Abstract
Let R be a noetherian ring, \({\mathfrak{a}}\) an ideal of R, and M an R-module. We prove that for a finite module M, if \({{\rm H}^{i}_{\mathfrak{a}}(M)}\) is minimax for all i ≥ r ≥ 1, then \({{\rm H}^{i}_{\mathfrak{a}}(M)}\) is artinian for i ≥ r. A local–global principle for minimax local cohomology modules is shown. If \({{\rm H}^{i}_{\mathfrak{a}}(M)}\) is coatomic for i ≤ r (M finite) then \({{\rm H}^{i}_{\mathfrak{a}}(M)}\) is finite for i ≤ r. We give conditions for a module which is locally minimax to be a minimax module. A non-vanishing theorem and some vanishing theorems are proved for local cohomology modules.
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Aghapournahr, M., Melkersson, L. Finiteness properties of minimax and coatomic local cohomology modules. Arch. Math. 94, 519–528 (2010). https://doi.org/10.1007/s00013-010-0127-z
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DOI: https://doi.org/10.1007/s00013-010-0127-z