2006/06/30 by Orlando E. Villamayor, Villamayor, Orlando · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.math/0606795
openalex publication_date 2006/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let V be a smooth scheme over a field k, and let \In, n≥ 0\ be a filtration of sheaves of ideals in \caloV, such that I0=\caloV, and Is⋅ It⊂ Is+t. In such case \bigoplus In 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(IN)⊂ IN-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.