On the Smoothness of the Noncommutative Pillow and Quantum Teardrops

Abstract

Recent results by Krähmer [Israel J. Math. 189 (2012), 237-266] 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 O(WPq(1,l)) [Comm. Math. Phys. 316 (2012), 151-170], quantum lens spaces O(Lq(l;1,l)) [Pacific J. Math. 211 (2003), 249-263], the quantum Seifert manifold O(Σ³q) [J. Geom. Phys. 62 (2012), 1097-1107], quantum real weighted projective planes O(RP²q(l;±)) [PoS Proc. Sci. (2012), PoS(CORFU2011), 055, 10 pages] and quantum Seifert lens spaces O(Σ³q(l;−)) [Axioms 1 (2012), 201-225] are homologically smooth in the sense that as their own bimodules they admit finitely generated projective resolutions of finite length.