Derived symmetries induced by relatively spherical contraction algebras

Purri Brant Godinho, Marina (2025) Derived symmetries induced by relatively spherical contraction algebras. PhD thesis, University of Glasgow.

Abstract

This thesis constructs homological symmetries (or, more precisely, derived autoequivalences) of algebras and varieties. In more detail, given a surjective ring morphism p: A → B, this thesis constructs endomorphisms T : D(Mod A) → D(Mod A) and C: D(Mod B) → D(Mod B) called the twist and cotwist around the extension of scalars functor induced by p. Moreover, we prove that these endomorphisms are equivalences in two settings: (1) twists for Gorenstein orders, and (2) twists induced by Frobenius exact categories. (1) When A is a Gorenstein order and B is self-injective, then the twist T and cotwist C are equivalences provided that B is perfect as an A-module and satisfies a certain Tor-vanishing condition. In fact, under these assumptions, C is a shift of the Nakayama functor of B. If, moreover, A is an order over a three-dimensional ring, then we prove that the Tor-vanishing condition is equivalent to the ring-theoretic condition that ker p = (ker p) 2 . (2) Given a Frobenius exact category E and an object x ∈ E, let A = EndE(x), B = EndE (x) so that there is a natural surjection p: A → B. In this setting, we show that the functors T and C are equivalences if B satisfies "hidden smoothness" and "spherical" criteria. We furthermore apply the technology developed in (1) and (2) to construct derived autoequivalences of varieties. More specifically, given a crepant contraction f : X → Y between varieties satisfying mild conditions, there is an associated epimorphism of OY - algebras π: A → Acon which, affine locally, induces a surjection of algebras. Therefore, using techniques in non-commutative geometry, we apply the technology of (1) and (2) to construct autoequivalences of D b (coh X). These results extend the construction of the noncommutative twist, introduced by Donovan and Wemyss, to more general settings. As a corollary, we also obtain that the noncommutative twist is in fact a spherical twist, and we discuss how our results extend previous works on spherical twists induced by crepant contractions.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics
• Supervisor's Name: Wemyss, Professor Michael
• Date of Award: 2025
• Depositing User: Theses Team
• Unique ID: glathesis:2025-85586
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 12 Nov 2025 11:54
• Last Modified: 12 Nov 2025 11:58
• Thesis DOI: 10.5525/gla.thesis.85586
• URI: https://theses.gla.ac.uk/id/eprint/85586