Let $A$ be an algebra, $S \subset A$ a subalgebra.* I was wondering:
- Is $A$ projective as an $S$-module?
I have a feeling that this cannot be true in this generality (the feeling comes from the statement being false on the level of rings), hence I should actually be asking
When is $A$ projective as an $S$-module?
When is it not aka what are some counterexamples?
It's true for group algebras because of Lagrange's theorem. More generally, the Nichols-Zoeller theorem shows that it's true for Hopf algebras and Hopf subalgebras.
*EDIT: I should add "over some ring", and the answers can/should also involve conditions on the ring. Also, in my examples I'm really thinking "finite-dimensional and over a field" just to be safe.