Homotopy invariants of singularity categories

TL;DR

This paper introduces a method to compute $A^1$-homotopy invariants of singularity categories for graded rings, providing explicit descriptions for certain algebraic categories and discussing invariants' behavior in specific cases.

Contribution

It develops a new approach to compute $A^1$-homotopy invariants of singularity categories for rings with suitable gradings, including applications to stable categories of self-injective algebras.

Findings

Computed $A^1$-homotopy invariants for singularity categories of graded rings.

Described invariants like homotopy K-theory for stable categories of self-injective algebras.

Noted vanishing of invariants for cluster categories of type $A_{2n}$ quivers.

Abstract

We present a method for computing $\mathbb{A}^1$-homotopy invariants of singularity categories of rings admitting suitable gradings. Using this we describe any such invariant, e.g. homotopy K-theory, for the stable categories of self-injective algebras admitting a connected grading. A remark is also made concerning the vanishing of all such invariants for cluster categories of type $A_{2n}$ quivers.