Coefficient theorems of Birancon-Skoda type

Abstract

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] The original Briancon-Skoda theorem, proved for the ring of convergent power series over the field $mathbb{C}$ of complex numbers, was later generalized to arbitrary regular local rings by Lipman and Sathaye, who showed that if (R, m) is a regular local ring and $I$ an ideal of $R$ generated by $ell$ elements, then for all $w geq 0$, $overline{I^{ell+w}} subseteq I^{w+1}.$ If (R,m) is regular local, $I subseteq R$ an ideal of analytic spread $ell$ and $J subseteq I$ any reduction, then Lipman and Sathaye's theorem implies that $overline{I^{ell+w}} subseteq J^{w+1}$, for all $w geq 0$. Set w=0 to conclude that $I^{ell} subseteq overline{I^{ell}} subseteq J$. Hence any element of $I^ell$ is a linear combination of the generators of $J$ with coefficients in $R$. In this thesis, we study the coefficients involved in the Briancon-Skoda theorem when $J$ is a minimal reduction, to show that, under some hypotheses, one can get some information on those coefficients. We also show that, in the case where the ring $R$ is Gorenstein, the power $ell$ of $I$ in $overline{I^ell} subseteq J$ can be reduced. Concretely, we give results on when the integral closure of $I^{ell-1}$ is contained in $J$.--From short.pdf.