Tilting theory of contraction algebras

Abstract

This thesis focuses on a class of finite dimensional symmetric algebras arising in geometry, known as contraction algebras. The main results presented here combine to give a complete description of the derived equivalence class of such an algebra, providing the first concrete evidence towards a key conjecture in the Homological Minimal Model Programme. More precisely, to each minimal model f : X → SpecR of a complete local isolated cDV singularity SpecR, Donovan{Wemyss associate a contraction algebra A. In this way, the collection of all minimal models of SpecR gives a collection of contraction algebras. We provide a new proof that these algebras are all derived equivalent, thus showing that the corresponding derived category is an invariant of the singularity SpecR. Donovan{Wemyss conjecture that this invariant actually provides a classification of such singularities. Given a contraction algebra A of a minimal model as above, we show that the two- term tilting complexes of A control the entire derived equivalence class of A. For the members of this class, we prove that the only basic algebras derived equivalent to A are the endomorphism algebras of these complexes. We further prove that these algebras precisely coincide with the collection of contraction algebras of SpecR, giving strong evidence to support the conjecture of Donovan{Wemyss. To understand the structure of maps between the members of this class, namely standard derived equivalences, we use the wall and chamber structure given by the two-term tilting theory of A. We prove that this wall and chamber structure coincides with a hyperplane arrangement arising from the geometry and that the chambers of this arrangement are naturally labelled by the collection of contraction algebras. Using our new proof that the contraction algebras of SpecR are derived equivalent, we establish that the combinatorics of the arrangement completely controls the structure of all the standard derived equivalences. This gives further evidence towards the Donovan{ Wemyss conjecture by demonstrating we can recover the group structure of certain derived symmetries arising from the geometry, known as ops, just from the derived category of the contraction algebras.