Approximations, ghosts and derived equivalences

TL;DR

This paper introduces symmetric approximation sequences in additive and weakly n-angulated categories, demonstrating they induce derived equivalences between certain quotient rings, thus unifying concepts in mutation theory across various mathematical fields.

Contribution

It defines symmetric approximation sequences in new categorical contexts and proves their role in producing derived equivalences, extending the understanding of mutation sequences in algebra and geometry.

Findings

Symmetric approximation sequences lead to derived equivalences.

Sequences include higher Auslander-Reiten sequences and mutation sequences.

Derived equivalences occur between quotient rings of endomorphism rings.

Abstract

Approximation sequences and derived equivalences occur frequently in the research of mutation of tilting objects in representation theory, algebraic geometry and noncommutative geometry. In this paper, we introduce symmetric approximation sequences in additive categories and weakly $n$-angulated categories which include (higher) Auslander-Reiten sequences (triangles) and mutation sequences in algebra and geometry,and show that such sequences always give rise to derived equivalences between the quotient rings of endomorphism rings of objects in the sequences modulo some ghost and coghost ideals.