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A classification of torsion classes in abelian categories. (English) Zbl 1411.18018

The authors first discuss monoform and a more general premonoform objects, properties situated between simple and indecomposable objects. Classes closed under extensions and quotients are termed nullity classes; a full subcategory is a torsion class if it is both a nullity class and is a coreflective subcategory. In some cases torsion classes coincide with nullity classes, for instance when \(\mathcal A\) is a Noetherian category and torsion class \(\mathcal T\) is a full subcategory of \(\mathcal A\). The authors exhibit a specific tensor algebra that is a premonoform, but not a monoform. If \(M\) is a module over a ring with unity, then the following implications for the properties hold: simple implies strongly uniform which is equivalent to monoform which implies both premonoform and uniform (which are not comparable) which both imply indecomposable. Examples of non-equivalencies are given. In the category of finitely generated modules over a commutative Noetherian ring with a unity, the notions of premonoform and monoform coincide. The main result is as follows: Let \(\mathcal A\) be an abelian category. Then the nullity classes of Noetherian objects are classified by the closed and extension closed subclasses of the Noetherian spectrum of \(\mathcal A\). Specifically, there is an order-preserving bijection Supp: \(\{ \mathcal N_{\text{noeth}}\mathcal A\}\overset{\sim}\rightleftharpoons \{S_c^{\text{ext}}\text{Spec}_{\text{noeth}}\mathcal A\} : \text{Supp}^{-1}\)where Supp\(^{-1}\Phi=\{M\in\mathcal A\, |\, M \text{is noetherian and}\, \text{Supp} M\in\Phi\}\).One consequence is that the torsion classes of \(\mathcal A\) are classified by the closed and extension closed subsets of the spectrum of \(\mathcal A\). Specifically, if \(\mathcal A\) is a Noetherian abelian category, then there is an order preserving bijection:Supp: \(\{ \mathcal T\mathcal A\}\overset{\sim}\rightleftharpoons \{S_c^{\text{ext}}\text{Spec}\mathcal A\} : \text{Supp}^{-1}\).

MSC:

18E40 Torsion theories, radicals
13D30 Torsion theory for commutative rings

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