Maximal Cohen-Macaulay Modules and Tate Cohomology

ISBN-13: 9781470453404

Ragnar-Olaf Buchweitz

Opis

This book is a lightly edited version of the unpublished manuscript Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings by Ragnar-Olaf Buchweitz.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.

Spis treści

• 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
• Bibliography
• Comments and errata
• Gorenstein Noether algebras
• Subsequent developments
• Additional bibliography
• Glossary
• Index