Cofiniteness with respect to a Serre subcategory

Abstract

Let Φ be a system of ideals in a commutative Noetherian ring R, and let Open image in new window be a Serre subcategory of R-modules. We set $$ H_\Phi ^i ( \cdot , \cdot ) = \mathop {\lim }\limits_{\overrightarrow {\mathfrak{b} \in \Phi } } Ext_R^i (R/\mathfrak{b}| \otimes R \cdot , \cdot ). $$ . Suppose that a is an ideal of R, and M and N are two R-modules such that M is finitely generated and N ∈ Open image in new window. It is shown that if the functor \( D_\Phi ( \cdot ) = \mathop {\lim }\limits_{\overrightarrow {\mathfrak{b} \in \Phi } } Hom_R (\mathfrak{b}, \cdot ) \) is exact, then, for any \( \mathfrak{b} \in \Phi ,Ext_R^j (R/\mathfrak{b},H_\Phi ^i (M,N)) \) ∈ Open image in new window for all i, j ≥ 0. It is also proved that if there is a nonnegative integer t such that \( H_\mathfrak{a}^i (M,N) \) ∈ Open image in new window for all i < t, then \( Hom_R (R/\mathfrak{a},H_\mathfrak{a}^t (M,N)) \) ∈ Open image in new window, provided that Open image in new window is contained in the class of weakly LaskerianR-modules. Finally, it is shown that if L is an R-module and t is the infimum of the integers i such that \( H_\mathfrak{a}^i (L) \) ∈ Open image in new window, then \( Ext_R^j (R/\mathfrak{a},H_\mathfrak{a}^t (M,L)) \) ∈ Open image in new window if and only if \( Ext_R^j (R/\mathfrak{a},Hom_R (M,H_\mathfrak{a}^t (L))) \) ∈ Open image in new window for all j ≥ 0.