A recollement approach to Brieskorn-Pham singularities

TL;DR

This paper develops a recollement framework for Brieskorn-Pham singularities, constructing tilting objects and exploring their derived categories, thereby extending known symmetries in singularity theory.

Contribution

It introduces a new recollement approach and constructs extended tilting n-cuboids, linking singularity categories to tensor products of Nakayama algebras and replicated algebras.

Findings

Constructed recollements and ladders for Brieskorn-Pham singularities.

Defined extended tilting n-cuboids with specific endomorphism algebras.

Established derived equivalences generalizing Happel-Seidel symmetry.

Abstract

In this paper, we construct recollements and ladders for Brieskorn-Pham singularities via reduction/insertion functors, and study the singularity categories of the Brieskorn-Pham singularities using these ladders. In particular, we construct a class of tilting objects, called the extended tilting $n$-cuboids, whose endomorphism algebras are $n$-fold tensor products of certain Nakayama algebras. Moreover, we show that such an endomorphism algebra is derived equivalent to a certain replicated algebra. This generalizes the Happel-Seidel symmetry to the context of Brieskorn-Pham singularities.