Commutative Algebra (De Gruyter Textbook)

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Foreword
Foreword to the second edition
Contents
Part I
1 Basic introductory theory
	1.1 Commutative rings and ideals
		1.1.1 Ideals, generators, residue classes
		1.1.2 Ideal operations
		1.1.3 Prime and primary ideals
		1.1.4 A source of examples: monomial ideals
	1.2 Algebras
		1.2.1 Polynomials and finitely generated algebras
		1.2.2 The transcendence degree
		1.2.3 Basic properties of the transcendence degree
	1.3 Historic note
		1.3.1 Terminology
		1.3.2 Early roots
	1.4 Exercises
2 Main tools
	2.1 Rings of fractions
		2.1.1 General properties of fractions
		2.1.2 Local rings and symbolic powers
	2.2 Integral ring extensions
		2.2.1 Preliminaries
		2.2.2 The Krull/Cohen–Seidenberg theorems
		2.2.3 Integral closure of ideals
	2.3 Krull dimension and Noether normalization
		2.3.1 Behavior in integral extensions
		2.3.2 Noether normalization and the dimension theorem
		2.3.3 Complements to Noether’s theorem
	2.4 Nullstellensatz
	2.5 Dimension theory I
		2.5.1 Noetherian and Artinian rings
		2.5.2 Associated primes
		2.5.3 Krull’s principal ideal theorem
		2.5.4 Dimension under extensions
	2.6 Primary decomposition
		2.6.1 The nature of the components
		2.6.2 The Lasker–Noether fundamental theorem
	2.7 Hilbert characteristic function
		2.7.1 Basics on the underlying graded structures
		2.7.2 First results
		2.7.3 More advanced steps
		2.7.4 The formula of van der Waerden
		2.7.5 Multiplicities galore
	2.8 Historic note
		2.8.1 Fractions
		2.8.2 Prüfer and the determinantal trick
		2.8.3 Noether and Krull
		2.8.4 Primary decomposition
		2.8.5 Hilbert and Artin
		2.8.6 The Lasker–Noether binary
		2.8.7 Hilbert function
	2.9 Exercises
3 Overview of module theory
	3.1 Exact sequences
	3.2 Internal properties
		3.2.1 Chain conditions
		3.2.2 Composition series
	3.3 External operations
		3.3.1 Modules of fractions
		3.3.2 The Hom operation
		3.3.3 Tensor product
		3.3.4 Exterior and symmetric powers
	3.4 Free presentation and Fitting ideals
	3.5 Torsion and torsion-free modules
	3.6 Historic note
		3.6.1 Composition series
		3.6.2 Fitting ideals
	3.15 Exercises
4 Derivations, differentials and Jacobian ideals
	4.1 Preliminaries
		4.1.1 Derivations of subalgebras
		4.1.2 Derivations with values in a larger ring
	4.2 Differential structures
		4.2.1 A first structure theorem
		4.2.2 The universal module of differentials
		4.2.3 The conormal exact sequence
		4.2.4 Kähler differentials
	4.3 The issue of regularity in algebra and geometry
		4.3.1 The Jacobian ideal
		4.3.2 Hypersurfaces
	4.4 Differents and ramification
		4.4.1 Ramification
		4.4.2 Purity
	4.5 Historic note
	4.6 Exercises
Part II
5 Basic advanced theory
	5.1 Dimension theory
		5.1.1 Annihilators, 1
		5.1.2 The Nakayama lemma
		5.1.3 The Krull dimension and systems of parameters
	5.2 Associated primes and primary decomposition
		5.2.1 Annihilators, 2
		5.2.2 Associated primes
		5.2.3 Primary decomposition
	5.3 Depth and Cohen–Macaulay modules
		5.3.1 Basic properties of depth
		5.3.2 Mobility of depth
	5.4 Cohen–Macaulay modules
		5.4.1 Special properties of Cohen–Macaulay modules
		5.4.2 Gorenstein rings
	5.5 Historic note
		5.5.1 Dimension
		5.5.2 Primary decomposition
		5.5.3 The depth behind the curtains
		5.5.4 The KruCheSam theorem
	5.6 Exercises
6 Homological methods
	6.1 Regular local rings
		6.1.1 Relation to basic invariants
		6.1.2 Properties
	6.2 The homological tool for Noetherian rings
		6.2.1 Projective modules
		6.2.2 Homological dimension
		6.2.3 Chain complexes
		6.2.4 Basics on derived functors
		6.2.5 Properties of injective modules
		6.2.6 Rees theorem and perfect ideals
	6.3 The method of the Koszul complex
		6.3.1 Preliminaries and definitions
		6.3.2 Long exact sequences of Koszul homology
		6.3.3 The theorem of Serre
	6.4 Variations on the Koszul complex: determinantal ideals
		6.4.1 The Eagon–Northcott complex
		6.4.2 The Scandinavian complex
		6.4.3 The Japanese–Polish complex
		6.4.4 The Osnabrück–Recife–Salvador complex
	6.5 Historic note
		6.5.1 Projective modules
		6.5.2 Homology
		6.5.3 Injective modules
		6.5.4 Determinantal ideals
	6.25 Exercises
7 Graded structures
	7.1 Graded preliminaries
	7.2 The symmetric algebra
		7.2.1 Torsion-freeness
		7.2.2 Ideals of linear type, I
		7.2.3 Dimension
	7.3 Rees algebras
		7.3.1 Geometric roots
		7.3.2 Dimension
		7.3.3 On the associated graded ring
		7.3.4 The fiber cone and the analytic spread
		7.3.5 Ideals of linear type, II
		7.3.6 Special properties (survey)
		7.3.7 A glimpse of specialization methods
	7.4 Hilbert function of modules
		7.4.1 Combinatorial preliminaries
		7.4.2 The graded Hilbert function
		7.4.3 Intertwining graded Hilbert functions
		7.4.4 The local Hilbert–Samuel function
	7.5 Historic note
		7.5.1 The Rees algebra
		7.5.2 The associated graded ring
		7.5.3 The symmetric algebra
		7.5.4 Artin–Rees lemma
		7.5.5 Associativity formulas
	7.6 Exercises
Bibliography
Index