# Symmetric approximation sequences, Beilinson-Green algebras and derived equivalences

## TL;DR

This paper introduces a new class of derived equivalences between certain algebra quotients and subalgebras using symmetric approximation sequences in n-exangulated categories, expanding the scope of algebraic equivalences.

## Contribution

It establishes derived equivalences for quotient algebras of locally Beilinson-Green algebras via symmetric approximation sequences, generalizing previous results and connecting higher exact sequences.

## Key findings

- Derived equivalences between quotient algebras of Beilinson-Green algebras.
- Derived equivalences between subalgebras of endomorphism algebras.
- Extension of Chen and Xi's results to higher exact sequences.

## Abstract

In this paper, we will consider a class of locally $\Phi$-Beilinson-Green algebras, where $\Phi$ is an infinite admissible set of the integers, and show that symmetric approximation sequences in $n$-exangulated categories give rise to derived equivalences between quotient algebras of locally $\Phi$-Beilinson-Green algebras in the principal diagonals modulo some factorizable ghost and coghost ideals by the locally finite tilting family. Then we get a class of derived equivalent algebras that have not been obtained by using previous techniques. From higher exact sequences, we obtain derived equivalences between subalgebras of endomorphism algebras by constructing tilting complexes, which generalizes Chen and Xi's result for exact sequences. From a given derived equivalence, we get derived equivalences between locally $\Phi$-Beilinson-Green algebras of semi-Gorenstein modules. Finally, from given graded derived equivalences of group graded algebras, we get derived equivalences between associated Beilinson-Green algebras of group graded algebras.