Commutative coherent rings
This book is the first book in the literature entirely devoted to commutative coherent rings; it gives a systematic and extensive treatment of most results on these rings, as well as new results never published before. The book consists of 8 chapters. In chapter 1 the author summarizes without proofs all the basic facts on projective and injective modules, flatness, homological dimensions, and classical results on rings of global and weak dimension \(\leq 1\), needed in the sequel. Chapter 2 introduces the reader to finitely presented modules and the main properties of coherent modules and rings. Chapter 3 presents the fundamental concepts which will be used in the next chapters: the behavior, in the context of coherence, of the homological dimensions under the change of rings, Zariski topology, projectivity and rank, associated prime ideals, Fitting invariants and Euler characteristic, Koszul complexes. --- The next five chapters are concerned with the main results in commutative coherent ring theory. Thus, chapter 4 presents topics in ring extensions such as the total ring of quotients, the maximal flat epimorphic extension of a ring, and trivial ring extensions; the presentation is given first as general as possible, followed then by the specific case of coherent rings. In the next chapter this manner of presentation (first the general case, and then the coherent case) is also used for the description of some ring constructions and overrings as: cartesian squares, \(D+M\) constructions, and a general overring approach to the study of the integral closure. Chapter 6 is concerned with several particular coherent rings: uniformly coherent rings, regular and super regular coherent rings, coherent rings of global and weak dimension 2, rings of small Ng dimension. Chapter 7 begins with the presentation of the notion of non-Noetherian grade, following Alfonsi's treatment; then, the most studied question in coherent ring theory, namely the question of stable coherence is investigated. This question is concerned with the analogue of Hilbert's basis theorem for coherent rings. Finally, the last chapter of the book deals with the coherence of power series ring over a coherent ring, of commutative group rings, and of symmetric algebras. The reader is assumed to have the knowledge of basic commutative and homological algebra. This book provides an up-to-date, clear and systematic account of commutative coherent rings. It is recommended as a reference book for researchers interested in this topic, as well as a textbook for a second-year graduate course in algebra.
scientific article; zbMATH DE number 42537