| Chapter | Title | Key topics | |---------|-------|-------------| | 1 | | Nakayama’s lemma, Krull’s intersection theorem, associated primes, primary decomposition (Noetherian rings) | | 2 | Valuation Rings | Valuation rings, discrete valuation rings (DVRs), Chevalley’s theorem, the valuative criterion of separatedness/properness (brief) | | 3 | Completion | Inverse limits, completions of rings and modules, exactness of completion for Noetherian rings, Cohen’s structure theorem | | 4 | Dimension Theory | Krull dimension, height, catenary rings, dimension of polynomial rings, dimension of fibers | | 5 | Regular Rings | Regular local rings, Auslander–Buchsbaum theorem (regular local rings are UFDs), Jacobian criterion, complete intersections | | 6 | Flatness | Flat modules, faithfully flat modules, flatness and completions, the miracle flatness theorem, descent of properties | | 7 | Derivations and Differentials | Module of Kähler differentials, separability, smooth and unramified homomorphisms, Jacobian criterion for smoothness | | 8 | Formal Smoothness | André–Quillen homology (brief), regular homomorphisms, formally smooth maps in characteristic 0 | | 9 | The Principal Ideal Theorem and Generalized Cohen–Macaulay Rings | Krull’s PID theorem, systems of parameters, Buchsbaum rings, local cohomology (introduced) | | 10 | Duality | Matlis duality, canonical modules, Gorenstein rings, Grothendieck’s local duality (brief) |
Matsumura frequently uses geometric language: local rings of varieties, smoothness, completions (formal neighborhoods), the Jacobian criterion, and the valuative criterion. This makes the book essential for algebraic geometers.
Matsumura states in the preface: “This book is intended as a textbook for graduate students and as a reference work for researchers in commutative algebra and related fields.” It assumes familiarity with basic abstract algebra (groups, rings, fields, modules) but no prior commutative algebra. matsumura commutative ring theory pdf
The latter, Commutative Ring Theory (CRT), is the one most widely circulated today as a PDF, often found on academic websites, institutional repositories, and private file-sharing networks. It is a legally free book, but its widespread availability as a scanned PDF reflects its enduring value as a reference.
: Discusses the algebraic theory of differentials, providing the algebraic background for the study of smooth varieties. Key Features | Chapter | Title | Key topics |
There are two famous books by Matsumura.
This book is widely regarded as a bridge between standard graduate algebra and research-level commutative algebra and algebraic geometry. The latter, Commutative Ring Theory (CRT), is the
Unlike some textbooks that are good for learning but poor for reference, Matsumura is dense. It is not a "read cover-to-cover" book for most; it is a "consult this chapter for the exact technical lemma you need" book.