Singularity categories, preprojective algebras and orthogonal decompositions

TL;DR

This paper constructs an embedding of graded singularity categories into derived categories of coherent sheaves for certain Gorenstein algebras, linking algebraic and geometric perspectives through semi-orthogonal decompositions.

Contribution

It introduces a new embedding of graded singularity categories into derived categories, connecting preprojective algebra constructions with Orlov's semi-orthogonal decompositions.

Findings

Established an embedding of singularity categories into derived categories.

Connected algebraic constructions with geometric semi-orthogonal decompositions.

Extended results to Gorenstein algebras of parameter 1.

Abstract

In this note we use results of Minamoto and Amiot, Iyama, Reiten to construct an embedding of the graded singularity category of certain graded Gorenstein algebras into the derived categories of coherent sheaves over its projective scheme. These graded algebras are constructed using the preprojective algebras of $d$-representation infinite algebras as defined by Herschend, Iyama and Oppermann. We relate this embedding to the construction of a semi-orthogonal decomposition of the derived category of coherent sheaves over the projective scheme of a Gorenstein algebra of parameter 1 described by Orlov.