Spherical Twists for Gorenstein Orders and $G$-Hilb

TL;DR

This paper develops new derived autoequivalences for Gorenstein orders using spherical twists, with applications to skew group algebras and G-Hilbert schemes.

Contribution

It introduces conditions under which spherical twists yield autoequivalences of Gorenstein orders and applies this to explicit algebraic examples.

Findings

Constructed derived autoequivalences for Gorenstein orders.

Connected cotwists to shifts of Nakayama functors.

Applied theory to skew group algebras and G-Hilbert schemes.

Abstract

This paper constructs derived autoequivalences of Gorenstein orders as twists around spherical functors. More precisely, given a Gorenstein order $A$ and a quotient $p \colon A \to B$, then we specify natural conditions on $B$ under which the twist around the corresponding derived restriction of scalars functor is a derived autoequivalence of $A$. In the process, we show that the associated cotwist is a shift of the Nakayama functor of $B$. These results, together with local-to-global technology, are then used construct new derived autoequivalences for skew group algebras and $G$-Hilbert schemes, and we apply this theory to explicit examples.