Let
R
R
denote a commutative Noetherian (not necessarily local) ring and
I
I
an ideal of
R
R
of dimension one. The main purpose of this paper is to generalize, and to provide a short proof of, K. I. Kawasaki’s theorem that the category
M
(
R
,
I
)
c
o
f
\mathscr {M}(R, I)_{cof}
of
I
I
-cofinite modules over a commutative Noetherian local ring
R
R
forms an Abelian subcategory of the category of all
R
R
-modules. Consequently, this assertion answers affirmatively the question raised by R. Hartshorne in his article Affine duality and cofiniteness [Invent. Math. 9 (1970), 145-164] for an ideal of dimension one in a commutative Noetherian ring
R
R
.
Let M be a non-zero finitely generated module over a commutative Noetherian local ring (R, m). In this paper we consider when the local cohomology modules are finitely generated. It is shown that if t ≥ 0 is an integer and p ∈ Supp H t p (M ), then H * The first and second authors were in part supported by IPM (No. 89130048, 89130053).
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