On standard derived equivalences of orbit categories

TL;DR

This paper introduces the concept of standard G-equivalences in orbit categories, explores their properties, and applies the theory to compute the derived Picard group of Frobenius Nakayama algebras.

Contribution

It defines and analyzes standard G-equivalences between derived categories with group actions, establishing maps between these and their orbit categories, with applications to Frobenius Nakayama algebras.

Findings

Constructed maps between standard G-equivalences and standard equivalences of orbit categories.

Analyzed properties of these maps and their implications.

Determined the generating set of the derived Picard group for Frobenius Nakayama algebras.

Abstract

Let $\kk$ be a commutative ring, $\AAA$ and $\BB$ -- two $\kk$-linear categories with an action of a group $G$. We introduce the notion of a standard $G$-equivalence from $\Kb\BB$ to $\Kb\AAA$. We construct a map from the set of standard $G$-equivalences to the set of standard equivalences from $\Kb\BB$ to $\Kb\AAA$ and a map from the set of standard $G$-equivalences from $\Kb\BB$ to $\Kb\AAA$ to the set of standard equivalences from $\Kb(\BB/G)$ to $\Kb(\AAA/G)$. We investigate the properties of these maps and apply our results to the case where $\AAA=\BB=R$ is a Frobenius $\kk$-algebra and $G$ is the cyclic group generated by its Nakayama automorphism $\nu$. We apply this technique to obtain the generating set of the derived Picard group of a Frobenius Nakayama algebra over an algebraically closed field.