Degenerating 0 in Triangulated Categories

Abstract

In previous work, based on the work of Zwara and Yoshino, we defined and studied degenerations of objects in triangulated categories analogous to the degeneration of modules. In triangulated categories ${\mathcal{T}}$ , it is surprising that the zero object may degenerate. We show that the triangulated subcategory of ${\mathcal{T}}$ generated by the objects that are degenerations of zero coincides with the triangulated subcategory of ${\mathcal{T}}$ consisting of the objects with a vanishing image in the Grothendieck group $K_{0}({\mathcal{T}})$ of ${\mathcal{T}}$ .