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\title{Normalization of a Poisson algebra is Poisson}

\author{D. Kaledin}

\date{{\em To Vassily Alexeevich Iskovskikh, on his 70-th birthday}}

\begin{document}

\maketitle

%\tableofcontents

\section*{Introduction}

It is well-known that functions on a smooth symplectic manifold $M$
are equipped with a canonical skew-linear operation called the
Poisson bracket. The bracket is compatible with multiplication in a
certain precise way. Formalizing this structure, one obtains the
notion of a Poisson algebra (see Definition~\ref{poi.def}).

The definition of a Poisson algebra is quite general; among other
things, it involves no assumption of smoothness. Recently there
appeared good reasons to study Poisson algebras in full
generality. In particular, they seem to be quite useful in the study
of the so-called symplectic singularities initiated by A. Beauville
\cite{B}.

However, while non-trivial Poisson structures on smooth manifolds
have been under close scrutiny for fifty years or more, the general
theory is much less developed. It seems that even the simplest facts
are not known, or at least, not easy to find in the existing
literature. 

The goal of the present note is to prove one of these simple facts
-- namely, we prove that under some natural assumptions, the
integral closure of a Poisson algebra is again Poisson
(Theorem~\ref{main}). The exposition is essentially
self-contained. We need a couple of preliminary lemmas which are
definitely not new, but not quite standard, either. For the
convenience of the reader, we have taken the liberty of re-proving
them from scratch.

\section{Statements and definitions.}

Fix once and for all a base field $k$ of characteristic $\cchar k =
0$.

\begin{defn}\label{poi.def}
A {\em Poisson algebra} over the field $k$ is a commutative algebra
$A$ over $k$ equipped with an additional skew-linear operation
$\{-,-\}:A \otimes A \to A$ such that
\begin{equation}\label{poi}
\{a,bc\} = \{a,b\}c + \{a,c\}b\quad, \quad
0 = \{a,\{b,c\}\} + \{b,\{c,a\}\} + \{c,\{a,b\}\},
\end{equation}
for all $a,b,c \in A$. An ideal $I \subset A$ is called a {\em
Poisson ideal} if $\{i,a\} \in I$ for any $i \in I$, $a \in A$.
\end{defn}

Additionally, we will always assume that a Poisson algebra $A$ has a
unit element $1 \in A$ such that $\{1,a\}=0$ for every $a \in A$.

\begin{defn}
A {\em Poisson scheme} over $k$ is a scheme $X$ over $k$ equipped
with a skew-linear bracket in the structure sheaf $\calo_X$
satisfying \eqref{poi}.
\end{defn}

\begin{lemma}\label{loc}
Let $A$ be a Poisson algebra.
\begin{enumerate}
\item For any multiplicative system $S \subset A$, the localization
$A[S^{-1}]$ carries a canonical Poisson algebra structure.
\item Any associated prime ideal $\pp \subset A$ is a Poisson ideal.
\item The radical $J \subset A$ of the algebra $A$ is a Poisson
ideal.
\end{enumerate}
\end{lemma}

\proof{} For \thetag{i}, set
\begin{align*}
\{a_1s_1^{-1},a_2s_2^{-1}\} &= \{a_1,a_2\}(s_1s_2)^{-1} -
\{a_1,s_2\}a_2\left(s_1s_2^2\right)^{-1}\\
&\quad - \{s_1,a_2\}a_1\left(s_1^2s_2\right)^{-1} +
a_1a_2\{s_1,s_2\}\left(s_1^2s_2^2\right)^{-1}.
\end{align*}
For \thetag{ii}, note that $\pp \subset A$ is the kernel of the
canonical Poisson map from the Poisson algebra $A$ to the fraction
field $A_\pp$. For \thetag{iii}, note that $J$ is the intersection
of all the associated primes.
\endproof

Lemma~\ref{loc}~\thetag{i}, in particular, means that the spectrum
of a Poisson algebra is a Poisson scheme. We also note the following
geometric corollary.

\begin{corr}
Let $X$ be a Poisson scheme over $k$. Then the reduction $X_{red}$
of the scheme $X$ is a Poisson scheme, and so is every irreducible
component $X_0$ of the reduction $X_{red}$.\endproof
\end{corr}

Our main result is the following.

\begin{theorem}\label{main}
Let $A_0$ be an excellent Noetherian domain over $k$, and let $A$ be
its integral closure in its fraction field.
\begin{enumerate}
\item Every derivation $\xi$ of the algebra $A_0$ extends to a
derivation of the algebra $A$.
\item Every Poisson bracket $\{-,-\}$ on the algebra $A_0$ extends
to a Poisson bracket on the algebra $A$.
\end{enumerate}
\end{theorem}

Note that both derivations and Poisson brackets extend naturally and
uni\-que\-ly to the fraction field $\Frac A_0 = \Frac A$. The point
is that both preserve the integral closure $A \subset \Frac A$. The
first claim is well-known; nevertheless, we will prove it, because
it is needed in the proof of \thetag{ii}.

The geometric corollary (in fact, an equivalent geometric
formulation) of Theorem~\ref{main} is the following.

\begin{corr}\label{geom}
Let $X_0$ be an excellent Noetherian integral scheme over $k$, and
let $X$ be its normalization. Then every vector field $\xi$ on $X_0$
and every Poisson scheme structure on $X_0$ extend to $X$.
\end{corr}

\section{Discrete valuation rings.}

To prove Theorem~\ref{main}, we first study the situation in
codimension $1$. In this section, assume given an excellent local
Noetherian algebra $A_0$ over $k$ of dimension $1$. Let $K_0$ be its
residue field. Let $A$ be the integral closure of the algebra
$A_0$. Since $A_0$ is excellent, $A$ is finite over $A_0$; therefore
it is a semilocal ring with a finite number $l$ of maximal ideals
$\m_i \subset A$, $1 \leq i \leq l$. For every such $\m_i$, the
localization $A_{\m_i}$ is a normal local ring of dimension $1$,
thus a discrete valuation ring whose residue field $K_i = A/\m_i$ is
a finite extension of the residue field $K_0$. Denote the valuation
on $A_{\m_i}$ by $v_i$, and fix uniformizing elements $\pi_i \in
A_{\m_i}$, $v_i(\pi_i) = 1$. The ring $A$ is regular and coincides
with the intersection
\begin{equation}\label{cap}
A = \bigcap_{1 \leq i \leq l}A_{\m_i} \subset \Frac(A)
\end{equation}
in the fraction field $\Frac(A) = \Frac(A_0)$.

\begin{lemma}\label{gen}
In the assumptions above, for any $i$, $1 \leq i \leq l$, there
exists a single element $x \in A_{\m_i}$ generating $A_{\m_i}$ over
$A_0$.
\end{lemma}

\proof{} By the Primitive Element Theorem, the residue field $K_i$
is generated over $K_0$ by a single element, say $\overline{x}$. Let
$P(x)$ be the minimal polynomial for $\overline{x}$ over $K_0$.
Lift $\overline{x}$ to an element $x \in A{\m_i}$ and consider
$$
y = P(x) \in A.
$$
By definition, we have $y=0 \mod \pi_i$, so that $v_i(y) > 0$. If
$v_i(y) = 1$, we are done: $x$ and $y$ generate $A_{\m_i}$ over
$A_0$, and $y = P(x)$. If not, replace $y$ with
$$
y' = P(x + \pi_i).
$$
By the binomial formula, we have
$$
y' = P'(\overline{x})\pi_i \mod \pi_i^2.
$$
Since the polynomial $P$ is minimal, its derivative $P'$ satisfies
$P'(\overline{x}) \neq 0$. Therefore $v_i(y') = 1$, and we are done:
$A_{\m_i}$ is generated over $A_0$ by $x + \pi_i$.  \endproof

\begin{lemma}\label{vf}
Every derivation $\xi_0:A_0 \to A_0$ of the algebra $A_0$ extends to a
derivation of the algebra $A$.
\end{lemma}

\proof{} Consider the formal power series algebra $B = A[[t]]$ in
one indeterminate $t$. Since $A$ is finite over $A_0$, its fraction
field
$$
\Frac(B) \subset \Frac(A)((t)) = \Frac(A_0)((t))
$$
coincides with $\Frac(A_0[[t]])$. Moreover, $B$ a regular local
algebra, in particular, it is integrally closed (see, for example,
\cite[Prop. 14]{AC}). Therefore it is the integral closure of the
power series algebra $B_0=A_0[[t]]$. By functoriality, every
automorphism of the algebra $B_0$ extends to an automorphism of its
integral closure $B$. Consider the automorphism $\sigma_0:B_0 \to
B_0$ given by
$$
\sigma_0(t) = t \qquad \sigma_0(a) = \exp(t\xi)(a) \text{ for }a \in
A_0 \subset B_0.
$$
Extend it to an automorphism $\sigma:B \to B$ of the algebra
$B$. Setting
$$
\xi(a) = \frac{\6}{\6 t}\sigma(a) \mod t
$$
gives a derivation $\xi:A \to A=B/tB$ extending the given derivation
$\xi_0$.
\endproof

\begin{remark}
Apparently, this result was first proved by A. Seidenberg \cite{S}
back in 1966 (moreover, he did not need the assumption of
excellence). However, it seems that this is not universally
known. In particular, and I am grateful to M. Lehn and D. van
Straten for bringing this to my attentuon, the result also appears
as Lemma~2.33 on page 36 of SGA7.2 (with essentially the same proof
as here).
\end{remark}

\begin{lemma}\label{nrm}
Assume that the algebra $A_0$ is equipped with a Poisson bracket
$\{-,-\}$. Then this bracket extends uniquely to the algebra $A$.
\end{lemma}

\proof{} Extend the bracket to the fraction field $\Frac A$. We have
to prove that $\{f,g\} \in A$ for every $f,g \in A$. By \eqref{cap},
it suffices to prove it for each of the $A_{\m_i}$ instead of
$A$. Let $x \in A_{\m_i}$ be the generator provided by
Lemma~\ref{gen}. It suffices to prove that $\{x,x\} \in A_{\m_i}$
and $\{x,f\} \in A_{\m_i}$ for every $f \in A_0$. But $\{x,x\} = 0$
tautologically, and $\{x,f\} \in A_{\m_i}$ by Lemma~\ref{vf} (define
$\xi_0:A_0 \to A_0$ by $\xi_0(a) = \{a,f\}$, and note that any
derivation $\xi:A \to A$ preserves all the localizations $A_{\m_i}
\subset \Frac(A)$).
\endproof

\section{Proof of the Theorem.}

We can now prove Theorem~\ref{main}. It is more convenient to
approach it in the geometric form of the Corollary~\ref{geom}. Thus,
let $X_0$ be a Noetherian integral scheme, and let $X$ be its
normalization. By Lemma~\ref{vf} and Lemma~\ref{nrm},
Corollary~\ref{geom} holds for the open complement $U \subset X$ to
a subscheme $Z \subset X$ of codimension $\codim Z \geq
2$. Therefore we have a derivation and/or a Poisson bracket on the
structure sheaf $\calo_U$. This induces a derivation and/or a
Poisson bracket on the sheaf $j_*\calo_U$, where $j:U
\hookrightarrow X$ is the embedding. Since $\codim Z \geq 2$, and
$X$ is normal, we have $\calo_X \cong j_*\calo_U$.\endproof

\subsection*{Acknowledgements.} It is a great pleasure to dedicate
the paper to Vassily Alexeevich Iskovskikh; I hope to be able to do
the same in ten years. I would like to thank E. Amerik,
R. Bezrukavnikov, A. Kuznetsov, M. Rovinsky and M. Verbitsky for
stimulating discussions. I am extremely gratefull to M. Lehn for
re-visiting this paper some time after it was first posted to the
web, indicating some serious gaps in the argument, and helping me to
fix them. The reference to Seidenberg's paper was communicated to me
by an anonymous referee in ``Functional analysis and Applications''
(who rejected the paper since one new result per four pages is not
enough for that journal).

\begin{thebibliography}{666}

\bibitem[AC]{AC} N. Bourbaki, {\em Alg\`ebgre Commutative, Ch. 5},
Hermann, Paris, 1964.

\bibitem[B]{B} A. Beauville, {\em Symplectic singularities}, Invent.
Math. {\bf 139} (2000), 541--549.

\bibitem[S]{S} A. Seidenberg, {\em Derivations and integral
  closure}, Pacific J. of Math. {\bf 116} (1966), 167--173.

\end{thebibliography}

\bigskip

\noindent
{\sc Steklov Math Institute\\
Moscow, USSR}\\
{\em E-mail\/}: {\tt kaledin@mccme.ru}

\end{document}
