Cofinite modules and local cohomology Original Research Article

We show that if M is a finitely generated module over a commutative Noetherian local ring R and I is a dimension one ideal of R (i.e., dimR/I = 1), then the local cohomology modules HIi(M) are I-cofinite; that is, ExtRj(R/I, HIi(M)) is finitely generated for all i,j. We also show that if R is a complete local ring and P is a dimension one prime ideal of R, then the set of P-cofinite modules form an abelian subcategory of the category of all R-modules. Finally, we prove that if M is an n-dimensional finitely generated module over a Noetherian local ring R and I is any ideal of R, then HIn(M) is I-cofinite.