Let $(R,\mathfrak {m})$ be a complete Noetherian local ring, $I$ an ideal of $R$ and $M$ a nonzero Artinian $R$-module. In this paper it is shown that if $\mathfrak p$ is a prime ideal of $R$ such that $\dim R/\mathfrak p=1$ and $(0:_M\mathfrak p)$ is not finitely generated and for each $i\geq 2$ the $R$-module ${\rm Ext}^i_R(M,R/\mathfrak p)$ is of finite length, then the $R$-module ${\rm Ext}^1_R(M,R/\mathfrak p)$ is not of finite length. Using this result, it is shown that for all finitely generated $R$-modules $N$ with $\operatorname {Supp}(N)\subseteq V(I)$ and for all integers $i\geq 0$, the $R$-modules ${\rm Ext}^i_R(N,M)$ are of finite length, if and only if, for all finitely generated $R$-modules $N$ with $\operatorname {Supp}(N)\subseteq V(I)$ and for all integers $i\geq 0$, the $R$-modules ${\rm Ext}^i_R(M,N)$ are of finite length.
Let $(R,\mathfrak m)$ be a commutative Noetherian regular local ring of dimension $d$ and $I$ be a proper ideal of $R$ such that ${\rm mAss}_R(R/I)={\rm Assh}_R(I)$. It is shown that the $R$-module $H^{{\rm ht}(I)}_I(R)$ is $I$-cofinite if and only if
${\rm cd}(I,R)={\rm ht}(I)$. Also we present a sufficient condition under which this condition the $R$-module $H^i_I(R)$ is finitely generated if and only if it vanishes., Jafar A'zami, Naser Pourreza., and Obsahuje bibliografické odkazy
Let $R$ be a commutative Noetherian ring with identity and $I$ an ideal of $R$. It is shown that, if $M$ is a non-zero minimax $R$-module such that $\dim \mathop {\rm Supp} H^i_I (M) \leq 1$ for all $i$, then the $R$-module $H^i_I(M)$ is $I$-cominimax for all $i$. In fact, $H^i_I(M)$ is $I$-cofinite for all $i\geq 1$. Also, we prove that for a weakly Laskerian $R$-module $M$, if $R$ is local and $t$ is a non-negative integer such that $\dim \mathop {\rm Supp} H^i_I (M)\leq 2$ for all $i<t$, then ${\rm Ext}^j_R (R/I, H^i_I (M))$ and ${\rm Hom}_R(R/I, H^t_I(M))$ are weakly Laskerian for all $i<t$ and all $j \geq 0$. As a consequence, the set of associated primes of $H^i_I (M)$ is finite for all $i\geq 0$, whenever $\dim R/I \leq 2$ and $M$ is weakly Laskerian.