Remarks on a Categorical Definition of Degeneration in Triangulated Categories

TL;DR

This paper explores a categorical approach to module degeneration in triangulated categories, extending classical module theory concepts and analyzing how these degenerations behave under functors.

Contribution

It provides a scheme-theoretic interpretation of degeneration in triangulated categories and studies its behavior under functors, generalizing known module degeneration results.

Findings

Degeneration can be characterized via distinguished triangles.

The concept extends to triangulated categories from module categories.

Behavior under functors preserves certain degeneration properties.

Abstract

This work reports on joint research with Manuel Saorin. For an algebra A over an algebraically closed field k the set of A-module structures on k d forms an affine algebraic variety. The general linear group Gl d (k) acts on this variety and isomorphism classes correspond to orbits under this action. A module M degenerates to a module N if N belongs to the Zariski closure of the orbit of M. Yoshino gave a scheme-theoretic characterisation, and Saorin and Zimmermann generalise this concept to general triangulated categories. We show that this concept has an interpretation in terms of distinguished triangles, analogous to the Riedtmann-Zwara characterisation for modules. In this manuscript we report on these results and study the behaviour of this degeneration concept under functors between triangulated categories.