In addition to being an interesting and profound subject in its own right, commutative ring theory is important as a foundation for algebraic geometry and complex analytical geometry. Matsumura covers the basic material, including dimension theory, depth, Cohen-Macaulay rings, Gorenstein rings, Krull rings and valuation rings. More advanced topics such as Ratliff's theorems on chains of prime ideals are also explored. The work is essentially self-contained, the only prerequisite being a sound knowledge of modern algebra, yet the reader is taken to the frontiers of the subject. Exercises are provided at the end of each section and solutions or hints to some of them are given at the end of the book.

Let G G be a finite group of automorphisms of a nonsingular three-dimensional complex variety M M , whose canonical bundle ω M \omega _M is locally trivial as a G G -sheaf. We prove that the Hilbert scheme Y = G Y = G - Hilb ⁡ M \operatorname {Hilb}M parametrising G G -clusters in M M is a crepant resolution of X = M / G X=M/G and that there is a derived equivalence (Fourier–Mukai transform) between coherent sheaves on Y Y and coherent 𝐺-sheaves on M M . This identifies the K theory of Y Y with the equivariant K theory of M M , and thus generalises the classical McKay correspondence. Some higher-dimensional extensions are possible.

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