A Generalization of Formal Local Cohomology Modules
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Let $\fa$ and $\fb$ be two ideals of a commutative Noetherian ring $R$, $M$ a finitely generated $R$-module and $ i $ an integer. In this paper we study formal local cohomology modules with respect to a pair of ideals. We denote the $ i $-th $ \fa $-formal local cohomology module $M$ with respect to $\fb$ by $ \mathfrak{F}_{\fa, \fb}^i(M) $. We show that if $ \mathfrak{F}_{\fa, \fb}^i(M) $ is artinian, then $\fa \subseteq \sqrt{(0: \mathfrak{F}_{\fa, \fb}^i(M))}$. Also, we show that $ \mathfrak{F}_{\fa, \fb}^{\di M} (M) $ is artinian and we determine the set $\Att_R\mathfrak{F}_{\fa, \fb}^{\di M} (M)$.
Abstract
Let $\fa$ and $\fb$ be two ideals of a commutative Noetherian ring $R$, $M$ a finitely generated $R$-module and $ i $ an integer. In this paper we study formal local cohomology modules with respect to a pair of ideals. We denote the $ i $-th $ \fa $-formal local cohomology module $M$ with respect to $\fb$ by $ \mathfrak{F}_{\fa, \fb}^i(M) $. We show that if $ \mathfrak{F}_{\fa, \fb}^i(M) $ is artinian, then $\fa \subseteq \sqrt{(0: \mathfrak{F}_{\fa, \fb}^i(M))}$. Also, we show that $ \mathfrak{F}_{\fa, \fb}^{\di M} (M) $ is artinian and we determine the set $\Att_R\mathfrak{F}_{\fa, \fb}^{\di M} (M)$.
참고문헌 (12)
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