Abstract:
Let R be a commutative Noetherian ring, I an ideal of R, M an
R-module (not necessary I-torsion) and K a finitely generated R-module
with SuppR(K) ⊆ V(i). It is shown that if M is I-ETH-cofinite
(i.e. ExtiR(R/I,M) is finitely generated, for all i ≥ 0) and dimM ≤ 1, then the R-module
ExtnR(M,K) is finitely generated, for all n ≥ 0. As a consequence it is shown that if
M is I-ETH-cofinite and FD≤1 (or weakly Laskerian), then the R-module
ExtnR(M,K) is
finitely generated, for all n ≥ 0 which removes I-torsion condition of M
from [3, Corollary 3.11] and [20, Theorem 2.8]. As an application to
local cohomology, let Φ be a system of ideals of R
and I∈Φ, if dim M/aM ≤ 1 (e.g., dim R/a ≤ 1) for all
a ∈ Φ, then the R-modules
ExtjR(HiΦ(M),K)
are finitely generated, for all i ≥ 0 and j ≥ 0. A similar result is true for local cohomology modules defined
by a pair of ideals.
Key Words: Local cohomology, FD≤n modules, cofinite modules,
ETH-cofinite modules, weakly Laskerian modules.
2010 Mathematics Subject Classification: Primary 13D45; Secondary 13E05.
