Singularity categories of skewed-gentle algebras

Xinhong Chen¹ ; Ming Lu²

¹ Department of Mathematics Southwest Jiaotong University Chengdu 610031, P.R. China
² Department of Mathematics Sichuan University Chengdu 610064, P.R. China

Colloquium Mathematicum, Tome 141 (2015) no. 2, pp. 183-198

Résumé

Let $K$ be an algebraically closed field. Let $(Q,Sp,I)$ be a skewed-gentle triple, and let $(Q^{sg},I^{sg})$ and $(Q^g,I^{g})$ be the corresponding skewed-gentle pair and the associated gentle pair, respectively. We prove that the skewed-gentle algebra $KQ^{sg}/\langle I^{sg}\rangle$ is singularity equivalent to $KQ/\langle I\rangle$. Moreover, we use $(Q,Sp,I)$ to describe the singularity category of $KQ^g/\langle I^g\rangle$. As a corollary, we find that $\operatorname{gldim} KQ^{sg}/\langle I^{sg}\rangle\infty$ if and only if $\operatorname{gldim} KQ/\langle I\rangle\infty$ if and only if $\operatorname{gldim} KQ^{g}/\langle I^{g}\rangle\infty$.

Détail

DOI : 10.4064/cm141-2-4

Keywords: algebraically closed field skewed gentle triple corresponding skewed gentle pair associated gentle pair respectively prove skewed gentle algebra langle rangle singularity equivalent langle rangle moreover describe singularity category langle rangle corollary operatorname gldim langle rangle infty only operatorname gldim langle rangle infty only operatorname gldim langle rangle infty