Symmetry of the Definition of Degeneration in Triangulated Categories

TL;DR

This paper investigates the concept of degeneration in triangulated categories, demonstrating that the inherent non-symmetry in its algebraic definition is essentially irrelevant under certain conditions, leading to a unified understanding.

Contribution

It proves that the non-symmetry in the algebraic definition of degeneration in triangulated categories is inessential, establishing the equivalence of different definitional choices.

Findings

The non-symmetry in the algebraic degeneration definition is inessential.

Under certain conditions, different choices in the definition lead to the same concept.

The result unifies the understanding of degeneration in triangulated categories.

Abstract

Module structures of an algebra on a fixed finite dimensional vector space form an algebraic variety. Isomorphism classes correspond to orbits of the action of an algebraic group on this variety and a module is a degeneration of another if it belongs to the Zariski closure of the orbit. Riedtmann and Zwara gave an algebraic characterisation of this concept in terms of the existence of short exact sequences. Jensen, Su and Zimmermann, as well as independently Yoshino, studied the natural generalisation of the Riedtmann-Zwara degeneration to triangulated categories. The definition has an intrinsic non-symmetry. Suppose that we have a triangulated category in which idempotents split and either for which the endomorphism rings of all objects are artinian, or which is the category of compact objects in an algebraic compactly generated triangulated K-category. Then we show that the non-symmetry in the algebraic definition of the degeneration is inessential in the sense that the two possible choices which can be made in the definition lead to the same concept.