Abstract
This paper focuses on the Ki-groups of two types of extensions of abelian categories, which are the trivial extension and the gluing of abelian categories. We prove that, under some conditions, Ki-groups of a certian subcategory of the trivial extension category is isomorphic to Ki-groups of the similar subcategory of the original category. Moreover, under some conditions, we show that the Ki-groups of a left (right) gluing of two abelian categories are isomorphic to the direct sum of Ki-groups of two abelian categories. As their applications, we obtain some results of the Ki-groups of the trivial extension of a ring by a bimodule (i∈N).
Highlights
A comma category ( F, B ) is isomorphic to the right trivial extension category e where Fe : A × B −→ A × B is the functor given by Fe(( A, B)) = (0, F ( A))
In order to describe the relation among the Ki -groups of abelian categories in the gluing more precisely, we introduce here two weaker forms of the gluing of abelian categories given as follows: A left gluing of abelian categories consists of three categories D, D 0, D 00 and four functors i∗, i∗, j!, j! in (2), satisfying the conditions (i∗, i∗ ) and ( j!, j! ) are adjoint pairs, i∗ and j! are fully faithful functors and Im (i∗ ) =
We mainly use the category theory to study the relation between Ki -groups of the two extension categories and Ki -groups of the original category
Summary
K-Groups of Trivial Extensions and Gluings of Abelian Categories. Since the category whose objects are all the finite generated projective modules over a ring is an exact category, Ki -groups of a ring is defined by. Fossum and Griffith [8] defined and studied the right (left) trivial extension of an abelian category by an endofunctor. They get some conclusions of a trivial extension of a ring by a bimodule.
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