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Let be an ideal of a commutative Noetherian ring . It is shown that the -modules are -cofinite for all finitely generated -modules and all if and only if the -modules and are -cofinite for all finitely generated -modules , and all integers .
Let R be a commutative Noetherian ring, I a proper ideal of R, and M be a finitely generated R-module. We provide bounds for the cohomological dimension of the R-module M with respect to the ideal I in several cases.
Let be a complete Noetherian local ring, an ideal of and a nonzero Artinian -module. In this paper it is shown that if is a prime ideal of such that and is not finitely generated and for each the -module is of finite length, then the -module is not of finite length. Using this result, it is shown that for all finitely generated -modules with and for all integers , the -modules are of finite length, if and only if, for all finitely generated -modules with and...
Let R be a Noetherian ring and I an ideal of R. Let M be an I-cofinite and N a finitely generated R-module. It is shown that the R-modules are I-cofinite for all i ≥ 0 whenever dim Supp(M) ≤ 1 or dim Supp(N) ≤ 2. This immediately implies that if I has dimension one (i.e., dim R/I = 1) then the R-modules are I-cofinite for all i,j ≥ 0. Also, we prove that if R is local, then the R-modules are I-weakly cofinite for all i ≥ 0 whenever dim Supp(M) ≤ 2 or dim Supp(N) ≤ 3. Finally, it is shown that...
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