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ABSTRACT. For a finite dimensional algebra A, we establish correspondences between torsion classes and wide subcategories in mod(A). In case A is representation finite, we obtain an explicit bijection between these two classes of subcategories. Moreover, we translate our results to the language of ring epimorphisms and universal localisations. It turns out that universal localisations over representation finite algebras are classified by torsion classes and support τ-tilting modules.

An important result in tilting theory states that a class of modules over a ring is a tilting class if and only if it is the Ext-orthogonal class to a set of compact modules of bounded projective dimension. Moreover, cotilting classes are precisely the resolving and definable subcategories of the module category whose Extorthogonal class has bounded injective dimension.In this article, we prove a derived counterpart of the statements above in the context of silting theory. Silting and cosilting complexes in the derived category of a ring generalise tilting and cotilting modules. They give rise to subcategories of the derived category, called silting and cosilting classes, which are part of both a tstructure and a co-t-structure. We characterise these subcategories: silting classes are precisely those which are intermediate and Ext-orthogonal classes to a set of compact objects, and cosilting classes are precisely the cosuspended, definable and co-intermediate subcategories of the derived category.Theorem. Let A be a ring, D(A) the derived category of A-Mod and V a full subcategory of D(A).(1) V is a silting class if and only if V is intermediate and V = S ⊥ >0 for a set S of compact objects.(2) V is a cosilting class if and only if V is cosuspended, co-intermediate and definable.Moreover, the first statement induces a bijection between co-intermediate and cosuspended subcategories of K b (A-proj) and silting complexes up to equivalence.This statement summarises Theorems 3.6 and 3.14. For the terminology used, we refer to the relevant sections. However, the parallel with the results concerning tilting and cotilting modules is evident.The proof of our main theorem involves some module categories built from the category of complexes. In fact, we reduce (co)silting problems in the derived category to (co)tilting problems in these module categories. This allows us to use the finite-type characterisation of tilting classes to conclude the compact generation of silting co-t-structures. Moreover, we use the fact that cotilting modules are pure-injective to conclude that so are cosilting complexes, from which we deduce the definability of cosilting classes.Finally, note that for the special case of two-term silting and cosilting complexes, the theorem above specialises to certain classification results recently obtained in the context of silting and cosilting modules. More precisely, part (1) translates to saying that a torsion class in the module category arises from a silting module if and only if it is divisible with respect to a set of maps between finitely generated projective modules (see [23, Theorem 6.3]). Part (2) of the theorem is, in this context, equivalent to stating that a torsionfree class in the module category arises from a cosilting module if and only if it is definable (see [2, Corollary 3.9] and the references therein).The structure of the paper is as follows. In Section 2, we set up the aforementioned module categories built from the category of complexes, and we show a useful correspondence between (...

Communicated by C.A. Weibel MSC: 16S85; 16G20; 16E30; 18E40We study universal localisations, in the sense of Cohn and Schofield, for finite dimensional algebras and classify them by certain subcategories of our initial module category. A complete classification is presented in the hereditary case as well as for Nakayama algebras and local algebras. Furthermore, for hereditary algebras, we establish a correspondence between finite dimensional universal localisations and finitely generated support tilting modules. In the Nakayama case, we get a similar result using τ -tilting modules, which were recently introduced by Adachi, Iyama and Reiten.

We show that silting modules are closely related with localizations of rings. More precisely, every partial silting module gives rise to a localization at a set of maps between countably generated projective modules and, conversely, every universal localization, in the sense of Cohn and Schofield, arises in this way. To establish these results, we further explore the finite-type classification of tilting classes and we use the morphism category to translate silting modules into tilting objects. In particular, we prove that silting modules are of finite type.