Absolute, Relative, and Tate Cohomology of Modules of Finite Gorenstein Dimension

We study finitely generated modules M over a ring R, noetherianon both sides. If M has finite Gorenstein dimension G-dim_RMin the sense of Auslander and Bridger, then it determines twoother cohomology theories besides the one given by the absolutecohomology functors . Relative cohomology functors are defined for all non-negative integers n; they treat the modules of Gorensteindimension 0 as projectives and vanish for n > G-dim_RM. Tatecohomology functors are defined for all integers n; all groups vanish if M or N has finite projective dimension. Comparisonmorphisms and 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 . 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.