Homological Methods in Commutative Algebra

Cover
Title page
Contents
Preface
	Conventions
Chapter 1. Categories
	1.1. Categories and functors
	1.2. Sets and classes
	1.3. Natural transformations
	1.4. Limits
	1.5. Adjoint pairs
	1.6. Exercises
Chapter 2. Abelian Categories
	2.1. Additive categories
	2.2. Abelian categories
	2.3. Sheaves
	2.4. Sites
	2.5. Standard lemmas
	2.6. Projectives and injectives
	2.7. Essential extensions
	2.8. The Freyd–Mitchell theorem
	2.9. Exercises
Chapter 3. Derived Functors
	3.1. Categories of complexes
	3.2. Derived functors
	3.3. Computing derived functors
	3.4. The Ext and Tor functors
	3.5. The limⁿ functors
	3.6. Exercises
Chapter 4. Spectral Sequences
	4.1. Exact couples
	4.2. Filtered complexes
	4.3. Double complexes
	4.4. Simple applications
	4.5. The Grothendieck spectral sequence
	4.6. The Ischebeck spectral sequences
	4.7. Exercises
Chapter 5. Projective and Injective Modules
	5.1. Projective and free modules
	5.2. Projective dimension
	5.3. Injective modules
	5.4. Injective dimension
	5.5. Global dimension of rings
	5.6. Free resolutions
	5.7. Stably free modules
	5.8. The Quillen–Suslin theorem
	5.9. Exercises
Chapter 6. Flatness
	6.1. Flat modules
	6.2. Flat morphisms
	6.3. Faithful flatness
	6.4. Criteria for flatness
	6.5. Flatness and freeness
	6.6. Exercises
Chapter 7. Koszul Complexes and Regular Sequences
	7.1. The Koszul complex
	7.2. Regular sequences
	7.3. Depth
	7.4. The Auslander–Buchsbaum formula
	7.5. Multiplicities revisited
	7.6. Macaulay resultants
	7.7. Computing the resultant
	7.8. Exercises
Chapter 8. Regularity
	8.1. Regular local rings
	8.2. Regularity and global dimension
	8.3. Change of rings
	8.4. Regularity and factorization
	8.5. Normal rings
	8.6. Regularity in characteristic