On the Smoothness of the Noncommutative Pillow and Quantum Teardrops

Abstract

Recent results by Kr\"ahmer [Israel J. Math. 189 (2012), 237-266, arXiv:0806.0267] on smoothness of Hopf-Galois extensions and by Liu [arXiv:1304.7117] on smoothness of generalized Weyl algebras are used to prove that the coordinate algebras of the noncommutative pillow orbifold [Internat. J. Math. 2 (1991), 139-166], quantum teardrops ${\mathcal O}({\mathbb W}{\mathbb P}_q(1,l))$ [Comm. Math. Phys. 316 (2012), 151-170, arXiv:1107.1417], quantum lens spaces ${\mathcal O}(L_q(l;1,l))$ [Pacific J. Math. 211 (2003), 249-263], the quantum Seifert manifold ${\mathcal O}(\Sigma_q^3)$ [J. Geom. Phys. 62 (2012), 1097-1107, arXiv:1105.5897], quantum real weighted projective planes ${\mathcal O}({\mathbb R}{\mathbb P}_q^2(l;\pm))$ [PoS Proc. Sci. (2012), PoS(CORFU2011), 055, 10 pages, arXiv:1203.6801] and quantum Seifert lens spaces ${\mathcal O}(\Sigma_q^3(l;-))$ [Axioms 1 (2012), 201-225, arXiv:1207.2313] are homologically smooth in the sense that as their own bimodules they admit finitely generated projective resolutions of finite length.