Noncommutative Images of Commutative Spectra

Abstract

We initiate a unified, axiomatic study of noncommutative algebras R whose prime spectra are, in a natural way, finite unions of commutative noetherian spectra. Our results illustrate how these commutative spectra can be functorially ``sewn together'' to form Spec R. In particular, we construct a bimodule-determined functor Mod Z -> Mod R, for a suitable commutative noetherian ring Z, from which there follows a finite-to-one, continous surjection Spec Z -> Spec R. Algebras satisfying the given axiomatic framework include PI algebras finitely generated over fields, noetherian PI algebras, enveloping algebras of complex finite dimensional solvable Lie algebras, standard generic quantum semisimple Lie groups, quantum affine spaces, quantized Weyl algebras, and standard generic quantizations of the coordinate ring of nxn matrices. In all of these examples (except for the non-finitely-generated noetherian PI algebras), Z is finitely generated over a field, and the constructed map of spectra restricts to a surjection Max Z -> Prim R.