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

Abstract

Let $V$ be a smooth scheme over a field $k$, and let ${In, n\\geq 0}$ 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 $\\bigoplus In$ 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.