Numerical Invariants of One Dimensional Local Rings
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Beschreibung
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.
Focus on the computational aspects Deep study of Hibert functions of one-dimensional local rings Present main results on Hilbert functions
Autor*in
Joan Elias
Themen in »Numerical Invariants of One Dimensional Local Rings«
One-dimensional local rings Effective computations Monotony of the Hilbert functions Monomial curve singularities Hilbert funtions and polynomials