Subcategories of extension modules by Serre subcategories (Q2845046)

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 \textit{Serre} if it is closed under taking submodules, quotients and extensions. A subset \(W\) of \(Spec R\) is said to be \textit{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 NEWLINE\[NEWLINE\mathcal{S}\longmapsto \bigcup_{M\in \mathcal{S}} \mathrm{Supp}_RMNEWLINE\]NEWLINE 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 NEWLINE\[NEWLINEW\longmapsto \{M\in R-mod~|~\mathrm{Supp}_RM\subseteq W\}.NEWLINE\]NEWLINENEWLINENEWLINELet \(\mathcal{S}_1\) and \(\mathcal{S}_2\) be two Serre subcategories of \(R-Mod\). Following the paper under review, we denote the full subcategory NEWLINE\[NEWLINE\{M\in R-Mod| \text{ there is an exact sequence} \;0\to S_1\to M\to S_2\to 0 \text{ with} \;S_i\in \mathcal{S}_i \;\text{for} \;i=1,2\}NEWLINE\]NEWLINE by \((\mathcal{S}_1,\mathcal{S}_2)\). The author investigates the question: When is \((\mathcal{S}_1,\mathcal{S}_2)\) a Serre subcategory?NEWLINENEWLINELet \(\mathcal{S}_1\) 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 \((\mathcal{S}_2,\mathcal{S}_1)\subseteq (\mathcal{S}_1,\mathcal{S}_2)\).NEWLINENEWLINENow, 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 \(R-mod\).