Rees algebras on smooth schemes: integral closure and higher differential operator

Abstract

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 $\mathcal{O}_V$, such that $I_0=\mathcal{O}_V$, 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 differential 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 differential algebra. There are two extensions of Rees algebras of interest in singularity theory: one defined by taking integral closures, and another by extending the algebra to a differential algebra. We study here some relations between these two extensions, with particular emphasis on the behavior of higher order differentials over arbitrary fields.