Torsion-free Classes Research Articles

The Jordan-Hölder property of torsion-free classes in a quiver of type a

ABSTRACT We investigate the torsion-free classes satisfying the Jordan-Hölder property in the representation category of a quiver of type A. Explicitly, we use reflection functors and c-sortable elements to give a combinatorial criterion for determining such torsion-free classes.

On torsion theories and open classes of linear modular lattices

Given a ring R, a bijection exists between torsion theories and idempotent radicals. Further, the class of hereditary torsion theories contains the skeleton of the open classes. In this work, we extend the notions and main results in R-Mod about torsion classes, torsion-free classes, and open classes to the category of linear lattices, whose objects are complete modular lattices and whose morphisms are linear morphisms.

Torsion and torsion-free classes from objects of finite type in Grothendieck categories

In an arbitrary Grothendieck category, we find necessary and sufficient conditions for the class of FPn-injective objects to be a torsion class. By doing so, we propose a notion of n-hereditary categories. We also define and study the class of FPn-flat objects in Grothendieck categories with a generating set of small projective objects, and provide several equivalent conditions for this class to be torsion-free. At the end, we present several applications and examples of n-hereditary categories in the contexts modules over a ring, chain complexes of modules and categories of additive functors from an additive category to the category of abelian groups. Concerning the latter setting, we find a characterization of when these functor categories are n-hereditary in terms of the domain additive category.

The Jordan-Hölder property and Grothendieck monoids of exact categories

We investigate the Jordan-Hölder property (JHP) in exact categories. First, we show that (JHP) holds in an exact category if and only if the Grothendieck monoid introduced by Berenstein and Greenstein is free. Moreover, we give a criterion for this which only uses the Grothendieck group and the number of simple objects. Next, we apply these results to the representation theory of artin algebras. For a large class of exact categories including functorially finite torsion(-free) classes, (JHP) holds precisely when the number of indecomposable projectives is equal to that of simples. We study torsion-free classes in a quiver of type A in detail using the combinatorics of symmetric groups. We introduce Bruhat inversions of permutations and show that simples in a torsion-free class are in bijection with Bruhat inversions of the corresponding c-sortable element. We use this to give a combinatorial criterion for (JHP).

Monobrick, a uniform approach to torsion-free classes and wide subcategories

For a length abelian category, we show that all torsion-free classes can be classified by using only the information on bricks, including non functorially-finite ones. The idea is to consider the set of simple objects in a torsion-free class, which has the following property: it is a set of bricks where every non-zero map between them is an injection. We call such a set a monobrick. In this paper, we provide a uniform method to study torsion-free classes and wide subcategories via monobricks. We show that monobricks are in bijection with left Schur subcategories, which contains all subcategories closed under extensions, kernels and images, thus unifies torsion-free classes and wide subcategories. Then we show that torsion-free classes bijectively correspond to cofinally closed monobricks. Using monobricks, we deduce several known results on torsion(-free) classes and wide subcategories (e.g. finiteness result and bijections) in length abelian categories, without using τ-tilting theory. For Nakayama algebras, left Schur subcategories are the same as subcategories closed under extensions, kernels and images, and we show that its number is related to the large Schröder number.

Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras

For an element w of a simply-laced Weyl group, Buan-Iyama-Reiten-Scott defined a subcategory of the module category over the preprojective algebra of Dynkin type. This paper studies categorical properties of using the root system. We show that simple objects in bijectively correspond to Bruhat inversion roots of w, and obtain a combinatorial criterion for to satisfy the Jordan-Hölder property (JHP). For type A case, we give a diagrammatic construction of simple objects, and show that (JHP) can be characterized via a forest-like permutation, introduced by Bousquet-Mélou and Butler in the study of Schubert varieties.

Tilting Torsion-Free Classes in the Category of Comodules

A well-known Assem–Smal $$\phi $$ theorem on tilting modules plays an important role in tilting theory. In this paper, we prove that a version of Assem–Smal $$\phi $$ theorem still holds in the category of comodules. Following this result, we characterize the tilting torsion-free classes in the category of comodules using the precover theory.

Torsion classes generated by silting modules

We study the classes of modules which are generated by a silting module. In the case of either hereditary or perfect rings, it is proved that these are exactly the torsion $\mathcal{T}$ such that the regular module has a special $\mathcal{T}$-preenvelope. In particular, every torsion-enveloping class in $\mathrm{Mod}\textrm{-}R$ are of the form $\mathrm{Gen}(T)$ for a minimal silting module $T$. For the dual case, we obtain for general rings that the covering torsion-free classes of modules are exactly the classes of the form $\mathrm{Cogen}(T)$, where $T$ is a cosilting module.

Dual closure operators and their applications

Departing from a suitable categorical description of closure operators, this paper dualizes this notion and introduces some basic properties of dual closure operators. Usually these operators act on quotients rather than subobjects, and much attention is being paid here to their key examples in algebra and topology, which include the formation of monotone quotients (Eilenberg–Whyburn) and concordant quotients (Collins). In fair categorical generality, these constructions are shown to be factors of the fundamental correspondence that relates connectednesses and disconnectednesses in topology, as well as torsion classes and torsion-free classes in algebra. Depending on a given cogenerator, the paper also establishes a non-trivial correspondence between closure operators and dual closure operators in the category of R-modules. Dual closure operators must be carefully distinguished from interior operators that have been studied by other authors.

Classifying $$\tau $$ τ -tilting modules over preprojective algebras of Dynkin type

We study support \(\tau \)-tilting modules over preprojective algebras of Dynkin type. We classify basic support \(\tau \)-tilting modules by giving a bijection with elements in the corresponding Weyl groups. Moreover we show that they are in bijection with the set of torsion classes, the set of torsion-free classes and many other important objects in representation theory. We also study \(g\)-matrices of support \(\tau \)-tilting modules, which show terms of minimal projective presentations of indecomposable direct summands. We give an explicit description of \(g\)-matrices and prove that cones given by \(g\)-matrices coincide with chambers of the associated root systems.

Flat covers in abelian and in non-abelian categories

A general existence theorem for flat covers in (e.g., quasi-abelian) locally finitely presented categories is obtained from an additive Ramsey type theorem. In the abelian case, it is shown that flat covers always exist. Applications to categories of separated presheaves or sheaves, localizations of Bousfield type, torsion-free classes of finite type, and categories of filtered objects or complexes, are given.

Locally finitely presented categories of sheaves

Let A be a locally finitely presented Grothendieck category. It is shown that a class of localizations of A in the sense of Bousfield is again locally finitely presented. The criterion is applied to torsion-free classes in A , sheaves and separated presheaves on a generalized ringed space, and representations of partially ordered sets.

On torsionfree classes which are not precover classes

In the class of all exact torsion theories the torsionfree classes are cover (pre-cover) classes if and only if the classes of torsionfree relatively injective modules or relatively exact modules are cover (precover) classes, and this happens exactly if and only if the torsion theory is of finite type. Using the transfinite induction in the second half of the paper a new construction of a torsionfree relatively injective cover of an arbitrary module with respect to Goldie’s torsion theory of finite type is presented.

Radical and Torsion Theory for Acts

After recapitulating the rudiments of the Kurosh-Amitsur radical theory of S-acts, hereditary radicals are discussed. The hereditary Hoehnke radical assignments r which designate a Rees congruence r(A) to each S-act A, are the hereditary Kurosh-Amitsur radical assignments. Then the corresponding radical as well as semisimple classes are characterized. Equivalence classes of injective S-acts determine hereditary torsion assignments t, these are just the hereditary Hoehnke radicals, but the congruence t(A) of A need not be a Rees congruence. Torsion and torsionfree classes are characterized; several hereditary torsion assignments may determine the same torsion class which is always a radical class closed under taking subacts. Examples show that a hereditary torsion assignment need not be a hereditary radical.

TORSION THEORIES FOR FINITE VON NEUMANN ALGEBRAS

ABSTRACT The study of modules over a finite von Neumann algebra 𝒜 can be advanced by the use of torsion theories. In this work, some torsion theories for 𝒜 are presented, compared, and studied. In particular, we prove that the torsion theory (T, P) (in which a module is torsion if it is zero-dimensional) is equal to both Lambek and Goldie torsion theories for 𝒜. Using torsion theories, we describe the injective envelope of a finitely generated projective 𝒜-module and the inverse of the isomorphism K 0(𝒜) → K 0 (𝒰), where 𝒰 is the algebra of affiliated operators of 𝒜. Then the formula for computing the capacity of a finitely generated module is obtained. Lastly, we study the behavior of the torsion and torsion-free classes when passing from a subalgebra ℬ of a finite von Neumann algebra 𝒜 to 𝒜. With these results, we prove that the capacity is invariant under the induction of a ℬ-module.

NATURAL CLASSES AND TORSION THEORIES

Conditions under which a natural class of right R-modules is closed under quotient modules are determined. In this case the unique complementary natural class of c(Δ) is closed under direct products. Hence (Δ, c(Δ)) is a hereditary torsion theory with torsion class Δ. In general a natural class [Formula: see text] of right R-modules is not closed under direct products (or quotient modules), yet it has been shown by Y. Zhou that each natural class [Formula: see text] determines a unique hereditary torsion theory τ. The torsion and torsion free classes of this torsion theory τ are studied, and in particular, their dependence on the original natural class [Formula: see text]. As an application, the resulting torsion theories τ are used to define the class of τ-simple, i.e., τ-cocritical modules. Most of the above is done more generally for M-natural classes in the category σ[M].

Tilting Preenvelopes and Cotilting Precovers

We relate the theory of envelopes and covers to tilting and cotilting theory, for (infinitely generated) modules over arbitrary rings. Our main result characterizes tilting torsion classes as the pretorsion classes providing special preenvelopes for all modules. A dual characterization is proved for cotilting torsion-free classes using the new notion of a cofinendo module. We also construct unique representing modules for these classes.

Cotilting and a Hierarchy of Almost Cotorsion Groups

We describe the structure of all cotilting and partial cotilting groups, and of all cotilting torsion-free classes of groups. Under V=L, we also describe all tilting and partial tilting groups. As a by-product, we introduce new classes of groups—called almost cotorsion groups. Unlike the well-known cotorsion groups, the almost cotorsion ones have a rather complex structure: we prove that any cotorsion free ring can be realized as an endomorphism ring of an almost cotorsion group.

Partial cotilting modules and the lattices induced by them

We study a duality between (infinitely generated) cotilting and tilting modules over an arbitrary ring. Dualizing a result of Bongartz, we show that a module P is partial cotilting iff P is a direct summand of a cotilting module C such that the left Ext-orthogonal class ⊥P coincides with ⊥C. As an application, we characterize all cotilting torsion-free classes. Each partial cotilting module P defines a lattice L = [Cogen P1P] of torsion-free classes. Similarly, each partial tilting module P′ defines a lattice L′ = [[Gen P′,P′⊥]] of torsion classes. Generalizing a result of Assem and Kerner, we show that the elements of L are determined by their Rejp-torsion parts, and the elements of L′ by their Trp-torsion-free parts.

Semisimple and torsionfree classes

Semisimple and torsionfree classes