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Your search keyword '"Torsion theory"' showing total 3 results
3 results on '"Torsion theory"'

1. Semi-localizations of semi-abelian categories.

Author

Gran, Marino and Lack, Stephen

Subjects
*LOCALIZATION (Mathematics), *ABELIAN functions, *MODULAR arithmetic, *MATHEMATICAL analysis, *RING theory
Abstract

A semi-localization of a category is a full reflective subcategory with the property that the reflector is semi-left-exact. There are many interesting examples of semi-localizations, as for instance any torsion-free subcategory of a semi-abelian category. By specializing a result due to S. Mantovani, we first characterize the categories which are semi-localizations of exact Mal'tsev categories. We then prove a new characterization of protomodular categories in terms of binary relations, allowing us to obtain an abstract characterization of the semi-localizations of exact protomodular categories. This result is very useful to study the (hereditarily)-torsion-free subcategories of semi-abelian categories. Some examples are considered in detail in the categories of groups, crossed modules, commutative rings and topological groups. We finally explain how these results extend the corresponding ones obtained in the abelian context by W. Rump. [ABSTRACT FROM AUTHOR]

Published
2016
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2. Dimension and torsion theories for a class of Baer *-rings

Author

Vaš, Lia

Subjects
*VON Neumann algebras, *RING theory, *MATHEMATICAL analysis, *TORSION theory (Algebra), *ISOMORPHISM (Mathematics)
Abstract

Abstract: Many known results on finite von Neumann algebras are generalized, by purely algebraic proofs, to a certain class of finite Baer *-rings. The results in this paper can also be viewed as a study of the properties of Baer *-rings in the class . First, we show that a finitely generated module over a ring from the class splits as a direct sum of a finitely generated projective module and a certain torsion module. Then, we define the dimension of any module over a ring from and prove that this dimension has all the nice properties of the dimension studied in [W. Lück, J. Reine Angew. Math. 495 (1998) 135–162] for finite von Neumann algebras. This dimension defines a torsion theory that we prove to be equal to the Goldie and Lambek torsion theories. Moreover, every finitely generated module splits in this torsion theory. If R is a ring in , we can embed it in a canonical way into a regular ring Q also in . We show that is isomorphic to by producing an explicit isomorphism and its inverse of monoids that extends to the isomorphism of and . [Copyright &y& Elsevier]

Published
2005
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3. Localization of triangulated categories and derived categories

Author

Jun-ichi Miyachi

Subjects
Subcategory, Ring theory, Pure mathematics, Algebra and Number Theory, Section (category theory), Functor, Mathematics::K-Theory and Homology, Mathematics::Category Theory, Torsion theory, Abelian group, Representation theory, Quotient, Mathematics
Abstract

The notion of quotient and localization of abelian categories by dense subcategories (i.e., Serre classes) was introduced by Gabriel, and plays an important role in ring theory [6, 131. The notion of triangulated categories was introduced by Grothendieck and developed by Verdier [9, 16) and is recently useful in representation theory [8,4, 143. The quotient of triangulated categories by epaisse subcategories is constructed in [16]. Both quotients were indicated by Grothendieck, and they resemble each other. In this paper, we will consider triangulated categories and derived categories from the point of view of localization of abelian categories. Verdier gave a condition which is equivalent to the one that a quotient functor has a right adjoint, and considered a relation between tpaisse subcategories [16]. We show that localization of triangulated categories is similarly defined, and have a relation between localizations and Cpaisse subcategories. Beilinson, Bernstein, and Deligne introduced the notion of t-structure which is similar to torsion theory in abelian categories [2]. We, in particular, consider a stable t-structure, which is an epaisse subcategory, and deal with a correspondence between localizations of triangulated categories and stable r-structures. And then recollement, in the sense of [2], is equivalent to bilocalization. Next, we show that quotient and localization of abelian categories induce quotient and localization of its derived categories. In Section 1, we recall standard notations and terminologies of quotient and localization of abelian categories. In Section 2, we define localization of triangulated categories, and consider a relation between localizations and stable r-structures (Theorem 2.6). In Section 3, we show that if A + A/C is a quotient of abelian categories, then D*(A) -+ D*(A/C) is a quotient of triangulated categories, where * = + , -, or b (Theorem 3.2). Moreover

Published
1991

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