## LSU Historical Dissertations and Theses

#### Title

On Prime Powers and Symbolic Prime Powers.

1988

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

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.

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