ON THE COFINITENESS OF INDIMENSION

Abstract

In this note, we prove the cofiniteness of local cohomology mod-
ules
H
i
I,J
(
N
) with respect to a pair of ideals (
I, J
) for all
i < t
and
the finiteness of (0 :
H
t
I,J
(
N
)
I
) and Ext
1
R
(
R/I, H
t
I,J
(
N
)) provided that
Ext
i
R
(
R/I, N
) is finitely generated for all
i
≤
t
+ 1 and
H
i
I,J
(
N
) is in
dimension
<
2 for all
i < t
, where
t
≥
1 is an integer (here,
N
is
not necessarily finitely generated over
R
). This extends the results of
Bahmanpour-Naghipour [5, Thm 2.6], Bahmanpour-Naghipour-Sedghi
[4, Thm 2.8] and H-N [16, Thm 1.1]