Commutative Coherent Rings
This book provides the first extensive and systematic treatment of the theory of commutative coherent rings. It blends, and provides a link, between the two sometimes disjoint approaches available in the literature, the ring theoretic approach, and the homological algebra approach. The book covers most results in commutative coherent ring theory known to date, as well as a number of results never published before. Starting with elementary results, the book advances to topics such as: uniform coherence, regular rings, rings of small homological dimensions, polynomial and power series rings, group rings and symmetric algebra over coherent rings. The subject of coherence is brought to the frontiers of research, exposing the open problems in the field. Most topics are treated in their fully generality, deriving the results on coherent rings as conclusions of the general theory. Thus, the book develops many of the tools of modern research in commutative algebra with a variety of examples and counterexamples. Although the book is essentially self-contained, basic knowledge of commutative and homological algebra is recommended. It addresses graduate students and researchers.
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absolutely flat ring algebra assume coherent domain coherent module coker f conclude conditions are equivalent COROLLARY countable defined DEFINITION denote dimºſ dimp direct summand element exact sequence exists faithfully flat field of quotients finite free resolution finitely generated free finitely generated ideal finitely presented finitely presented module flat epimorphism flat R module following conditions FP dim free R module grade Homº idempotent injective R module integrally closed isomorphism ker f Krull dim Krull dimension Lemma let f maximal ideal module and let Noetherian ring nonzero divisor obtain polynomial ring presented R module prime ideal principal ideal proj projective R module Proof Prüfer domain pure submodule rank G reduced ring RG module ring and let ring homomorphism satisfying semihereditary ring Spec(R supp surjective uniformly coherent ring valuation domain weak dimension