Torsion Classes Research Articles

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.

Hereditary torsion classes of s-systems

Torsion classes and their corresponding quasi-filters in S-systems will be investigated. Torsion theory, in particular the hereditary torsion theory of S-systems will be characterized by the conguences defined on a semigroup S. A number of results on torsion theory of semigroups in the literature will be generalized and extended.

Some properties of lattices of radical classes of lattice-ordered groups

Aradical class ℛ of lattice-ordered groups (l-groups) is a class closed under taking convexl-subgroups, joins of convexl-subgroups, andl-isomorphic images. Imposing various other closure conditions leads to many specific types of radical classes (e.g., torsion classes). For several of these types, the complete latticeT of radical classes of that type has been studied, and such latticesT are our object of study here. We give the characteristic properties of closed-kernel radical mappings and polar kernel radical mappings. We prove in many instances thatT isrelatively polarized, that is, for any\(U\)],ℛ eT with\(U\)] ⫅ℛ there exists a unique largest ℛ′e T such thatℛ ∩ ℛ′=\(U\)], and often we are able to explicitly identifyℛ′. By using these properties we characterize meet irreducibility in the latticeT of polar kernel radical classes.

Countably valued lattice-ordered groups

A countably valued lattice-ordered group is a lattice-ordered group in which every element has only countably many values. Such lattice-ordered groups are proven to be normal-valued and, though not necessarily special-valued, every element in a countably valuedl-group must have a special value. The class of countably valuedl-groups forms a torsion class, and the torsion radical determined by this class is anl-ideal that is the intersection of all maximal countably valued subgroups.

Tilting Modules and Tilting Torsion Theories

We generalize basic results about classical tilting modules and partial tilting modules to the infinite dimensional case, over an arbitrary ring R. The methods employed combine classical techniques of representation theory of finite dimensional algebras with new techniques of the theory of ∗-modules. Using a generalization of the Bongartz lemma, we characterize tilting torsion theories in Mod-R, i.e., torsion theories induced by (infinitely generated) tilting modules. We investigate lattices [Gen(P), P⊥] of torsion classes induced by partial tilting modules P. Applying our results to tilting torsion classes, we prove a version of the Brenner-Butler theorem, and a generalization of the Assem-Smalø theorem to the case when R is artinian.

Engel lattice-ordered groups

An Engel l-group generating a proper normal-valued l-variety is shown to be o-approximable. It is also established that for every proper normal-valued l-varietyF, the class ɛ(F) of Engel l-groups from F is a torsion class.

Torsion classes and a universal constraint on Donaldson invariants for odd manifolds

This paper studies the topology of the gauge group and givesmod2\bmod \,2universal relations along Donaldson polynomials of smooth44-manifolds, generalizing Y. Ruan’s previous related result.

Torsion Classes and a Universal Constraint on Donaldson Invariants for Odd Manifolds

This paper studies the topology of the gauge group and gives $\bmod 2$ universal relations along Donaldson polynomials of smooth $4$-manifolds, generalizing Y. Ruan’s previous related result.

Discriminants of Casson–Gordon invariants

Casson–Gordon invariants were first used to prove that certain algebraically slice knots in S3 are not slice knots [2, 3]. Since then they have been applied to a wide range of problems, including embedding problems and questions relating to boundary links [2, 10, 21, 25]. The most general Casson–Gordon invariant takes its value in L0(ℚ(ζd)(t)) ⊗ ℚ; here ζd denotes a primitive dth root of unity. Litherland [20] observed that one could usually tensor with ℤ(2) instead of ℚ, and in this way preserve the 2-torsion in the Witt group. He then constructed new examples of non-slice genus two knots which were detected with torsion classes in L0(ℚ(ζd)) ⊗ ℤ(2) modulo the image of L0(ℚ(ζd)) ⊗ ℤ(2).

Elasticity of factorizations in integral domains

For an atomic integral domain R, define ϱ( R)= sup{ m⧸ n| x 1⋯ x m = y 1⋯ y n , each x i , y j ϵ R is irreducible}. We investigate ϱ( R), with emphasis for Krull domains R. When R is a Krull domain, we determine lower and upper bounds for ϱ( R); in particular, ϱ( R)≤ max{| Cl( R)|⧸ 2, 1}. Moreover, we show that for any real numbers r≥1 or r=∞, there is a Dedekind domain R with torsion class group such that ϱ( R)= r.

ON A CLASSIFICATION OF PRIME RINGS

Abstract Let m be a positive cardinal. We denote by Pr(m) (resp. P t(m)) the class of all rings R for which m is the least cardinal such that all nonzero elements of R possess right (resp. left) insulators of cardinality less than m + 1. We also set P r(m) = Un ≤ m Pr(n). The classes Pr(m),> m 0, partition the class of all prime rings. Various descriptions of these classes are obtained. In particular if m is regular then P r(m) contains just those rings R such that t(R) = 0 for all proper torsion preradicals t on Mod -R whose torsion classes are closed under direct products of fewer than m modules. Examples are provided which show that P r(m) is non-empty for all m > 0 and which partially answer the question: for which cardinals m, n is P r(m) ∩ P t(n) nonempty?

Almost Bézout domains

An integral domain R is said to be an almost Bézout domain (respectively, almost GCD-domain) if for x, y ϵ R − {0}, there exists an n with ( x n , y n ) (respectively, ( x n , y n ) v ) principal. In this paper we continue the investigation of AGCD-domains begun by the second author and introduce the notion of an almost Bézout domain. We show that R is an almost Bézout domain if and only if R ̄ , the integral closure of R, is a Prüfer domain with torsion class group and for every x ϵ R ̄ , there exists an n with xsn ϵ R.

Very large subgroups of a lattice-ordered group

This article introduces the concept of a very large subgroup in the theory of lattice-ordered groups. The existence of a minimal very large subgroup is connected to some previously known structure theory, but it is also linked to conditions not studied before. Very large subgroups are useful in studying torsion and radical classes, and among other things, extension of lattice-ordered groups using very large kernels yields an intriguing completion operation for torsion classes. In the final section there is a new contruction which produces a lattice-ordered group in which every value is essential, having no special values.

Torsion classes in the cohomology of congruence subgroups

For any prime number p, let Γn, p denote the congruence subgroup of SLn(ℤ) of level p, i.e. the kernel of the surjective homomorphism fp: SLn(ℤ) → SLn(p) induced by the reduction mod p (Fp is the field with p elements). We defineusing upper left inclusions Γn, p ↪ Γn+1, p. Recall that the groups Γn, p are homology stable with M-coefficients, for instance if M = ℚ, ℤ[1/p], or ℤ/q with q prime and q ╪ p: Hi(Γn, p; M) ≅ Hi(Γp; M) for n ≥ 2i + 5 from [7] (but the homology stability fails if M = ℤ or ℤ/p).

The topology of torsion classes

The topology of torsion classes

Compact modules

Recently, Herrlich, Salicrup, and Strecker [HSS] have shown that Kuratowski’s Theorem, namely, that a space X is compact if and only if for every space Y, the projection π2X×Y → Y is a closed map, can be interpreted categorically, and hence generalized and applied in a wider setting than the category of topological spaces. It is the purpose of this work to set up and apply this categorical interpretation of compactness in categories of R—modules. We obtain a theory of compactness for each torsion theory T, and in the case that the torsion theory T is hereditary, and R is, for example, noetherian and left hereditary, obtain a characterization of T-compact modules: a module G is T-compact provided G/TG is a T-injective module. Furthermore, under relatively mild assumptions on the ring R, the class of T—compact modules forms a torsion class for a torsion theory which we identify in the lattice of all torsion theories.

A characterization of ℵ1-free abelian groups and its application to the chase radical

A groupA is an ℵ1-free abelian group iffA is a subgroup of the Boolean power Z(B) for some complete Boolean algebraB. The Chase radicalvA=Σ{C≦A: Hom(C, Z)=0 &C is countable). The torsion class {A:vA=A} is not closed under uncountable direct products.

A vanishing theorem for the conformal anomaly in dimension >2

We examine the conformal anomaly of quantum field theory in the light of Atiyah and Singer’s recent work on the chiral anomaly. We show that the bundle over the space of conformal structures defined on a given manifold, constructed as the index of the conformally invariant family of Laplace operators, is a torsion class in real K K -theory, Indeed, its Chern forms are zero. These Chern forms transgress to define classes in certain local cohomology groups calculated recently by Bonora, Cotta-Ramusino, and Reina; a corollary is that, although these obstruction groups are nonzero, the obstruction to the existence of a gauge-covariant propagator for the Laplace operator is the trivial element of obstruction group.

THE LARGEST NON-TRIVIAL TORSION CLASS OF MODULES

ABSTRACT In this paper we investigate the following two classes of left R-modules: N(P) ={A|A has no non-zero direct summand P e P} and H(p) = {A} if B ⋚ A with B e N(P), then B = 0}, where P is a class of projective R-modules. We demonstrate that N(p) is, in general, not a torsion class but that H(P) is always a torsionfree class. We also investigate those classes P and rings R for which N(P) is the largest non-trivial torsion class of R-modules.

Torsion Theories Induced by Tilting Modules

Let k be a commutative field, and A a finite-dimensional k-algebra. By a module will always be meant a finitely generated right module. Following [8], we shall call a module TA a tilting module if (1) pdTA ≦ 1, (2) Ext1A(T, T) = 0 and (3) there is a short exact sequencewith T’ and T” direct sums of direct summands of T. Given a tilting module TA, the full subcategories andof the category modA of A -modules are respectively the torsion-free class and the torsion class of a torsion theory on modA[8]. The aim of the present paper is to find conditions on a torsion theory in order that it be induced by a tilting module.