Author(s): Mohamed Barakat
Series: Lecture notes
Edition: version 2018-01-17
Year: 2018
Language: English
Commentary: Downloaded from https://algebra.mathematik.uni-siegen.de/barakat/Lehre/SS17/CA/Skript/CA.pdf
Gröbner bases
Monomials and monomial ideals
Monomial orderings
Division with remainder
Gröbner bases and Buchberger's algorithm
First applications: Submodule membership and syzygies
Iterated syzygies and Hilbert's Syzygy Theorem
Elimination
Constructive Abelian categories
Categories
The category of modules
Objects
Morphisms
Subobjects and factor objects
Kernels and cokernels, images and co-images
Direct sums and direct products
Pull-backs and push-outs as kernels and cokernels
More categorical notions
Additive and Abelian categories
Computable rings
A constructively Abelian category of matrices
Functors
Natural transformations
Exact sequences, homology, and cohomology
Free resolutions
Localization and tensor products
Preliminaries
Nilradical and Jacobson Radical
The Cayley-Hamilton Theorem and Nakayama's Lemma
Tensor products of modules over commutative rings
Localization of rings
Localization of modules
Local properties
Chain conditions
Artinian modules and rings
Primary decomposition
Definition and Existence in Noetherian rings
Uniqueness results
Integral ring extensions
Basic definitions
Lying over and going up
Going down
Krull Dimension, Noether Normalization, Hilbert’s Nullstellensatz and Krull’s Principle Ideal Theorem
Definition of Krull dimension
Noether Normalization
Properties of Krull dimension
Hilbert's Nullstellensatz
Krull's Principal Ideal Theorem and regular local rings
Valuation rings and Dedekind domains
Valuation rings
Dedekind domains
