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

We study finitely generated modules M over a ring R , noetherian on both sides. If M has finite Gorenstein dimension G‐dim RM in the sense of Auslander and Bridger, then it determines two other cohomology theories besides the one given by the absolute cohomology functorsE x t R n ( M ,   ) . Relative cohomology functorsE x tG n ( M ,   ) are defined for all non‐negative integers n ; they treat the modules of Gorenstein dimension 0 as projectives and vanish for n > G‐dim RM . Tate cohomology functorsE x t ^ R n ( M ,   ) are defined for all integers n ; all groupsE x t ^ R n ( M , N ) vanish if M or N has finite projective dimension. Comparison morphisms ε G n : E x tG n ( M ,   ) → E x t R n ( M ,   ) and ε R n : E x t R n ( M ,   ) →E x t ^ R n ( M ,   ) link these functors. We give a self‐contained treatment of modules of finite G‐dimension, establish basic properties of relative and Tate cohomology, and embed the comparison morphisms into a canonical long exact sequence 0 → E x tG 1 ( M ,   ) → ⋯ → E x tG n ( M ,   ) → E x t R n ( M ,   ) →E x t ^ R n ( M ,   ) → E x tGn + 1 ( M ,   ) → ⋯ . We show that these results provide efficient tools for computing old and new numerical invariants of modules over commutative local rings. 2000 Mathematical Subject Classification : 16E05, 13H10, 18G25.