*-Modules, Tilting, and Almost Abelian Categories (original) (raw)

Communications in Algebra, 2001

Abstract

The concept of *-module arose from a remarkable converse of the tilting theorem due to Menini and Orsatti [25] who essentially proved that for suitable full subcategories G R-Mod and H S-Mod, any equivalence H! G is of the form G ffi V s : H! G and H ffi HomRðV ; Þ : G! H with some bimodule RVS . The conditions for G;H are that G is closed with respect to direct sums and epimorphic images (i.e. G is a pretorsion class ), and H is closed with respect to products and submodules (i.e. H is a pretorsionfree class ). Then G 1⁄4 Im G, the class of R-modules isomorphic to some GðNÞ, and H 1⁄4 Im H . Menini and Orsatti [25] already showed that the conditions on G;H can be stated as a property of the adjoint pair G a H , namely: the unit Z : 1! HG has to be epic, and the counit e : GH ! 1

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