In the last two decades Cohen-Macaulay rings and modules have been central topics in commutative algebra. This book meets the need for a thorough, self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules. A separate chapter is devoted to Hilbert functions (including Macaulay's theorem) and numerical invariants derived from them. The authors emphasize the study of explicit, specific rings, making the presentation as concrete as possible. So the general theory is applied to Stanley-Reisner rings, semigroup rings, determinantal rings, and rings of invariants. Their connections with combinatorics are highlighted, e.g. Stanley's upper bound theorem or Ehrhart's reciprocity law for rational polytopes. The final chapters are devoted to Hochster's theorem on big Cohen-Macaulay modules and its applications, including Peskine-Szpiro's intersection theorem, the Evans-Griffith syzygy theorem, bounds for Bass numbers, and tight closure. Throughout each chapter the authors have supplied many examples and exercises which, combined with the expository style, will make the book very useful for graduate courses in algebra. As the only modern, broad account of the subject it will be essential reading for researchers in commutative algebra.
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acyclic Artinian assume big Cohen—Macaulay modules canonical module characteristic Cohen—Macaulay local ring Cohen—Macaulay ring cohomology complete intersection containing a ﬁeld Corollary deﬁne deﬁnition degree denote depth dimR direct summand domain equation exact sequence Exercise exists extension F rational ﬁltration ﬁnite R-module ﬁrst ﬂat free resolution functor G Spec Gorenstein ring grade(I grade(I,M graded ring hence Hilbert function Hilbert series Hochster Hodge algebra homogeneous elements homology homomorphism Huneke hypothesis implies induction integral invariants isomorphism k-vector Koszul complex Lemma Let R,m,k linear M-sequence Macaulay Math maximal Cohen—Macaulay maximal ideal minimal prime ideals monomials multiplicity Noetherian local ring Noetherian ring non-zero normal semigroup polynomial ring polytope positively graded proj dimM PROOF Proposition prove rank regular local ring regular rings residue class ring satisﬁes semigroup rings simplicial complex submodule subset Suppose system of parameters tight closure vector yields Z"-graded