On the cofiniteness of generalized local cohomology modules with respect to the class of modules in dimension less than a fixed integer

Let n and t be non-negative integers, R a commutative Noetherian ring with dim (R) ≤n+2, a an ideal of R, M and N finite R-modules, and X an arbitrary R-module. We prove that if ExtRi (M/aM, X) is an FD<n R-module for all i≤t, then Hai (M, X) is an (FD<n, a) -cofinite R-module for all i < t, HomR (R/a, Hat (M, X) ) is an FD<n R-module, and p∈AssR (Hai (M, X) ): dim (R/p) ≥n is a finite set for all i≤t. It shows that Hai (M, N) is (FD<n, a) -cofinite and p∈AssR (Hai (M, N) ): dim (R/p) ≥n is finite for all i. In particular, Hai (M, N) is a-cofinite for all i whenever dim (R) ≤2. Also, AssR (Hai (M, N) ) is finite for all i when R is semi-local with dim (R) ≤3.