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 SpecR. In particular, we construct a bimodule-determined functor ModZ → ModR, for a suitable commutative noetherian ring Z, from which there follows a finite-to-one, continous surjection SpecZ → SpecR. 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 co-ordinate ring of n×n 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 MaxZ → PrimR. 1