Basic commutative algebra (Q2786366)

The book under review provides exactly what its title indicates, namely an introduction to the basic concepts, methods, and results of commutative algebra. As the author points out, the text grew out of various courses taught by him at several instructional schools, at several institutions and at different levels: masters to fresh graduate to advanced graduate. Therefore the book is geared toward students at these levels, be it as an accompanying course book or for profound self-study. Equally, the book can be used by instructors as a well-structured textbook for appropriate courses in commutative algebra.NEWLINENEWLINE On the part of the reader, only a rudimentary knowledge of basic general algebra is assumed, notably the elementary facts about groups, rings, fields, and field extensions.NEWLINENEWLINE As for the contents, the material is organized in twenty-one chapters covering the following standard topics in commutative algebra and its related homological aspects:NEWLINENEWLINE Chapter 1 deals with the basic ideal theory in commutative rings, including the spectrum of a ring and its Zariski topology. Chapter 2 introduces modules and algebras, with separate sections on localizations, graded rings and modules, exact sequences, and homogeneous prime and maximal ideals.NEWLINENEWLINE Chapter 3 briefly discusses polynomial and power series rings, whereas Chapter 4 provides the first set of homological tools comprising categories and functors, exact functors, the functor Hom and tensor products of modules, base change, direct and inverse limits, and -- finally -- the basics of injective, projective and flat modules. Chapter 5 explains the fundamental concepts of multilinear algebra for modules, inclusive of anticommutative and alternating algebras. Chapter 6 turns to finiteness conditions, thereby introducing modules of finite length, Noetherian rings and modules, Artinian rings and modules as well as locally free modules.NEWLINENEWLINE Chapter 7 treats the primary decomposition of Noetherian modules, the support of a module, and the concept of Krull dimension of a ring. Filtrations, topological rings, and completions are the topics of Chapter 8, culminating in the description of the completion of a finitely generated module over a Noetherian local ring.NEWLINENEWLINE Chapter 9 analyzes the most important numerial functions in commutative algebra, in particular the Hilbert function of a graded module and the Hilbert-Samuel function over a local ring. Krull's Principal Ideal Theorem and the dimension of a local ring are studied in Chapter 10, whereas the subsequent Chapter 11 is devoted to integral ring extensions and their ideal theory.NEWLINENEWLINE Chapter 12 touches upon unique factorization domains, normal domains, discrete valuation rings, Dedekind domains, and extensions of Dedekind domains. Then, after an insertion of transcendental field extensions and Lüroth's Theorem in Chapter 13, finitely generated affine algebras over a field are discussed in Chapter 14. The reader meets here the Noether Normalization Lemma, Hilbert's Nullstellensatz, and the dimension theory of affine algebras, graded rings, and standard graded rings.NEWLINENEWLINE Chapter 15 introduces derivations and differentials in commutative algebra, together with their basic properties and applications. Valuation rings and valuations appear in Chapter 16, including Hensel's Lemma and completions via real valuations.NEWLINENEWLINE Chapter 17 provides the second set of homological tools, thereby focussing on derived functors, complexes and homology, resolutions of a module, the functors Ext and Tor, local cohomology, and (co-)homology of groups.NEWLINENEWLINE Chapter 18 deals with the various concepts of homological dimension for rings and modules, with a special view toward finitely generated modules over a Noetherian local ring.NEWLINENEWLINE Chapter 19 is dedicated to regular sequences and the notion of depth of a module, with Cohen-Macaulay rings and modules being the central objects of study in this chapter.NEWLINENEWLINE Chapter 20 analyzes regular local rings from various viewpoints. This includes a homological characterization of regular Noetherian local rings, a differential criterion for regularity, and the Jacobian criterion for geometric regularity. Finally, a short proof of the factoriality of a regular local ring is given.NEWLINENEWLINE The concluding Chapter 21 may be seen as another particular highlight of the present textbook. The author discusses here divisor class groups of normal domains and their fundamental functorial properties, where the case of Galois descent under the action of a finite group is treated in some greater detail.NEWLINENEWLINE Each chapter comes with a large series of related exercises, many of which provide illustrating examples or additional results.NEWLINENEWLINE Altogether, this is an excellent introduction to modern basic commutative algebra. Written in a very lucid, detailed and streamlined fashion, the text appears to be rather comprehensive and tailor-made for beginners in the field. Some well-known classical results are presented from a new perspective, which makes this textbook highly interesting and useful for researchers and teachers, too.NEWLINENEWLINE Also, the rich collection of exercises adds up to the many particular features of this well-polished primer of commutative algebra.