On the vanishing and finiteness properties of generalized local cohomology modules

Details On the vanishing and finiteness properties of generalized local cohomology modules On the vanishing and finiteness properties of generalized local cohomology modules

2009-09-07

arxiv.org/abs/0909.1131v1

Mathematics

Commutative Algebra

5 pages

Scientific paper

Let $R$ be a commutative noetherian ring, $\fa$ an ideal of $R$ and $M,N$ finite $R$--modules. We prove that the following statements are equivalent. \begin{enumerate} \item[(i)] $\lc^{i}_{\fa}(M,N)$ is finite for all $i< n$. \item[(ii)] $\Coass_R(\lc^{i}_{\fa}(M,N)) \subset \V{(\fa)}$ for all $i< n$. \item[(iii)] $\lc^{i}_{\fa}(M,N)$ is coatomic for all $i< n$. \end{enumerate} If $\pd M$ is finite and $r$ be a non-negative integer such that $r>\pd M$ and $\lc^{i}_{\fa}(M,N)$ is finite (resp. minimax) for all $i\geq r$, then $\lc^{i}_{\fa}(M,N)$ is zero (resp. artinian) for all $i\geq r$.

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