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1. Dimension of a local ring. 1.1. Nakayama's lemma. 1.2. Prime ideals. 1.3. Noetherian modules. 1.4. Modules of finite length. 1.5. Hilbert's basis theorem. 1.6. Graded rings. 1.7. Filtered rings. 1.8. Local rings. 1.9. Regular local rings -- 2. Modules over a local ring. 2.1. Support of a module. 2.2. Associated prime ideals. 2.3. Dimension of a module. 2.4. Depth of a module. 2.5. Cohen-Macaulay modules. 2.6. Modules of finite projective dimension. 2.7. The Koszul complex. 2.8. Regular local rings. 2.9. Projective dimension and depth -- 2.10. [symbol]-depth. 2.11. The acyclicity theorem. 2.12. An example -- 3. Divisor theory. 3.1. Discrete valuation rings. 3.2. Normal domains. 3.3. Divisors. 3.4. Unique factorization. 3.5. Torsion modules. 3.6. The first Chern class. 3.7. Regular local rings. 3.8. Picard groups. 3.9. Dedekind domains -- 4. Completion. 4.1. Exactness of the completion functor. 4.2. Separation of the [symbol]-adic topology. 4.3. Complete filtered rings. 4.4. Completion of local rings. 4.5. Structure of complete local rings -- 5. Injective modules. 5.1. Injective modules. 5.2. Injective envelopes. 5.3. Decomposition of injective modules. 5.4. Matlis duality. 5.5. Minimal injective resolutions. 5.6. Modules of finite injective dimension. 5.7. Gorenstein rings -- 6. Local cohomology. 6.1. Basic properties. 6.2. Local cohomology and dimension. 6.3. Local cohomology and depth. 6.4. Support in the maximal ideal. 6.5. Local duality for Gorenstein rings -- 7. Dualizing complexes. 7.1. Complexes of injective modules. 7.2. Complexes with finitely generated cohomology. 7.3. The evaluation map. 7.4. Existence of dualizing complexes. 7.5. The codimension function. 7.6. Complexes of flat modules. 7.7. Generalized evaluation maps. 7.8. Uniqueness of dualizing complexes -- 8. Local duality. 8.1. Poincare series. 8.2. Grothendieck's local duality theorem. 8.3. Duality for Cohen-Macaulay modules. 8.4. Dualizing modules. 8.5. Locally factorial domains. 8.6. Conductors. 8.7. Formal fibers -- 9. Amplitude and dimension. 9.1. Depth of a complex. 9.2. The dual of a module. 9.3. The amplitude formula. 9.4. Dimension of a complex. 9.5. The tensor product formula. 9.6. Depth inequalities. 9.7. Condition Sr of Serre. 9.8. Factorial rings and condition Sr. 9.9. Condition S[symbol]. 9.10. Specialization of Poincare series -- 10. Intersection multiplicities. 10.1. Introduction to Serre's conjectures. 10.2. Filtration of the Koszul complex. 10.3. Euler characteristic of the Koszul complex. 10.4. A projection formula. 10.5. Power series over a field. 10.6. Power series over a discrete valuation ring. 10.7. Application of Cohen's structure theorem. 10.8. The amplitude inequality. 10.9. Translation invariant operators. 10.10. Todd operators. 10.11. Serre's conjecture in the graded case -- 11. Complexes of free modules. 11.1. McCoy's theorem. 11.2. The rank of a linear map. 11.3. The Eisenbud-Buchsbaum criterion. 11.4. Fitting's ideals. 11.5. The Euler characteristic. 11.6. McRae's invariant. 11.7. The integral character of McRae's invariant
This book is a comprehensive treatment of the representation theory of maximal Cohen-Macaulay (MCM) modules over local rings. This topic is at the intersection of commutative algebra, singularity theory, and representations of groups and algebras. Two introductory chapters treat the Krull-Remak-Schmidt Theorem on uniqueness of direct-sum decompositions and its failure for modules over local rings. Chapters 3-10 study the central problem of classifying the rings with only finitely many indecomposable MCM modules up to isomorphism, i.e., rings of finite CM type. The fundamental material--ADE/simple singularities, the double branched cover, Auslander-Reiten theory, and the Brauer-Thrall conjectures--is covered clearly and completely. Much of the content has never before appeared in book form. Examples include the representation theory of Artinian pairs and Burban-Drozd's related construction in dimension two, an introduction to the McKay correspondence from the point of view of maximal Cohen-Macaulay modules, Auslander-Buchweitz's MCM approximation theory, and a careful treatment of nonzero characteristic. The remaining seven chapters present results on bounded and countable CM type and on the representation theory of totally reflexive modules.
The papers in this volume contain results in active research areas in the theory of rings and modules, including non commutative and commutative ring theory, module theory, representation theory, and coding theory.
Commutative algebra is a rapidly growing subject that is developing in many different directions. This volume presents several of the most recent results from various areas related to both Noetherian and non-Noetherian commutative algebra. This volume contains a collection of invited survey articles by some of the leading experts in the field. The authors of these chapters have been carefully selected for their important contributions to an area of commutative-algebraic research. Some topics presented in the volume include: generalizations of cyclic modules, zero divisor graphs, class semigroups, forcing algebras, syzygy bundles, tight closure, Gorenstein dimensions, tensor products of algebras over fields, as well as many others. This book is intended for researchers and graduate students interested in studying the many topics related to commutative algebra.
Ideal for graduate students and researchers, this book presents a unified treatment of the central notions of integral closure.
This book is a lightly edited version of the unpublished manuscript Maximal Cohen–Macaulay modules and Tate cohomology over Gorenstein rings by Ragnar-Olaf Buchweitz. The central objects of study are maximal Cohen–Macaulay modules over (not necessarily commutative) Gorenstein rings. The main result is that the stable category of maximal Cohen–Macaulay modules over a Gorenstein ring is equivalent to the stable derived category and also to the homotopy category of acyclic complexes of projective modules. This assimilates and significantly extends earlier work of Eisenbud on hypersurface singularities. There is also an extensive discussion of duality phenomena in stable derived categories, extending Tate duality on cohomology of finite groups. Another noteworthy aspect is an extension of the classical BGG correspondence to super-algebras. There are numerous examples that illustrate these ideas. The text includes a survey of developments subsequent to, and connected with, Buchweitz's manuscript.