JournalsrmiVol. 24, No. 1pp. 213–242

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

  • Orlando Villamayor U.

    Universidad Autónoma de Madrid, Spain
Rees algebras on smooth schemes: integral closure and higher differential operator cover
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Abstract

Let VV be a smooth scheme over a field kk, and let {In,n0}\{I_n, n\geq 0\} be a filtration of sheaves of ideals in OV\mathcal{O}_V, such that I0=OVI_0=\mathcal{O}_V, and IsItIs+tI_s\cdot I_t\subset I_{s+t}. In such case In\bigoplus I_n is called a Rees algebra. A Rees algebra is said to be a differential algebra if, for any two integers N>nN > n and any differential operator DD of order nn, D(IN)INnD(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.

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