The Cohomological Brauer Group of a Torsion ${\mathbb{G}}_{m}$-Gerbe

Let $S$ be a scheme and let $\pi : \mathcal{G} \to S$ be a ${\mathbb{G}}_{m,S}$-gerbe corresponding to a torsion class $[\mathcal{G}]$ in the cohomological Brauer group ${\operatorname{Br}}^{\prime}(S)$ of $S$. We show that the cohomological Brauer group ${\operatorname{Br}}^{\prime}(\mathcal{G})$ of $\mathcal{G}$ is isomorphic to the quotient of ${\operatorname{Br}}^{\prime}(S)$ by the subgroup generated by the class $[\mathcal{G}]$. This is analogous to a theorem proved by Gabber for Brauer–Severi schemes.

Further Properties of the Lattice of Torsion Classes of Abelian Cyclically Ordered Groups

The notion of torsion class of abelian cyclically ordered groups has been introduced and fundamental properties of the collection T of all such classes, ordered by the class-theoretical inclusion, have been proved by the second author in 2011. The present paper can be considered as a continuation of the above mentioned one. We describe all atoms of T , show that T does not have any dual atom and prove complete distributivity of T .

ON SOME RELATIONS BETWEEN THE LATTICES R-nat, R-conat AND R-tors AND THE RINGS THEY CHARACTERIZE

This article consists of two sections. In the first one, the concepts of spanning and cospanning classes of modules, both hereditarily and cohereditarily, are explained, and some closure properties of the class of modules hereditarily cospanned by a conatural class are established, which amount to its being a hereditary torsion class. This gives a function from R-conat to R-tors and it is proven that its being a lattice isomorphism is part of a characterization of bilaterally perfect rings. The second section begins considering a description of pseudocomplements in certain lattices of module classes. The idea is generalized to define an inclusion-reversing operation on the collection of classes of modules. Restricted to R-nat, it is shown to be a function onto R-tors, and its being an anti-isomorphism is equivalent to R being left semiartinian. Lastly, another characterization of R being left semiartinian is given, in terms solely of R-tors.

Torsion classes of generalized Boolean algebras

Torsion classes and radical classes of lattice ordered groups have been investigated in several papers. The notions of torsion class and of radical class of generalized Boolean algebras are defined analogously. We denote by T g and R g the collections of all torsion classes or of all radical classes of generalized Boolean algebras, respectively. Both T g and R g are partially ordered by the class-theoretical inclusion. We deal with the relation between these partially ordered collection; as a consequence, we obtain that T g is a Brouwerian lattice. W. C. Holland proved that each variety of lattice ordered groups is a torsion class. We show that an analogous result is valid for generalized Boolean algebras.

Torsion classes of abelian cyclically ordered groups

We introduce the notion of torsion class of abelian cyclically ordered groups; the definition is analogous to that used in the theory of lattice ordered groups. The collection T of all such classes is partially ordered by the class-theoretical inclusion. Though T is a proper class, we can apply the usual terminology for this partial order. We prove that T is a complete, infinitely distributive lattice having infinitely many atoms.

New torsion theory in unital abelian ℓ-groups

This paper introduces the notion of a functorial torsion class (FTC): in a concrete category $\mathfrak{C}$ which has image factorization, one considers monocoreflective subcategories which are closed under formation of subobjects.Here the interest is in FTCs in the category of abelian lattice-ordered groups with designated strong order unit. The FTCs $\mathfrak{T}$ consisting of archimedean latticeordered groups are characterized: for each subgroup A of the rationals with the identity 1, either $\mathfrak{T} = \mathfrak{S}\left( A \right)$, the class of all lattice-ordered groups of functions on a set X which have finite range in A, or $$\mathfrak{T} = \mathbb{T}\left( A \right)$$, the class of all subgroups of A with 1.As for FTCs possessing non-archimedean groups, it is shown that if $\mathfrak{T}$ is an FTC containing a subgroup A of the reals with 1, of rank two or greater, then $\mathfrak{T}$ contains all ℓ-groups of the form $A\vec \times G$, for all abelian lattice-ordered groups G. Finally, the least FTC that contains a non-archimedean group is the class of all $\mathbb{Z}\vec \times G$, for all abelian lattice-ordered groups G.

HIGHER-ORDER CLASS GROUPS AND BLOCK MONOIDS OF KRULL MONOIDS WITH TORSION CLASS GROUP

Extensions of the notion of a class group and a block monoid associated to a Krull monoid with torsion class group are introduced and investigated. Instead of assigning to a Krull monoid only one abelian group (the class group) and one monoid of zero-sum sequences (the block monoid), we assign to it a recursively defined family of abelian groups, the first being the class group, and do alike for the block monoid. These investigations are motivated by the aim of gaining a more detailed understanding of the arithmetic of Krull monoids, including Dedekind and Krull domains, both from a technical and conceptual point of view. To illustrate our method, some first arithmetical applications are presented.

Pure Projective Approximations

A notion of good behavior is introduced for a definable subcategory of left R-modules. It is proved that every finitely presented left R-module has a pure projective left -approximation if and only if the associated torsion class of finite type in the functor category (mod-R, Ab) is coherent, i.e., the torsion subobject of every finitely presented object is finitely presented. This yields a bijective correspondence between such well-behaved definable subcategories and preenveloping subcategories of the category Add(R-mod) of pure projective left R-modules. An example is given of a preenveloping subcategory ⊆ Add(R-mod) that does not arise from a covariantly finite subcategory of finitely presented left R-modules. As a general example of this good behavior, it is shown that if R is a ring over which every left cotorsion R-module is pure injective, then the smallest definable subcategory (R-proj) containing every finitely generated projective module is well-behaved.

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].