These lecture notes grew out of a Columbia seminar on Grothendieck's Bourbaki talk on duality and his SGA talk on flat, étale and smooth morphisms. The materials are divided into four parts. The first part (Chapter I), presupposing the others, discusses general features of Grothendieck's duality theory. The second part, consisting of Chapters II, III and IV, is devoted to the proof of the duality theorem. The third part (Chapters V, VI, VII) studies smooth morphisms aiming for general familiarity. Finally, in the last part (Chapter VIII) the authors treat the theory of curves. The traditional duality theorem and the theorem of residues are reconstructed from the modern point of view. The details are as follows: In Chapter I, they first give exact formulations of the duality theorems of Serre and Grothendieck and then there is some discussion of the dualizing sheaf ωX, which is naturally isomorphic to Ωr(X/k) when X is a smooth proper r-dimensional scheme over a field k. The following two chapters are devoted to an exposition of the necessary materials from commutative and homological algebras. They contain many topics which range from the elementary results such as completions of modules by ideal-adic topology, primary decomposition of noetherian modules, notions of depth, dimension, etc., to elaborate results such as Serre's characterization of regular local rings by the finiteness of global homological dimension. They are well and compactly written and may be used as a nice introductory course for advanced students. Chapter IV is devoted to the proof of the duality theorem stated in Chapter I. There a knowledge of spectral sequences is presupposed. It is in a sense quite reasonable to do so, but it seems not to be consistent with the style of the foregoing chapters. In Chapter V they treat first (faithful) flat modules, and then these results are interpreted in terms of schemes and morphisms. The local criteria of flatness are discussed next. Constructive sets play the main role in § 4, and in § 5 it is proved that the set of points x where the morphism f:X→Y is flat forms an open subset of X. Chapter VI is devoted to the exposition of étale (flat and unramified) morphisms. Here the sheaf of Kähler differentials plays the fundamental role. Purity of the branch locus for flat morphisms is proved. In the addenda (unpublished) they give a proof of purity of branch loci for a quasi-finite S-morphism f:X→Y with a smooth scheme Y. Chapter VII deals with smooth morphisms. The notions of Cartier divisors and Weil divisors are introduced. The factoriality of a regular local ring is proved here. Some results on regular rings and smooth morphisms are summarized. The Jacobian criterion for smoothness of a morphism and the differential theoretic characterization for an algebraic k-scheme to be smooth are the last topics in this chapter. Chapter VIII deals with the theory on curves. They first survey the geometry on a proper scheme of dimension 1 over an Artinian ring. The name of pseudo-differential is attributed to the notion which corresponds to the classical notion of differentials, i.e., linear functionals on the space of repartitions. The dualizing sheaf ωX is defined here as a sheaf of germs of regular pseudo-differentials. The results of Rosenlicht-Gorenstein are also recovered. The remaining parts are devoted to the proof of residue theorems following Tate's idea. As a whole this book is nicely and compactly written, though it contains a number of careless mistakes. It will also be helpful to those who study algebraic geometry of scheme theoretic type. Compared with the contents, the title seems to be a little too specific. {Reviewer's remark: No assumption is made concerning the ring k of Chapter VIII, § 2. But it seems that it is necessary to assume that k is Artinian for the remaining discussion.}
Reviewer: Nakai, Y. [form MathSciNet]