Document Type : Original Manuscript


Department of Mathematics, Payame Noor University, Tehran, Iran.


‎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 $R$‎.


 1. H. Bamdad and A. Vahidi, Extension functors of Cousin cohomology modules, bBull. Iranian Math. Soc., 44 (2018), 253–267.
2. M. P. Brodmann and R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge University Press, Cambridge, 1998.
3. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, Cambridge, 1998.
4. R. Hartshorne, Residues and Duality, Springer, Berlin, 1966.
5. A. M. Riley, R. Y. Sharp, and H. Zakeri, Cousin complexes and generalized fractions, Glasgow Math. J., 26 (1985), 51–76.
6. J. J. Rotman, An Introduction to Homological Algebra, Academic Press, San Diego, 1979.
7. R. Y. Sharp, A Cousin complex characterization of balanced big Cohen-Macaulay modules, Q. J. Math., 33 (1982), 471–485.
8. R. Y. Sharp, Gorenstein modules, Math. Z., 115 (1970), 117–139.
9. R. Y. Sharp, Local cohomology and the Cousin complex for a commutative Noetherian ring, Math. Z., 153 (1977), 19–22.
10. R. Y. Sharp, The Cousin complex for a module over a commutative Noetherian ring, Math. Z., 112 (1969), 340–356.