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Subcategories of extension modules by Serre subcategories. (English) Zbl 1273.13018
Let $$R$$ be a Noetherian commutative ring and $$R-Mod$$ denote the category of $$R$$-modules. Let $$R-mod$$ denote the full subcategory of finitely generated $$R$$-modules. A full subcategory $$\mathcal{S}$$ of $$R-Mod$$ is called Serre if it is closed under taking submodules, quotients and extensions. A subset $$W$$ of $$Spec R$$ is said to be specialization closed if for any two prime ideals $$\mathfrak p\subseteq \mathfrak q$$ with $$\mathfrak p\in W,$$ it follows that $$\mathfrak q \in W$$. By a celebrated result of Gabriel, the assignment $\mathcal{S}\longmapsto \bigcup_{M\in \mathcal{S}} \mathrm{Supp}_RM$ establishes a lattice isomorphism between the set of Serre subcategories of $$R-mod$$ and the set of specialization closed subsets of $$\mathrm{Spec} R$$. The inverse map is given by $W\longmapsto \{M\in R-mod~|~\mathrm{Supp}_RM\subseteq W\}.$
Let $$\mathcal{S}_1$$ and $$\mathcal{S}_2$$ be two Serre subcategories of $$R-Mod$$. Following the paper under review, we denote the full subcategory $\{M\in R-Mod| \text{ there is an exact sequence} \;0\to S_1\to M\to S_2\to 0 \text{ with} \;S_i\in \mathcal{S}_i \;\text{for} \;i=1,2\}$ by $$(\mathcal{S}_1,\mathcal{S}_2)$$. The author investigates the question: When is $$(\mathcal{S}_1,\mathcal{S}_2)$$ a Serre subcategory?
Let $$\mathcal{S}_1$$ and $$\mathcal{S}_2$$ be two Serre subcategories of $$R-Mod$$. The main result of this paper asserts that $$(\mathcal{S}_1,\mathcal{S}_2)$$ is a Serre subcategory if and only if $$(\mathcal{S}_2,\mathcal{S}_1)\subseteq (\mathcal{S}_1,\mathcal{S}_2)$$.
Now, assume that $$\mathcal{S}_1$$ and $$\mathcal{S}_2$$ are two Serre subcategories of $$R-mod$$. In view of the above mentioned result of Gabriel, one can easily see that $$(\mathcal{S}_2, \mathcal{S}_1)=(\mathcal{S}_1,\mathcal{S}_2)$$. Hence, $$(\mathcal{S}_1,\mathcal{S}_2)$$ is a Serre subcategory of $$R-mod$$.

##### MSC:
 13C60 Module categories and commutative rings 13D45 Local cohomology and commutative rings
##### Keywords:
Extension module; local cohomology; Serre subcategory
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##### References:
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