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Paper IPM / M / 8292  


Abstract:  
The study of the cohomological dimension of algebraic varieties
has produced some interesting results and problems in local
algebra. Let \fraka be an ideal of a commutative Noetherian
ring R. For finitely generated Rmodules M and N, the
concept of cohomological dimension cd_{\frak a}(M,N) of M
and N with respect to \fraka is introduced. If 0→ N′→ N" → 0 is an exact sequence of finitely
generated Rmodules, then it is shown that cd_{\frak a}(M,N)
= max {cd_{\fraka}(M,N′),cd_{\fraka}(M,N")} whenever
proj dim M < ∞. Also, if L is a finitely generated
Rmodule with Supp(N/Γ_{\fraka}(N)) ⊆ Supp
(L/Γ_{\fraka}(L)), then it is proved that cd_{\fraka}(M,N) ≤ max{cd_{\fraka}(M,L),proj dim M}.
Finally, as a consequence, a result of Brodmann is improved.
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