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Singularity categories of skewed-gentle algebras. (English) Zbl 1357.16027

Summary: 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 \(\operatorname{gldim} KQ^{g}/\langle I^{g}\rangle<\infty\).

MSC:

16G10 Representations of associative Artinian rings
16E35 Derived categories and associative algebras