We introduce a categorical closure operator g in the category of topological abelian groups (and continuous homomorphisms) as a Galois closure with respect to an appropriate Galois correspondence defined by means of the Pontryagin dual of the underlying group. We prove that a topological abelian group G is maximally almost periodic if and only if every cyclic subgroup of G is g-closed. This generalizes a property characterizing the circle group from (Studia Sci. Math. Hungar. 38 (2001) 97-113. A characterization of the circle group and the p-adic integers via sequential limit laws, preprint). and answers an appropriate version of a question posed in (A characterization of the circle group and the p-adic integers via sequential limit laws, preprint)
Abstract. Let T R be a right n-tilting module over an arbitrary associative ring R. In this paper we prove that there exists an n-tilting module T R equivalent to T R which induces a derived equivalence between the unbounded derived category D(R) and a triangulated subcategory E ⊥ of D(End(T )) equivalent to the quotient category of D(End(T )) modulo the kernel of the total left derived functor − ⊗ L S T . If T R is a classical n-tilting module, we have that T = T and the subcategory E ⊥ coincides with D(End |(T )), recovering the classical case.
Given two rings R and S, we study the category equivalences T T ¡ Y Y, where T T is a torsion class of R-modules and Y Y is a torsion-free class of S-modules. These Ž . equivalences correspond to quasi-tilting triples R, V, S , where V is a bimodule R S which has, ''locally,'' a tilting behavior. Comparing this setting with tilting bimodules and, more generally, with the torsion theory counter equivalences introduced by Colby and Fuller, we prove a local version of the Tilting Theorem for quasi-tilting triples. A whole section is devoted to examples in case of algebras over a field.
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