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, $\mathfrak {a}$ an ideal of $R$, $M$ an $R$-module and $t$ a non-negative integer. In this paper we show that the class of minimax modules includes the class of $\mathcal {AF}$ modules. The main result is that if the $R$-module ${\rm Ext}^t_R(R/\mathfrak {a},M)$ is finite (finitely generated), $H^i_\mathfrak {a}(M)$ is $\mathfrak {a} $-cofinite for all $i<t$ and $H^t_\mathfrak {a}(M)$ is minimax then $H^t_\mathfrak {a}(M)$ is $\mathfrak {a} $-cofinite. As a consequence we show that if $M$ and $N$ are finite $R$-modules and $H^i_\mathfrak {a}(N)$ is minimax for all $i<t$ then the set of associated prime ideals of the generalized local cohomology module $H^t_\mathfrak {a}(M,N)$ is finite.