Date of Award

1988

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Augusto Nobile

Abstract

We study the conditions under which the power of a prime ideal is equal to the corresponding symbolic prime power. We begin by extending a result of Villamayor(h). We consider a smooth k-algebra S(S$\sp\prime$) which is the localization of a finite k-algebra where k is a field of characteristic zero. For a prime ideal P (P$\sp\prime$) we show that if $S\over P$ $\cong$ $S\sp\prime\over P\sp\prime$ then P$\sp{n}$ = P$\sp{(n)}$ if and only if P$\sp\prime\sp{n}$ = P$\sp\prime\sp{(n)}$, for n $\geq$ 1. In the proof we use a generalization of the notion of a truncated cotangent complex introduced by Illusie. We then continue on by using the notions developed in the course of the proof to construct a new class of cohomological objects ${\cal U}\sp{n{,}i}$ which play an analogous role for the higher order differentials to the role played by the cotangent complex of Lichtenbaum and Schlessinger in the case of the ordinary Kahler differentials.

Pages

45

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