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| Paper IPM / M / 8292 |
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| Abstract: | |
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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 R-modules 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 R-modules, 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
R-module 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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