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 Cech-formal local cohomology modules with respect to a pair of ideals.