An Axiomatic Approach for Degenerations in Triangulated Categories

Abstract

Abstract. We generalise Yoshino's definition of a degeneration of two Cohen Macaulay modules to a definition of degeneration between two objects in a triangulated category. We derive some natural properties for the triangulated category and the degeneration under which the Yoshino-style degeneration is equivalent to the degeneration defined by a specific distinguished triangle analogous to Zwara's characterisation of degeneration in module varieties.

Symmetry of the Definition of Degeneration in Triangulated Categories

“…In Section 1 we give a summary of the contents of reference [10], in order to provide the vocabulary needed to understand the proof of the main result in the main body of the paper, without being obliged to go into the full details of that reference. In Section 2 we give the relevant background, facts and definitions of degenerations of objects in module categories, as well as in triangulated categories as it was shown in our earlier papers [4,11]. Section 3 then proves the main result Theorem 1 under the hypothesis (a), i.e.…”

“…The main purpose of [11] was to define a geometric notion of degeneration along the lines of [15], and to prove that this notion is equivalent with the notion of degeneration in the triangle sense. More precisely we gave the following definition.…”

“…[4] concentrated on partial order properties of ≤ ∆ . Further conditions guaranteeing partial order properties of ≤ ∆ can be found for various situations in [17], [16] and [11]. In this latter reference a geometric setting was developed replacing the module variety mod(A, d) for general triangulated categories, mimicking for this purpose Yoshino's scheme theoretic approach [15].…”

Triangle Order ≤△ in Singular Categories

“…[6,7,12]) has been studied. In a triangulated category we say for two objects X and Y that X ≤ Y if there is an object Z and a distinguished triangle…”

“…For the background on the degeneration theory of modules and triangulated categories, we refer to [6,7,[11][12][13][14]18] and [19]. For the definition of singular category, we refer to [1,10] and [17].…”