This book is a lightly edited version of the unpublished manuscript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.
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