COFINITE MODULES AND COFINITENESS OF LOCAL COHOMOLOGY MODULES

Letnbe a n on-negative integer,R a commutative Noetherianring,aan ideal ofR,Ma finitely generated R-module, and X an arbitrary R-module. In this paper, we first prove that if dim( R)(M)<= n+ 2, then H-a(i)(M)is an (FD<n,a)-cofinite R-module and {p is an element of Ass(R)(H-a(i)(M)) : dim(R/p)>= n}is a finite set for alli. As a consequence, it follows that AssR(H-a(i)(M))is a finite set for all i when R is a semi-local ring and dim(R)(M)<= 3.Then, we show that if dim(R/a)<= n+ 1, then Ext(R)(i)(R/a, X) is an FD<n R-module for alliwhenever ExtiR(R/a, X) is an FD<nR-module for all i <= dim(R)(X)-n. Finally, in the case that dim(R/a)<= 2,X is a-torsion, and n >0 or Supp(R)(X)boolean AND Var(a)boolean AND Max(R) is finite, we prove that X is an(FD<n,a)-cofiniteR-module when Ext(R)(i)(R/a, X) is an FD<n R-module for all i <= 2-n. We conclude with some ordinarya-cofiniteness results for localcohomology modules H-a(i)(X).