Bulletin of the
Korean Mathematical Society BKMS

Let $R$ be a commutative Noetherian ring and $I$ an ideal of $R$. In this paper, we study colocalization of generalized local homology modules. We intend to establish a dual case of local-global principle for the finiteness of generalized local cohomology modules. Let $M$ be a finitely generated $R$-module and $N$ a representable $R$-module. We introduce the notions of the representation dimension $r^I (M, N)$ and artinianness dimension $a^I (M, N)$ of $M,N$ with respect to $I$ by $r^I (M, N)= \inf\{i\in \mathbb{N}_0 : H^I_i(M,N) \text{ is not representable}\}$ and $a^I (M, N)= \inf\{i\in \mathbb{N}_0 : H^I_i(M,N)\text{ is not artinian}\}$ and we show that $a^I (M, N)=r^I (M, N)$ $=\inf\{r^{IR_{\mathfrak{p}}}(M_{\mathfrak{p}},_{\mathfrak{p}}N) : \mathfrak{p}\in \operatorname{Spec}(R)\}\geq \inf\{a^{IR_{\mathfrak{p}}}(M_{\mathfrak{p}},_{\mathfrak{p}}N) : \mathfrak{p}\in \operatorname{Spec}(R)\}$. Also, in the case where $R$ is semi-local and $N$ a semi discrete linearly compact $R$-module such that $N/\bigcap_{t>0} I^tN$ is artinian we prove that $\inf\{i: H^I_i(M,N) \text{ is not minimax}\}\!=\!\inf\{r^{IR_{\mathfrak{p}}}(M_{\mathfrak{p}},_{\mathfrak{p}}N) : \mathfrak{p}\in \operatorname{Spec}(R)\setminus\operatorname{Max}(R)\}.$

Keywords: Colocalization, generalized local homology modules, representable modules

MSC numbers: 13D45, 13E99