On the vanishing, artinianness and finiteness of local cohomology modules

Details On the vanishing, artinianness and finiteness of local cohomology modules On the vanishing, artinianness and finiteness of local cohomology modules

2009-03-12

arxiv.org/abs/0903.2235v1

Mathematics

Commutative Algebra

10 pages

Scientific paper

Let $R$ be a noetherian ring, $\fa$ an ideal of $R$, and $M$ an $R$--module. We prove that for a finite module $M$, if $\LC^{i}_{\fa}(M)$ is minimax for all $i\geq r\geq 1$, then $\LC^{i}_{\fa}(M)$ is artinian for $i\geq r$. A Local-global Principle for minimax local cohomology modules is shown. If $\LC^{i}_{\fa}(M)$ is coatomic for $i\leq r$ ($M$ finite) then $\LC^{i}_{\fa}(M)$ is finite for $i\leq r$. 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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