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Let denote an ideal of a commutative Noetherian ring R, and M and N two finitely generated R-modules with pd M < ∞. It is shown that if either is principal, or R is complete local and is a prime ideal with dim R/ = 1, then the generalized local cohomology module is -cofinite for all i ≥ 0. This provides an affirmative answer to a question proposed in [13].
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...
We study series of the form , where is a commutative local ring, is a non-negative integer, and the summation extends over all finite -modules , up to isomorphism. This problem is motivated by Cohen-Lenstra heuristics on class groups of number fields, where sums of this kind occur. If has additional properties, we will relate the above sum to a limit of zeta functions of the free modules , where these zeta functions count -submodules of finite index in . In particular we will show that...
We study when the modifications of the Cohen-Macaulay vertex cover ideal of a graph are Cohen-Macaulay.
Let R be a commutative multiplication ring and let N be a non-zero finitely generated multiplication R-module. We characterize certain prime submodules of N. Also, we show that N is Cohen-Macaulay whenever R is Noetherian.
Let denote an ideal in a Noetherian ring R, and M a finitely generated R-module. We introduce the concept of the cohomological dimension filtration , where c = cd(,M) and denotes the largest submodule of M such that . Some properties of this filtration are investigated. In particular, if (R,) is local and c = dim M, we are able to determine the annihilator of the top local cohomology module , namely . As a consequence, there exists an ideal of R such that . This generalizes the main results...
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