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Numerical Invariants of One Dimensional Local Rings
Hardcover
Series: Rsme Springer, Book 19
ISBN13: 9783032420916
Publisher: Springer
Published: Feb 24 2027
Pages: 136
Language: English
The purpose of this monograph is to present the main results on Hilbert functions of one-dimensional Cohen-Macaulay local rings and related numerical invariants, covering algebraic, geometric, and computational aspects. The study of Hilbert functions of local rings is a classical topic, originating in Samuel's extension of Hilbert's notion of characteristic functions to the local setting.
We focus on the one-dimensional case, where the theory is better understood and the construction of explicit examples is more accessible.
The main examples of one-dimensional Cohen-Macaulay local rings considered in these notes arise from curve singularities, with special emphasis on monomial curve singularities.
From our point of view, one-dimensional local rings provide a natural and effective starting point for the study of Hilbert functions in higher dimensions.
Although the main goal of this monograph is the study of Hilbert functions of one-dimensional Cohen-Macaulay local rings, in the second chapter we present a basic toolkit of results and techniques on Hilbert functions for higher-dimensional local rings. The results of chapter two could form the core of a graduate course on Hilbert functions, complemented by some results on Hilbert functions of one-dimensional local rings presented in other chapters of this monograph.
This monograph is intended for graduate students and researchers interested in local algebra and its computational and geometric aspects.
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