Artinianness and finiteness of formal local cohomology modules with respect to a pair of ideals

Abstract

Let $(R,\mathfrak {m})$ be a commutative Noetherian local ring, M be a finitely generated R-module and $\mathfrak {a}$, I and J are ideals of R. We investigate the structure of formal local cohomology modules of $\mathfrak {F}^i_{\mathfrak {a},I,J}(M)$ and $\check{\mathfrak {F}}^i_{\mathfrak {a},I,J}(M)$ with respect to a pair of ideals, for all $i\ge 0$. The main subject of the paper is to study the finiteness properties and artinianness of $\mathfrak {F}^i_{\mathfrak {a},I,J}(M)$ and $\check{\mathfrak {F}}^i_{\mathfrak {a},\mathfrak {m},J}(M)$. We study the maximum and minimum integer $i\in \mathbb {N}$ such that $\mathfrak {F}^i_{\mathfrak {a},\mathfrak {m},J}(M)$ and $\check{\mathfrak {F}}^i_{\mathfrak {a},\mathfrak {m},J}(M)$ are not Artinian and we obtain some results involving cosupport, coassociated and attached primes for formal local cohomology modules with respect to a pair of ideals. Also, we give an criterion involving the concepts of finiteness and vanishing of formal local cohomology modules and Čech-formal local cohomology modules with respect to a pair of ideals.