  Let $V$ be a smooth scheme over a field $k$, and let $\{I_n, n\geq 0\}$ be a
filtration of sheaves of ideals in $\calo_V$, such that $I_0=\calo_V$, and
$I_s\cdot I_t\subset I_{s+t}$. In such case $\bigoplus I_n$ is called a Rees
algebra.
  A Rees algebra is said to be a Diff-algebra if, for any two integers $N>n$
and any differential operator $D$ of order $n$, $D(I_N)\subset I_{N-n}$. Any
Rees algebra extends to a smallest Diff-algebra.
  There are two ways to define extensions of Rees algebras, and both are of
interest in singularity theory. One is that defined by taking integral closures
(in which a Rees algebra is included in its integral closure), and another
extension is that defined, as above, in which the algebra is extended to a
Diff-algebra.
  Surprisingly enough, both forms of extension are compatible in a natural way.
Namely, there is a compatibility of higher differential operators with integral
closure which we explore here under the assumption that $V$ is smooth over a
perfect field.
