Coefficient theorems of Birancon-Skoda type
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[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] The original Brian con-Skoda theorem, proved for the ring of convergent power series over the field 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 l elements, then for all ω≥0, Il+w ⊆ Iω+1. If (R;m) is regular local, I ⊆ R an ideal of analytic spread l and J ⊆ I any reduction, then Lipman and Sathaye's theorem implies that Il+ω ⊆ Jω+1, for all ω≥0. Set ω=0 to conclude that Il ⊆ Il ⊆ J. Hence any element of Il is a linear combination of the generators of J with coe fficients in R. In this thesis, we study the coe fficients 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 l of I in Il ⊆ J can be reduced. Concretely, we give results on when the integral closure of Il-1 is contained in J.
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