Unbounded ladders induced by Gorenstein algebras

TL;DR

This paper demonstrates that the derived categories of Gorenstein triangular matrix algebras have unbounded ladders, which restrict to various subcategories, and establishes a period 1 ladder structure for the singularity category.

Contribution

It introduces the concept of unbounded ladders in derived categories of Gorenstein algebras and shows their restriction to subcategories, also linking to singularity categories.

Findings

Derived category $D({\rm Mod}A)$ admits an unbounded ladder.

Ladders restrict to $D^-({\rm Mod})$, $D^b({\rm Mod})$, $D^b({\rm mod})$, $K^b({\rm proj})$.

Singularity category admits a ladder of period 1.

Abstract

The derived category $D({\rm Mod}A)$ of a Gorenstein triangular matrix algebra $A$ admits an unbounded ladder; and this ladder restricts to $D^-({\rm Mod})$ {\rm(}resp. $D^b({\rm Mod})$, $D^b({\rm mod})$, $K^b({\rm proj})${\rm)}. A left recollement of triangulated categories with Serre functors sits in a ladder of period $1$; as an application, the singularity category of $A$ admits a ladder of period $1$.