambda$ as dictated by rotational invariance
in isotropic media – this implies that typical interfaces are up-down
asymmetric. Interfaces moving up consist essentially of (noisy) cap-convex
arcs, while downwards moving interfaces are chains of cup-convex arcs.

More general behavior can result from non-trivial anisotropy of the medium and
of the microscopic mechanism. Phenomenologically this can be modelled by
replacing the quadratic term $\lambda(\nabla h)^{2}/2$ by some other non-
linear function $G(\nabla h)$, and a corresponding velocity dependence. One
expects KPZ scaling whenever $G(x)$ has a quadratic minimum (or maximum) at
$x=0$, while EW scaling is expected when $G(x)$ is flat in an entire interval
$-\epsilon<x<\epsilon$. More interesting is the case where
$G^{\prime\prime}(x)=0$ just at $x=0$. The specific case where $G(x)$ has an
inflection point at $x=0$ was studied in Devillard , where EW scaling was
found with possible logarithmic corrections.

In the present letter, we shall study another special case of non-linearity
where the mean velocity of growth is periodic in the ‘tilt’ $a$,

$v(a)=v(a+1),\qquad{\rm where}\;\;a=\tan(\phi).$ (3)

To be precise, we study the following model: We have a 1-d lattice of even
size $L$ and with discrete time $t=0,1,\ldots$. An integer variable $n(x,t)$
is defined on each even site at even times, and on odd sites at odd times.
Initial conditions are tilted periodic, $n(x+L,0)=n(x,0)+aL$, with integer
$aL$. The dynamics is synchronous, with the updating rule

$n(x,t+1)=\frac{1}{2}n_{\rm in}(x,t+1)+\eta(n_{\rm in}(x,t+1))$ (4)

where

$n_{\rm in}(x,t+1)=n(x-1,t)+n(x+1,t).$ (5)

Here, $\eta(n_{\rm in})$ is the simplest non-trivial noise term that depends
only locally on $n_{\rm in}$ and preserves the integer nature of $n(x,t)$ DMG
,

$\displaystyle\eta(n_{\rm
in})=\left\\{\begin{array}[]{r@{\quad}l@{\quad}l}\pm\frac{1}{2}&\mbox{with
prob's}\;(q,1-q)\;\mbox{for odd}\;n_{\rm in}\\\ \pm 1,0&\mbox{with
prob's}\;(p_{\pm 1},p_{0})\;\mbox{for even}\;n_{\rm in}\\\ \end{array}\right.$
(8)

where $p_{-1}+p_{0}+p_{1}=1$.

Theorem: For this model, the velocity depends periodically on the tilt with
period 1, i.e. $v(a)=v(a+1)$.

Proof: The instantaneous propagation velocity $v(a,t)$ of an interface ${\cal
I}_{t}=\\{n(x,t)\\}$ at time $t$ is defined as

$v(a,t)=\frac{2}{L}\sum_{x}[\langle n(x,t+1)\rangle-n(x,t)],$ (9)

and $v(a)$ is just its forward time average. Due to Eq.(4), we have

$\displaystyle v(a,t)$ $\displaystyle=$
$\displaystyle\frac{2}{L}\sum_{x}\langle\eta(n_{\rm in}(x,t+1))\rangle$
$\displaystyle=$ $\displaystyle\frac{2}{L}[(q-1/2)\langle m_{\rm
odd}(t+1)\rangle$ $\displaystyle+(p_{1}-p_{-1})\langle m_{\rm
even}(t+1)\rangle],$

where $m_{\rm odd}$ and $m_{\rm even}$ are the numbers of sites where $n_{\rm
in}$ is even and odd, respectively.

Assume now that $n^{\prime}(x,t)=n(x,t)+x$. If $n(x,t)$ has average tilt $a$,
then $n^{\prime}(x,t)$ has average tilt $a^{\prime}=a+1$. On the other hand,
both have the same instantaneous (and thus also average) velocities, because
even-ness and oddness are not changed when going from $n_{\rm in}(x,t)$ to
$n^{\prime}_{\rm in}(x,t)$ and, since $\eta^{\prime}=\eta$, also
$n^{\prime}(x,t+1)=n(x,t+1)+x.\;\;$ QED.

Figure 1:  (color online) Average asymptotic interface velocity for tilted
interfaces with tilt $a$. Parameters are such that interfaces without tilt are
critical, i.e. $v(0)=0$. The curve resembles a cosine, but is clearly distinct
from it. The heavy dots indicate inflection points where the curvature is
zero.

To illustrate this, $v(a)$ is plotted in Fig. 1 against $a$ for one particular
set of control parameters. These are such that flat interfaces are ‘critical’,
$v(0)=0$, and slightly tilted ones move down, $v^{\prime\prime}(0)<0$.
Therefore, in this case, typical globally flat interfaces consist of chains of
cup-convex arcs, and we expect that interfaces that are strictly flat at time
$t=0$ (having $n(x,0)=0$ for all even $x$) move up during a transient,

$v(0,t)\sim c\;t^{-2/3}$ (11)

with $c>0$ KPZ , such that the average height of the interface should scale as

$\langle n(t)\rangle=\frac{3c}{2}t^{1/3}+const.$ (12)

Finally, its width should scale as KPZ

$W^{2}(L,t)\equiv{\rm Var}[n(x,t)]\sim L^{2\alpha}f(t/L^{z})$ (13)

with $f(u)\sim 1$ for $u\gg 1,f(u)\sim u^{2\beta}$ for $u\ll
1,\alpha=1/2,\beta=1/3$, and $z=3/2$.

In the following we shall, for simplicity, stick to the critical case, i.e. we
shall assume that $v(0)=0$ (the situation is basically the same for $v(0)\neq
0$, but the discussion is more complicated). One then finds that
$v^{\prime\prime}(0)<0$ whenever $q<1/2$ (as in Fig. 1), while
$v^{\prime\prime}(0)>0$ if $q>1/2$. Indeed, for $q=1/2$ the model is critical
for $p_{1}=p_{-1}$, and is in the EW universality class due to up-down
symmetry. In that case, $\langle n(t)\rangle=const,\;\alpha=1/2,\;\beta=1/4$,
and $z=2$.

Figure 2:  (color online) Average interface height for initially flat
interfaces starting at height $n(t=0)=0$. Parameters
$[p_{0}=0,q=0.45,p_{1}=0.525295]$ are again such that interfaces are critical,
i.e. $v(0)=0$. Panel (a) shows a log-log plot of $\langle n(t)\rangle$ versus
$t$, while panel (b) shows $\langle n(t)\rangle-c$, where the constant $c$ is
chosen such that the scaling region with $\langle n(t)\rangle-c\sim t^{x}$ is
maximal, with the exponent $x$ fitted together with $c$. In both panels, the
improved estimator (smooth curve) is compared to the much more noisy direct
measurement.

In order to verify Eqs. (11-13) numerically, we use large scale simulations.
In these simulations we can either measure $n(t)$ directly, or we can measure
$m_{\rm odd}$ and $m_{\rm even}$ and use Eq. (Kardar-Parisi-Zhang type
dynamics with periodic tilt dependence of the propagation velocity in 1+1
dimensions). The latter is always more precise (using it, we are not affected
by random number fluctuations when actual up/down steps are made guided by
$m_{\rm odd}$ and $m_{\rm even}$ footnote1  ), but the difference depends
grossly on the control parameters. In some cases variances are reduced roughly
by a factor 2 when using Eq. (Kardar-Parisi-Zhang type dynamics with periodic
tilt dependence of the propagation velocity in 1+1 dimensions), but when $q$
is close to $1/2$ the reduction is dramatic (see Fig. 2; variances are reduced
by more than 2 orders of magnitude). Indeed, the present analysis would have
been impossible using the direct measurement of $\langle n(t)\rangle$.

In the following we shall only show results for $t\ll L^{3/2}$ (with
$L=2^{19}$), in which case the variance is independent of $L$ and Eq. (13)
simplifies to

$\displaystyle{\rm Var}[n(x,t)]\sim
t^{2\beta}\sim\left\\{\begin{array}[]{r@{\quad}l@{\quad}l}t^{2/3}&\;\mbox{for
}q\neq 1/2\\\ t^{1/2}&\;\mbox{for }q=1/2\;.\\\ \end{array}\right.$ (16)

Finally, we shall always choose the parameters such that $v(0)=0$. This then
leaves us with two control parameters, for which we can use e.g. $p_{0}$ and
$p_{1}-p_{-1}$, or $p_{0}$ and $q$. For fixed $p_{0}$, the difference
$p_{1}-p_{-1}$ decreases monotonically with $q$, being minimal for $q=1$ and
maximal for $q=0$ (see Fig. 3). Due to the up-down symmetry of Fig. 3, we
shall in the following restrict our discussion to $0\leq q\leq 1/2$.

Figure 3: (color online) Contour lines $q=0,0.1,0.2,...1.0$, plotted with
$p_{0}$ on the $x-$axis versus $p_{1}-p_{-1}$ on the $y-$axis. The triangular
regions at the right (with $p_{1}<0$ or $p_{1}>1$) are unphysical, as are the
regions with $q>1$ and $q<0$. EW scaling is expected at $q=1/2$, and KPZ
everywhere else.

We shall first discuss variances of $n(x,t)$, since we do not need for them
precise estimates of ‘critical’ parameter values, while such precise estimates
are needed in discussing $\langle n(t)\rangle$. In Fig. 4 we show ${\rm
Var}[n(x,t)]/t^{1/2}$ versus $t$, for $p_{0}=0.4$ and for ten values of $q$.
All curves for $q\leq 0.25$ bend upwards for large $t$, indicating thereby
that we do not see EW scaling. But for no value of $q$ the rise of ${\rm
Var}[n(x,t)]/t^{1/2}$ is as strong as we would expect from pure KPZ scaling,
indicating that we are in a cross-over regime. For $q\geq 0.3$, however, the
curves seem to be perfectly flat for large $t$, suggesting very clean EW
scaling. But we cannot rule out, of course, that at least some of these curve
would bend uppwards for even larger $t$ – and observing the trend for $q\leq
0.25$, this looks indeed very plausible. Thus we conclude that KPZ is observed
for small $q$, while there is a cross-over to another universality class –
maybe EW – either at $q=1/2$ or at a critical value $q_{c}<1/2$. Similar
results for other values of $p_{0}$ are summarized in Fig. 5, where we see
that ${\rm Var}[n(x,t)]/t^{2/3}$ seems to tend towards zero in a very large
part of control parameter space.

Figure 4: (color online) Log-log plots of $t^{-1/2}\;{\rm Var}[n(x,t)]$
against $t$ for $p_{0}=0.4$ and for ten values of $q$. Figure 5: (color
online) Amplitudes $c=\lim_{t\to\infty}{\rm Var}[n(x,t)]/t^{2/3}$ of the
expected KPZ scaling law, plotted against the two control parameters $p_{0}$
and $p_{1}-p_{-1}$. In a wide region (magenta and green points) $c$ is
compatible with zero. Crosses indicate clear cross-overs.

Let us now discuss the scaling of $\langle n(t)\rangle$. Precise parameter
values where $v(0)=0$ are obtained as in Fig. 2b by making log-log plots of
$\langle n(t)\rangle-const$ versus $t$, where the constant is chosen such that
the plots show the cleanest possible straight lines for large $t$. A priori we
would have expected KPZ scaling for all $q\neq 1/2$, i.e. $\langle
n(t)\rangle-const\sim t^{1/3}$, while the above scaling of variances would
suggest $\langle n(t)\rangle=const$ for $q\approx 1/2$. Neither is seen in
Fig. 6, where such plots are shown, again for $p_{0}=0.4$. Nothing resembling
EW scaling is seen for any value of $q$. Rather, for all $q>0.2$ the curves
are much closer to $\langle n(t)\rangle-const\sim bt^{1/2}$ than to KPZ
scaling. We shall call this “pseudo-EW” scaling. As far as we know there is no
known generalization of proper EW where $\langle n(t)\rangle\sim
bt^{1/2}+O(1)$. Clear deviations from pseudo-EW towards KPZ are only seen for
$q<0.2$. Indeed, even for $q=0$ we see no clean KPZ scaling, but we see a
clear cross-over which suggests that KPZ scaling would hold asymptotically.
Similar results are seen also for all other values of $p_{0}$. For
$q=1/2-\epsilon$ and $\epsilon\to 0$, the amplitude $b$ scales roughly as
$b\sim\epsilon$.

Figure 6: (color online) Log-log plots of $\langle n(t)\rangle-const$ against
$t$ for $p_{0}=0.4$. The values of $q$ the same ten as in Fig. 4, in addition
to $q=0.48$. The uppermost curve is for $q=0$. For each $q$, the constant is
fitted such that the curve becomes, for large $t$, closest to a straight line.
Also shown are the straight lines corresponding to pure power laws $t^{x}$
with $x=\beta=1/3$ (KPZ; upper line) and $x\approx 1/2$ (“pseudo-EW”; lower
line). The exponent $0.489$ of the latter is the best overall fit, but we
propose that the exact value is $0.50\pm 0.2$.

It could well be that – as suggested by the last plots – there is indeed a
phase transition between KPZ scaling and a new “pseudo-EW” universality class
at values of $q<1/2$. But we believe that this is not really the case. Rather
we suggest that this transition occurs at $q=1/2$, i.e. for $q\approx 1/2$
there are extremely slow cross-overs which would be impossible to see with
present-day computers. In any case, there is a new universality class.

In addition, we also looked at the widths and average speeds of tilted
interfaces. As expected, we found EW scaling at and only at inflection points
of $v(a)$ Devillard . Thus it is indeed only the curvature
$v^{\prime\prime}(a)$ and not the linear dependence $v^{\prime}(a)$ which
determines whether we see EW scaling or not. But, in contrast to conventional
wisdom and in agreement with the above findings, we do not see KPZ scaling
when we approach the inflection point, i.e. when $v^{\prime\prime}(a)\to 0$.
Rather we see again pseudo-EW scaling with leading behaviors $\langle
n(t)\rangle-const\sim t^{1/2}$ and ${\rm Var}[n(x,t)]\sim t^{1/2}$.

The intuitive reason for this transition to a new universality class – either
for some very small but non-zero value of $v^{\prime\prime}(a)$, or in the
limit $v^{\prime\prime}(a)\to 0$ – is the following. As we said in the
introduction, a 1-dimensional KPZ interface consists, for large times $t$,
essentially of chains of noisy arcs. As time goes on, these arcs keep their
curvatures, but they become wider and thus also deeper. But for the present
model this scenario cannot go on forever. As the arcs become deeper, they
become also steeper at their wings. Thus sooner or later they would typically
have slopes $>1$. But before this could happen, a point would be encountered
where interfaces tilted with this slope feel the opposite curvature of $v(a)$,
and thus the arcs would stop becoming steeper. It remains an open challenge to
verify this scenario mathematically. It is also not yet clear how it
generalizes to higher dimensions. Finally, the above heuristic arguments would
suggest that the same novel universality class would not only be seen for
periodic $v(a)$ dependence, but also when $v(a)$ is a shallow “Mexican hat”
function. Whether this is indeed true is another open question.

Acknowledgment: I deeply want to thank Deepak Dhar and Pradeep Mohanty for
numerous discussions. Indeed, this work was performed as part of a larger
joint project with them, the main part of which is still to be published. I am
also much indebted to Jan Meinke for help in vectorizing the algorithm.

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