When are the local cohomology modules finitely generated?

摘要

Let R be a commutative Noetherian ring, alpha an ideal of R, and M an R-module. We prove that for a fixed non-negative integer n, the nth local cohomology module H-alpha(n)(M) is finitely generated if and only if H-alpha(n)(M) is alpha-cofinite and alpha subset of root 0:(R) Hn alpha(M). This enables us to establish the Noetherian property of local cohomology modules in several cases. Finally, we obtain a new characterization of the cohomological dimensions.