## LSU Historical Dissertations and Theses

#### Title

Multiplicities and Transforms of Ideals.

1995

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

Augusto Nobile

#### Abstract

Let (R, M$\sb{\rm R})$ be a regular local ring of dimension 3 of the form k (x,y,z) $\sb{\rm (x,y,z)},$ where k is an algebraically closed field and let I be an M$\sb{\rm R}$-primary ideal that admits generators. We prove that if I$\sb1$ is the proper transform of I to a quadratic transform (A, M$\sb{\rm A})$ of(R, M$\sb{\rm R})$ such that the analytic spread of I$\sb1$ is 3 and the generators of I$\sb1$ induced by those of I satisfy certain divisibility conditions, then the inequality of multiplicities$$\rm e\sb{A}(M(I\sb1)) < e\sb{R}(I)$$is valid, where M $\rm(I\sb1) \supseteq I\sb1$ is an M$\sb{\rm A}$-primary ideal associated to I$\sb1$ (the ideal I$\sb1$ may not be M$\sb{\rm A}$-primary if dim (R) = 3) through an operation M that we define for ideals in a regular local ring.

67

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