On the cominimaxness of generalized local cohomology modules

The local cohomology theory plays an important role in commutative algebra and algebraic geometry. The I-cofiniteness of local cohomology modules is one of interesting properties which has been studied by many mathematicians. The I-cominimax modules is an extension of I-cofinite modules which was introduced by Hartshorne. An R-module M is I-cominimax if Supp_R(M)\subseteq V(I) and Ext^i_R(R/I,M) is minimax for all i\ge 0. In this paper, we show some conditions such that the generalized local cohomology module H^i_I(M,N) is I-cominimax for all i\ge 0. We show that if H^i_I(M,K) is I-cofinite for all i<t and all finitely generated R-module K, then H^i_I(M,N) is I-cominimax for all i<t and all minimax R-module N. If M is a finitely generated R-module, N is a minimax R-module and t is a non-negative integer such that dim Supp_R(H^i_I(M,N))\le 1 for all i<t then H^i_I(M,N) is I-cominimax for all i<t. When dim R/I\le 1 and H^i_I(N) is I-cominimax for all i\ge 0 then H^i_I(M,N) is I-cominimax for all i\ge 0.

Some homological properties of generalized local cohomology modules

Let [Formula: see text] be a Noetherian local ring and [Formula: see text], [Formula: see text] be two finitely generated [Formula: see text]-modules. In this paper, it is shown that [Formula: see text] and [Formula: see text] for each [Formula: see text] and each integer [Formula: see text]. In particular, if [Formula: see text] then [Formula: see text]. Moreover, some applications of these results will be included.

On cominimaxness of generalized local cohomology modules

This research introduces and focuses on (I, M)-cominimax modules. The paper shows that if t is an nonnegative integer, M is a finitely generated projective R-module and N is an R-module such that is minimax and is (I, M)-cominimax for all then is minimax and is finite.