Narrow Results

We study some properties of formal local cohomology modules ${{ℱ}}_a^i(M)$ in Serre subcategories. As a consequence, we obtain some results on the minimax modules. We also describe the sets ${\mathrm{Cos}}_R({{ℱ}}_a^i(M))$ and ${\mathrm{Coass}}_R({{ℱ}}_a^i(M))$.

Let ${𝒵}$ be a specialization closed subset of $\mathrm{\text{Spec}}R$ and ${𝒮}$ be a Serre subcategory of $ModR$. As a generalization of the notion of cofiniteness, we introduce the concept of ${𝒵}$-coserreness with respect to ${𝒮}$ (see Definition 4.1). First, as a main result, for some special Serre subcategories ${𝒮}$, we show that an $R$-module $M$ with ${dim}_{R}M\le 1$ is ${𝒵}$-coserre with respect to ${𝒮}$ if and only if ${Ext}_R^{0,1}(R/𝔞,M)\in {𝒮}$ for all ideals $𝔞\in F({𝒵})$. Indeed, this result provides a partial answer to a question that was recently raised in [K. Bahmanpour, R. Naghipour and M. Sedghi, Modules cofinite and weakly cofinite with respect to an ideal, J. Algebra Appl. 16 (2018) 1–17]. As an application of this result, we show that the category of ${𝒵}$-coserre $R$-modules $M$ with ${dim}_{R}M\le 1$ is a full Abelian subcategory of $ModR$. Also, for every homologically bounded $R$-complex $X$ whose homology modules belong to ${𝒮}$ we show that the local cohomology modules ${\mathrm{\text{H}}}_{{𝒵}}^i(X)$ for all $i$, are ${𝒵}$-coserre in all the cases where ${dim}_R{𝒵}\le 1$, ${dim}_{R}X\le 2-supX$ and $cd({𝒵},X)\le 1-supX$.

In this paper, we first describe the gradual modules and the homomorphisms between two gradual modules. Next, we introduce the category G-Mod, whose objects are all gradual modules and morphisms are all homomorphisms between two gradual modules. Also, according to the definition of the gradual module’s structure induced on each homomorphism $f$, we consider the gradual module’s structure on ker $f$ and $\mathrm{\text{Coker}}f$. We show that being a monomorphism (respectively, epimorphism) in the category G-Mod is equivalent to a monomorphism (respectively, epimorphism) in the category $R$-Mod. As the main result, we prove that in any short exact sequence $0\to M^{\prime }\to M\to M^{″}\to 0$, in $R$-Mod, if the $R$-module $M$ has a gradual module’s structure, then there are gradual module’s structure on the modules $M^{\prime }$ and $M^{″}$ where the short exact sequence created is a short exact sequence in the category G-Mod.

Let R be a Noetherian ring, I and J two ideals of R, M an R-module and t an integer. Let S be a Serre subcategory of the category of R-modules satisfying the condition C_I, and N be a finitely generated R-module with Supp_RN=V(𝔞) for some . It is shown that if for all i < t and all j < t-i, then for all i < t. Let S be the class of all R-modules N with dim_R N ≤ k, where k is an integer. It is proved that if for all i < t and all , then for all i < t. It follows that .