| Homological smoothness is a kind of homological property for associative algebras.As a noncommutative version of smoothness in the commutative sense,it plays an important role in the fields of noncommutative algebraic geometry,quantum group,operator algebra,mathematical physics,and so on.Many homologically smooth algebras have a duality between their Hochschild homology and cohomology,i.e.,the Van den Bergh duality.Generally speaking,it is hard to judge an algebra to be homologically smooth or not.This master dissertation is dedicated to the study of homological smoothness of a class of generalized Weyl algebras of Gelfand-Kirilov dimension three.The notion of generalized Weyl algebras was introduced by Bavula in 1992,motivated by the study of algebras analogous with the classical Weyl algebra.The class of generalized Weyl algebras W discussed in the dissertation,containing the polynomial ring k[z1,z2]in two variables as a subalgebra,are parameterized by an algebra automorphism σa of affine type on k[z1,z2]and a nonzero polynomial φ=φ(z1,z2).By constructing a homotopy double complex for W,we first obtain a projective resolution of W over We,and then find a sufficient condition for W to be homologically smooth.More precisely,we prove that W is homologically smooth if the ideal generated by φ,αφ/αz1,αφ/αz2 is equal to k[z1,z2]itself.As applications,we prove that the quantum groups Oq(SL2)and U(sl2)are both homologically smooth,coinciding with a result of K.A.Brown and J.J.Zhang in 2008,and we also study the algebra M(1,q)defined by H.X.Chen in 1999,whose quotient by a normal regular element is proved to be homologically smooth too.Some outlooks are given at the end of the dissertation. |
