Document Type : Original Manuscript
Authors
Department of Mathematics, Payame Noor University, Tehran, Iran.
Abstract
Let $R$ be a commutative Noetherian ring with non-zero identity and $\mathcal{F}$ a filtration of $\operatorname{Spec}(R)$. We show that the Cousin functor with respect to $\mathcal{F}$, $C_R(\mathcal{F},-):\mathcal{C}_{\mathcal{F}}(R)\longrightarrow\operatorname{Comp}(R)$, where $\mathcal{C}_{\mathcal{F}}(R)$ is the category of $R$-modules which are admitted by $\mathcal{F}$ and $\operatorname{Comp}(R)$ is the category of complexes of $R$-modules, commutes with the formation of direct limits and is right exact. We observe that an $R$-module $X$ is balanced big Cohen-Macaulay if $(R,\mathfrak{m})$ is a local ring, $\mathfrak{m}X\neq X$, and every finitely generated submodule of $X$ is a big Cohen-Macaulay $R$-module with respect to some system of parameters for $R$.
Keywords
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