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

### Orlando Villamayor U.

Universidad Autónoma de Madrid, Spain

## Abstract

Let $V$ be a smooth scheme over a field $k$, and let ${I_{n},n≥0}$ be a filtration of sheaves of ideals in $O_{V}$, such that $I_{0}=O_{V}$, and $I_{s}⋅I_{t}⊂I_{s+t}$. In such case $⨁I_{n}$ 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 $n$, $D(I_{N})⊂I_{N−n}$. 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.

## Cite this article

Orlando Villamayor U., Rees algebras on smooth schemes: integral closure and higher differential operator. Rev. Mat. Iberoam. 24 (2008), no. 1, pp. 213–242

DOI 10.4171/RMI/534