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Absolute, Relative, and Tate Cohomology of Modules of Finite Gorenstein Dimension
Authors:Avramov  Luchezar L; Martsinkovsky  Alex
Institution:Department of Mathematics, Purdue University West Lafayette, IN 47907, USA
Department of Mathematics, Northeastern University Boston, MA 02115, USA; e-mail: alexmart{at}neu.edu
Abstract:We study finitely generated modules M over a ring R, noetherianon both sides. If M has finite Gorenstein dimension G-dimRMin the sense of Auslander and Bridger, then it determines twoother cohomology theories besides the one given by the absolutecohomology functors Formula. Relative cohomology functors Formula are defined for all non-negative integers n; they treat the modules of Gorensteindimension 0 as projectives and vanish for n > G-dimRM. Tatecohomology functors Formula are defined for all integers n; all groups Formula vanish if M or N has finite projective dimension. Comparisonmorphisms Formula and Formula link these functors. We give a self-contained treatmentof modules of finite G-dimension, establish basic propertiesof relative and Tate cohomology, and embed the comparison morphismsinto a canonical long exact sequence Formula. We show that these results provide efficient tools for computingold and new numerical invariants of modules over commutativelocal rings. 2000 Mathematical Subject Classification: 16E05, 13H10, 18G25.
Keywords:Gorenstein dimension  proper resolution  complete resolution  Betti numbers  Bass numbers  delta invariants
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