Singularity categories of skewed-gentle algebras

TL;DR

This paper investigates the singularity categories of skewed-gentle algebras, establishing their equivalence with related gentle algebras and characterizing conditions for finite global dimension.

Contribution

It proves the singularity equivalence between skewed-gentle and gentle algebras and describes the singularity category using skewed-gentle triples.

Findings

Skewed-gentle algebra is singularity equivalent to associated gentle algebra.

Singularity category of gentle algebra described via skewed-gentle triples.

Finite global dimension conditions are equivalent for skewed-gentle, gentle, and associated algebras.

Abstract

Let $K$ be an algebraically closed field. Let $(Q,Sp,I)$ be a skewed-gentle triple, $(Q^{sg},I^{sg})$ and $(Q^g,I^{g})$ be its corresponding skewed-gentle pair and associated gentle pair respectively. It proves that the skewed-gentle algebra $KQ^{sg}/< I^{sg}>$ is singularity equivalent to $KQ/< I>$. Moreover, we use $(Q,Sp,I)$ to describe the singularity category of $KQ^g/< I^g>$. As a corollary, we get that $\mathrm{gldim} KQ^{sg}/< I^{sg}><\infty$ if and only if $\mathrm{gldim} KQ/< I><\infty$ if and only if $\mathrm{gldim} KQ^{g}/< I^{g}><\infty$.