Let be a commutative ring and let be a -algebra. In this post we’ll investigate a condition on which generalizes the condition that is a finite separable field extension (in the case that is a field). It can be stated in many equivalent ways, as follows. Below, “bimodule” always means “bimodule over .”
Definition-Theorem: The following conditions on are all equivalent, and all define what it means for to be a separable -algebra:
- is projective as an -bimodule (equivalently, as a left -module).
- The multiplication map has a section as an -bimodule map.
- admits a separability idempotent: an element such that and for all (which implies that ).
(Edit, 3/27/16: Previously this definition included a condition involving Hochschild cohomology, but it’s debatable whether what I had in mind is the correct definition of Hochschild cohomology unless is a field or is projective over . It’s been removed since it plays no role in the post anyway.)
When is a field, this condition turns out to be a natural strengthening of the condition that is semisimple. In general, loosely speaking, a separable -algebra is like a “bundle of semisimple algebras” over .