Derived symmetries induced by relatively spherical contraction algebras

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