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The present book is an English translation of Algebre Locale - Multiplicites published by Springer-Verlag as no. 11 of the Lecture Notes series. The original text was based on a set of lectures, given at the College de France in 1957-1958, and written up by Pierre Gabriel.
- Jean-Pierre Serre
The goal of these notes is to provide an overview of some facts from local algebra, and more importantly, how they relate to algebraic geometry. The content is based on the course Math 233B. Theory of Schemes, taught by Dennis Gaitsgory in Spring 2010 at Harvard1.
16. Okt. 2023 · Let $A$ be a unital, associative commutative algebra over a field $k$. In particular, $A$ is a ring and we call $A$ local if it is local as a ring. Namely, call $A$ local if $A$ has a unique maximal ideal.
Local algebra is the branch of commutative algebra that studies commutative local rings and their modules. In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal. The concept of local rings was introduced by Wolfgang Krull in 1938 under the name Stellenringe.
Ein lokaler Ring ist im mathematischen Gebiet der Ringtheorie ein Ring, in dem es genau ein maximales Links- oder Rechtsideal gibt. Lokale Ringe spielen in der algebraischen Geometrie eine wichtige Rolle, um das „lokale Verhalten“ von Funktionen auf algebraischen Varietäten und Mannigfaltigkeiten zu beschreiben. Das Konzept des ...
It is necessary to first recall some basie results of local algebra: primary decomposition, Cohen-Seidenberg theorems, normalization of polynomial rings, Krull dimension, characteristie polynomials (in the sense of Hilbert Samuel). Homology comes next, when we consider the multiplicity e q (E, r) of an ideal of definition q = (Xl, ... ,
Local rings are the bread and butter of algebraic geometry. Definition 10.18.1. A local ring is a ring with exactly one maximal ideal. The maximal ideal is often denoted mR in this case. We often say “let (R,m, κ) be a local ring” to indicate that R is local, m is its unique maximal ideal and κ = R/m is its residue field.
