Betti numbers and the integral closure of ideals

Sangki Choi, Purdue University

Abstract

We study the nondecreasing, strict increasing and exponential growth of Betti numbers. Tools are provided for measuring these. Let $(S,\underline m)$ be a local ring with maximal ideal $\underline m$ and $R = S/I$ for an ideal I of S. Put $J = (I:\underline m)$. If the integral closure $\bar I$ of I is properly contained in the integral closure $\bar J$ of J, then for any finitely generated R-module M the sequence $b\sbsp{i}{R} (M)$ of Betti numbers of M is nondecreasing. If dim $\underline mJ/\underline mI \geq 2$, then for any finitely generated non-free R-module M the sequence $b\sbsp{i}{R} (M)$ has strong exponential growth with a lower exponential bound $\sqrt{{\rm dim}\underline mJ/\underline mI}.$ For each artinian local ring R we derive an invariant $B(R)$ such that if $B(R) > 1$, then for any finitely generated non-free R-module M the sequence $b\sbsp {i}{R} (M)$ is strictly increasing and has strong exponential growth with a lower exponential bound A for any $1 < A < B(R)$.

Degree

Ph.D.

Advisors

Huneke, Purdue University.

Subject Area

Mathematics

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