Tensor homological minimal algebras, global dimension of the tensor product of algebras and of generalized Weyl algebras
An algebra \(A\) over a field is called tensor-\(d\)-minimal with respect to some class of algebras \(\Omega\), if \(d(A\otimes B)=d(A)+d(B)\) for each \(B\in\Omega\) where \(d\) is one of the following dimensions of rings: the weak dimension (wd), the left homological dimension (lgd), the Krull dimension (kd). The author gives sufficient conditions for an algebra to be tensor \(d\)-minimal. In particular, it is proved that the Weyl algebra \(A_n\) over an algebraically closed uncountable field is tensor lgd-minimal with respect to the class of all finitely generated left Noetherian algebras. It should be noted also that the author computes the global dimension of the generalized Weyl algebra of degree one with basic Dedekind ring.
finitely generated left Noetherian algebras
scientific article; zbMATH DE number 915280