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Maximal Cohen-Macaulay modules and Tate cohomology / Ragnar-Olaf Buchweitz ; with appendices by Luchezar L. Avramov, Benjamin Briggs, Srikanth B. Iyengar, Janina C. Letz.

By: Buchweitz, Ragnar-Olaf, 1952- [author.].
Contributor(s): Avramov, L. L. (Luchezar L.), 1948- [contributor.] | Briggs, Benjamin (Mathematician) [contributor.] | Iyengar, Srikanth, 1970- [contributor.] | Letz, Janina C [contributor.].
Material type: TextTextSeries: Mathematical surveys and monographs: no. 262.Publisher: Providence, Rhode Island : American Mathematical Society, [2021]Copyright date: ©2021Description: xii, 175 pages : USD 125.00 illustrations ; 26 cm.Content type: text Media type: unmediated Carrier type: volumeISBN: 9781470453404.Subject(s): Cohen-Macaulay modules | Modules (Algebra) | Homology theory | Modules de Cohen-Macaulay | Modules (Algèbre) | Homologie | Cohen-Macaulay modules | Homology theory | Modules (Algebra) | Commutative algebra -- Theory of modules and ideals -- Cohen-Macaulay modules [See also 13H10] | Commutative algebra -- Homological methods {For noncommutative rings, see 16Exx; for general categories, see 18Gxx} | Commutative algebra -- Local rings and semilocal rings -- Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05] | Associative rings and algebras {For the commutative case, see 13-XX} -- Homological methods {For commutative rings, see 13Dxx; for general categories, see 18Gxx}Genre/Form: Festschriften.DDC classification: 512.44 Other classification: 13C14 | 13Dxx | 13H10 | 16Exx Also issued online.
Contents:
Notations and conventions -- Perfect complexes and the stable derived category -- The category of modules modulo projectives -- Complete resolutions and the category of acyclic projective complexes -- Maximal Cohen-Macaulay modules and Gorenstein rings -- Maximal Cohen-Macaulay approximations -- The Tate cohomology -- Multiplicative structure, duality and support -- First examples -- Connection to geometry on projective super-spaces -- Applications to singularities and hypersurfaces.
Summary: "This book is a lightly edited version of the unpublished manuscript Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings by Ragnar-Olaf Buchweitz.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." -- Provided by publisher.
List(s) this item appears in: 2023-03-06
Item type Current location Call number Status Date due Barcode Item holds
Book Chennai Mathematical Institute
General Stacks 		512.44 BUC (Browse shelf) Available 11038
Total holds: 0

Includes bibliographical references (pages 155-165) and index.

Notations and conventions -- Perfect complexes and the stable derived category -- The category of modules modulo projectives -- Complete resolutions and the category of acyclic projective complexes -- Maximal Cohen-Macaulay modules and Gorenstein rings -- Maximal Cohen-Macaulay approximations -- The Tate cohomology -- Multiplicative structure, duality and support -- First examples -- Connection to geometry on projective super-spaces -- Applications to singularities and hypersurfaces.

"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." -- Provided by publisher.

Also issued online.