Let $\mathfrak{a}$ be an ideal of a commutative noetherian ring $R$ and $M$ $R$-module with Cosupport in $\mathrm{V}(\mathfrak{a})$. We show that is $\mathfrak{a}$-coartinian if only $\mathrm{Ext}_{R}^{i}(R/\mathfrak{a},M)$ artinian for all $0\leq i\leq \mathrm{cd}(\mathfrak{a},M)$, which provides computable finitely many steps to examine $\mathfrak{a}$-coartinianness. also consider the duality...

The purpose of this note is to characterize the finite Hilbert functions which force all of their artinian algebras to enjoy the Weak Lefschetz Property (WLP). Curiously, they turn out to be exactly those (characterized by Wiebe in [W i]) whose Gotzmann ideals have the WLP. This implies that, if a Gotzmann ideal has the WLP, then all algebras with the same Hilbert function (and hence lower Bett...

 Let $R=oplus_{nin Bbb N_0}R_n$ be a Noetherian homogeneous ring with local base ring $(R_0,frak{m}_0)$, $M$ and $N$ two finitely generated graded $R$-modules. Let $t$ be the least integer such that $H^t_{R_+}(M,N)$ is not minimax. We prove that $H^j_{frak{m}_0R}(H^t_{R_+}(M,N))$ is Artinian for $j=0,1$. Also, we show that if ${rm cd}(R_{+},M,N)=2$ and $tin Bbb N_0$, then $H^t_{frak{m}_0R}(H^2_...