Degenerations for derived categories

TL;DR

This paper develops a theory of degenerations for derived categories, establishing algebraic and geometric equivalences, with applications to tilting complexes and their module structures.

Contribution

It introduces a novel degeneration framework for derived categories, paralleling module variety degenerations, and applies it to characterize two-term tilting complexes.

Findings

Algebraic and geometric degenerations are equivalent.

Two-term tilting complexes are determined by their graded modules.

Framework extends degeneration concepts to derived categories.

Abstract

We propose a theory of degenerations for derived module categories, analogous to degenerations in module varieties for module categories. In particular we define two types of degenerations, one algebraic and the other geometric. We show that these are equivalent, analogously to the Riemann-Zwara theorem for module varieties. Applications to tilting complexes are given, in particular that any two-term tilting complex is determined by its graded module structure.