K-Groups of Trivial Extensions and Gluings of Abelian Categories

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 is an element of N).

Automated summary

This paper focuses on the Ki-groups of two types of extensions of abelian categories, the trivial extension and the gluing of abelian categories, showing isomorphisms of Ki-groups under certain conditions. The results provide a foundation for future research in related fields.