Let ðR,mÞ be a complete commutative Noetherian local ring, I an ideal of R, M an R-module (not necessarily I-torsion) and N a finitely generated Rmodule with SuppRðNÞ VðIÞ. It is shown that if M is I-ETH-cominimax (i.e. Exti RðR=I,MÞ is minimax (or Matlis reflexive), for all i 0) and dimM 1 or more generally M 2 FD1, then the R-module ExtnR ðM, NÞ is finitely generated, for all n 0. As an application to local cohomology, let U be a system of ideals of R and I 2 U, if dimM=aM 1 (e.g., dim R=a 1) for all a 2 U, then the R-modules Extj RðHi UðMÞ, NÞ are finitely generated, for all i 0 and j 0. Similar results are true for local cohomology defined by a pair of ideals and ordinary local cohomology modules.
