On the Smoothness of the Noncommutative Pillow and Quantum Teardrops

[摘要] 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 ${\mathcal O}({\mathbb W}{\mathbb P}_q(1,l))$ [ Comm. Math. Phys. 316 (2012), 151-170], 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], quantum real weighted projective planes ${\mathcal O}({\mathbb R}{\mathbb P}_q^2(l;\pm))$ [ PoS Proc. Sci. (2012), PoS(CORFU2011), 055, 10 pages] and quantum Seifert lens spaces ${\mathcal O}(\Sigma_q^3(l;-))$ [ 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.

[关键词] smooth algebra;generalized Weyl algebra;strongly graded algebra;noncommutative pillow;quantum teardrop;quantum lens space;quantum real weighted projective plane