Bimodule duality and Artin's problem for skew field extensions

Abstract

Emil Artin asked whether the left and right dimensions of an extension of skew fields must agree. Cohn and Schofield gave counterexamples through elaborate constructions. It has been a long-standing open question whether there is any division ring, infinite-dimensional over its center, for which the left and right indices agree whenever at least one is finite. We give the first such examples. Call a division ring index-symmetric if every division subring with at least one finite index has equal finite indices. Using a duality formula for bimodules over division algebras, we prove that every twisted Calabi--Yau division algebra is index-symmetric. The examples include the Weyl skew fields and, more generally, the quotient division rings of rings of differential operators on smooth affine varieties in characteristic zero. Further examples are the quotient division rings of enveloping algebras of finite-dimensional Lie algebras, Ore extensions of twisted Calabi--Yau noetherian domains, quantum affine spaces, and group algebras of torsion-free polycyclic-by-finite groups.