Classifying subcategories of modules (Q2716134)

Let \(R\) be a ring. A full subcategory \({\mathcal C}\) of R-mod is called wide if whenever \(f:M\to N\) is a map of \({\mathcal C}\), then the kernel and cokernel of \(f\) are in \({\mathcal C}\) and \({\mathcal C}\) is closed under extensions. In this paper, the author shows that the above notion is the analogue of thick subcategory in the derived category \({\mathcal D}(R)\) and classifies wide subcategories of finitely presented modules over a large class of commutative coherent rings, using the classification of the thick subcategories of small objects in \({\mathcal D}(R)\).NEWLINENEWLINENEWLINEThe idea is to construct an adjunction \(f:{\mathcal L}_{\text{wide}}(R)\to {\mathcal L}_{\text{thick}} ({\mathcal D}(R))\), where \({\mathcal L}_{\text{wide}}(R)\) is the lattice of wide subcategories of \({\mathcal C}_0\), \({\mathcal C}_0\) the wide subcategory generated by \(R\) and \({\mathcal L}_{\text{thick}} ({\mathcal D}(R))\) is the lattice of thick subcategories of small objects in \({\mathcal D}(R)\). It turns out from the classification of thick subcategories in \({\mathcal D} (R)\) that \(f\) is surjective for all commutative rings \(R\) and when \(R\) is a regular coherent commutative ring then \(f\) is an isomorphism.NEWLINENEWLINENEWLINEThe author also studies the relationship between wide subcategories of \(R\)-modules and wide subcategories of \(R/ \alpha\)-modules where \(\alpha\) is a two sided ideal in \(R\) and he extends his classification theorem for all rings \(R\) that are quotients of regular commutative coherent rings by finitely generated ideals.NEWLINENEWLINENEWLINEFinally, using Neeman's classification of localizing subcategories of \({\mathcal D}(R)\) for a Noetherian commutative ring \(R\), the author deduces a classification of wide subcategories of \(R\)-modules closed under direct sums.