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Absolute, Relative, and Tate Cohomology of Modules of Finite Gorenstein Dimension
Author(s) -
Avramov Luchezar L.,
Martsinkovsky Alex
Publication year - 2002
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611502013527
Subject(s) - mathematics , functor , dimension (graph theory) , cohomology , morphism , noetherian ring , commutative property , combinatorics , local cohomology , discrete mathematics , pure mathematics , finitely generated abelian group
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.