TWO PROPERTIES OF COUSIN FUNCTORS

‎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.