Commutative Algebra: Sommersemester 2017

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